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

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.

Figure 1

A few examples of deformed dielectric cavities. (a) Side-view and top-view scanning electron microscope (SEM) images of a quantum cascade laser made of a flattened quadrupolar-shaped GaAs cylinder. From [148]. (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 $200\mu \mathrm{m}$. From [215]. (d) A microwave cavity made of a Teflon disk on a brass ground plate with dimensions $380\times 260{\mathrm{mm}}^2$. From [334].

Figure 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 $n=1.5$ at 170° with respect to the incoming plane wave with wave number $k=\omega /c$, where $c$ is the speed of light in vacuum. The scattering intensity shows narrow peaks (resonances) at the scaled complex frequencies $\mathrm{\Omega }=\omega R/c=kR$ 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 [400].

Figure 3

Ray dynamics in a billiard; $s$ is the arclength coordinate along the boundary of the cavity and $\chi$ is the angle of incidence with respect to the boundary normal $\overrightarrow{\nu }$. (a) The solid line $1\to 2\to 3$ is a counterclockwise traveling ray in real space and the dashed line $1^{\prime }\to 2^{\prime }\to 3^{\prime }$ is a clockwise traveling ray. (b) The same dynamics in the Poincaré surface of section. The coordinate $s$ is normalized to the total circumference of the boundary $s_{\max }$. In this representation the angle $\chi$ is conventionally defined negative for clockwise traveling rays.

Figure 4

Example of a simple mixed phase space. Numerically computed Poincaré surface of section for a desymmetrized mushroom billiard showing regular and chaotic regions in phase space; the size of the billiard is scaled such that $s_{\max }=\pi /2$. From [22].

Figure 5

Poincaré surface of section of a quadrupole billiard (12) for $ϵ=0.072$. The boundary is here parametrized by the polar angle $\phi$. The direction $\phi =0$ corresponds to the right part of the horizontal axis in the three real-space plots. From [401].

Figure 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 $R^{\mathrm{TM},\mathrm{TE}}I$ and a transmitted ray with intensity $(1-R^{\mathrm{TM},\mathrm{TE}})I$. $\overrightarrow{\nu }$ is the outward normal vector. The angle of the reflected ray $\eta$ is related to the angle of the incident ray $\chi$ by Snell’s law. The emission direction can be described by the polar angle $\phi$ which equals asymptotically the angle $\varphi$ between the $x$ axis and the emitted ray.

Figure 7

The leaky region in phase space of a dielectric cavity. (a) Poincaré surface of section with leaky region $|\sin \chi |\le 1/n$ in which the condition for the total internal reflection is not met. (b) Reflection coefficient $R^{\mathrm{TM},\mathrm{TE}}(\sin \chi )$ for a planar dielectric interface with the incident plane wave coming from the high-index medium ($n=2$). The low-index medium is air with $n=1$.

Figure 8

Husimi function in phase space. (a) Poincaré surface of section (dots, horizontal lines with $|\sin \chi |>0.6$), critical lines (horizontal lines with $|\sin \chi |\approx 0.3$), and emerging Husimi function (shaded regions) of a mode in an annular cavity. (b) Mode in real space.

Figure 9

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 cylinder 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 [311].

Figure 10

(a) Scanning electron micrograph (SEM) of a silica microsphere at the end of the preform wire. Its diameter is $70\mu \mathrm{m}$. No surface defect was observed on a 30 nm scale. From [90]. (b) SEM of a silica microtoroid. From [11].

Figure 11

(a) Optical microscope image of a spiral microcavity made of a DCM-doped polysmethylmethacrylated film. From [33]. (b) Side view SEM of a InGaAsP microdisk on top of an InP pedestal. The disk diameter is $3\mu \mathrm{m}$. From [258]. (c) SEM of a GaAs/AlGaAs microstadium laser with a metal electrode on the top for current injection. From [132]. (d) SEM of a limaçon-shaped micropillar with a vertical cavity formed by two Bragg mirrors. Image courtesy of S. Reitzenstein, TU Berlin.

Figure 12

Numerically computed standing-wave modes in a dielectric microdisk; $n=3.3$ (GaAs), TM polarization. (a) Radial mode number $l=1$ and azimuthal mode number $m=19$, scaled frequency $\mathrm{\Omega }=kR=7.02783-i2.99188\times {10}^{-13}$; (b) $l=3$, $m=12$, and $\mathrm{\Omega }=7.0175-i6.29188\times {10}^{-5}$.

Figure 13

Kolmogorov-Arnol’d-Moser transition to chaos in the limaçon cavity (23). The left-hand side shows the Poincaré surface 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.

Figure 14

Four chaotic whispering-gallery rays in the phase space of the quadrupole billiard (12) for $ϵ=0.072$ followed for 100–200 reflections. Superimposed are the adiabatic curves (24) for different values of $\alpha$. The thick lines mark the border of the leaky region for two different refractive indices $n$. From [282].

Figure 15

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 [148].

Figure 16

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 contribution (dashed curve) based on the fictitious integrable system, and numerical data (filled circles) for azimuthal mode number $m=7,\dots ,21$. From [24].

Figure 17

Using chaos-assisted tunneling for channeling rays into waveguides for efficient collection of light emission from microcavities. (a) Poincaré surface of section of a quadruple billiard, Eq. (12), at $ϵ=0.08$. Squares mark a period-4 orbit 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-4 orbit and the chaotic trajectory. Inset: Scanning electron microscope image of the experimental realization. From [376].

Figure 18

Light emission along unstable manifolds of short periodic orbits. (a) Emitted-ray intensity (color scale) overlaid on the Poincaré surface 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 Poincaré surface of section (available data are restricted to $\phi \in [-\pi /2,\pi /2]$). From [348].

Figure 19

Light emission along the unstable manifold of the chaotic saddle. (a), (b) Calculated Husimi functions of two different modes in the quadruple cavity (12) for $ϵ=0.16$ and refractive index $n=1.361$. The leaky region below the critical line $\sin {\chi }_c=1/n$ (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 [229].

Figure 20

Dynamical localization in optical microcavities with strong boundary roughness. (a) Scanning electron microscope image of rough microdisk. From [303]. (b) Computed high $Q$ mode dynamically localized in angular momentum space.

Figure 21

Scarring in optical microcavities. Computed optical mode in the quadruple cavity (12) which is scarred by the triangular periodic ray trajectories depicted in the inset; $ϵ=0.12$ and refractive index $n=2.65$. From [319].

Figure 22

Illustration for the perturbation theory of deformed microdisk cavities [circle (solid curve) and a deformed circle (dashed)]. The horizontal line represents a symmetry axis. Grey regions mark the area $\delta a$ where the refractive index of the deformed cavity differs from the one of the circular cavity.

Figure 23

Hexagonal-shaped microcavities. (a) Scanning electron microscope image of a zeolitic aluminophosphate microcavity. (b) Computed mode structure. (c) Four members of the family of periodic-6 ray trajectories. The ray picture breaks down for the rightmost orbit which hits the corners of the hexagon. Adapted from [64].

Figure 24

(a) Calculated optical mode in the spiral (41) with $ϵ=0.1$, $n=2.6$, and $nkR\approx 200$. (b) Experimental (open circles) and calculated (solid and dashed) far-field pattern. From [83].

Figure 25

An avoided crossing of two energy levels $E_{\pm }(\mathrm{\Delta })$ in a closed system. Equation (43) has been used with $E_1$ being linear dependent on the parameter $\mathrm{\Delta }$, whereas $E_2$ and $V=W^{*}$ are kept constant. The arrows visualize the eigenvectors for certain values of $\mathrm{\Delta }$.

Figure 26

Illustration of an avoided resonance crossing in the weak (left) and strong (right) coupling regimes in the case of internal coupling. Small (large) $|\mathrm{Im}(E)|$ corresponds to a long (short) lifetime. The arrows indicate eigenvectors of the system which hybridize near the region of resonant coupling.

Figure 27

An avoided resonance crossing in the strong coupling regime for the case of external coupling. The resonant formation of a long-lived mode moving close to the real energy axis can be clearly seen.

Figure 28

Calculated near-field intensity of modes in a dielectric ellipse with refractive index $n=3.3$. Modes (a) and (b)  [(e) and (f)] are on the left-hand (right-hand) side of an avoided crossing. The rectangular-shaped mode (c) and the diamond-shaped mode (d) at the center of the avoided crossing show a localization of intensity along periodic ray trajectories (dashed lines). From [407].

Figure 29

Complex-square-root topology of eigenvalue surfaces $E=E_{\pm }(\mathrm{\Delta },|V|)$ with a branch point singularity at the exceptional point (EP). The black curves result from a double loop in the parameter space $(\mathrm{\Delta },|V|)$.

Figure 30

Measured eigenfrequency surfaces ${\nu }_l-{\nu }_{\text{ref}}$ of modes in a smoothly deformed liquid-jet microcavity near an exceptional point (EP). The two independent parameters consist of a deformation parameter and a quasicontinuous internal parameter, the mode number $n$. From [226].

Figure 31

Chirality of a mode in a microdisk perturbed by two small bumps. The disk is optically excited and the directionality of the light outcoupled to a waveguide is measured (squares) as a function of the normalized angle between the bumps. Positive (negative) values mean transmission mainly to the left (right) indicating a finite chirality of the excited mode. The calculated chirality shown as circles is positive (negative) for mainly counterclockwise (clockwise) rotation of the mode. Inset: SEM of the perturbed microdisk with the coupling waveguide. From [198].

Figure 32

(a) Family of interior whispering-gallery-type trajectories. (b) Far-field intensity distribution, showing unidirectional emission. The index of refraction of the dielectric disk is $n=3$. Adapted from [31].

Figure 33

Normalized frequencies $\mathrm{\Omega }=\omega R/c$ and quality factors vs $d$ for a high $Q$ WGM and a low $Q$ mode with directed emission in the annular cavity, a microdisk of radius $R$ with an air hole of radius $R_2=0.22R$ located at the distance $d$ to the disk’s boundary. The index of refraction for TM polarization is $n=3.3$. The coupling of the two modes leads to unidirectional emission in the far field (lower right).

Figure 34

Ray simulations of unidirectional emission from a limaçon cavity with refractive index $n=3.3$ and deformation parameter $ϵ=0.43$. (a) Far-field intensity patterns are normalized so that the integrated intensity is unity. The solid (dashed) curve is for TE (TM) polarization. (b) Survival probability distribution for an ensemble of 50 000 rays starting uniformly in the phase space with identical intensity for TE polarization. The arclength $s$ is normalized to the cavity’s perimeter. The Fresnel weighted unstable manifold in the leaky region ($|\sin \chi |<1/n$) is concentrated on very few high-intensity spots, giving highly directional output. From [433].

Figure 35

Calculated intensity distribution of an even-parity TE mode (a) inside and (b) outside a limaçon cavity with refractive index $n=3.3$ and deformation parameter $ϵ=0.43$. The mode has a normalized frequency $\mathrm{\Omega }=\omega R/c=26.0933$, and a $Q$ factor of 185 000. The mode is confined spatially near the cavity boundary and emits predominantly to the direction $\phi =0$. (c) Husimi function of the TE mode in (a), exhibiting enhanced mode intensity around an unstable periodic ray trajectory illustrated in the inset. The dots mark the bouncing points of the periodic ray trajectory from the cavity boundary. The horizontal lines $\sin {\chi }_c=1/n$ enclosing the leaky region are indicated. (d) Magnified Husimi function in the leaky region agrees to the unstable manifold in Fig. 34, confirming its responsibility for the directional emission. Adapted from [433].

Figure 36

Ray dynamics in the face cavity made of silica. (a) The Poincaré SOS. The diamonds mark the unstable period-4 orbit. Its unstable manifold is shaped by the islands around the stable periodic-4 orbit. The horizontal line is the critical line. (b) The cavity boundary and a ray trajectory close to the stable periodic-4 orbit. (c) The far-field pattern obtained by ray simulations with modified Fresnel’s law. Adapted from [462].

Figure 37

(a) Illustration of the notched-elliptical cavity. The arrows indicate that light is scattered by the notch and collimated as a parallel beam by the right-side boundary of the cavity. The boundary of the cavity (solid curve) is designed to best approximate that of the auxiliary ellipse (dashed curve) within the largest possible range. The notch is located at one of the foci of the auxiliary ellipse. The optimal long-to-short aspect ratio of the elliptical cavity is 1.2 for the refractive index of 3.2. (b) Ray simulation of the collimation effect: a number of rays are started at the position of the notch with different outgoing angles, simulating a scattering process. The solid rays leave the notch under relatively smaller outgoing angles; they hit the right boundary of the cavity and get refracted out. The dash-dotted ray leaves the notch at a high outgoing angle and is relaunched into a whispering-gallery mode. (c) A single ray is started at some position along the cavity boundary with an initial angle of incidence larger than the critical angle. It is then specularly reflected many times until at some point it hits the notch and gets reflected to the opposite boundary, refracted out, and leaves the cavity parallel to the horizontal axis due to the collimation effect. (d) Scanning electron microscope image of the notched-elliptical cavity. (e) Experimental and simulated far-field intensity profiles. Adapted from [418].

Figure 38

Goos-Hänchen shift and Fresnel filtering at a dielectric interface. (a) Goos-Hänchen shift near or above the critical angle of incidence: the center of a beam (solid lines) with finite beam waist (curve) reflected at a dielectric interface is spatially shifted by $\mathrm{\Delta }s$ with respect to the geometric-optics prediction (dashed line). (b) Fresnel filtering near or below the critical angle of incidence: deviations from the law of reflection (angular shift $\mathrm{\Delta }\chi$ of reflected beam) and Snell’s law (angular shift $\mathrm{\Delta }\eta$ of transmitted beam).

Figure 39

Calculated optical mode in a dielectric ellipse with refractive index $n=3.3$. The mode pattern is localized along an optical beam (beam center is indicated by the dashed lines) including the spatial Goos-Hänchen shift (highlighted by the bars); cf. the corresponding geometric-optics ray in Fig. 28.

Figure 40

Numerical simulation of a wavelength-scale deformed dielectric cavity of refractive index $n=3.13$. The cavity boundary is given by $r(\phi )=R(1+ϵ\cos \phi )(1-{ϵ}_1\cos 2\phi )+d$ in polar coordinates, where $R=890\mathrm{nm}$, $ϵ=0.28$, ${ϵ}_1=0.06$, and $d=60\mathrm{nm}$. (a) Calculated $Q$ factor and (b) directionality $U$ vs $kR$ for the high $Q$ modes (HQMs) (dots) and the low $Q$ modes (LQMs) (squares). The coupling of the HQM and LQM at $kR\simeq 7$ leads to a drop of $Q$ for the HQM and an increase of its $U$. (c)–(e) Calculated magnetic field intensity for the modes labeled 1, 5, 4 in (a), revealing the fact that mode 4 is a hybrid of 1 and 5. Adapted from [377].

Figure 41

Spatial separation of the period-3 orbit for clockwise (CW) and counterclockwise (CCW) propagating rays in a wavelength-scale deformed cavity due to beam shifts. The cavity is the same as that in Fig. 40, and the orbit corresponds to the LQM. The three bounce points are labeled i, ii, and iii. The arrow outside the cavity represents the direction of dominant emission from the bounce point where the angle of incidence is the smallest. Both CW and CCW rays emit in the same direction. From [379].

Figure 42

Calculated spatial intensity distribution of the (a) CW and (b) CCW waves that constitute a high $Q$ mode at $kR=3.2$ in a limaçon cavity. The intensity maxima for the CW and CCW waves are spatially separated. The solid lines depict the path for the CW beam in (a) and CCW beam in (b), reconstructed from the incident and emergent Husimi functions. The split in the CW and CCW orbits is due to beam shifts. (c) The CW wave intensity is enhanced outside the cavity to show that the emission is predominantly from bounce 1. Similarly, the CCW wave (a mirror image with respect to the horizontal axis) emits from bounce 3 into the same direction (not shown). (d) A polar plot of the experimentally measured far-field pattern of laser emission (thin dotted line) which agrees well with the calculated output of the high $Q$ mode at the same wavelength (thick solid line). From [318].

Figure 43

A wavelength-scale deformed microdisk coupled to a waveguide. (a) Calculated intensity distribution in a waveguide coupled deformed microdisk, showing directional coupling to a waveguide positioned at polar angle $\theta ={45}^{°}$. (b) Directionality of waveguide coupling $V$ as a function of the coupling position $\theta$ on the cavity boundary. $V>0$ ($V<0$) corresponds to stronger coupling in the CCW (CW) direction. The crosses represent the experimental data points which agree well with the numerical simulation (solid line). The inset shows the orbits for the CW and CCW beams for this mode. (c) Top-view SEM image of a GaAs disk coupled to a waveguide. The disk is supported on an ${\mathrm{Al}}_{0.7}{\mathrm{Ga}}_{0.3}\mathrm{As}$ pedestal at the center, and the GaAs waveguide is free standing in air and supported by two ${\mathrm{Al}}_{0.7}{\mathrm{Ga}}_{0.3}\mathrm{As}$ pedestals at the ends. The background shows the residual ${\mathrm{Al}}_{0.7}{\mathrm{Ga}}_{0.3}\mathrm{As}$ on the GaAs substrate after selective etching of the ${\mathrm{Al}}_{0.7}{\mathrm{Ga}}_{0.3}\mathrm{As}$. (d) Local chirality $W(\theta )$ for the same mode in the absence of waveguide coupling. Locally, the CW and CCW intensities are not equal, leading to directional output to the waveguide shown in (b). Only half of the cavity boundary is plotted in (b) and (d); the other half can be obtained by mirror symmetry. From [318].

Figure 44

Length spectrum of a circular microwave resonator. The solid (dashed, dotted) curve shows the measured (numerically calculated, semiclassical) spectrum. The arrows mark the lengths of the depicted periodic ray trajectories and the circumference of the boundary. Except for the square-shaped ray trajectory good agreement can be observed. From [45].

Figure 45

Sagnac effect in deformed microcavities. The cavity boundary is closed, and the refractive index inside is equal to 1. The cavity modes under consideration are TM polarized. (a) A rotating quadrupole cavity (12) with $ϵ=0.12$, and $R=R_0=6.2866\mu \mathrm{m}$. Calculated frequency difference $R\mathrm{\Delta }\omega /c$ of a pair of quasidegenerate modes at $nkR\simeq 49.338$ as a function of the (dimensionless) angular velocity $R\mathrm{\Omega }/c$. The frequency difference does not change when the angular velocity is below the threshold $R{\mathrm{\Omega }}_{\text{th}}/c\sim 5\times {10}^{-8}$; above the threshold the frequency difference increases linearly with the angular velocity. From [392]. (b) A rotating microcavity of symmetry $C_{3\nu }$. The cavity boundary is defined by $r(\phi )=R(1+ϵ\cos 3\phi )$, where $ϵ=0.065$. Calculated frequency splitting $R\mathrm{\Delta }\omega /c$ of a pair of degenerate modes at $nkR=50.220$ vs $R\mathrm{\Omega }/c$. The frequency splitting increases linearly with the angular velocity with no threshold. Adapted from [392].

Figure 46

Comparison of rotation-induced changes in frequency and $Q$ factor of resonant modes in a circular microdisk. The disk radius is $R=590\mathrm{nm}$ and refractive index is $n=3$. Calculated relative change in the $Q$ factor $\mathrm{\Delta }Q/Q_0$ (squares) and the normalized frequency splitting $\mathrm{\Delta }\omega /{\omega }_0$ (circles) of a pair of WGMs with $l=1$ and $m=\pm 7$ as a function of the rotation speed $\mathrm{\Omega }$. The lines are linear fits. The slope for $\mathrm{\Delta }Q/Q_0$ is 20 times higher than that for $\mathrm{\Delta }\omega /{\omega }_0$, indicating the $Q$ factor is 20 times more sensitive to rotation than the resonant frequency. From [333].

Figure 47

Numerical simulation of rotation-induced change in the far-field emission pattern of an asymmetric limaçon cavity. The cavity boundary is defined in the polar coordinates as $r(\phi )=R[1+{ϵ}_1\cos (\phi )+{ϵ}_2\cos (2\phi +\delta )]$, where $R=591\mathrm{nm}$, ${ϵ}_1=0.1$, ${ϵ}_2=0.075$, and $\delta =1.94\mathrm{rad}$. The refractive index inside the cavity is $n=3.0$, and outside the cavity $n=1.0$. Without rotation, a pair of quasidegenerate modes at $\lambda =598\mathrm{nm}$ consists mainly of CW traveling waves. Their near-field patterns are similar, one of them is shown in (a). The intensity outside the cavity is enhanced to illustrate the main emission direction. At the normalized rotation frequency $\mathrm{\Omega }R/c=0.001$, one of the two modes is converted to CCW traveling waves (c), the other remains as a CW wave (b). Their emission directions are very different. (d) Angular distribution of far-field emission intensity for the two modes in (b) and (c). The rotation causes a dramatic change in the far-field pattern of the mode in (b). From [332].