## Abstract

Dielectric cavity systems, which have been studied extensively so far, have uniform refractive indices of their cavities, and Husimi functions, the most widely used phase space representation of optical modes formed in the cavities, accordingly were derived only for these homogeneous index cavities. For the case of the recently proposed gradient index dielectric cavities (called as transformation cavities) designed by optical conformal mapping, we show that the phase space structure of resonant modes can be revealed through the conventional Husimi functions by constructing a reciprocal virtual space. As examples, the Husimi plots were obtained for an anisotropic whispering gallery mode (WGM) and a short-lived mode supported in a limaçon-shaped transformation cavity. The phase space description of the corresponding modes in the reciprocal virtual space is compatible with the far-field directionality of the resonant modes in the physical space.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Maxwell's equations, the governing equations for electromagnetic fields in space-time including media, have fundamental symmetries under various transformations such as well-known Lorentz transformation and U(1) gauge transformation. Maxwell's equations are also form-invariant under general coordinate transformations [1–3]. Exploiting the form invariance of Maxwell's equations for general coordinate transformations, Pendry *et al.* theoretically presented transformation optics (TO) [4] which is a general methodology for designing electromagnetic materials including optical invisibility cloaks, and various photonic devices which manipulate the path of light waves [5, 6]. Particularly, very recently, applying the TO to the design of optical dielectric resonators, Kim *et al.* reported 2-dimensional (2D) gradient index dielectric cavities called transformation cavities by using conformal TO [7–9]. Surprisingly, in their transformation cavities, directional whispering gallery modes (WGMs) with very high quality factors (Q-factor) are possible, providing a great potential for cutting-edge optical device design.

Conventional 2D optical dielectric cavities with uniform refractive indices have been regarded as an open system version of closed quantum billiard systems and also as a potential candidate of low-threshold micro-lasers, so there have been vast and intensive works on these platform systems over the last twenty years [10, 11]. In order to analyze the characteristics of optical modes supported in dielectric cavities, it is often more useful to represent the optical modes in phase space than merely depict the mode intensity patterns in real physical space. Husimi function, one of the popular phase space representations of a wave function can be regarded as a quasi-probability distribution in phase space. It is defined by the overlap of the wave function with a coherent state of a minimum uncertainty wave packet [12]. The so-called Poincaré Husimi functions on the billiards boundary have been widely used when studying quantum-classical correspondence in quantum billiards [13,14]. For the case of optical dielectric cavities, Hentschel *et al.* have derived four different Husimi functions in a reduced phase space, i.e., on the Poincaré surface of section (PSOS) at the dielectric interfaces by using a stationary phase approximation [15].

However, since the Husimi functions of Hentschel *et al.* are valid exclusively for homogeneous dielectric interfaces, it is inappropriate to apply their Husimi functions directly to the transformation cavities to see the phase space structure of an optical mode formed there. In this paper, we show that the Husimi functions can still be useful for optical modes supported in transformation cavities with inhomogeneous refractive indices. To this end, we construct a virtual space which we call a reciprocal virtual (RV) space, through the inverse of the conformal mapping, then both polarized resonant modes supported in the transformation cavity of the physical space and the corresponding virtual modes in the unit disk cavity of the RV space turn out to be identical. Besides, the refractive index of the unit disk cavity in the RV space is uniform. Thus, based on these facts, one can obtain a phase space description of the internal waves in a transformation cavity through the conventional Poincaré Husimi functions calculated in the RV space.

This paper is organized as follows. In section 2, starting from a uniform index unit disk cavity in a fictitious space (called as an original virtual space) we describe the gradient index transformation cavity model. In section 3, RV space is introduced to apply conventional Husimi functions for the internal waves in a uniform index cavity. As an illustration of our method, Husimi functions for a directional high-Q WGM and a directional low-Q mode in a limaçon-shaped transformation cavity are calculated and analyzed in section 4. In the last section comes our conclusion.

## 2. Gradient index cavities designed by optical conformal mapping

We start with a homogeneous dielectric cavity of an infinite cylinder with a unit disk cross section in an original virtual (OV) space with Cartesian coordinates $\left(u,v,w\right)$, taking $w$ -axis along the axis of the cylinder as shown in Fig. 1(a). In the OV space, the uniform refractive index of the cavity is $n$ and the refractive index of the exterior region is 1 as the refractive index of vacuum or practically that of air. In the physical space with Cartesian coordinates $\left(x,y,z\right)$, a realizable 2D gradient index cavity model (hereafter called ‘transformation cavities’) can be built by applying a conformal coordinate transformation only to the inside region (${\text{\Omega}}_{1}^{OV}$) of the unit disk cavity in the OV space and assigning the uniform refractive index of ${n}_{0}\text{}=\text{}1\text{}$ for the outside region $({\text{\Omega}}_{0}$) of the transformed cylindrical cavity ($z=w$) with a deformed cross section as shown schematically in Fig. 1(b). For these transformation cavities with deformed shapes, it was demonstrated that directional WGMs called ‘conformal WGMs’ (cWGMs) can be formed [7]. The conformal coordinate transformation from the OV space to the physical space is given by an analytic function $\zeta =f\left(\eta \right)=x\left(u,v\right)+iy\left(u,v\right)$ of a complex variable $\eta =u+iv$, that satisfies the Cauchy–Riemann equations,

Due to the translation symmetry along the $z$ -axis of the aforementioned cylindrical geometry, Maxwell’s equations simplify to a 2D scalar wave equation for the two polarization of light with respect to the $z$ -axis [10]. Optical modes in a transformation cavity with a deformed boundary are found by solving the following 2D scalar wave equation,

where $\Delta \equiv {\partial}_{x}^{2}+{\partial}_{y}^{2}$, position vector, $r=\left(x,y\right)=(r\mathrm{cos}\theta ,r\text{sin}\theta )$ and the refractive index $n\left(r\right)$ is given by## 3. Poincaré Husimi function at transformation cavities

The Husimi function, a quasiprobability distribution in phase space is originally defined as the overlap of the wave function with a coherent state that represents a minimal-uncertainty wave packet. Crespi *et al.* derived so-called a Poincaré Husimi function at the boundary of 2D closed billiard systems by projecting the conventional Husimi function from full 4-dimensional phase space onto a 2D reduced phase space, namely on the PSOS at the system boundary [13]. It has been widely used in 2D closed billiard systems in the context of quantum chaos, especially in order to study the semiclassical regime [14, 17]. For the case of dielectric systems with piecewise constant refractive indices, Hentschel, *et al.* have derived four different Poincaré Husimi functions, corresponding to incident and emerging waves on inside and outside of the interface by using a semiclassical (saddle point) approximation with a Green’s function technique [15]. Among them, the Husimi functions for the internal waves have been useful when exploring ray-wave correspondence in optical dielectric cavities.

Their expression for circular cavity can be readily extended to cavities with deformed boundaries as below. Four different Husimi functions which represent the intensities of the incident and emerging waves from arc length $s$ along the cavity boundary $\text{\Gamma}$ in the directions ${\chi}_{j}$ ($j\text{}=\text{}1$; interior region, $j\text{}=\text{}0$; exterior region), respectively are (see Fig. 2)

The boundary Husimi function of each region is valid when the refractive index is constant in each region. Therefore, in the case of the transformation cavities the Eq. (6) can be used for the exterior region where the refractive index is 1, but the formula cannot be directly applied to the interior region. However, fortunately, if we construct a virtual space through inverse conformal transformation, a transformation cavity is mapped back to the original unit disk cavity with a constant refractive index, so the boundary Husimi function for the resonant modes in the interior region can be calculated there by the above formula. In other words, it becomes possible to look at the phase space structure that shows the emission direction of the resonant modes and the location on the boundary where the major emission leaks out.

So, we introduce the reciprocal virtual (RV) space with Cartesian coordinates $\left(u,v,w\right)$, which is obtained from the physical space by the inverse conformal mapping, $\eta ={f}^{-1}\left(\zeta \right)$, as shown in Fig. 1(c). It is noted that the inverse conformal mapping is not a one-to-one mapping and therefore the RV space corresponding to the OV space should be selected. Functions, differential operators, and relevant symbols in the RV space will be expressed with tildes. Under the inverse conformal mapping, the scalar wave Eq. (2) is transformed to

In the RV space, the Husimi functions for the incident and the emerging waves in the unit disk cavity with uniform index $n$ can be written as below.

In order to calculate the Husimi function in the RV space, it is necessary to obtain $\tilde{\psi}$, ${\tilde{\partial}}_{\nu}\tilde{\psi}$ on the boundary of the unit disk cavity. To obtain these from the $\psi $,${\tilde{\partial}}_{\nu}\psi $ in physical space, we need to know the transformation rules between them. According to TO [4,18], 3-dimensional transformation rules for fields and gradient operators between the physical space and the RV space can be expressed as follows,

First of all, for TM/TE polarizations, wave functions in the physical space and their counterpart functions in the RV space are identical as can be shown by using the above field transformation rules;

Thus, normal derivative transforms as

## 4. Examples: a limaçon-shaped transformation cavity

As examples, we calculate the Husimi function for incident waves in the unit disk cavity in the RV space for a high-Q resonance (cWGM) and a low-Q resonance in a limaçon-shaped transformation cavity. The conformal mapping which transforms the unit circle to the limaçon [7,20], is given by

where $\eta =u+iv$ and $\zeta =x+iy$ are complex variables that denote positions in the respective complex planes, $\u03f5$ is a deformation parameter, and $\beta $ is a positive scaling factor. We use $\u03f5=0.23$, $\beta =0.685$ and $n=1.8$, the refractive index of unit disk cavity in the OV space. Wave functions of resonant modes formed in the transformation cavity can be calculated by a boundary element method (BEM) exploiting the RV space [7,21] or finite element method (FEM) based electromagnetic field solvers e.g., COMSOL Multiphysics.Using the BEM, we obtained a high-Q TM mode which is identified as a cWGM with a complex wave number $k=12.2333-i0.0022$. The Q-factor of the resonance is 2776.23. The mode intensity distribution, the far field pattern and the intracavity Husimi function for incident waves are depicted in Fig. 3 ($\tilde{s}$ is normalized with $2\pi $, the total arc length of the unit disk cavity). As can be seen in Fig. 3(a), the cWGM is well confined by total internal reflection of light around the rim of the limaçon-shaped transformation cavity and this feature is reflected in the Husimi plot as the two band-type intensities located in the regions $\left|\mathrm{sin}{\tilde{\chi}}_{1}\right|{\left|d\eta /d\zeta \right|}_{\tilde{s}(\tilde{r})}^{-1}\text{}/n$ where the r.h.s. of the inequality represents the critical lines of total internal reflection depicted by two yellow solid curves in Fig. 3(c) (the $\pm $ signs in $\mathrm{sin}{\tilde{\chi}}_{1}$ denote counterclockwise (CCW) and the clockwise (CW) circulations of light waves, respectively). The bi-directional far field emission shown in Fig. 3(b) can be explained by the Husimi plot in the RV space in Fig. 3(c); the closest position between the intensity bands and critical lines in $\tilde{s}$ is $\tilde{s}=0$ or 1 which corresponds to the position $\zeta =($ 0.84255, 0) on the boundary of the limaçon-shaped transformation cavity where maximum evanescent leakage or tunneling emissions come out in tangential directions i.e., parallel to the $y$ -axis [7].

Also, we obtained a low-Q TM mode with $k=12.5960-i0.1302$ whose intensity pattern is depicted in Fig. 4(a). The Q-factor of the resonance is 48.37. The far field emission shown in Fig. 4(b) is refractive emission as can be seen clearly from the Husimi plot in Fig. 4(c) where the intensity peaks of CW and CCW waves are located below the critical lines, i.e., in the regions $\left|\mathrm{sin}{\tilde{\chi}}_{1}\right|<{\left|d\eta /d\zeta \right|}_{\tilde{s}(\tilde{r})}^{-1}/n$. To analyze this refractive emission approximately, we resort to Snell’s law, ${n}_{1}\mathrm{sin}{\chi}_{1}=\mathrm{sin}{\chi}_{0}$ in the physical space. Because of conformality of the transformation, one can take $\mathrm{sin}{\chi}_{1}=\mathrm{sin}{\tilde{\chi}}_{1}~-0.385$, the peak position of lower intensity in the Husimi plot. The peak position in $\tilde{s}$ coordinate is $\tilde{s}=0.1256$ which can be converted into $s=0.1732$, normalized value with the whole boundary length of the limaçon. The intracavity refractive index at this position is ${n}_{1}$ = 1.9314. So, the refraction angle, ${\chi}_{0}=-{48.04}^{\xb0}$ is obtained from $\mathrm{sin}{\chi}_{0}~-0.7436$. In polar coordinate, angular coordinate of the main emission point on the boundary of the limaçon-shaped transformation cavity is 0.929 (rad) or ${53.226}^{\xb0}$ and the outward normal direction at the point is ${58.93}^{\xb0}$. Thus, we can conclude that the main refractive emission comes out at the boundary point of angular coordinate ${53.226}^{\xb0}$ in the direction ${10.89}^{\xb0}$. By the mirror symmetry with respect to the $x$ -axis, the peak of upper intensity in the Husimi plot corresponds to the refractive emission at the boundary point of angular coordinate $-{53.226}^{\xb0}$ in the direction $-{10.89}^{\xb0}$. Thus the far field pattern in Fig. 4(b) can be understood as the interference of the two dominant (out of phase) refractive emission.

## 5. Conclusion

A method to find a widely used phase space representation i.e., the Husimi representation for the internal waves on the boundary of a transformation cavity was presented by constructing a virtual space with a uniform dielectric cavity *via* inverse conformal transformation. As a verification of the method, Husimi plots for a high-Q and a low-Q resonance were obtained in the virtual space and their emission characteristics could be revealed from the plots. The phase space description of the corresponding modes in the virtual space agrees well with the far-field directionality of the resonant modes of a transformation cavity in the physical space. We expect that our method will be useful in studying waves in transformation cavities.

## Funding

National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIP) (No. 2017R1A2B4012045 and No. 2017R1A4A1015565); Center for Advanced Meta-Materials (CAMM) funded by Korea Government (MSIP) as Global Frontier Project (CAMM 2014M3A6B3063709); Institute for Basic Science in Korea (IBS-R024-D1).

## Acknowledgments

We appreciate our previous collaborator, late Professor Dr. S.-Y. Lee.

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**21. **In preparation (to be published elsewhere).