2015 RevModPhys：Dielectric microcavities： Model systems for wave chaos and non-Hermitian physics

Dielectric microcavities:Model systems for wave chaos and non-Hermitian physics

Hui Cao*

Department of Applied Physics,Yale University,

New Haven,Connecticut06520-8482,USA

Jan Wiersig?

Institut für Theoretische Physik,Otto-von-Guericke-Universit?t Magdeburg,

Magdeburg,Postfach4120,D-39016Magdeburg,Germany

(published22January2015)

This is a review on theoretical and experimental studies on dielectric microcavities,which play a significant role in fundamental and applied research.The basic concepts and theories are introduced.

Experimental techniques for fabrication of microcavities and optical characterization are described.

Starting from undeformed cavities,the review moves on to weak deformation,intermediate deformation with mixed phase space,and then strong deformation with full ray chaos.Non-Hermitian physics such as avoided resonance crossings and exceptional points are covered along with various dynamical tunneling phenomena.Some specific topics such as unidirectional output,beam shifts, wavelength-scale microcavities,and rotating microcavities are discussed.The open microdisk and microsphere cavities are ideal model systems for the studies on wave chaos and non-Hermitian physics.

DOI:10.1103/RevModPhys.87.61PACS numbers:42.55.Sa,05.45.Mt,42.60.Da,42.25.?p

CONTENTS

I.Introduction61

A.Motivation61

B.Scope63 II.Theoretical Model and Experimental Techniques64

A.Mode equation and wave simulations64

B.Ray model65

C.Husimi functions67

D.Cavity fabrication68

1.Liquid droplets and microjets68

2.Solid microspheres and microtoroids69

3.Microdisks and micropillars70

E.Optical characterization71

1.Passive cavities71

2.Active cavities71 III.Overview of Nondeformed Dielectric Microcavities71

A.Whispering-gallery modes71

B.Optical losses and quality factors72

https://www.sodocs.net/doc/9d3354504.html,sing in whispering-gallery cavities72

D.Evanescent field coupling73 IV.Smooth Deformation73

A.Weak deformation:Nearly integrable ray dynamics73

B.Moderate deformation:Mixed phase space74

1.Adiabatic curves and dynamical eclipsing74

2.Gaussian modes based on stable periodic orbits75

3.Dynamical tunneling75

https://www.sodocs.net/doc/9d3354504.html,rge deformation:Predominantly chaotic dynamics76

1.Chaotic saddle and its unstable manifold76

2.Dynamical localization and scar modes79

3.Level statistics80

4.Partial barriers and turnstile transport81

D.Perturbation theory82 V.Cavity with Sharp Corner or Boundary Defect83

A.Polygonal cavity83

B.Boundary defect84 VI.Mode Coupling85

A.Avoided resonance crossings85

B.Exceptional points87 VII.Unidirectional Free-space Light Emission from

Deformed Microlasers89

A.Spiral-shaped cavity89

B.Interior whispering-gallery modes90

C.Annular cavity90

D.Lima?on cavity91

E.“Face”cavity92

F.Ellipse with a notch93 VIII.Beam Shifts and Semiclassical Approaches95

A.Beam shifts95

B.Wavelength-scale microcavities97

C.Semiclassical approaches99 IX.Rotating Microcavities100

A.Sagnac effect in microcavities100

B.Wave chaos in rotating cavities101

C.Rotation-induced changes of quality factors

of open microcavities102

D.Far-field patterns from rotating microcavities

of deformed shape103 X.Summary and Prospects104 Acknowledgments104 References104 I.INTRODUCTION

A.Motivation

Optical microcavities can greatly enhance light-matter interactions by storing optical energy in small volumes

*hui.cao@https://www.sodocs.net/doc/9d3354504.html,

?jan.wiersig@ovgu.de

REVIEWS OF MODERN PHYSICS,VOLUME87,JANUARY–MARCH2015

0034-6861=2015=87(1)=61(51)61?2015American Physical Society

(Chang and Campillo,1996;Vahala,2004).The ability to

concentrate light is important not only to fundamental physics

studies,but also to practical device applications(Vahala,

2003).Instead of using metals which are usually lossy at

optical frequencies,most microcavities are made of trans-

parent dielectrics.In vertical cavities with distributed Bragg

mirrors or photonic-crystal defect cavities,optical confine-

ment is achieved through constructive interference of multiply

reflected or scattered light.An alternative scheme is total

internal reflection from a dielectric interface,which occurs

when light is incident from the higher refractive index(n1)

medium to the lower index(n0)one with an angle χ≥arcsinen0=n1T.Consider a light beam propagating in a circular disk or a sphere via consecutive reflections from the

boundary,the rotational symmetry of the cavity shape keeps

the angle of incidence constant,and the condition for total

internal reflection is maintained.The phase delay for light

traveling one circle along the boundary must be equal to 2πm(m?1;2;3;…),so that the returning field has the same phase as the original field and a steady state is reached. Consequently,only light at certain frequencies can be con-fined in a cavity,and these frequencies are called cavity resonant frequenciesωm.The corresponding electromagnetic modes are whispering-gallery modes(WGMs),in analogy with the acoustic wave propagating along the smooth surface of a circular gallery(Rayleigh,1945).They have also been referred to as“morphology-dependent resonances.”The first observation of stimulated emission into optical WGMs was reported soon after the invention of laser in solid spheres of diameter1–2mm(Garrett,Kaiser,and Bond,1961). Since then,WGMs have been studied in a range of micron-sized cavities,from liquid droplets and jets to solid spheres, cylinders,disks,and rings.The optical confinement is, however,not perfect.Because of the curvature of the cavity boundary,light escapes out of the cavity via evanescent leakage,the optical analog of quantum tunneling.In addition, the surface roughness introduces scattering loss,and there is residual absorption in the bulk material and at the surface. They all contribute to a finite lifetimeτof light in a WGM, which leads to a spectral widthδω?1=τ.The quality factor is defined as Q?ωmτ.

Compared to other microcavity resonances,the WGMs

have extraordinarily high Q and small volume,which lead to

diverse applications in linear and nonlinear optics as well as

quantum optics.Ilchenko and Matsko(2006)reviewed the

applications of dielectric whispering-gallery resonators to

optical devices such as filters,modulators,switches,sensors,

lasers,and frequency mixers,as well as to microwave

photonics.Next we will mention a few recent developments.

The extremely long lifetime of light in a WGM makes it

sensitive to the adsorption of a single molecule or virus onto

the cavity surface(V ollmer and Arnold,2008).Discrete

changes in the resonance frequency have been observed

due to the binding events of individual molecules or virons,

allowing real-time label-free detection(Armani et al.,2007;

V ollmer,Arnold,and Keng,2008).A further enhancement of

the sensitivity is realized using whispering-gallery microlasers

(He et al.,2011).The Purcell enhancement of the optical

density of states(DOS)by the WGM dramatically increases

light emission and scattering(Chang and Campillo,1996;Vahala,2004).Strong coupling of a single emitter(atom or quantum dot)to a WGM of a microdisk or a microtoroid has been achieved(Peter et al.,2005;Aoki et al.,2006;Srinivasan and Painter,2007),facilitating the studies of cavity quantum electrodynamics.The strong buildup of intracavity optical field greatly enhances nonlinear coupling of light with matter(Chang and Campillo,1996).For example,the Kerr-nonlinearity induced optical parametric oscillation in ultra-high Q WGMs produces optical frequency combs with high repetition rate,permitting applications in astronomy,micro-wave photonics,and telecommunications[see Kippenberg, Holzwarth,and Diddams(2011)and references therein].The whispering-gallery resonators also play a crucial role in the emerging field of cavity optomechanics(Kippenberg and Vahala,2008;Aspelmeyer,Kippenberg,and Marquardt, 2014).As light is reflected from the cavity boundary,it exerts radiation pressure on the cavity wall,inducing a mechanical flex of the cavity structure.The intense circulating field of a WGM produces strong radiation pressure and excites vibra-tional resonances.An interesting example is the acoustic WGMs excited via stimulated Brillouin scattering of optical WGMs in a microsphere(Carmon and Vahala,2007).The optomechanical coupling may lead to amplification or cooling of mechanical motion(Schliesser et al.,2006;Bahl et al., 2011;Bahl et al.,2012).

Nearly perfect confinement of light also implies the difficulty of coupling light into or out of a WGM. Consequently,whispering-gallery microlasers do not provide adequate output power despite a low lasing threshold. Moreover,the rotational symmetry of a sphere or a circular disk leads to isotropic emission to free space,making it impossible to collect all the output.This is a serious problem for certain applications, e.g.,single photon emitters.To increase the collection efficiency,a coupler such as a prism, a waveguide,or a fiber is often placed in close proximity to the cavity to extract the evanescent field(Matsko and Ilchenko, 2006).High precision is required in positioning the coupler with respect to the cavity boundary in order to obtain sufficient output while avoiding a dramatic reduction of the quality factor(Q spoiling).An alternative way of increasing the collection efficiency is to make the emission to free space directional by modifying the cavity boundary.Shortly after the first realization of semiconductor microdisk lasers,Levi et al. (1993)achieved directional output by introducing a tab or patterning a grating on the disk circumference.This kind of defect,however,also caused a serious Q spoiling.To minimize this problem,N?ckel and Stone(1997)proposed smooth deformation of cavity shape to break the rotational symmetry and achieve anisotropic emission.They called such cavities“asymmetric resonant cavities.”Even before their work,lasing in nonspherical liquid droplets was reported by Chang and co-workers(Qian et al.,1986).The laser emission was confined to the liquid-air interface,confirming the surface nature of the lasing https://www.sodocs.net/doc/9d3354504.html,ter,Gmachl et al.(1998)used semiconductor microcylinders with a deformed cross section as laser resonators and achieved high-power directional out-put.In the favorable directions of the far field,a power increase of up to3orders of magnitude over the conventional circularly symmetric lasers was obtained.Following these works,various shapes of deformed cavities were studied and

62Hui Cao and Jan Wiersig:Dielectric microcavities:Model systems for…Rev.Mod.Phys.,V ol.87,No.1,January–March2015

fabricated (some examples are shown in Fig.1),most of them either produce multidirectional output beams or have rela-tively low Q factor (Wiersig,Unterhinninghofen et al.,2011).The goal of combining unidirectional emission with high Q has been reached recently with a deformed microdisk whose boundary is described by the lima?on of Pascal (Wiersig and Hentschel,2008;Shinohara et al.,2009;Song,Fang et al.,2009;Yan et al.,2009;Yi,Kim,and Kim,2009;Albert et al.,2012).In addition,the reverse process,i.e.,free-space excitation of directional high-Q modes in a deformed cavity,is made efficient with an appropriate choice of the pump beam direction and impact position.This has been utilized for nonresonant optical pumping of microcavity lasers (Lee et al.,2007a ;Yang et al.,2008)and cryogenic cooling of opto-mechanical resonators (Park and Wang,2009).

From the fundamental physics perspective,deformed microcavities have become a model system for the studies of nonlinear dynamics,quantum chaos,and non-Hermitian physics (Stone,2001;Tureci et al.,2005).For a classical chaotic system,the particle ’s trajectory depends with expo-nential sensitivity on the initial conditions.A common example is a two-dimensional (2D)billiard with reflecting walls and negligible friction,in which a point mass moves in a straight line until it hits the boundary and bounces back with the angle of reflection equal to the angle of incidence.Chaotic motion can be induced by proper shaping of the billiard;physicists and mathematicians have learned a great deal about

chaotic motion and its onset by studying dynamical billiards with varied shapes.If the billiard becomes very small and the point mass is a quantum particle,the dynamics is governed by quantum mechanics.The quantum billiard has been a focus of theoretical study on the quantum manifestation of classical chaos,but it is difficult to realize experimentally.For example,quantum dots were investigated as chaotic quantum systems,but the interactions of electrons complicate the dynamics.A breakthrough came in the 1990s when the microwave cavities were used as quantum billiards,with the recognition of the formal analogy between the wave properties of quantum particles and classical electromagnetism.The electromagnetic fields of Maxwell ’s equations are the analog of the wave functions of the Schr?dinger equation;thus quantum chaos can be studied in the context of wave chaos for electromagnetic fields.The “classical limit ”corresponds to the limit of geo-metric optics where the wavelength is much smaller than the cavity size.Statistical analysis of the eigenfrequencies and eigenfunctions in 2D microwave cavities of varied shapes illustrated the differences between classical chaotic and non-chaotic systems.Reviews on microwave billiards can be found in the book of St?ckmann (2000)and the review of Richter (1999).Interesting effects studied in quantum billiards are,for example,dynamical localization and dynamical tunneling.Dynamical localization is the suppression of chaotic diffusion by destructive interference (Fishmann,Grempel,and Prange,1982;Frahm and Shepelyansky,1997).Dynamical tunneling is a generalization of conventional tunneling which allows pas-sage not only through an energy barrier but also through other kinds of dynamical barriers in phase space (Davis and Heller,1981).While most microwave billiards are closed systems with reflecting boundaries,dielectric cavities have open boundaries through which waves may escape.The openness makes the effective Hamiltonian of the system non-Hermitian.This leads to various interesting phenomena such as an increase of the quality factor at avoided resonance crossings (Persson et al.,2000),chirality of mode pairs (Wiersig,Kim,and Hentschel,2008)and exceptional points (Heiss,2000),which are branch point singularities of eigenvalues and eigenvectors of a non-Hermitian matrix.Therefore,deformed dielectric microcavities are ideal models for the fundamental studies of open chaotic systems (N?ckel and Stone,1997)and non-Hermitian quantum physics (Lee et al.,2009a ).

B.Scope

We review the experimental and theoretical studies of dielectric microcavities as open chaotic systems in the past two decades.There are previous short reviews on this topic (Stone,2001;Tureci et al.,2005;Xiao et al.,2010;Harayama and Shinohara,2011),which focus on either specific cavity shapes [e.g.,quadrupolar deformation in Stone (2001)and Tureci et al.(2005)]or certain features [e.g.,output direction-ality in Xiao et al.(2010)].Here we will cover a variety of cavity shapes,from smooth deformations of circle or sphere to polygons and cavities with boundary defects.We explain how the shape of the cavity boundary determines the intracavity ray dynamics and how light escapes out of the cavity.Depending on the type and degree of shape deformation,the intracavity ray dynamics can be regular,chaotic,or partial

chaotic.

FIG.1(color online).

A few examples of deformed dielectric

cavities.(a)Side-view and top-view scanning electron micro-scope (SEM)images of a quantum cascade laser made of a flattened quadrupolar-shaped GaAs cylinder.From Gmachl et al.,1998.(b)Optical image of a liquid microjet which traps light on one cross section by total internal reflection from the liquid-air interface.Courtesy of Kyungwon An,Seoul National University.(c)Optical image of a deformed fused-silica sphere with the long axis equal to 200μm.From Lacey and Wang,2001.(d)A microwave cavity made of a Teflon disk on a brass ground plate with dimensions 380×260mm 2.From Sch?fer,Kuhl,and St?ckmann,2006.

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Correspondingly,there is a diversity of cavity modes with rich

spatial structures, e.g.,whispering-gallery modes,chaotic

modes,scar modes(localized on unstable periodic ray

trajectories),etc.We discuss how these modes are formed

and explore their characteristics such as quality factor and far-

field pattern.

In this review we will not cover the nonlinear interactions of

light with gain materials or the interactions of multiple lasing

modes via the active media,which have been reviewed by

Harayama and Shinohara(2011).Instead we will focus on

linear wave optics and(nonlinear)ray optics in the following

three regimes:(i)the“classical”regime nkR>103(n is the

intracavity index of refraction k?2π=λ,andλis the vacuum

wavelength),where the ray dynamics rules;(ii)the“semi-

classical”regime nkR～102?103,where wave corrections emerge;and(iii)the“quantum”regime nkR～10?102,

where wave effects become dominant.We will explore the

ray-wave correspondence and emphasize the consequence of

cavity openness,e.g.,the nonorthogonality of cavity modes

which leads to excess quantum noise.In addition to stationary

cavities,we will review wave chaos in rotating microcavities

and explain how the rotation will affect the resonance

frequency,quality factor,and far-field pattern.

II.THEORETICAL MODEL AND EXPERIMENTAL TECHNIQUES

A.Mode equation and wave simulations

The aim of this section is to give the definition of

electromagnetic(optical)modes in passive dielectric cavities,

to introduce the corresponding mode equation with emphasis

on the deformed disk,and to review the numerical schemes to

solve it.

The geometry of a dielectric cavity is determined by the

spatial profile of the refractive index ne~rT.For a given profile

an electromagnetic mode is defined as a time-harmonic

solution of Maxwell’s equations with frequencyω,in the

same way as a quantum mechanical energy eigenfunction is a

solution of the Schr?dinger equation with fixed eigenenergy.

However,dielectric cavities are open systems as light leaks

out of the cavity.Hence,a mode in a dielectric cavity is a

quasibound state or quasinormal mode(Gamow,1928;Kapur

and Peierls,1938)decaying exponentially in time with life-

timeτ.This can be conveniently expressed by a complex-

valued frequencyω,where the imaginary part is related to the

lifetime viaτ??1=e2ImeωTTwith ImeωT<0.The quality

factor Q compares the lifetimeτwith the oscillation period of

the light T?2π=ReeωT,

Q?2πτ

T

??ReeωT

2ImeωT:e1T

The quasibound states are connected to the peak structure

in scattering spectra[see,e.g.,Landau(1996)]as illustrated in Fig.2.

To derive the mode equation one has to substitute the

complex representation of time-harmonic electric field ~Ee~r;tT?~Ee~rTexpe?iωtTand magnetic field~He~r;tT?

~He~rTexpe?iωtTinto Maxwell’s equations for nonmagnetic,dielectric materials in the absence of free charges and currents. As most dielectric cavities consist of one or several homo-geneous regions the refractive index ne~rTis often a piecewise constant function.In that case one arrives at

?2tn2e~rT

ω2

c2

~Ee~rT

~He~rT

?0e2T

provided that~r is not a boundary point.If~r is on a boundary separating two regions1and2with constant refractive indices n1and n2,the fields are subject to the continuity relations

~νe~rT×?~E1e~rT?~E2e~rT ?0;e3T

~νe~rT·?n21~E1e~rT?n22~E2e~rT ?0;e4T

~νe~rT×?~H1e~rT?~H2e~rT ?0;e5T

~νe~rT·?~H1e~rT?~H2e~rT ?0;e6T

where~νe~rTis the local normal vector.The appropriate boundary condition at infinity is the“outgoing wave”con-dition(Sommerfeld radiation condition).Together with this boundary condition,Eqs.(2)–(6)define the modes in a dielectric cavity.

In the case of a(deformed)disk cavity,the mode equation and the boundary conditions can be significantly simplified by replacing the disk by an infinite dielectric cylinder with an arbitrary cross section.The translation symmetry along the z axis of this idealized geometry allows the ansatz(Jackson, 1962;Tureci et al.,2005

)

FIG.2.Resonances and long-lived optical modes.The back panel shows the intensity scattered off a dielectric circular disk of radius R and refractive index n?1.5at170°with respect to the incoming plane wave with wave number k?ω=c,where c is the speed of light in vacuum.The scattering intensity shows narrow peaks(resonances)at the scaled complex frequenciesΩ?ωR=c?kR which are closest to the real axis.These are the frequencies of the long-lived modes;the short-lived modes contribute to broader peaks and the scattering background.From Tureci et al.,2005.

64Hui Cao and Jan Wiersig:Dielectric microcavities:Model systems for…Rev.Mod.Phys.,V ol.87,No.1,January–March2015

~E e~r T?~E

ex;y Texp eink z z Te7T

and analog for ~H

.The particular case of k z ?0corresponds to light propagation in the ex;y Tplane only.For this case the mode equation can be written as the scalar Helmholtz equation

?2tn 2ex;y Tω

2c

2 ψex;y T?0;

e8T

with ?now restricted to the x and y coordinates.The complex-valued wave function ψequals E z in the case of transverse magnetic (TM)polarization (H z ?0).For trans-verse electric (TE)polarization (E z ?0)the wave function ψequals H z .The other electric and magnetic field components can be computed from E z and H z ,respectively (Tureci et al.,2005).Admissible solutions of the mode equation in Eq.(8)are those which remain finite everywhere inside the cavity.The continuity relations (3)–(6)in the ex;y Tplane simplify to

ψ1?ψ2;?νψ1??νψ2TM polarization ;e9Tψ1?ψ2;

?νψ1n 21

??νψ2

n 22TE polarization :

e10T

?νis the normal derivative defined as ?ν?~ν

e~r T·?.Note the structural equivalence of mode equation (8)and the stationary Schr?dinger equation of a quantum particle in a piecewise constant potential.In polar coordinates er;φTthe outgoing wave condition in two dimensions for large r can be written as

ψ～ψout ?h eφ;k T

exp eikr T???r

p :e11T

Because of this boundary condition the solution of the mode equation has to decay in time.It is therefore a quasibound state with complex-valued frequency ω;Im eωT<0.Moreover,ψdiverges as exp ??Im ek Tr with Im ek T?Im eωT=c <0as the radial coordinate r tends to infinity.Hence,the quasibound state ψis strictly speaking not normalizable.The divergence does not affect the angular distribution of the emitted light h eφ;k T.

In practice a microdisk has a finite vertical extension which is usually taken into account within the effective-index approximation ;see,e.g.,Smotrova et al.(2005).The central assumption is that the separation ansatz (7)is still valid,ignoring a weak mixing of TM and TE polarizations.The resulting equation for the z direction leads to a series of quantized values of k z .Usually it is sufficient to consider the smallest one.Associated with this value of k z is the mode equation (8)and the continuity relations (9)and (10)with n replaced by an effective index of refraction n eff ?n ???????????????????????

1?ek z =k T2p inside the cavity and n eff ?1outside the cavity.

The effective-index approximation cannot be justified rigorously and no error estimates can be given.However,many publications have confirmed that this approach works well in terms of eigenfrequencies for different kinds of planar geometries such as photonic crystals (Qiu,2002),annular Bragg cavities (Scheuer et al.,2005),and microdisks

(Michael,2009).Even microcavities supported by a pedestal are described in sufficient accuracy (Lozenko et al.,2012).Moreover,the near-field pattern (Fang,Cao,and Solomon,2007;Redding,Ge,Song et al.,2012)and far-field pattern (Schwefel et al.,2004;Shinohara et al.,2009)of deformed microdisks computed in the effective-index approximation agree with experimental data.The validity of the effective-index approximation for dielectric disks was questioned recently by Bittner et al.(2009).However,the observed deviations of typically below 1%can be considered as being small,keeping in mind that the bulk refractive index is often known with less accuracy.

As most problems in electrodynamics do not allow for an analytical treatment,much effort has been put into the development of numerical schemes.The most prominent one is the finite-difference time-domain (FDTD)method (Taflove and Hagness,2000)which is perfectly suited to simulate the dynamics of light propagation in complex environments.It can also be used to determine light confine-ment in dielectric cavities (Kim et al.,2004;Fang,Yamilov,and Cao,2005;Srinivasan et al.,2006;Fang,Cao,and Solomon,2007),but for long-lived modes in the semiclassical regime (short-wavelength regime),i.e.,when the wavelength is small compared to the characteristic length scales of the system,it requires immense computational power.In this case it is more convenient to work in the frequency domain.This is,in particular,advantageous if the frequency dependence of the refractive index has to be included.For the frequency domain several approaches can be applied to quasi-two-dimensional geometries,such as the finite-difference frequency-domain (FDFD)method (Shainline et al.,2009),wave-matching method (N?ckel and Stone,1995;Hentschel and Richter,2002),internal scattering quantization approach (Tureci et al.,2005),volume element methods (Martin et al.,1999),boundary element methods (Wiersig,2003a ;Zou et al.,2011),and related methods based on boundary integral equations (Boriskina et al.,2004).The FDFD and the volume element methods are restricted to small structures because of the limited computational power that is available today.The wave-matching method based on the expansion of the wave function into a basis of Bessel and Hankel functions is more efficient and can be applied to large structures.However,usually the expansion is around a single point in position space (single pole method).In this case the method relies on the Rayleigh hypothesis which fails for strongly deformed disks (van den Berg and Fokkema,1979).This same is true for the highly efficient internal scattering quantization approach.No such problem exists for the boundary element methods which are also efficient (Zou et al.,2011),in particular,in combination with the harmonic inversion technique (Wiersig and Main,2008).

B.Ray model

The short-wavelength limit of wave optics is geometrical (ray)optics.In the semiclassical regime much understanding about the wave dynamics in dielectric cavities can be gained by studying the dynamics of rays inside the given structure.In the following we describe the basic ray model introduced by N?ckel,Stone,and Chang (1994),Mekis et al.(1995),N?ckel

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and Stone (1995,1997),and N?ckel et al.(1996),which is nowadays commonly used for dielectric cavities.

First we address ray dynamics in a closed cavity with perfectly reflecting walls.This problem is mathematically equivalent to a classical particle moving freely along straight lines in a two-dimensional planar domain (billiard)with specular reflections at the boundary.Figure 3(a)depicts the elementary aspects of the billiard dynamics.According to the law of reflection,the incident ray and the reflected ray make

the same angle χwith respect to the inward normal vector ~ν

at the boundary point of the reflection.Clearly,the shape of the boundary determines the dynamical properties of the billiard.The real-space trajectories in a typical billiard can be very complicated,so it is more appropriate to study the trajectories in phase space.The phase space of a 2D billiard is four dimensional consisting of 2spatial degrees of freedom and two conjugate momenta.However,due to conservation of the particle ’s energy,the motion actually takes place on a three-dimensional surface.A further reduction of dimensionality can be achieved by the Poincarésurface of section (SOS)(Lichtenberg and Lieberman,1992).For billiards,it is a plot of the intersection points of a set of trajectories with the cavity ’s boundary.This is illustrated in Fig.3.When a ray or particle is reflected at the cavity ’s boundary,its position in terms of the arclength coordinate along the boundary s and the quantity p ?sin χare recorded.We follow here the convention that sin χ>0means counterclockwise (CCW)rotation and sin χ<0means clockwise (CW)rotation;cf.Figs.3(a)and 3(b).The quantity p ∈??1;1 can be interpreted as the tangential component of the normalized momentum with respect to the boundary curve at a given position s ∈?0;s max .The coordinate s and its canonical conjugate momentum p are called Birkhoff coordinates.This pair is the natural set of coordinates since the mapping from bounce to bounce es i ;p i T→es i t1;p i t1Tis area pre-serving (Birkhoff,1927);see also Berry (1981).

In the special case of the circular billiard,the angle of incidence χis not changed by the billiard mapping.Hence,rays are confined to two-dimensional surfaces of constant sin χand constant energy.The topology of such invariant

surfaces is that of a two-dimensional torus (Arnol ’d,1978).In the SOS these tori are lines sin χ?const.The dynamics on these lines can be periodic or quasiperiodic.

A more complicated example,the mushroom billiard,is shown in Fig.4.This exotic class of geometries has attracted much attention because the phase space of such a system is sharply divided into regular and chaotic parts (Bunimovich,2001).In a regular region,the dynamics is similar to the case of the circular billiard with χbeing a constant of motion.In contrast,the dynamics in a chaotic region exhibit an expo-nential sensitivity on the initial conditions (Lichtenberg and Lieberman,1992).Moreover,the dynamics is ergodic,i.e.,a single trajectory eventually comes arbitrarily close to any point in the given chaotic region and as a result it covers a finite fraction of the SOS.The phase space of such a partially chaotic system is called “mixed phase space.”A mushroom-shaped optical microcavity has been studied by Andreasen et al.(2009).

In a generic billiard,the coexistence of regular and chaotic dynamics in the mixed phase space is much more involved.Regular regions (called “islands ”)embedded in a chaotic region (which is usually referred to as “sea ”to complete the analogy)are surrounded by a chain of smaller islands which in

s/s

max

s i n χ

(b)FIG.3.Ray dynamics in a billiard;s is the arclength coordinate

along the boundary of the cavity and χis the angle of incidence

with respect to the boundary normal ~ν

.(a)The solid line 1→2→3is a counterclockwise traveling ray in real space and the dashed line 10→20→30is a clockwise traveling ray.(b)The same dynamics in the Poincarésurface of section.The coordinate s is normalized to the total circumference of the boundary s max .In this representation the angle χis conventionally defined negative for clockwise traveling

rays.

FIG.4(color online).

Example of a simple mixed phase space.

Numerically computed Poincarésurface of section for a desym-metrized mushroom billiard showing regular and chaotic regions in phase space;the size of the billiard is scaled such that s max ?π=2.From B?cker,Ketzmerick,L?ck,Robnik et al.,2008

.

FIG.5.Poincarésurface of section of a quadrupole billiard (12)

for ε?0.072.The boundary is here parametrized by the polar angle φ.The direction φ?0corresponds to the right part of the horizontal axis in the three real-space plots.From Tureci et al.,2002.

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Rev.Mod.Phys.,V ol.87,No.1,January –March 2015

turn are surrounded by even smaller islands and so on,leading

to an infinite hierarchy of islands.Figure5shows as an

example for a generic case of the SOS of a quadrupole billiard

(N?ckel et al.,1996;Tureci et al.,2002)with boundary given

in polar coordinates by

reφT?Re1tεcos2φT:e12TThe ray dynamics is solely determined by the deformation

parameterε≥0and independent on the average radius R.

Only trajectories with positiveχare shown in Fig.5.The SOS

is symmetric with respect toχ→?χdue to the time-reversal

symmetry of the billiard system.It can be clearly seen that

the ray dynamics can be regular or chaotic depending on the

initial conditions.Figures5(a)–5(c)depict a quasiperiodic

whispering-gallery ray,a periodic ray from the center of an

island,and a chaotic ray.

A ray in a billiard system never leaves the interior of the

domain enclosed by the boundary.In a dielectric cavity,

however,a ray can leave the cavity via refractive escape.

Figure6illustrates the fact that the ray partially leaves the

cavity when the angle of incidenceχis smaller than the critical

angleχc for total internal reflection;sinχc?1=n assuming that air surrounds the cavity.Hence,a dielectric cavity

can be considered as an“open billiard”(N?ckel and Stone,

1995,1997).

In the SOS of such an open billiard the region between the “critical lines”sinχ??1=n is called the“leaky”region;see Fig.7(a).The size of the leaky region increases with decreas-ing refractive index n.Chaotic systems with a leaky region in phase space have been given a lot of attention in recent years; for a review,see Altmann,Portela,and Tél(2013).When a ray inside the dielectric cavity hits the leaky region then,in the crudest approximation,the ray is lost for the internal ray dynamics and the transmitted ray contributes to the far-field intensity pattern feφTaccording to Snell’s law n sinχ?sinη(Jackson,1962);cf.Fig.6.A more sophisticated scheme is to account for the partial leakage in the leaky region by assigning an initial intensity I to a given ray.Whenever it hits the cavity’s boundary the intensity is reduced according to

I i→I it1?R TM;TEesinχiTI i;e13Twhere R TM;TEesinχT≤1is the polarization-dependent reflec-tion coefficient(Schwefel et al.,2004).The simplest choice for R TM;TE is according to Fresnel’s laws for a planar dielectric interface(Jackson,1962);see Fig.7(b).Note that in the case of TE polarization the reflection coefficient goes down to zero at Brewster’s angle.

Tracing a single ray is not sufficient for the computation of the far-field intensity pattern feφT.What is needed is a properly chosen ensemble of rays which establishes a link to the modes of the dielectric cavity.There is no general recipe for constructing these ensembles as it depends on the geometry of the cavity,so we postpone this issue to Sec.IV. The reflection coefficient R TM;TE can be used to incorporate tunneling into the ray model(N?ckel and Stone,1997). Tunneling is the main decay channel of the(weakly deformed) circular cavity as refractive escape is forbidden due to con-servation of the angle of incidence.In this case,the tunneling can be related to a modified reflection coefficient at curved dielectric interfaces(Hentschel and Schomerus,2002).Other extensions of the ray model are discussed in Sec.VIII. Ray tracing has also been performed in deformed dielectric spheres.In the special case of an axisymmetric deformed sphere the conservation of angular momentum reduces this problem effectively to a two-dimensional billiard with cen-trifugal potential.Such a case has been studied by N?ckel and Stone(1995).Ray dynamics in a nonaxisymmetric deformed dielectric sphere has been analyzed by Lacey and Wang (2001).Here the SOS is four dimensional;therefore the ray trajectories have to be laboriously visualized and analyzed in a number of different projections.

C.Husimi functions

In this section we discuss a powerful tool for the compari-son of ray and wave properties,the Husimi function for dielectric cavities.The Husimi function is one of the simplest quasiprobability distributions of a quantum state in phase space(Husimi,1940).It is obtained from the overlap of the wave function with a coherent state that represents a minimal-uncertainty wave packet.

FIG.6.Refractive escape from a dielectric cavity with refractive index n>1.A ray with intensity I is split into a reflected ray with intensity R TM;TE I and a transmitted ray with intensity e1?R TM;TETI.~νis the outward normal vector.The angle of the reflected rayηis related to the angle of the incident rayχby Snell’s law.The emission direction can be described by the polar angleφwhich equals asymptotically the angle?between the x axis and the emitted

ray.

(b)

FIG.7.The leaky region in phase space of a dielectric cavity.

(a)Poincarésurface of section with leaky region j sinχj≤1=n in

which the condition for the total internal reflection is not met.

(b)Reflection coefficient R TM;TEesinχTfor a planar dielectric

interface with the incident plane wave coming from the high-

index medium(n?2).The low-index medium is air with n?1. Hui Cao and Jan Wiersig:Dielectric microcavities:Model systems for (67)

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The Husimi function was adapted to billiards by Crespi,Perez,and Chang(1993).It can be considered as a representation of the quantum state on the SOS at the boundary of the billiard.This approach was applied also quite exten-sively to dielectric cavities even though the boundary conditions(9)and(10)are different.For TM boundary conditions(9)Hentschel,Schomerus,and Schubert(2003) have derived Husimi functions by using a saddle point approximation valid in the semiclassical regime.In total four different Husimi functions have been obtained,two for incident and emerging waves inside the cavity and two for those outside the cavity.The Husimi functions for the internal waves have been widely used,so we focus on them in the following.The incident and the emerging Husimi functions are

H inceemTes;pT?nk

2π

F h

ψ

es;pT?etTi

k F

h?ψes;pT

2e14T

with weighting factor F?

?????????????????????

n

?????????????

1?p2

p

q

.The function

h ges;pT?Z

s max

ds0ges0Tξes0;s;pTe15T

is the overlap of the wave function(g?ψ)or its normal derivative(g??νψ)on the cavity’s boundary with the minimal-uncertainty wave packet

ξes0;s;pT?eσπT?1=4

X∞

l??∞exp

h

?

es0ts max l?sT2

2σ

?inkpes0ts max lTi

:e16T

The wave packetξes0;s;pTis centered aroundes;pT.The relative uncertainty in s and p can be controlled with the parameterσ.

Figure8shows as an example the Husimi function of a mode in an annular cavity[a microdisk with an internal disk-shaped area of different refractive index;see, e.g., Hentschel and Richter(2002),Schomerus,Wiersig,and Hentschel(2004),Wiersig and Hentschel(2006),and Preu et al.(2013)]superimposed on the SOS of the outer boundary.It can be seen that the mode is localized in the chaotic region of phase space.The contribution in the leaky region deter-mines the emission properties.

There is an independently developed approach to visualize the mode in the leaky region of the SOS,the so-called intensity flux distribution which is based on a coarse-grained Poynting vector(Shinohara and Harayama,2007).It has been proven that the flux distribution coincides with the difference between the incident and the emerging Husimi function (Shinohara and Harayama,2011).

D.Cavity fabrication

In this section,we introduce various types of dielectric microcavities and describe briefly how they are fabricated. For more information,we refer the interested reader to several reviews(Ilchenko and Matsko,2006;Chiasera et al.,2010; Xiao et al.,2010;Righini et al.,2011;He,?zdemir,and Yang,2013).

1.Liquid droplets and microjets

Liquid droplets are3D microcavities formed by surface tension forces.In the early days they were generated by Berglund-Liu piezoelectric vibrating-orifice aerosol genera-tors(Qian et al.,1986).As shown in Fig.9,

immediately FIG.8(color online).Husimi function in phase space.(a)Poincaré

surface of section(dots,horizontal lines with j sinχj>0.6),critical

lines(horizontal lines with j sinχj≈0.3),and emerging Husimi

function(shaded regions)of a mode in an annular cavity.(b)Mode

in real

space.

FIG.9(color online).A series of photographs of laser emission

from the droplet stream within the first few millimeters of the

vibrating orifice.(Left)The upper portion of the stream showing

the periodically perturbed,continuously connected liquid cylin-

der and the development of separate,highly distorted droplets.

(Right)The lower portion of the stream,showing the transition

from oscillating prolate-to-oblate spheroids to a stream of

monodisperse,equally spaced spherical droplets.From Qian

et al.,1986.

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below the vibrating orifice is a periodically perturbed,con-tinuously connected liquid cylinder,which develops to sep-arate highly distorted droplets that oscillate between prolate spheroids and oblate spheroids.Farther away,the stream transits to monodisperse,equally spaced spherical droplets. The radii of the droplets,which depend on the size of the orifice aperture,are typically a few tens of microns.Instead of flying in air,the droplets may also be suspended in liquids or placed on solid substrates(He,?zdemir,and Yang,2013). The total internal reflection of light at the liquid-air inter-face leads to the formation of WGMs in the droplet.Despite a low refractive index contrast(commonly used liquids have refractive index between1.3and1.4),the droplet has a very smooth surface,which minimizes the scattering loss.Light emitting or amplifying materials can be easily incorporated into the liquid droplets,e.g.,by adding dye molecules or quantum dots to a solution before creating https://www.sodocs.net/doc/9d3354504.html,sing was realized in dye-doped liquid droplets with optical exci-tation of dye molecules(Tzeng et al.,1984).Microsized liquid droplets were also used for cavity enhanced spectroscopy (Symes,Sayer,and Reid,2004).Liquid microjet can create a

continuous and stationary column of liquid with a precise control of the hydrodynamics of the jet[see Fig.1(b)].Light propagating perpendicular to the axis of the column may be trapped in one cross section by total internal reflection at the liquid-air interface.Thus the microjet was used in the study of 2D microcavities(Moon et al.,1997).Moreover,the cavity size and shape can be varied continuously along the stream. By deforming the orifice aperture(Yang et al.,2006)or applying a lateral gas flow(Moon et al.,1997),the cross section of the microjet column is distorted from the circle. The exact surface profile may be deduced from the optical diffraction pattern(Moon et al.,2008).The typical dimension of the cross section is a few tens of microns,and the surface roughness induced scattering is as weak as that in the droplets.

2.Solid microspheres and microtoroids

Solid microspheres have been produced from a large variety of materials,organic and inorganic,amorphous and crystal-line.The widely used amorphous microspheres are fabricated with two techniques based on a melting process and sol-gel chemistry.A detailed description of these techniques is given in the reviews of Chiasera et al.(2010)and Righini et al. (2011).As an example,we next describe a common and effective method of making a glass microsphere by melting the end of a glass fiber.The heating source can be an oxygen-butane torch,a high-power CO2laser,or an electric arc as in a commercial fiber splicer.Upon heating the distal tip of a silica fiber,the glass reflows to form a spherical volume under the influence of surface tension.Because of high viscosity of silica,the reflowed structure becomes extremely uniform and highly spherical(eccentricities<3%).The sphere diameter varies from10to100μm,depending on the original diameter of the fiber tip.Smaller spheres are produced by first tapering the fiber to reduce the diameter of the tip.The silica micro-sphere remains attached to the fiber stem from which it was formed,making it easy to handle[Fig.10(a)].Typically one excites the WGMs that lie in the equatorial plane and have very small overlap with the stem;thus the effect of the stem on the WGMs is negligible.The surface roughness is extremely low,on the order of1nm;thus high quality factors can be reached for the WGMs.

Asymmetric microspheres[Fig.1(c)]have also been fabricated by fusing two silica spheres together with a CO2 laser beam(Lacey and Wang,2001).Alternatively,a single spherical microsphere can be deformed by reheating with one or two laser beams incident on different sides(Xiao et al., 2007,2009).Microbottle resonators were made from optical fibers in a two-step heat-and-pull process by sequentially tapering the fiber in two adjacent locations to form the bottle (Poellinger et al.,2009).To facilitate sensing applications, liquid core resonators were fabricated by blowing a silica microbubble(Sumetsky,Dulashko,and Windeler,2010). The process is similar to the traditional glass blowing,a gas pressure is applied while a glass capillary is heated.

To achieve on-chip integration,silica microtoroid cavities were fabricated on silicon wafers(Armani et al.,2003).First, silica microdisks are made by photolithography and dry etching.Then the underneath silicon is selectively etched to form a post that supports the silica disk.Finally,a CO2laser beam irradiates a silica disk to melt the silica along the rim,and a toroidlike structure is formed by surface tension[Fig.10(b)]. The reflow of silica produces a nearly atomic-scale surface finish,greatly enhancing the Q factor.The dimension of the toroid is determined by that of the initial disk and the reflow process.The toroid diameter is typically between20and 100μm,and the toroid thickness is a few microns.

To make microspheres and microtoroids optically active, various approaches have been developed,such as fabricating the resonators from materials doped with active media,coat-ing the resonators with light emitters,doping the resonators with gain material by ion implantation,etc.More detail about these approaches can be found in the review by He,?zdemir, and Yang(2013).

Single crystals have also been used to make spherical and toroidal cavities,and they are expected to have less loss and stronger nonlinear response than amorphous

materials FIG.10.(a)Scanning electron micrograph(SEM)of a silica microsphere at the end of the preform wire.Its diameter is 70μm.No surface defect was observed on a30nm scale.From Collot et al.,1993.(b)SEM of a silica microtoroid.From Armani et al.,2003.

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(Ilchenko and Matsko,2006).The fabrication of crystalline spheres and toroids involves mechanical cutting,drilling,and polishing.The typical diameter exceeds1mm.It is extremely difficult,if not impossible,to make microscale resonators with crystalline materials.

3.Microdisks and micropillars

Well-developed microfabrication and nanofabrication tech-nologies,such as photolithography,electron-beam lithogra-phy,chemical and physical etching,have been adopted to make microdisk and microcylinder resonators,allowing a precise control of the cavity shape and size.The commonly used materials are semiconductor and polymer.The latter can be either a passive polymer doped or coated with active material,e.g.,dye-doped poly(methyl methacrylate) (PMMA),or an active polymer such as poly(para-phenylene vinylene)(PPV)or poly(para-phenylene)(PPP).The polymer is first dissolved in a solvent and then spin coated on a glass substrate.The layer thickness is a few hundred nanometers to 1μm,depending on the spin speed and the concentration of the solution.To guide light in the polymer layer,its refractive index must be higher than that of the substrate.In the case the substrate has a large refractive index,a low-index material is deposited on the substrate first and then the polymer is spin coated on top of it(Chern et al.,2004;Lebental et al.,2006). The disk patterns are written on a resist layer covering the polymer by photolithography or e-beam lithography and then transferred to the polymer layer via wet or dry etching.

Alternatively,microdisks may be made by direct photolithog-raphy or e-beam lithography with polymers or monomers that are active to UV light or electron beam(Fang and Cao,2007; Djellali et al.,2009).An additional bake may follow to reflow the polymer and smooth the disk edges.Figure11(a)shows a dichloromethane(DCM)-doped PMMA disk of spiral shape. Semiconductor microdisks and microcylinders have been made with GaAs/AlGaAs,InP/InGaAsP,Si,and GaN/InGaN. They are single crystals grown by molecular beam epitaxy (MBE)or metal-organic chemical vapor deposition (MOCVD).The semiconductor disks have large refractive index contrast with the surrounding air,leading to strong light confinement even in small disks.It enables lasing in disks that are merely a few microns or even submicrons in diameter (Zhang,Yang,Liu et al.,2007;Song,Cao et al.,2009).The disk thickness is typically a few hundred nanometers.Gain materials such as quantum wells(McCall et al.,1992), quantum dots(Cao et al.,2000),or nanocrystals(Liu et al., 2004)are embedded in the disk layer or deposited on top of the disk.To isolate a disk from the high-index substrate,selective etching of the substrate forms a pedestal underneath the disk (Liu et al.,2004).If this is not possible,e.g.,the substrate is made of the same material as the disk,another semiconductor layer is grown between the disk and the substrate,and it is selectively etched to form a pedestal(McCall et al.,1992),as shown in Fig.11(b).The WGMs that are located near the edge of a disk are barely affected by the presence of the pedestal. Alternatively,a lower index semiconductor layer is sandwiched between the higher index disk layer and the substrate,enabling index guiding of light in the disk layer(Gmachl et al.,1998; Fukushima and Harayama,2004).

In addition to the planar cavities,vertical cavities can be formed by stacking two Bragg mirrors.Standard lithography and etching have been used to make micropillars that are a few microns in height.Quantum wells or dots are embedded in the cavity.Figure11(d)shows a micropillar with the lima?on-shaped cross section(Albert et al.,2012).Since the spacing of the two Bragg mirrors is on the order of one wavelength,only one longitudinal mode of the cavity falls in the emission spectra.However,if the cross section of the cavity is large(a few tens of microns in diameter),multiple transverse modes exist,and they may produce complex field patterns(Huang et al.,2002;Gensty et al.,2005).

One advantage of the semiconductor microdisk or micro-cylinder lasers is that they can be pumped electrically with current injection[Fig.11(c)],while previously discussed microcavities are optically pumped by another laser. However,their sidewall roughness,which is created during the fabrication process,is much larger than that of the surface-tension-formed microcavities.Since the melting temperature of GaAs/AlGaAs,InP/InGaAsP,or GaN/InGaN is very high, at which the quantum wells or quantum dots would be damaged,one cannot reflow the semiconductor to remove the sidewall roughness.Light scattering due to sidewall roughness reduces the quality factor,making the lasing threshold higher.One way of reducing the scattering loss is to make the sidewall wedge shaped to push the optical modes away from the rough lithographic edge(Kippenberg et al., 2006).Another solution is to replace the sidewall with the atomic-flat facets.This can be done with the bottom-up approach,e.g.,which makes crystalline microdisks or micro-needles(Zhu et al.,2009;Gargas et al.,2010).The

cross FIG.11.(a)Optical microscope image of a spiral microcavity made of a DCM-doped polysmethylmethacrylated film.From Ben-Messaoud and Zyss,2005.(b)Side view SEM of a InGaAsP microdisk on top of an InP pedestal.The disk diameter is3μm. From McCall et al.,1992.(c)SEM of a GaAs/AlGaAs micro-stadium laser with a metal electrode on the top for current injection.From Fukushima and Harayama,2004.(d)SEM of a lima?on-shaped micropillar with a vertical cavity formed by two Bragg mirrors.Image courtesy of S.Reitzenstein,TU Berlin.

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sections of these cavities are polygons,so light may diffract from the sharp corners(Wiersig,2003b).

E.Optical characterization

In this section,we describe the experimental techniques used to probe the microcavity resonances,e.g.,their frequen-cies,quality factors,intracavity intensity distributions,and far-field patterns.The characterization has been done on both passive cavities and active cavities that contain light emitting or amplifying media.

1.Passive cavities

To probe the resonances of a passive cavity,light must be efficiently coupled into the cavity.Several schemes have been developed.In terms of free-space coupling,a tightly focused Gaussian light beam passing outside but near a spherical cavity preferentially excites specific WGMs,depending on its distance from the cavity center(Lin et al.,1998).For a deformed cavity,the modes with directional output can be efficiently excited by external beams in reversed directions.In addition to free-space coupling,cavity resonances may be excited with evanescent field couplers such as prisms,wave-guides,or tapered fibers.Typically,the input light is swept in frequency and a dip in the transmission spectrum gives the resonance frequency.The spectral width of the dip reflects the quality factor of the resonance.By varying the coupling position and/or direction,the intracavity mode profile may be inferred or confirmed(Gao et al.,2007).

2.Active cavities

With light emitters embedded inside the microcavity or coated on its surface,the cavity resonances,especially the ones with high quality factors,appear as peaks in the spontaneous emission spectrum.The position and width of each peak tell the frequency and quality factor of the corresponding resonance.At a high pump level,stimulated emission and lasing oscillation may occur.

Pumping can be either electrical or optical.The optical pumping is usually nonresonant,i.e.,the frequency of the pump light differs from that of the emission.Thus the pump light and emitted light couple to different cavity modes.The pump beam may be incident onto the cavity from free space or coupled evanescently.To enhance the pump efficiency,ray and wave chaos were used to trap the pump light inside deformed microcavities(Lee et al.,2007a).

A broadband emission will couple to multiple cavity modes.The emission is collected either in free space or via an evanescent field coupler.Near-field imaging of light scattered at the cavity boundary reveals the locations where most emission escapes from the cavity.The directions of the emission can be measured by placing a photodetector in the far-field zone and moving it around the cavity.A bandpass filter may be used to select one particular mode.To measure the spectra of emission into different directions,the detector is replaced by a fiber or fiber bundle connected to a spectrom-eter.The angle-resolved emission spectra give the far-field patterns of individual modes that appear in the spectra. Alternatively,a large ring may be fabricated around a microdisk,and the in-plane emission from the disk reaches the ring and is scattered vertically.By imaging the intensity of scattered light along the ring from above the sample,one may infer the output directions(Song,Fang et al.,2009).A simultaneous measurement of the emission direction and location on the cavity boundary is possible by imaging the intensity profile from the sidewall of a micropillar as viewed from different angles(Schwefel et al.,2004).

Finally we briefly discuss the microwave dielectric cavities. Most optical processes in passive microcavities can be studied in microwave cavities with higher precision(Richter,1999; Sch?fer,Kuhl,and St?ckmann,2006;Bittner et al.,2009; Kuhl,Sch?fer,and St?ckmann,2011).The much longer wavelength makes the microwave cavity much larger[see Fig.1(d)]and thus much easier to fabricate.Moreover,both the amplitude and phase of the electromagnetic field can be readily measured at the microwave frequency by using antennas,which are difficult to realize in optics.

III.OVERVIEW OF NONDEFORMED DIELECTRIC MICROCAVITIES

This section briefly reviews the properties of nondeformed WGM cavities.

A.Whispering-gallery modes

For a circular microdisk with refractive index n and radius R the solution of the mode equation(8)with outgoing wave condition(11)and with the requirement for a finite wave function inside the cavity is

ψer;φT?

a

m

J menkrTe imφif r≤R;

b m He1TmekrTe imφotherwise;

e17T

where m∈Z is the azimuthal mode number,J m and He1Tm are the m th order Bessel function and first-kind Hankel function. The boundary conditions(9)and(10)lead to the“quantization condition”

S mekRT?0e18T

with

S mekrT?

n

μ

J0menkrT

J menkrT

?H

e1T0

m

ekrT

H mekrT

;e19T

whereμ?1e?n2Tfor TM(TE)polarization,and the 0denotes the first derivative with respect to the argument. For given m Eq.(18)is to be solved numerically for the discrete values of k?k ml∈C labeled by the radial mode number l∈N.

Modes with azimuthal mode number m≠0are twofold degenerate.The mode with m>0(m<0)is a counterclock-wise(clockwise)traveling wave in the azimuthal direction. Linear superpositions of these two modes are also modes of the cavity.A particular superposition is standing waves in the azimuthal direction with sin mφand cos mφdependence with m restricted to positive integers.

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Figure 12shows exemplarily two standing-wave modes in the circular microdisk.Modes with small radial mode number l are called whispering-gallery modes.Beside the internal modes (Feshbach resonances)shown in Fig.12there are also external modes (shape resonances,above-barrier resonances)with a much lower Q factor which are located mostly outside the cavity (N?ckel,1997;Bogomolny,Dubertrand,and Schmit,2008;Dubertrand et al.,2008;Dettmann et al.,2009a ).Only the internal modes become bound states in the small opening limit n →∞.

A straightforward calculation based on the stationary phase approximation shows that the Husimi function (14)of a mode in a circular microdisk (17)is strongly localized around

sin χ?

m nkR

;e20T

where kR is here understood as the real part of kR .This relation between ray and wave properties of the dielectric disk was first derived by N?ckel and Stone (1995)using the eikonal approximation.

For microspheres analytical solutions of the mode equation are available in terms of vectorial spherical harmonics;see,e.g.,the review of Chiasera et al.(2010)on spherical WGM microresonators.

B.Optical losses and quality factors

In the ideal situation the quality factor (1)of an optical mode is determined solely by its radiation losses through the curved boundary of the cavity,Q ?Q rad .Asymptotic for-mulas for these losses are given by McCall et al.(1992),N?ckel (1997),Apalkov and Raikh (2004),and Dubertrand et al.(2008)for microdisks and by Chiasera et al.(2010)for microspheres.In practice,however,also absorption and Rayleigh scattering in the bulk material as well as scattering upon rough surfaces or contaminants contribute to the decay of light.According to Slusher et al.(1993)the total quality factor can be written as

1Q total

?

1Q rad t1Q mat t1Q surf

:e21T

The quantity Q mat is related to the material absorption coefficient αby

Q mat ?

2πn

λα

;e22T

where the dispersion of the refractive index n is ignored;λis the vacuum wavelength.The coefficient αcan also describe Rayleigh scattering in the bulk material which,however,can be significantly altered by the modified optical density of states in the presence of the microcavity (Gorodetsky,Pryamikov,and Ilchenko,2000).These internal losses in the material can be alternatively taken into account by the mode equation using a complex-valued refractive index ~n

?n ti λα=4π.Also the surface roughness Q surf can be directly modeled by the mode equation provided that fluctua-tions in the boundary function ρ?ρeφTare taken explicitly into account;see,e.g.,Rahachou and Zozoulenko (2003).The maximal total Q factor achievable in microcavities depends on the size and refractive index (determining the radiation losses)and the quality of the material (determining the internal losses and surface scattering).For semiconductor microdisks the highest Q factors can be achieved for “large ”silicon cavities.Here the Q ranges from 3×106to 6×107with disk radius of 20?60μm (Borselli,Johnson,and Painter,2005;Kippenberg et al.,2006;Soltani,Yegnanarayanan,and Adibi,2007).For AlGaAs disks with much smaller radius 2.25μm the quality factor is lower but can be still high Q ≈3.5×105(Srinivasan et al.,2005).For a GaAs disk with a small radius 361nm the quality factor is still around 4000(Song,Cao et al.,2009).For AlN/AlGaN microdisks of radii 2?5μm the quality factor ranges from 5000to 7300(Mexis et al.,2011).For polymer-based microdisks a quality factor around 6000has been reported (Lozenko et al.,2012).

The Q factors in microspheres are usually larger.For silica microspheres the record Q is around 8×109(Gorodetsky,Savchenkov,and Ilchenko,1996).For spherical droplets made of rhodamine 6G in water solution a quality factor of about 108has been measured (Lin,1992).In microtoroid cavities the quality factors can be also very high,e.g.,108for a toroid made of silica (Armani et al.,2003).

Optical gain may be introduced to microcavities to com-pensate the optical losses mentioned earlier.Coherent ampli-fication of light via stimulated emission effectively increases the photon lifetime and reduces the mode linewidth.When optical amplification fully compensates the total loss,self-sustained oscillation occurs in the cavity,which corresponds to the onset of lasing action.

https://www.sodocs.net/doc/9d3354504.html,sing in whispering-gallery cavities

Because of the high quality factors and the small mode volumes,WGM microcavities are excellent resonators for low threshold and narrow linewidth lasers.This section briefly reviews the pioneering works on WGM microlasers.More details can be found in a recent review on this topic by He,?zdemir,and Yang (2013).

Lasing in whispering-gallery cavities was first observed in spheres with diameter between 1and 2mm made of CaF 2doped with Sm t2(Garrett,Kaiser,and Bond,1961).Later,stimulated emission in liquid ethanol droplets

containing

FIG.12(color online).Numerically computed standing-wave

modes in a dielectric microdisk;n ?3.3(GaAs),TM polariza-tion.(a)Radial mode number l ?1and azimuthal mode number m ?19,scaled frequency Ω?kR ?7.02783?i 2.99188×10?13;(b)l ?3,m ?12,and Ω?7.0175?i 6.29188×10?5.

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rhodamine6G dye as active medium was observed by Tzeng et al.(1984).Sandoghdar et al.(1996)reported on the realization of a WGM laser based on neodymium-doped silica microspheres.A Raman laser with ultralow threshold based on a microsphere was fabricated by Spillane, Kippenberg,and Vahala(2002).

Lasing in microdisks was first observed in semiconductor disks made of InP/InGaAsP with InGaAs quantum wells as active medium(McCall et al.,1992).Stimulated emission from microdisks with InAs quantum dots as active medium has been reported by Cao et al.(2000).Liu et al.(2004)fabricated ultraviolet microdisk lasers on silicon substrates.The first room-temperature continuous-wave lasing in GaN/InGaN microdisks was observed by Tamboli et al.(2007).Lasing in submicron disks was achieved by Zhang,Yang,Hong et al. (2007),Shainline et al.(2009),and Song,Cao et al.,2009.The first quantum cascade microdisk laser was demonstrated by Faist et al.(1996).Kuwata-Gonokami et al.(1995)achieved laser emission from polymer microdisk lasers. Microlasers based on microtoroids covered by erbium-doped sol-gel films was fabricated by Yang,Armani,and Vahala(2003).WGM lasing in electrically driven quantum-dot micropillars was achieved by Albert et al.(2010).

D.Evanescent field coupling

To couple light into and out of a WGM,an evanescent field coupler is often used.It provides efficient energy transfer through the evanescent field of a guided wave in a fiber or channel waveguide or the evanescent wave produced by total internal reflection of light at the surface of a dielectric prism or side-polished fiber(Matsko and Ilchenko,2006;Chiasera et al.,2010).

We consider a waveguide or fiber positioned parallel to the boundary of a microdisk or microsphere.To couple light from the waveguide to a WGM in the cavity requires the phase synchronism,i.e.,the tangential component of the wave vector of the guided wave matches that of the WGM.This can be achieved by adjusting the waveguide width or orientation. Complete energy exchange between the waveguide and the resonator is possible when the coupling strength matches the intrinsic loss of the resonator(Yariv,2000).This is called critical coupling,a notion that was developed earlier in radio frequency(rf)engineering.By changing the distance from the waveguide to the cavity,the coupling strength is varied and the critical coupling may be reached for the lowest-order mode of the waveguide(Cai,Painter,and Vahala,2000).Parasitic coupling to higher-order waveguide modes and radiation modes is quantified by the“ideality”—the ratio of power coupled to a desired mode by power coupled or lost to all modes.An ideality of99.97%was shown with the coupling of a tapered fiber to a silica microsphere(Spillane et al.,2003). Next we discuss the prism coupler by considering a microsphere placed on the surface of a dielectric prism (Gorodetsky and Ilchenko,1994).A laser beam is directed into the prism and undergoes total internal reflection at the prism surface.The resulting evanescent optical field at the prism surface may be coupled to a WGM of the microsphere. The phase matching is obtained by adjusting the incident angle of the input light.

The evanescent field coupler has also been used as the output coupler for the WGMs.A detailed analysis of the coupling can be found in Gorodetsky and Ilchenko(1999) and Chiasera et al.(2010).

IV.SMOOTH DEFORMATION

In this section we discuss the properties of smoothly deformed microdisk cavities.The degree of deformation is classified here in terms of the chaoticity of the internal ray dynamics.To illustrate this concept,we consider a specific boundary curve,the lima?on of Pascal which reads in polar coordinateser;φT

reφT?Re1tεcosφT:e23TThe limiting case of vanishing deformation parameterεis the circle with radius R.An experimental realization is shown in Fig.11(d).

Forε<0.5the lima?on shape is a smooth convex defor-mation of the circle.The ray dynamics in billiards with such a boundary obey the Kolmogorov-Arnol’d-Moser(KAM)theo-rem(Kolmogorov,1954;Moser,1962;Arnol’d,1963).It states that for a sufficiently smooth perturbation of an integrable system some of the invariant tori survive,while others are destroyed giving rise to partially chaotic dynamics.Figure13 illustrates this so-called KAM transition to chaos for the lima?on billiard by varying the deformation parameterεfrom small to large values.For a small but nonzero value ofεmost of the invariant curves survive with their shape sightly distorted. The others are replaced by chains of stable and unstable periodic orbits as dictated by the Poincaré-Birkhoff theorem (Lichtenberg and Lieberman,1992;Ott,1993).The stable periodic orbits are surrounded by new invariant curves which form small islands;see Fig.13(a).The unstable periodic orbits are located in small chaotic layers not visible in Fig.13(a).A boundary deformation which leads to such a nearly integrable ray dynamics is here classified as weak deformation. Increasing the deformation parameter leads to the disap-pearance of more invariant curves and to an increase of the chaotic layers;cf.Fig.13(b).The remaining invariant curves prevent rays from exploring the whole SOS.These curves act as barriers for the ray dynamics and divide the phase space into disjoint regions.This situation of a mixed phase space is classified here as moderate deformation.Note that as long as the billiard boundary is convex and sufficiently smooth, there is always an infinite family of invariant curves in the whispering-gallery region j sinχj≈1.This fact is implied by Lazutkin’s theorem(Lazutkin,1973).

For large deformation these invariant curves and most of the others have been broken up and therefore the dynamics is predominately chaotic as shown in Fig.13(c).A ray starting in the region well above the critical line can diffuse to the leaky region.

A.Weak deformation:Nearly integrable ray dynamics Considering the ray dynamics in smoothly deformed microcavities,it seems that the case of weak deformation is not interesting.A ray starting well above the critical line for

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total internal reflection j p j ?j sin χj >1=n [see the invariant curve in the upper part of Fig.13(a)]is not able to enter the leaky region and therefore no light is emitted.However,this reasoning is not the full picture for several reasons to be discussed in this section.

For systems with more than 2degrees of freedom,e.g.,deformed microspheres,the KAM invariant curves no longer divide the phase space into disjoint regions,leading to the possibility of diffusion over large distances in phase space.This phenomenon is called Arnol ’d diffusion (Arnol ’d,1964).Lacey and Wang (2001)attempted to explain directional emission from deformed fused-silica microspheres by Arnold diffusion [Fig.1(c)].

In the absence of Arnol ’d diffusion the light output of a weakly deformed cavity is dominated by evanescent leakage (tunneling)of waves.It came as a surprise to observe experimentally directed emission even in this situation (Lacey et al.,2003).This sensitivity to small shape deforma-tions had been explained by preferential tunneling from the local minima of the invariant curves j p es Tj .Later,however,Creagh (2007)provided a toy model which clearly demon-strated that the distinctness of the local minima of the invariant curves is not correlated with the degree of directionality of light emission.Based on this observation,Creagh and White (2010,2012)introduced a more sophisticated explanation using the complex Wentzel-Kramers-Brillouin (WKB)approxima-tion and canonical perturbation theory for weakly deformed microcavities.

The sensitivity of the emission directionality to weak boundary deformations can be further enhanced by a strong mixing of nearly degenerate modes induced by the deforma-tion (Ge,Song,Redding,Ebersp?cher et al.,2013;Ge,Song,Redding,and Cao,2013).

B.Moderate deformation:Mixed phase space 1.Adiabatic curves and dynamical eclipsing

In the case of moderate deformation a considerable amount of rays is still confined by invariant curves;see the upper part of Fig.13(b).A ray starting in a sufficiently large chaotic part of phase space,however,can diffuse toward the leaky region and escape refractively;see around sin χ≈0.5in Fig.13(b).For moderate deformation the phase-space dif-fusion can be rather slow,so that the reduction of the Q factor of the corresponding optical mode (Q spoiling)is not serious (N?ckel,Stone,and Chang,1994;Mekis et al.,1995).Another consequence of the slow diffusion in sin χis that refractive escape typically occurs near the border of the leaky region,i.e.,at the critical angle for total internal reflection χc ,implying that the ray is emitted almost tangentially to the boundary of the cavity.

Later N?ckel et al.(1996)and N?ckel and Stone (1997)showed that a ray in the diffusive part of phase space of a moderately deformed cavity follows for some time the adiabatic curve (see Fig.14)

sin χes T?????????????????????????

1?ακes T2=3q ;

e24T

with αbeing an adiabatic constant and κes Tbeing the curvature of the boundary curve at position s .Equation (24)is based on an adiabatic approximation intro-duced by Robnik and Berry (1985)for billiards in magnetic fields.For longer times the chaotic whispering-gallery ray diffuses to the leaky region by going through a continuous sequence of adiabatic curves (24)with slowly increasing α.When the adiabatic curve touches the critical angle χc the ray can escape tangentially.From Eq.(24)it can be seen that the mimina of the adiabatic curves occur at the points of maximum curvature.This is consistent with the intuitive expectation that the escape of light happens primarily near the points of maximum curvature.The localization in the spatial coordinate (at the maximum of the curvature)and

in

FIG.13(color online).

Kolmogorov-Arnol ’d-Moser transition

to chaos in the lima?on cavity (23).The left-hand side shows the Poincarésurface of section for parameter (a)ε?0.1,(b)ε?0.3,and (c)ε?0.43.The shaded region indicates the leaky region for n ?3.3.The right-hand side shows the marked trajectories in real space.

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the angle (light is emitted tangentially)results in strong emission maxima in the far field in directions tangent to the highest-curvature points (N?ckel et al.,1996;N?ckel and Stone,1997).

Moreover,using the adiabatic curves allows one to derive an approximate quantization of the system via the semi-classical Einstein-Brillouin-Keller (EBK)quantization scheme (N?ckel et al.,1996;N?ckel,1997;N?ckel and Stone,1997).In this way a correspondence is made between a set of optical modes and a set of initial conditions for the rays in phase space.This correspondence is needed to set up a ray model to describe quantitatively the properties of optical modes,as mentioned in Sec.II .

The prediction based on Eq.(24)concerning the tangential emission from the highest-curvature points fails if regular islands are located at the critical angle right at the highest-curvature points;cf.Fig.14for the low refractive index n ?1.54.As the rays cannot enter the regular islands,they do not escape at the maximum of the curvature but mainly at two points separated by roughly the size of the islands.This effect is called dynamical eclipsing (N?ckel et al.,1996).It leads to a splitting of the emission peaks in the far field.The first experimental demonstration of dynamical eclipsing of chaotic WGMs has been done for prolate-deformed lasing micro-droplets (Chang et al.,2000),see https://www.sodocs.net/doc/9d3354504.html,ter,dynamical eclipsing was also observed in moderately deformed cylin-drical polymer lasers (Schwefel et al.,2004).

2.Gaussian modes based on stable periodic orbits

In moderately deformed microcavities there exist not only chaotic WGMs but also other types of long-lived modes depending on boundary shape and refractive index.A par-ticular important example is the bow tie mode (see Fig.15),first observed by Gmachl et al.(1998)in the flattened quadrupole

r eφT?R ???????????????????????????

1t2εcos 2φp :

e25T

In phase space the bow tie mode is localized inside a regular island centered around a stable periodic ray with the shape of a bow tie.The periodic ray is born in a period-doubling

bifurcation as the deformation parameter εis increased through the critical value of about 0.1.For sufficiently high refractive index,n ≈3.3,and ε≈0.15the regular island is located right on the border of the leaky region which results in directed emission based on refractive escape (in mainly four directions;cf.Fig.15)and moderate Q factors.Gmachl et al.(1998)demonstrated high-power directional emission from such a bow tie mode in a semiconductor quantum cascade microlaser [Fig.1(a)]with R ?30?50μm at a wavelength of around 5.2μm.This experiment can be considered as a milestone as it allowed for the first time to systematically vary the shape of a microdisk cavity in a controlled manner.Optical modes based on regular islands in phase space can be analytically described in a generalized Gaussian optical approach based on the parabolic equation approximation (Tureci et al.,2002).

3.Dynamical tunneling

Dynamical tunneling is a wave phenomenon which couples two distinct regions of ray-dynamical phase space (Davis and Heller,1981);see also B?cker,Ketzmerick,L?ck,and Schilling (2008)and L?ck et al.(2010).An example is the tunneling from a regular to the chaotic region in the phase space of the mushroom billiard (see Fig.4),as studied theoretically and experimentally in a microwave mushroom billiard by B?cker,Ketzmerick,L?ck,Robnik et al.(2008).Tunneling between regular islands that are separated by a chaotic sea can be enhanced by the presence of the chaotic part of phase space (Tomsovic and Ullmo,1994;Doron and Frischat,1995;Podolskiy and Narimanov,2003).This chaos-assisted tunneling can be considered as a three-step process:(i)dynamical tunneling from the initial island into the chaotic sea,(ii)(classical)ray propagation through the chaotic sea to the border of the other island,and (iii)dynamical tunneling into the island.Chaos-assisted tunneling was first experimentally observed in a microwave billiard (Dembowski et al.,2000).

In open systems such as dielectric microcavities,however,chaos-assisted tunneling may also appear as a two-step process (N?ckel and Stone,1997):(i)dynamical

tunneling

FIG.14.Four chaotic whispering-gallery rays in the phase

space of the quadrupole billiard (12)for ε?0.072followed for 100–200reflections.Superimposed are the adiabatic curves (24)for different values of α.The thick lines mark the border of the leaky region for two different refractive indices n .From N?ckel et al.,1996

.

FIG.15(color online).

Calculated bow tie mode localized on a

stable periodic orbit in a flattened quadrupole (25)with TM polarization,refractive index n ?3.3,and deformation parameter ε?0.15.Inset:Measured far-field pattern for ε?0(triangles),ε?0.14(open circles),and ε?0.16(filled circles)compared to calculated data for ε?0.15(dashed line).Adapted from Gmachl et al.,1998.

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from the initial island into the chaotic sea and (ii)ray propagation through the chaotic sea into the leaky region where the ray escapes from the cavity.Effects of this two-step chaos-assisted tunneling have often been discussed for the annular microcavity where outside the leaky region a clear separation of two regular whispering-gallery regions and a chaotic region can be observed;see Fig.8(a).While rays in these whispering-gallery regions stay in the cavity forever,the rays in the chaotic region can diffuse to the leaky region and leave the cavity.Hackenbroich and N?ckel (1997)showed that modes in this kind of cavity can show strong fluctuations of the quality factor due to dynamical tunneling between the different phase-space regions;for a general theory of this effect,see Podolskiy and Narimanov (2005).Moreover,dynamical tunneling in this cavity can be utilized to achieve unidirectional light emission from high Q modes (Wiersig and Hentschel,2006).For the annular cavity,the quantitative connection of the quality factors to the dynamical tunneling was established by B?cker et al.(2009).Based on the concept of the fictitious integrable system (B?cker,Ketzmerick,L?ck,and Schilling,2008),analytical expressions for the tunneling rates from the regular whispering-gallery region to the chaotic sea [see SOS for the annular cavity in Fig.8(a)]can be derived.If rays in the chaotic region leave the cavity quickly,the dynamical tunneling rates approximate the cavity losses and therefore allow one to compute the Q factors.The approximation can be improved by including the rates for direct tunneling along the radial degree of freedom to the exterior of the cavity;see Fig.16.

Shinohara et al.(2010,2011)were the first to provide clear experimental evidence for dynamical tunneling in optical microcavities.They used a cavity whose ray-dynamical phase space consists of a dominant chaotic region and an island chain,supporting a rectangular-shaped ray orbit fully confined by total internal reflection.Light emission from the corre-sponding optical mode happens via dynamical tunneling from the island chain to the chaotic sea.In such a situation,measuring the near and far fields of the light emission

unambiguously proves the mechanism of dynamical tunneling (Podolskiy and Narimanov,2005).

Another clear experimental demonstration of dynamical tunneling in optical microcavities has been achieved by free-space excitation of a liquid-jet cavity [Fig.1(b)](Yang et al.,2010).Here the light couples from outside to the chaotic sea inside the cavity and from there the light tunnels into regular islands which supports high Q modes for lasing.This scheme is of practical use as the pump efficiency of this microcavity laser is increased by 2orders of magnitude.The same scheme has been used to demonstrate experimentally tunneling-induced transparency in a chaotic microcavity similar to the case of electromagnetically induced transparency (Xiao et al.,2013).

Chaos-assisted tunneling as a three-step process has been discussed to determine the frequency splitting of nearly degenerate bow tie modes (Fig.15)in the quadrupole cavity (Tureci et al.,2002;Podolskiy and Narimanov,2003).

Another variant of dynamical tunneling is resonance-assisted tunneling (Ozorio de Almeida,1984;Brodier,Schlagheck,and Ullmo,2001;L?ck et al.,2010).Here island structures in phase space (also called nonlinear resonances)can enhance dynamical tunneling rates.Kwak et al.(2013)demonstrated resonance-assisted tunneling in a liquid-jet microcavity [Fig.1(b)]by measuring avoided resonance spectral gaps which are proportional to the square of the phase-space area associated with the given island chain.Chaos-assisted tunneling can be exploited for channeling rays into waveguides for efficient collection of light emission from microcavity lasers (Song et al.,2012).Figure 17shows that an attached waveguide introduces a vertical exit window in the phase space of the microcavity.This exit window seriously spoils the quality factor of (chaotic)WGMs but only mildly influences the quality factor of the modes related to the island chain around the diamond-shaped period-4orbit.In a laser based on this waveguide-cavity system these modes reach the lasing threshold first.Their emission is efficiently collected by the waveguide because emission is due to dynamical tunneling from the island chain into the chaotic sea from which most chaotic rays diffuse laterally to the exit window as illustrated in Fig.17instead of vertically down to the critical line.By using this scheme more than 95%of the emission can be collected by the waveguide.

https://www.sodocs.net/doc/9d3354504.html,rge deformation:Predominantly chaotic dynamics

In the case of large boundary deformation the ray dynamics is predominantly chaotic.From a phase-space plot such as in Fig.13(c)one could naively expect that modes in a strongly deformed cavity should be short lived and exhibit a rather diffuse far-field pattern.This is,however,not necessarily the case as we see in the following.

1.Chaotic saddle and its unstable manifold

In nonlinear dynamics it has been known for quite some time that the long-time behavior of an open chaotic system with time reversible dynamics is governed by the so-called chaotic saddle and its stable and unstable manifolds;see,e.g.,Lichtenberg and Lieberman (1992)and Lai and Tél (2010)

.

FIG.16(color online).

Quality factors and dynamical tunneling

rates for the annular microcavity.Shown is the theoretical prediction (solid curve)which is the sum of the direct tunneling contribution (dotted curve)and the dynamical tunneling contri-bution (dashed curve)based on the fictitious integrable system,and numerical data (filled circles)for azimuthal mode number m ?7;…;21.From B?cker et al.,2009.

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The chaotic saddle is the set of points in phase space that never visits the leaky region in both forward and backward time evolutions.The stable (unstable)manifold of a chaotic saddle is the set of points that converges to the saddle in forward (backward)time evolution.The unstable manifold of the chaotic saddle therefore describes the route of escape from the chaotic system.The stable (unstable)manifold of the chaotic saddle is also called the forward (backward)trapped set.The intersection of both sets is the chaotic saddle.The concept of the chaotic saddle and its manifolds had been applied to several physical systems both classical and quantum mechani-cal;see,e.g.,Gaspard and Rice (1989a ,1989b).In the field of optics of deformed microcavities this knowledge has been reinvented to a large part as discussed next.

In experiments on polymer microlasers with various shapes,Schwefel et al.(2004)demonstrated that light emission from microcavities with predominately chaotic ray dynamics can be highly directional.This unexpected finding was explained by the numerical observation that typical rays escape the cavity by following the unstable manifolds of short periodic orbits close to the boundary of the leaky region;see Fig.18(a).The stable (unstable)manifold of a periodic orbit is defined as the set of points in phase space which converge to the periodic orbit in the forward (backward)time evolution.The numerical simulation of intensity-weighted ray dynamics (13)shown in Fig.18(a)revealed that the asymptotic escape behavior of initially randomly chosen rays above the critical line is well approximated by the unstable manifolds of short periodic orbits.This was nicely confirmed by wave simu-lations and by a reconstruction of light intensity in the leaky region of phase space by using experimental far-field data of multimode fields;cf.Figs.18(a)and 18(b).

In the same year Lee et al.(2004)introduced the survival probability distribution (SPD)of intensity of rays inside the microcavity to explain the spatial localization of optical modes inside spiral-shaped cavities [an example is shown in Fig.11(a)].The SPD is defined as the probability P es;p;t Twith which a ray with Birkhoff coordinates es;p Tcan survive in the cavity at time t .In strongly chaotic systems,this distribution decays exponentially in time,and the dependence

on es;p Tis independent on initial conditions (Ryu et al.,2006).

The SPD of Lee et al.(2004)and the computed asymptotic behavior of initially randomly chosen rays by Schwefel et al.(2004)are equivalent to the unstable manifold of the chaotic saddle extended by the intensity-weighted ray dynamics (13)as first noted by Wiersig and Hentschel (2008).A systematic and clear discussion of this extended version of the chaotic saddle and its relation to the ergodic theory of transient

chaos

FIG.17(color online).

Using chaos-assisted tunneling for channeling rays into waveguides for efficient collection of light emission

from microcavities.(a)Poincarésurface of section of a quadruple billiard,Eq.(12),at ε?0.08.Squares mark a period-4orbit in the center of an island chain.Dots indicate a typical chaotic trajectory out of the island chain.Vertical lines mark the exit window due to the attached waveguide.(b)Real-space representation of the period-4orbit and the chaotic trajectory.Inset:Scanning electron microscope image of the experimental realization.From Song et al.,2012

.

FIG.18(color online).Light emission along unstable manifolds

of short periodic orbits.(a)Emitted-ray intensity (color scale)overlaid on the Poincarésurface of section of the quadruple cavity for ε?0.18,Eq.(12),and refractive index n ?1.49.The curve is the unstable manifold of a rectangular periodic orbit.(b)Experimental far-field data (color scale)projected onto the Poincarésurface of section (available data are restricted to φ∈??π=2;π=2 ).From Schwefel et al.,2004.

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can be found in Altmann(2009)and Altmann,Portela,and Tél (2013).Altmann(2009)pointed out that the unstable mani-folds of short periodic orbits(which are part of the chaotic saddle)close to the critical line as discussed by Schwefel et al. (2004)are parallel to the unstable manifold of the chaotic saddle and therefore lead to nearly the same far-field emission. Often the term“chaotic repeller”instead of“chaotic saddle”is used to describe the light emission from dielectric cavities. However,as emphasized by Altmann(2009),the term chaotic saddle is more appropriate as the dynamics is time reversible.

A chaotic repeller appears in noninvertible dynamical systems and possesses only unstable manifolds(Lai and Tél,2010). The emission mechanism along the unstable manifold of the chaotic saddle indicates that all long-lived modes in a given strongly deformed microcavity exhibit a similar far-field pattern;see Fig.19.This universal output directionality of single modes was proven without ambiguity in experiments on a liquid-jet microcavity(Lee et al.,2007b).Using

this FIG.19(color online).Light emission along the unstable manifold of the chaotic saddle.(a),(b)Calculated Husimi functions of two different modes in the quadruple cavity(12)forε?0.16and refractive index n?1.361.The leaky region below the critical line

sinχc?1=n(horizontal line)is magnified in(c)and(d),respectively.Superimposed is the unstable manifold of the chaotic saddle.

(e)Measured far-field pattern of individual modes in a liquid-jet cavity of the same shape and refractive index as in(a)–(d).Adapted from Lee et al.,2007b.

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concept Wiersig and Hentschel(2008)provided numerical evidence that all long-lived modes in the lima?on cavity(23) with deformation parameterε≈0.43and refractive index between2.7and3.9exhibit the universal and unidirectional light emission.This was confirmed experimentally by a number of groups(Shinohara et al.,2009;Song,Fang et al., 2009;Wang et al.,2009;Yan et al.,2009;Yi,Kim,and Kim, 2009;Albert et al.,2012).

In the case of mixed phase space the chaotic saddle is divided into hyperbolic and nonhyperbolic components (Altmann,2009).The mechanism of escape of electromag-netic radiation along the unstable manifold works also in this case as demonstrated by experiments on microwave cavities of quadrupolar shape(Sch?fer,Kuhl,and St?ckmann,2006). Moreover,a combination with dynamical tunneling is pos-sible.As discussed in Sec.IV.B.3Shinohara et al.(2010,

2011)demonstrated experimentally and theoretically that modes localized in an island chain(a nonhyperbolic compo-nent)can tunnel into the chaotic sea(the hyperbolic compo-nent).From there,the rays follow the unstable manifold of the chaotic saddle.

The details of the relation between optical modes and the chaotic saddle of the ray dynamics are still not fully understood. For open quantum maps it has been rigorously proven that in the semiclassical limit the right eigenvectors of the nonunitary time evolution matrix U are supported by the corresponding classical unstable manifold of the chaotic saddle(Keating et al., 2006).Long-lived states are localized on the chaotic saddle (which is part of the unstable manifold).The localization, however,is not uniform because of quantum fluctuations.In the case of microcavities these fluctuations can have a significant impact on the far-field emission pattern(Shinohara, Fukushima,and Harayama,2008;Shinohara et al.,2009). 2.Dynamical localization and scar modes

It is natural to expect that modes in chaotic microdisks have low Q factors.This Q spoiling(N?ckel,Stone,and Chang, 1994;N?ckel and Stone,1995)would limit the possible applications of deformed microdisks considerably.However, wave localization effects discovered in the field of quantum chaos provide the possibility of high Q modes in chaotic microcavities.For example,wave packets mimic to some extent the chaotic ray diffusion in phase space.However, destructive interference suppresses the chaotic diffusion on long time scales(Casati et al.,1979;Fishmann,Grempel,and Prange,1982;Borgonovi,Casati,and Li,1996;Frahm and Shepelyansky,1997).This dynamical localization in phase space is closely related to real-space Anderson localization in disordered solids(Fishmann,Grempel,and Prange,1982). The first experimental observation of dynamical localization was reported by Moore et al.(1994)using ultracold atoms placed in a modulated standing wave of a near-resonant laser. Another experimental verification of dynamical localization used a microwave circular billiard with boundary roughness (Sirko et al.,2000).In the regime of dynamical localization, the angular momentum l～sinχin such a“rough billiard”is exponentially localized around a mode-dependent value.The localization in sinχis of interest for optical microcavities as it suppresses the diffusion into the leaky region and therefore allows for modes with a high quality factor even in the regime of fully chaotic ray dynamics(Frahm and Shepelyansky, 1997).The first theoretical study of dynamical localization in optical microdisks with boundary roughness has been performed by Starykh et al.(2000).They showed that the dynamical localization leads to a log-normal distribution of the modes’linewidths and decay rates.The direct observation of lasing action from dynamically localized modes was reported by Podolskiy et al.(2004)and Fang et al.(2005) using GaAs-InAs microdisks with enhanced boundary rough-ness;see Fig.20.

Another wave localization phenomenon known from closed chaotic systems is scarring(Heller,1984).It refers to the existence of a small fraction of quantum eigenstates with strong concentration along unstable periodic orbits of the underlying classical system.In optical microcavities,the localization of wave intensity along unstable periodic ray trajectories has been observed experimentally first in liquid-jet microlasers(Lee et al.,2002)and shortly after in GaN microlasers(Rex et al.,2002)and in GaAs/GaInAs/GaInP quantum well microlasers(Gmachl et al.,2002).The observed modes,such as the one shown in Fig.21,can have high

quality FIG.20.Dynamical localization in optical microcavities with strong boundary roughness.(a)Scanning electron micro-scope image of rough microdisk.From Podolskiy et al.,2004.

(b)Computed high Q mode dynamically localized in angular momentum

space.

FIG.21(color online).Scarring in optical https://www.sodocs.net/doc/9d3354504.html,-puted optical mode in the quadruple cavity(12)which is scarred by the triangular periodic ray trajectories depicted in the inset;ε?0.12and refractive index n?2.65.From Rex et al.,2002.

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factors since the corresponding short periodic orbit is located entirely outside the leaky region and is therefore part of the chaotic saddle.

A well-studied system in the field of quantum chaos is the stadium billiard given by two semicircles and two parallel segments.This system is not a smooth deformation of the circle as the radius of curvature changes discontinuously at the points of connection.For this billiard it is rigorously proven that the ray dynamics is fully chaotic,i.e.,there are no regular regions in phase space(Bunimovich,1974).A microcavity of stadium shape is shown in Fig.11(c).Theoretical analyses of such a microcavity revealed a localization along multiple periodic orbits(Harayama et al.,2003;Fang,Yamilov,and Cao,2005).Associated with this localization is a nonmono-tonic decrease of the Q factor with increasing deformation because of interference of waves propagating along different constituent orbits(Fang,Yamilov,and Cao,2005).This interference effect has been discussed in terms of a peri-odic-orbit-sum formula by Fukushima,Harayama,and Wiersig (2006).These theoretical findings have been confirmed exper-imentally in GaAs/AlGaAs(Harayama et al.,2003),GaAs (Fang,Cao,and Solomon,2007),and polymer microstadia (Fang and Cao,2007).The observation of localization along multiple periodic orbits is consistent with a recent study of an open three-disk system(Weich et al.,2014)which relates this phenomenon to the formation and interaction of resonance chains in the complex frequency plane.

Numerical simulations indicate that scarring in optical microcavities with strongly chaotic ray dynamics is rather the rule than the exception;see,e.g.,Lee et al.(2002,2004,2005, 2007b),Rex et al.(2002),Harayama et al.(2003),Fang, Yamilov,and Cao(2005),Wiersig(2006),Fang and Cao (2007),Fang,Cao,and Solomon(2007),Wiersig and Hentschel (2008),and Wiersig et al.(2010).This conclusion can also be drawn from studies of open quantum maps(Wisniacki and Carlo,2008;Ermann,Carlo,and Saraceno,2009).

An interesting phenomenon not observed in any closed system is the appearance of quasiscarred modes showing a strong localization on simple geometric structures with no underlying periodic ray(Lee et al.,2004;Lee,Rim et al., 2008).Lasing on quasiscarred modes has been successfully realized for spiral-shaped InGaAsP microcavity lasers(Kim et al.,2009).Quasiscars find a natural explanation in terms of an extended ray dynamics as discussed in Sec.VIII.A.

3.Level statistics

A central topic of quantum chaos is the analysis of the statistical properties of energy levels in quantum systems whose classical counterpart is chaotic(St?ckmann,2000).In the last decade,the focus has shifted from closed to open systems.For a review of unsolved problems in this field see Nonnenmacher(2011).

One particularly interesting aspect is the fractal Weyl law for long-lived states in open fully chaotic systems.This conjecture,based on the work of Sj?strand(1990)and Zworski(1999),is an extension of the well-known Weyl’s formula for closed systems.Weyl’s formula states that the number of energy levels NekTwith wave number k m≤k,or more precisely the smooth part of it,ˉNekT,behaves asymptotically as～k2for the particular case of a two-dimensional system which scales with the energy such as quantum billiards.For an open system the number of resonances with complex wave numbers k m can be defined as N CekT?f k m∶Imek mT>?C;Reek mT≤k g:e26T

The cutoff constant C>0ensures that only long-lived states are taken into account;fast decaying states are ignored.The fractal Weyl law for an open chaotic system(which again scales with the energy)can be written as

ˉN

C

ekT～kα:e27TIt is conjectured that the noninteger exponent in Eq.(27)is

α?Dt1

2

?dt2

2;e28Twhere D is the fractal dimension of the chaotic saddle or repeller(Lin and Zworski,2002);d?D?1is the dimension of the saddle in a properly chosen SOS.

The fractal Weyl law has been numerically confirmed for a number of physical model systems:a three-bump scattering potential(Lin,2002;Lin and Zworski,2002),a three-disk system(Lu,Sridhar,and Zworski,2003),open quantum maps (Schomerus and Tworzydlo,2004;Nonnenmacher,2006; Shepelyansky,2008),a Hénon-Heiles Hamiltonian with Coriolis term(Ramilowski et al.,2009),and a four-sphere system(Ebersp?cher,Main,and Wunner,2010).The asymp-totic form(27)has been rigorously proven only for a simplified variant of the open quantum baker’s map (Nonnenmacher and Zworski,2007).Experimental evidence for the fractal Weyl law has been obtained for a five-disk microwave system(Potzuweit et al.,2012).

For dielectric cavities the situation is more complicated than in the above examples.First,a dielectric cavity possesses internal and external modes(Bogomolny,Dubertrand,and Schmit,2008;Dubertrand et al.,2008;Dettmann et al., 2009a);see the discussion in Sec.III.However,the latter are extremely short lived and are therefore conveniently withdrawn from the counting process by the cutoff constant C.Second,the partial leakage of intensity according to Fresnel’s laws has an important implication.Consider a dielectric cavity(n finite)and the corresponding closed billiard system(n→∞).The states in the billiard system have zero decay rate and their number satisfies the conventional Weyl law.When the openness of the system is gradually increased by reducing n,each mode acquires a nonzero but finite decay rate because the transmission through the boundary is not complete(except at Brewster’s angle for TE polarization). Therefore,“no mode can disappear to infinity”along the imaginary direction in complex frequency space.This implies that the total number of internal modes of a dielectric cavity fulfills the conventional k2law as pointed out by Bogomolny, Dubertrand,and Schmit(2008).This,however,is not in contradiction with the fractal Weyl law which applies to the long-lived modes within the set of internal modes.

The fractal Weyl law for dielectric microcavities has been tested only for the stadium-shaped cavity(Wiersig and

80Hui Cao and Jan Wiersig:Dielectric microcavities:Model systems for…Rev.Mod.Phys.,V ol.87,No.1,January–March2015