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Review of matter

Diffraction theory and its application, quantum optics, problems in radiation coherence, light sources, holography and its application, scientific photography , methods of image reconstruction, oplical application o f Fourier transform, theory of oplical systems, criteria of oplical image evaluation, optical materials, technology o f manufacturing oplical elements, aspheric optics, optical properli es o f solids and thin film s, lasers and their application, photo- and radiometry, problemsin spectroscopy, nonlinear optics, optical data processing, oplical measurements, fibre optics, optical instrumentation, interferometry, microscopy, non-visible optics, automation of optical computing, optoelectronics, colorimetry, optical detectors, ellipsametry and photoelasticity, optical modulation, optics of electron beams, biooptics, optometry .

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© Copyright by Oficyna Wydawnicza Politechniki Wrocławskiej, Wrocław 2008

Drukarnia Olicyny Wydawniczej Polilechniki Wrocławskiej. Za m. nr l 040/2008.

OPTICA APPLICATA

HONORARY EolTOR IN CHIEF

EDITOR IN CHIEF

VICE-EDITOR

EDITING COUNSELLOR

TOPICAL EDITORS

KRZYSZTOF ABRAMSKI, Wrocław University ojTechnology, Poland

TADEUSZ PUSTELNY, Silesian University ojTechnology, Gliwice, Po/and

TOMASZ SZOPLIK, Warsaw University, Poland

HENRYK KASPRZAK, Wrocław University ojTechnology, Poland

EwA WEINERT-RĄCZKA, Szczecin University ojTechnology, Poland

INTERNATIONAL ADVISORY BOARD

The quarterly o f the lnstitute o f Physics Wrocław University ofTechnology, Poland

PL ISSN 0078-5466 lndex 367729

MIRONGAJ wACŁAw URBAŃCZYK

AGNIESZKA POPIOłEK-MASAJADA

IRENEUSZ WILK

Fiber optics and optical communication, spectroscopy, lasers and their applications

Integrated optics, acoustooptics, microoptics, optical instrumentation, optical measurements, optical sensing

Nanooptics, plasmonics, optical imaging, optical computing, optical data storage and processing

Holography, diffraction and gratings, biooptics, merlical optics, optometry, optical imaging, Fourier optics

Nonlinear optics, oplical waveguides, photonic crystals

0LEG V. ANGELSKY, Chernivtsy University, Ukraine

SIEGFRIED BOSECK, University oj Bremen, Germany

ROMAN S. INGARDEN, Nicolm1s Copernicus University, Toruń, Poland

EuGENIUSZ JAGOSZEWSKI (Chairman), Wroclaw University ojTeclrnology, Poland

ROM UALD JÓŹWICKI , Warsaw University ojTechnology, Poland

FRANCISZEK KACZMAREK, Adam Mickiewicz University, Poznań, Poland

BoLESt.A w KĘDZIA, Poznań Universi ty oj Medical Sciences, Poland

MAŁGORZATA KUJ A WIŃSKA, Warsaw University ojTechnology, Poland

MIROSLAV MILER, !nstitute ojPhotonics and Electronics, v. v.i., A.S. C. R., Prague, Czech Republic

JAN MISIEWICZ, Wroclaw University ojTechnology, Poland

WŁODZ IMIERZ NAKWASKI, Technical University oj Łódź, Poland

JAN PERINA, Palaclry University, Olomouc. Czech Republic

COLIN SHEPPARD, National University ojSingapore

TADEUSZ STACEWICZ, University oj Warsaw, Poland

TOMASZ WOLIŃSKI , Warsaw University ojTechnology, Poland

JAN WÓJCIK, Maria Curie-Skłodowska University in Lublin, Poland

OPTICA APPLICATA

Contents

Inteńerometry

Vol. XXXVIII (2008) No. 3

Mullipie polarized beam interferometers for array generation with improved efficiency PATRA A .S., KHARE A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 495

Characteristic analysis on a n interleaver wit h a fiber lo o p resonator by using a signal jlow grap h method LI W., SUN J. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 503

Diffraction

Beam shaping based on intermedia/e zone diffraction o f a micro-aperture DANY AN ZENG, ZHIJUN SuN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . • . . . . . . . . . 511

Ray tracing

A n iterative programmable graphics process unit based on ray casting approach for virtual endoscopy system FEINIU Y UAN . . . . . . . . . . . . . • . . . . . . . . . . . . . . . . . . • . . . . . • . . . . . . . . . . • • . . . . . . . • . . . 519

Medical optics

Serum glycoproteins in diabetic and non-diabetic patients with and without cataract GuL A., RAHMAN M .A ., AHMED N. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 531

Seattering

Lyapunov exponent o f t he oplical radialian scallered by the Brownian partie/es GAVRYLYAK M .S., MAKSIMY AK O.P., MAKSIMY AK P .P . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 539

Partial coherence in turbolent media

Propagation properli es o f partially coherent beams through turbuleni media with coherent modes representation ALAVINEJAD M ., ASHIRI F ., GHAFARY B. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 549

Semiconductors characterization

Evidence for metaslabie behavior o f Ga-doped CdTe PLACZEK-POPKO E., GUMIENNY Z ., TRZMIEL J., SZATKOWSKI J. . . . . . . . . . . . . . . . . . . . . . . . . . 559

494

Laser-induced damage in crystals

Morplwlogy o f laser-induced damage o f lilhium niobale and KDP cryslals KRUPYCH 0 ., 0YACHOK Y., SMAGA L, VLOKH R .O . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 567

Thin films

Distribulion o f electronic states in amorphous Zn-P lhinfilms on t he bas i s o f oplicni measurements JARZĄBEK 8 ., W ESZKA J. , CISOWSKI J. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 575

Absorptive CdTe films oplical pnrameters andfilm thickness delermination by the e/lipsom elric method EvMENOVA A.Z., OoARYCH V .A ., Srzov F.F., VuiCHYK M . V . . . . . . • . . . . . . . . • • . . . . . • • . . 585

Fiber sensing

Fiber Bragg grating and /ong period grating sensor for simultaneous measurement and discrimination o f strain and tempera/U re effects SRIMANNARAYANA K ., SA l SHANKAR M., SAl PRASAD R .L.N ., KRISHNA MOHAN T.K., RAMAKRISHNA S. , SRIKANHI G., RAVI PRASAD RAO S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 601

Rcfractive index measurcments

Clwracterization ofthe refractive index in gradient-index e/ements WYCHOWANIEC M ., LITWIN D. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 609

Optica Applicata, Vol. XXXVIII, No. 3, 2008

Multiple polarized beam interferometers for array generation with improved efficiency

ARDHENDU S. PATRA1, ALIKA KHARE2*

1Department of Electro-Optical Engineering, National Taipei University of Technology, Taipei 10608, Taiwan

2Department of Physics, Indian Institute of Technology Guwahati, Guwahati 781039, India

*Corresponding author: [email protected]

A highly efficient multiple polarized beam interferometer for the generation of hexagonal array isreported. An expression for the intensity distribution is worked out using Jones’ calculus andcomputed pattern is compared with the experimental results. The array pattern could be scannedover large longitudinal distances without loss of distortion. Fringe visibility of interferogramshas been studied as a function of relative state of polarization of the interfering beams. Some ofthe potential applications of such arrays are also proposed.

Keywords: hexagonal array, interferometry, polarization.

1. Introduction

Tiny light spots in the hexagonal array structures have been commonly used in imageprocessing [1], fiber couplers [2], atom lithography [3] and wave front sensing [4], etc.Hexagonal tiny arrays of equal illuminating light spots have been reported using a directthree beam interferometer [3], a three wave lateral shearing interferometer [5, 6], aswell as using eight randomly polarized beams from the three interferometers intandem [7]. The interference pattern and hence the arrays are highly localized in directthree beam interferometer [3]. In the lateral shearing interferometer [5, 6], the arraypattern is independent of the plane of observation along the longitudinal direction butthe scope of on line control on the dimensions and the geometry are limited. An N beaminterferometer is reported [8] for generation of various array patterns but it requiresthe designing of special aperture/grating making the on line control on array geometrylimited. The eight beam randomly polarized interferometer [7] offers the advantage ofon line control on the array geometry and pattern can be scanned over large longitudinaldistances making it delocalized in longitudinal direction. The efficiency of such

496 A.S. PATRA, A. KHARE

configuration [9] is only around 0.09. The reason for poor efficiency is the repeatedsplitting of the beams at the beam splitters after reflection from mirrors of the Michelsoninterferometers used. This efficiency can be improved by using the polarized beamand polarizing beam splitters along with quarter wave plate’s (QWP) and polarizerplaced at appropriate locations in the configuration similar to Michelson interferometer.One such configuration was reported theoretically [10] for the formation of squarearray. Recently, we reported experimentally such configuration [11] for the generationof square arrays using polarized beams having light coupling efficiency of 44.3%which is about twice the efficiency of randomly polarized setup [7]. In the presentpaper, we report the hexagonal arrays of small light spots by using the interference ofeight polarized beams coming out from the three interferometers in tandem. The lightcoupling efficiency of such system is four times higher than that of randomlypolarized setup [9]. The resultant intensities at the output plane are worked outusing Jones’ calculus for the polarized light and computed patterns are compared withthe experimental observations. Several possible applications of this array have beenproposed and discussed.

2. Experimental set-up

The experimental set-up is shown in Fig. 1a. A collimated beam was passed througha polarizer P1 and launched into an interferometric setup consisting of threeinterferometers in tandem as shown in the figure. The polarizer P1 (03FPG003 – MellesGriot) was aligned at 45° to ensure 50-50 splitting of light from polarizing cube beamsplitter PBS1 (03PBB005 – Melles Griot). The reflected beam from PBS1 (onlys-polarized light) was passed through quarter-wave plate Q1 (02WRM015 – MellesGriot), and converted into left circularly polarized light (l.c.p.) for Q1 oriented at 45°(fixed throughout the experiment). The l.c.p. beam was reflected from mirror M1 andchanged into right circularly polarized (r.c.p.) and again passed through Q1. It wasturned into p-polarized light and completely transmitted through PBS1 in the outputof the first stage of the interferometer. The transmitted beam (only p-polarized light)from PBS1 was passed through quarter-wave plate Q2 and changed into r.c.p. lightfor Q2 oriented at 45°. The r.c.p. beam was reflected from mirror M2 and convertedinto l.c.p. and again passed through Q2. It turned into s-polarization and reflectedfrom PBS1. Thus, the two orthogonally linear polarized beams came out from PBS1.After passing through the quarter-wave plate Q3 these two orthogonally polarizedbeams converted into l.c.p. and r.c.p. The two beams were launched into another similarsetup with one polarizing beam splitter (PBS2), two quarter-wave plates (Q4 and Q5)and two mirrors (M3 and M4). The output of the second stage consists of two pairs oforthogonally polarized beams. These four beams were passed through a quarter-waveplate Q6 and subjected into another similar interferometric setup comprising onepolarizing beam splitter (PBS3), two quarter-wave plates (Q7 and Q8) and two mirrors(M5 and M6). The four pairs of orthogonally polarized beams (total of eight polarized

Multiple polarized beam interferometers ... 497

beams) came out from the final stage. These polarized beams after passing throughthe quarter-wave plate Q9 became circularly polarized and produced non-observablefringes. The fringes were observed using a polarizer P2 at the output plane. The Q1–Q9and P2 were oriented in such a way that the optic axis of each made 45° with respectto the plane of polarization of the wave. Mirrors M1, M3 and M5 were kept for normalincidence and mirrors M2, M4 and M6 were adjusted by giving tilt for the interferencepattern of individual stages. To generate the regular hexagonal pattern mirrors M2,M4 and M6 were given the tilt such that the interferograms from each stage wereoriented 60° with each other and had same spatial frequencies. The use of polarizedbeam splitter does not bring about the problem of repeated splitting of the beams inany arm and hence the Michelson like geometry has the advantage as it requires lessnumber of optical components compared to the Mach–Zehnder configuration.

Fig. 1. Experimental setup for generation of hexagonal arrays, P – polarizers, PBS – polarizing cube beamsplitters, Q – quarter-wave plates, CCD – charged coupled device – a; Schematic of the path of all eightbeams – b; Location of the center of eight beams in the transverse plane, defining φp – c.

a

bc


The complex field distribution of the entire set of eight beams at the output planeis given by:

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

The resultant intensity of the eight beams is given by

(9)

where μ1 = 2πsin(θ1)/λ, μ2 = 2πsin(θ2cos(φ2))/λ, μ3 = 2πsin(θ3cos(φ3))/λ, ν2 == 2πsin(θ2sin(φ2))/λ, ν3 = 2πsin(θ3cos(φ3))/λ, a2 is the intensity of the incidentbeam, with θp (p = 1, 2, 3) being an angle between the two beams of the p-th stageinterferometer (Fig. 1b), and φp (p = 2, 3) an angle between the y-axis and the linejoining two beams of the p-th stage (Fig. 1c).

3. Results and discussion

The output, at any interferometric stage in the present experiment, i.e., after Q3, Q6or Q9 shows non-observable fringes. The fringes can be observed by placing a polarizerafter Q3, Q6 or Q9. When patterns from individual stages were oriented with respectto each other at 60° and were of exactly the same frequency, then only the resultantinterference pattern was of regular hexagonal geometry. The near field hexagonalpattern of regular geometry recorded onto CCD is shown in Fig. 2a. The measuredvalues of the angle between the beams were θ1 = θ2 = θ3 = 1.1 mrad and angle

u1a

16---------- 1 i+[ ] 1

1=

u2a

16---------- 1 i+[ ] 1

1 iμ1 y–( )exp=

u3a

16---------- 1 i+[ ] 1

1 i μ2 y ν2 x+( )–exp=

u4a

16---------- 1 i+[ ] 1

1 i μ1 y μ2 y ν2 x+ +( )–exp=

u5a

16---------- 1 i+[ ] 1

1 i μ3 y ν3 x–( )–exp=

u6a

16---------- 1 i+[ ] 1

1 i μ1 y μ3 y ν3 x–+( )–exp=

u7a

16---------- 1 i+[ ] 1

1 i μ2 y μ3 y ν2 x ν3 x–+ +( )–exp=

u8a

16---------- 1 i+[ ] 1

1 i μ1 y μ2 y μ3 y ν2 x ν3 x–+ + +( )–exp=

I Uii 1=

8

∑2

=


between the y-axis and the line joining two beams is φ2 = φ3 = 60°. The correspondingcomputed hexagonal array pattern from Eq. (9) is shown in Fig. 2b. It is in goodagreement with the experimental results. The appearance of faint spurious peaks inFig. 2a can be attributed to the front surface reflections from various opticalcomponents. The arrays were scanned to a distance of 2.5 meter without observingany significant loss of distortion and loss of contrast. The array pattern recorded ata distance of 2.5 m is shown in Fig. 3.

The variation of fringe visibility as a function of quarter-wave plate Q9 is shownin Fig. 4, where one can see that fringe visibility is sensitive in the range of orientationof Q9 from 0° to 20° and from 70° to 90°.

The efficiency of interferometric configuration was estimated by comparingthe output power in the plane of observation after P2 and the input power to

Fig. 2. Hexagonal array pattern at the out put of third stage of interferometer (a) recorded onto CCD,(b) computed pattern.

a b

Fig. 3. Hexagonal array pattern at a distance of 2.5 m.

Fig. 4. Fringe visibility curve for the orientationof the quarter-wave plate Q9.0 20 40 60 80 100

0.0

0.2

0.4

0.6

0.8

1.0

Frin

ge v

isib

ility

ρ1


the interferometers after P1. The ratio of the output power to the input power was0.384. In comparison with the eight-beam interferometers [9] of randomly polarizedsetup the efficiency of polarized setup is ~ 4 times better.

4. Several possible applications

There are widespread applications of the arrays. Some of the possible applications ofthese arrays are discussed here.

The interferometric array illuminators have non-uniform light intensity distributionwithin each spot, the spot densities and width of which can be varied in real time,which allows some possible applications in modifying the trajectories of the atoms.An atom placed in such a non-uniform optical field experiences the dipole force[12, 13]. Due to the interaction of induced dipole moment in the presence of field withnon-uniform light distribution the dipole force is generated [12]. The magnitude ofthis force depends on the intensity gradient and the amount of detuning from the atomicresonance frequency. This dipole force can modify the trajectories of atoms and bycarefully choosing the parameters it can lead to the focused spots of atoms of the orderof tens of nanometers [14], with periodicity down to λ /8 [15]. The interference fringescan also be used as optical tweezers for the alignment and manipulation of low-indexspheres [16]. Optical tweezer techniques found widespread application in biologicalresearch as well.

The delocalized arrays reported in the present paper can be assumed to be travelingin the form of optical channel. There is an intensity variation in any transverse planeof the interferometric arrays. If a nonlinear medium with an intensity dependentnon-linear refractive index is placed in the path of the arrays, an array of light channelswill be established in the medium. These optical channels in the medium will behavelike a light induced graded-index polarization dependent optical planar waveguide.The state of polarization within the spots of arrays can be changed from linearlypolarized light to circularly polarized light or elliptical polarized light. Hence, thisset up may find applications in light induced graded index optical channel [17], atomicbeam channels [4] and polarization optical switching. The polarized arrays can findapplication in measuring the properties of birefringent media [18].

The interference of the polarized beams can also be used to generate the periodicmicrostructures. Thus, an array illuminator can be used to write down the three--dimensional photonic crystal [19]. By controlling the relative polarization of eightinterfering beams and their angular separations, the periodicity and the geometry ofthe arrays can be controlled on line. Therefore, the band gap of such engineeredmaterials can be controlled. By changing the geometry and spatial frequencies ofarrays, periodic multiple structures can also be generated which allows further finetuning on to the band gap. These photonic band gap materials have promisingapplication in fabricating optical waveguide, high capacity data storage devices, digitaloptical computing [20], and in other related areas. The dependence of fringe visibilityon the orientation of the quater-wave plate Q9, can be used in on line optical testing.


5. Conclusions

We have demonstrated the formation of regular hexagonal arrays of tiny light spotswith good contrast using interference of eight polarized beams. The efficiency ofthe polarized setup is four times more than that of the randomly polarized setup.The interference patterns were observed without any significant loss of quality todistances of 2.5 m, confirming the formation of delocalized arrays. The computedpattern shows good agreement with experimental results. Hence, using computercontrolled motorized optomechanical mounts, an array pattern can be obtained inthe predetermined geometry or can be controlled on line in a programmable way.In the present set-up, we reported the improved efficiency compared to randomlypolarized set-up but the efficiency is low compared to some of the diffraction basedtechniques. However, this technique has the major advantage in terms of on line controlof the array geometry, no requirement of specialized aperture/grating, and the patterncan be scanned in the longitudinal direction. In the present paper, the analysis wasfocused on the bright spot or, in other words, on the intensity distribution, a simplerecordable parameter for the interference where the phase information is embedded inthe interference field. There have also been reports on alternative approach of opticalvortices seeded in the dark spot of the interference field for the analysis of phasesingularity [21, 22]. Some of the potential applications are also listed in the article.

References[1] LIU L., LIU X., CUI B., Optical programmable cellular logic array for image processing, Applied

Optics 30(8), 1991, pp. 943–49.[2] MORTIMORE D.B., ARKWRIGHT J.W., Monolithic wavelength-flattened 1×7 single-mode fused fiber

couplers: theory, fabrication, and analysis, Applied Optics 30(6), 1991, pp. 650–9.[3] SANDOGHDAR V., DRODOFSKY U., SCHULZE TH., BREZGER B., DREWSEN M., PFAU T., MLYNEK J.,

Lithography using nano lens arrays made of light, Journal of Modern Optics 44(10), 1997,pp. 1883–98.

[4] CHANTELOUP J.C., Multiple-wave lateral shearing interferometry fro wave-front sensing, AppliedOptics 44(9), 2005, pp. 1559–71.

[5] PRIMOT J., Three-wave lateral shearing interferometer, Applied Optics 32(31), 1993, pp. 6242–9.[6] PRIMOT J., SOGNO L., Achromatic three-wave (or more) lateral shearing interferometer, Journal of

the Optical Society of America A: Optics, Image Science and Vision 12(12), 1995, pp. 2679–85.[7] PATRA A.S., KHARE A., Interferometric array generation, Optics and Laser Technology 38(1), 2006,

pp. 37–45.[8] GUERINEAU N., PRIMOT J., Nondiffracting array generation using an N-wave interferometer, Journal

of the Optical Society of America A: Optics, Image Science and Vision 16(2), 1999, pp. 293–8.[9] PATRA A.S., Multiple Beams Interferometry for Array Illuminator and its Application, Ph.D. Thesis,

2005. [10] SENTHILKUMARAN P., Interferometric array illuminator with analysis of nonobservable fringes,

Applied Optics 38(8), 1999, pp. 1311–6.[11] PATRA A.S., KHARE A., Generation and fringe visibility studies of a non-observable array illuminator

using polarized beams, Journal of Optics A: Pure and Applied Optics 7(10), 2005, pp. 535–9.[12] COHEN-TANNOUDJI C., DUPONT-ROC J., GRYNBERG G., Atom-Photon Interaction, Wiley, New York

1992.


[13] DALIBARD J., RAIMOND J.M., ZINN-JUSTIN J., Fundamental Systems in Quantum Optics, NorthHolland, Amsterdam 1992.

[14] GUPTA R., MCCELLAND J.J., JABBOUR Z.J., CELOTTA R.J., Nanofabrication of two-dimensional arrayusing laser-focused atomic deposition, Applied Physics Letters 67(10), 1995, pp. 1378–80.

[15] BREZER B., SCHULZE TH., SCHMIDT P.O., MERTENS R., PFAU T., MLYNEK J., Polarization gradientlight masks in atom lithography, Europhysics Letters 46(2), 1999, pp. 148–53.

[16] MACDONALD M.P., PATERSON L., SIBBETT W., DHOLAKIA K., BRYANT P.E., Trapping andmanipulation of low-index particles in a two-dimensional interferometric optical trap, OpticsLetters 26(12), 2001, pp. 863–5.

[17] NEMOTO S., KIDA J., Reflector using gradient-index rods, Applied Optics 30(7), 1991, pp. 815–22.[18] BORWIŃSKA M., POPIOŁEK-MASAJADA A., KURZYNOWSKI P., Masurement of birefringent media

properties using optical vortex birefringence compensator, Applied Optics 46(25), 2007,pp. 6419–26.

[19] CAMPBELL M., SHARP D.N., HARRISON M.T., DENNING R.G., TURBERFIELD A.J., Fabrication ofphotonic crystals for the visible spectrum by holographic lithography, Nature 404(6773), 2000,pp. 53–6.

[20] LOHMANN A.W., What classical optics can do for the digital computer, Applied Optics 25(10), 1986,pp. 1543–9.

[21] MASAJADA J., POPIOLEK-MASAJADA A., LENIEC M., Ceation of vortex lattices by a wavefront division,Optics Express 15(8), 2007, pp. 5196–207.

[22] BORWINSKA M., KURZYNOWSKI P., Generation of vortex-type marker in a one-wave setup, AppliedOptics 46(5), 2007, pp. 676–9.

Received January 14, 2008in revised form February 21, 2008


Characteristic analysis on an interleaver with a fiber loop resonator by using a signal flow graph method

WEIBIN LI1, 2 , JUNQIANG SUN1

1Wuhan National Laboratory for Optoelectronics, School of Optoelectronic Science and Engineering, Huazhong University of Science and Technology, Wuhan 430074, Hubei, P.R. China

2Department of Physics and Chemistry, Henan Polytechnic University, Jiaozuo 454000, Henan, P.R. China

We propose an interleaver based on the Mach–Zehnder interferometer with a resonatorincorporated in one arm. By using the method of signal flow graph, we get the simple closed-formexpressions for transmission of this interleaver. The result indicates that the widths of 0.5 dBpassband and 25 dB stopband of the interleaver are improved remarkably, which are much widerthan that of the conventional Mach–Zehnder interferometer. The interleaver proposed has an idealrectangular spectral response. Finally, we have analyzed the influence exerted by the parameterson its transimission characteristics and found that the transmission spectrum of this device dependshighly on the intensity coupling coefficient of the coupler and fiber loop length.

Keywords: interleaver, resonator, signal flow graph, transmission.

1. Introduction

The dense wavelength division multiplexing (DWDM) is an effective method to increasethe transmission capacity for optical communications. If one wishes to decreasethe channel spacing in order to enlarge the transmission capacity, a so-calledinterleaver can be used to separate the input DWDM wavelengths into two interleavedsequences, and consequently, the channel spacing for each sequence is doubled ascompared to the original DWDM channel spacing. This is a more cost-effective wayto realize the DWDM technology with a narrow channel spacing [1, 2]. Because ofthis, it receives much attention in optical communication community. As regardthe interleavers developed so far, most designs focus on Gires–Tournois based(GT-based) Michelson interferometer [1, 3], arrayed-waveguide router (AWG) [4, 5],ring resonator type filters [6], Mach–Zehnder interferometer (MZI) filters made withplanar waveguides or fused fiber [7–9], and lattice filter employing birefringent

504 W. LI, J. SUN

crystals [10], etc. Among those filters, the MZI filters are potential candidates, sincethey are inherently low loss, spectrally extremely selective, and potentially low cost.Furthermore, the MZIs have other important applications in optical nonlinearitymeasurement [11] and optical fiber sensing field [12]. So, the MZIs have attractedconsiderable interest.

In applications, the filters should have an ideal rectangular spectral response,i.e., large passband width with a flat top spectrum, and steep slope to achieve highisolation between adjacent channels over a large stopband region, which can relaxthe requirements on wavelength control for lasers. However, the spectral response ofa conventional MZI is sinusoid, due to which the interleaver is limited in the bandwidthof passband/stopband. So, the conventional MZI is not appropriate for use. In orderto obtain a flat-top passband, an interleaver based on cascaded MZIs is used [13].A similar flat passband filter with 2×2 and 3×3 couplers has also been reported [14].In this paper, we design a new modified project. An interleaver based on MZI witha fiber loop resonator in one arm is presented. The result indicates that the interleaverin this paper can simultaneously improve the widths of 0.5 dB passband and 25 dBstopband, which are much wider than those of the conventional Mach–Zehnderinterferometer. The filtering performance of this novel interleaver, which achievesa nearly square spectrum response, is much better than that of the conventional MZI.

2. Theory and simulation analysis

Figure 1 shows a configuration of the interleaver proposed. It consists of an MZIwith a ring resonator in one arm. The difference between this device and conventionalMZI lies in a ring resonator used in one of the MZI arms. This can cause a multi-lightbeam interference effect, which results in the new change of phase between the twoarms of MZI.

Various techniques have been used for the analysis of fiber optic and filter [15].There are two main classes of such techniques. The first class comprises a set ofanalytical methods including the scattering matrix method, the transfer matrix/chainmatrix algebraic method, and the method of solving the field equations. In the secondclass, use is made of a graphical approach; it is called the signal flow graph (SFG)method as proposed by MASON [16]. This method has originally been used in theelectrical circuits, which is not a common practice in the analysis of optical circuits.The advantage of this method is that relations between unknown variables could bedisplayed explicitly and it provides better insight into the interaction of the systemcomponents. In addition, a set of equations corresponding to the graph can be easilyderived by the association rule. This paper thus aims to employ this approach inthe analytical derivation for the optical transfer functions of the structure proposed.

The SFG of the device is shown in Fig. 2. To determine the transmittance fromnode 1 to node 10, we examined all the paths going from node 1 to node 10 in the SFG.

Characteristic analysis on an interleaver ... 505

By inspection, there are three forward paths: P1, 1 2 4 7 8 10; P2, 1 2 4 5 f4 6 7 8 10;P3, 1 3 9 10. Because these paths are the product of their directed branches,respectively, P1, P2 and P3 can be written as:

(1)

where xi = (1 – ki)1/2, yi = j (ki)1/2 (i = 1, 2, 3), ki is the intensity coupling coefficient ofthe corresponding coupler; and fi = exp(iβLi), Li is the length of the fiber shown inFig. 1, f4 = exp(iβL), L is the fiber length of the fiber loop. In addition, we examinedthe SFG, with only one feedback loop, which is presented by nodes: 5 f4 6 5. Itsvalue is y2 f4. Also, we can see that only P2 touches this feedback loop from the SFG.So, the Δi can be separately written as:

(2)

P1 f1 f2 x1 x3 y2=

P2 f1 f2 f4 x1 x22 x3=

P3 f3 y1 y3=⎩⎪⎪⎨⎪⎪⎧

Δ1 1 y2 f4–=

Δ2 1=

Δ3 1 y2 f4–=⎩⎪⎪⎨⎪⎪⎧

6 7

541

2

3

8 10

9 11

K

L

E1

E2 L3

L2 E3

E4

L1

Fig. 1. Configuration of the MZI with a fiber loop resonator.

Fig. 2. SFG of the MZI with a ring resonator.

506 W. LI, J. SUN

Using the Mason rule, the transmission of the node 1 to 10 can be written as:

(3)

with Δ = Δ1. Using the above formulas, we obtain

(4)

We assume 3 dB couplers are used in MZI. The optical fiber and coupler lossesare neglected. The coefficient k is the intensity coupling coefficient of the coupler inthe fiber loop resonator. ΔL = L3 – L1 – L2 is the pathlength difference of the twoarms of MZI. Having satisfied the resonating condition, ΔL, Δλ and L should satisfythe following relationships [8]:

(5)

with λ0, neff and Δλ standing for the center wavelength, the fiber effective index andthe channel spacing, respectively.

Figure 3 shows the transmission spectra of MZI with a ring resonator as a functionof frequency, where ΔL is set so as to obtain channel spacing of approximately 0.8 nmat center frequency f0 ≈ 193.4 THz (corresponding center wavelength λ0 = 1550 nm).The intensity coupling coefficient k is 0.25 and neff is 1.47. It can be seen from Fig. 3that the transmission spectrum of T is characterized by a series of equally spacedtransmission peaks in the frequency domain, and the transfer function of rectangular--shaped frequency is accomplished.

For better understanding the filter characteristics of this device, the simulationresults of a 25 dB stopband in 50 GHz channel spacing application are shown in Fig. 4.To have a closer look at the top portion of the spectra, Fig. 4a is enlarged and its topportion is shown in Fig. 4b. In these figures, the spectral response of the interleaverproposed is marked with solid line, and the dashed line represents a conventional MZI.It is clear that the flat-top ripple is less than 1.3×10–2 dB and the 0.5 dB passband widthis about 42 GHz, which is near 84% of 50 GHz channel spacing. The 25 dB stopbandwidth is found to be about 34 GHz (being nearly 68% of 50 GHz channel spacing).

tE3

E1----------

P1Δ1 P2Δ2 P3Δ3+ +Δ

--------------------------------------------------------= =

T E3

E1----------

3

12--- 1 k+( ) k βΔL( )sin k βL( )sin+– 1

2--- β ΔL L–( )cos– k

2--- β ΔL L+( )cos+

1 k 2 k βL( )sin+ +---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------

=

=

L 2ΔLλ0

4neff----------------–=

ΔLλ0

2

neff Δλ---------------------=

⎩⎪⎪⎨⎪⎪⎧


The channel isolation is, at least, to be better than 25 dB between the adjacent channels.The conventional MZI’s 0.5 dB passband and 25 dB stopband widths are 21.4 GHz,3.6 GHz, respectively. They correspondingly account for the 50 GHz channel spacing42.8% and 7.2%. From Fig. 4, one can see that the interleaver proposed providesthe 25 dB stopband and 0.5 dB passband widths, which are larger than those ofconventional MZI. It is clear from the figures that the proposed interleaver is attractivein flat passbands. In addition, the 0.5 dB passband widths of the interleaver reportedin [13, 14] are about 26 GHz and 33 GHz, and the 25 dB stopband widths are 24 GHz,31 GHz, respectively. From this comparison, we can see that the interleaver proposedin this paper offers better filter effect.

From Eq. (4), we can see two new parameters, k and L, introduced in the deviceproposed. It is necessary to study the influence of these parameters on transimissioncharacteristics of the interleaver proposed.

In Figure 5, transmission spectra of T versus frequency, with different k, arepresented. The B, C, D, E, F correspond to the values of intensity coupling coefficientk for 1/4, 1/2, 3/4, 7/8 and 1. It is clear from the figure that all the curves intersect at

Fig. 3. Plot of transmission T versus frequency.

193.38 193.4 193.42 193.44–0.5

–0.4

–0.3

–0.2

–0.1

0

Frequency [THz]

Tran

smis

sion

[dB]

193.25 193.3 193.35 193.4 193.45 193.5–30

–20

–10

0

Frequency [THz]

Tran

smis

sion

[dB

]

Fig. 4. Simulantion results of a 25 dB stopband (a) and a 0.5 dB passband (b) of the interleaver in 50 GHzchannel spacing. Solid line: the interleaver proposed; dashed line: conventional MZI interleaver.

a b

508 W. LI, J. SUN

half maximum intensity 1/2, and the intensity in the passband and that in the stopbandof the interleaver are complementary. Note that the curve with k = 1 restores to that ofconventional MZI. In addition to this, when k = 0, the transmission spectra are alsothose of the conventional MZI. This is because the multilight beam interferencehas not been produced in the fiber loop resonator, and thus the proposed interleaveris equivalent to the conventional MZI. From the figure, it can also be seen that asthe value of k increases, one peak and one trough symmetrically appear on either sideof the center frequency; these extreme points move outwards and the trough getsdeeper, resulting in a wider bandwidth but with a large ripple. Besides, with an increaseof k, the side lobe is enhanced, which results in the channel isolation reduction. So, inthe real-world application, k should take a small value to satisfy a given isolationrequirement.

Fig. 5. Transmission spectrum T versus frequency with different k; B, C, D, E, F correspond to the valuesof k for 1/4, 1/2, 3/4, 7/8 and 1.

Fig. 6. The transmission spectra of interleaver with different L; B, C, D, E correspond to the change of L,which are 0, 100, 200, 300 nm, respectively.


Because the transmission spectrum of interleaver is sensitive to the length of fiberloop, the system has strict request for the optical fiber loop length. In order to showhow big this kind of influence is, we have drawn transmission spectra of the interleaverproposed with different L in Fig. 6. The parameters in the figure are: ΔL ≈ 2.05 mm,L ≈ 4.1 mm, λ0 = 1550 nm, k = 0.25 and Δλ = 0.8 nm; B denotes the curve when L is4.1 mm; C, D, E are the curves when the length changes are 100 nm, 200 nm and300 nm, respectively. One can see in the figure one trough appearing on the right-handside of the center wavelength with L increasing. This causes the ripple to increase.It can also be seen that as the value of L increases, the side lobe is enhanced, whichresults in the channel isolation reduction. Furthermore, the center wavelength showsthe drifting along the long wave direction. The cause of these phenomena is thatmatching condition, L = 2ΔL –λ0/(4neff), has been destroyed with the change of L. Inbrief, when L changes, the shape of the curve is distorted. In order to achieve the bestsystem performance, the L drifting should be smaller than 100 nm by our calculation.

3. Conclusions

In this paper, we have proposed a novel interleaver based on MZI with a fiber loopresonator in one arm. The interleaver can have the widths of 0.5 dB passband and 25 dBstopband simultaneously improved, which are much wider than those of conventionalMach–Zehnder interferometer. The interleaver proposed has an ideal rectangularspectral response, i.e., large passband bandwidth with a flat top spectrum, steep slopeand high isolation. This performance can relax the requirements on wavelengthcontrol for lasers in a DWDM system. Finally, we have analyzed the influence exertedby the parameters on its transimission characteristics and found that the transmissionspectrum of this device depends highly on the intensity coupling coefficient ofthe coupler and fiber loop length. The calculation and analysis in this paper aim atthe optical fiber, and the interleaver may be realized by using the plane waveguidetechnology.

References[1] HSIEH C.-H., WANG R., WEN Z.J., MCMICHAEL I., YEH P., LEE C.-W., CHENG W.-H., Flat-top

interleavers using two Gires–Tournois etalons as phase-dispersive mirrors in a Michelsoninterferometer, IEEE Photonics Technology Letters 15(2), 2003, pp. 242–44.

[2] MIZUNO T., KITOH I., SAIDA I., OGUMA M., SHIBATA T., HIBINO Y., Dispersionless interleave filterbased on transversal form optical filter, Electronics Letters 38 (19), 2002, pp. 1121–22.

[3] DINGEL B.B., IZUTSU M., Multifunction optical filter with a Michelson–Gires–Tournoisinterferometer for wavelength-division-multiplexed network system applications, Optics Letters23(14), 1998, pp. 1099–101.

[4] OKAMOTO K., SUGITA A., Flat spectral response arrayed-waveguide grating multiplexer withparabolic waveguide horns, Electronics Letters 32(18), 1996, pp. 1661–2.

[5] AMERSFOORT M.R., SOOLE J.B.D., LEBLANC H.P., ANDREADAKIS N.C., RAJHEL A. CANEAU C., Passbandbroadening of integrated arrayed waveguide filters using multimode interference couplers,Electronics Letters 32(5), 1996, pp. 449–51.

510 W. LI, J. SUN

[6] KOHTOKU M., OKU S., KADOTA Y., SHIBATA Y., YOSHIKUNI Y., 200-GHz FSR periodic multi/demultiplexer with flattened transmission and rejection band by using a Mach–Zehnderinterferometer with a ring resonator, IEEE Photonics Technology Letters 12(9), 2000, pp. 1174–6.

[7] OGUMA M., JINGUJI K., KITOH T., SHIBATA T., HIMENO A., Flat-passband interleave filter with200 GHz channel spacing based on planar lightwave circuit-type lattice structure, ElectronicsLetters 36(15), 2000, pp. 1299–300.

[8] JINGUJI K., OGUMA M., Optical half-band filters, Journal of Lightwave Technology 18(2), 2000,pp. 252–9.

[9] WANG Q.J., ZHANG Y., SOH Y.C., All-fiber 3×3 interleaver design with flat-top passband, IEEEPhotonics Technology Letters 16(1), 2004, pp. 168–70.

[10] DAMASK J.N., DOERR C.R., Polarization diversity for birefringent filters, US Patent 6252711, B1.2001.

[11] VINEGONI C., WEGMULLER M., GISIN N., Measurements of the nonlinear coefficient of standard, SMF,DSF, and DCF fibers using a self-aligned interferometer and a Faraday mirror, IEEE PhotonicsTechnology Letters 13(12), 2001, pp. 1337–9.

[12] SONG M., YIN S., RUFFIN P.B., Fiber Bragg grating strain sensor demodulation with quadraturesampling of a Mach–Zehnder interferometer, Applied Optics 39 (7), 2000, pp. 1106–11.

[13] HUAI-WEI LU, YU-E ZHANG, GUAN-WEI LUO, Study of all-fiber flat-top passband interleaver basedon 2×2 and 3×3 fiber couplers, Optics Communications 276(1), 2007, pp. 116–21.

[14] HUAI-WEI LU, BAO-GE ZHANG, MIN-ZHI LI, GUAN-WEI LUO, A novel all-fiber optical interleaver withflat-top passband, IEEE Photonics Technology Letters 18(13), 2006, pp. 1469–71.

[15] YUPAPIN P.P., SAEUNG P., LI C., Characteristics of complementary ring-resonator add/drop filtersmodeling by using graphical approach, Optics Communications 272(1), 2007, pp. 81–6.

[16] MASON S.J., Feedback theory – further properties of signal flow graphs, Proceedings of the IRE44(7), 1956, pp. 920–6.

Received February 26, 2008


Beam shaping based on intermediate zone diffraction of a micro-aperture

DANYAN ZENG, ZHIJUN SUN*

Department of Physics, Xiamen University, Xiamen 361005, China


We analyze optical diffraction of a micro-aperture (slit or hole) in a metal screen in the intermediatezone and report its application for beam focusing and collimating in micro-optics. Both finite--difference time-domain simulations and Rayleigh–Sommerfeld diffraction formula were appliedto calculate the intermediate-zone diffraction patterns. It is shown that, by controlling the aperturesize, the focal length and depth can be adjusted in a very wide range, from subwavelength to tensof wavelengths, while the focal width maintains in an order of wavelength.

Keywords: aperture, diffraction, beam shaping.

1. Introduction

The study of optical diffraction/scattering of apertures has a long history [1–4].Considering the size of an aperture (a << λ, a ~ λ, a >> λ) and position of an observationpoint (near field zone R << λ, intermediate field zone R ~ λ, far field zone R >> λ),diffraction can be categorized according to the different kinds. Being of interest, farfield diffractions have been elaborately studied for both large apertures (known asFraunhofer diffraction or Fresnel diffraction [4]) and small apertures (a ~ λ) [5, 6].Theories based on assumptions and proper approximations accurately describe farfield diffractions in elegant expressions [4–7]. For a tiny aperture (a << λ), near fieldelectro-optical interactions are in the focus of the study, where finite conductivity ofthe metal of the opaque plane has to be considered [2, 6, 8]. And recently, opticaltransmission through such a tiny aperture or periodically arrayed apertures has beenthe subject of extensive research [9–11]. Nowadays, dimensions of integrated opticaldevices have reached a scale of tens of micrometers in micro-optical systems;further miniaturization will involve the intermediate zone diffraction of small apertures(~ micrometers). In analyzing such diffractions, effects of near field interactions willhave to be considered.

In this work, firstly we analyze the diffraction of a small slit in a metal plane withfinite-difference time-domain (FDTD) numerical method, which rigorously solves

512 DANYAN ZENG, ZHIJUN SUN

the problem based on Maxwell’s equations and Drude model of metal. Both TE (electricfield is parallel with slit) and TM (magnetic field is parallel with slit) polarizations atnormal incidence are simulated and compared with those obtained with the scalar-fieldRayleigh–Sommerfeld (R–S) diffraction formula [12]. Diffraction patterns with localmaxima of different orders are observed in the intermediate zone. Of particularinterest is the 0th-order diffraction in this zone, which offers strong potential for beamfocusing or collimating applications in micro-optics. The effect/function can then beextended to micro-apertures considering practically insensitivity of field patterns topolarization in the intermediate zone.

2. Optical diffraction of a micro-aperture in the intermediate zone

Figure 1 shows FDTD calculated distribution of average intensity of diffracted lightnormally incident on a slit of varying widths (a = λ, 2λ, 3λ, 4λ) in a metal (Ag) filmin TE/TM polarizations, which are compared with those calculated with scalar-fieldR–S diffraction formula. In the FDTD analysis, finite conductivity of the metal (Ag)is embodied in a Drude model fit of its experimentalpermittivity [13]. The thickness of the metal screen is 100 nm, discretized withthe grid size of 10 nm. An optical plane wave of 600 nm wavelength is normallyincident onto the screen. Since normalized units [14] and sameincidence field amplitude of unity have been used, the calculated FDTD absolutevalues of intensity in Fig. 1 should be comparable for both polarizations. Incalculations with scalar R–S formula [12], the kernel function was replaced with G0 == for the 2-dimensional cylindrical waves, i.e., the scalar field

(1)

at the observation points. The corresponding intensity is I = φφ*. Here, k = 2π/λ,R is a vector from point (x', 0) in the slit plane to the observation point (x, y) (z directionis perpendicular to the paper plane), n' is the unit normal vector of the slit plane.The field magnitude φ (x', 0) within the slit is considered as that of an unperturbedincidence wave as usual. But it will be shown that a field distribution equivalent tothe perturbed one in reality can be self-constructed by evolution of the unperturbedone within a very short spatial shift.

From Figure 1, one can observe similarity of the local diffraction patterns inthe intermediate zone for TE/TM polarizations and those calculated from scalar R–Sformula. A further examination indicates larger transmissivity for TE polarizationincidence. It is contrary to the case for very narrow slits (a << λ), in which surfacewave of plasmons excited at the slit enhances transmission of TM polarized incidencelight, while it is within a cut-off of waveguiding for TE polarization. Here, the inducedEy-field in the TM case generates surface waves on the metal plane, which bring moreabsorption and result in the transmission reduction (shown in Fig. 2 as an example). If

ε ω( ) ε∞ ω p2 ω2 iγω+( )⁄–=( )

E ε0 μ0⁄ E=( )

ikR( )exp R⁄

φ x y,( ) kπi

--------- φ x' 0,( ) n' R⋅R

------------------ 1 i2kR

-------------+⎝ ⎠⎜ ⎟⎛ ⎞ ikR( )exp

R---------------------------- dx'

y 0=∫=

Beam shaping based on intermediate zone diffraction of a micro-aperture 513

neglecting the near field effects, diffraction of a slit can be considered as a superpositionof forward transmitting waves and perturbation waves induced by the edges. Inthe results shown in Fig. 1, the perturbed field near the slit exit has M = Int(a/λ)maxima of intensity, different from that of standing wave type interference forming

Fig. 1. FDTD simulated average intensity distributions of light diffracted by slits of varying widths ina metal (Ag) film for TE and TM polarizations, compared with those calculated with scalar R–Sformula. In FDTD simulations, the thickness of the metal film is 100 nm, and the plane wave of 600 nmwavelength is normally incident onto both the screen and the slit. Images of each column have the sameslit widths as indicated above of the set of photos, and each row corresponds to TE or TM polarizationin FDTD simulations or scalar analysis with R–S formula. Each colorbar is applied to images of the samerow. Sizes of all the images are the same (5×5 μm2).

Fig. 2. Distributions of Hy (or Ey) field amplitude for a slit of width 3λ under illumination of TE (or TM)polarized light. The plane wave of wavelength λ = 600 nm is incident from the bottom side of each image.


between the edge boundaries, which correspond to 2M maxima in principle. As itpropagates forward through the slit, it evolves into degressive numbers (M – 1,M – 2, ..., 1) of maxima in the intermediate zone, which relate to far field diffractionbeams of the same orders. Among the local maxima in the intermediate zone,the 0th-order shows strong confinement of transmitted power in the transversedirection and there is a notable extent in the longitudinal direction. This property mightbe used for beam shaping in the micro-optics regime.

We also calculated phase distributions of partial fields with FDTD simulations inFig. 3. It is interesting to find that the geometric symmetry results in antisymmetricphase distributions of the induced fields Ey (for TM) and Hy (for TE) with respect tothe middle line. There are directions that transverse differentiation of phase of the fieldEy (for TM) or Hy (for TE) is large or abrupt (singular). It is shown, in Fig. 2, that

Fig. 3. Instant phase distributions of partial fields for a slit of width 3λ under illumination of TE (or TM)polarized light. The plane wave of wavelength λ = 600 nm is incident from the bottom side of each image.

Fig. 4. A zoom-in line plot of the diffraction intensity distribution near the secondary source line of a slitcalculated with R–S formula, which shows self-construction of equivalent perturbed field in evolutionof the secondary source with finite width. The corresponding slit width is 1.8 μm and the wavelength is600 nm.


amplitudes of the partial fields appear minimal in such directions; while diffractionintensity in these directions takes its maxima in far field.

As for calculation with the R–S formula, it neglects perturbation on initialfield within the aperture and near field interactions at the metal screen surfaces.The dependence of diffraction on polarization is subject to the near field effects. Asthe slit is larger than incidence wavelength, the near field effects are not dominant,especially for the 0th-order diffractive transmission. It is also found that, althoughan unperturbed field in the slit is assumed in the calculations, an equivalent edgediffraction effect can result by the assumed finite secondary sources (see Fig. 4).Self-construction of “perturbed” field shows effect immediately in an infinitesimalshift from the source line. Notice that it is just the result of implementation ofR–S formula (Eq. (1)); for the far field (R >> λ) case, the equation evolves intoan approximation of Fourier transform. Self-construction of the perturbed field inthe aperture discloses the historical issue of how false assumptions of the diffractiontheory (Kirchhoff’s theory, or in Rayleigh–Sommerfeld form) lead to correct predictions.From comparison with FDTD simulations, it also suggests that the scalar formulacannot only be applied for diffraction of large apertures and far field diffractions, italso works well in the intermediate zone for a small aperture of dimension in the orderof wavelength.

3. Performance of a micro-slit/hole lens for beam forming

We further evaluated the beam focusing of the 0th-order maxima for wider slits ofinterest (up to ten wavelengths). It was found, as shown in Fig. 5, that the focal length(distance of maxima to slit screen) can extend to a range of tens of wavelengths, whilethe focal width (defined with full-width-half-maximum, FWHM) maintains withina few wavelengths. The focal depth (defined as the spacing between a 10% drop fromthe focal maximum in the forward direction) also varies in a range from subwavelengthto tens of wavelengths with the widening of the slit. Thus, for a narrower slit, it worksbetter as a focusing lens; for a wider slit, it can also work as a collimator. The collimatedbeam was shown to have nearly plane wave fronts and Gaussian distribution ofintensity in transverse direction. In extrapolation, for large slits (a >> λ), the focal depthapproaches infinity, the diffraction principal maxima of which in far field appeartransversely well confined and nearly uniform in propagation direction. But for smallslits, far field diffraction is far away from the maxima near slit, thus it is highlyattenuated.

Note that, although the above analysis is based on the wavelength of 600 nm,the results can be scaled with the wavelength and generalized in that the near fieldeffects are negligible for intermediate zone diffraction from the point of view ofapplications. The scalar analysis can also be applied to apertures of rectangular andcircular shape, where the kernel equation is taken for spherical waves [12], G0 == exp(ikR)/R. Well-defined beam focusing is observed, as shown in Fig. 6. Notice that,


for rectangular aperture of size within a few wavelengths, it needs to be square to avoidmismatching of focus for orthogonal directions of diffraction. It is worth mentioningthat such intermediate zone diffraction pattern of a small aperture is very similar tothat of a small sphere, which also shows some focusing effect [15]. But the smallaperture shows obvious advantages in flexibilities of design, manufacturing andfunctioning, e.g., in integration with other photonic or optoelectronic arrayed devices.For conventional micro-optic lens, as its dimensions are reduced to a scale of

Fig. 5. Dependence of beam focusing parameters on slit width calculated with R–S formula. A colormapof intensity along the middle line of forward propagating beam for different slit widths (a). Dependenceof focal length (FL) on slit width (b). Dependence of FWHM focal width on slit width (c). The label Δyin (a) indicates focal depth of 11 μm for slit width of 5 μm. The reference wavelength is 600 nm incalculations.

a

b c

Fig. 6. Sliced 3D images of intermediate zone diffraction patterns of rectangular (1.8×1.8 μm2) – a,and circular (1.8 μm in diameter) – b apertures. Wavelength of normal incidence plane wave is 600 nm.Inset of each image is a transverse cross-section display of intensity distribution at the focal point.

a b


micrometers, it fails to function because of edge diffraction and micro-cavity effects.If it is to achieve the same level of performance with high numerical aperture (NA)micro-optic lens, the integration density will have to be reduced, far less comparablewith a lens array of just micrometers-spaced apertures.

4. Conclusions

By analyzing the intermediate zone diffraction of a small aperture, we proposed sometype of optical beam shaping lens consisting of just a micro-aperture for beam focusingand collimating for ultra-compact micro-optic applications. By controlling the aperturesize, the focal length and depth can be adjusted in a very wide range, fromsubwavelength to tens of wavelengths, while the focal width maintains narrow withinan order of wavelength. The functions can also be extended to other spectrum regimeof electromagnetic waves.

Acknowledgements – This work was supported by National Natural Science Foundation of China(No. 60707012), Natural Science Foundation of Fujian Province of China (No. A0710019) andthe NCETFJ program in China.

References[1] RAYLEIGH L., On the passage of waves through apertures in plane screens, and allied problems,

Philosophical Magazine 43, 1897, pp. 259–72.[2] BETHE H.A., Theory of diffraction by small holes, Physical Review 66(7–8), 1944, pp. 163–82.[3] BOUWKAMP C.J., Diffraction theory, Reports on Progress in Physics 17, 1954, pp. 35–100.[4] BORN M., WOLF E., Principles of Optics, 7th Ed., Cambridge University Press, 1999.[5] STRATTON J.A., CHU L.J., Diffraction theory of electromagnetic waves, Physical Review 56(1), 1939,

pp. 99–107. [6] SMYTHE W.R., The double current sheet in diffraction, Physical Review 72(11), 1947, pp. 1066–70.[7] JACKSON J.D., Classical Electrodynamics, 3rd Ed., Wiley, New York, 1998, Chap. 10.[8] SCHOUTEN H.F., VISSER T.D., LENSTRA D., BLOK H., Light transmission through a subwavelength

slit: Waveguiding and optical vortices, Physical Review E 67(3), 2003, p. 036608.[9] GIRARD C., DEREUX A., Near-field optics theories, Reports on Progress in Physics 59(5), 1996,

pp. 657–99.[10] EBBESEN T.W., LEZEC H.J., GHAEMI H.F., THIO T., WOLFF P.A., Extraordinary optical transmission

through sub-wavelength hole arrays, Nature 391(6668), 1998, pp. 667–9.[11] SUN Z., JUNG Y.S., KIM H.K., Role of surface plasmons in the optical interaction in metallic gratings

with narrow slits, Applied Physics Letters 83(15), 2003, pp. 3021–3.[12] JACKSON J.D., Classical Electrodynamics, 3rd Ed., Wiley, New York, 1998, p. 481.[13] PALIK E.D. [Ed], Handbook of Optical Constants of Solids, Academic, New York, 1998.[14] SULLIVAN D.M., Electromagnetic Simulation Using the FDTD Method, IEEE Press, New York, 2000.[15] KOFLER J., ARNOLD N., Axially symmetric focusing as a cuspoid diffraction catastrophe: Scalar and

vector cases and comparison with the theory of Mie, Physical Review B: Condensed Matter andMaterials Physics 73(23), 2006, p. 235401.

Received January 17, 2008


An iterative programmable graphics process unit based on ray casting approach for virtual endoscopy system

FEINIU YUAN

School of Information Technology, Jiangxi University of Finance and Economics, Nanchang 330013, Jiangxi, China; e-mail: [email protected]

State Key Lab of Fire Science, University of Science and Technology of China, Hefei 230027, Anhui, China

In this paper, a fast graphics process unit (GPU) based ray casting algorithm is presented to improveimage quality. A linear interpolation is used to estimate the intersection between a ray andisosurfaces. Thus, resampling artefacts is greatly reduced and the performance is not influenced.An iterative estimation is presented to further improve image quality. According to the distancethe ray goes across, z values in the z-buffer are modified to implement hiding of hybrid scenes.Experimental results show that the algorithm can produce high quality images at interactive framerates and implement hiding of hybrid scenes very well.

Keywords: virtual endoscopy, linear interpolation, medical image processing, graphics process unit(GPU), ray casting, isosurface.

1. IntroductionVirtual endoscopy is a new diagnostic and surgery planning method that is non-invasiveand reusable compared with traditional optical endoscopy. There are two main groupsof visualization techniques including surface rendering and volume rendering. Insurface rendering techniques, intermediate polygons of isosurfaces must be extractedfrom the 3D volumetric data set and then the polygons are rendered using the traditionalcomputer graphics rendering algorithms. It can obtain interactive rendering rates.However, the quality of the rendered images is not very high because of much detailedinformation lost during the process of extraction of polygons. While in volumerendering techniques, extraction of polygons is not required and the volume data setis directly rendered according to a specific transfer function, so it can produce highquality images. But it is memory and computation intensive. Ray casting algorithmis a volume rendering technique, which belongs to image space rendering techniques.

520 FEINIU YUAN

It can generate high quality images and often be applied in virtual endoscopy systems.It can obtain high frame rates on high end workstations and special volume renderinghardware (such as VolumePro [1], etc.) and so on. However, it is unable to acquireinteractive frame rates on the popular PC platform without special purpose hardware.

There are many acceleration techniques to speed up the brute force ray castingalgorithm. One kind of these techniques is the acceleration technique based on spaceleaping, which can avoid many empty samplings and does not degrade the image quality,such as cylindrical approximation of tubular organs presented by VILANOVA et al. [2],spherical approximation of tubular organs proposed by SHARGHI and RICKETTS [3].Another kind of these techniques is the acceleration technique with a fundamentaltradeoff between image quality and rendering speed, such as two-phase perspectiveray casting for interactive volume navigation presented by BRADY et al. [4], screen andobject adaptive sampling, etc.

A third kind of acceleration techniques is based on consumer level graphicshardware. There are two distinct approaches.

The first approach which originally presented by CULLIP et al. [5] and furtherdeveloped by CABRAL et al. [6] is directly exploiting the texture mapping capabilitiesof graphics hardware by resampling proxy surfaces. Proxy surfaces are either axis--aligned [7] with three 2D textures or view-aligned [8] with one 3D texture, as shownin Fig. 1. This technique can obtain interactive frame rates, but produce relatively lowquality images.

The second approach is to implement a ray caster in the fragment shader ofthe GPU, as proposed by KRUGER and WESTERMANN [9]. In the algorithm, the datasetis stored as a 3D-texture to take advantage of the built-in tri-linear interpolation ingraphics hardware [10]. And then a bounding box for the dataset is created where allthe coordinates of 8 corner points are within the range from 0.0 to 1.0. Supposed eachcorner point (x, y, z) of the bounding box has the color (r, g, b), and then r, g, b areequal to x, y, z. That is to say, the position on the surface of the box is encoded inthe color channel, as shown in the Fig. 2a. The viewing vector at any given pixel canbe easily computed by subtracting the color of the front surfaces of the color cube atthis pixel from the color of the back surfaces at this pixel, as shown in Fig. 2. The colorof the front surfaces is also regarded as the start point at each pixel for the ray castingprocess. And the color of the back surfaces is regarded as the end point at each pixelfor the ray casting process. KIM et al. [11] presented vertex transformation streamsbased on GPU, which addressed the input geometry bandwidth bottleneck forinteractive 3D graphics applications. Flexibility of GPU programming improvesparallel computing performance in many time-critical applications [12, 13].

In this paper, an optimized approach is presented, in order to avoid artifacts andimprove rendering performance. Modification of z values in the fragment shaderprogram can simply implement hybrid visualization of polygons and volumetricobjects. Our approach is similar to the algorithm that NEUBAUER et al. [14] presented.But our approach is different from Neubauer’s method in three aspects. First,the approach is based on GPU and it can obtain interactive frame rates. Second, it uses

An iterative programmable graphics process unit ... 521

regular ray traversal strategy with linear estimation of intersection, so it does not needany special data structure. Third, z modification implements hybrid visualization.

This paper is organized as follows. In Section 2, an initial plane is created togenerate start points of all the rays. Section 3 discusses the iterative linear interpolationof intersection to avoid artifacts and speedup rendering. In Section 4, modificationof z values is used to implement hiding of polygons and volumes. At last, someexperiments are performed and conclusions are drawn.

2. Entering into the volume

For virtual endoscopy applications, the viewpoint is usually located in the volume togenerate similar views to those produced by traditional optical endoscopic diagnoses.In the GPU based ray casting algorithm presented by KRUGER and WESTERMANN [9],front and back surfaces of the bounding box of the volume are successively renderedto produce entry and exit points of the ray cast from the viewpoint. However, whenthe viewpoint moves into the volume, the front surfaces of the bounding box wouldbecome invisible. In this case, the color of the front surfaces usually becomes solidcolor and does not contain any position information. So, we cannot directly computethe viewing vector by subtracting the color of the front surfaces from the color ofthe back surfaces. In order to solve this problem, when the virtual camera is locatedin the volume, an initial plane is created in the neighborhood of the near clipping plane

Fig. 1. Axis-aligned (a) and view-aligned resampling slices (b).

Fig. 2. Front (a) and back (b) faces of the bounding box encoding the position.

522 FEINIU YUAN

that replaces the front surface of the bounding box as the proxy geometry where startpoints of all the rays are located in, as shown in Fig. 3. And end points of all the raysare still located in the back surfaces of the bounding box.

The initial plane is parallel to the near clipping plane and located in the viewingvolume to assure that the initial plane is visible all the time, as shown in Fig. 4.And the initial plane is very close to the near clipping plane. Following the processes

Fig. 3. The viewpoint in the volume.

Fig. 4. Location of the initial plane.

Fig. 6. Linear interpolation of intersection.

Fig. 5. Virtual endoscopic views.


mentioned above, the viewing ray can be directly computed by subtracting the colorof the initial plane from the color of the back surfaces of the bounding box.

The GPU based ray casting algorithm for virtual endoscopic applications isdescribed as follows:

1. Render back surfaces of the bounding box into an intermediate texture.2. Render the initial plane, subtract the color of initial plane from the color of back

surfaces to get a direction vector, store the normalized vector together with the lengthof the vector in a direction texture. The length of the vector is just the maximumdistance that the ray goes across. The normalized viewing vector and the distance areencoded in the color channel and the alpha channel, respectively.

3. Regard the color of initial plane as an inputted start point of the ray for the fragmentshader program. Cast a ray along the viewing vector stored in the direction texture,and composite color and opacity of each resampling. Terminate the ray if the distancethe ray goes across is greater than the distance stored in alpha channel of the directiontexture or the opacity has reached a certain threshold (early ray termination).

Figure 5 illustrates two experimental results by the algorithm. From renderedimages, we can find that colon and chine cavity of human are clearly displayed byGPU based ray casting algorithm.

3. Linear interpolation of intersection

In many virtual endoscopic diagnoses, rendering of one or multiple isosurfaces is usuallyadopted. High accurate computation of intersection between the ray and isosurfaces isvery important for correct rendering. But high accurate computation of intersectionwill greatly increase the searching time of isosurfaces and markedly decrease framerates.

In order to improve the performance, a linear interpolation of intersection isproposed to approximate the actual intersection. As shown in Fig. 6, sn–1 and Pn–1 arethe intensity value and position of resampling at the (n–1)-th step, respectively; sn andPn are the intensity value and position of resampling at the n-th step, respectively.T is a threshold value for an isosurface and P is the estimated intersection betweenthe ray and isosurface. When sn–1 is less than T and sn is greater than T, there must bean isosurface between these two resampling points. So, the estimated intersection Pcan be computed by the following equation:

(1)

where

Pn = P0 + L dn (2)

dn = nd (3)

Psn T–

sn sn 1––--------------------------- Pn 1–

T sn 1––sn sn 1––

--------------------------- Pn+=

524 FEINIU YUAN

In the above Eqs. (1)–(3), d is the resampling step size, P0 – the start point inthe initial plane, L – the normalized vector of a ray and n the count of steps.

According to the estimated intersection, the Phong lighting effects are computedfor current pixel. Linear interpolation of intersection with a relatively larger resamplingdistance can improve frame rates without obviously degrading image quality. Searchingcodes of isosurfaces in the fragment shader program can be written as follows:

/* P0 is the start point in the initial plane. DirectionTexture isthe direction texture containing the normalized viewing ray and themaximum distance the ray goes across */

vec4 L=texture3D(DirectionTexture, P0);vec3 Pn, P;vec4 vol;float dn=0.0;float dp;float sn, sn_1=0.0;while (dn<L.a){ Pn =P0 +L.xyz*dn;

vol=texture3D(VolumeTexture, Pn); //T is the threshold for isosurfacesif(vol.a<T){ sn_1=vol.a; dn=dn+d; //d is the resampling step size}else{ sn=vol.a; break;}

}if(dn>=L.a) discard;if((sn- sn_1)>=0.0001){

dp=dn-d*(sn-T)/ (sn- sn_1);}else{

dp=dn;}P=P0+L.xyz*dp; //P is the estimated intersection

After the above searching with linear interpolation, the estimated intersection isreasonably accurate. So, using the estimated intersection, we can produce high qualityimages. We can repeat the linear interpolation of intersection again using newlyestimated positions P and Pn or Pn–1, in order to obtain more accurate estimation ofintersection between the ray and isosurfaces. Experiments show that only 1 to 3 timesare enough to obtain very good estimation of intersection. So, it is still useful foracceleration of rendering. Iterative searching codes in the fragment shader programare given as follows:

int i=0;float sp,dn_1;dn_1=dn-d;while(i<3)


{ vol=texture3D(VolumeTexture, P);sp=vol.a; /*When the estimation error is very small, iteration can be stoppedin advance*/if(abs(sp-T)<0.0001) break;if(sp<T)

{ /*The resampling step d is the half distance between points P andPn */d=(dn-dp)/2.0;sn_1=sp;dn_1=dp;

}else{ /*The resampling step d is the half distance between points Pn_1

and P */d=(dp-dn_1)/2.0;sn=sp;dn=dp;

}dp=dn_1+d;P=P0 +L.xyz*dp;vol=texture3D(VolumeTexture, P);if(vol.a>T) { //Linear interpolation between Pn_1 and P

sn=vol.a;dp=dp-d*(sn-T)/ (sn- sn_1);

}else{ //Linear interpolation between P and Pn

sn_1=vol.a;dp=dn-d*(sn-T)/ (sn- sn_1);

}//P is the estimated intersection at current time

P=P0+L.xyz*dp;i=i+1;

}

In the above iterative program, if the estimation error is very small, for example,it is less than 0.0001, the iteration can be early terminated in order to improveperformance, otherwise, the linear estimation must be performed for this time. Andthen, if the resampling value at the estimated position P is less than the threshold T,the linear interpolation of intersection should be performed between positions P and Pn.

Fig. 7. Iterative linear interpolation of intersection.

526 FEINIU YUAN

Accordingly, the step d is the half distance between P and Pn. Otherwise, the linearinterpolation should be done between points Pn–1 and P, and the step d is set to thehalf distance between Pn–1 and P.

The iterative process can be shown as Fig. 7. In Figure 7, a quadrangle stands forthe estimated intersection for the first time, a pentagon for the second time, and a circlefor the third time. As we can see, the accuracy is rapidly improved while times ofiteration increase.

4. Hybrid visualization

In virtual endoscopy systems, volumetric datasets and traditional polygons usuallyexist together. Traditional primitives often assist doctors to complete the diagnosis,for example, polygons may be used to indicate the orientation and location of the virtualcamera in the 3D datasets. So, the problem must be solved of how to correctly renderthe hybrid scenes.

In the z-buffering algorithm, hiding is naturally implemented according to depthsof screen pixels and the principle is very simple. So, it is very easy to be implementedin hardware. Nearly all the consumer-level graphics hardware supports z-buffering.Due to the mathematics involved, the generated value z in a z-buffer is not distributedevenly across the z-buffer range (typically, 0.0 to 1.0, inclusive)

(4)

As shown in Eq. (4), r is the actual distance between the viewpoint and the estimatedintersection, zn is the distance between the viewpoint and near clipping plane, and zfis the distance between the viewpoint and far clipping plane. Corresponding z valuein the z-buffer can be computed according to Eq. (4). So, in the fragment shaderprogram, we add the following codes before writing the color to frame buffer:

float z=10.0/(10.0-0.002)*(1.0-0.002/r);if(z<gl_FragDepth) discard;gl_FragDepth=z;

In our implementation, zn and zf are equal to 0.002 and 10.0, respectively, andthe variable r is the actual distance between the viewpoint and the estimatedintersection. In Figure 8, the hybrid scene includes an MRI head volume and polygonsof the bounding box. Figure 8a illustrates no hiding effects and Fig. 8b showsthe hiding effects after adding the above codes in the fragment shader program. Wecan see that modification of z values can visualize the hybrid scene correctly.

5. Experiments

We have implemented the proposed and traditional GPU based ray casting algorithmsusing VC++ and OpenGL, and several experiments were performed on a Pentium D/3.0GHz PC with a GeForce FX7300 graphics card. Compared with traditional 3D

zzf

zf zn–------------------- 1

zn

r---------–

⎝ ⎠⎜ ⎟⎛ ⎞

=


texture based methods, our optimized method produces high quality images atinteractive frame rates.

First, navigate into an abdomen volumetric dataset (512×512×86) with the algorithmproposed. As shown in Fig. 9, we can find that the rendered images have high quality.Those images were generated using the isosurface ray casting based on GPU. We can

Fig. 8. Visualization of hybrid scene including an MRI volume and its bounding box, a – no hiding,b – hiding.

Fig. 9. Rendered images by the algorithm whileinteractively navigating through human colon.

528 FEINIU YUAN

observe highlight effects on the colon due to enabling the Phong illumination modelafter classification.

Second, navigate into another volumetric dataset (512×512×112) with the algorithm.As shown in Fig. 10, we can find that the rendered images have high quality. We caninteractively alter the transfer function by simply regenerating the 2D lookup textureor directly specifying different ambient, diffuse and specular colors. So, our algorithmis very convenient in cases where users need to frequently modify material property.

We also implemented traditional GPU based ray casting algorithm. The Tablegives frame rates. In the Table, our optimized searching approach can greatlyaccelerate rendering speed without degrading image quality when the resampling stepin the optimized algorithm is 4 times the step in traditional algorithms. As shown inFig. 11a, although large steps can accelerate rendering speed, the traditional algorithm

Fig. 10. A series of images rendered while inter-actively navigating through human trachea.

Fig. 11. A comparison of image quality: a – traditional method, b – our method.

a b

T a b l e. Frame rates.

Datasets Traditional GPU ray casting Our optimized GPU ray castingHuman colon 12.5 fps 30.2 fpsHuman trachea 16.8 fps 39.3 fps


produces severe artifacts due to large steps. As shown in Fig. 11b, our method is ableto avoid such artifacts and obtain obvious improvements of computation. In order tofurther speedup visualization, space leaping based acceleration techniques is verysuitable for being implemented using GPU shader programs.

6. Conclusions

GPU based ray casting in the fragment shader program has been presented that allowsboth orthogonal and perspective projection and enables the user to move the viewpointinto the dataset for virtual endoscopic visualization. An optimized GPU based raycasting algorithm is used in virtual endoscopy applications. To avoid artifacts, a linearinterpolation of intersection is presented to estimate the actual intersection betweenthe viewing ray and the isosurface. Thus, resampling artifacts is greatly avoided andthe performance is not influenced. A larger resampling step with the linearinterpolation of intersection can speed up rendering. Using the accurate estimatedintersection, we can produce high quality images. Iterative linear interpolation ofintersection is presented in order to obtain more accurate estimation of intersectionbetween the ray and isosurfaces. Experiments show that only 1 to 3 times are enoughto obtain very good estimation of intersection. So, it is still useful for acceleration ofrendering. The author takes advantage of currently available programmable graphicshardware and OpenGL Shading Language to enable Phong lighting model with a pointlight source on the fly. Thus, it is possible that we can acquire high quality renderedimages as well as interactive frame rates. The z values in the z-buffer are modified toimplement visualization of hybrid scenes that usually exist in virtual endoscopyapplication. Several experiments show that the optimized GPU based ray castingalgorithm is very useful for virtual endoscopy.

Acknowledgements – This project was supported by Natural Science Foundation of Jiangxi Province(2007GQS0076), Foundation of Education Department of Jiangxi Province (2007[272]), Key project ofJiangxi University of Finance and Economics, Open Foundation of the State Key Lab of Fire Science(HZ2006-KF03) and China Postdoctoral Science Foundation (20070410792).

References

[1] PFISTER H., Architectures for real-time volume rendering, Future Generation Computer Systems15(1), 1999, pp. 1–9.

[2] VILANOVA I BARTROLÍ A., KÖNG A., GRÖLLER M.E., Cylindrical Approximation of Tubular Organs forVirtual Endoscopy, Technical Report TR-186-2-00-02, February 2000, Abteilung für Computer-graphik, Institut für Computergraphik und Algorithmen, Technische Universität Wien, Austria.

[3] SHARGHI M., RICKETTS I.W., A novel method for accelerating the visualization process used in virtualcolonoscopy, Proceedings of Fifth International Conference on Information Visualisation, 2001, SanDiego, California, October 22–23, 2001, pp. 167–72.

[4] BRADY M., JUNG K., NGUYEN H.T., NGUYEN T., Two-phase perspective ray casting for interactivevolume navigation, IEEE Visualization’97 Conference, Phoenix, Arizona, October 19–24, 1997,pp. 183–9.

530 FEINIU YUAN

[5] CULLIP T., NEUMANN U., Accelerating volume reconstruction with 3D texture mapping hardware,Technical Report TR93-027, Department of Computer Science, University of North Carolina,Chapel Hill, 1993.

[6] CABRAL B., CAM N., FORAN J., Accelerated volume rendering and tomographic reconstructionusing texture mapping hardware, Proceedings of IEEE Symposium on Volume Visualization 1994,pp. 91–8.

[7] REZK-SALAMA C., ENGEL K., BAUER M., GREINER G., ERTL T., Interactive volume rendering onstandard PC graphics hardware using multi-textures and multi-stage rasterization, [In] Proceedingsof Graphics Hardware 2000, pp. 109–18.

[8] WESTERMANN R., ERTL T., Efficiently using graphics hardware in volume rendering applications,[In] Proceedings of SIGGRAPH’98, 1998, pp. 169–78.

[9] KRUGER J., WESTERMANN R., Acceleration techniques for GPU-based volume rendering, IEEEVisualization 2003, pp. 287–92.

[10] STEGMAIER S., STRENGERT M., KLEIN T., ERTL T., A simple and flexible volume rendering frameworkfor graphics-hardware based raycasting, Fourth International Workshop on Volume Graphics, 2005,pp. 187–241.

[11] YOUNGMIN KIM, CHANG HA LEE, AMITABH VARSHNEY, Vertex–transformation streams, GraphicalModels 68(), 2006, pp. 371–83.

[12] FIALKA O., CADK M., FFT and Convolution Performance in Image Filtering on GPU, TenthInternational Conference on Information Visualization, 2006, pp. 609–14.

[13] KRUGER J., WESTERMANN R., Linear Algebra Operators for GPU Implementation of NumericalAlgorithms, SIGGRAPH 2003, pp. 908–16.

[14] NEUBAUER A., MROZ L., HAUSER H., WEGENKITTLE R., Cell-based first-hit ray casting, Proceedingsof the Symposium on Data, Visualization 2003, 2002, pp. 77–86.

Received December 11, 2007in revised form January 22, 2008


Serum glycoproteins in diabetic and non-diabetic patients with and without cataract

ANJUMAN GUL1*, M. ATAUR RAHMAN2, NESSAR AHMED3*

1Department of Biochemistry, Ziauddin University, Shah rah-e- Ghalib, Clifton, Karachi-75600, Pakistan

2International Centre for Chemical and Biological Sciences, HEJ Research Institute of Chemistry, University of Karachi, Karachi-75270, Pakistan

3School of Biology, Chemistry and Health Science, Manchester Metropolitan University, Manchester M1 5GD, UK

*Corresponding authors: A. Gul – [email protected], N. Ahmed – [email protected]

This study describes the changes in serum glycoproteins from type 2 diabetic and non-diabeticpatients with and without cataract. A total of 85 subjects were selected for the study and dividedinto four groups. The first group consisted of 21 healthy subjects, the second group consisted of21 diabetic patients with no chronic complications, the third group consisted of 20 diabeticpatients with cataract, and the fourth group had 23 non-diabetic patients with age related cataract.The patients with and without cataract were selected on clinical grounds from the ZiauddinUniversity and Jinnah Postgraduate Medical Centre in Karachi, Pakistan. As expected diabeticpatients with and without cataract had significantly higher levels of fasting plasma glucose,glycated haemoglobin, glycated plasma proteins and serum fructosamine. In addition to theseparameters, the levels of hexosamine, sialic acid and serum total protein were higher in diabeticcompared to non-diabetic subjects with age related cataract and healthy subjects. Analysis ofthe protein fractions showed that alpha-1-globulins and alpha-2-globulins were higher in diabeticpatients without complications compared to non-diabetic subjects with age related cataract andhealthy subjects. Serum alpha-1-globulin, alpha-2-globulin, beta globulins and gamma globulinswere all significantly higher in diabetic patients with cataract compared to healthy subjects but notserum albumin. In conclusion, the levels of beta globulins and gamma globulins were significantlyhigher in diabetic patients with cataract and non-diabetic age related patients with cataractcompared to healthy subjects. Thus, mechanisms other than hyperglycaemia are responsible forthe development of cataract in these patients.

Keywords: glycoprotein, diabetes mellitus, cataract, fructosamine, sialic acid, hexosamine.

1. Introduction Diabetes mellitus is a common endocrine disorder characterized by hyperglycemia,metabolic abnormalities and long-term complications afflicting the eyes, kidneys, nerves

532 A. GUL, M.A. RAHMAN, N. AHMED

and blood vessels [1]. Worldwide projections suggest that more than 220 millionpeople will have diabetes by the year 2010 and the majority of these will suffer fromtype 2 diabetes mellitus [2]. In Pakistan, diabetes mellitus is a major health problemaffecting more than 16% of people over the age of 25 years in some areas with a further10% suffering from impaired glucose tolerance [3].

Cataract is one of the major causes of impaired vision and blindness worldwideand often referred to as senile cataract to indicate that it is more common in advancedage. It is also a serious consequence of long-term diabetes and according to the WHO,affects about half of the 45 million blind people worldwide [4]. Senile cataract ischaracterized by opacification of the eye lens affecting mainly the nuclear, cortical,and posterior subcapsular regions. Pathological studies of cataractous lenses haverevealed that cataracts are composed of protein aggregates that precipitate in the lensof the eye. The insoluble aggregates obstruct the passage of light through the lenspreventing it from reaching the photoreceptors in the retina [5]. The lens crystallinsmay be divided into α -, β - and γ -crystallins. Diabetic cataract occurs much earlierthan senile cataract and causes opacification of the lens and eventual loss of vision. Ithas been suggested that increased non-enzymatic glycosylation (glycation) of lenscrystallins may cause conformational changes resulting in exposure of thiol groups tooxidation and cross-link formation [6–8]. Furthermore, the lens crystallins havevirtually no turnover and are ideal candidates for accumulation of glycation-derivedcross-links. Thus, increased cross-linking of lens crystallins may cause aggregationproducing the high molecular weight material responsible for opacification. Increasedglycation of serum proteins could cause an increase in the serum of circulatingadvanced glycation endproducts (AGEs). These AGEs are responsible for glycation--induced cross-linking of structural proteins and believed to underlie the pathogenesisof diabetic complications [9]. Previous studies have also shown an increase inenzymatically glycosylated proteins such as alpha-2 glycoprotein fractions which wereincreased in diabetic patients [10].

The aim of this study was to investigate changes in serum glycoproteins (bothenzymatic and non-enzymatic) in diabetic and non-diabetic patients with and withoutcataract.

2. Materials and methods

Patients over 50 years of age were selected on clinical grounds from ZiauddinUniversity Hospital, Karachi and Jinnah Postgraduate Medical Centre, Karachi.The study included a total of 85 subjects which were divided into four groups. Groupone consisted of 21 apparently healthy subjects who had no history of diabetes, cataractor any other major illness, like macrovascular disease, retinopathy, tuberculosis,rheumatoid arthritis, liver disease or malignancy. Group two consisted of 21 type 2diabetic patients without any clinical evidence of chronic diabetic complications,whereas group three had 20 type 2 diabetic patients with cataract. Finally, group four

Serum glycoproteins in diabetic and non-diabetic patients ... 533

consisted of 23 non-diabetic patients suffering from age related cataract. The age, sex,weight, duration of diabetes and treatment received were recorded. Drugs werestopped 48 hours prior to any sample collection. Physical examination includingmeasurement of blood pressure and any family medical history was recorded.Individuals were classified as having diabetes mellitus if they had a fasting plasmaglucose concentration ≥ 7.0 or random plasma glucose level ≥ 11.1 mmol/L accordingto established criteria [11]. Patients with a history of blurred vision or double visionand spots were examined with a slit lamp to determine the type of cataract. The localethical committee approved the protocol and blood samples were collected fromsubjects after completing a consent form for each patient and explaining the nature ofthe study.

Blood glucose was determined by the glucose oxidase method [12]. The reagentswere obtained from glucose enzymatic PAP 7500 kit of bioMerieux. For the estimationof glycated haemoglobin, haemolysate was prepared by treating blood with a detergentin a buffered medium and removal of the labile fraction. Haemoglobins are retainedby a cationic exchange resin. Haemoglobin A1C is specially eluted after washing awaythe haemoglobin A1A and haemoglobin A1B fractions and is quantified by directphotometric reading at a wavelength of 415 nm. The kit was obtained from Bio SystemsReagents and Instruments, Spain and based on an established procedure [13].Fructosamine was detected by the nitro-blue tetrazolium reaction. The kit was obtainedfrom Quimica Clinia Aplicada, Spain. Serum hexoamine was determined by Cessi andPillego’s method [14], total serum protein by the Biuret method of Reinhold [15],sialic acid by Natelson method [16], and glycosylated proteins by the method ofMa et al. [17]. Glycated plasma proteins were hydrolysed with oxalic acid to release5-hydroxymethyl furfural which was detected by reaction with thiobarbituric acid.This method gives overall estimation of ketoamine linkages. Serum proteinelectrophoresis [18] was carried out by Helena Electrophoretic System, using a kitmethod (Titan III Cat. No. 3023 obtained from Helena Laboratories).

3. Statistical analysis

Epi-Info was used for statistical analysis of the data. Epi-Info is a statistical packageavailable from the US Centre for Disease Control and Prevention. The statisticalsignificance of the difference between two means of various parameters betweendifferent groups was evaluated by Student’s t test. The difference was regarded ashighly significant if the P value was less than 0.001, statistically significant if the Pvalue was less than 0.05, and non-significant if the P value was greater than 0.05.

4. Results

The mean age of non-diabetic and diabetic patients with cataract was significantlyhigher (P < 0.05) as compared with control subjects (the Table). Fasting plasma

534A

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T a b l e. The age, weight and concentration of blood analytes in non-diabetic and diabetic patients with and without cataract. (The values are expressedas mean ± SEM. The units and number of cases are shown in parentheses.)

a – significant as compared with control subjects; b – significant as compared with non-diabetic patients with cataract

Parameters Control subjects (21) Diabetic patients without any complications (21)

Non-diabetic patients with cataract (20)

Diabetic patients with cataract (23)

Age (years) 53.81±1.20 54.71±1.40 59.83±1.69a 57.50±1.58Sex (F/M) 10/11 10/11 10/13 10/10Weight (kg) 64.30±1.57 64.24±1.62 65.22±1.45 67.78 ±1.55a

Duration of diabetes (years) — 9.29±0.50 — 9.00±1.00Fasting plasma glucose (mmol/L) 5.04±0.13 7.83±0.32a 5.34±0.18 9.32±0.34ab

% Glycosylated haemoglobin (HbA1C) 4.98±0.11 9.30±0.37a 5.04±0.09 8.80±0.34ab

Serum fructosamine (mmol/L) 2.25±0.08 3.72±0.17a 1.98 ± 0.07b 3.05±0.21ab

Glycosylated plasma protein (absorbance/g) 6.20±0.12 7.90±0.30a 5.85±0.12 8.90±0.34ab

Hexosamine (mg/dL) 67.86±3.12 102.94±3.63a 77.52±3.31 118.80±3.43ab

Sialic acid (mg/dL) 35.36±1.34 49.66±1.78 39.22±1.38 50.49±1.76ab

Total serum protein (gm%) 7.32±0.12 7.94±0.17a 6.94±0.20 7.97±0.12ab

Serum albumin (gm%) 4.01±0.10 4.03±0.11 3.01±0.09 3.57±0.12b

Alpha-1 globulin (gm%) 0.16±0.02 0.38±0.06a 0.17±0.02 0.36±0.09ab

Alpha-2 globulin (gm%) 0.77±0.03 0.96±0.05a 0.87±0.05 1.54±0.56a

Beta globulin (gm%) 1.00±0.03 0.92±0.06 1.08±0.07a 1.14±0.07a

Gamma globulin (gm%) 1.48±0.07 1.67±0.09 1.89±0.12a 2.01±0.11a


glucose, HbA1C, serum fructosamine, glycosylated plasma protein, serum hexosamineand serum sialic acid levels were significantly higher (P < 0.05) in all diabetic patientswith or without cataract as compared with control subjects (the Table). Theseparameters did not change (P > 0.05) in non-diabetic patients with age related cataractas compared with control subjects (the Table). However, they were higher in diabeticpatients when compared to non-diabetic patients with age related cataract (P < 0.001;the Table). Total serum protein, alpha-1- and alpha-2 globulins were significantlyhigher (P < 0.05) in diabetic patients with or without cataract as compared with controlsubjects (the Table). Beta globulin and gamma globulin were significantly higher(P < 0.05) in non-diabetic and diabetic patients with cataract as compared with controlsubjects (the Table).

The correlation between fasting plasma glucose and HbA1C in control subjects wasr = 0.284, between fasting plasma glucose and HbA1C in diabetic patients withoutcataract r = 0.478, between fasting plasma glucose and HbA1C in non-diabetic patientswith age related cataract r = 0.267 and between fasting plasma glucose and HbA1C indiabetic patients with cataract r = 0.467.

5. Discussion

Diabetes mellitus and its complications constitute an important health problem in bothdeveloping and developed countries. Cataract remains the commonest cause of blindnessworldwide. In the present study, patients were selected with cataract and the possibilityof other complications was excluded by the absence of any sign and symptoms onphysical examination. Changes in protein concentration and increased enzymaticglycation of various proteins in diabetic patients have been correlated withhyperglycemia, which in turn causes early functional alterations in different tissues.In the present study, non-diabetic patients with age related cataract have normal valuesexcept beta and gamma globulin which are increased in non-diabetic patients with agerelated cataract. Glycaemic related analytes increased in all diabetic patients with andwithout cataract and the levels did not change in non-diabetic patients with age relatedcataract and control subjects (the Table). The presence of cataract in non-diabeticpatients did not affect the glycaemic related analytes, and increase in diabetic patientswithout complications, reflect that the diabetes was uncontrolled. They may bedeveloping changes at the subclinical level which later on present a complication.

The correlation between fasting plasma glucose and HbA1C in control subjects wasr = 0.284, between fasting plasma glucose and HbA1C in diabetic patients withoutcataract r = 0.478, between fasting plasma glucose and HbA1C in non-diabetic patientswith age related cataract r = 0.267, and between fasting plasma glucose and HbA1Cin diabetic patients with cataract r = 0.467. Our results show that fasting plasmaglucose is directly proportional to HbA1C level. The correlation among HbA1C, bloodglucose concentrations and late complications has been established over the last 30years [19–21]. Serum fructosamine and glycated plasma protein concentrations haveclose correlation with HbA1C because they reflect glycaemic control within the last 2


to 3 weeks and HbA1C reflects glycaemic control for the last 4 to 6 weeks [22, 23]. Inthe present study, serum fructosamine and glycated plasma proteins in diabetic patientsalso have a close correlation with HbA1C. The degree of glycation of plasma proteins,as an alternative index of control and as reflection of possible structural alterations oftissue proteins leading to complications was associated with the diabetic state [22].STRATTON et al. [23] suggested that in patients with type 2 diabetes, the risk of diabeticcomplications was strongly associated with previous hyperglycemia. Any reductionin HbA1C is likely to reduce the risk of complications, with the lowest risk being inthose with HbA1C in the normal range. In the present study, the coefficient of variationof HbA1C was higher in diabetic patients with cataract as compared with non-diabeticpatients with age related cataract. It seems that there could be different mechanismsfor the development of cataract in diabetic and non-diabetic patients. Serumhexosamine and sialic acid levels were significantly increased in all diabetic patientsand were non-significant in non-diabetic patients with age related cataract as comparedwith control subjects (the Table). Other workers have made similar observations[24, 25]. HANGLOO et al. [26] found that age and sex had no influence on serum sialicacid levels. As sialic acid is incorporated into carbohydrate chains of glycoproteinsand glycolipids in serum and tissues, the degree of incorporation of sialic acid has beenreported to affect transvascular permeability and accumulation of lipid in the arterialwall. Sialic acid is conjugated with constituents of acute phase reactants, which arehighly concentrated on surface of endothelial cells [26]. One likely hypothesis mightbe that the relationship between clinical condition and sialic acid concentration isdue to the activity of a current inflammatory atherosclerotic process and/or to a directdamage to vascular endothlium causing sialic acid into the circulation [27]. Serumhexosamine levels rise due to hexosamine biosynthetic pathway, which is involvedin the pathogenesis of insulin resistance in patients with type 2 diabetes mellitus[24, 28, 29]. In the present study, the values of hexosamine and sialic acid innon-diabetic patients with age related cataract were in normal limits, but the valueswere increased in diabetic patients with cataract and without complications.

The uniform increase in fasting plasma glucose, glycosylated haemoglobin (HbA1C),serum fructosamine, glycosylated plasma protein, serum hexosamine and serum sialicacid levels in diabetic patients indicates that the process of glycosylation depends uponhyperglycemia. The parameters do not rise in non-diabetic patients, and hence someother underlying mechanism may be responsible for the development of complicationsin these patients. However, at present we cannot decide whether or not there is relevantdysfunction of the β - and γ -globulin and what role any such dysfunction would playin the development of cataract. A limitation of this study was the low number ofpatients and if the study is carried out on a larger sample of patients with other factors,which is necessary for definite conclusion.

Acknowledgement – We are grateful to Dr. Arif Hussain, in charge of Clinical Laboratory, ZiauddinUniversity Hospital, Pakistan for his cooperation and assistance with the laboratory work.


References[1] NATHAN D.M., Long-term complications of diabetes mellitus, New England Journal of Medicine

328(23), 1993, pp. 1676–85.[2] ALBERTI K.G.M.M., ZIMMET P.Z., Defination, diagnosis and classification of diabetes mellitus and

its complication. Part 1: diagnosis and classification of diabetes mellitus provisional report ofa WHO consultation, Diabetic Medicine 15(7), 1998, pp. 539–53.

[3] SHERA A.S., RAFIQUE G., KHAWAJA I.A., ARA J., BAQAI S., KING H., Pakistan national diabetessurvey: prevalence of glucose intolerance and associated factors in Shikarpur, Sindh Province,Diabetic Medicine 12(12), 1995, pp. 1116–21.

[4] The world health report 1998: life in the 21st century: a vision for all, World Health Organization,1998.

[5] BLOEMENDAL H., DE JONG W., JAENICKE R., LUBSEN N.H., SLINGSBY C., TARDIEU A., Ageing andvision: structure, stability and function of lens crystallins, Progress in Biophysics and MolecularBiology 86(3), 2004, pp. 407–85.

[6] HORWITZ J., The function of α-crystallin in vision, Seminars in Cell and Developmental Biology11(1), 2000, pp. 53–60.

[7] MACRAE T.H., Structure and function of small heat shock/α-crystallin proteins: established conceptsand emerging ideas, Cellular and Molecular Life Sciences 57(6), 2000, pp. 899–913.

[8] MONNIER V.M., STEVENS V.J., CERAMI A., Nonenzymatic glycosylation, sulfhydryl oxidation andaggregation of lens proteins in experimental sugar cataracts, Journal of Experimental Medicine150(5), 1979, pp. 1098–107.

[9] MONNIER V.M., Nonenzymatic glycosylation, the maillard reaction and the aging process, Journalof Gerontology 45(4), 1990, pp. B105–11.

[10] RAHMAN M.A., ZAFAR G., SHERA A.S., Changes in glycosylated proteins in log-term complicationsof diabetes mellitus, Biomedicine and Pharmacotherapy 44(4), 1990, pp. 229–34.

[11] GABIR M.M., ROUMAIN J., HANSON R.L., BENNETT P.H., DABELEA D., KNOWLER W.C.,IMPERATORE G., The 1997 American Diabetes Association and 1999 World Health Organizationcriteria for hyperglycemia in the diagnosis and prediction of diabetes, Diabetes Care 23(8), 2000,pp. 1108–12.

[12] TIETZ N.W., Clinical Guide to Laboratory Tests, 3rd Ed., WB. Saunders Company, Philadelphia,PA, 1995, pp. 268–73.

[13] ROCHMAN H., Hemoglobin A1C and diabetes mellitus, Annals of Clinical and Laboratory Science10(2), 1980, pp. 111–5.

[14] CESSI C., PILIEGOF., The determination of amino sugars in the presence of amino acids and glucose,Biochemical Journal 77, 1960, pp. 508–10.

[15] VARLEY H., GOWENLOCK A.H., BELL M., Practical Clinical Biochemistry, 5th Ed., WilliamHeinemann Medical Books Ltd, London, 1980, pp. 545–7.

[16] NATELSON S., Microtechniques of Clinical Chemistry for the Routine Laboratory, 2nd Ed., ThomasSpringfield, Illinois, 1961, pp. 378–80.

[17] MA A., NAUGHTON M.A., CAMERON D.P., Glycosylated plasma proteins: a simple method forthe elimination of interference by glucose in its estimation, Clinica Chimica Acta 115(2), 1981,pp. 111–7.

[18] RALLI E.P., BARBOSA X., BECK E.M., LAKEN B., Serum electrophoretic patterns in normal anddiabetic subjects, Metabolism 6(4), 1957, pp. 331–8.

[19] MCLELLAN A.C., THORNALLEY P.J., BENN J., SONKSEN P.H., Glyoxalase system in clinical diabetesmellitus and correlation with diabetic complications, Clinical Science 87(1), 1994, pp. 21–9.

[20] THORNALLEY P.J., The glyoxalase system: new developments towards functional characterizationof a metabolic pathway fundamental to biological life, Biochemical Journal 269(1), 1990,pp. 1–11.


[21] BROWNLEE M., CERAMI A., VLASSARA H., Advanced glycosylation end products in tissue and thebiochemical basis of diabetic complications, The New England Journal of Medicine 318(20), 1988,pp. 1315–21.

[22] SKEIE S., THUE G., SANDBERG S., Interpretation of hemoglobin A1C (HbA1C) values among diabeticpatients: Implications for quality specifications for HbA1C, Clincal Chemistry 47(7), 2001,pp. 1212–7.

[23] STRATTON I.M., ADLER A.I., NEIL H.A., MATTHEWS D.R., MANLEY S.E., CULL C.A., HADDEN D.,TURNER R.C., HOLMAN R.R., Association of glycaemia with macrovascular and microvascularcomplication of type 2 diabetes (UKPDS 35): prospective observational study, BMJ 321(7258),2000, pp. 405–12.

[24] CROOK M., The determination of plasma or serum sialic acid, Clinical Biochemistry 26(1), 1993,pp. 31–8.

[25] MARSHALL S., BACOTE V., TRAXINGER R.R., Discovery of a metabolic pathway mediatingglucose induced desensitization of the glucose transport system: Role of hexosamine biosynthesisin the induction of insulin resistance, Journal of Biological Chemistry 266(8), 1991, pp. 4706–12.

[26] HANGLOO V.K., KAUL I., ZARGARH.U., Serum sialic acid levels in healthy individuals, Journal ofPostgraduate Medicine 36(3), 1990, pp. 140–2.

[27] LINDBERG G., EKLUND G.A., GULLBERG B., RASTAM L., Serum sialic acid concentration andcardiovascular mortality, BMJ 302(6769), 1991, pp. 143–6.

[28] SPAN P.N., POUWELS M.J., OLTHAAR A.J., BOSCH R.R., HERMUS A.R., SWEEP C.G., Assay forhexosamine pathway intermediates (uridine diphosphate-N-acetyl amino sugars) in small samplesof human muscle tissue, Clinical Chemistry 47(5), 2001, pp. 944–6.

[29] HAWKINS M., BARZILAI N., LIU R., HU M., CHEN W., ROSSETTI L., Role of the glucosamine pathwayin fat-induced insulin resistance, Journal of Clinical Investigation 99(9), 1997, pp. 2173–82.

Received November 5, 2007in revised form January 12, 2008


Lyapunov exponent of the optical radiation scattered by the Brownian particles

MYKHAYLO S. GAVRYLYAK*, OLEKSANDER P. MAKSIMYAK, PETER P. MAKSIMYAK

Department of Correlation Optics, Chernivtsi National University, Chernivtsi 58012, Ukraine


The computer and physical simulation of light scattering by the system of Brownian particles hasbeen carried out. Temporary fluctuations of field intensity have been found to save chaoticproperties of driving particles. Empirical diagnostic links have been retrieved of the largestLyapunov exponent of fluctuations of field intensity with parameters of the dispersive media.

Keywords: light scattering, Lyapunov exponent, sulfur hydrosols, Brownian particles.

1. Introduction

The light scattering of coherent optical radiation on Brownian particles causesa complicated space–time modulation of field intensity as a result of interferencecomposition of partial waves with random amplitudes and phases. Time correlationof scattered radiation field is defined with the help of particle motion speed andexperiment geometry, which is the subject of research in Doppler spectroscopy [1, 2].Gorelik was the first scientist to propose the method of optical detection in 1947 [3].Forrester’s, Town’s, Cummins’s, Pike’s, and other scientists’ experiments beingdiscussed in detail [4] are considered to be the basis of a new trend, i.e., optical mixingspectroscopy which is successfully used in physical and chemical investigation,biology and medicine.

At present the application of optical mixing spectroscopy has a definite criticalstate. On the one hand, the coefficient determination of translation diffusion ofmacromolecules, eritrocites, colloid particles, viruses and others has become a standardand rather reliable measuring method [5]. On the other hand, when investigating morecomplicated systems (polydispersed, with high particle concentration, inhomogeneous)there appear some problems requiring the development of theory and mastering ofexperimental techniques.

According to modern views Brownian motion of particles is either random or chaotic.Moreover, it possesses fractal properties [6, 7]. That is why one can use the theory of

540 M.S. GAVRYLYAK, O.P. MAKSIMYAK, P.P. MAKSIMYAK

stochastic and chaotic fluctuations for describing Brownian motion [8]. There arisesan important question concerning the character of space-time chaotization of scatteredradiation field, quantitative diagnostic of relationship between stochastic characteristicfeatures of medium and field, advantages of stochastic approach in determiningstructural and dynamic characteristic features of media with Brownian particles.

We will consider the mathematical modeling of Brownian particle motion andcalculate the optical radiation field scattered on them. We will carry out experimentalinvestigation of light scattering by sulfur hydrosols. We will use the largestLyapunov exponent as an example for researching the time chaotization of parametersof light-scattering medium. We will show the possibility of taking into accountthe errors of defining Brownian particle sizes with the help of measured largestLyapunov exponent.

2. The modeling of Brownian particle motion

The replacement of Brownian particle along axis OX at time t is given by the normalprobability distribution:

(1)

The succession of such random values {xi} is a set of independent random numberswith Gaussian distribution and dispersion [9]:

(2)

where D – diffusion coefficient, defined by astringent medium resistance. Forspherical particles of radius R:

(3)

where k – Boltzmann constant, T – absolute temperature, μ – medium ductility.Particle coordinate at axis x at time moment t = nτ is as follows:

(4)

In an extreme case for great n and small τ the set of n random numbers are gradedinto a random function X (t ) [7] having the same properties as the replacement x.

p x τ,( ) 14π Dτ

---------------------- x2

4 Dτ----------------–

⎝ ⎠⎜ ⎟⎛ ⎞

exp=

x2⟨ ⟩ x2p x τ,( )dx∞–

+∞

∫ 2 Dτ= =

D kT6πμR

--------------------=

X t( ) xii 1=

n

∑=

Lyapunov exponent of the optical radiation ... 541

A program for modeling N Brownian particle motion been elaborated. The initialparticle motion was given with the help of equiprobable distribution of theircoordinates (X, Y, Z ) within volume L3 being investigated. Particles size was given byGaussian distribution function with an average volume R0 and size dispersion σ.Particle replacement x, y, z was given under a normal law (1), and the replacementdispersion of particle (2Dτ )1/2 depended on particle size as well. Particle coordinatesafter each replacement step were determined with the help of relationship (4).

For calculating scattered radiation field distribution the Rayleigh–Gans–Debyelight scattering model was chosen [10]. For monodisperse ensemble such particlesscatter the intensity, which is proportional to R [6] at the same angle. The fieldamplitude at an arbitrary point of space (z0, ξ, ζ ) is defined as a sum of complexfield amplitudes scattered by all Brownian particles:

(5)

where ri = [(zi – z0)2 + (xi –ξ )2 + ( yi – ζ )2]1/2 is the distance from i-Brownian particleto the point in the observation plane; z0 – the distance between the plane where thereis a scattering volume and observation plane, xi, yi, zi, ξ, ζ, z0 – rectangular coordinatesin object plane and observation plane, respectively; k = 2π /λ – a wave number, whereλ is a wavelength.

The complex field amplitude was calculated and written down in the form ofU = ReU + i ImU. It was used for determining amplitude A, phase ϕ and fieldintensity I.

3. Objects of experimental investigation

Sulfur hydrosols were chosen for physical modeling objects [11]. They are obtainedby mixing l – normal solutions of hydrochloric acid and sodium thiosulfate. Then,molecularly dispersion sulfur is condensed in the form of the drops of overcooledsulfur, which are equally increased in size with the rate of sol deterioration. Sulfurrefraction index is equal to 1.44 relative to water. In the absence of extraneousnuclei of condensation the drops are formed after gaining the definite solutionsupersaturation. Then, their initial radii will be of the order of 0.01 μm.

When using sulfur hydrosols in physical experiment they have to be calibrated,i.e., the definite particle size is to be brought to correspondence with the definite timeof sol growth. Assuming that particle sizes of sulfur hydrosols (0.01–2 μm) correspondto Rayleigh–Gans–Debye particles, we used light scattering tables edited by Shifrinfor their calibration [12]. The essence of calibration consists in comparing scatteringindicatrices, measured experimentally, to theoretical ones, obtained from the tables,

U ξ ζ z0, ,( )4π2I0

λ2------------------

n12 n2

2–

n12 2n2

2+---------------------------

Ri3 ik ri z0+( )–exp

ri-------------------------------------------------------

i 1=

N

∑=


for the definite parameters of sulfur particles. The main criterion of the comparativeestimation was the coincidence of diffraction extremes in scattering indicatrices.

Figure 1 shows the calibrated dependence of particle sizes of sulfur hydrosolson their growth time. Under inclination of environment temperature ±5 °C it leads tothe inclination of hydrosol sizes not greater than 10%.

4. Definition of Brownian particle sizes

The correlation function of radiation field scattered by Brownian particle system onceis as follows [13]:

(6)

where the light scattering vector .

For Gaussian distribution spectral density of scattered radiation is determined asfollows [4]:

(7)

Laurents’s counter with a halfwidth Δω1/2 = u2DT and center ω = ωR is described bymeans of this expression.

According to (6) and (7) for Gaussian distribution one has [4]:

(8)

Measuring the inclination angle of line (in linear approximation)to axis t we found the values of diffusion coefficient and particle size.

Experimental investigation of sulfur hydrosols was carried out for the growth timefrom 2 to 25 hours. This corresponded to sizes of sulfur particles from 0.01 to 1 μmas one can see in the calibrated diagram of Fig. 1. The values of initial concentration

FS u t,( ) u2DT t–( )exp=

u k kR– 4πλ

----------- n θ2

------⎝ ⎠⎜ ⎟⎛ ⎞

sin= =

S u ω,( ) N A 2 u2DT π⁄

ωR ω–( )2 u2DT⎝ ⎠⎛ ⎞

2+

----------------------------------------------------------=

K R2( ) t( ) 1 2u2DT t–( )exp+=

K R2( ) t( ) 1–ln

Fig. 1. Calibrated dependence of particle sizesof sulfur hydrosols on their growth time.

0 4 8 12 16 20 24 280.0

0.2

0.4

0.6

0.8

1.0

1.2

R [

m]

μ

t [h]


of condensed sulfur particles in forming the hydrosols in various sources aredifferent [14], being of the order of 3×109–1010 particles per 1 mm3. We decreasedthe initial concentration a hundred times. The smallest concentration under study wasapproximately 5×107 particles per 1 mm3, and it was defined by photoelectronicmultiplier capability to register weak scattered streams. However, this limitcorresponds to sulfur particles less than 0.2 μm only, as the intensity of light scatteringincreases as R with the particle radius growth R6.

Having a halfwidth of power spectrum, according to relation (7), we definedthe coefficient of translation diffusion DT , which we used for calculating particleradius.

Figure 2 represents the dependence of halfwidth of power spectrum Δω onparticle concentration for sulfur hydrosols of three sizes (R = 0.2, 0.5 and 0.9 μm) withthe following parameters of the experiment: d = 0.1 mm, θ = 0.02, z0 = 100 mm,ω = 0.001.

At small concentrations of sulfur particles spectrum halfwidth cannot be practicallychanged, and at big concentrations it increases. Obviously, this is connected withthe appearance of multiple scattering effects and the appropriate spectrum widening.From rectilinear parts of concentration dependence of power spectrum halfwidth wedefined Brownian particle size. It was equal to 0.31, 0.72 and 1.15 μm, respectively,and it exceeded by 35% the values obtained when calibrating the process ofhydrosol growth. The reason for that may be the error of experimental measurements,temperature instability of hydrosol growth. However, the main reason can bethe increase of a real size of sulfur particles at the expense of the formation ofthe transitional layer of water molecules [15]. Hydrodynamic radius at light scatteringis impossible to be studied, and the real movability of particles increases.

A more complicated situation is observed when investigating the concentration ofparticles greater than 5×109 per 1 mm3. Figure 2 shows that halfwidth of powerspectrum Δω increases abruptly. This is connected with the chaotization growth ofscattered radiation field upon the growth of scattering multiplicity. For qualitativeestimation of chaotization degree of scattered radiation intensity we used the largestLyapunov exponent [8].

Fig. 2. Dependence Δω on particleconcentration for sulfur hydrosols ofthree sizes.

2 10× 9 4 10× 9 6 10× 9 8 10× 9 1 10× 10

50

100

150

200

C [1/mm ]3

Δω [H

z]

0.2 mμ

0.5 mμ

0.9 mμ


5. Investigation of the largest Lyapunov exponent of radiation field scattered by Brownian particles

Lyapunov exponents play an important role in studying dynamic systems. Theycharacterize the average velocity of exponential divergence of close phasetrajectories. Taking the initial distance d0 between two initial points of phasetrajectories, the distance between trajectories, coming off these points, at time t willbe as follows:

d (t ) = d0exp(λ t ) (9)

The value λ is called Lyapunov exponent [8]. Each dynamic system is characterizedby Lyapunov exponent spectrum λ i (i = 1, 2, ..., n), where n – quantity of differentialequations which are necessary for system description. For experimental data, obtainedat observing dynamic systems, the availability of positive Lyapunov exponent can bethe proof of chaos existence in the system. Generally speaking, chaotic system ischaracterized by the divergence of phase trajectories in similar directions and theirconvergence in others, i.e., there are both positive and negative Lyapunov exponentsin chaotic system. The sum of all the indices is negative, i.e., the trajectory convergencedegree exceeds that of divergence. If this condition is not met, dynamic system isinstable, and the behavior of such a system is recognized easily. Thus, in most cases,it is sufficient to calculate the largest Lyapunov exponent only. The positive value ofthe largest Lyapunov exponent gives the possibility of chaos existing in the system,and the value of this index characterizes chaoticness intensity.

Most algorithms for calculating the largest Lyapunov exponent have somedisadvantages. For example, a great quantity of experimental data is required, there isa relative complexity of algorithm program realization, and numerous calculations aretime consuming [8]. Using theoretical conclusion, described in paper [15], we haveelaborated the algorithm and calculation program of the largest Lyapunov exponent,which is free of the above-mentioned disadvantages.

The first step of the algorithm consists in reconstructing the phase trajectory.The latter is represented in the form of matrix X, each column of which is a vector inphase space:

where Xi – system condition at time moment i. For a series of N measurements{x1, x2, …, xN} any X is defined as follows:

where J – time delay (reconstruction delay), m – embedding dimension.

X X1 X4 … XM

T=

Xi xi xi J+ … xi m 1–( )J+=


Embedding dimension is usually estimated based on Takens’ theorem, accordingto which m > 2n, where n is the system order. However, the method described enablesobtaining correct result at lesser value m. Reconstruction delay is chosen as being equalto time at which autocorrelation function decreases by 1–1/e compared to its initialvalue.

After reconstructing the phase trajectory by the algorithm the search is expectedof the nearest “neighbor” for each trajectory point. The point Xl is considered to bethe nearest “neighbor” with minimum distance dj (0) from it to the basic point Xj:

A couple of “neighbor” points diverges exponentially in some period of time:

where λ1 – the largest Lyapunov exponent. It can be found as a line inclination definedby the formula:

(10)

where means the average value for all j.This algorithm was used as a basis for compiling a program in Pascal. For testing

program the model and experiment chaotic signals were used. Henon’s mapping wasregarded as a model signal: xi+1 = 1 – a + yi ; yi+1 = bxi ; a = 1.4; b = 0.3.

The calculation results for Henon’s mapping are demonstrated in Fig. 3. Diagraminclination was calculated with the help of the least squares, and there wereobtained the values λ1 = 0.403. The latter almost corresponds to the theoretical valueλ1 = 0.418.

Experimental investigation of the largest Lyapunov exponent did not testify tothe availability of radiation intensity fluctuations scattered by sulfur hydrosols anddid not find its dependence on sulfur particle sizes. This can be explained by that fact

dj 0( ) minXj

Xj Xj–=

dj i( ) Cj λ1iΔt( )exp≈

y i( ) 1Δt

---------- dj i( )[ ]ln⟨ ⟩=

…⟨ ⟩

xi2

Fig. 3. Calculation results y (i ) for Henon’smapping.0 5 10 15 20 25 30

2

3

4

5

6

7

ln[d

(i)]

i


that the size change of sulfur monodisperse particles leads to the change of generalintensity of scattered radiation field, and not to its distribution. However, the largestLyapunov exponent depended essentially on particle concentration C and angle ofscattering θ.

Figure 4 shows the dependence of the largest Lyapunov exponent on concentrationfor particle sizes 0.2, 0.5 and 0.9 μm for geometrical experimental parameterscorresponding to Fig. 2.

One can observe practically a similar dependence for the above-mentioned sizeparticles. Moreover, the dependences given in Figs. 2 and 4 appear to have similarcourse. This fact was taken as a basis for increasing the measurement accuracy ofBrowne’s particle sizes with the help of the method of correlation spectroscopy.Dependences, obtained by standardization curves in Fig. 2 for coefficients ,do not practically depend on concentration of Brownian particles and give the valuesof particle sizes which are closer to real ones: 0.25, 0.59 and 1.04 μm. And theseexceed by only 20% the results of optical measurements at calibrating the hydrosolgrowth process.

λ1 λ1max⁄

Fig. 4. Dependence λ1 on concentration forparticles with the sizes of 0.2, 0.5 and 0.9 μm.

1 10× 8 1 10× 9 1 10× 10

2.0

2.5

C [1/mm]

0.2 mμ

λ1

0.5 mμ 0.9 mμ

Fig. 5. Dependence λ1 on the angle of scatteringθ for three types of concentration: 108, 109 and1010 particles per mm3.

0.00 0.05 0.10 0.15 0.20

1.0

1.5

2.0

2.5

θ [rad]

λ1

c = 108

c = 109

c = 1010


The decrease of the angle of scattering influences λ1 behavior sufficiently. Figure 5demonstrates dependences of the largest Lyapunov exponent on the angle of scatteringθ for three types of concentration: 108, 109 and 1010 particles per mm3. One can observeλ1 to increase with θ upon its next saturation. And for great saturation concentrationsthe process is realized quicker.

Computer modeling enabled us to consider situations which are impossible torealize in a real, physical experiment. It has been found out with the help of computermodeling that the scattering of Brownian particles by sizes has not essential influenceon the correlation integral course essentially. The normal distribution dispersion waschanged from 0 to 0.1 μm for the average particle size R = 0.3 μm.

6. ConclusionsThe results of carrying out the computer and physical modeling of light scattering byBrownian particle system consist in the fact that temporal intensity fluctuationsof scattered radiation field have a chaotic character, as largest Lyapunov exponent ispositive.

Stochastic field parameters actually do not depend on particle sizes. The increaseof Brownian particle concentration and of the angle of scattering leads to the chaotizationincrease of temporal intensity fluctuations of scattered radiation field.

There have been determined the empirical diagnostic relationship of largestLyapunov exponent and the widening of the spectrum of temporal intensity fluctuations.This gave the possibility to widen the concentration range and increase the measurementaccuracy of Brownian particle sizes.

References[1] VIKRAM C.S., Particle Field Holography, Cambridge University Press, Cambridge 1992.[2] SYVITSKI P.M., Principles, Methods and Application of Particle Size Analysis, Cambridge University

Press, Cambridge 1991.[3] GORELIK G.S., On the possibility of quick-response fotometry and demodulating analysis of a light,

Reports Academy of Science of the USSR, 58, 1947, pp. 45–47.[4] CUMMINS H.Z., PIKE E.R., [Eds.], Photon Correlation and Light Beating Spectroscopy, Plenum, New

York 1974.[5] DUBNISHCHEV YU.N., RINKEVICHUS B.S., The Techniques of Laser Doppler’s Anemometry, Nauka,

Moscow 1982.[6] BROWN R., On the existence of active molecules in organic and inorganic bodies, Philosophical

Magazine 4, 1828, pp. 162–73.[7] FEDER E., Fractals, Plenum, New York 1988.[8] NEUMARK YU.I., LANDA P.S., Stochastic and Chaotic Fluctuations, Nauka, Moscow 1987.[9] EINSTEIN A., Über die von der molekularkinetischen Theorie der Wärme geforderte Bewegung

von in ruhenden Flüssigkeiten suspendierten Teilchen, Annalen der Physik 322(8), 1905,pp. 549–60.

[10] VAN DE HULST H.C., Light-Scattering by Small Particles, John Wiley and Sons, New York; Chapmanand Hall, London 1957.


[11] ASANO S., SATO M., Light scattering by randomly oriented shperoidal particles, Applied Optics19(6), 1980, pp. 962–74.

[12] SHIFRIN K.S., ZELMANOVICH I.L., The Tables of Light Scattering, 2. Gidrometizdat, Leningrad 1968.[13] KELIKH S., Molecular Non-Linear Optics, Nauka, Moscow 1981.[14] FROLOV YU.G., The Course of Colloid Chemistry, Khimiya, Moscow 1982.[15] ROSENSTEIN M.T., COLLINS J.J., DE LUCA C.J., A Practical Method for Calculating Largest Lyapunov

Exponents from Small Data Sets, MA 02215, Boston University, 1992.

Received November 12, 2007


Propagation properties of partially coherent beams through turbulent media with coherent modes representation

M. ALAVINEJAD*, F. ASHIRI, B. GHAFARY*

Photonics Laboratory, Physics Department, Iran University of Science and Technology, Tehran, Iran

*Corresponding authors: M. Alavinejad – [email protected], B. Ghafary – [email protected]

The partially coherent beams propagating through turbulent atmosphere have been studied inthe past using coherent mode representation. In this research, the propagation of any modes ofHermite–Gaussian beam in a turbulent atmosphere is investigated and analytical formula forthe average intensity of these beams is derived. The power in bucket (PIB) for any modes is alsoexamined. The number of modes which exist in a partially coherent beam with known degree ofglobal coherence (ratio of correlation length and the waist width of the Gaussian–Schell model(GSM) beam) is determined and the PIB for partially coherent beams is investigated using coherentmode representation.

Keywords: atmospheric turbulence, partially coherent beam, Hermite–Gaussian beam, power inbucket (PIB).

1. Introduction

Propagation of a laser beam through random media is the topic that has beentheoretically and experimentally studied for a long time, as is evident from the numberof books and papers written on the subjects [1, 2]. In many practical applications,such as remote sensing, tracking, and long-distance optical communication, etc.,the propagation properties of laser beam through atmospheric turbulence are of greatimportance and atmospheric turbulence has essential effects in such applications.Most research is about the spreading of a laser beam in a turbulent atmosphere [3–6].Recently, Eyyuboglu and Baykal investigated the properties of cos-Gaussian,cosh-Gaussian, Hermite-sinusoidal-Gaussian and Hermite-cosine-Gaussian laserbeams in a turbulent atmosphere [7–9]. Cai and He have studied the spreadingproperties of an elliptical Gaussian beam in a turbulent atmosphere [10]. In this paper,the average intensity of Hermite–Gaussian modes propagating in turbulent media isinvestigated and the effects of turbulence on the properties of any arbitrary mode areanalyzed.

550 M. ALAVINEJAD, F. ASHIRI, B. GHAFARY

The number of full coherent modes constituting a specific partially coherent beamhas been determined and it is shown that the intensity profile and PIB of a propagatedpartially coherent beam are equal to the sum of intensity profile and PIB for fullcoherent modes constituting it. It means that if we expand a specific partially coherentbeam into a number of fully coherent modes, then calculating intensity and PIB of theconstituting modes and integrating the results will lead to intensity and PIB of thatpartially coherent beam.

2. Propagation of Hermite–Gaussian mode in turbulent atmosphereField distribution of a Hermite–Gaussian laser beam on the source plane is as follows [5]:

(1)

where Hm and Hn are Hermite polynomials, w0 is the spot size at the source plane,(ρx, ρy ) is a two-dimensional vector in the source plane and Bmn are normalizationcoefficients represented in the following form [5]:

(2)

Assume that the medium is statistically homogeneous and isotropic, then accordingto the paraxial form of extended Huygens–Fresnel principle the relation betweenelectrical field before and after propagation is [1, 2]:

(3)

where E (ρ, z = 0) and E (ρ', z) represent the amplitude of incident field and the fieldafter propagating a distance z through the turbulent medium, respectively, and ψ isthe phase function that depends on the properties of the medium.

The average intensity at the receiver plane is given by where denotes the ensemble average over the medium

statistic. From Eq. (3), we obtain

(4)

φmn0( ) ρx ρy 0, ,( ) Bmn Hm

2w0-------- ρx⎝ ⎠⎜ ⎟⎛ ⎞

Hn2

w0-------- ρy⎝ ⎠⎜ ⎟⎛ ⎞ ρ x

2 ρ y2+

w02------------------------–

⎝ ⎠⎜ ⎟⎛ ⎞

exp=

Bmn1

w0 π2m n 1–+ m!n!------------------------------------------------------=

E ρ' z,( ) ik ikz( )exp2πz

------------------------------- E ρ( ) ik ρ' ρ– 2

2z------------------------ ψ ρ' ρ z, ,( ) d2ρexpexp∫∫–=ρ ρρ ρ

ρ ρ

I ρ' z,( )⟨ ⟩ =ρE ρ' z,( ) E * ρ' z,( )⟨ ⟩= ρ ρ ⟨ ⟩

I ρ' z,( )⟨ ⟩ k2πz

--------------⎝ ⎠⎜ ⎟⎛ ⎞2

d2ρ 1 d2ρ 2 E ρ1( )E* ρ2( ) ikρ' ρ1– 2 ρ' ρ2– 2–

2z------------------------------------------------------–exp∫∫∫∫

ψ ρ' ρ1 z, ,( ) ψ * ρ' ρ2 z, ,( )+exp⟨ ⟩×

=ρ ρ ρρ ρ ρ ρ

ρρρρ

ρ ρ

Propagation properties of partially coherent beams ... 551

The last term in the integrand of Eq. (4) can be expressed as [3–5]

(5)

where Dψ (|ρ1 – ρ2|) is the phase structure function in Rytov’s representation andΦ = (0.545 k 2z)6/5 is the coherence length of a spherical wave propagating inthe turbulent medium with being the structure constant. In order to obtainan analytical result the quadratic approximation (see, e.g., Eqs. (14) and (15) in [11])for Rytov’s phase structure function is used here since this quadratic approximationhas been widely investigated and shown to be reliable [3–5].

To evaluate Eq. (4), it is convenient to introduce the new variable of integration,

, (6)

and Eq. (4) reduces to

(7)

This is the basic formula that will be used to study the propagation of Hermite–Gaussianmodes through atmospheric turbulence.

Substituting Eq. (1) and Eq. (2) in Eq. (7) and calculating the related integralresults:

(8)

ψ ρ' ρ1 z, ,( ) ψ * ρ' ρ2 z, ,( )+exp⟨ ⟩ 0.5D– ψ ρ1 ρ2–( )exp

Φ ρ1 ρ2– 2–exp

=

=

ρ ρ ρ ρ ρ ρ

ρ ρ

Cn2

Cn2

uρ1 ρ2+

2------------------=ρ ρ

v ρ1 ρ2–= ρ ρ

I ρ' z,( )⟨ ⟩ k2πz

--------------⎝ ⎠⎜ ⎟⎛ ⎞2

d2u d2v E u v2

------+⎝ ⎠⎜ ⎟⎛ ⎞

E* u v2

------–⎝ ⎠⎜ ⎟⎛ ⎞

i kz

------- u v⋅–

i kz

-------ρ' v⋅ Φ v2–expexp×

exp∫∫∫∫=ρ

ρ

Imn ρ' z,( ) k2πz

-------------⎝ ⎠⎜ ⎟⎛ ⎞2

Bmn2 π

2-------w0

2 2m n+

1–( )l l'+ m!n!( )2

m l–( )! n l'–( )l!l!l'!l'!------------------------------------------------------------

j' 1=

l'

∑l' 0=

m

∑j' 1=

l

∑l 0=

n

∑×

η l l'+ k2

4z2ζ 2------------------- ρ 'x2 ρ 'y2+⎝ ⎠⎛ ⎞–exp× π

22 l 2 l'+ ζ 2 l l'+( ) 2+------------------------------------------------ Bj 2n, Bj' 2n,

ikρx'2ζ z

----------------–⎝ ⎠⎜ ⎟⎛ ⎞2 l 2 j–

×ikρy'2ζ z

----------------–⎝ ⎠⎜ ⎟⎛ ⎞2 l' 2 j'–

=ρ

ρ ρ


whereBm, n +1 = nBm – 1, n – 1 + Bm, n, B1, n = 1, Bn, n = 0, B1, 1 = 1 (9)

(10)

(11)

The propagation properties of Hermite–Gaussian modes in a turbulent atmospherecan be investigated using Eqs. (8)–(11). Figure 1 shows the behavior of the normalized

η 1

w02----------

k2w02

4z2-----------------+=

ζ 2 Φ 1

2w02

--------------k2w0

2

8z2-----------------+ +=

Fig. 1. Normalized intensity distribution of Hermite–Gaussian modes in free space and turbulence media,with w0 = 0.005 m, k = 107 m–1, = 10–14, l0 = 0.01 m, σs = 5 mm, z = 5000 m, m = n = 1 andm = n = 2.

Cn2


intensity distribution of Hermite–Gaussian modes in atmospheric turbulence and infree space, with w0 = 0.005 m, k = 107 m–1, = 10–14, m = n = 1 and m = n = 2 andz = 5 km. As it is shown in Fig.(1) the peaks of intensity of Hermite–Gaussian modesin atmospheric turbulence and in free space. As is shown in Fig. 1, the peaks of intensityof Hermite–Gaussian modes in atmospheric turbulence are more spread out andoverlapped in comparison to those in free space.

3. Explanation of the partially coherent mode in terms of fully coherent mode

The initial field of a partially coherent beam takes the typical form of Gaussian–Schellmodel (GSM) beam, whose cross-spectral density function W (0)(ρ1, ρ2, z) at the sourcez = 0 is [13, 14]

(12)

where A is a constant, ρ1 = (ρ1x , ρ1y) and ρ2 = (ρ2x , ρ2y) denote two different pointsat the source plane and σs and σμ are the waist width and correlation length of the GSMbeam, respectively.

It supposes that the same conditions for turbulent atmosphere as mentioned inthe pervious section, the propagation equation of intensity of GSM beams turns outto be [5]

(13)

where

(14)

The cross-spectral density function of such a source assumed, for simplicity, to berectangular, may be represented in the form [5]:

(15)

where φmn represent the different coherent modes such as Hermite–Gaussianpolynomials. This expansion is called coherent mode representation. Expansioncoefficient of Eq. (15) is as follows [5]:

Cn2

W 0( ) ρ1 ρ2 z 0=, ,( ) Aρ1

2 ρ22+

4σ s2

-----------------------------------–ρ1 ρ2– 2

2σμ2

----------------------------–expexp=ρ ρρ ρ ρ ρ

I ρ' z,( ) A

Δ2 z( )------------------ ρ' 2

2kσ s2Δ2 z( )

-------------------------------–⎝ ⎠⎜ ⎟⎜ ⎟⎛ ⎞

exp=

Δ2 z( ) 1 1

k2w02

---------------- 1

4w02

--------------- 1

σμ2

------------+⎝ ⎠⎜ ⎟⎜ ⎟⎛ ⎞

z2 Φ

σ s2

----------- z3+ +=

W 0( ) ρ1 ρ2 ω, ,( ) βmn ω( )φmn0( ) ρ1 ω,( )φmn

0( ) ρ2 ω,( )n∑

m∑=ρρ ρ ρ

ρ ρ

ρ ρ


(16)

where lg = σμ /σs is the degree of global coherence. Evidently, in the limit ofthe complete coherence (lg → ∞) and in the incoherence limit (lg → 0). This impliesthat an infinite number of modes are needed to represent a spatially non-coherentsource.

To determine the number of modes in a special partially coherent beam, first usingEq. (13) we calculate the intensity profile of that partially coherent beam and thenname it profile one.

Then using Eqs. (8)–(11) each propagated mode is calculated and replacedin right-hand side of Eq. (15). We choose the modes whose respective expansion

βmn ω( )β00 ω( )

---------------------- 1

lg2

2-------- 1 lg

lg

2-------⎝ ⎠⎜ ⎟⎛ ⎞2

1++ +

------------------------------------------------------------------

m n+

=

φmn0( )

Fig. 2. Intensity distribution of partial coherence beam (a), intensity distribution of partial coherencebeam by using coherent modes representation (b) with w0 = 0.01 m, k = 107 m–1, = 10–14, l0 = 0.01 m,σs = 5 mm, z = 5000 m.

Cn2

a b

Fig. 3. Intensity profile of partially coherentbeams (45 modes) and the behavior of theintensity distribution for limited number ofmodes.


coefficients are greater than 0.001 (βmn > 0.001) and calculate corresponding intensityprofile and name it profile two.

Finally, comparing profiles one and two, one can find the number of fully coherentmodes constituting supposed partially coherent beam.

In Figure 2a, intensity profiles of various partially coherent beams are plottedusing Eq. (13). The behavior of the intensity distribution using coherent modesrepresentation and Eqs. (8)–(11) is illustrated in Fig. 2b.

Comparing Figs. 2a and 2b, we have determined corresponding number of fullcoherent modes for any lg, as mentioned before; the result of this determination areshown in the Table.

It is worth mentioning that, as is evident from Fig. 2, the number of modes involved,increases with decreasing lg.

One goal of this paper is to find the number of full coherent modes required forconstruction of any arbitrary partially coherent beam. In practice, it is very difficult tocombine more than several modes which will lead us to a remarkable error, as is clearlyshown in Fig. 3.

Figure 3 shows the intensity profile of partially coherent beams (45 modes) andthe behavior of the intensity distribution using coherent modes representation forlimited number of modes. Although it seems that combining 45 modes is impracticablebut deceasing the number of modes will lead to a remarkable error, which is clearlyshown in this figure.

4. Power in bucket

As suggested by Siegman, PIB is a measure of laser power focus ability in the farfield [14]. It indicates the amount of beam power within a given bucket. The PIB isdefined as [15]

(17)PIBI ρ z,( )ρ dρ dθ

0

α

∫0

2π

∫

I ρ z,( )ρ dρ dθ0

∞

∫0

2π

∫---------------------------------------------------------=

ρ

ρ

T a b l e. Number of modes with full coherence which exist in partially coherent beam.

lg Fully coherent modes100 φ00

2 φ00, φ10, φ01, φ11, φ20, φ02, φ21, φ12, φ30, φ03

1 φ00, φ10, φ01, φ11, φ20, φ02, φ21, φ12, φ22, φ30, φ03, φ31, φ13, φ32, φ23, φ33, φ40, φ04, φ41, φ14, φ42, φ24, φ50, φ05, φ51, φ15, φ60, φ06

0.7 φ00, φ10, φ01, φ11, φ20, φ02, φ21, φ12, φ22, φ30, φ03, φ31, φ13, φ32, φ23, φ33, φ40, φ04, φ41, φ14, φ42, φ24, φ43, φ34, φ44, φ50, φ05, φ51, φ15, φ52, φ25, φ53, φ35, φ60, φ06, φ61, φ16, φ62, φ26, φ70, φ07, φ71, φ17, φ80, φ08


where α is the radius of the bucket. The PIB for any Hermite–Gaussian mode wascalculated using Eqs. (8) and (17).

Figures 4a and 4b show the behavior of PIB for Hermite–Gaussian modes in freespace and turbulent atmosphere respectively. These figures show that PIB for higherorder modes is more extended. It is also evident that PIB in turbulence for any arbitrarymode is more broadened in comparison with PIB in free space.

Figure 5a shows the behavior of PIB for partially coherent beam calculated fromEqs. (13), (17) and Fig. 5b shows the behavior of PIB calculated by the methodof coherent mode representation. The accurate conformity of these two figures

Fig. 4. PIB for Hermite–Gaussian modes in free space (a), PIB for Hermite–Gaussian modes in turbulentmedia (b) with w0 = 0.01 m, k = 107 m–1, = 10–14, l0 = 0.01 m, σs = 5 mm, z = 5000 m.Cn

2

a b

Fig. 5. PIB for partially coherent beam in turbulent media (a), PIB for partially coherent beam in turbulentmedia by using the sum of PIB of modes which exist in the partially coherent beam (b) with w0 = 0.01 m,k = 107 m–1, = 10–14, l0 = 0.01 m, σs = 5 mm, z = 5000 m.Cn

2

a b


demonstrates that our procedure of determining the number of fully coherent beamsinvolved in a specific partially coherent beam (specific lg) is quite reliable.

5. Conclusions

In this article, the analytical formula for average intensity of Hermite–Gaussian modespropagating in turbulent media was investigated. We have noticed that peaks ofHermite–Gaussian beams propagating through turbulent atmosphere are distributedin plane and are overlapped. It has been found that the number of full coherent modesconstituting a specific partially coherent beam can be determined by calculatingintensity profile and PIB of the partially coherent beam by two distinct methods andcomparing the results of them. It has also been found that PIB for Hermite–Gaussianmode propagating through atmospheric turbulence is more broadened in comparisonwith that in free space, and calculating PIB of the constituting modes of a partiallycoherent beam and summing up the results, will lead to PIB of that partially coherentbeam.

This procedure is implemented in this article by comparing the intensity profilesand validated by calculation of corresponding PIBs.

References[1] FANTE R.L., Wave propagation in random media: a system approach, [In] Progress in Optics,

E. Wolf [Ed.], Elsevier, Amsterdam, 1985, Vol. 22, pp. 341–98.[2] ANDREWS L.C., PHILLIPS R.L., Laser Beam Propagation through Random Media, SPIE Press,

Bellingham, Washington 1998.[3] WANG S.C.H, PLONUS M.A., Optical beam propagation for a partially coherent source in

the turbulent atmosphere, Journal of the Optical Society of America 69(9), 1979, pp. 1297–304.[4] LEADER J.C., Atmospheric propagation of partially coherent radiation, Journal of the Optical Society

of America 68 (2), 1978, pp. 175–85.[5] SHIRAI T., DOGARIU A., WOLF E., Mode analysis of spreading of partially coherent beams

propagating through atmospheric turbulence, Journal of the Optical Society of America A: Optics,Image Science and Vision 20(6), 2003, pp. 1094–102.

[6] YANGJIAN CAI, SAILING HE, Propagation of a partially coherent twisted anisotropic GaussianSchell-model beam in a turbulent atmosphere, Applied Physics Letters 89(4), 2006, p. 041117.

[7] EYYUBOGLU H.T., BAYKAL Y., Analysis of reciprocity of cos-Gaussian and cosh-Gaussian laserbeams in turbulent atmosphere, Optics Express 12(20), 2004, pp. 4659–74.

[8] EYYUBOGLU H.T., BAYKAL Y., Average intensity and spreading of cosh-Gaussian laser beams inthe turbulent atmosphere, Applied Optics 44(6), 2005, pp. 976–83.

[9] BAYKAL Y., Correlation and structure functions of Hermite-sinusoidal-Gaussian laser beams in aturbulent atmosphere, Journal of the Optical Society of America A: Optics, Image Science andVision 21(7), 2004, pp. 1290–9.

[10] YANGJIAN CAI, SAILING HE, Average intensity and spreading of an elliptical Gaussian beampropagating in a turbulent atmosphere, Optics Letters 31(5), 2006, pp. 568–70.

[11] EYYUBOGLU H.T., ARPALI C., BAYKAL Y.K., Flat topped beams and their characteristics in turbulentmedia, Optics Express 14(10), 2006, pp. 4196–207.


[12] MANDEL L.M., WOLF E., Optics Coherence and Quantum Optics, Cambridge University Press,Cambridge, England 1995.

[13] FRIBERG A.T., SUDOL R.J., Propagation parameters of Gaussian Schell-model beams, OpticsCommunications 41(6), 1982, pp. 383–7.

[14] SIEGMAN A.E., How to (maybe) measure laser beam quality, OSA TOPS, Vol. 17, 1998, p. 184.[15] HAI XING YAN, SHU SHAN LI, DE LIANG ZHANG, SHE CHEN, Numerical simulation of an adaptive

optics system with laser propagation in the atmosphere, Applied Optics 39(18), 2000, pp. 3023–31.

Received December 11, 2007in revised form January 22, 2008


Evidence for metastable behavior of Ga-doped CdTe

EWA PŁACZEK-POPKO, ZBIGNIEW GUMIENNY, JUSTYNA TRZMIEL, JAN SZATKOWSKI

Institute of Physics, Wrocław University of Technology, Wybrzeże Wyspiańskiego 27, 50-370 Wrocław, Poland

In this paper, we report for the first time on persistent photoeffects in gallium doped CdTe.Persistent photoconductivity and photoinduced persistent absorption were observed at 77 K. Botheffects quenched above 120 K. The photoeffects have been attributed to the metastable behaviorof gallium in CdTe.

Keywords: semiconductors, semiconducting compounds, CdTe, DX centers, persistent photoeffects.

1. Introduction

CdTe based mixed crystals have been widely applied in semiconducting devices withgallium commonly used as the n-type dopant. However the operation of the devices isseriously affected by the deep–shallow metastability of gallium. According to the largelattice relaxation model of DX centers introduced by Park and Chadi, gallium dopantin CdTe and CdTe based Cd1–xMnxTe semiconducting alloys forms DX centers [1].The experimental findings concerning the material are comparable to the well-knownDX behavior of, e.g., Si in AlxGa1–xAs [2]. When the atom of Ga-dopant replaces Cdin the lattice it behaves as a shallow donor. At low temperature, after capturing anelectron the dopant passes to the interstitial position. Along with this new position,another deep, localized state of the DX center appears. Thus, the different charge statesof the DX center are related to the different locations in the crystal lattice. The transitionfrom one state to another is accompanied by a very significant lattice reconfiguration,the so-called large lattice relaxation. At low temperature, the photoionization ofthe DX center accompanies the deep–shallow transition and return to the “dark”ground state is not possible unless the system possesses enough energy. This energyis called the capture barrier.

The photoionization of DX centers is accompanied by an increase of excess carrierdensity in the conduction band followed by the conductivity rise. The presence of thebarrier for capture prevents electrons from returning to the DX center and the highervalue of conductivity persists even after termination of light excitation. Thisphenomenon is called persistent photoconductivity. Persistent photoconductivity has

560 E. PŁACZEK-POPKO et al.

been commonly recognized as a finger print for the presence of metastable defects inthe materials under study.

Another effect related to the existence of metastable defects is the persistentphotoinduced change of the absorption coefficient. The photoinduced absorptionrelated to the materials with DX centers has been reported only in a few papers [3–6],the most significant effect being found in CdF2 [4] and AlSb [6].

One of the crucial parameters for the operation of semiconducting devices such asphotodiodes, lasers, etc., is their time constant. The devices are expected to be very fastwith the time constants on the order of nanoseconds and less. Obviously, the presenceof metastable defects strongly affects this parameter, making the devices useless.Therefore, there is a strong need to study the properties of the defects responsible forthe degradation of the devices.

So far the DX behavior of gallium was confirmed only for gallium dopedCd1–xMnxTe [7–10], mostly by persistent photoconductivity and photocapacitance.In this paper, we report for the first time on persistent photoeffects in gallium dopedCdTe. Persistent photoconductivity and photoinduced persistent absorption wereobserved at 77 K. Thus, the presence of DX centers in the material has been proved.

2. Experiment

2.1. Samples and experimental methods

Experiments were performed on the samples of gallium doped CdTe grown bythe Bridgman method. Prior to the measurements the samples were annealed at 600 °Cin cadmium vapor for one week to reduce the level of compensation in the materialdue to cadmium vacancies. Slices of the material were prepared by mechanicalpolishing followed by the chemical etching in 2% Br2 in methanol solution to removethe remaining damaged surface layer. The Au-CdTe Schottky diodes were realised byevaporation of gold in the vacuum of 10–6 torr on the chemically prepared surface andwere situated on the front side of the samples. Ohmic contacts were produced bysoldering indium onto the fresh backside surface.

For the conductivity measurements a four point method was applied. A Keithleyconstant current source was used and the voltage drop across the sample measured atconstant current was equal to 10 μA. The temporal kinetics of conductivity wasmeasured at 77 K. The samples were initially cooled down in the darkness to the lowtemperature to achieve the ground state of the DX centers. The conductivity transientswere recorded in the darkness, after illumination, until the conductivity saturated anduntil the equilibrium state was reached. A shutter was used to turn the light on/off.The monochromatic light beam coming out of the monochromator was focused withthe help of a fiber optic onto the sample, mounted in the sample holder immerseddirectly in the liquid nitrogen. The conductivity transients were recorded by means ofmultimeters and a computer. The monochromatic light of photon energy equal to1.24 eV was chosen.

Evidence for metastable behavior of Ga-doped CdTe 561

Absorption measurements were made at room temperature and at 77 K. Persistentphotoinduced absorption was investigated at 77 K. Persistent photoinduced absorptionwas observed for the 77 K measurements run for 10 min after terminating a 30 minhalogen light excitation.

2.2. Characterization of the samples

The results of optical measurements run at room temperature and at 77 K for the samplesunder study are given in Fig. 1. In the figure, the square of the absorption coefficientα2 as a function of photon energy hν is shown. CdTe is a direct band gap material,therefore the value of energy gap can be determined from the linear part of the α2 == f (hν ) plot. The values of energy gap obtained from the linear fit of the solid lineslopes in Fig. 1 are equal to 1.38 eV at 300 K and 1.45 eV at 77 K. Both, of the obtained

Fig. 1. The square of the absorption coefficient α2 as a function of photon energy hν for CdTe:Ga atroom temperature (open circles) and at 77 K (open squares). The solid lines are the best square linear fitsto the experimental data.

0.9 1.1 1.3 1.5 1.7 1.9 2.1

0.0

5.0 10× 2

1.0 10× 3

1.5 10× 3

2.0 10× 3

2.5 10× 3

3.0 10× 3

E [eV]

α2

2 [c

m]

–

Fig. 2. Capacitance–voltage characteristic for CdTe:Ga measured at 300 K. The solid line is the bestsquare linear fit to the experimental data. The fit yields the value of built in voltage equal to 0.33 eV anddonor net concentration equal to 1017 cm–3.

–1.2 –0.8 –0.4 0.0 0.4 0.80.0

5.0 10× 5

1.0 10× 6

1.5 10× 6

2.0 10× 6

2.5 10× 6

3.0 10× 6

U [V]

S/C

[m/F

]2

24

2

Vbi


values of energy gap and linear behavior of the dependence show good quality ofthe CdTe samples [11].

The donor net concentration in the samples was estimated from the capacitance–voltage measurement. A sample capacitance–voltage characteristic taken at 300 K isshown in Fig. 2. The solid line is the least square linear fit to the experimental data.The room temperature net donor concentration found from the fit is equal to 1017 cm–3.


In the gallium doped CdTe under study, persistent photoeffects were observed at lowtemperatures. A typical PPC behavior is presented in Fig. 3. In the figure, a sampletemporal kinetic of photoconductivity measured at 77 K for photon energy equal to1.24 eV is shown. Both the conductivity build-up after turning on the light and its

Fig. 3. Conductivity build-up and decay for CdTe:Ga at 77 K. The sample was illuminated with photonsof energy equal to 1.24 eV.

0 500 1000 1500 2000 2500

140

150

160

170

180

190

200

210

220

230

Light off

Light on

t [s]

1/R

[]

Ω–1

Fig. 4. Optical absorption for CdTe:Ga at 77 K (open triangles) and after preliminary 30 min illuminationwith halogen light (open squares). The latter measurements were performed 10 min after terminationof light.

0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.51

2

3

4

5

6

7

8

9

10

α [c

m]

–1

E [eV]


decay after turning off the light exhibit slow kinetics. After light terminationthe conductivity of the sample does not return to the value it had before illumination.The higher value persists for a long time after termination of light.

A spectrum of photoinduced absorption is shown in Fig. 4. Illumination attemperatures less than 120 K produced an increase in the optical absorption overa spectral range extending from about 0.62–1.4 eV. In Figure 5, the difference betweenthe dark and illuminated transmission spectra corresponding to those given in Fig. 4is shown. A well defined maximum corresponding to the energy equal to 1 eV can beobserved. The photoinduced absorption is persistent, i.e., it remains almost unchangedafter terminating the light.

It has been found that both persistent photoconductivity and photoinducedpersistent absorption are quenched for temperature above 120 K, approximately.

The results shown in Figs. 3–5 give direct evidence for the presence of metastabledefects in the material under investigation. As regards the temporal kinetics of photo-conductivity shown in Fig. 3, it can be explained by the presence of DX centers inthe material. Upon cooling, the sample DX centers get occupied by electrons andundergo a shallow–deep transformation. Illumination at sufficiently low temperature(here, 77 K) leads to the photoionization of DX centers if the photon energy exceedstheir photoionization energy. The photoionization energy of DX centers for Ga dopedCd0.99Mn0.01Te was found to be equal to 1.1 eV [11]. It may be expected thatthe photoionization energy for the DX centers in CdTe is close to this value.The kinetics presented in Fig. 3 was taken for photon energy equal to 1.24 eV, higherthan 1.1 eV, we may therefore assume that the DX centers have become photoionized.Once photoionized, the DX centers undergo a deep–shallow transformation. Asa result the occupation of deep levels decreases, and subsequently, the density ofelectrons in conduction band increases leading to an increase in conductivity.

After termination of light initial fast kinetics is followed by a very slow decay.The barrier for capture prevents DX centers from returning to their ground state. Atlow temperature the electrons have not got enough energy to surpass the barrier.

Fig. 5. The difference ΔT between the transmission spectra measured in darkness and after illuminationat 77 K, related to the data shown in Fig. 4.

0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.510–1

100

101

E [eV]

ΔT


Warming above 120 K leads to the quenching of the photoeffect as electrons areprovided with energy to overcome the barrier for capture.

The presence of the barrier for capture between the shallow, metastable state andthe deep ground state of DX center does not explain however the observedphotoinduced persistent absorption increase. Usually, in materials with DX centersa photoinduced persistent decrease in absorption is observed within the energyrange corresponding to the DX photoionization energy and simultaneous increase inthe infrared range [3–6]. The former is due to the lack of electrons which alreadyleft the photoionized DX ground state. The latter is the result of absorption byelectrons occupying the shallow metastable hydrogenic Ga donor level. In Figure 5,the difference between the transmissions measured in darkness without and withpreliminary illumination is shown. In this way, a broad absorption band with maximumaround 1 eV was found. There is a possibility that as in the case of Al xGa1–xAs:Tethe band may be associated with another metastable state of DX center [3], the DX0

state or a new kind of metastable defect created during iluminaton. This is an openquestion and needs further investigation. The photoinduced absorption has undoubtedlyits origin in a metastable state.

4.Conclusions

The presence of metastable defects in gallium doped CdTe was confirmed bythe observation of persistent photoconductivity and photoinduced persistent absorptionat 77 K. It was found that both effects are quenched at temperatures above 120 K,approximately. The two-fold nature of DX centers with deep–shallow transformationaccompanied by large lattice relaxation explains the persistent photoconductivityeffect. As for the photoinduced persistent absorption it cannot be understood withinthe frame of this model. There has to be another metastable state responsible forthe effect. No matter whether the origin of the observed persistent photoeffects isthe DX center or any other metastable defect, its formation in the materialis undesirable. Metastable defects act as carrier traps and thus may seriously affect,e.g., carrier densities. Their presence is a crucial factor which may deteriorateparameters of semiconductor devices made of the gallium doped CdTe material.

References

[1] PARK C.H., CHADI D.J., First-principles study of DX centers in CdTe, ZnTe, and CdxZn1–xTe alloys,Physical Review B: Condensed Matter 52(16), 1995, pp. 11884–90.

[2] DOBACZEWSKI L., KACZOR P., Photoionization of DX(Te) center in AlxGa1–x As: Evidence fora negative U-character of the defect, Physical Review B: Condensed Matter 44(16), 1991,pp. 8621–32, and references therein.

[3] MORI Y., YOKOTA T., OHKURA H., Metastable states observed by optical absorption of DX centers inAlxGa1–x As:Te, Japanese Journal of Applied Physics, Part 2: Letters 31(8A), 1992, pp. L1005–8.

[4] RYSKIN A.I., SHCHEULIN A.S., ONOPKO D.E., DX centers in ionic semiconductor CdF2:Ga, PhysicalReview Letters 80(13), 1998, pp. 2949–52.


[5] PINHEIRO M.V.B., KRAMBROCK K., Experimental evidence for the distinction between metastabilityand persistence in optical and electronic properties of bulk GaAs and AlGaAs, Brazilian Journal ofPhysics 29(4), 1999, pp. 806–9.

[6] BECLA P., WITT A., LAGOWSKI J., WALUKIEWICZ W., Large photoinduced persistant opticalabsorption in selenium doped AlSb, Applied Physics Letters 67(3), 1995, pp. 395–7.

[7] SEMALTIANOS N.G., KARCZEWSKI G., WOJTOWICZ T., FURDYNA J.K., Persistent photoconductivity andphotoionization of deep electron traps in Ga-doped Cd1–x Mn xTe, Physical Review B: CondensedMatter 47(19), 1993, pp. 12540–9.

[8] STANKIEWICZ O., YARTSEV V.M., Photoionization of electron traps in Ga-doped Cd1–x Mn xTe, SolidState Communications 95(2), 1995, pp. 75–8.

[9] PŁACZEK-POPKO E., NOWAK A., SZATKOWSKI J., SIERAŃSKI K., Capture barrier for DX centers ingallium doped Cd1–x MnxTe, Journal of Applied Physics 99 (8), 2006, p. 083510-1.

[10] PŁACZEK-POPKO E., SZATKOWSKI J., BECLA P., Photoionization of DX-related traps in indium- andgallium-doped Cd1–x MnxTe, Physica B: Condensed Matter 340–342, 2003, pp. 886–9.

[11] MADELUNG O., SCHULZ M., WEISS H. [Eds], Landolt–Bornstein, Numerical Data and FunctionalRelationships in Science and Technoloy, Vol. 17 – Semiconductors, Springer Verlag Berlin 1982.

Received January 2, 2008


Morphology of laser-induced damage of lithium niobate and KDP crystals

OLEH KRUPYCH, YAROSLAV DYACHOK, IGOR SMAGA, ROSTYSLAV O. VLOKH*

Institute of Physical Optics, 23 Dragomanov Str., 79005 Lviv, Ukraine


The regularities of laser-induced damage are studied for LiNbO3 and KDP single crystals. It isshown that the shape of damage in the dielectric crystals depends on the elastic symmetry of crystaland the propagation direction of laser beam. When the beam propagates along the optic axis ofthose crystals, the figures of laser damage are six-point stars for LiNbO3 and four-point ones forKDP crystals. For the beam direction parallel to the X and Y axes in the KDP crystal, the damageinitially has a cross-like configuration, with splitting of Z-oriented crack into two cracks duringits further evolution.

Keywords: optical damage, anisotropy, single crystals, KDP, LiNbO3.

1. Introduction

Studies of the bulk laser-induced damage of optical materials have usually beendirected towards measuring its threshold [1–4] and analyzing mechanisms ofthe damage [5–7]. Only few researchers have studied bulk damage morphology, whenirradiating the crystals with high-power laser pulses [8–10]. It has been noted that, inthe case of light beam propagation along the optic axis, the patterns of laser-induceddamage have a star-like shape, if observed along the optic axis. These stars wouldgenerally correspond to the order of the corresponding symmetry axis. When we dealwith the laser beam propagation directions perpendicular to the higher-fold symmetryaxis, the literature data become ambiguous. For example, YOSHIDA et al. [10] haveshown that the laser beam directed along the X axis in potassium dihydrogenphosphate crystals (KH2PO4 or KDP) yields in eight-point damage stars. However,YOSHIMURA et al. [9] have observed six-point damage stars in CsLiB6O10 crystals forthe identical experimental geometry, though those crystals have exactly the samesymmetry as KDP (point group 42m). The goal of this work is to study regularitiesof the laser-induced damage in anisotropic materials by the example of dielectriccrystals.

568 O. KRUPYCH et al.

2. Experimental results and discussion

For studies of the optical damage, we used a setup described earlier in [4] and a pulsedNd3+ laser (the pulse duration τ = 6 ns and the output radiation energy of 60 mJ).The laser beam was focused onto the spot of about 108 μm in diameter, correspondingto the intensity level 1/e2. The focus plane was positioned at a depth of 3–5 mm fromthe sample face. For each laser beam direction, we performed several laser shots,shifting the sample after each shot in order to avoid damage accumulation. To studythe bulk damage anisotropy, we used lithium niobate (LiNbO3 or simply LN) and KDPcrystals.

Laser beam intensities used for damage were fitted experimentally for bothcrystals, while the threshold level has not been determined because it is not the aim ofthe present study (readers may refer to studies [10–12]). The laser beam has passedthrough the crystalline sample parallel to the optic axis. We have increased the radiationenergy gradually and observed whether damage occurs. After passing the damagethreshold, the damage shape and dimensions have been analyzed. For small intensitiesexceeding E = I/Ithr over the threshold level Ithr (E ≤ 2), the damage shape has beenmainly irregular, with a central dark spot and randomly oriented small cracks. Regularcracks have been feebly marked. Damage dimensions have had the size of the orderof 0.1 mm or less. For the excess values E > 2, the damage stars have been well marked,while the central spot dimensions have varied slightly and remained of the order of0.1 mm. For the intensity exceeding the value E ≈ 3 we have observed clear damagestars with the cracks of the order of 0.5–0.6 mm. Further laser energy increase leadsto enlarging the damage dimensions and joining and overlapping the neighbourdamaged site’s cracks. Therefore, the most convenient laser beam intensitycorresponds to the case of intensity excess over the threshold level E ≈ 3. The intensityvalues corresponding to the specified excess over the threshold level have been usedin the case of laser-induced damage presented below for the crystals under test.

The LN sample was cut out perpendicular to the principal {100} directions. It hada shape of parallelepiped with dimensions of 8a×10b×8.3c mm3. The light propagatedalong the Z axis. The resulting damage is shown in Fig. 1. The damage stars obtainedfor the LN crystals, manifest nearly hexagonal shape when viewed along the optic axis(see Fig. 1a). This configuration of cracks corresponds to the point symmetry 3m ofthe LN. The damage channel has an arrowhead shape, with a wide part located nearthe light focus point and a thinning oriented into the sample (Fig. 1b).

The KDP crystal sample had nearly cubic shape, with dimensions of 11.2a×11.2b××10.8c mm3 and the faces (100), (010) and (001). We performed damage experimentsusing the light directed along all of the three principal axes. When the laser beampropagates along the optic axis (the Z direction), the damage stars have the shape ofprecise rectangular cross, with the cross hairs directed parallel to the X and Y axes(Fig. 2a). Our results differ from those reported in [10], where the cross-type starsrotated by 45° around the Z axis have been observed, with the crack planes (110) and(110). The difference might be explained by different choices of crystallographic frame

Morphology of laser-induced damage of lithium niobate ... 569

of reference. In our case, X and Y axes are oriented along the two-fold symmetry axes.At the same time, one may find in the literature that the X and Y axes in KDP are chosenas being perpendicular to the mirror planes, which are rotated just by 45° with respectto the two-fold axes. Nevertheless, the damage stars correspond to the tetragonal group42m in both cases.

YOSHIDA et al. [10] have observed eight-point stars from the crystallographicplane a for the laser beam directed along the X axis. The crack projections on the a planeobserved in this experiment subtend the angles 20°, 47° and 66°. This fact seeminglyagrees with neither the orientation of symmetry elements nor the experimentalgeometry. However, we have noticed an interesting fact. If the authors mentionedabove used the alternative crystallographic reference frame and the elementary cellparameters a = b = 10.543 Å and c = 6.959 Å [13], then it would be easy to calculatethat the angle between the cell diagonals is equal to 66.8°. This correlates well withthe angle between the crack projections reported by YOSHIDA et al. [10].

When the laser beam in our experiments is directed along the X axis (direction), we obtain the damage stars close to regular six-point ones (see Fig. 3).The orientation of the axes, as well as the reference hexagonal star, is shown in Fig. 3a.One can see that one crack plane is perpendicular to the Z axis and the two others aresymmetric with respect to it (as well as the plane (010) which contains the Z axis).However, we have noticed in this case that the damage has initially a cross-type form

100⟨ ⟩

Fig. 1. View of the damage tracks for Z-cut LN crystals (a) and the same damage track observed fromY-plane (b). The light propagates along the Z axis.

a

b

Fig. 2. View of the damage tracks for Z-cut KDP crystal (a) and the same damage track observed from(010) plane (b). The light propagates along the Z axis.

a b


a b c

de

Fig. 3. View of the damage tracks for X-cut KDP crystal (a–d) and the same track observed from (001)plane (e). The focus plane corresponds to: exactly the damage origin (a) and 0.15 mm (b), 0.30 mm (c)and 0.45 mm (d) behind the origin. The light propagates along the X axis.

Fig. 4. View of the damage tracks for Y-cut KDP crystal (a–d) and the same track observed from (001)plane (e). The focus plane corresponds to: exactly the damage origin (a) and 0.15 mm (b), 0.30 mm (c)and 0.45 mm (d) behind the origin. The light propagates along the Y axis.

a b c

d e


(Fig. 3a), with the cross hairs parallel to the Y and Z axes. The damage evolutiondeep into the sample leads to the splitting of Z-oriented crack into two cracks (seeFigs. 3a–3d). In this manner, the initially orthogonal-type damage transforms toa hexagonal-type one. Quite similar damage behaviour is observed for the case of laserbeam directed along the Y axis (see Fig. 4). Then, the initially Z-oriented crack splitsinto two and forms a hexagonal star, too.

The damage channels for the KDP crystals observed along perpendiculars to the laserbeam directions are shown in Figs. 2b, 3e and 4e. Unlike the LN crystals, the damagetraces in the KDP crystals have almost symmetric shape along the laser beam path.Moreover, the damage channels in the KDP possess noticeable periodically situatednodes, thus indicating a self-focusing character of the damage.

In order to explain the observed shapes of damage, we have built indicative surfacesof the “expansion factor” E–1 (with E being Young’s modulus) and its stereographicprojections for the crystals under study (see Figs. 5 and 6), using the coefficients ofelastic compliances taken from [14]. From the experiment, the damage star in the LNcrystals, obtained for the laser light directed along the Z axis, has the points parallel

18.2

55.7

Y

Z

315 45

135

180

225

270

a b

c

Fig. 5. Symmetry of the “expansion factor” E–1 forthe LN crystals: the indicative surface (a), itsstereographic projection (b) and X-cut (c).


to the X axis (Fig. 1a). However, the expansion factor in this direction does not acquireits extreme values. It would be suggested that the damage cracks are perpendicular tothe directions of extreme values of the expansion factor (i.e., Y axis in our case).

The analysis of the damage stars and the elastic symmetry for the KDP crystalsallows one to make it clear which extrema of the expansion factor might be associatedwith the directions of cracks. As seen from Fig. 6b, the expansion factor has its minimain the directions of X and Y axes and the maxima in the directions and .The experiment (see Fig. 2a) gives us the orientations of the crack planes (100) and(010), when the laser beam is directed along the Z axis. This means that the cracks areperpendicular to the directions of minimal values of the expansion factor.

The conclusion could be confirmed by the following fact. As seen from the X-cutof the indicative surface of expansion factor for the LN crystals (Fig. 5c), the directionof minimal value is inclined by 18.2° to the plane XY. Thus, the cracks obtained withthe laser beam directed along the Z axis must be oblique to it. It leads to wide starpaths observed experimentally on the Z-cut of LN crystals (Fig. 1a). Unlike in the caseof the LN, the directions of minimal values of the expansion factor in KDP are

110⟨ ⟩ 110⟨ ⟩

Z

Y

42.3

60

330

300

270

240

210

180

150

120

30

a b

c

Fig. 6. Symmetry of the “expansion factor” E –1 forthe KDP crystals: the indicative surface (a), itsstereographic projection (b) and X-cut (c).


perpendicular to the Z axis (see Fig. 6). Consequently, the cracks really observed onthe Z-cut of KDP crystals are sharp enough (see Fig. 2a).

Therefore, the results mentioned above enable us to conclude that the shape ofthe damage related to the laser beam directed along the optic axis in optically uniaxialcrystals is explained by the elastic symmetry of those crystals and corresponds tothe order of the symmetry axis. As some directions in crystals are more “compliant”and the other “stiffer”, the crack orientations are not random. As a result of symmetry,such directions repeat N times per revolution, where N is the order of the symmetryaxis (see Figs. 5b and 6b). Thus, the laser-induced damage cracks form symmetricstars defined eventually by the elastic symmetry of crystal.

When the laser beam propagates perpendicular to the optic axis in optically uniaxialcrystal, the figures of damage cracks are more complicated. First, the damage tracksdepend on the direction of laser beam ( or ). Second, the symmetry ofthe damage pattern does not correspond to the symmetry of crystal in the givendirection. Although the damage begins with appearance of two perpendicular cracks,its development leads to splitting one of the cracks (which is parallel to the Z axis) intotwo cracks and forming a hexagonal star.

3. Conclusions

Our experimental results indicate that the shape of laser-induced damage of the dielectricsingle crystals depends on the elastic symmetry of crystal and the propagation directionof the laser beam. When the latter propagates along the optic axis in optically uniaxialcrystals, the figures of the laser damage are star-like, with the number of points definedby the highest-order symmetry axis available in the crystal structure.

References[1] SWAIN J., STOKOWSKI S., MILAM D., RAINER F., Improving the bulk laser damage resistance of

potassium dihydrogen phosphate crystals by pulsed laser irradiation, Applied Physics Letters 40(4),1982, pp. 350–2.

[2] NAKATANI H., BOSENBERG W.R., CHENG L.K., TANG C.L., Laser-induced damage in beta-bariummetaborate, Applied Physics Letters 53(26), 1988, pp. 2587–9.

[3] FURUKAWA Y., MARKGRAF S.A., SATO M., YOSHIDA H., SASAKI T., FUJITA H., YAMANAKA T., NAKAI S.,Investigation of the bulk laser damage of lithium triborate, LiB3O5, single crystals, Applied PhysicsLetters 65(12), 1994, pp. 1480–2.

[4] VLOKH R., DYACHOK YA., KRUPYCH O., BURAK YA., MARTYNYUK-LOTOTSKA I., ANDRUSHCHAK A.,ADAMIV V., Study of laser-induced damage of borate crystals, Ukrainian Journal of PhysicalOptics 4(2), 2003, pp.101–4.

[5] KOLDUNOV M.F., MANENKOV A.A., POKOTILO I.L., Thermoelastic and ablation mechanisms of laserdamage to the surfaces of transparent solids, Quantum Electronics 28(3), 1998, pp. 269–73.

[6] KOLDUNOV M.F., MANENKOV A.A., POKOTILO I.L., Efficiency of various mechanisms of the laserdamage in transparent solids, Quantum Electronics 32(7), 2002, pp. 623–8.

[7] EMEL’YANOV V.I., ROGACHEVA A.V., Spatial self-organisation of a defect generation wave and laser--induced formation of ordered and crystallographic-oriented regions of optical damage in crystals,Quantum Electronics 34(6), 2004, pp. 531–6.

100⟨ ⟩ 110⟨ ⟩


[8] SALO V.I., ATROSHENKO L.V., GARNOV S.V., KHODEYEVA N.V., Structure, impurity composition, andlaser damage threshold of the subsurface layers in KDP and KD*P single crystals, Proceedings ofthe SPIE 2714, 1996, pp. 197–201.

[9] YOSHIMURA M., KAMIMURA T., MURASE K., INOUE T., MORI Y., SASAKI T., YOSHIDA H.,NAKATSUKA M., Bulk laser damage in CsLiB6O10 crystal, Proceedings of the SPIE 3244, 1998,pp. 106–10.

[10] YOSHIDA H., JITSUNO T., FUJITA H., NAKATSUKA M., YOSHIMURA M., SASAKI T., YOSHIDA K.,Investigation of bulk laser damage in KDP crystal as a function of laser irradiation direction,polarization, and wavelength, Applied Physics B: Lasers and Optics 70 (2), 2000, pp. 195–201.

[11] WOODS B., RUNKER M., YAN M., STAGGS M., ZAITSEVA N., KOZLOWSKI M., DE YOREO J.,Investigation of Damage in KDP Using Scattering Techniques, Report LLNL, UCRL-JC-125368(1996); https://e-reports-ext.llnl.gov/pdf/230901.pdf.

[12] ABRAHAMS S.C., MARSH P., Defect structure dependence on composition in lithium niobate, ActaCrystallographica Section B: Structural Science 42(1), 1986, pp. 61–8.

[13] BACON G.E., PEASE R.C., A neutron diffraction study of potassium dihydrogen phosphate by Fouriersynthesis, Proceedings of the Royal Society of London. Series A, Mathematical and PhysicalSciences 220(1142), 1953, pp. 397–421.

[14] SHASKOLSKAYA M.P., Acoustic Crystals, Nauka, Moscow 1982 (in Russian).

Received October 28, 2007in revised form January 15, 2008


Distribution of electronic states in amorphous Zn-P thin films on the basis of optical measurements

BOŻENA JARZĄBEK1*, JAN WESZKA1, JAN CISOWSKI1, 2

1Centre of Polymer Chemistry, Polish Academy of Sciences, P.O. Box 20, 41-819 Zabrze, Poland

2Institute of Physics, Cracow University of Technology, ul. Podchorążych 1, 30-084 Cracow, Poland


Transmission and fundamental reflectivity studies, completed on amorphous Zn-P thin films,allowed us to obtain parameters describing the fundamental absorption edge, i.e., the opticalpseudogap EG , Urbach energy EU and exponential edge parameter ET . All these data, togetherwith the results of earlier transport measurements, have been utilized in developing simple modelsof electronic structure (distribution of electronic states) for amorphous Zn-P thin films oftwo compositions, i.e., Zn57P43 (near stoichiometry of Zn3P2) and Zn32P68 (near stoichiometryof ZnP2).

Keywords: amorphous semiconductors, thin films, absorption coefficient, model of electronic structure.

1. Introduction

Amorphous films of the Zn-P system are of interest due to their potential applicationsin solar cells [1], similarly to the Zn3P2 crystalline counterpart [2]. StoichiometricZn3P2 thin films have been prepared by various techniques, including electron beamevaporation [3, 4], thermal vacuum evaporation [5–8], and reactive r.f. sputtering ofzinc in a PH3-containing argon atmosphere [9, 10]. All these films have beenprepared in both the crystalline and amorphous forms near the stoichiometric Zn3P2ratio. The optical properties of thermally evaporated amorphous films belonging tothe Zn-P system within a broader Zn to P ratio have been presented in [11].

Optical measurements are one of the simplest and most direct experimentalmethods used to investigate the electronic structure of semiconductors, also inamorphous phase [12]. In the case of crystals, the band structure models can beobtained by theoretical methods and verified by experimental results, while the opticalinvestigations allow one to deduce the energy, character and direction of the opticaltransitions, the width and type of the forbidden gap EG , as well as existence of

576 B. JARZĄBEK, J. WESZKA, J. CISOWSKI

the excitons or impurity levels. For amorphous semiconductors, the electronic structureappears to be a very complicated problem because the long-range order (LRO) is absentin their atomic structure and the forbidden gap is not clearly defined. In spite ofthat, the optical investigations carried out on amorphous semiconductors revealedthe presence of a characteristic energy value, called the pseudogap , which maybe connected with the short-range order (SRO) [13], i.e., atomic order on a length ofa few inter-atomic distances (1–2 nm).

Based on the experimental results and the theoretical works of ANDERSON [14] andMOTT [15], some phenomenological models of the electronic structure have beenproposed for amorphous semiconductors [16–19] and these schematic density of statesdiagrams are well known as Cohen–Fritshe–Ovsinsky [16], Mott–Davis [17] andMarshall–Owen [18] or “real” glass with defect states models [19]. All these modelsand parameters such as: the optical pseudogap , mobility gap , Fermi levelEF, Urbach energy EU , exponential edge parameter ET and E04 energy are shortlydescribed in [12].

The aim of this paper is to present the simple models of electronic statesdistribution in amorphous Zn-P (a-ZnP) thin films, based on our optical data andthe phenomenological models mentioned above. The fundamental absorption edgesalong with the optical parameters obtained by us for the a-ZnP thin films system havebeen discussed and compared with the optical parameters presented by other authors[3, 6–10]. Moreover, the earlier transport data of the a-ZnP films [9, 10], are alsoutilized to create these models.

2. Experimental

Amorphous Zn-P thin films were prepared by thermal vacuum evaporation of bulkpolycrystalline material of either Zn3P2 or ZnP2 compositions from one source ontoborosilicate glass substrates held at 300 K. During the deposition process, the vacuumwas maintained at a level of 10–3 Pa and the film deposition rate νs was 10 nm/s.The thickness of the films, determined with an interference microscope, was inthe range 0.5–3.5 μm (±0.05 μm). The compositions of evaporated films (determinedby photometric analysis, with accuracy ±2 at%) were Zn57 P43 and Zn32 P68 and haveappeared to be dependent on the source material.

The optical properties were studied at room temperature, using both transmissionand fundamental reflectivity measurements within the 0.4–2.0 eV photon energyinterval.


3.1. Optical absorptionThe recorded transmission and reflectivity spectra of as-prepared a-ZnP thin filmswere used to calculate the absorption coefficient α according to the formulae givenelsewhere [11]. The resulting fundamental absorption edges α vs. the photon energy

EGopt

EGopt EG

m

Distribution of electronic states in amorphous Zn-P thin films ... 577

E for all investigated a-films deposited from bulk Zn3P2 and ZnP2 are presented inFig. 1, while the method used to obtain optical gaps for these films is shown in Fig. 2.The spectral dependences for these two types of a-ZnP films are presented togetherbecause of the similarity of the values of their optical pseudogaps. Figures 1a and 2ashow the spectra recorded on thinner films (d = 0.5 μm), while Figs. 1b and 2billustrate the spectra taken on the 3.5 μm thick films.

The spectra presented reveal quite a high absorption level, being of the order of104 cm–1 (Fig. 1b) and even reaching 105 cm–1 for thinner films (see Fig. 1a). Sucha high level of absorption coefficient is typical of amorphous thin films, especiallyof very thin ones, where various defects such as voids and dangling bonds in theinterfacial film-substrate and film-surface areas play an important role in the absorption

Fig. 1. Absorption coefficient α vs. photon energy for a-ZnP thin films ( , , – Zn57P43 films;, – Zn32P68 films): d = 0.5 μm (a), d = 3.5 μm (b).

0.5 1.0 1.5 2.0

103

104

105

h [eV]ν

α [c

m]

–1

0.5 1.0 1.5 2.0102

103

104

h [eV]ν

α [c

m]

–1

a b

Fig. 2. Absorption edges, obtained from the Tauc dependence, for a-ZnP thin films ( , , – Zn57P43films; , – Zn32P68 films): d = 0.5 μm (a), d = 3.5 μm (b).

0.0 0.5 1.0 1.5 2.00

125

250

375

500

h [eV]ν

(h

) [e

V/cm

]α

ν1/

21/

2

0.0 0.5 1.0 1.5 2.00

50

100

150

h [eV]ν

(h

) [e

V/cm

]α

ν1/

21/

2

a b


process. The defect related localized states connected with interfacial areas haveconsiderably stronger impact on the bulk properties of thinner films than thicker ones.The values of the standard deviation of thickness of the a-ZnP thin films (σw), obtainedby the interference spectroscopy from the optical transmittance and reflectanceinterferences, as well as the average surface roughness (Ra) evaluated from the atomicforce microscopy (AFM) studies were comparable, i.e., ≅ 20 nm [20, 21] and appearedto be independent of the film thickness. This means that the localized states, due todefects on the surface play a considerably greater role for thinner films than for thickerones. Additionally, such defects as voids and dangling bonds on the surface causedisorder in the thetrahedric coordination and weak molecular bonds or bipolarons areformed [22]. The higher the density of defects, the larger the wave function overlap,which in turn leads to an increase in matrix element of optical transitions in the vicinityof the mobility gap. Therefore, the increasing level of the absorption coefficientobserved in the range of the absorption edge for thinner Zn-P films may be explainedby an increase in the density of localized defect states and also by higher matrixelements for these films. This seems to show that the f-sum rule may give differentresults when applied to films with different thickness.

The shapes of nearly all absorption edges presented in Fig. 1 are at the first sight,similar to those suggested by Tauc for an idealized amorphous semiconductor [23–25];two exponential parts with different slopes are seen for almost all α (E ) curves in thisfigure. The low energy exponential absorption edges follow the α ∝ exp(E /ET)dependence, with parameter ET obtained for nearly all absorption curves, as seen inFig. 1. The fast increasing linear part of absorption in the higher energy regionfollowing the Urbach relation: α ∝ exp(E /EU) [26] and the Urbach energy EU havebeen found for all a-ZnP films under investigation.

Figure 2 shows, typical of amorphous semiconductors, dependence (α E )1/2 ∝∝ (E – ) proposed by Tauc to obtain the optical gap. The linear approximationof (α E )1/2 vs. E (in the high absorption region, where E > ) allows us to findthe values of pseudogaps for all the films investigated. For thicker films,the parameter E04 (corresponding to the energy when the absorption coefficient α == 104 cm–1 and the level of absorption is almost constant) has also been found (seeFig. 1b). All the optical parameters of a-ZnP films investigated are gathered in the Table.

At low energy region, a nearly exponential behavior of absorption is attributed tothe optical transitions between the defect states, localized inside the optical gap. Thesestates are mainly due to the voids, which provide additional boundary surfaces fordangling bonds. These defects play a significant role in the structure of filmsprepared by the vapor deposition technique. The absorption curves, presented in Fig. 1,indicate that the defect states are distributed over a large part of the gap, similar as inIII–V compounds [22], and the dangling bonds at the void surface form bondingstates at P atoms and anti-bonding states at Zn atoms. It would be interesting to comparethe influence of interatomic bonding of a-ZnP films of both compositions on theirdefect states. These films reveal tetrahedral coordination but there are differencesbetween the atomic structures, as found in structural investigations [27, 28].

EGopt

EGopt

EGopt


The structure of amorphous P-rich films (near ZnP2 composition) resemblesthe tetragonal ZnP2 crystal, where the hybridization is realized through the promotionof p electron from two different P atoms to one Zn atom. Thus, each Zn atom iscoordinated to four P atoms, while each P atom is linked by two different P atoms andtwo Zn atoms. The Zn and P atoms are binding more through sp3-type bonds whilep-type P–P bonds are mostly covalent with the P spiral chains characteristic of bothamorphous and crystalline ZnP2 films [27]. A different situation is in the case ofa-Zn3P2 films with homopolar bonds (between Zn atoms and between P atoms) sincetheir atomic structure is clearly different from the Zn3P2 crystal structure and similarto the deformed CdAs mixed rhomb and regular Si III structures [28]. A higherstructural disorder of Zn3P2 amorphous films causes a decrease of the slope ofthe linear part of absorption edge in the law energy region (Fig. 1) and values of ETare higher than for the amorphous films of near ZnP2 composition.

Structural disorder influences also the defect states in the band tails and the Urbachedge slope. Thus, instead of an abrupt absorption edge, we can observe the edges withsmaller slopes, as seen in the case of a-ZnP films (Fig. 1). It is thought that lower valuesof the Urbach energies for a-ZnP2 films are the result of lesser structural disorder thanfor a-Zn3P2 and the existence of P chains like in the ZnP2 crystalline structure.

Comparing the values of ET and EU for films with different thickness, one can seethat these parameters are higher for thinner films independently of composition,indicating a greater number of localized defect states inside the gap and localized statesin the Urbach tails for very thin films.

The pseudogaps of the a-ZnP films investigated cover the range 0.9–1.1 eVfor a-Zn3P2 films and 1.3–1.45 eV for a-ZnP2 films. The absorption edges, followingthe Tauc power law, are due to the interband transitions between the extended statesbeyond the mobility gap as well as to transitions between localized states in the vicinityof the mobility edge in one band and the delocalized states at the mobility edge inthe other band [22–25]. Therefore, for amorphous semiconductors, the optical gap isalways smaller than mobility gap. In most cases, the value of mobility gap can beapproximately represented by the E04. From Fig. 1b, we have the values E04 ≅ 1.55 eVfor Zn57 P43 films and 1.75 eV for Zn32 P68 film. The limiting value of the mobility gapfor amorphous semiconductors seems to be the energy gaps for their crystalcounterparts [29], i.e., 1.5 and 2.0 eV for Zn3P2 and ZnP2, respectively. These two

EGopt

T a b l e. Optical parameters of a-ZnP films under investigation.

Material d [μm] ET [meV] (±5 meV)

EU [meV] (±2 meV)

[eV] (±0.01 eV)

E04 [eV] (±0.01 eV)

a-Zn32P68 0.56 660 171 1.45 —3.50 455 142 1.30 1.75

a-Zn57P43 0.56 — 572 0.90 —3.50 575 145 1.15 1.553.50 1265 313 1.05 1.55

EGopt


compounds are wide-gap semiconductors, so for their amorphous counterparts,the energy gap is smaller than for crystalline phase, according to Vorliček’s rule [29].

3.2. Models of electronic structure

The results of optical measurements presented above and gathered in the Table,together with earlier transport parameters [9, 10], have been used to create models ofdistribution of electronic states for the a-ZnP films of compositions near Zn3P2 andZnP2. We have utilized the phenomenological models, mainly the “real” glass andthe Mott–Davis models, as a basis of our proposition. The models proposed are shownin Figs. 3 and 4, for a-Zn3P2 and a-ZnP2, respectively; the localized states are markedby dashed lines and the optical transitions which suit the optical gap are shown asarrows.

3.2.1. Amorphous Zn-P films evaporated from Zn3P2

In developing the model of distribution of electronic states for the a-Zn3P2 films(Fig. 3) we have used the mobility gap = 1.6 eV and the activation energyEa = 0.69 eV obtained by WEBER et al. [9, 10]. The Fermi level is located ~0.1 eVbelow the middle of the mobility gap [9, 10], making these films p-type semiconductors.This value of the mobility gap obtained from transport measurements is comparableto E04 = 1.55 eV, determined by us from the absorption spectra. A distinct differencebetween the mobility gap = 1.6 eV and the optical gap ≅ 1.1 eV, and relativelygreat value of the Urbach energy, indicate that localized states in the band tails playimportant role in optical transitions which are responsible for the existence of opticalgap. The optical transitions within the optical gap in a-Zn3P2 can occur not onlybetween the extended states but also from the localized states below the Fermilevel and above the mobility edge of the valence band to the extended states nearthe mobility edge of the conduction band (see the arrow with ≅ 1.1 eV, in Fig. 3).

EGm

EGm EG

opt

EGopt

Fig. 3. Model of the electronic structure fora-Zn3P2 thin films.

EC

V

EF

ECm

0.1 eV

EVm

g(E)

E 1.6 eVG ≈m

E 1.55 eV04 ≈

E 1.1 eVG ≈opt


Greater values of the optical gap (1.25–1.6 eV) for a-Zn3P2 thin films reported inworks [6–9] may be due to the differences in technology, but ≅ 1.95 eV in [3]indicate rather a-ZnP2 films because this value is near to the crystalline ZnP2, i.e.,

≅ 2.0 eV.The relatively large values of energies EU and ET for the a-Zn3P2 thin films indicate

the existence of the localized states in the band tails and inside the gap andthe overlapping band tails, as seen in Fig. 3. Transitions between the localized statesinside the gap may occur between states lying under the Fermi level and abovethe mobility edge of the valence band. These localized states are due to defects createdduring the evaporation process, and result in a smaller slope of the low energyabsorption edges of a-Zn3P2 than that of a-ZnP2 films (see Fig. 1).

3.2.2. Amorphous Zn-P films evaporated from ZnP2

In the case of a-ZnP2, we have not found any information about the transportparameters, so the proposed model of distribution of the electronic states (shown inFig. 4) is based on our optical measurements. As a mobility gap, we have takenthe value of energy E04 ≅ 1.75 eV and situated the Fermi level a little below the middleof the mobility gap ( p-type semiconductor). A lower value of the Urbach energy fora-ZnP2 than for a-Zn3P2 indicates a lesser number of the localized states in the bandtails and a smaller overlapping of these tails, as compared to the a-Zn3P2 (see Fig. 3).Additionally, a lower energy ET for a-Zn3P2 than for a-Zn3P2 indicates a lesser numberof the localized states inside the gap. This is a consequence of a lesser structuraldisorder in a-ZnP2 than in a-Zn3P2 and the existence of a part of P chains in the formerfilms, as confirmed by structural investigations [27]. Optical transitions responsible forthe existence of the optical gap ≅ 1.45 eV in a-ZnP2, take place from the localizedstates in the valence band tail under the Fermi level up to the extended states nearthe mobility edge of the conduction band (as shown by the arrow in Fig. 4).

EGopt

EGopt

EGopt

Fig. 4. Model of the electronic structurefor a-ZnP2 thin films.

EC

V

EF

g(E)

E E 1.75 eV04 G≈ ≈m

EGm

E 1.45 eVG ≈opt

EVm


4. Conclusions

Analysis of the optical data obtained for amorphous Zn-P films has yielded the valuesof optical parameters, such as energy E04, pseudogap , Urbach energy EU andparameter ET , which together with accessible transport data providing the mobilityedge and the activation energy Eact, have allowed us to create the simple modelsof distribution of extended and localized states and the optical transitions for thesematerials. The greater values of both ET and EU energies for amorphous filmsevaporated from Zn3P2 than for a-ZnP2 films indicate a greater number of the localizedstates inside the gap and in the band tails and more distinct overlapping of these tailsin the case of a-Zn3P2 films. Also, a greater difference between the values of and confirms the role of the localized states of a-Zn3P2 films. Transitions betweenthe localized states in the valence band tail under the Fermi level and the extendedstates near the mobility edge of conduction band energy are responsible forthe pseudogap in the films studied ( p-type semiconductors). These considerationsconfirm once more the important role of the optical measurements in creating modelsof the electronic structure for amorphous semiconductors.

References[1] LOUSA E., BERTRAN E., VARELA M., MORENZA J.L., Deposition of Zn3P2 thin films by coevaporation,

Solar Energy Materials 12(1), 1985, pp. 51–6.[2] MISIEWICZ J., SZATKOWSKI J., MIROWSKA N., GUMIENNY Z., PŁACZEK-POPKO E., Zn3P2 – a new

material for optoelectronic devices, Materials Science and Engineering: B 9(1–3), 1991, pp. 259–62.[3] RAO V.J., SALVI M.V., SAMUEL V., SINHA A.P.B., Structural and optical properties of Zn3P2 thin

films, Journal of Materials Science 20(9), 1985, pp. 3277–82.[4] NAYAK A., RAO D.R., BANERJEE H.D., Optical studies on electron-beam-deposited Zn3P2 thin films,

Journal of Materials Science Letters 10(7), 1991, pp. 403–5.[5] BRYJA L., JEZIERSKI K., MISIEWICZ J., Optical properties of Zn3P2 thin films, Thin Solid Films 229(1),

1993, pp. 11–3.[6] DEISS J.L., ELIDRISSI B., ROBINO M., WEIL R., Amorphous thin films of Zn3P2: preparation and

characterization, Applied Physics Letters 49 (15), 1986, pp. 969–70.[7] DEISS J.L., ELIDRISSI B., ROBINO M., TAPIRO M., ZIELINGER J.P., WEIL R., Amorphous thin films of

Zn3P2, Physica Scripta 37(4), 1988, pp. 587–92.[8] ARSENAULT C.J., BRODIE D.E., Crystallization of Zn-rich and P-rich amorphous Zn3P2 thin films,

Canadian Journal of Physics 66(5), 1988, pp. 373–5.[9] WEBER A., Thesis, Zurich 1993.

[10] WEBER A., SUTTER P., VON KAENEL H., Optical, electrical, and photoelectrical properties of sputteredthin amorphous Zn3P2 films, Journal of Applied Physics 75(11), 1994, pp. 7448–55.

[11] JARZĄBEK B., WESZKA J., BURIAN A., POCZTOWSKI G., Optical properties of amorphous thin films ofthe Zn-P system, Thin Solid Films 279(1–2), 1996, pp. 204–8.

[12] JARZĄBEK B., WESZKA J., CISOWSKI J., Distribution of electronic states in amorphous Cd-As thin filmson the basis of optical measurements, Journal of Non-Crystalline Solids 333(2), 2004, pp. 206–11.

[13] WEAIRE D., THORPE M.F., Electronic properties of an amorphous solid. I. A simple tight-bindingtheory, Physical Review B: Solid State 4 (8), 1971, pp. 2508–20.

[14] ANDERSON P.W., Absence of diffusion in certain random lattices, Physical Review 109(5), 1958,pp. 1492–505.

EGopt

EGm

EGopt

EGm


[15] MOTT N.F., Conduction in non-crystalline systems. IV. Anderson localization in a disordered lattice,Philosophical Magazine 22(175), 1970, pp. 7–29.

[16] COHEN M.H., FRITSCHE H., OVSHINSKY S.R., Simple band model for amorphous semiconductingalloys, Physical Review Letters 22(20), 1969, pp. 1065–8.

[17] DAVIS E.A., MOTT N.F., Conduction in non-crystalline systems. V. Conductivity, optical absorptionand photoconductivity in amorphous semiconductors, Philosophical Magazine 22(179), 1970,pp. 903–22.

[18] MARSHALL J.M., OWEN A.E., Drift mobility studies in vitreous arsenic triselenide, PhilosophicalMagazine 24(192), 1971, pp. 1281–305.

[19] OWEN A.E., SPEAR W.E., Electronic properties and localised states in amorphous semiconductors,Physics and Chemistry of Glasses 17(5), 1976, pp. 174–92.

[20] JARZĄBEK B., JURUSIK J., CISOWSKI J., NOWAK M., Roughness of amorphous Zn-P thin films, OpticaApplicata 31(1), 2001, pp. 93–101.

[21] JARZĄBEK B., Thesis, Zabrze 1997.[22] CONNELL G.A., [In] Amorphous Semiconductors, [Ed.] M.H. Brodsky, Springer, Berlin 1979, p. 79.[23] TAUC J., MENTH A., States in the gap [of chalcogenide glasses], Journal of Non-Crystalline Solids

8–10, 1972, pp. 569–85.[24] MOTT N.F., DAVIS E.A., Electronic Processes in Non-Crystalline Materials, 2nd Ed., Clarendon,

Oxford 1978.[25] Cody D., [In] Semiconductors and Semimetals, [Ed.] J.I. Pankove, Vol. 21, Part B, Academic Press,

New York 1984, p. 11.[26] URBACH F., The long-wavelength edge of photographic sensitivity and of the electronic absorption

of solids, Physical Review 92(5), 1953, p. 1324.[27] LECANTE P., MOSSET A., GALY J., BURIAN A., Structural studies of amorphous Zn-P films, Journal

of Materials Science 27(12), 1992, pp. 3286–92.[28] BURIAN A., LECANTE P., MOSSET A., GALY J., Extended X-ray absorption fine-structure studies of

short-range order in amorphous Zn-P films, Philosophical Magazine B: Physics of CondensedMatter, Electronic, Optical and Magnetic Properties 66(6), 1992, pp. 727–36.

[29] VORLICEK V., Optical absorption edge of CdAs2-based glasses, Physica Status Solidi B 67(2), 1975,pp. 731–42.

Received December 21, 2007in revised form July 14, 2008


Absorptive CdTe films optical parameters and film thickness determination by the ellipsometric method

ANNA Z. EVMENOVA1*, VOLODYMYR A. ODARYCH2*, FEDIR F. SIZOV1, MYKOLA V. VUICHYK1

1V.E. Lashkariov Institute of Semiconductor Physics NAS of Ukraine, pr. Nauky, 45, Kyiv, Ukraine, 03028

2Taras Shevchenko Kyiv National University, Physical Department, pr. Gloushkova, 2, Kyiv, Ukraine, 03022

*Corresponding authors: A.Z. Evmenova – [email protected]; V.A. Odarych – [email protected]

Ellipsometric detective method of refractive index, absorptive index and thickness of the filmdeposited on the substrate with some optical parameters has been developed. This method isapplied for optical parameters and film thickness detecting in visible and near IR spectrum.Refraction index and film thickness dispersion has been studied. It has been determined that filmrefractive index (2.6 on average) is by 7% less than that of monocrystalline CdTe.

Keywords: ellipsometry, passivation CdTe films, methods of calculating film parameters.

1. Introduction

Semiconductor films are used in micro- and optoelectronics for producing sun energyprocessing devices, and signal transmitting devices. Cadmium telluride-based devicesare used in particular as electroluminescent radiators or for surface passivation ofthe solid-state sun radiation receiver in the near IR spectral region. The protectiveability of films appreciably depends both on the structure and thickness of the films.

Knowledge of the film optical constants and its thickness, film thicknessproportional distribution in area extent, film structure radiation and chemical durabilityare the actual issues of the film technology.

The existing examples of cadmium telluride application indicate the appreciabledependence of the electric and optical properties, and surface structure of CdTe filmson the method and technological conditions of its passivation [1–9]. It has been found[3–6] that the value of the refractive index of the film is smaller than that of

586 A.Z. EVMENOVA et al.

monocrystalline material. In particular, we have found [6] that refractive index of CdTefilms deposited on the monocrystalline silicon substrate is considerably smaller thanthe refractive index of monocrystalline CdTe. This might be connected withconsiderable difference between the crystal grate constants of silicon and cadmiumtelluride. In reference [6], the fact of the refractive index value decreasing is explainedby the film porous structure.

So, at the present stage, such technological conditions of film passivation aresearched for that would permit obtaining films with perfect structure, homogeneousover the entire working surface of the film. It is, therefore, necessary to have a suitable,accurate and high-precision no contact control method.

Methods based on measures of transmission and reflective spectra at soundingbeam normal incidence on the object being studied are widely used for determinationof optical parameters and film thickness [1–4, 9]. These methods are used ifthe substrate is clear, but its accuracy is not high (Δn = 0.01 [1]). Measuringphotoconductivity as an absorption coefficient function we can find thickness witherror up to 5% [10].

In the case where substrate is opaque, reflecting methods are used for determinationof optical parameters and thickness of CdTe films [4–8], in particular, reflectingellipsometry, which is based on examination of polarization characteristics of lightwave reflected from the sample being studied.

Ellipsometry in some cases allows optical parameters and film thickness to bedetermined. At the same time, application of computing mathematics methods ofmeasured results calculation to the ellipsometric function, which is nonlinear andtranscendental relative to the determinative parameters of the reflective system, is notalways successful because of differences between the iteration procedures used orbecause such determinations appear to be ambiguous and wrong. That is why searchingfor a suitable algorithm for determination of system parameters from experimentresults is relevant in ellipsometry.

The aim of the paper was to develop a method of determination of opticalparameters and thickness of semiconductor films, and using this method, to determinateparameters and their heterogeneity in area of CdTe film deposited on the surface ofthe composite monocrystalline CdHgTe substrate whose lattice constant is close tothat of the CdTe lattice.

2. Objects under study and details concerning the experiment

CdTe films were deposited on the surface of monocrystalline CdHgTe fusion bythe epitaxy vacuum “hot-wall” method. Before loading into the installation sampleswere chemically treated in HF acid and washed in acetone for oxide and impurityremoving. CdTe film formation was conducted in high vacuum of 10–7 mmHg.For obtaining thin film growth conditions that are to the greatest possible extentapproximated to the balanced ones, evaporation is conducted in semi-closed spaceformed by the source, wall and substrate that are kept at different temperatures.

Absorptive CdTe films optical parameters and film thickness determination ... 587

The values of temperature, required for setting conditions of the thin CdTe filmgrowth were the following: source Tsource = 380 °C, wall Twall = 400 °C, substrateTsub = 50–80 °C. Nominal thickness of CdTe layers was obtained by the continuousgrowth of a thin layer during the whole time of evaporation, which made upa continuous interval of 8 minutes. We have produced and studied five samples.

Some properties of the given type of the film obtained under the same conditionsand at the same plant are described in [11, 12]. In particular, X-ray investigations haveshown that CdTe films deposited on CdHgTe have high quality monocrystallinestructure with (111) orientation. The data obtained by means of atomic powermicroscope indicate the flat surface with roughness of 15–100 nm on the base lengthof 50 μm.

Ellipsometric measurements were conducted on a 632.8 nm wavelength of LEF-3Mcompensatory zero ellipsometer and on the wavelength of a 579, 546, 435, 405 and366 nm spectrum of mercury lamp radiation of non-standard photometric ellipsometer,which was calibrated with the help of monocrystalline silicon plate. The ellipsometeris constructed according to the non-compensatory analyzer–sample–polarizer scheme.Measurements were conducted at the incidence angle of ϕ = 65°.

The wavelengths were chosen so that on part of them cadmium telluride wouldhave relatively weak absorption (632, 579 and 546 nm) and on the other (435, 405 and366 nm) would have strong absorption.

Ellipsometric measurements were conducted sequentially on several areas of eachsample (up to seven areas) with different interference color, or with different thicknessof the film.

In the applied modification of photoelectric ellipsometric method [13] the measuredparameters were cosΔ and tgψ, where Δ is the phase difference between p- ands-components of electric vector of reflected light wave, and tgψ is the ratio of reflectivecoefficient in incident plane (p-plane) and in the perpendicular s-plane. Ellipsometricparameters could be found by measuring the intensities of the radiation reflectedfrom the sample at four azimuths of analyzer: 0°, 45°, 90°, and –45° in relation tothe plane of incidence and at the fixed polarizer azimuth. Before measurementsthe ellipsometer was adjusted according to work [13] and was calibrated with the helpof monocrystalline silicon plane in order to get the well known optical constants forsilicon.

3. Method of determining film parameters

In the papers presented, ellipsometric measurements were conducted on several areasof the same film, with the thickness being different but unknown. One can expectthat the substrate on these areas was the same and its optical parameters ns and κswe obtained from the ellipsometric measurements conducted earlier on the film-freesubstrate.

From ellipsometric measurements we obtain two parameters, i.e., ellipsometricangles ψ and Δ (tgψ and cosΔ on the photometric ellipsometer) or principal angle Φ


(incidece angle, Δ = 90°) and ellipticity (value of tgψ at the principal angle). Usingtwo values of ellipsometric parameters we can determine two unknown parametersof the film being studied, but in our case, we have to determine a bigger number ofindeterminate values (refractive index n, absorptive index κ, and also film thicknesson each area under study).

Different calculation procedures based on using ellipsometric function that is setby the basic equation of ellipsometry are used for the determination of reflective systemparameters from ellipsometric measurements. General review of numerous papers canbe found, for instance, in [14, 15].

A method of obtaining a complete solution to the inverse problem of ellipsometryfor one layer isotropic absorptive system relative to the layer and substrate parametersis described in [16]. The method is based on the ellipsometric measurements of Δ andψ at a constant incidence angle during the process of the film growth at four consistentmoments of time, whereby the value of the film thickness increasing (or decreasing)must be known. In our case, this method cannot be used, and that is why one ofthe aims of the paper was to develop a program for determining system parametersfrom ellipsometric data.

We have earlier created in [17] a package of programs of ellipsometric datacalculation that is based on the Newton iteration method of two equation systemdetermination. The base of the package comprises the ellipsometric function of a two--layer reflective system. Each program permits finding two unknown parameters ofthe system if the rest of the parameters are known using two values of ellipsometricparameters obtained at the fixed angle of incidence. The existence of such programsallows a method for finding more than two parameters to be created.

This method could be used for calculating ellipsometric data obtained in severalareas of the same film with different unknown thickness or for several films withdifferent unknown thickness, deposited on the same substrate with known value ofoptical constants (refractive index ns and absorptive index κ s).

The method includes calculation and graphic procedures; with the help of one ofthe package programs [17] optical parameters n and κ of the film are calculated forseveral thickness values (which are chosen arbitrarily within the expected value limits)using a measured ellipsometric parameter couple. Then obtained couples of opticalparameters are plotted on the diagram, where, for example, values of the refractiveindex n are plotted on the vertical scale and values of the absorptive index κ areplotted on the horizontal scale. As a result, a curve is plotted for each area that connectsthe points obtained and along which the film thickness changes. Let us call such curvesthickness curves. If optical parameters of the film are equal in all areas, then the curveswould intercross at one point, and the couple of optical parameters corresponding tothis point would be general for all the curves. The cross point position on each curvewould give us the film thickness on the corresponding areas.

This method could be illustrated by the model calculations. First of all, ellipsometricparameters for a certain one-layer system with optical constants that are close to opticalparameters of CdTe film and CdHgTe hybrid crystals, and film thickness in the range


of 0–110 nm were calculated. Results of calculations are presented in Fig. 1 as twocurves of constant values of optical parameters.

It is seen that the curves look like spiral and they are twisted with the film thicknessincreasing, with the curve twisting faster when the film absorptive index is bigger.

Fig. 1. Theoretical curves ellipticity tgψ –principal angle Φ calculated for different values of opticalconstants n and κ and film thickness. Film thickness in nm is shown by numbers near the curve marks.

64 66 68 70 72 74 76

–0.05

0.00

0.05

0.10

0.15

0.20

0.25

0.30

(CdHgTe)0

10080

6040

20

100

8070

60

50

40

20

3

21

32

1

λ579

1( ) n = 3.22 = 0.75κ2( ) n = 2.7 = 0.4κtg

ψ

Φ [deg]

Fig. 2. Curves of probable solutions of film optical parameters obtained for different film thicknesses forthree areas of the first (a) and second (b) curves. Effect of the error of measurement of ellipsometricparameters ΔΦ = 0.2, Δtgψ = 0.01 on the thickness curves of graphic method (c, d).

0.25 0.30 0.35 0.40 0.45 0.50

2.55

2.60

2.65

2.70

2.75

2.80

2.85

2.90

80

100

90

11090

70

60

5070

0.30 0.35 0.40 0.45 0.502.50

2.55

2.60

2.65

2.70

2.75

2.80

2.85

110

106

102

98

94

90

90

86

82

78

74

70

70

66

62

58

54

50

n

0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85

3.10

3.15

3.20

3.25

3.30

3.35

110

102

90

9080

70

70

50

60

κ0.55 0.60 0.65 0.70 0.75 0.80 0.85

3.10

3.15

3.20

3.25

3.30

3.35

90 80

70100

90

110

70

60

50

n

Areas 1 2 3

λ579

κ

a c

b d


Three areas with fixed thickness of 60 nm (area 1), 80 nm (area 2) and 100 nm (area3) were chosen on each curve. Then the inverse problem of ellipsometry was solved(film parameters were calculated with the help of the method described earlier) usingthe couple of values Φ and tgψ that characterized each point.

Calculated results of the optical parameters of the two spirals (Fig. 1) are presentedin Figs. 2a and 2b. In both cases, it was noted that solution curves really intercross atone point, which gives optical parameters of the film and its thickness which coincidewith true values of these parameters. Obviously, curves of probable solutions couldbe noncrossing, they could touch each other at one point, as observed for the secondspiral in Fig. 2b.

In conditions of real experiment, values of ellipsometric parameters obtained areaffected by an experiment error. The experimental error, being typical of each areaunder study, would lead to the systematic error of detected optical parameters ofthe film. So, the thickness curves obtained in data calculation would be dislocated andwould not intercross at one point in the diagram of detected values, they wouldconverge in a bigger or smaller local region according to the permissible error. Thisfact is presented in Figs.2c and 2d.

The values of ellipsometric parameters in areas 1, 2 and 3 were perturbed atΔΦ = ±0.2° and Δtgψ = ±0.01, which are close to the experimental error and evensomewhat bigger. Then thickness curves, similar to those in Figs. 2a and 2b forthe system presented, were calculated with the help of present method using perturbedvalues of principal angle and ellipticity as analogue of experiment date.

As a result of the error effect, generic intersection points of all thickness curvesdisappear; curves intercross in pairs, but certain region of the curves converging exists.Under such circumstances solution is found by averaging film parameters obtainedfrom all intersection points in the region of the maximal convergence of the curves.

After averaging the value of the film refractive index would be estimated at 0.04and thickness value at 3–5 nm for the system with considerably small absorption(curve 1 in Fig. 2). In the case of the system with considerably large absorption(curve 2 in Fig. 2) the film refractive index would be estimated at 0.02 and thicknessvalue at 1–3 nm relative to the true values of these parameters.

So, the error of ellipsometric parameters measured by photometric method can givean error of about 0.05 in the value of refractive index and the error of 5 nm at the mostin the value of film thickness.

Model calculations have shown that in some cases several possible solutions mayexist for a certain pair of ellipsometric parameters with regard to film optical constants.This conforms to the situation in which in absorptive film model several curvesrepresenting constant values of optical parameters could pass through each point inthe diagram of the measured values. When we assign film thickness the programchooses that one from possible curves which corresponds to this thickness. However,values of optical constants would be different for each curve. In the model experimentpresented, up to three possible solutions were obtained for each area.


So, while searching for solutions with the help of the program one has to determineall possible solutions by modifying initial values of optical parameters within a certainlimit. False solutions could be distinguished from the true ones by the fact that theirthickness curves have no common intersection point. As a result, experimentalmeasurements must be conducted on as many as possible different thickness areas ofthe same film.

4. Results of the examination of CdTe/CdHgTe samples

4.1. Optical parameters of CdHgTe substratePrior to the deposition of CdTe film on one of the samples ellipsometric measurementswere conducted on the substrate of monocrystalline Cd0.2Hg0.8Te and optical constantsof the substrate were determined by measured ellipsometric parameters. The obtainedvalues of optical parameters were further used in the program for finding parametersof the films that were deposited on this substrate.

The substrate used for the CdTe film deposition consisted of hybrid three-compoundcrystal CdHgTe film with relatively big thickness that was deposited on the CdZnTeby epitaxial method. During ellipsometric measurements, it was radiation reflectedfrom the sample surface that was studied, whereas reusable reflection was neglectedbecause of the strong matter absorption. The measurement was conducted at a fixedangle of 65°. Optical constants of matter (refractive index n and absorptive index κ )were found from the ellipsometric measurements. The oxide layer with the thicknessof several nanometers was neglected in the calculation.

Measurements of ellipsometric parameters were conducted by studying the beamreflected from different areas of the sample in order to control the similarity ofthe substrate properties on the sample surface. The obtained values of optical parametersare presented in Fig. 3.

The high-strength doublet maximum is observed in the spectrum of absorptiveindex, and a wide region of anomaly dispersion that corresponds to this absorptivemaximum is observed in the spectrum of refractive index.

Fig. 3. Refractive index n and absorptive index κ of mixed CdHgTe crystals.

300 350 400 450 500 550 600 6501.82.02.22.4

2.62.83.0

3.23.4

3.63.8

n

λ [nm]300 350 400 450 500 550 600 650

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

2.0Cd Hg Te0.2 0.8

k Areas

1 2 3

λ [nm]


According to the customary interpretation of the semiconductor properties whichare crystallized in the zinc blend structure [18], the obtained spectrum of the opticalconstants conforms to intrinsic electron transition from the valence band that is splitby the spin-orbit interaction to the lower conduction band. Such transitions occur inthe wide region of the quasi-momentum towards the Brilluen zone Λ. The refractiveindex spectrum splitting conforms to the spin-orbit splitting of the valence band inthe corresponding direction.

Values of the optical constants that we obtained coincide with the accuracy of5–10% with values presented in [19] for the matter of the given composition.Deviation could be explained by the imperfection of sample surface and it indicatesthat the optical constants obtained have physical meaning of some effective parameters,which characterizes not only the substrate matter but geometry of the reflective surface.

From the analysis of the data obtained one can draw conclusion that the differencein the values of optical constants in different areas of the sample surface does notexceed 1% that is close to the experimental error. The obtained values of the substrateoptical parameters were used for calculating the parameters of CdTe films depositedon this substrate.

4.2. CdTe film parameters

This series contains 4 samples obtained by depositing CdTe film on the substrate fromthe hybrid monocrystals of CdHgTe under the same conditions of the hot-wall epitaxymethod; time of the evaporation time was the same and equalled 8 minutes. Up to 7areas with different interference color and different thickness of deposited film werechosen on the sample surface.

Measurements were conducted at the wavelength of 632.8 nm and incidence angleof 60°. In ellipsometric data calculation film parameters were detected for each sampleseparately by the method of thickness curves that was described earlier. Thicknesscurves of one of the samples that illustrate film parameters detection are presented inFig. 4 as an example.

It was found that for each couple of ellipsometric parameters the program gavetwo possible solutions: two couples of film optical constants for the area under study.So, two couples of thickness curves, one of which contained true solution and anothercontained false solutions were obtained for each area.

All solutions, both true and false are presented in Fig. 4a. The region wherethe greatest number of curves intercross is marked with rectangle. Thickness curvesthat correspond to the false solutions are mostly almost parallel to each other. Curvesin the outlined region where film parameters were obtained are presented in Fig. 4b.Optical constants and film thickness were detected as the average values of thoseobtained from the diagram as intersection points of all thickness curves of the sample.The average dispersion of parameters was detected, it possible (provided thatthe number of intersection points was sufficient). Results of calculation are presentedin Tab. 1.

Absorptive CdTe film

s optical parameters and film

thickness determination ...

593

T a b l e 1. Refractive index n and absorptive index κ of the film, and its thickness in different areas of the sample, thickness scatter in terms of samplesurface (as a degree of film homogeneity).

Sample n κFilm thickness in areas [nm]

Δd [nm]1 2 3 4 5 6 7 8100-2 2.5±0.04 0.23±0.03 52±4 57±4.5 68±4 73 67±5 — — — 21104-4 2.14±0.02 0.47±0.005 78±1 88.5 92 76±2 97.5 — — — 21.581-3 2.52±0.02 0.385±0.01 50.5 63 71.5 68 66 — 58 57 2181-5 2.66±0.02 0.31±0.01 44±1 52±2 54.5 59 61±1 57 60 — 17

Fig. 4. Finding refractive index n and absorptive index κ by the methodof thickness curves. Arrows on the curves (a) show direction of the filmthickness increasing over the interval of 40–85 nm, numbers near eachcurve show the area number and solution number over dash. Curves inthe region of the greatest number of their intercrosses are presented in(b), numbers near the marks show the film thickness in nm.

a b


The obtained values of optical parameters range (0.01–0.05) and thickness (1–4 nm)are close to those burdened with the measuring error of the ellipsometric parameters.

It was found that values of the film optical parameters for different samples differfrom each other by the value that exceeds the experimental error. Particularly, filmrefractive indexes of three samples (100-2, 81-3, 81-5) are almost similar and lie withinthe limit of 2.6, whereas the film refractive index of sample 104-4 is appreciablysmaller and equals 2.14.

Experimental data distribution for different areas of the samples under study andtheoretical curves that describe this distribution are presented in Fig. 5. Curves areplotted according to the values of film parameters obtained, which are presented inTab. 1. It is obvious that theoretical curves pass close to the experimental points.

Ellipsometric measurements were conducted at several wavelengths of the mercurylamp radiation on one of the samples (time of evaporation was 8.5 minutes) in orderto determine optical parameters dispersion. These measurements were conducted withthe use of the photometric ellipsometer by the method described in Section 3.

Measurements were conducted on several areas with different thickness at fixedincident angle of 65° and in a wide region of incident angles. Principal angle wasdetected from the multiangular measurements. Optical constants and film thicknesswere determined by the method described earlier, using ellipsometric parameters ata fixed angle and applying the values of principal angle and ellipticity.

Fig. 5. Comparison of experimental values of principal angle Φ and ellipticity tgψ with theoreticalcalculations. Film thickness in nm is shown by numbers near the theoretical curves marks.

CdTe:CdHgTe

0.0 0.1 0.2 0.3 0.4 0.5–1.0–0.8–0.6–0.4–0.20.00.20.40.60.81.0 Sample 100-2

65

62

60

50

9070 80 20

40

5

4

3

2

1

0

Experimental Theoretical

n = 2.5 κ = 0.23

cosΔ

0.0 0.1 0.2 0.3 0.4 0.5

–1.0

–0.8

–0.6

–0.4

–0.2

0.0

0.2

0.4

90

80 70

100

60

50

403

4

25

1

0

Sample 104-4

0.10 0.20 0.30 0.40 0.50

–1.0

–0.9

–0.8

–0.7

–0.6

–0.5

–0.4

070

80 9065

60

5550

40

20

456

8

2

7

3

1

Sample 81-3

tgψ0.10 0.20 0.30 0.40 0.50

–1.0

–0.9

–0.8

–0.7

–0.6

–0.5

90807060

55

50

40

30

20

54

7 6

3

2

1

0

Sample 81-5


n = 2.14κ = 0.47


n = 2.522κ2 = 0.385

cosΔ

tgψ


n = 2.66κ = 0.31


n = 2.52κ = 0.385


An example of data calculation for the principal angle, where sensibility ofthe ellipsometric parameters to the film characteristics is somewhat higher than forthe angle ϕ = 65°, is presented in Fig. 6 (the data for an angle of 65° are similar tothose of the principal angle, so a diagram that describes it is not presented in the paper).These data are effective for estimating of the accuracy of the method.

From the data of Fig. 6 it is obvious that thickness curves in the shortwave regionbecome twisted and diverge like in Fig. 2d. So, the film thickness for the wavelengthsof 405 nm and 366 nm was not exactly determined. Such a character of the curves isexplained by the fact of the film absorption being rather high in the short-wave regionand considerably big film thickness. As a result, experimental points obtained at these

Fig. 6. Determination of film parameters by principal angle and ellipticity.

0.30 0.35 0.40 0.45 0.50 0.55 0.602.30

2.35

2.40

2.45

2.50

2.55

2.60

2.65

2.70

2.75

50

7080

605040

λ632

n

Areas1234

0.30 0.35 0.40 0.45 0.50 0.55

2.4

2.5

2.6

2.7

2.8

62

80

50

60

50

42

λ579

CdTe:CdHgTe; Sample 66-1

0.6 0.7 0.8 0.9 1.0 1.1 1.22.6

2.7

2.8

2.9

3.0

3.1

3.2

3.3

3.4

55

50

25

80

66

50

λ435

κ1.0 1.1 1.2 1.3 1.4 1.5 1.6

2.75

2.80

2.85

2.90

2.95

3.00

3.05

3.10

50

50

75

6025

λ405

n

κ

T a b l e 2. Comparison of 66-1 sample optical parameters obtained by different methods.

λ [nm]Measurements at angle of 65° Measurements at principal anglen κ n κ

632.8 2.68±0.015 0.308±0.005 2.56±0.02 0.45±0.01579 2.62±0.01 0.422±0.002 2.66±0.01 0.41±0.003546 2.51±0.01 0.423±0.001 — —435 3.03±0.02 0.994±0.01 3.02±0.025 0.93±0.015405 2.83±0.03 1.34±0.015 2.90 1.34366 2.35 1.4 — —


wavelengths were found to lie close to the pole of corresponding spiral theoreticalcurve (curve 1 in Fig. 1), and sensitivity of ellipsometric parameters particularly tothe thickness in different areas appeared to be small.

The obtained values of film parameters are presented in Tables 2 and 3; besides,the values of optical parameters obtained at a fixed and principal angles are comparedin Tab. 2. The coincidence of optical parameters is satisfactory.

Comparison of experimental data with theoretical curves in the principal angle–ellipticity diagram, which are calculated using the data from Tab. 2, is presented in

T a b l e 3. Film thickness in different areas.

AreaFilm thickness [nm]

Measurements at angle of 65° Measurements at principal angle1 68±2 622 51±4 44±33 64±3.5 6624 44±5 39.5±5Thickness dispersion [nm] 24 26

Fig. 7. Comparison of experiment values of principal angle Φ and ellipticity tgψ with theoreticalcalculations. Film thickness in nm is shown by numbers near the theoretical curves marks. Numbers nearsquares indicate the numbers of areas on the sample surface.

62 64 66 68 70 72 74 76

0.00

0.05

0.10

0.15

0.20

0.25

0.30

CdHgTe

80

60

50

40

20λ632

70

3 1

2

4

tgψ

62 64 66 68 70 72 74 760.00

0.05

0.10

0.15

0.20

0.25

0.30

CdHgTe

1

324

70 8060

50

40

20λ579

68 70 72 74

0.10

0.15

0.20

0.25

0.30

0.35 CdHgTe

60

50

4030

201

3

24

λ435

Φ [deg]

CdTe:CdHgTe; Sample 66-1

70 71 72 73 74

0.20

0.25

0.30

0.35

0.40

70 60

40

3020

3

24

CdHgTeλ405


n = 2.56κ = 0.45


n = 2.66κ = 0.41

tgψ


n = 3.02κ = 0.93

Φ [deg]


n = 2.90κ = 1.34


Fig. 7. Clearly, the experimental points fit well appropriate places on the theoreticalcurves.

Values of the film thickness in different areas and also thickness dispersion onthe sample surface calculated as a difference between maximal (the center) andminimal (the periphery) thicknesses are presented in Tab. 3. Thickness scattering ofthe sample surface depends on technological conditions of film evaporation in a givendevice. Particularly, in our case, we can see that the heat field in conditions of filmdepositing by the hot-wall epitaxy was not sufficiently isotropic in the region ofsubstrate allocation, hence the film growth with the thickness being heterogeneous.

5. Discussion of the results

As can be easily seen from the data of Fig. 2, the error relative to film parametersinduced by the error of ellipsometric parameters measurement by means of the methodpresented amounts to Δn = ±0.05, Δκ = ±0.02, Δd = ±2 nm. Values of optical constantsdispersion obtained experimentally are located within the above-mentioned limits (asis seen from data presented in Tabs. 1 and 2).

So, the present method of absorptive film parameters detection was found to besuitable for CdTe films on the surface of mixed crystals CdHgTe. In all the cases, theseparameters were found by thickness curves obtained for different areas of the samplesurface. This means that the film obtained by the hot-wall epitaxy under conditions ofevaporation described above is homogenous enough; in certain limits opticalparameters are similar within the surface extent.

Moreover, the film thickness is different in different areas of the same sampleand varies within the limits of 20 nm increasing from the periphery to the center ofthe sample. Besides, this error of the thickness determination in the same sample(Tab. 3) was somewhat greater than the value mentioned above, which characterizesthe error of the method presented. This means that the film is heterogeneous as regardsthe thickness.

In the method applied, film thickness is determined by the temperature field andthe time of evaporation. So, it is expected that certain temperature distribution inthe working volume exists in a given setup for film deposition; the temperature (mostprobably of the substrate) increasing from the periphery to the center.

The refractive index of the CdTe films is appreciably smaller than that of the mono-crystalline cadmium telluride (~2.8 in the visible spectrum [20, 21]) when absorptiveindexes for both systems are equal.

As refractive index value is sensitive to the technological conditions of filmdepositing, its deflection from the refractive index of the monocrystalline matter couldbe the measure of perfection of the film obtained under given conditions.

Thickness dispersion of the sample surface (which changes in the limit of 20 nm,as seen from Tabs. 2 and 3) could be a certain characteristic of the film depositionprocess. It can be seen that the thickness dispersion for sample 66-1 is 17 nm which


is the smallest value obtained for all the samples from this series. The refractive indexvalue of this sample is the nearest to the refractive index of the monocrystalline CdTe.This fact could be explained by the quality of the film obtained for this sample.

In [3, 4], it was also observed that in the case of glass, the CdTe films haverefractive index which is smaller than that of the monocrystalline CdTe; this might becaused by the polycrystalline structure of the film. In [6], we have found that refractiveindex of CdTe film on the monocrystalline Si in the visible spectrum makes up 2.1,which is appreciably smaller than average value 2.5 of CdTe films on the CdHgTesubstrate obtained under similar conditions. Such refractive index decreasing isexplained by the varied structure of the film (which among single grids of basicmatter contains porous ones), which is the result of big difference in lattice constants(5.4282 Å for Si and 6.477 Å for CdTe).

In the present work, measurements were conducted for high-quality films withmonocrystalline structure, which is testified by the X-ray diffraction [11, 12]. Biggervalues of refractive index which, however, are smaller than refractive index ofmonocrystalline CdTe were obtained, though lattice constants of CdTe and solidsolution of Cd0.2 Hg 0.8Te (6.464) are practically equal. This means that refractive indexvalues obtained from ellipsometric data are affected by different factors.

Roughness of the film surface is one of the factors that can appreciably changevalues of film parameters, and particularly refractive index that is obtained fromellipsometric data. As is shown in [12], films of such type, obtained under the sameconditions as ours, have roughness equal to 50–100 nm. Neglecting roughness inellipsometric data calculation could lead to a decrease in the values of refractiveindex obtained. In this case, film parameters have physical meaning of someeffective parameters which characterizes not only the film matter but the geometry ofthe reflective surface. So, ellipsometric data calculation has to be conducted inthe model with roughness boundaries of the layers. This task is rather hard because ofthe incompleteness of the theory of electromagnetic wave formation by the matter withroughness surface.

6. Conclusions

A method of the determination of optical constants and absorptive film thickness bythickness curves, each containing a great number of possible solutions for filmarea (or films) with its thickness, was found to be suitable for determination ofthe parameters of monocrystalline CdTe films, obtained on monocrystalline CdHgTesubstrate. Experiments and model calculations have proved that in conditions ofmeasuring error existing the thickness curves have no generic intersection point butthey intercross in pairs. In such circumstances, refractive index n, absorptive index κ,and film thickness d are found by averaging all intersection points of thickness curves.In our case, the other film parameter errors were obtained: Δn = ±0.02–0.04,Δκ = ±0.01, Δd = ±1–4 nm.


This method allows elimination of false solutions that could appear as a result ofthe main ellipsometric equation properties – thickness curves of false solutions do notintercross and/or are situated far from the basic.

The films under study are homogeneous with respect to refractive index andheterogeneous with respect to thickness which decreases from the center to the periphery.Refractive index of CdTe film deposited on CdHgTe substrate is somewhat bigger thanrefractive index of films obtained on Si substrate in the same conditions, but is 7%smaller than refractive index of monocrystalline cadmium telluride.

Decreasing of refractive index of the film on the substrate could be explained bythe imperfect film surface, roughness presence in particular.

References[1] LAAZIZ Y., BENNOUNA A., CHAHBOUN N., OUTZOURHIM A., AMEZIANE E.L., Optical characterization

of low optical thickness thin films from transmittance and back reflectance measurements, ThinSolid Films 372(1–2), 2000, pp. 149–55.

[2] MUTHUKUMARASAMY N., BALASUNDARAPRABHU R., JAYAKUMAR S., KANNAN M.D.,RAMANATHASWAMY P., Compositional dependence of optical properties of hot wall depositedCdSexTe1–x thin films, Physica Status Solidi A 201(10), 2004, pp. 2312–8.

[3] RUSU M., RUSU G.G., On the optical properties of evaporated CdTe thin films, Physics of LowDimensional Structures No. 3/4, 2002, pp. 105–15.

[4] THUTUPALLI G.K.M., TOMLIN S.G., The optical properties of thin films of cadmium and zinc selenidesand tellurides, Journal of Physics D: Applied Physics 9(11), 1976, pp. 1639–46.

[5] PAULSON P.D., XAVIER MATHEW, Spectroscopic ellipsometry investigation of optical and interfaceproperties of CdTe films deposited on metal foils, Solar Energy Materials and Solar Cells 82(1–2),2004, pp. 279–90.

[6] KORNIENKO K.N., ODARYCH V.A., POPERENKO L.V., VUICHIK M.V., Determination of opticalparameters of CdTe films by principal angle ellypsometry, Functional Materials 13(1), 2006,pp. 179–82.

[7] MEHTA B.R., KUMAR S., SINGH K., CHOPRA K.L., Application of spectroscopic ellipsometry to studythe effect of surface treatments on cadmium telluride films, Thin Solid Films 164, 1988, pp. 265–8.

[8] PEIRIS F.C., WEBER Z.J., CHEN Y., BRILL G., Optical properties of CdSexTe1–x epitaxial films studiedby spectroscopic ellipsometry, Journal of Electronic Materials 33(6), 2004, pp. 724–7.

[9] BHATTACHARYA D., CHAUDHURI S., PAL A.K., Determination of optical constants and band gaps ofbilayered semiconductor films, Vacuum 46(3), 1995, pp. 309–13.

[10] POHORYLES B., MORAWSKI A., Photoconductivity – a novel method of evaluation of thinsemiconducting film thickness, Thin Solid Films 301(1–2), 1997, pp. 122–5.

[11] BILEVYCH YE.O., BOKA A.I., DARCHUK L.O., GUMENJUK-SICHEVSKA J.V., SIZOV F.F., BOELLING O.,SULKIO-CLEFF B., Properties of CdTe thin films prepared by hot wall epitaxy, Semiconductor Physics,Quantum Electronics and Optoelectronics 7(2), 2004, pp. 129–32.

[12] BILEVYCH YE., SOSHNIKOV A., DARCHUK L., APATSKAYA M., TSYBRII Z., VUYCHIK M., BOKA A.,SIZOV F., BOELLING O., SULKIO-CLEFF B., Influence of substrate materials on the properties of CdTethin films grown by hot-wall epitaxy, Journal of Crystal Growth 275(1–2), 2005, pp. e1177–e1181.

[13] ODARYCH V.A., , Zavod. Labor. 43(9), 1977, p. 1093 (in Russian).[14] AZZAM R.M.A., BASHARA N.M., Ellipsometry and Polarized Light, North-Holland Publ.,

Amsterdam, New York, Oxford, 1977, p. 583.[15] GORSHKOV M.M., Ellipsometry, Sov. Radio, Moscow, 1974, p. 200 (in Russian).


[16] DAGMAN E.E., Complete solution of the inverse problem in ellipsometry for a single-layer systemwith variations in film thickness, Optics and Spectroscopy 66(1), 1989, pp. 101–4; (original: Optikai Spektroskopiya 66(1), 1989, pp. 174–9).

[17] ODARYCH V.A., PANASYUK V.I., STASCHUK V.S., Zhurnal Prikladnoi Spektroskopii 56(5/6), 1992,p. 827 (in Russian); BYATEC M.A., KUSCH V.T., ODARYCH V.A., PANASYUK V.I., Visn. Kiev Univ.(Fiz.-Mat.) No. 7, 1992, p. 7 (in Ukrainian).

[18] CHADI D.J., WALTER J.P., COHEN M.L., PETROFF Y., BALKANSKI M., Reflectivities and electronic bandstructures of CdTe and HgTe, Physical Review B: Solid State 5(8), 1972, pp. 3058–64.

[19] JOHS B., HERZINGER C.M., DINAN J.H., CORNFELD A., BENSON J.D., Development of a parametricoptical constant model for Hg1–xCdxTe for control of composition by spectroscopic ellipsometryduring MBE growth, Thin Solid Films 313/314(1–2), 1998, pp. 137–42.

[20] MARPLE D.T.F., EHRENREICH H., Dielectric constant behavior near band edges in CdTe and Ge,Physical Review Letters 8(3), 1962, pp. 87–9.

[21] ADACHI S., KINURA T., SUZUKI N., Optical properties of CdTe: experiment and modeling, Journal ofApplied Physics 74(5), 1993, pp. 3435–41.

Received June 13, 2007


Fiber Bragg grating and long period grating sensor for simultaneous measurement and discrimination of strain and temperature effects

KAMINENI SRIMANNARAYANA1*, MADHUVARASU SAI SHANKAR1*, RAVINUTHALA L.N. SAI PRASAD2*, T.K. KRISHNA MOHAN2, S. RAMAKRISHNA2, G. SRIKANTH2, SRIRAMOJU RAVI PRASAD RAO3*

1Photonics Labs, Department of Physics, National Institute of Technology, Warangal-506 004, India

2Department of Physics, National Institute of Technology, Warangal-506 004, India

3Department of Physics, Kamala Institute of Technology and Science, Huzurabad, Karimnagar-505 486, India

*Corresponding authors: K. Srimannarayana – [email protected], M. Sai Shankar – [email protected], R.L.N. Sai Prasad – [email protected], S. Ravi Prasad Rao – [email protected]

Fiber Bragg grating sensors are novel in the determination of various physical parameters. Inthe case of fiber Bragg grating, on a single measurement of wavelength shift, it is impossible todifferentiate between the effects of changes in strain and temperature. The simulation of fiberBragg grating, long period grating characteristics and use of the different strain-temperatureresponse of combined sensor for the simultaneous measurement and discrimination of strain andtemperature in C band are demonstrated.

Keywords: fiber Bragg grating, long period grating, discrimination of strain and temperature.

1. IntroductionFiber Bragg grating (FBG) consists of periodically spaced modulated refractive indexzones in an optical fiber. The Bragg wavelength of a grating λB, is a function ofthe effective index of guided mode neff and period of index modulation Λ given byλB = 2neffΛ [1]. The reflected wavelength is also a function of the parameters liketemperature, strain, pressure, etc., which upon proper calibration, enables accuratemeasurement of these parameters. However, the technical drawback of this FBG sensoris its inability to distinguish between the shifts in wavelength caused by the strain ortemperature.

The second class of Bragg gratings are long period gratings (LPGs), which havethe period in the range of hundreds of micrometers. The transmitted peak wavelength

602 K. SRIMANNARAYANA et al.

is given by where n01 is the effective index of the core mode and is the effective index of the i-th axially symmetric cladding mode [2]. LPG has

the larger temperature and smaller strain response compared to those of FBG. Inprinciple, a sensor with an LPG and single FBG can be realized for the measurementof strain and temperature, but firstly, the LPG has larger bandwidth leading touncertainty in the determination of its central wavelength. The second drawback liesin the complex setup for determining the transmission spectrum of LPG, compared tothe interrogation of FBGs narrow reflection spectrum. Hence, in this study, a complexsensor consisting of an LPG followed by two FBGs for simultaneous measurementand discrimination of strain and temperature effects has been employed.

2. Simulation of FBG and LPG and their characteristics

Theoretically, the normalized reflection produced by an FBG is given by [3]

(1)

where L is the grating length, κ is the coupling coefficient, δ is the detuning parameterand δ /κ is the detuning ratio. The detuning parameter for Bragg grating of periodΛ is δ = Ω – (π /Λ) where Ω = 2πneff/λ. For sinusoidal variation in the indexperturbation, the coupling coefficient for the 1-st order for unblazed Bragg grating isκ = πη /λB [4], where η is the overlap integral between the forward and reversepropagating guided modes calculated over the fiber core of Bragg grating. In this case,η = ΔnF, F is the fractional modal power in the core given by F = [1 – (1/V 2)] whereV is the normalized frequency and Δn is the amplitude of induced refractive indexperturbation. The normalized power transmitted by the fundamental guided modethrough the LPG is given by [5]

(2)

where m is the mode number and all the specified parameters in the above case nowdepend on m.

Using Equation (1), the reflection spectrum of FBG is simulated and drawn usingMATLAB. The variations in these spectra were demonstrated by varying the parameters

λ i n01 ncladi( )–( )Λ,=

ncladi( )

R

sinh2 κL 1 δκ

-------⎝ ⎠⎜ ⎟⎛ ⎞2

–

cosh2 κL 1 δκ

-------⎝ ⎠⎜ ⎟⎛ ⎞2

– δκ

-------⎝ ⎠⎜ ⎟⎛ ⎞2

–

------------------------------------------------------------------------------------=

T

cos2 κ m( )L 1 δ m( )

κ m( )---------------⎝ ⎠⎜ ⎟⎛ ⎞

2

+δ m( )

κ m( )---------------⎝ ⎠⎜ ⎟⎛ ⎞

2

+

1 δ m( )

κ m( )---------------⎝ ⎠⎜ ⎟⎛ ⎞

2

+

---------------------------------------------------------------------------------------------------------=

Fiber Bragg grating and long period grating sensor ... 603

like grating length and index difference for an FBG with Δ = 0.0036, ncore = 1.45, coreradius of 4.5 μm and grating period of 0.534 nm. It is observed that the spectralbandwidth of the gratings decreased with an increase in the grating length. For a 10 mmlong uniform grating, the bandwidth is approximately 0.16 nm and for 20 mm longgrating, the bandwidth is reduced to 0.084 nm, as shown in Fig. 1. Further, ifthe refractive index change is varied, keeping the length of the grating constant, it isobserved that, by changing the index of refraction to half the value of the first grating,the reflectivity decreased to approximately 60% and the bandwidth decreasedconsiderably, as shown in Fig. 2. It is observed that the bandwidth approachesa minimum value and remains constant for further decrease in the index of refractionchange Δn. The transmission spectrum of the LPG is simulated using Eq. (2) and isshown in Fig. 3.

Fig. 1. Spectral profiles for uniform fiber Bragg gratings with different grating lengths.

Fig. 2. Spectral profiles for uniform fiber Bragg gratings with different refractive index change.


In the case of the FBGs, the wavelength shift induced by the applied strain ε ata constant temperature is given by [1]

ΔλB = λB(1 – Pe)ε (3)

where Pe is the photoelastic coefficient of the fiber. For a temperature change ΔT,the corresponding wavelength shift is given by [1]

(4)

where α is the thermal expansion coefficient and ξ is the thermooptic coefficient.The response of LPG depends on the change in the grating period and on

the differential change in core and cladding indices of the refraction and it also dependson fiber type [2].

When a sensor having one LPG and two FBGs is strained or is under temperaturefluctuations, a change in reflected power peaks of FBGs R1 and R2 occurs. A changein strain leads to a small decrease in R1 and small increase in R2, because the shiftin λLP (peak transmission wavelength of LPG) lags the shift in λB1 and λB2. However,a change in temperature produces a large increase in R1 and large decrease in R2,because the shift in λLP leads the shifts in λB1 and λB2. Figures 4 and 5 show typicalchanges in the reflected powers of FBG peaks due to strain and temperaturevariations. To analyze the reflectance signals, the function F(R1, R2) is calculated usingthe relation

(5)

Since the LPG transmission vs. wavelength is approximately linear over the regionthat the FBG and LPG overlap, the difference of and is linearly proportionalto the amount by which λLP leads or lags λB1 and λB2. The square root is required

ΔλB λB1Λ

-------- ∂Λ∂T

------------ 1n

------- ∂n∂T

-----------+ ΔT λB α ξ+( )ΔT= =

F R1 R2,( )R1 R2–

R1 R2+------------------------------------=

R1 R2

Fig. 3. Transmission spectrum of the LPG.


because the light passes two times through the LPG. The denominator is normalizationconstant. This minimizes the fluctuations caused in the total optical power reachingthe interrogator. For ease of calculation, a system of two equations

ΔF = AΔε + BΔT

Δλ = CΔε + DΔT

relating the change in F and the change in one of the FBG wavelengths due to strainand temperature changes (A, C and B, D are the strain and temperature coefficients,respectively) Δε and ΔT can be solved by

3. Experimental details

A hybrid FBG/LPG sensor formed on SMF-28 supplied by O/E Land, Canada, withthe specifications LPG: L = 15 mm, λLP ~ 1554 nm; FBG1: L = 12 mm, λB1 ~ 1548 nm;and FBG2: L = 12 mm, λB2 ~ 1560 nm, has been used in this experimental study.The light from the broadband source coupled into the SM fiber with the above sensors,passes through the LPG and gets reflected back from the two FBGs (see Fig. 6).

ΔFΔλ

A BC D

ΔεΔT

=

Fig. 4. Change in the reflected powers of FBGpeaks with strain.

Fig. 5. Change in the reflected powers of FBGpeaks with temperature.


The wavelengths of the two FBGs were chosen at nearly 50% of the transmission peakof the LPG on either side, so that the variation of relative intensities of the FBGswill be linear to the change of the LPG. Upon reflection, the light retraces the LPGand the reflected spectrum is observed on the Interrogator Sensing system(OEFSS-100 supplied by O/E Land, Canada). The two normalized reflected peaksR1 and R2 were found by averaging the heights with stored trace of the source spectrum,so that any false variations in R1 and R2 are checked due to the change in sourcespectrum vs. wavelength.

In the experimental setup, a microcontroller operated temperature chamber whichcan be maintained at the desired temperature for the specific time requirement isemployed and one end of the fiber is mounted on a motorized actuator (Newport) withcomputer interface, for applying the strain. The sensor is calibrated by applying knowntemperatures and strains, and the shift in one of the FBG wavelength and change inF(R1, R2) are determined. In order to minimize the effect of thermal expansion ofthe fiber, the latter was heated locally, while the strain was applied over the longer

Fig. 6. Experimental setup for simultaneous application of strain and temperature.

Fig. 7. Variation of F (R1, R2) and FBG2 wavelength with strain.


section of it. The experimental results were plotted in Figs. 7 and 8 for one of the Braggwavelengths λB2 depicting the variation of the function F(R1, R2) and the shift ofthe Bragg peak λB2 with the strain and temperature, respectively. From the averagevalues of the slopes and also the values of F (R1, R2) and λB2 at zero strain and 0 °C,a set of two linearly independent equations were developed which can be convenientlywritten in the form:

In the experiment, the applied strain was increased from 100–1600 microstrainwhile the applied temperature was varied between 20–60 °C. To show the quality of

F R1 R2,( ) 0.35108–

λB2 1559.97–4.8– 10 5–× 6.52 10 3–×

0.0946 10 3–× 8 10 3–×

ΔεΔT

=

Fig. 8. Variation of F (R1, R2) and FBG2 wavelength with temperature.

Fig. 9. The rms deviation of the applied and measured strain (a), the rms deviation of the applied andmeasured temperature (b).

a b


the sensor over this wide range of temperature and strain, measured strain versusapplied strain was plotted along with simultaneously measured temperature versusapplied temperature in Figs. 9a and 9b, respectively. The deviation of the measuredstrain from the applied one was found to be ±11 microstrain. Similarly, the measuredand the applied temperatures were compared and the deviation was found to be ±2 °C.

4. Conclusions

A better approach for the accurate measurement of strain and temperaturesimultaneously with high accuracy has been simulated and demonstrated. The hybridsensor has the advantage of using the properties of both the FBG and LPG sensors,which in turn, improves its calibration. However, the sensor performance depends onthe precision of the grating parameters and the accuracy of the applied strain andtemperature for better calibration and the use of sophisticated interrogator system.

Acknowledgement – This work was funded by the Department of Information Technology (DIT) ofthe Ministry of Communication and Information Technology, the Government of India, New Delhi,India.

References[1] HILL K.O., MELTZ G., Fiber Bragg grating technology fundamentals and overview, Journal of

Lightwave Technology 15(8), 1997, pp. 1263–76.[2] VENGSARKAR A.M, LEMAIRE P.J., JUDKINS J.B., BHATIA V., SIPE J.E., ERDOGAN T., Long-period fiber

gratings as band-rejection filters, Journal of Lightwave Technology 14(1), 1996, pp. 58–65.[3] SYMS R., COZENS J., Optical Guided Waves and Devices, McGraw-Hill, England 1992.[4] KASHYAP R., Photosensitive optical fibers: devices and applications, Optical Fiber Technology 1(1),

1994, pp. 17–34.[5] HALL D.G., Theory of waveguides and devices, [In] Integrated Optical Circuits and Components,

[Ed.] L.D.Huctheson, Marcel-Dekker, New York 1987.

Received June 22, 2007


Characterization of the refractive index in gradient-index elements

MAREK WYCHOWANIEC, DARIUSZ LITWIN

Institute of Applied Optics, Kamionkowska 18, 03-805 Warsaw, Poland

The paper is focused on measurement techniques of gradient-index elements particularlyuseful for slightly inhomogeneous glasses from small laboratory melts. A construction ofthe measurement cuvette and dependence of the refractive index of an immersion liquid on itsparameters is discussed. The dependence of the refractive index of α-bromonaphtalene ontemperature for λ = 0.6328 μm is described. A measuring method of the refractive index profileof gradient-index elements is presented.

Keywords: GRIN lenses measurements, refractive index measurement, GRIN glass.

1. Introduction

In gradient-index (GRIN) technology the key issue is to measure a refractive indexprofile across an element in question. There are no references (to the best of the authors’knowledge) dealing with measurement techniques suitable for GRIN elements fromsmall melts of glass, with some amount of inhomogeneity.

Non-destructive measurement techniques applied to cylindrical elements of highdegree of symmetry have been more extensively treated in the scientific literature. Thispaper presents a method of measurement of refractive index for elements for whichone cannot use a non-destructive method. The following sections of the paper discusstheory, a construction of the cuvette and measurement results of refractive index ofα-bromonaphtalene, a popular immersion liquid, against temperature. The paper isconcluded with comments and remarks on measurement of gradient-index elements.

2. The dependence of a refractive index of immersion liquid on parameters of the cuvette

Consider an arrangement of two coherent beams of light of wavelength λ with planewave fronts which intersects at an angle θ. The beams interfere with each other andstraight interference fringes are produced. The fringes make the angle equal to θ /2with direction of each of the beams. The spacing between fringes is Λ (Fig. 1).

610 M. WYCHOWANIEC, D. LITWIN

In Figure 1, if x is the length of a fragment of a fringe, marked with the thick dashedline, and λ is a wavelength, then:

(1)

If Λ is the distance between fringes then:

(2)

In terms of the above expressions, the formula for the spacing Λ becomes:

(3)

The distance between fringes depends on wavelength λ and the angle between bothbeams θ. In the above relation, λ is a wavelength in air.

Consider a cuvette which is built of two plane-parallel walls, and a prism ofthe known angle, between the walls (Fig. 2). If a beam of light falls perpendicularlyto the front side of the cuvette, it crosses the immersion liquid of the refractive indexnb without refraction, provided the walls of the cuvette are made plane-parallelwith high accuracy. The prism with the refractive index nc is in optical contact with“rear” side of the cuvette. In Figure 2, for simplicity, the real thickness of the walls isneglected.

The beam of light refracts at the boundary between the immersion liquid andthe prism and again when it leaves the prism (and the cuvette). If C is an angle of

λx

-------- θ2

-------cos=

x 2⁄Λ

-------------- θ2

-------tan=

Λ λ

2 θ2

-------sin------------------------=

Fig. 1. Two plane-parallel beams of light of wavelength λ which intersect at the angle θ and the systemof interference fringes. Thin solid line marks maximum amplitude of light and thin dashed line – minimum.Thick solid and dashed line marks places of interference fringes; Λ – the distance between fringes.

Characterization of the refractive index in gradient-index elements 611

incidence of the beam of light at the surface of the prism, C' is the refraction angle inthe prism, as marked in Fig. 2, then:

C = C' + C'' (4)

The angle C is equal to the angle of the prism.For both the immersion liquid–prism and the prism–air boundaries we can apply

Snell’s law:

nb sinC = ncsinC' (5)

ncsinC'' = np sinC''' (6)

where angles C, C', C'', C''' are marked in Fig. 2, nb is the refractive index ofthe immersion liquid, nc – refractive index of the prism, and np – refractive index ofair. From Eq. (6) we have:

ncsin (C – C' ) = npsinC''' (7)

(8)

Substituting (5) in this equation:

ncsinCcosC' – nb sin CcosC = np sinC''' (9)

From Equation (5):

(10)

(11)

nc C C'cossin Ccos Csin '– np C'''sin=

nb2 sin2C nc

2 sin2C'=

nb2 sin2C nc

2 1 cos2C'–=

Fig. 2. A plane-wave propagated through the cuvette.


So, we have:

(12)

where sign “–” before the square root is not used (because C' > 0 and C' < 90°).Substituting the last equation in (9) we have:

(13)

When the cuvette with the prism is put in one arm of the Mach–Zehnderinterferometer (which is arranged so that the interference field is homogeneous) [1, 2],as in Fig. 3, then an auxiliary lens images the “rear” side of the cuvette with backgroundinterference fringes (the one which is in optical contact with the prism, as in Fig. 2)on the CCD sensor. These fringes are produced through interference of the wave passingthrough the prism and the wave propagated in the second arm of the interferometer.

The angle between the beam which falls on the prism in the cuvette and the beamwhich leaves the prism is C''' (Fig. 2). In this case, interference fringes observed inthe mirror Zw (Fig. 3) are inclined to the normal line to the wall of the cuvette atan angle C'''/2 (Fig. 4).

nc C'cos nc2 nb

2 sin2C–=

C nc2 nb

2 sin2C–sin nb C Ccossin– np C'''sin=

Fig. 3. The cuvette in the Mach–Zehnder interferometer.

Fig. 4. Virtual interference fringes observed afterrefraction of the beam at the prism in the cuvette.


The distance between fringes Λ can be expressed as in formula (3):

(14)

If k is the number of fringes in segment BD which is perpendicular to the fringes,d is the length of the scale AB, then, as in Fig. 4:

(15)

Joining both of these expressions we have:

(16)

(17)

(18)

We can substitute expression (18) into formula (13):

(19)

Assuming nc > nb, which means that the refractive index of the prism is greaterthan the refractive index of the immersion oil (with directions of beams as in Fig. 2)and the refractive index of air np = 1, and the indices of glass and immersion oil aremeasured in air we have:

(20)

If the right-hand side of this expression we denote by A, we will get the expressionfor the index of the immersion oil nb in the form:

(21)

(22)

Formula (22) can be converted into a classic quadratic equation, the solution ofwhich is as follows:

Λ λ

2 C'''2

------------sin------------------------------=

kΛd

----------- C'''2

-------------cos=

kλ

2d C'''2

-------------sin--------------------------------- C'''

2-------------cos=

kλd

----------- 2 C'''2

------------- C'''2

-------------cossin=

kλd

----------- C'''sin=

C nc2 nb

2 sin2C–sin nb C Ccossin– npkλd

-----------=

nc2 nb

2 sin2C– nb Ccos– kλd Csin

--------------------=

A kλd Csin

--------------------=

nc2 nb

2 sin2C– nb Ccos– A=


(23)

A negative nb has not any physical context in this case. For nc > nb (the refractive indexof the prism is higher than the index of the immersion oil):

(24)

And, in a similar way, we get for nc < nb:

(25)

Equations (24) and (25) express dependence of the refractive index of the immersionoil on the parameters of the prism, i.e., the refractive index of the prism nc , angle ofthe prism C, length of the scale d, the number of fringes on the scale k, the wavelengthλ. It is valid on the strict assumption that the beam of light falls exactly perpendicularlyupon the front side of the cuvette.

3. Construction of the cuvetteFor measurements of a gradient-index distribution in glass and indices of immersionliquids a glass cuvette has been designed (Fig. 5).

It has been built of two plane-parallel plates of glass (BK7) of dimensions50×55 mm and the thickness of 5 mm. These plates are polished with high accuracy,with the flatness of the surfaces being better than λ /10, and they are in optical contact

nb nc2 kλ

d-----------⎝ ⎠⎜ ⎟⎛ ⎞2

– kλd

-----------ctgC–±=

nb nc2 kλ

d-----------⎝ ⎠⎜ ⎟⎛ ⎞2

– kλd

-----------ctgC–=

nb nc2 kλ

d-----------⎝ ⎠⎜ ⎟⎛ ⎞2

– kλd

-----------ctgC+=

Fig. 5. The cuvette for interference measurements of gradient-index materials.

Fig. 6. The homogenous dark interference field in the Mach–Zehnder interferometer with the emptycuvette.


with the side walls of the cuvette, i.e., two cuboids with a 12×15 mm base andthe height of 50 mm. The flatness of the side walls is the same as that of the “front”plane-parallel plates and their thickness is such that it is possible to obtain ina homogenous interference field, in the Mach–Zehnder interferometer, a homogenouscolor of light passing through two such plates – walls of the cuvette (Fig. 6).

The above setup allows us to measure polished slices cut from gradient-indexelements with the diameter smaller than 25 mm or elements of this size andthe thickness smaller than 12 mm (it is the interior width of the cuvette).

In the lower part of the cuvette there are two prisms mounted by means of opticalcontact with plane-parallel “front” plate of dimensions 20×10×(3–10) mm and withexactly measured angles (about 18°) – Fig. 7. The prisms are made of glass BK7(nd = 1.51498) and PSK3 (nd = 1.55168). A beam of laser light which fallsperpendicularly on the lower part of the cuvette passes through an immersion liquidand refracts at the prisms (Fig. 7) in the cuvette. Scales for measurements are placedat half height of the prisms.

When the cuvette is inserted in one of the arms of the Mach–Zehnder interferometerperpendicularly to the beam of light, like in Fig. 8, one can see, by means of the lensS, the scale of the cuvette and an image of the interference fringes orientedperpendicularly to the line of the scale (Fig. 9).

The number of interference fringes between extreme lines of the scale depends onthe refractive index of the immersion liquid and the wavelength. The scale and

Fig. 7. A view of the cuvette from the top fornb < nc (refractive index of immersion is smallerthan index of prism).

Fig. 8. The Mach–Zehnder interferometer and cuvette for observation of gradient-index elements.


the fringes are recorded utilizing a computerized image acquisition system. The numberof fringes is counted by a dedicated software. The refractive index of the immersionliquid can be calculated according to formulae (24) or (25).

In the cuvette there are two prisms of different indices of refraction. For countingthe fringes we apply the prism that produces a smaller number of fringes. In addition,it is possible to estimate a fraction of the distance between fringes near the edges ofthe scale.

4. Measurement of the dependence of the refractive index of α -bromonaphtalene against temperature

The cuvette described above has been applied to measure the dependence ofthe refractive index of α-bromonaphtalene against temperature. The index can differ

T a b l e. The measured number of fringes on the scale of the cuvette and the corresponding refractiveindex of α-bromonaphtalene against temperature (λ = 0.6328 μm).

Lower prism Upper prismTemperature [°C]

Number of fringes

Refractive index of immersion oil

Temperature [°C]

Number of fringes

Refractive index of immersion oil

16.77 1455.4 1.65277 16.66 1068.7 1.6527517.04 1453.1 1.65225 16.98 1067.6 1.6526417.66 1451.0 1.65236 17.58 1064.7 1.6523617.79 1450.0 1.65226 17.65 1063.5 1.6522517.95 1449.9 1.65225 17.86 1062.5 1.6521518.3 1447.8 1.65205 17.86 1063.7 1.6522718.37 1446.7 1.65194 18.07 1062.5 1.6521519.21 1443.6 1.65165 18.45 1061.2 1.6520319.49 1441.4 1.65144 19.07 1058.8 1.6517926.02 1410.9 1.64853 19.22 1057 1.65162

19.54 1055.2 1.6514526.1 1027.8 1.64882

Fig. 9. Interference fringes at the end of the scale ofthe measurement prism. The immersion liquid is notα-bromonaphtalene here.


depending on the date of production. The difference is usually of the order of 3×10–4

(according to the Central Office of Measurements in Warsaw, Poland).The temperature has been measured with the accuracy of 0.1 °C. More accurate

values are estimated in the Table (with ±0.02 °C). For counting the number offringes on a scale there were photographed 20 pictures of the scale of the cuvette.The temperature of the immersion liquid was recorded before and after countingthe fringes. It has been shown that, in general, the difference is below 0.1 °C. The meanvalue has been used in further calculations. The results are presented in the Table.

Using a standard procedure of the Microsoft Excel one can give an experimentaldependence of the refractive index of α-bromonaphtalene against the temperature t(in °C): ni = –0.000435 t + 1.65999.

The estimate error of a factor of variable t is 10–5, but the estimate error ofthe constant is less than 2×10–4. The calculation has been done for points given inthe Table. For instance, for 20 °C calculated value of ni is equal to 1.6513 and for25 °C it is equal to 1.6491. The estimate error of the measurement is less than 0.0002.

5. Measurement of the distribution of refractive index for gradient-index elements

In order to measure the distribution of the refractive index in a gradient-index elementwe cut out (from a cylinder or a plate) a flat section plate perpendicularly to the surfaceof the GRIN element [3–5]. This measurement can be performed for any section ofthe glass plate or cylinder. Especially, it is necessary for elements made of glass oflow homogeneity. For these glasses – cylinders, a non-destructive method cannot beused. In such a method, the cylinder is submerged in an immersion oil and the beamof light falls perpendicularly onto the surface of the cylinder. The beam deflectsaccording to the index profile which is the subject of investigation.

The thickness of the polished glass plate was about 0.3 mm. The plate was cutalong the diameter in such a way as to make a wedge of an angle close to 60°.The glass plate prepared in this way was inserted in the cuvette, filled with an immersionliquid and put in one arm of the Mach–Zehnder interferometer (Fig. 5). A specialholder was designed enabling rotation and moving the sample inside the cuvette.The plate measured in the homogenous interference field and an interferogram ofthe plate was recorded by means of the computerized image acquisition system.

In Figure 10, the arrow points to the place where the refractive index ofthe surrounding immersion oil is the same as that of the GRIN plate. In this case,the fringe in the plate connects with the surroundings.

When in the interference pattern of the section plate there is no such fringe forwhich the refractive index is equal to the index of the immersion liquid, the value ofthe refractive index in any place in the plate can be calculated by adding (or subtracting)the value λ /d multiplied by the number of fringes between the background and theplace in question to the refractive index of the immersion liquid. It is relatively easy


to check by visual inspection whether the element (the section plate) is positive ornegative.

For instance, if we know that the element is negative and after putting in immersionliquid we see in the center of the plate (with wedge of the edge) a smaller numberof fringes on the wedge than near the edge of the plate, then the range of indices ofthe plate is higher than index of the immersion liquid (Fig. 11). If we see, for the sameelement, in the center of the plate a larger number of fringes on the wedge than nearthe edge of the plate, then the range of indices of the element is lower than index ofthe immersion liquid.

It is worth mentioning that slices of cylinders can also be measured when theirsurfaces and wedges of edges are matt – not polished at all. In immersion liquid onecan observe interference fringes through such matt slice on its surface and on the wedgeof the edge.

6. Conclusions

The method measuring the parameters of gradient-index elements is specially usefulfor glasses with some (but not too high) inhomogeneity, obtained from small laboratorymelts. Such melts of the weight of the order of 0.5–1 kg have been made forthe Institute of Applied Optics (INOS) in the Institute for Electronic Materials(ITME) in Warsaw. Gradients of refractive index have been produced in ion diffusionprocesses to glass. All the work associated with the production of plates or cylinders

Fig. 10. A cross-section plate cut from a gradient-index cylinder and immersed in α-bromonaphtalene, inthe Mach–Zehnder interferometer in the dark homogenous interferometric field. Arrows show the placesand the ring in the plate for which the refractive index is the same as that for the immersion oil.

Fig. 11. Refractive index for the second from the edge of the plate black “vertical” interferometricfringe (fringe A) differs from the refractive index of surroundings (immersion liquid) by 5(λ /d ), but forthe first of “vertical” fringes from the edge (fringe B) – by 6(λ /d ). The plate is plane-parallel withan edge of a wedge (the wedge is along lower edge of the plate). The element disperses the light and itsrange of indices of refraction is higher than the index of immersion liquid.


of glass, together with carrying on the diffusion processes and measurements of platescut from glass elements were done in the Institute of Applied Optics in Warsaw.

References[1] MALACARA D., Optical Shop Testing, Wiley, New York 1991.[2] HARIHARAN P., Optical Interferometry, Academic Press, New York 2003.[3] KINDRED D.S., Gradient-index silver alumina phosphate glasses by exchange of Na+ for Ag+, Applied

Optics 29(28), 1990, pp. 4051–5.[4] WYCHOWANIEC M., Gradient-index elements produced in Instytut Optyki Stosowanej and methods of

measuring their index of refraction, Biuletyn „Optyka”, InOS, No. 3–4/1998, p. 67 (in Polish).[5] WYCHOWANIEC M., Phosphate glass for gradient-index lenses, Optical Engineering 36(6), 1997,

pp. 1622–4.

Received May 21, 2007in revised form June 20, 2007

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