Ultra-Narrow Faraday Rotation Filter at the Rb D1 Line

Joanna A. Zielińska1,2, Federica A. Beduini1, Nicolas Godbout3 and Morgan W. Mitchell1,4,∗1ICFO-Institut de Ciencies Fotoniques, Mediterranean Technology Park, E-08860 Castelldefels (Barcelona), Spain

2Instytut Fizyki Doświadczalnej, Uniwersytet Warszawski, PL-00-681 Warszawa, Poland3COPL, Département de Génie Physique, École Polytechnique de Montréal, C.P. 6079, Succ. Centre-ville, Montréal (Québec)

H3C 3A7, Canada4ICREA, Passeig Lluís Companys, 23, E-08010 Barcelona, Spain

∗Corresponding author: morgan.mitchell@icfo.es

Compiled October 31, 2018We present a theoretical and experimental study of the ultra-narrow bandwidth Faraday anomalous dispersionoptical filter (FADOF) operating at the rubidium D1 line (795nm). This atomic line gives better performancethan other lines for the main FADOF figures of merit, e.g. simultaneously 71% transmission, 445MHzbandwidth and 1.2GHz equivalent noise bandwidth. c© 2018 Optical Society of America

OCIS codes: 020.1335, 010.3640

Ultra-narrow bandwidth optical filters are key el-ements in laser remote sensing (LIDAR), observa-tional astronomy, free-space communications and quan-tum optics. Relative to conventional interference fil-ters, FADOFs offer high background-rejection, mechani-cal robustness, imaging capability and high transmission.FADOFs have been developed for several alkali atom res-onances – Cs D2 [1] and 6S1/2 → 7P3/2 [2] lines, Rb D2

line [3, 4], K (three lines) [5], Na D lines [6], and forCa [7].

We demonstrate a FADOF on the D1 line of Rb (wave-length 795 nm). This line, efficiently detected with Sidetectors, accessible with a variety of laser technolo-gies, and with large hyperfine splittings, is a favoritefor coherent and quantum optics with warm atomic va-pors. Applications include electromagnetically-inducedtransparency [8], stopped light [9], optical magnetom-etry [10, 11], laser oscillators [12], polarization squeez-ing [13, 14], quantum memory [15] and high-coherenceheralded single photons [16,17].

Here we show that the Rb D1 line provides supe-rior FADOF performance. We demonstrate a FADOFsurpassing other atoms and other Rb transitions forkey figures of merit, including peak transmission Tmax,transmission bandwidth and equivalent noise bandwidthENBW = T−1

max

´T (ν)dν, where T (ν) is the filter trans-

mission versus frequency ν [4].A FADOF, shown schematically in Figure 1 (inset),

consists of an atomic vapor cell between two crossed po-larizers. A homogeneous magnetic field along the propa-gation direction induces circular birefringence in the va-por. The crossed polarizers block transmission away fromthe absorption line, while the absorption itself blocks res-onant light. Nevertheless, Faraday rotation just outsidethe Doppler profile can give high transmission for a nar-row range of frequencies. FADOF is simple and robust,but performance depends critically on optical proper-ties of the atomic vapor. We model these with a first-principles calculation and find excellent agreement with

experiment, as shown in Figure 1.

Rb cell

BP

1P

2

A

B

C

D

E

Figure 1. Absorption spectrum for Rb D1: theory(green), experiment (black dashed). FADOF transmis-sion spectrum: theory (blue), experiment (red). Condi-tions: T = 365K, B = 4.5mT. Inset - FADOF setup.

Experiment

We use a rubidium cell, with anti-reflection coated win-dows, a 10 cm internal path, no buffer gas or wall coat-ings and natural abundance rubidium (Technical Glass,Inc.). An oven and solenoidal field coil allow the cell to bemaintained at constant temperature in a uniform axialfield. Probe light at 795 nm from a Littrow-configurationexternal-cavity diode laser (Toptica DL100) is spatially-filtered with a fiber and prepared in an adjustable lin-ear polarization using quarter- and half-waveplates. Af-ter the cell, the beam is polarization-filtered with a linearpolarizer (colorPol VIS IR) and detected with a low-noiseamplified photodiode.

Saturated absorption spectroscopy was used to cali-brate field strength produced by the solenoid current, asshown in Figure 2. Labeling the eigenstates by the to-tal angular momentum quantum numbers (F,mF ), the

1

arX

iv:1

110.

2362

v2 [

phys

ics.

optic

s] 1

4 Ju

l 201

6

Frequency reference 361.58(17) MHz

crossover

F=3 F'=3

F=3 F'=2

Δ

mF= -3 m'

F= -2

mF=3 m'

F=2

I=0 A

Sat

. Abs

. Tra

nsm

issi

on [a

rb.u

.]

Frequency

I=1 A

I=2 A

I=5 A

I=4 A

I=3 A

Figure 2. Saturated absorption spectra of the Doppler-broadened line (85Rb transitions from the F = 3 groundstate) for different coil currents. Splitting ∆ betweenZeeman-split lines provides a precise measure of the field.

zero-field splitting between the (3,m) → (2,m)′ and(3,m) → (3,m)′ transitions of 85Rb provides a precisereference for the frequency scale, while the splitting ∆between (3, 3) → (2, 2)′ and (3,−3) → (2,−2)′ is max-imally sensitive to magnetic field. Comparing the ob-served splittings to the first-principles model, we findthe field with ±0.6 % accuracy. Absorption spectra, ac-quired at the same time as the FADOF spectra, are usedto calibrate the frequency axis, which otherwise would benon-linear due to piezo hysteresis. These same spectraindicate the temperature with an uncertainty of ± 2K.

The FADOF transmission was measured under a va-riety of temperature and field conditions in the range T= 355–380K and B = 3.3–6.0mT. Typical results, takenat 4.5mT and 365K, are shown in Figure 1 and show anagreement with the model at the few-percent level. Thisfield/temperature combination gives a single dominantpeak with transmission of 70% at +2.8GHz relative toline center with FWHM of 445MHz. Other peaks, oftransmission 49%, 17% and 20% are also present, givingan ENBW of 1.2GHz, considerably better than reportedfor other atoms. As shown in Table 1, the Rb D1 FADOFalso achieves narrower bandwidths than other transitionsand can achieve transmission up to 92% consistent withultra-low ENBW.

The superior performance of the Rb D1 line appears tobe a fortunate accident of the hyperfine splittings. Foreither pure 85Rb or pure 87Rb, the FADOF transmis-sion at these field strengths shows four peaks, with thestrongest ones at the extremes of the spectrum and withlong tails. The strong 87Rb peaks are visible as peak Aand E of Figure 1. The strong 85Rb peaks include one at−2.5GHz, completely obscured by the 87Rb absorption,and the peak D. The long tail of peak D is blocked byabsorption from the 87Rb 1 → 1 transition, improvingENBW.

Conclusions

We have demonstrated a Faraday anomalous dispersionoptical filter (FADOF) operating at the rubidium D1

Table 1. Comparison of reported FADOF transmis-sion for different atoms and wavelengths. Tmax: peaktransmission. BT: full-width at half-maximum band-width of main transmission peak. BN: Equivalent-noisebandwidth (ENBW). - value not reported. Reference [5]also shows a K 770 nm FADOF curve very similar to K766 nm.

Atom λ[nm] Ref. Tmax BT[GHz] BN[GHz]K 405 [5] 0.93 1.2 6Ca 423 [7] 0.55 1.5 -Cs 455 [2] 0.96 0.9 3.3Na 589 [6] 0.85 1.9 5.1Na 590 [6] 0.37 10.5 8.3K 766 [5] 0.96 0.9 5Rb 780 [4] 0.93 1.3 4.7Cs 852 [1] 0.90 0.6 -T[K] B[mT]353 18.0 0.92 0.48 2.1378 5.9 0.91 1.10 2.7365 4.5 0.71 0.45 1.2345 2.0 0.04 0.32 0.8

line. The filter gives high transmission and narrow band-width, with typical numbers being 0.5 GHz bandwidth,maximum transmission of 0.7 and equivalent-noise band-width of 1.2GHz, surpassing similar filters using otheratoms and/or other optical transitions. The spectrumcan be optimized for different figures of merit by adjust-ing the temperature and magnetic field conditions of theatomic vapor. A theoretical model shows excellent agree-ment with experimental results. The simplicity and highnoise-rejection may make the new filter attractive forLIDAR and free-space communications and introducesFADOF for the D1 line, widely preferred for coherentand quantum optical applications.

Acknowledgements

We thank Y. A. de Icaza Astiz for discussions and assis-tance. This work was supported by the Spanish Ministryof Science and Innovation under projects FIS2008-01051,FIS2011-27806, and the Consolider-Ingenio 2010 Project“Quantum Optical Information Technologies.”

Appendix: FADOF spectra calculation

For completeness, we present the full theoretical modelof the Rb D1 FADOF. A Mathematica notebook to per-form the calculations is available with this document athttp://arxiv.org/.

The Rb D transitions connect the 52SJ=1/2 groundstates to the 52PJ=1/2 (D1) and 52PJ=3/2 (D2) excitedstates. Within each manifold, the Zeeman/hyperfine

2

structure is determined by the Hamiltonian

H = EF + HZe + HHF (1)HZe = −gJµB(B · J)⊗ I2I+1

−gIµBI2J+1 ⊗ (B · I) (2)HHF = AHFDHF +BHFQHF (3)DHF = J◦I (4)

QHF =

{3(J◦I)2+ 3

2 J◦I−I(I+1)J(J+1)

2I(2I−1)J(2J−1) J > 1/2

0 otherwise(5)

where EF is the level energy including the fine struc-ture contribution, AHF and BHF are magnetic dipole andquadrupole energies, J and I are vector operators rep-resenting the total electronic and nuclear angular mo-menta, respectively, J◦I ≡ Jx⊗ Ix + Jy⊗ Iy + Jz⊗ Iz, gJand gI are total electronic and nuclear gyromagnetic fac-tors, respectively, In is the n-dimensional identity matrixand µB is the Bohr magneton. All constants are takenfrom Steck [18,19].

The transition electric dipole operator D, with circularcomponents Dq, q = 0,±1, is given by

DJ,mJ ,J

′,m′J

q =⟨J ||D||J ′

⟩(−1)J

′−1+mJ

×√

2J + 1

(J ′ 1 Jm′J q mJ

)(6)

where last expression (:::) is the Wigner 3-j symbol. Thereduced dipole moment

⟨J ||D||J ′

⟩for each isotope is

given by Steck [18,19].For 85Rb (I = 5/2) or 87Rb (I = 3/2), and for the

ground and excited state manifolds, the Hamiltonianof Eq. (1) is diagonalized to find transition frequenciesωba ≡ ωb − ωa and their corresponding dipole matrix el-ements Dab

q = 〈a| Dq |b〉, between all pairs of ground andexcited states |a〉 , |b〉. The (linear) electric susceptibilitytensor χ is diagonal in the circular (q = ±1) basis, andwe compute its diagonal entries (in S.I. units) as

χqq(ω) =∑

Z∈{85,87}

NZ(T )

ε0~∑a,b

|Dabq |2(ρbb − ρaa)

×V (σ,Γ

2, ωba − ω) (7)

where NZ(T ) is the atomic number density for the iso-tope Z, T is the temperature, ε0 is the vacuum permit-tivity, ρ ∝ exp[−H/(kBT )] is the atomic density matrixand kB is the Boltzmann constant. The Voigt profile is

V (σ,Γ

2, ωba − ω) = i

√π

2σ2ex

2

Erfc(x), (8)

where Erfc is the complementary error function, x ≡[Γ2 + i(ωba − ω)]/(

√2σ), and σ = ωba

√kBT/(MZc2).

Atomic masses MZ and natural linewidths Γ are givenin [18,19].

Atom number density is calculated as NZ(T ) =

cZN(pure)Z (T ), where cZ are the relative abundances, the

single-isotope densities are N (pure)Z (T ) = PZ(T )/(RT )

where R is the ideal-gas constant and the single-isotopevapor pressures PZ(T ) are from Steck [18,19]. We assumenatural abundance c85 = 0.7217 and c87 = 0.2783.

The transmission of the filter is

T = |E†det ·Mcell · Ein|2, (9)

where Ein = (1, 1)T /√

2 is the Jones vector, in thecircular left/right basis, of the linearly polarized in-put, Mcell = diag(exp[inLkL], exp[inRkL]) is the trans-mission matrix of the cell, nq =

√1 + χqq, Edet =

(1,−1)/√

2 is the Jones vector of the detected polariza-tion, k = ωlaser/c and L is the cell internal path length.

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