Designing arbitrary-shaped whispering-gallery cavities based on transformation optics

Sang-Jun Park; Inbo Kim; Jinhang Cho; Yushin Kim; Muhan Choi

doi:10.1364/OE.27.016320

1. Introduction

Dielectric microcavities have been intensively studied for the last two decades due to their role as an important platform in theoretical aspect as well as their high potential in device applications [1]. Particularly, dielectric disk cavities can support high-Q whispering gallery modes (WGMs) due to the total internal reflection (TIR) along their boundaries. However, owing to the rotational symmetry, emission directionality cannot be attained, which is a serious drawback in device applications. To remedy this drawback, since the first demonstration of the WGM disk laser [2], considerable effort has been made to break the rotational symmetry without seriously spoiling Q-factor [3, 4]. One approach is the introduction of local defects such as hole [5], line [6], or notch [7, 8]. Another popular one is to deform the cavity shape smoothly. Such cavities, called asymmetric resonant cavities, have attracted great interest because of their link with quantum chaos [9, 10], non-Hermitian physics [11], and low-threshold microlasers [12] which are regarded as a potential candidate for directional light sources in photonic integrated circuits.

Recently, Kim et al. [13] reported another kind of approach handling above problem. They utilized the concept of transformation optics which can manipulate the wave propagation at will by spatially varying the constitutive parameters where the effect of coordinate transformation is encoded [14, 15]. Their cavities, called transformation cavities (TCs), therefore have gradient index profiles in contrast to the conventional ones. Because conformal mappings are reflected in their ray trajectories, incident angles to the TC boundaries are maintained regardless of cavity deformation. Thus, TCs support high-Q conformal whispering gallery modes (cWGMs) that inherently possess emission directionality as verified by limaçon-shaped and (rounded) triangular-shaped TCs [13].

While there are a plethora of possibilities for other shapes of TC, a general design methodology for arbitrary-shaped TCs is not proposed yet. In this regard, the major problem is to find a smooth conformal mapping from the unit circle domain in a virtual space to a desired TC domain in the physical space. Even when a conformal mapping connecting a circle with other shape that is a TC boundary is found, it may contain unwanted singularities inside. In this paper, we show that arbitrary-shaped TCs can be designed by virtue of a quasi-conformal mapping (QCM) method without suffering such problems by representative examples. The QCM method in transformation optics [16] was first devised to find a nearly conformal mapping in all-dielectric carpet cloaking, while in our case, the resulting mappings are exactly conformal due to the fully sliding boundary condition which are distinguished from the former.

Fig. 1 Schematic illustration of a limaçon-shaped transformation cavity in the physical space and its associated two virtual spaces: (a) a uniform unit disk cavity in an original virtual (OV) space, (b) a limaçon-shaped transformation cavity in the physical space, and (c) its reciprocal virtual (RV) space obtained by inverse conformal mapping. Note that curved grids are not coordinate grids but transformed images of Cartesian coordinate grids. (Colorbar for refractive index profiles is shared.)

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2. Quasi-conformal mapping method for transformation cavities

Under a general coordinate transformation from $(u, v, w)$ to $(x, y, z)$ , the form of Maxwell’s equations is preserved by following constitutive parameters [14]:

ε^{'} = \frac{Λ ε Λ^{T}}{| detΛ |}, μ^{'} = \frac{Λ μ Λ^{T}}{| detΛ |},

where

Λ = \frac{\partial (x, y, z)}{\partial (u, v, w)}

is the Jacobian matrix. Especially for conformal mapping

ζ = f (η)

between two-dimensional spaces (

η = u + i v

,

ζ = x + i y

and

w = z

), the Cauchy-Riemann equations

\frac{\partial x}{\partial u} = \frac{\partial y}{\partial v}, \frac{\partial x}{\partial v} = - \frac{\partial y}{\partial u},

highly simplify above tensorial relations (Eq. (1)) to the following refractive index profile:

n^{'} = n {| \frac{d ζ}{d η} |}^{- 1}

for transverse electric (TE) or transverse magnetic (TM) polarized electromagnetic waves. TCs exploit not only the angle-preserving property of conformal mapping, but also this remarkable simplification like other conformal transformation optics-based devices [17]. We note that to obtain a TC in the physical space, Eq. (3) should be applied to only the inside of the uniform unit disk cavity (UUDC) with refractive index n₀ in the original virtual (OV) space. That is, the refractive index function in physical space is given as

n (x, y) = {\begin{array}{l} n_{0} {| \frac{d ζ}{d η} |}^{- 1}, & inside TC, \\ 1, & outside TC . \end{array}

Thus, one can construct another space called reciprocal virtual (RV) space which is obtained by inverse conformal mapping from the physical space. The refractive index function in the RV space is written as

\tilde{n} (u, v) = {\begin{array}{l} n_{0}, & inside UUDC, \\ {| \frac{d η}{d ζ} |}^{- 1}, & outside UUDC . \end{array}

Fig. 2 (a) Ring-shaped cavity with refractive index 1.8 in OV space. (b) Quadrupole-shaped TC in the physical space constructed by $f_{q} (η) = β_{q} [η + \frac{α_{q}}{2} (η^{3} + \frac{1}{η})]$ where $α_{q} = 0.1$ and $β_{q} = 0.7341$ . Index profiles are expressed by the shared color bar. The gray range represents the index cut-off over 3.3. It reaches infinity at the two ends of the branch-cut (magenta-colored). In (a), the dashed green region inside the magenta-colored closed curve spans the entire physical space by f_q. (f_q is not injective in the unit circle domain.)

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As an example, Fig. 1 illustrates OV space, physical space, and RV space for lima $c ç$ on conformal mapping $f_{l} (η) = β_{l} (η + α_{l} η^{2})$ where $α_{l} = 0.155$ , $β_{l} = 0.7634$ and $n_{0} = 1.8$ . The blue grid in the OV space (Fig. 1(a)) is transformed to the curved blue grid in the physical space (Fig. 1(b)) by f_l. Note that to obtain transformation cavity in the physical space (Fig. 1(b)) from OV space (Fig. 1(a)), the transformation is applied only for the cavity region where the blue grid covers. In contrast, to obtain the RV space from physical space, inverse transformation throughout the entire space is required as depicted in the Fig. 1(b) and Fig. 1(c) by the gray grids.

Then, resonant modes formed in TCs can be explored by two equivalent Helmholtz equations with complex eigenvalue k:

[\partial_{x}^{2} + \partial_{y}^{2} + n^{2} (x, y) k^{2}] ψ (x, y) = 0

with outgoing-wave condition

ψ (x, y) \sim h_{k} (θ) e^{i k | ζ |} / \sqrt{| ζ |} for | ζ | \to \infty

in physical space and

[\partial_{u}^{2} + \partial_{v}^{2} + {\tilde{n}}^{2} (u, v) k^{2}] \tilde{ψ} (u, v) = 0

with outgoing-wave condition

\tilde{ψ} (u, v) \sim g_{k} (ϕ) e^{i k | η |} / \sqrt{| η |} for | η | \to \infty

in RV space, where

x + i y = | ζ | e^{i θ}

,

u + i v = | η | e^{i ϕ}

(

0 \leq θ, ϕ < 2 π

) and

h_{k} (θ)

,

g_{k} (ϕ)

are far field distributions in the physical space and the RV space, respectively. This equivalence enables the boundary element method (BEM) and phase space description for TCs [13, 18].

Despite the simple formation of TCs, it is hardly to be expected that the desired mappings can be written in closed form like lima $c ç$ on and rounded triangle. In addition, there may be unwanted singularities inside the cavity. As an example of these pathological situations, we provide the well-known quadrupole shape [19] usually given in polar form as $r (θ) = β_{q} [1 + α_{q} cos (2 θ)]$ . One can easily show that

f_{q} (η) = β_{q} [η + \frac{α_{q}}{2} (η^{3} + \frac{1}{η})]

maps the boundary of the unit circle to that of the quadrupole shape. Unfortunately, however, f_q does not map the unit circle domain (Fig. 2(a)) onto the quadrupole domain (Fig. 2(b)), leaving the branch-cut (magenta-colored). Furthermore, singularities implying infinite refractive index occur at the two ends of the branch-cut. Instead of cutting off the enormously high index profile near the singularities inside, we focus on finding inherently singularity-free conformal mappings for any smooth boundary shapes of TCs. To this end, we consider the QCM method first introduced in transformation optics by Li and Pendry in the context of carpet cloaking [16]. Based on the grid generation technique [20], they found an optimal QCM that is notexact but close to the conformal mapping. This process can be conducted by finding an extremal of the Winslow functional

Φ = \iint \frac{Tr (g)}{\sqrt{det (g)}} D [u, v]

where

g = {ΛΛ}^{T}

; this functional produces unfolded grids which means that the optimal QCM has no singularity. Alternatively, to obtain an optimal QCM, one can solve the following inverse Laplace equations

\frac{\partial^{2} u}{\partial x^{2}} + \frac{\partial^{2} u}{\partial y^{2}} = 0, \frac{\partial^{2} v}{\partial x^{2}} + \frac{\partial^{2} v}{\partial y^{2}} = 0

which are the Euler-Lagrange equations of the Winslow functional in the physical space with sliding boundary condition [21].

Fig. 3 Schematic illustration of the QCM method: (a) Unit circle in OV space and (b) quadrupole (identical to Fig. 1(b)) in the physical space with two boundary constraints ( $| η | = 1$ , $\partial_{⊥} ϕ = 0$ , black), inner point constraint ( $| η | = 0$ , red) at the origin, and boundary point constraint (ϕ = 0, blue) at $(x, y) = (β_{q} (1 + α_{q}), 0)$ for Eq. (10). (c) UUDC with refractive index 1.8 in OV space, (d) quadrupole-shaped TC in physical space obtained by the QCM method. Color bar is shared for (c) and (d). The TIR condition (min $[n (x, y)] \geq 1.8$ ) is satisfied in (d).

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Fig. 4 Index profiles of conformal-cubic TC with $α_{c} = 0.08$ and $β_{c} = 0.8064$ obtained by (a) analytical and (b) QCM methods. (c) Index profiles of (a) and (b) on the x and y axes. Index profiles of lima $c ç$ on-shaped TC with $α_{l} = 0.25$ , $β_{l} = 0.7337$ , and $d = 0.3$ obtained by (d) analytical and (e) QCM methods. (f) Index profiles of (d) and (e) on the x and y axes. Color bar is shared for (a), (b), (d) and (e). These TCs commonly originate from the UUDC with refractive index 1.8 and satisfy the TIR condition ( $min [n (x, y)] \geq 1.8$ ).

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Hence, we numerically find the solution of Eq. (10) imposing the constraints shown in Fig. 3(b). Two boundary constraints ( $| η | = 1$ and $\partial_{⊥} ϕ = 0$ at the TC boundary; $\partial_{⊥}$ means the normal derivative in physical space) act as a free sliding boundary condition that connects the boundary of TC in the physical space with that of UUDC in the OV space. Under this boundary condition, above functional (Eq. (9)) is minimized at an exact conformal mapping. The point constraints ( $| η | = 0$ at an internal point and ϕ = 0 at a point on the boundary of TC), as will be shown by example shortly after, uniquely specify the map. Therefore, the resulting map is a unique smooth conformal map and it can be explained by the Riemann mapping theorem [22]. This theorem means that there exists a unique, singularity-free, smooth conformal mapping which maps a simply connected domain (in our case, this is unit disk) to an arbitrary simply connected domain, when satisfying specific point constraints. Our point constraints in Fig. 3(b) were given accordingly. Applying this method to the quadrupole (identical to Fig. 2(b)), a singularity-free quadrupole-shaped TC in physical space (Fig. 3(d)) is obtained from UUDC in OV space (Fig. 3(c)). The branch-cut and singularities emerged in Fig. 2(b) are automatically prohibited because the Riemann map is biholomorphic which is compatiable with the grid generation perspective as the functional diverges for folded grids.

To verify the aforementioned method, we consider two special cases whose conformal mappings can be written in simple closed forms. Firstly, we present what we call the conformal-cubic TC given by the following mapping:

f_{c} (η) = β_{c} (η + α_{c} η^{3}) .

Figure 4(a) shows the analytically obtained refractive index profile of the conformal-cubic TC. Despite its slight deviation from the quadrupole shape obtained by f_q, the result is largely different because the singularities are now located outside the TC. For the same geometry, we performed the QCM method, putting the inner point constraint ( $| η | = 0$ ) at the origin and boundary point constraint (ϕ = 0) at $(x, y) = (β_{c} (1 + α_{c}), 0)$ . As a result, both refractive index profiles obtained from analytic mapping and QCM method shown in Figs. 4(a) and 4(b) are well matched at every point. In Fig. 4(c) we compare them on the x-axis cut and y-axis cut for verification. Secondly, we present a lima $c ç$ on-shaped TC. At this time, we used composite mapping, $f_{l} \circ f_{m} (η)$ of lima $c ç$ on conformal mapping, $f_{l} (η) = β_{l} (η + α_{l} η^{2})$ and a Möbius transform

f_{m} (η) = \frac{η + d}{1 + d^{*} η},

which maps the unit circle domain to itself with a shifted center at d (

| d | < 1

). Accordingly, the inner point constraint and boundary point constraint were placed at

(x, y) = (β_{l} (d + α_{l} d^{2}), 0)

and

(x, y) = (β_{l} (1 + α_{l}), 0)

, respectively. Again, the refractive index profile obtained by the QCM method (Fig. 4(e)) shows good agreement with the result of the analytical method (Fig. 4(d)). Because in this way infinite number of composite conformal mappings for certain given shape can be made by the center-shifting Möbius transform f_m, the two point constraints are necessary for the uniqueness of the Riemann map. Particularly, the boundary point constraint (ϕ = 0) removes the rotational degree of freedom of the mapping but it does not affect the refractive index profile.

Fig. 5 Refractive index profiles of TCs: (a) Ellipse-shaped TC with major axis of length 1.5652 and minor axis of length 1.2806, (b) half-circle-half-conformal-cubic TC (its junction is represented by the green dashed line) with $β_{c} = 0.7462$ and $α_{c} = 0.1$ and (c) stadium-shaped TC with length of linear section = 0.4513 and radius = 0.4513. All TCs originate from the UUDC with refractive index 1.8 and satisfy the TIR condition ( $min [n (x, y)] \geq 1.8$ ). For all cases, inner point constraints ( $| η | = 0$ ) and boundary point constraints (ϕ = 0) are placed at the origins and right endpoints, respectively.

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Thus, in general, arbitrary-shaped TCs with smooth boundaries can be designed by the QCM method solving inverse Laplace equations under the free sliding boundary condition and the point constraints. Particularly, in the following three examples, we show that seemingly difficult cases can be resolved by this method. At first, Fig. 5(a) represents the singularity-free refractive index profile of an ellipse-shaped TC obtained by the QCM method. In contrast, Joukowski mapping $f_{e} (η) = β_{e} (η + \frac{a l p h a_{e}}{η})$ known to be a conformal mapping which maps unit circle to ellipse leads to singularities and branch-cut inside, similar to f_q. Secondly, for hybrid structures like Fig. 5(b) which consists of half circle and half conformal-cubic cavity shape, artificial attachment of half part of each TC (simple scaling and f_c) is useless because of the mismatch at the junction represented by the green dashed line. Lastly, it appears to be much more challenging to find the mapping for the stadium shape (Fig. 5(c)) [23]. Over again, as shown in Fig. 5(b) and Fig. 5(c), the QCM method provides the unique Riemann map without singularities and mismatch.

Fig. 6 Q-factor variation of resonances in TCs (red lines) and homogeneous cavities (blue lines) with quadrupole shapes as a function of deformation parameter α_q. Among them, near field intensity patterns of three representative cases ( $(α_{q}, β_{q})$ = (0.1, 0.7340), (0.2, 0.53) and (0.3, 0.3829)) are shown below the graph. ((i)∼(iii) for TCs and (iv)∼(vi) for the corresponding homogeneous cavities)

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Fig. 7 (a) Husimi function and (b) far field intensity pattern of cWGM (m=16) in quadrupole-shaped TC with $α_{q} = 0.1$ . In (a), the upper and lower bands represent intensity distributions of counterclockwise and clockwise wave components, respectively. Yellow lines are critical lines for TIR and white dotted lines are angles given by the semiclassical relation (sin $\tilde{χ} = m / n_{0} Re (k) R = \pm 0.8072$ , where R is the radius of UUDC in the RV space, i.e., R = 1).

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3. Resonances in quadrupole-shaped transformation cavities

In this section, we analyze resonances of so-designed quadrupole-shaped TCs (Eq. (8)) using commercial software (COMSOL Multiphysics). The graph in Fig. 6 shows the Q-factor variations of TM (out of plane electric field component) resonant modes in quadrupole-shaped TCs and homogeneous cavities as a function of deformation parameter α_q. The scale parameter β_q was selected to be its maximum for each α_q satisfying the TIR condition ( $n_{0} = 1.8 = min [n]$ ). Three representative cases ( $α_{q} =$ 0.1, 0.2, and 0.3) for each cavity are shown below the graph. Regardless of deformation, cWGMs (azimuthal mode number m = 16) in TCs clearly display the feature of standing mode including nodes along boundaries as with WGMs. The Q-factor of cWGM is sustained to some extent even under the severe deformation. On the other hand, the Q-factor of the homogeneous cavity is degraded much faster as α_q increases, and is finally overwhelmed by chaos ( $α_{q} =$ 0.3). In the ray dynamics picture, such chaotic behavior is expected because there is no whispering gallery-like caustic in the non-convex boundary ( $α_{q} \geq$ 0.2). The survival probability time distribution also exhibits a distinctive exponential decay for $α_{q} = 0.2$ compared to the algebraic decay for $α_{q} = 0.1$ [24]. To describe the wave functions in phase space, Poincaré Husimi function has been widely used, which is obtained by overlapping the wave function with minimum uncertainty wave packets on the cavity boundary [25]. Commonly used expression of the Husimi function for homogeneous cavities can also be used similarly for TCs by introducing the RV space [18]. However, in our case, the QCM method provides mappings only inside the TCs, i.e., the index profile outside the cavity in the RV space remains unknown. The absence of grid lines outside the TCs in Fig. 4(b), Fig. 4(e), and Fig. 5, in contrast to Fig. 4(a) and Fig. 4(d), describes this situation. Nevertheless, because the acquired mapping is conformal, at least we know that on the boundaries of TC and UUDC,

ψ (s) = \tilde{ψ} (\tilde{s}), \partial_{⊥} ψ (s) = {| \frac{d ζ}{d η} |}^{- 1} {\tilde{\partial}}_{⊥} \tilde{ψ} (\tilde{s})

hold. Here,

ψ (s)

,

\tilde{ψ} (\tilde{s})

,

\partial_{⊥} ψ (s)

and

{\tilde{\partial}}_{⊥} \tilde{ψ} (\tilde{s})

denote boundary wave functions and their normal derivatives at each space. (The transformation rule for arc length coordinates s and

\tilde{s}

in each space is given by the mapping obtained by the QCM method.) Thus, the method introduced in [18] can also be applied in this case. If it is not a conformal mapping, even when it has a small anisotropy, the relation regarding the normal derivatives of wave functions cannot be used. Figure 7(a) shows the Husimi function of quadrupole-shaped TC with α_q

= 0.1. We note that the critical lines (yellow lines) for TIR transformed from $sin χ = 1 / n (s)$ fluctuates in accordance to the refractive index along the boundary. Considering that the upper (lower) band of the Husimi function which represents the intensity of counterclockwise (clockwise) wave component lies above (below) the critical line, we can say that TIR is well-sustained for the cWGM. Dominant evanescent leakage occurs near the two opposite positions, $\tilde{s} / {\tilde{s}}_{m a x} = 0, 0.5$ (i.e., $θ = 0, π$ ), and it leads to bi-directional far field emission with the interference pattern shown in Fig. 7(b).

4. Conclusion

In conclusion, we showed that arbitrary-shaped TCs without singularity can be designed by the QCM method. This methodology was verified in the context of the Riemann mapping theorem with several examples. Near field patterns and far field distribution of cWGMs in so-designed quadrupole-shaped TC were obtained numerically. Additionally, a phase space description was given in order to analyze emission directionality. We believe that the proposed scheme will broaden the realm of TCs and will contribute to TC-based device researches.

Funding

National Research Foundation of Korea (NRF) grant funded by the Korean government (MSIP) (No. 2017R1A2B4012045 and No. 2017R1A4A1015565)

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Designing arbitrary-shaped whispering-gallery cavities based on transformation optics

Abstract

1. Introduction

2. Quasi-conformal mapping method for transformation cavities

3. Resonances in quadrupole-shaped transformation cavities

4. Conclusion

Funding

References

Cited By

Figures (7)

Equations (13)

Optics Express