Berry phase magnification in opticalmicrocoil resonators
Timothy Lee,1,* Neil G. R. Broderick,2 and Gilberto Brambilla1
1Optoelectronics Research Centre, University of Southampton, Southampton, SO17 1BJ, United Kingdom2Physics Department, University of Auckland, Private Bag 92019, Auckland 1142, New Zealand
Optical microcoil resonators (OMRs) have recently at-tracted much research interest owing to their high qualityfactor (Q-factor) and extinction ratio (ER), as well as lowinput and output coupling losses and a large evanescentfield [1,2]. The highly compact resonator architecture isformed by wrapping a tapered fiber around a rod toallow evanescent coupling between adjacent turns as inFig. 1. With an increasingly diverse range of applications,such as sensing [3,4] and signal processing [5], devicedesign remains important and indeed the optical proper-ties have been studied in the linear [1,2,6,7] and nonlinear[2] regimes.However, the models published so far do not incorpo-
rate the generation of a Berry phase ϕB, which couplesthe orthogonal polarizations, because the OMR radiusR ¼ 10−3 − 10−2 m was assumed to be much larger thanthe pitch between turns p ≈ 10−6 m. For applications inwhich this geometric assumption is invalid, the effects ofϕB on the OMR’s polarization sensitive transmission be-come significant near resonance due to the recirculationof light (we note that similar resonantly enhanced Berryphase amplification was recently reported using ring re-sonators [8]). The emerging class of OMR refractometricsensors are a particularly pertinent example where thiswould apply—since these rely on resonance wavelengthshift, it is often desirable to maximize the dynamic rangeby reducing R, thus increasing the free spectral range.Light traversing a helical path in space acquires a Berry
phase per turn given by ϕB ¼ τS, where the helix torsionτ ¼ 2πp=S2 measures how sharply the fiber is coiled, thesingle turn length is S ¼ ðp2 þ 4π2R2Þ1=2, and p is mea-sured between the centers of two adjacent turns [9–11].The Berry phase therefore increases linearly along a uni-form helix and for a plane polarized input this is mani-fested classically as a rotation of the polarization plane,which can be modeled as an optical activity by couplingthe orthogonal x and y polarizations according to the tor-sion [10]. However, in a microcoil the coupling betweenturns must also be taken into account. Letting Ax
j andAyj denote the field amplitudes along the local normal
and binormal axes in the jth turn of the coil as in Fig. 1,the OMR coupled mode equations (ODEs) [1] are modi-fied as follows:
dAxj
ds¼ iκxðAx
j−1 þ Axjþ1Þ þ τAy
j
þ�iΔβb − i
C2
2β − α�Axj ; ð1aÞ
dAyj
ds¼ iκyðAy
j−1 þ Ayjþ1Þ − τAx
j − αAyj ; ð1bÞ
and are solved under boundary conditions imposed byfield continuity between the n turns of the coil:
Ax;yj ð0Þ ¼
�Ax;yin j ¼ 1
Ax;yj−1ðSÞ expðiβSÞ j ¼ 2…n: ð2Þ
κx;y are the coupling coefficients between two x or y po-larized modes in neighboring turns, and α is the loss coef-ficient. Because of the geometry, κy is slightly larger thanκx, and both parameters were calculated numerically
Fig. 1. (Color online) Uniform microcoil formed by wrappingmicrofiber around a rod. Inset shows the local x‒y fiber axes.
using a commercial finite element method solver soft-ware (COMSOL). For the first and last turns, Eq. (1) isrewritten to couple only with the second and penultimateturns, respectively. Of the two propagation mismatchterms, the first arises geometrically from the curvatureC ¼ 4π2R=S2 [10], and the second from bend-stress bire-fringence Δβb ¼ βx − βy as calculated by adapting Eq. 4in [12] for a microfiber. These terms sum to ≈ − 50m−1,which is 2 orders of magnitude smaller than κx;y, butnonetheless are included for completeness.Note that the difference between κx and κy, although
small, can significantly affect the transmission. As the re-sonance strength depends on both β and κ, small polar-ization dependent variations of either parameter willdramatically alter the cavity Q, and so the OMR will be-have differently for different input polarizations. In addi-tion there is a slight shift between the x and y resonancefrequencies, and the overall line shape is broader andasymmetric due to the fiber birefringence.Although the equations can be solved numerically for
any geometry, we restrict our attention to a three turnOMR with an air-clad silica microfiber of radius r ¼0:5 μm operating at a wavelength of λ ¼ 1:55 μm, sincethese parameters correspond to OMRs that have beenfabricated for use as refractometric sensors. However,as the Berry phase is a purely geometric effect, similarresults would be seen in OMRs with a different numberof turns or made out of different materials.Solving Eqs. (1) and (2) with a pitch of p ¼ 2:50 μm
produces the transmission spectrum in Fig. 2. Note thatAx and Ay would be uncoupled if the Berry phase termwas neglected (τ ¼ 0), and hence the Berry phase effectsare clearly seen in the coupling of light from one ortho-gonal state of polarization to another. This is apparentin Fig. 2(a) where an x-polarized input propagatingthrough a lossless coil produces an output with up toη ¼ Py=PTotal > 97% of the power in the y polarization.
For this value of p, κx is closer to the critical couplingκc than κy (physically at the critical coupling κc light ata resonant wavelength λc is trapped indefinitely in thecoil). The y-polarized light being further from resonancecouples out of the coil quicker than the x polarization,which contributes to the high value of η.
By comparison, an uncoupled helix (κx;y ¼ 0) with thesame geometry only has a Berry phase of ϕB ¼ 0:025 radand η0 ¼ 6 × 10−4, because the light only makes a singlepass through the coil. The high degree of opticalactivity in an OMR is only observed near resonance;off-resonance however, η < η0 since the light is simplycoupled up the turns and out of the OMR.
In the more realistic lossy case of Fig. 2(b), η is margin-ally lower but remains large enough to be easily detect-able experimentally, despite the τ term being over 100times smaller than κx;y. Decreasing R to increase torsionand using a narrower microfiber in conjunction with awider pitch would further maximize the generation of theBerry phase.
Finally, in Figs. 2(c) and 2(d) we show the effect ofchanging the input state of polarization on the outputpower. Figure 2(c) shows the case for a y-polarized in-put with a significantly smaller polarization rotation atresonance than before. This is due to two reasons: first,the y resonance is intrinsically weaker, and second, anylight coupled into the x polarization will be stored forlonger within the coil, during which it will be partiallycoupled back into the y polarization. When the inputcontains both x and y polarizations as in Fig. 2(d), theresulting spectrum varies strongly with the geometry.The elasto-optic effects, though minor, induce a bendbirefringence which shifts the x and y resonances apartby 10 pm, which is comparable to the resonance line-width, and hence the total power PT spectrum is gener-ally asymmetric.
An important consequence of the cross-polarizationcoupling is a reduction in the ER as summarized inFig. 3(a), which plots PT near resonance for τ from 0 →
40m−1 while keeping the other coil parameters constant.Physically, this is equivalent to changing the coil diam-eter from D ¼ ∞ → 0:2mm while maintaining a constantdimensionless coupling parameter K ¼ κS, such that anychange in resonance quality is attributed to a change in τrather than deviation from critical coupling. The τ ¼ 0case models an OMR in the planar limit, under whichϕB ¼ 0 and the ER is maximum. As τ increases, morelight couples into Ay, which spends less time circulating
0
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1
Nor
mal
ised
Pow
er
1551.7 1551.75 1551.80
0.2
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0.8
1
1551.7 1551.75 1551.8Wavelength (nm)
PX
PY
PTotal
)b()a(
)d()c(
Fig. 2. (Color online) OMR output spectrum showing thecoupling of light between the x and y polarization states dueto Berry phase effects for an input polarzed along (a) the x axis,with no loss, (b) the x axis, with loss, (c) the y axis, and (d) atπ=4 rad. For (b)–(d), the loss is α ¼ 4:6m−1. Parameters: threeturn OMR, jAinj ¼ 1, r ¼ 0:5 μm, R ¼ 0:1mm, p ¼ 2:50 μm,κx ¼ 5770m−1, κy ¼ 6163m−1, τ ¼ 40m−1.
0 10 20 30 400
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0
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Nor
mal
ised
Pow
er
τ = 0m−1
τ = 40m−1
increasing torsion
Q f
acto
r (x
10 )5
(a) (b)
Fig. 3. (Color online) (a) Output power against detuningfrom resonance for helix torsion values τ from 0 to 40m−1.(b) Resonance ER (dashed) andQ factor (solid) against τ. Otherparameters are the same as Fig. 2.
in the coil than Ax, and hence less light is absorbed. Theoverall ER of the OMR is limited by that of the y reso-nance, and Fig. 3(b) shows the ER can be reducedby up to 80%.Similarly, the Q-factor (defined as Q ¼ ΔλFWHM=λ
where ΔλFWHM is the full width at half maximum line-width) falls by 32% by the same mechanism. This beha-vior is both geometry and polarization dependent—inparticular, if a y-polarized input was used or if the pitch
was chosen such that jκy − κcj < jκx − κcj, the aforemen-tioned trends would be reversed so both the Q-factor andER would increase with τ.
The choice of a 1 μm microfiber diameter in these si-mulations ensures that at λ ¼ 1:55 μm the fundamentalmode has a large evanescent field, so that the interturncoupling remains high even for pitches several timeswider than the microfiber. However, the exponentialdependence of κx;y on p indicates that even small pitchvariations can noticeably alter η, especially when κ isclose to the critical coupling condition given by κc ¼ffiffiffi2
p ðmπ � sin−1ð1= ffiffiffi3
p Þ=S [1], which for m ¼ 1 evaluatesto κc ¼ 5686m−1. Figure 4 illustrates this for a range ofpitches near 2:5 μm, where κx;y vary from 4500 to6900m−1. When κ ≈ κc, the value of η is close to 100%,but when far from critical coupling the resonance is tooweak to observe the effects of Berry phase because neg-ligible power is coupled into y polarization. Interestingly,η ¼ 0when the coupling is precisely critical at p ¼ pxc andpyc—at these two points, respectively, the x and y reso-nances are in fact absent when solving the ODEs [6].
In conclusion, we have shown that the generation ofBerry phase near resonance in an OMR can lead to a sig-nificant exchange of power between the two states ofpolarization which would otherwise be uncoupled. Alongwith the resonance ER and Q-factor, this behavior variesstrongly with the coil’s pitch and torsion. The values usedin the simulations are similar to OMRs that have alreadybeen fabricated, and thus we hope that experimentalconfirmation of this effect should be observed in thenear future.
References
1. M. Sumetsky, Opt. Express 12, 2303 (2004).2. N. G. R. Broderick, Opt. Express 16, 16247 (2008).3. F. Xu and G. Brambilla, Appl. Phys. Lett. 92, 101126 (2008).4. J. Scheuer, Opt. Lett. 34, 1630 (2009).5. N. G. R. Broderick and T. Ng, IEEE Photon. Technol. Lett.
21, 444 (2009).6. M. Sumetsky, Opt. Express 13, 4331 (2005).7. T. Lee, N. G. R. Broderick, and G. Brambilla, Opt. Commun.
284, 1837 (2011).8. I. Golub, T. Audet, and L. Imobekhai, J. Opt. Soc. Am. B 27,
1698 (2010).9. S. G. Lipson, Opt. Lett. 15, 154 (1990).10. M. V. Berry, Nature 326, 277 (1987).11. A. Tomita and R. Y. Chiao, Phys. Rev. Lett. 57, 937 (1986).12. R. Ulrich, S. C. Rashleigh, and W. Eickhoff, Opt. Lett. 5,
273 (1980).
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Detuning δλ (pm)
Pitc
h p
(µm
)
Normalised Power
(b) P(a) P yx
2.46 2.48 2.5 2.52 2.54 2.56 2.580
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Pitch p (µm)
η
Cou
plin
g κ
(mm
)-1
p xc
p yc
κ = κc
(c)
κy (p)κ
x (p)
Fig. 4. (Color online) Output power against pitch in the (a) xpolarization and (b) y polarization. (c) The fraction of outputpower in the y polarization, η, and coupling coefficients as afunction of pitch. Input is x polarized, and the other parametersare the same as Fig. 2.
