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1260 J. Opt. Soc. Am. B/Vol. 27, No. 6 /June 2010 Y. Zhao and L. Zhu

Coaxial hybrid plasmonic nanowire waveguides

Yunsong Zhao* and Lin Zhu

Department of Electrical and Computer Engineering, Center for Optical Materials Science and EngineeringTechnologies, Clemson University, Clemson, South Carolina 29634, USA

*Corresponding author: yunsonz@g.clemson.edu

Received January 12, 2010; revised April 20, 2010; accepted April 21, 2010;posted April 21, 2010 (Doc. ID 122653); published May 19, 2010

We investigate two different types of coaxial hybrid plasmonic nanowire waveguides. The first type consists ofa metal cladding, a sandwiched low index dielectric layer, and a high index dielectric core. The second type isthe reverse version of the first type. We analyze their modal properties by the use of the mode hybridizationconcept and calculate the dispersion relations, normalized mode areas, and confinement factors. For the firstkind of hybrid waveguide, we can obtain less loss with similar confinement and slightly less overlap betweenoptical modes and the core region. For the second kind of hybrid waveguide, we can obtain strong confinementand enhanced optical fields in a low refractive index region. © 2010 Optical Society of America

OCIS codes: 240.6680, 230.7370.

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. INTRODUCTIONemiconductor nanowires can offer unique electrical andptical properties for nanophotonic applications, such asaveguides, sensors, photodetectors, and lasers [1–5].he size and dimensions of nanowires can be feasibly con-rolled by the use of modern fabrication techniques.hrough their high aspect ratios, nanowires can effec-ively bridge nanoscale and microscale devices. However,ue to the limited index contrast, optical modes of a di-lectric nanowire are not well confined when the nano-ire radius is much smaller than the operation wave-

ength. Weak confinement results in a poor overlapetween the optical mode and nanowire and leads to higheflection losses at the end facet. This makes it difficult tobtain a deep subwavelength scale nanowire laser withltrasmall mode volumes and wire radius. In order to ad-ress these challenges, there has been a great amount ofnterest in using plasmonic nanowire structures, in which

etallic materials are used to support surface plasmonodes and provide strong confinement [6–9]. The main

roblem associated with using plasmonic nanowireaveguides is the loss that mainly comes from metal ab-

orption in the visible and near infrared wavelengthanges.

Recently, a novel hybrid plasmonic waveguide that con-ists of a dielectric nanowire waveguide and a planaretal substrate has been proposed for nanolaser applica-

ions [10–13]. Due to the coupling between the lossless op-ical mode of the dielectric waveguide and the surfacelasmon mode at the planar metal-dielectric interface,he hybrid plasmonic waveguide exhibits less loss withhe same degree of confinement in comparison with otherlasmonic waveguides. A deep subwavelength plasmonicaser has also been experimentally demonstrated by these of the hybrid nanowire-metal waveguide [12]. In thisaper, we investigate two different coaxial hybrid plas-onic nanowire waveguides and compare them with their

orresponding conventional cylindrical plasmonic wave-uides. The first coaxial hybrid plasmonic waveguide con-

0740-3224/10/061260-6/$15.00 © 2

ists of a metal cladding, a sandwiched low index dielec-ric “cushion” layer, and a high index dielectric core [Fig.(c)]. The coupling between the surface plasmon modes lo-alized at the circular metal–low-index-dielectric inter-ace and the optical modes inside the high index dielectricore determines modal properties of this coaxial hybridaveguide. The thickness of the low index layer can effec-

ively control the coupling coefficient. We can obtain lessoss with similar confinement and a slightly less overlapetween the optical modes and the core region by usinghis hybrid structure in comparison with a correspondingonventional plasmonic waveguide shown in Fig. 1(a).hus this hybrid design is very important for ultrasmallanowire laser applications. In [14], a similar structureith different dielectric profile is used for microwave ap-lications. The second coaxial hybrid plasmonic wave-uide is the reverse version of the first one, consisting of aigh index dielectric cladding, a sandwiched low index di-lectric cushion layer, and a metal core [Fig. 1(d)]. Thehysics behind this structure is different since there is noonfined mode located at the interface between two differ-nt dielectric layers. The propagation constant of a con-ned mode of this hybrid waveguide should be larger thanhe free space wave number in the high index region. Thiseads to a strong transverse confinement in the low indexegion, and thus the energy is effectively confined in thisegion. In comparison with a corresponding conventionallasmonic waveguide shown in Fig. 1(b), the most inter-sting property of this hybrid waveguide is that it can en-ance and confine optical fields in a low index region. Theole of plasmonic modes in the metal-dielectric-metal co-xial structures is also analyzed in [15].

. THEORY AND METHODecause all the waveguides discussed in this paper have

he circular symmetry, we can take advantage of the ana-ytical solutions of the wave equations in cylindrical coor-

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Y. Zhao and L. Zhu Vol. 27, No. 6 /June 2010/J. Opt. Soc. Am. B 1261

inates. The direction z is the propagation direction. Theolutions of the longitudinal components in a generalorm [16] are as follows:

Ez = �AJn�kt�� + BHn�kt���e−jn�e−jkzz,

Hz = �CJn�kt�� + DHn�kt���e−jn�e−jkzz, �1�

here Jn and Hn are the Bessel function of the first kindnd the Hankel function, respectively; kt is the transverseropagation constant; kz is the longitudinal propagationonstant which satisfies kt

2+kz2=k2; and n is the azi-

uthal mode number which is an integer in our cases.The transverse field components, such as E�, E�, H�,

nd H�, can be obtained from the vertical and horizontalelations of electromagnetic fields which are as follows:

Et =− j

k2 − kz2 ����t � Hz + kz�tEz�,

Ht =j

k2 − kz2 ����t � Ez − kz�tHz�, �2�

here Et and Ht can be denoted by E��̂+E��̂ and H��̂

E��̂, respectively. �t� is the curl operation and �t is theradient operation both in the transverse plane. Ez andz in normal font weight represent the magnitude of the z

omponent of the electric and magnetic fields.By applying the boundary conditions, in which the tan-

ential components of the electric and magnetic fieldsust be continuous at every interface, respectively, a

ranscendental equation can be obtained. Solving thisquation gives the dispersion relation and propagationonstant which can be expressed as �+ i�. The imaginaryart, �, represents the modal loss and 1/ �2�� is defined ashe propagation length. Furthermore, we can also obtainhe mode area and confinement factor. The normalizedode area is defined as A /A [10], where A

b

d(c) (d)

(a) (b)

ig. 1. (Color online) Cross sections of the waveguides discussedn this paper. (a) Structure A: a dielectric core surrounded by a

etal cladding ; (b) structure B: a metal core surrounded by aielectric cladding; (c) structure C: from inside to outside, a highndex dielectric core, sandwiched low index dielectric layer, and

etal cladding; (d) structure D: from inside to outside, a metalore, sandwiched low index dielectric layer, and high index di-lectric cladding.

m 0 0

�0 / �2nindex��2 and Am=�W�r�dV /max�W�r��. W�r� ishe electromagnetic energy density, which in a dispersiveaterial can be written as

W�r� =1

2Re�d��r��

d���E�r��2 +

1

2�0�H�r��2. �3�

he conventional definition of the confinement factor ���s the integral of the electric energy within the gain me-ium over the whole space integral of the electric energy,hich can be written as

� =

gain

�E�2dV

�

�E�2dV

. �4�

The mode area is to measure the confinement of thelectromagnetic energy and the field enhancement [11]nd the confinement factor is to show the overlap betweenhe optical mode and the gain medium. However, this pa-er is not aimed to show the influence of the gain mediumn the modal profile, but focused on the propagation prop-rties of these hybrid waveguides. So in our simulations,e do not include any gain medium. In order to properlyefine the confinement factor, we replace the numeratorn the definition above with the integral of the electric en-rgy within the high index dielectric core for structures And C, and the integral within the low index dielectricushion layer for structure D. Together with the propaga-ion constant, we will investigate the modal properties ofwo different types of coaxial hybrid waveguides.

We run our simulations with the operation wavelengtho be 700 nm, the index of high index dielectric to be 3.6,he index of low index dielectric to be 2.4, and the metal toe gold. Accordingly, the permittivity of gold [17] is16.6473+1.0467i at the chosen operation wavelength.

. RESULT AND ANALYSISe first analyze the conventional plasmonic waveguide

hown in Fig. 1(a). This structure can support well con-ned optical modes at the cost of high losses when theanowire radius is much smaller than the wavelength.e focus on the fundamental HE11 mode since it exhibits

he lowest loss. When the wire radius is very small, a lotf optical fields penetrate into the metal region, and theode is mostly confined in the metal region. Therefore,igh loss is unavoidable in this situation as is shown inig. 2(a). As the radius gets larger, more of the energy wille confined in the dielectric region. That is why � experi-nces the minimum value and � experiences the maxi-um value at the same radius. However, as the radius

rows further, the increasing �, � and dropping � implyhat the mode becomes more localized at the interface be-ween the metal and the dielectric, getting similar to aurface plasmon mode at the planar metal-dielectric in-erface.

We aim to suppress loss without reducing confinementy the use of a hybrid design. We put a low index cushionayer between the high index dielectric core and the metal

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1262 J. Opt. Soc. Am. B/Vol. 27, No. 6 /June 2010 Y. Zhao and L. Zhu

ladding as shown in Fig. 1(c). The coupling between theaveguide modes in the high index dielectric core and the

urface plasmon modes at the circular metal–low-index-ielectric interface gives the hybrid nature to the eigen-odes of this structure. By adjusting the thickness of this

ushion layer, t, we can obtain different coupling efficien-ies. The immediate benefit of this structure is the extrareedom to control the loss and confinement. Consider twoxtreme situations: one is that when t is equal to zero,his structure becomes the conventional plasmonic wave-uide in Fig. 1(a) and exhibits high losses with smallode volumes; and the other one is that when t is ap-

roaching infinity, the structure becomes a cylindrical di-lectric waveguide with no losses and large mode vol-mes. Therefore, by selecting a suitable thickness t, wean operate in a regime in which the hybrid waveguidehows less loss with the same degree of confinement.

The fundamental mode of the hybrid waveguide in Fig.(c) is a HE11-like mode as shown in Fig. 3(a). We show itsispersion relation, normalized mode area, and confine-ent factor in Fig. 4. Figures 4(a) and 4(b) show the real

nd imaginary parts of the propagation constant, respec-ively. The variations of � and � with respect to t can bexplained by the analysis above. � gets smaller and theoss gets lower when the cushion layer thickness getsarger. The loss of all the waveguides increases quickly

0 0.05 0.1 0.15 0.2 0.25 0.3 0.350

5

10

β/k 0

r/ λ

(a)

0 0.05 0.1 0.15 0.2 0.25 0.3 0.350

1

2

α/k 0

βα

ig. 2. (Color online) The results of the fundamental mode of stmaginary part. (b) The mode area and the confinement factor.

b(a) (b)ig. 3. (Color online) Mode profiles of structures C and D. (a)E11-like fundamental mode of structure C; (b) TM01-like funda-ental mode of structure D.

round r /=0.04. Thus for practical considerations, theanowire radius should be kept larger than 0.04 in plas-onic nanolasers for this operation wavelength. Theode area of the hybrid waveguide only increases a little

it in comparison with the conventional plasmonic wave-uide when the cushion layer is a few nanometers thicks shown in Fig. 4(c). Since all the hybrid waveguidesave much lower loss than the conventional plasmonicaveguide, we can obtain less loss with the similar degreef confinement. This is not surprising in that the couplingetween the waveguide mode and surface plasmon modeesults in a relatively large amount of electromagnetic en-rgy confined in the thin low index region, as shown inig. 3(a). Similar to the mode area, the plot of � alsohows certain degree of compromise for the hybrid wave-uide. Here, according to our definition above for struc-ure C, only the electric energy within the high index di-lectric region contributes to �. So when part of opticalelds are now located in the cushion layer, � drops a littleit. The situation when t=50 nm is different. Here, t ishick enough (compared with the operation wavelength)o support a well confined waveguide mode in the core re-ion where the core radius is large and the coupling iseak. Thus, as the radius increases, much more energy is

onfined in the high index dielectric core in this condition.Another common cylindrical plasmonic waveguide is

hown in Fig. 1(b). In this structure, all confined modesre located along the metal-dielectric interface. Comparedith the waveguide in Fig. 1(a), optical modes in this

tructure have lower losses and larger mode volumes. Theundamental mode in this structure is TM01 mode. Struc-ure B has been widely used for transiting terahertzaves [18,19]. We can apply the same procedure, by in-

erting a low index cushion layer to structure B, to gettructure D shown in Fig. 1(d). As we have pointed out,he physics behind this structure is different. Since theres no confined mode at a low-index/high-index dielectricnterface, the interpretation by the coupled-mode theoryails. In order to obtain a confined mode, the propagationonstant has to be greater than the free space wavenum-er in the high index dielectric, which results in the decay

0 0.05 0.1 0.15 0.2 0.25 0.3 0.350

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(b)

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e A. (a) The propagation constant. � is the real part and � is the

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Y. Zhao and L. Zhu Vol. 27, No. 6 /June 2010/J. Opt. Soc. Am. B 1263

f transverse fields in the high index region. This actuallyeads to much stronger confinement and enhanced opticalelds in the low index region.To fulfill the mode existing condition, we have to select

he operation wavelength and the indices of dielectric ma-erials carefully. In fact, we can apply an approximateondition to obtain a non-radiative mode in this cylindri-al multilayer structure,

Re�ksp� nhk0, �5�

here ksp is the propagation constant of the planar sur-ace plasmon mode localized at the metal–low-index-ielectric interface at the chosen operation wavelength,h is the index of the high index dielectric, and k0 is the

ree space wavenumber in air. The TM01-like fundamen-al mode profile is shown in Fig. 3(b). We show its disper-ion relation, normalized mode area, and confinement fac-or in Fig. 5. The most interesting property of thistructure is its ability to confine optical fields in the lowndex region. Therefore, when we calculate the confine-

ent factor, we consider the ratio between the electric en-rgy in the low index region over the total electric energy.igure 5(a) shows the real part of the propagation con-tant with respect to the radius of the metal core, r, andhe thickness of the low index dielectric layer, t. The dash

0 0.1 0.2 0.3 0.40

1

2

3

4

5

6

7

8

r/ λ

β/k 0

(a)

t=0nmt=2nmt=6nmt=10nmt=20nmt=50nm

0 0.1 0.2 0.3 0.40

0.5

1

1.5

2

2.5

3

3.5

4

r/ λ

A m/A

0

(c)

t=0nmt=2nmt=6nmt=10nmt=20nmt=50nm

ig. 4. (Color online) Simulation results of the fundamental modart of propagation constant �. (c) Normalized mode area. (d) Co

ine represents the results for structure B. It is clear thatll the mode effective indices are well larger than nhdashed-dotted line in Fig. 5(a)] except t=50 nm. At t50 nm, the mode effective index is smaller than the re-

ractive index of the outmost dielectric material when theire radius is larger than 0.2. In this situation, trans-erse energy leakage happens [20]. This explains why theode area increases rapidly and the confinement factor

rops dramatically for these modes in Figs. 5(c) and 5(d).n the same way, to analyze the influence of the cushionayer in this structure, we consider two extreme situa-ions: one is that when the thickness t is equal to zero, thetructure becomes a metal core surrounded by a high in-ex dielectric. The other one is that when t is equal to in-nity, the structure becomes a metal core surrounded by a

ow index dielectric. As is known, surface plasmon modest the metal–low-index-dielectric interface have lowerosses and less confinement than those at the metal–high-ndex-dielectric interface [21]. Therefore, the introductionf the cushion layer in this structure decreases loss � ashown in Fig. 5(b), but increases the mode area as shownn Fig. 5(c). The ability of confining optical fields in a thinow index region is demonstrated in Fig. 5(d). It showshat more than half of the total electric energy can be con-ned in a 6 nm thin low index region. When the radius

0 0.1 0.2 0.3 0.40

0.5

1

1.5

r/ λ

α/k 0

(b)

t=0nmt=2nmt=6nmt=10nmt=20nmt=50nm

0 0.1 0.2 0.3 0.40

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

r/ λ

Γ

(d)

t=0nmt=2nmt=6nmt=10nmt=20nmt=50nm

ructure C. (a) Real part of propagation constant �. (b) Imaginaryent factor.

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1264 J. Opt. Soc. Am. B/Vol. 27, No. 6 /June 2010 Y. Zhao and L. Zhu

ncreases, the confinement factor drops a little bit sincehe effective index becomes smaller.

. CONCLUSIONe have investigated the optical properties of two coaxialultilayer plasmonic nanowire waveguides. One consists

f a metal cladding, a low index dielectric cushion layer,nd a high index dielectric core and the other one is theeverse version. Simulation results including the propa-ation constant, the normalized mode area, and the con-nement factor are presented. The results show that wean obtain less loss with the similar degree of confine-ent for the first type of hybrid waveguide compared withconventional plasmonic nanowire waveguide. So this

ind of waveguide can be used for ultrasmall nanowire la-er applications. In the second waveguide, we demon-trate that the energy can be effectively confined in a veryhin low index dielectric layer.

CKNOWLEDGMENTShis work is partially supported by a sub-award from009 SC NSF EPSCOR RII Track-I Award (EPS-0903795)nd 2010 ORAU Ralph E. Powe Junior Faculty Enhance-

0 0.1 0.2 0.3 0.40

2

4

6

8

10

12

14

16

18

20β/k 0

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(a)

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(C)

0 0.1 0.2 0.3 0.40

0.5

1

1.5

A m/A

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(c)

ig. 5. (Color online) Simulation results of the fundamental modart of propagation constant �. (c) Normalized mode area. (d) Co

ent Award.

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0 0.1 0.2 0.3 0.40

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