Numerical study of light propagation via whispering gallery modes in microcylinder coupled resonator optical waveguides

By using a discontinuous spectral element method, we analyze evanescent wave coupling of whispering gallery modes (WGMs) in microcylinder coupled resonator optical waveguides (CROWs). We demonstrate successful light propagation by WGMs through a chain of coupled cylinder resonators, and that the speed of such propagation is strongly dependent on the inter-resonator gap sizes. Our simulations also show that light propagates slower by WGMs with bigger azimuthal numbers than by those with smaller azimuthal numbers. On the other hand, the light propagation by WGMs of the same azimuthal number appears to have the same speed in CROWs regardless of the size and the material of the resonators, indicating that the tail (the part of a WGM outside the resonator) determines inter-resonator coupling strength. © 2004 Optical Society of America OCIS codes: (000.4430) Numerical approximation and analysis; (230.5750) Resonators; (230.7370) Waveguides References and links 1. A. Yariv, Y. Xu, R. K. 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Introduction
The concept of coupled resonator optical waveguides (CROWs) [1]- [4] emerged a few years ago as a new way of integration of coupled cavities on a single chip.This concept attracted significant interest of photonic community due to the possibilities of manipulating light paths and photonic dispersions on a microscopic scale [5,6].Due to the weak evanescent coupling between adjacent high-Q resonators, the group velocity of light can be significantly reduced in such structures, which leads to applications in optical buffering and controllable delay lines.The coupled cavity waveguides (CCWs) were realized in a three-dimensional photonic crystal for a microwave regime [7], in two-dimensional photonic crystal waveguides [8]- [10] and in coupled semiconductor microcavities for a near-IR regime [11].CROWs are not limited to coupled cavities in photonic crystals and can be realized in a coherent coupling of whispering gallery modes (WGMs) in microrings [3], microdisks, dielectric bispheres [12,13] and long chains of microspheres [14].
The optical transport properties of such systems can be understood within the Tight Binding Method (TBM) for modes confined in dielectric "atoms" due to the total internal or Bragg reflection.In this model, the probability of optical tunnelling or photon "hopping" between adjacent resonators is determined by the spatial overlap between evanescent fields of the resonant modes.In the strong coupling regime, these modes form molecular states with bonding and antibonding properties [12,13].A particularly interesting case for practical applications is represented by the systems with the size and positional disorders where the uncoupled resonances are randomly detuned, but can be effectively coupled together if the average detuning is comparable with the normal mode splitting.This leads to efficient broad band optical transport observed recentlyin long chains of touching dielectric microspheres [14].The applications in optical delay lines however require a weak coupling [1] between adjacent cavities.This regime of coupling can be achieved by increasing the gap size between the cavities or by changing the refractive index contrast.A qualitative description of this regime has been achieved by coupled wave theory [1][15]- [17] based on introducing a phenomenological coupling constant.However, from a designer's point of view, more quantitative and self-consistent study of light propagation in CROWs is needed to address various issues such as propagation losses, propagation speed and efficiency of coupling of light in-and out-of such circuits.
For the applications of optical buffering, the propagation speed, the arrival time and the phase of signals are critical parameters to have.Thus, a high order and phase-preserving numerical technique is crucial for any numerical simulations.It has been well known that spectral element methods [18]- [21], in which approximations are based on high order orthogonal polynomials such as Legendre or Chebyshev polynomials, have the least numerical phase error among all numerical methods including Yee's scheme [22] and finite volume methods.Therefore, in the present work we apply a high order and phase-preserving discontinuous spectral element method (DSEM), a discontinuous Galerkin method [21] [23,24], to study the light propagation in CROW devices formed by identical dielectric resonators with circular symmetric properties supporting WGM resonances.The numerical simulations are performed for the case of dielectric microcylinders; however, this method can be readily extended to the cases of microrings, microdisks and microspheres.In contrast to previous numerical study by the finite difference Yee's scheme [22], the DSEM can provide high order accuracy to gauge the influence of the separation and size variation of microcylinders on the optical energy transport.For examples, with the DSEM the optical coupling between input/output waveguides and coupled microrings has been studied in Ref. [24], while the optical coupling between two microcylinders has been studied in Ref. [25].In this paper, we study optical transport phenomena and light propagation by WGMs in long chains of microcylinders.The results demonstrate not only successful optical energy transport by WGMs in microcylinder CROWs but also strong dependency of such energy transport on the resonator size, the inter-resonator separation and the refractive index of the resonators, and thereby provide valuable insight for designing and optimizing circuits containing CROW devices.
The remaining part of the paper is organized as follows.In Section 2, we give the definition of the cylindrical WGMs, the Maxwell's equations for coupled cylinders, and the discontinuous spectral element method.In section 3, results and discussions of numerical simulations are presented to investigate the dependence of light slowing down via WGMs through microcylinder CROWs on the azimuthal numbers of the WGMs, the inter-resonator gap sizes, and the material refractive indices.In Section 4, conclusions are given regarding the light propagation by WGMs through microcylinder CROWs, and future research plan is outlined.

Cylinder fields of whispering gallery modes
We shall consider electromagnetic WGMs of a circular dielectric cylinder of radius a and infinite length with dielectric constant ε 1 and magnetic permeability µ 1 , which is embedded in an infinite homogeneous medium of material parameters ε 2 and µ 2 .With respect to a cylindrical coordinate system (r, θ , z), for a time factor exp(−iωt), the components of the magnetic field H = (H r , H θ , H z ) and the electric field E = (E r , E θ , E z ) of the WGMs are given by the following equations [26] where F n = exp(inθ + ihz − iωt) with h being the axial propagation constant.The function G n ≡ J n for r < a and H (1) n for r > a, where J n is the Bessel function of the first kind and H (1) n is the Hankel function of the first kind.Prime denotes differentiation with respect to the argument λ r.Also, for r < a, k The coefficients a n and b n are determined by the boundary condition that, at the cylindrical boundary r = a, the tangential components of the fields are continuous.For a nontrivial solution, the axial propagation constant h shall satisfy the following eigenvalue equation [27] where u = λ 1 a and v = λ 2 a.For a given mode number n, Eq. (3) does not have a unique solution and the electromagnetic WGMs are represented by solutions of Eq. (3) when n is of the order of λ 1 a.Note that the mode number n is also the number of maxima in the field intensity in the azimuthal direction and is thus called the azimuthal number of the WGMs.
In this paper, we will confine ourselves to WGMs with an axial propagation constant h between k 1 and k 2 , i.e., k 1 > h > k 2 .In this case, λ 2 = −i|λ 2 | and λ 1 = |λ 1 |, which prevents any ohmic losses and the WGMs would be unattenuated along a perfectly straight cylinder [27].

Maxwell's equations for coupled microcylinders
In order to investigate light propagation by WGMs in microcylinder CROWs, we shall turn to the Maxwell's equations.For a WGM with the axial propagation constant h, the magnetic field H = (H x , H y , H z ) and the electric field E = (E x , E y , E z ) in a rectangular coordinate system (x, y, z) may be expressed as Substituting them into the Maxwell's equations we obtain the following hyperbolic system of equations in matrix form where

Discontinuous spectral element method
The finite difference time domain (FDTD) method has been used extensively in computational electromagnetics [28,29].The algorithm enforces Gauss's Law, handles material interface naturally, and is easy to code.However, it has large phase errors, works well only for simple geometries or Cartesian grids, and cannot implement any local refinement.Also, a large number of grid points are needed for FDTD methods to have numerical solutions of high order accuracy.For instance, if we use a second order finite difference method to solve a wave equation, more than 30 grid points per wavelength will be needed to have a numerical solution with 1% error [30].
( Because of its high order accuracy and phase-preserving nature, in our simulations the discontinuous spectral element method is employed for solving the Maxwell's equations to analyze light propagation by WGMs in microcylinder CROWs.This algorithm is briefly summarized in this section.For more details the readers may consult Refs.[23] and [24]. To approximate the Maxwell's equations (6) in the time domain, we write them in a conservation form as where the flux F = (AQ, BQ).To solve Eq. ( 10) in a two-dimensional geometry, the physical domain under consideration is divided into non-overlapping quadrilaterals.Each physical element is then mapped onto a reference element, i.e., the standard square ] by an isoparametric transformation x = χ(ξ ), where x = (x, y) and ξ = (ξ , η) are the coordinates in the physical element and the reference element, respectively.In cases that curvilinear elements are involved, the method of blending function [31] can be used to construct appropriate transformations.
Under the transformation x = χ(ξ ), Eq. ( 10) on each element becomes The new variables in Eq. ( 11) are where J is the Jacobian of the transformation, i.e., J = ∂ χ ∂ ξ .In the discontinuous spectral element method, the solution Q is approximated by a linear combination of the basis functions of the approximation space on the reference element, and the approximation is not required to be continuous across the element boundary.In this paper, the approximation space on the standard square Q 0 is chosen as P M,M = P M × P M , where P M represents the space of polynomials of degree M or less.An orthogonal basis for P M,M is the set of tensor products of Lagrange interpolating polynomials where with the nodal points We then approximate the solution Q element-by-element in terms of the basis functions as where Qmn (t) are time dependent coefficients.The residual of the approximation is required to be orthogonal to the approximation space locally within each element Ω, yielding the following equations where (u, v) = Ω uv dξ represents the usual L 2 inner product, ∂ Ω the element boundary, and n the outward unit normal to the element boundary.
The integrals in Eq. ( 16) are calculated numerically by quadratures, depending on the element type and the basis functions, and the discretization requires the evaluation of the fluxes along the element boundaries.However, the approximation is not continuous across the element boundaries.The difference is resolved by solving a local Riemann problem for the numerical normal flux.The Riemann problem for the Maxwell's equations is discussed in detail in Refs.[23] and [32].In a three-dimensional space, given two states Q − and Q + , the numerical normal flux for a dielectric interface or continuous medium can be written as In Eq. ( 17), Z ± and Y ± are the local impedance and admittance, respectively, and are defined as Equation ( 16) is a system of ordinary differential equations for the time dependent coefficients Qmn (t), m, n = 0, 1, 2, ••• , M, which can be solved by the Leap Frog method, Runge-Kutta methods, etc.
Appropriate boundary conditions shall be posted at the artificial numerical boundaries of the computational domain to prevent outgoing waves from reflecting off the boundaries.Otherwise, the reflections may falsify the computational results.In practical simulations, absorbing boundary conditions (ABC) such as perfectly matched layer (PML) boundary conditions [23] [33,34] are widely used.Since most of the electromagnetic energy of a WGM is confined inside the cylinder and fields decay fast away from the cylindrical boundary, a simple matched layer (ML) technique introduced in Ref. [35] will be sufficient.The ML approach solves the following damped Maxwell's equations in the matched layer where σ x and σ y signify the lossy absorbing material parameters.

Light propagation by WGMs in microcylinder CROWs
We shall first consider a chain of 6 coupled cylinder resonators to demonstrate successful light propagation by evanescent wave coupling of WGMs in CROWs.The identical cylinders are assumed to have infinite length.The radius of the cylinders is r = 4.023µm, the material index is n = 3.2, and the cylinders are separated to each other by a gap with size w = 4%r.The external medium is vacuum.
To see whether light can be transmitted through the CROW by WGMs, we assume that initially in the most left cylinder there exists a WGM circulating counterclockwisely around its boundary, and no fields exist inside all other cylinders.We choose a WGM with azimuthal number 16 in this simulation.The time frequency is always assumed to be ν = 100THz.In this case, the eigenvalue equation (3) has a solution 13.6657 between k 1 = 6.4π and k 2 = 2π.
To show the temporal dynamics of the light propagation by WGMs through the chain of cylinders, in Fig. 1 we display the snapshots of the field component E z at five different times.As shown in Fig. 1, the fact that the light has been transmitted all the way through the chain to the most right cylinder clearly demonstrates the successful wave propagation by WGMs in the CROW.Moreover, since the rotation of the WGM in the first cylinder is set in the counterclockwise direction, due to the phase matching, the mode in the odd numbered cylinders should be counterclockwise, while the mode in the even numbered ones should be automatically clockwise.This has been observed in the simulation as well [25].
For more quantitative study of the light propagation by WGMs through the CROW, we shall consider how the optical energy is transferred through the chain.To this end, we monitor the electromagnetic energy distribution in individual microcylinders from t = 0 to t = 14080fs.The electromagnetic energy stored in an electromagnetic field in a volume V is defined as Figure 2 shows the temporal history of the energy distribution in individual microcylinders during the light propagation through the CROW described in Fig. 1.Since our configuration does not include either an input or an output waveguide at either end of the CROW, several light propagation phenomena are observed in the simulation.First, the total electromagnetic energy in the system is conserved as the WGM is assumed to be lossless.Second, as indicated by the separated main energy peaks in individual cylinders in Fig. 2, the light appears to travel through the CROW at a constant rate.Third, we have observed spatial dispersion of the pulse along the chain of the resonators.There is however another potential factor of attenuation associated with curvature loss, which is not considered in our lossless WGMs.On the other hand, due to the absence of an output waveguide at the right end of the CROW, the most right cylinder can accumulate as much as 85% of the electromagnetic energy in the system at t = 12100fs.And last, after the most right cylinder reaches its maximum field intensity, the light will start to travel backward from the right end to the start end of the CROW.As a confirmation of the above propagation phenomena, we consider another 10 cylinder long CROW.The index and the inter-cylinder gap size remain unchanged, but the radius of the cylinders is changed to r = 2.283µm.Also, we choose an initial WGM in the most left cylinder with a smaller azimuthal number 10. Figure 3 shows the corresponding temporal history of the energy distribution in individual microcylinders from t = 0 to t = 6000fs.Again, the successful light propagation through the CROW and the aforementioned phenomena are clearly observed.
For physical realizations of CROWs, one is more interested in CROWs with input and output coupling.To see whether an input or output waveguide will affect the propagation speed of light through CROWs, we consider the CROW described in Fig. 3 again but with an output waveguide being attached to its end.Figure 4  tion in individual cylinders.It is clearly found that the output waveguide does not affect the propagation speed of light in the CROW.
In the following, we shall study the dependence of light propagation speed on various physical parameters of CROWs and WGMs.

Azimuthal number of WGMs
In general, optical coupling strength and thus light propagation speed may depend on many factors including resonator sizes, resonator dielectric constants, the inter-resonator separation width, and variations in resonator sizes.For convenience, in this paper we define the speed of light propagating through a CROW as the ratio of the distance between two cylinders' centers and the time between the main energy peaks in the two corresponding cylinders.Under this definition, for the first simulation where the azimuthal number of the initial WGM is 16, the speed of light in the CROW is slowed down to 1/92 of the speed of light in the vacuum, and for the second simulation where the initial WGM is assumed to have a smaller azimuthal number 10, the speed of light in the CROW is slowed down to just 1/32 of that of light in the vacuum.Therefore, it is fair to say that, given the same dielectric material and the same inter-resonator gap size, the optical transport velocity via evanescent WGMs in individual resonators decreases as the azimuthal number increases.Such a dependence of the optical transport velocity on the azimuthal number of the WGM can be partially understood in terms of the waveguiding mechanism in CROWs.Compared with WGMs with smaller azimuthal numbers, WGMs with bigger azimuthal numbers will have more energy confined inside the resonators, and thus the spatial overlap of the evanescent WGMs of two neighboring resonators will be smaller.For example, for the WGM with azimuthal number 16 described in Fig. 1, 98.5% of the electromagnetic energy is confined inside the cylinder, while for the WGM with azimuthal number 10 described in Fig. 3, only 96.2% of the energy is confined inside the cylinder.Figure 5  Fig. 4. Temporal history of energy distribution in individual microcylinders during the light propagation through the CROW described in Fig. 3.However, light will be coupled out of the CROW via an output waveguide attached to the end of the CROW.The energies are normalized to the electromagnetic energy of the initial WGM in the most left cylinder.cross-center sections of the normalized field components E z for the above two WGMs, respectively.In other words, our numerical simulation indicates that the energy velocity is reduced for higher azimuthal numbers, which implies that the observed behavior is consistent with one of the factors (confinement).The total treatment of the dependence of energy velocity on the confinement and effective interaction length however needs further study.and 3, respectively.The fields are normalized to the maximum field amplitude in the system.For the WGM described in Fig. 1, 98.5% of the electromagnetic energy is confined inside the cylinder, while for the WGM described in Fig. 3

Inter-resonator gap size
Next we shall investigate how the inter-resonator gap size will affect optical coupling strength and the light propagation speed in CROWs.For this reason, we analyze the speed of light propagating through CROWs of different inter-resonator gap sizes.The material index and the radius of the cylinders are n = 3.2 and r = 1.7325µm, respectively.In this set of simulations, we choose to use a WGM with a very small azimuthal number 8 as the initial WGM in the most left cylinder, for which around 93.7% of the electromagnetic energy is confined inside the cylinder.As shown in Fig. 6, the inter-cylinder gap size strongly affects the speed of light propagating through a CROW.For instance, when the gap size is 15% of the cylinder radius, the speed of the light in the CROW is slowed down to 1/18 of the speed of light in the vacuum, and when the gap size is increased to 40% of the cylinder radius, the propagation speed of light through the CROW is slowed down remarkably to 1/138 of that of light propagating in the vacuum.(WGMs with small azimuthal numbers will become distorted as light propagates in CROWs with small inter-resonator gap sizes due to strong coupling.And that is for this reason that we only simulate CROWs with the gap sizes greater than 15%r in this set of simulations.)

Refractive index n
Finally, we consider the wave propagation through CROWs of different materials but the same inter-cylinder gap size.The azimuthal number of the initial WGM is fixed at 8 and the intercylinder gap size is always assumed to be 16% of the corresponding cylinder radius.However, the material index of refraction n varies from 1.6 to 3.6, and accordingly the cylinder radius varies from 4.05µm to 1.53µm.As indicated in Fig. 7, the light travels at approximately the same speed through these CROWs, i.e., around 1/45 of the speed of light in the vacuum.Such a phenomenon can be partially understood again in terms of the waveguiding mechanism in CROWs.We use WGMs with a fixed azimuthal number 8 in this set of simulations.Although the material index of refraction and the radius of the cylinders vary, the percentage of the electromagnetic energy of the WGMs confined inside the cylinders is always around 94% for all cases.The fact that the energy velocity is the same for the similar confinement is interesting because our modeling takes self-consistently into account the changes to the evanescent coupling occurring in two different planes: YZ and XY planes.This result indicates that the phenomenological coupling constant in the Yariv's theory [1] is actually less for larger cylinders with the same azimuthal number.This can be seen from the formula (7) in Ref. [1] for the group velocity is proportional to Rκ where R is the period (the size of a unit cell) and κ is the coupling constant.The same velocity for larger resonators/smaller indices implies that the coupling constant κ becomes less as R is increased.

Conclusions
In this paper, to provide insight for building CROW devices with coupled microresonators or microcavities, we have applied the discontinuous spectral element method to investigate evanescent wave coupling of WGMs in microcylinder CROWs.We have demonstrated the successful wave propagation by WGMs in microcylinder CROWs and the strong dependence of the speed of such propagation on the inter-resonator gap size and the overlap of the evanescent WGMs of two neighboring resonators.Further work is under way to address the light propagation through microdisk or microsphere CROWs.

Fig. 1 .
Fig. 1.DSEM simulation of light propagation through 6 coupled microcylinder resonators.The field plotted here is the E z component.The radius of the cylinders is r = 4.023µm, the material index is n = 3.2, and the inter-resonator gap size is w = 4%r.The initial state of the system is represented by a WGM of azimuthal number 16 in the most left cylinder, and the frequency of the WGM is assumed to be 100THz.The five sequential snapshots demonstrate the successful wave propagation all the way through the chain to the most right cylinder.(a) t = 3520fs; (b) t = 5500fs; (c) t = 7480fs; (d) t = 8800fs; and (e) t = 12100fs.

Fig. 2 .
Fig. 2. Temporal history of energy distribution in individual microcylinders during the light propagation through the CROW described in Fig. 1.The energies are normalized to the electromagnetic energy of the initial WGM in the most left cylinder.

Fig. 3 .
Fig. 3. Temporal history of energy distribution in individual microcylinders during the light propagation by a WGM through a 10 cylinder long CROW.The radius of the cylinders is r = 2.283µm, the material index is n = 3.2, and the inter-resonator gap size is w = 4%r.The initial WGM in the most left cylinder has azimuthal number 10 and time frequency 100THz.The energies are normalized to the electromagnetic energy of the initial WGM in the most left cylinder.

Fig. 5 .
Fig.5.The cross-center sections of the field component E z of the two WGMs described in Figs.1 and 3, respectively.The fields are normalized to the maximum field amplitude in the system.For the WGM described in Fig.1, 98.5% of the electromagnetic energy is confined inside the cylinder, while for the WGM described in Fig.3, only 96.2% of the energy is confined inside the cylinder.

Fig. 6 .
Fig.6.The speed of light propagating through 10 cylinder long CROWs.The radius and the index of refraction of the cylinders are r = 1.7325µm and n = 3.2, respectively.The initial WGM's azimuthal number is 8.And the inter-cylinder gap size varies from 15%r to 40%r.

Fig. 7 .
Fig.7.The speed of light propagating through 10 cylinder long CROWs.The initial WGM's azimuthal number and the inter-cylinder gap size are fixed at 8 and w = 16%r, respectively.However, the material index of refraction varies from n = 3.6 to n = 1.6, and accordingly the radius of the cylinder varies from r = 1.53µm to r = 4.05µm.