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˖ڍመ᝶஠ڙጲ

http://www.paper.edu.cn

自由空间耦合高Q回音壁模式的理论

邹长铃

，舒方杰

，孙方稳

，贡赵军

，韩正甫

，郭光灿

中国科技大学物理学院光学与光学工程系中科院量子信息重点实验室，合肥，230026

商丘师范学院物理系，商丘，476000

摘要：A theoretical study of free space coupling to high-Q whispering gallery modes both in

circular and deformed microcavities are presented. In the case of a circular cavity, both

analytical solutions and asymptotic formulas are derived. The coupling eﬃciencies at diﬀerent

coupling regimes for cylinder incoming wave are discussed, and the maximum eﬃciency is

estimated for the practical Gaussian beam excitation. In the case of a deformed cavity, the

coupling eﬃciency can be higher if the excitation beam can match the intrinsic emission well

and the radiation loss can be tuned by adjusting the degree of deformation. Employing an

abstract model of slightly deformed cavity, we found that the asymmetric and peak like line

shapes instead of the Lorentz-shape dip are universal in transmission spectra due to

multi-mode interference, and the coupling eﬃciency can not be estimated from the absolute

depth of the dip. Our results provide guidelines for free space coupling in experiments,

suggesting that the high-Q ARCs can be eﬃciently excited through free space which will

stimulate further experiments and applications of WGMs based on free space coupling.

关键词：自由空间耦合，回音壁模式，变形微腔，圆柱波，高斯光束

中图分类号： O431.2

Theory of free space coupling to high-Q

whispering gallery modes

ZOU Chang-Ling

,SHU Fang-Jie

,SUN Fang-Wen

, GONG Zhao-Jun

,HAN

Zheng-Fu

,GUO Guang-Can

Key Laboratory of Quantum Information and Department of Optics and Optical

Engineering, University of Science and Technology of China, Chinese Academy of Sciences,

Hefei, Anhui 230026, China

Department of Physics, Shangqiu Normal University, Shangqiu, Henan, 476000, China

Abstract: A theoretical study of free space coupling to high-Q whispering gallery modes

both in circular and deformed microcavities are presented. In the case of a circular cavity,

基金项目： National Natural Science Foundation of China (Grant Nos.11004184,11204169),National Basic Re-

search Program of China (Grant Nos. 2011CB921203, 2011CBA00200), Research Fund for the Doctoral Program of

Higher Education of China (Grant No. 20103402120009),the Foundation of He’nan Educational Committee (Grant

No. 2011A140021)

作者简介： Chang-Ling Zou （1989-），male，PhD candidate，major research direction：Nano-photonics and

Quantum Optics. Correspondence author： Fang-Wen Sun（1979-）， male，professor， major research direction：

Quantum Optics and Quantum Information.

- 1 -

˖ڍመ᝶஠ڙጲ

http://www.paper.edu.cn

both analytical solutions and asymptotic formulas are derived. The coupling eﬃciencies at

diﬀerent coupling regimes for cylinder incoming wave are discussed, and the maximum

eﬃciency is estimated for the practical Gaussian beam excitation. In the case of a deformed

cavity, the coupling eﬃciency can be higher if the excitation beam can match the intrinsic

emission well and the radiation loss can be tuned by adjusting the degree of deformation.

Employing an abstract model of slightly deformed cavity, we found that the asymmetric and

peak like line shapes instead of the Lorentz-shape dip are universal in transmission spectra

due to multi-mo de interference, and the coupling eﬃciency can not be estimated from the

absolute depth of the dip. Our results provide guidelines for free space coupling in

experiments, suggesting that the high-Q ARCs can be eﬃciently excited through free space

which will stimulate further experiments and applications of WGMs based on free space

coupling.

Key words: Free space coupling, Whispering-Gallery mode, Deformed micro-cavity,

Cylinder wave, Gaussian beam

0 Introduction

Whispering gallery modes (WGMs) were ﬁrst explained by Lord Rayleigh [1] in the case of

sound propagation in St Paul’s Cathedral circa in 1878. WGMs also exist in light waves which

are almost perfectly guided round by optical total internal reﬂection in low loss resonators [2].

High quality (Q) factor excessing 10

has been achieved. Combining the ultra-small mode

volume with the high-Q factor, the light-matter interaction can be greatly enhanced in WGM

microcavities. Therefore, optical WGMs are gaining growing attentions in a wide range of

ﬁelds including ultra-sensitive sensors [3], low threshold lasers [4], frequency combs [5], cavity

quantum electrodynamics (QED) [6], and quantum optomechanics [7, 8]. The WGMs also

aﬀect the scattering on spherical particles [9], which was found to play an important role in

the glory phenomenon [10]. In the last twenty years, the high-Q optical WGMs have been

reported in various microcavities, such as liquid droplet [11], microsphere [12], microdisk [13],

microcylinder [14] and microtoroid [15].

The excitation and collection of the WGMs are essential issues in practical applications.

It is taken for granted that free space coupling to WGMs is very ineﬃcient since the radi-

ation loss of circular shaped microcavity is isotropic and low, which brings the diﬃculty of

energy transference between outside and WGMs. Near ﬁeld couplers, such as prism [16], ﬁber

taper[17, 18] and perpendicular waveguide [19], enable high eﬃcient excitations and collections

to microresonators. Therefore, these near ﬁeld couplers are widely adapted in experiments.

The underlying mechanism of coupling process between WGMs and couplers have been well

understood[20, 21, 22] and many phenomena in the spectrum are reported, such as the ana-

- 2 -

˖ڍመ᝶஠ڙጲ

http://www.paper.edu.cn

logue of electromagnetically induced transparent (EIT) [23, 24, 25], asymmetric Fano line shape

[19, 26, 27, 28] and ringing phenomena [29]. However, there are some limitations in the near ﬁeld

couplers, such as large footprint, stabilization requirement, and power limit (high excitation

energy would cause strong nonlinear eﬀects in ﬁber tapers).

On another hand, the asymmetric resonant cavities (ARCs) have been demonstrated to be

able to support high-Q WGMs and give highly directional emission [30, 31, 32, 33, 34, 35, 36, 37,

38]. The underlying principle and interesting chaotic ray dynamics in such an open billiard have

been studied extensively [39, 40, 41, 42, 43]. Especially, the unidirectional emission cavities have

been successfully designed [44, 45, 46, 47, 48, 51, 49, 50], which enable high eﬃcient collection

of high-Q WGMs through free space. As the reversal of emission, the focused beam in free

space can excite the WGMs, which have been studied in experiments [52, 53, 54, 55, 56, 57].

However, the basic features and potentials of this free space beam coupling strategy are not

fully explored.

In this paper, we present a theoretical study on free space coupling to high-Q WGMs in

both circular and deformed microcavities. First of all, the cylinder waves incident to a circular

cavity is solved analytically and asymptotically in the cylinder coordinates, and the analytical

equations are well consistent with the standard input-output formulation. Then, we consider

the excitation of the circular cavity by Gaussian beam for practical applications, the maximal

eﬃciency of about 20% can be achieved under phase matching condition. After that, the

excitation of deformed cavity by Gaussian beam is studied by an abstract model of multimode

interaction. The result shows that the ARCs not only gives good matching parameters, but

also can balance radiation and non-radiation losses, which can be well adapted in experiments

for high eﬃcient free space coupling. One interesting result is that the asymmetric spectra

is universal in the free space coupling as a result of multimo de interference, and the energy

transferring can not be deduced from the dip depth in the transmission spectrum.

1 Circular Cavity

1.1 Cylinder wave

We start with a two-dimensional circular shaped microcavity, where electromagnetic ﬁeld

(ψ) can be solved in the cylinder coordinates (r, ϕ) analytically. Any electromagnetic ﬁeld

can be decomposed in the basis of Bessel functions J

(kr) and Hankel functions H

(1(2))

(kr)

with integral m ∈ (−∞, ∞). Since the ﬁeld intensity is ﬁnite everywhere including the origin,

thus the ﬁeld inside the cavity should be represented by

∑

(nkr)e

imϕ

when r < r

with

is the radius of the cavity. The ﬁeld outside (r > r

) is represented by

∑

(2)

(kr)e

imϕ

and

∑

(1)

(kr)e

imϕ

, which correspond to the inward and outward traveling cylinder waves,

respectively.

- 3 -

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