## Abstract

When two microdisks are placed close to each other and the evanescent fields of their whispering gallery modes are overlapped, a strong coupling can be induced in the modes and lead to a doublet state. We attempt to evaluate the frequency splittings of the doublets by applying a semiclassical analysis in the regime of small wavelengths. Since a whispering gallery mode in a microdisk is a leaky mode, an established semiclassical method that deals with coupled closed systems is modified. As a result, we attain an analytic formula which can conveniently compute the frequency splittings of coupled whispering gallery modes. The derived formula is verified by demostrating a perfect agreement with numerical solutions of Maxwell’s equations.

© 2013 Optical Society of America

## 1. Introduction

A microdisk is a small two-dimensional optical cavity which is made out of a dielectric with a rotational symmetry [1]. Due to the rotational symmetry, one can imagine a ray trajectory which is eternally confined in a microdisk by total internal reflection. Along such a confined trajectory a mode can be formed. This kind of a mode is called a ‘whispering gallery mode (WGM)’, which is known to have an extremely high quality factor. Thus, various applications of the microcavity with whispering gallery modes have been developed, such as low-threshold lasers and optical sensors with high sensitivity.

If two microdisks are placed very closely to each other, two WGMs can be strongly coupled through the evanescent field. Such evanescent coupling leads to a doublet state which comprises a bonding and an anti-bonding mode (Fig. 1). Because the coupled WGMs of two microdisks have many valuable features for applications, the experimental technics to manipulate the evanescent coupling of WGMs have been continuously developed up to now [2, 3, 4]. Also numerous applications of the system are already established and show interesting results, such as the generation of slow-light [5], the optical realization of graphene [6] and the induction of optomechanical vibration [7].

The analysis for the experimentally realized evanescent coupling of WGMs has so far mostly resorted to numerical computation, using algorithms such as ‘Boundary/Finite element methods’ [8, 9] or ‘Finite Domain Time Difference method’ [10], which require quite a lot of computing cost. Using Bessel functions as basis, an analytic method is also available[11], and this method is advantageous for the precision.

In this work, we introduce a new analytic approach based on the semiclassical analysis, which can provide a convenient formula for the frequency splittings of coupled WGMs. Originally, semiclassical physics was derived from eikonal approximation, which is a ray-dynamics based approximation for optics in the regime that the wavelength is smaller than a characteristic system size. Since quantum mechanics was established, the eikonal approximation has been actively employed in many fields of quantum physics. Accordingly the remarkable development of theoretical methods has been made in semiclassical physics. As quantum mechanics and optics have the Helmholtz equation

as a common governing equation for a stationary state, we can carry over the semiclassical methods to optical systems.As the resonant wavelength of a WGM is much smaller than the radius of a microdisk in most of cases, the application of the semiclassical analysis is safely justified. In this work, we employ Wilkinson’s formula, a semiclassically derived equation which deals with integrable systems coupled via tunneling. Because a single microdisk is an integrable system from a ray dynamical point of view and the evanescent field can be viewed as a extension of a WGM in ray dynamically forbidden region, the evanescent coupling of WGMs can be handled by the Wilkinson’s formula. Hence, by customizing the formula for coupled WGMs, the frequency splitting of the coupled microdisks (radii *R*) can be derived as a function of the half distance between the centers of the two disks (*d*), the resonant wave number (*k*), and the azimuthal mode number (*m*),

In addition, the derivation of Eq. (2) can give a physical insight into the distribution of *k*-vectors around coupled WGMs, since the equation is derived on the basis of ray dynamics.

In this paper, the derivation of Eq. (2) is presented step by step. In the next section, it will be first outlined how to derive the Wilkinson’s formula in general. In section 3, it will be demonstrated how well a semiclassical analysis can reproduce properties of a single microdisk such as its resonance wave number and attenuation, so that we make sure that the semiclassical analysis works for microdisks, even though they have open boundary conditions and attenuation in their WGMs. In section 4 we will finally tackle the frequency splittings of coupled microdisks. By modifying and applying Wilkinson’s formula to semiclassically built-up WGMs, Eq. (2) will be derived, and be verified by comparing its result with the numerical computation.

## 2. Wilkinson’s formula

If two potential wells stand closely by each other in quantum mechanics and two respective stationary states of the potentials are resonantly coupled through tunneling, the coupled states get split to form a doublet. In such a case, the energy splitting of a doublet can be calculated by the mode overlap as follows:

*V*and

_{L}*V*are the potential wells on the left-hand and the right-hand side, and

_{R}*ψ*and

_{L}*ψ*are the corresponding stationary states. In Ref. [12], C. Herring showed that if the coupled systems are two-dimensional, Eq. (3) can be reduced to a line integral

_{R}Based on Eq. (4), M. Wilkinson derived an analytic formula which can calculate energy splittings of coupled integrable systems, using the semiclassical analysis. In this section, a brief sketch of Wilkinson’s formula will be presented. For the details of the formula, we refer readers to M. Wilkinson’s original work [13].

A stationary state of a two-dimensional integrable system can be semiclassically described with a two-dimensional vector of actions *I* in action-angle variables:

*S*is the generating function which has actions

**I**and coordinates

**r**as independent variables. Here, the generating function and the actions are defined in a classically forbidden region as well as in a classically allowed region.

By substituting Eq. (5) into Eq. (4), the semiclassical description for energy splittings in coupled integrable systems is derived as

*ħ*

^{2}) are neglected, because

*ħ*is supposed to be small enough.

For the same reason, the stationary phase approximation can be applied to Eq. (6), so that the integration is analytically evaluated. The physical meaning of the stationary phase approximation is that a trajectory satisfying the classical least action principle has the dominant contribution to wave function propagation in quantum mechanics. Then, the contribution of its neighboring trajectories can be factorized by taking the Taylor’s expansion of the phase *S/ħ*, and the integrand in Eq. (6) can be easily integrated by using Fresnel’s integral.

If the stationary phase approximation is applied to a trajectory which consist of two steps, then a composite trajectory which has the most contribution to wave function propagation should fulfill the following composite properties at the boundary between the steps represented by the coordinate *x*_{1}:

*x*

_{1}as the coordinate on it, the trajectory which has the most contribution to tunneling can be identified and the integral in Eq. (6) can be evaluated in the form

*ω*and

_{L}*ω*are frequencies of vertical motions to ∑, and {

_{R}*I*,

_{R}*I*} is the Poisson bracket of the actions.

_{L}S. Creagh et. al [15] have applied this formula to two coupled circular potential wells and derived a formula for the corresponding energy splittings. However, a microdisk has an essential difference to such closed systems, because WGMs in a microdisk have leakage through evanescent field. In terms of ray dynamics, this implies that two separate real manifolds are connected by a complex manifold in the classically forbidden region. Therefore, the classically forbidden region of a microdisk is spatially limited, whereas that of a circular closed potential well is infinitely stretched. For the same reason, a mode in a microdisk can not be a stationary state, but a quasistationary state which is characterized by a complex eigenvalue. Hence, there must also be a difference in the application of Wilkinson’s formula.

In the next section, we will make sure that the semiclassical analysis is available for a microdisk despite its openness, by applying the WKB method.

## 3. Validity of semiclassical analysis for a single microdisk

In order to apply the semiclassical analysis to a microcavity, we have to build up a proper ray dynamical model corresponding to a mode. Let us assume that a WGM with wavenumber *k* is excited in a microdisk with a homogeneous refractive index *n*. Taking advantage of the analogy with quantum mechanics, the momentum of a ray can be set as *nk* in the microdisk and as *k* outside the microdisk by putting *ħ* equal to 1. Then, the ray motion in the microdisk can be described by a particle dynamics in a shallow circular potential well, and its Hamiltonian is given by

*k*

^{2}, and the total energy of the ray, (

*nk*)

^{2}/2 always lies above the well. However, such a shallow potential well can confine a ray motion inside of it, when its angular momentum is large enough to fulfill the inequality

*l*>

*kR*, which is exactly the same as the total internal reflection condition.

Since the given microcavity has a cylindrical symmetry, the Hamiltonian in Eq. (11) can be separated into radial and azimuthal motions, taking its angular momentum *l* as a constant of motion. Then, the radial motion is projected onto one dimensional oscillation with the effective Hamiltonian

From Fig. 2(a), one can notice that the ray oscillates between *l/nk* and *R* in the effective potential well, and the region *R* < *r* < *l/k* is classically forbidden. However, if a WGM corresponding to this ray motion is excited, this mode is able to leak out by tunneling through the effective potential barrier in the forbidden region.

Based on the separation of degrees of freedom, the semiclassical quantization of ray trajectories can be performed. As remarked, the angular momentum of the internal ray motion is a constant, and can be represented by the rotational mode number *m* which is an integer, when the system is brought to wave mechanics. As the ray motion in the radial degree of freedom is oscillating, the trajectory forms an enclosed area in the corresponding phase space portrait. By setting the enclosed area in phase space equal to 2*π* times an integer, we can derive a semiclassical quantization condition according to the WKB theory. In this derivation, the phase shifts associated with the both turning points in the effective potential well have to be taken into account. Because the reflecting boundary on the right-hand side of the confining one-dimensional potential in Fig. 2(a) is smooth, a constant phase shift of *π*/2 is involved on this side. The phase shift on the other side is given by the Fresnel’s law as

*ν*is a parameter, determined by the polarization of a mode. If the polarization is given such that its electric field is parallel to the cavity plane (Transverse Electric Mode),

*ν*is given by

*n*

^{2}. In the opposite case (Transverse Magnetic Mode), it is given by 1;

*N*is assigned as a radial mode number, and characterize a resonant mode together with a rotational mode number

*m*.

The enclosed area *S*_{re} can be analytically derived in terms of a rescaled resonant wavenumber *nkR* [14]. By substituting the analytic formula of *S*_{re} into Eq. (17), the following formula is obtained for the resonant wave number *nkR*:

As a microcavity is an open system due to the tunneling, the leakage of a mode can be characterized by an attenuation rate, Γ, which is related to the imaginary part of a resonant wavenumber (= −Γ/(2*nkR*)). As done in [17], it is attempted to estimate the attenuation rate of a WGM by semiclassically computing the tunneling action *S*_{im} [14]:

*τ*is the period of the oscillatory motion in the effective potential well. As the tunneling action of the forbidden region in Fig. 2 can be expressed as

By reproducing the complex resonant wavenumber (= *nkR* − *i*Γ/(2*nkR*)), we have verified that the semiclassical analysis is applicable to microdisks.

Corresponding to the tunneling action in Eq. (20), a trajectory can be imagined in the forbidden region. By taking the absolute value of the imaginary radial momentum in the classically forbidden region and keeping the angular momentum as a constant of motion, the complex trajectory in phase space can be projected onto configuration space (Fig. 4(b)). The trajectory starts at a reflecting point of a ray on the microcavity boundary and runs around the cavity spirally. At the end of the forbidden region, the trajectory escapes the cavity following a straight line which is tangent to the outer boundary of the forbidden region.

By the forbidden region in this description, we can define the region around a microdisk where the evanescent field is formed [18]. This region corresponds to the effective potential barrier in Fig. 2(a) which lies in *R* < *r* < *m/k*. Hereafter, we will call it the evanescent region.

## 4. Energy splittings in two coupled microdisks

In this section, we will eventually tackle the energy splittings in coupled double microdisks by applying Wilkinson’s formula. As introduced previously, the physical situation we will deal with is that two microdisks are coupled through the overlapped evanescent fields of WGMs. If this situation is described in terms of ray dynamics, manifolds corresponding to the WGMs can be constructed inside the microdisks, and the complex extensions of them are intersecting as illustrated in Fig. 5. Here, we introduce a postulate about the coupling of two WGMs: The tunneling paths on the composite complex manifold have dominant contribution to the coupling of the two microdisks. Because other contributions such as scattering of radiation from one microdisk on the other carry tiny coupling, they are supposed to be negligible.

Figure 5 shows ray trajectories corresponding to two (21, 1)-WGMs in each microdisk. On the complex extension of them one can find two trajectories connected smoothly. This implies that they satisfy the composite property of the stationary phase approximation in Eq. (8) and make the most contribution to tunneling. The identified trajectory is highlighted by a red line in Fig. 5.

Since the profile of the refractive index is given by a step function (see Eq. (12)) at the boundary, the semiclassical approximation is not available for wave functions at the boundary. Therefore, in order to apply Wilkinson’s formula to double microdisks we need to modify the semiclassical approximation of the wave function in Eq. (5) using the following ansatz, as done in Ref. [15]:

*ψ*(

**r**) is an analytic solution and

*ψ*

_{sc}(

**r**) is its semiclassical approximation. In the evanescent region (

*R*<

*r*<

*m/k*), a WGM is written in the analytic form

*N*is a normalization factor. Conventionally, the norm of a WGM is defined by integrating the squared amplitude of the mode only in the interior of a microdisk: By means of Lommel’s integral [19], the integral in Eq. (25) can be evaluated as

_{m}First, we make use of the above asymptotic to further simplify the norm of a WGM. By applying Eq. (28) together with the boundary condition of a microdisk (Eq. (41) in Appendix 5), Eq. (26) is reduced to

Then, by substituting Eqs. (28) and (29) into Eq. (24) a WGM in the evanescent region can be finally written asMeanwhile, the determinant in the amplitude of Eq. (5) is evaluated for a WGM by canonical transformations as follows:

*ω*and

_{r}*k*are the frequency and the wavenumber of the radial ray motion. Then the semiclassically approximated wave function of a WGM is written as

_{r}^{2}must be multiplied to Wilkinson’s formula in Eq. (10).

The Poisson’s bracket in the Wilkinson’s formula is computed as

by setting two angular momenta*l*and

_{L}*l*as dynamical invariants, where

_{R}*d*is a half of the distance between two centers of microdisks. Thus the final form of the Wilkinson’s formula for coupled microdisks is written as

In addition, the validity of Eq. (35) is analyzed. For this purpose, the angular displacement of an imaginary trajectory up to the connecting point (e.g. Fig. 5) is first expressed as a function of *m*, *k* and *d*.

*m*Δ

*θ*> 1, we derive the minimum distance

*d*for TM(

_{min}*m*, 1)-WGMs, until which Eq. (35) is valid. As can be intuitively predicted, Fig. 6(b) shows that

*d*gets smaller with increase of the angular mode number

_{min}*m*in the range of

*m*> 5. However, when

*m*gets less than 5, a drastic decrease of

*d*is noticed in Fig. 6(b). This aspect of

_{min}*d*cannot be viewed as a physical fact, because in this range the wavelength is comparable to the cavity size and the semiclassical approach is accordingly not valid. In a such range, coupled WGMs can show a discrepancy to a semiclassical approach [11].

_{min}## 5. Conclusion and discussion

In this work, we addressed the frequency splittings of doublets in coupled optical microdisks, which arises when evanescent fields of WGMs are spatially overlapped. By implementing semi-classical analysis, a formula to calculate the frequency splittings of doublets was derived. The resultant formula was verified by demonstrating a perfect agreement with analytic and numerical solutions of Maxwell’s equations.

Additionally, we would like to discuss about the result that the splittings are real-numbered values. Strictly speaking, the splitting should be given by a complex number, although it is approximated as a real number in Eq. (35), because the channel mediating the coupling is open. The openness of the channel can be seen in the visualization of the complex trajectory. As Fig. 5 shows, the complex trajectory has a connection to the outside, and it can be imagined that a WGM can be coupled to an outgoing mode. The openness of the coupling channel manifests itself in some observable phenomena (see e.g. Ref. [2]). Elsewhere we will further discuss about this subject.

Recently, the evanescent field of a WGM in a deformed microcavity has been semiclassically analyzed [21] as well as their internal structure [22]. Thus, the theoretical approach proposed in this work can be extended to the coupling of deformed microcavities.

## Appendix

## Quasistationary modes of microdisks: solutions of Maxwell’s equations

Because of the rotational symmetry, a WGM of a microdisk can be described by using Bessel functions as basis. The internal modes of a bare microdisk can be represented by Bessel functions of the first kind *J _{m}*, because there is no source or sink in the cavity. Depending on the polarization of a mode, either internal electric field or magnetic field distribution of a mode can be set as

*ψ*(

**r**) =

*J*(

_{m}*nkr*)

*e*. At the boundary of the disk, this internal mode distribution and its normal derivative on the boundary should be equal to Hankel functions of the first kind ${H}_{m}^{(1)}$. With a specified rotational mode number

^{imϕ}*m*, the boundary matching equations are written as follows:

*ν*is the same as in Eq. (16). Then the rescaled wavenumbers

*kR*stisfying the following condition, give us the resonances of a microdisk.

*m*is fixed, a radial mode number is assigned to every resonance in order of the magnitude of

*kR*.

By using the recursive relation of Hankel functions

Eq. (39) can also be written as## Asymptotic form of Hankel functions in evanescent region

In this section, it is presented how to derive an asymptotic form of Hankel functions of the second kind in the evanescent region [23]. In the evanescent region of a WGM, the value of the mode function is approximately given by a pure imaginary number, although the mode distribution in the exterior of a microdisk is represented by a Hankel function. This implies that the real part of the Hankel function in the evanescent region, i.e. the component of the Bessel function of the first kind is close to zero. Therefore, the Hankel function in this case is well approximated by its imaginary part, i.e. the Bessel function of the second kind.

If*m*≫ 1, then a Bessel function of the second kind can be written in the form

*z*) is the Airy function of the second kind and

*ζ*(

*z*) is the solution of the differential equation The function

*ζ*(

*z*) is reduced to

*ζ*(

*z*) is related to

*f*(

*x*) in the main text (see Eq. (21)). The Airy function of the second kind Bi(

*z*) is the secondary solution of the differential equation

The Airy function of the second kind can be approximated by the first term of the Poincaré-type expansion in the limit of a large argument (*z* ≫ 1):

The substitution of Eqs. (45) and (47) into Eq. (43) leads to

## Acknowledgments

This work was financially supported by the
German Research Foundation (DFG) within the framework of the *Forschergruppe* FOR760.

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