## Abstract

We present a detailed study on the behavior of coupling-induced resonance frequency shift (CIFS) in dielectric microring resonators. CIFS is related to the phase responses of the coupling region of the resonator coupling structure, which are examined for various geometries through rigorous numerical simulations. Based on the simulation results, a model for the phase responses of the coupling structure is presented and verified to agree with the simulation results well, in which the first-order coupled mode theory (CMT) is extended to second order, and the important contributions from the inevitable bent part of practical resonators are included. This model helps increase the understanding of the CIFS behavior and makes the calculation of CIFS for practical applications without full numerical simulations possible.

©2009 Optical Society of America

## 1. Introduction

Dielectric microring resonators are key microphotonic circuit components [1–3], and several recent works have utilized them in both single-resonator and coupled-resonator architectures for realizing complex functionalities in a planar platform for applications like bandpass filters [4,5], slow-wave structures [6], on-chip lasers [7], and sensing elements [8]. In coupled-resonator structures, where the response is engineered by appropriate choice of the resonance frequencies of each single resonator and their mutual coupling strengths, one critical issue is the precise control of the resonance frequency, which depends on both the resonator material and the resonator structure. In addition, it has been shown that when it is coupled to access waveguides or adjacent resonators, the resonance frequency of a resonator will deviate from its original isolated value [9,13]. Such coupling-induced resonance frequency shift (CIFS), which is recently investigated more systematically [9], causes resonance frequency mismatches between individual resonators and thus significantly impacts the performance of the coupled-resonator device [9–11].

The work in Ref [9]. gives an excellent explanation on the origin of CIFS, especially, that CIFS is beyond the first-order coupled mode theory (CMT) and sometimes could be counterintuitive. Two interesting features of CIFS shown by Ref [9]. are: 1) despite the expectation of a negative CIFS from positive-index perturbations, CIFS can be of positive or negative sign, i.e., the resonance frequency of a resonator can be either blue shifted or red shifted by the coupling effect; and 2) two coupled microrings with different radii can experience CIFS of opposite signs, one being positive and the other being negative. However, Ref [9]. focuses on the qualitative discussions of the second-order, self-perturbing frequency shift term in the CMT formalism, and it concludes that quantitative calculation using the first-order CMT is not exact due to the existence of inevitable errors. As a result, currently CIFS calculation must be performed using laborious numerical simulations, where little physical insight is gained. In fact, the understanding of CIFS is so limited that for a given material system and resonator coupling structure, the estimation of the sign or the order of magnitude of CIFS is still difficult.

In this work, we present a detailed investigation of CIFS for dielectric microring resonators. As observed in Refs [9,10], we show that CIFS can be related to the phase responses of the coupling region in the resonator coupling structure. Due to this relationship, CIFS study is facilitated by developing a more accurate analysis tool for the coupling structures beyond the first-order CMT. To do this, we first simulate resonator coupling structures with efficient numerical simulations to obtain the phase responses for structures with different geometric properties. Subsequently a model is developed to help understand the simulation results, in which the CMT is naturally extended to second order. In addition, geometric effects are included, among which the effect of the inevitable bent part of practical resonators is revealed as an important contributor to the final phase responses. This model is then verified to agree with the simulation results by quantitative examinations, and finally a conclusion is given.

## 2. CIFS in microring resonators

For microring resonators coupled to access waveguides or adjacent resonators, the coupling region can be treated as a multi-port coupler. Figure 1
shows two such examples with different coupling geometries. Here *a*
_{1}, *b*
_{1} stand for the input and output signals travelling inside the resonator under investigation, respectively, and *a*
_{2}, *b*
_{2} stand for the input and output signals travelling inside the access waveguide or the adjacent resonator, respectively. To facilitate the CIFS simulation, waveguides and resonators are terminated beyond the coupling region by perfectly matched layers (PMLs), and the signal source is also implemented [12], as will be discussed later. We also assume that the reference planes for *a _{i}* and

*b*(

_{i}*i*= 1, 2) are far from the coupling region, and the signals are normalized in a way that |

*a*|

_{i}^{2}and |

*b*|

_{i}^{2}(

*i*= 1, 2) represent powers at each port. If the coupling is lossless and reflection is negligible, the output signals can be related to the input signals by a unitary 2 × 2 transfer matrix as [9],

*u*,

_{ij}*ϕ*(

_{ij}*i*,

*j*= 1, 2) are the amplitude and phase responses of each component of the transfer matrix

*U*, respectively. As a convention, in Eq. (1) we have assumed that the free propagation phase term, namely, the phase contribution resulting from wave propagation from the reference plane of

*a*

_{1}(

*a*

_{2}) to that of

*b*

_{1}(

*b*

_{2}) in the absence of coupling is excluded. In this sense, phase responses

*ϕ*

_{11},

*ϕ*

_{22}result from pure coupling interaction and will asymptotically approach zero when the coupling is weakened to a negligible extent.

The effect of coupling on the resonance frequency of the resonator is then modeled as the added phase to the signal travelling inside the resonator as it passes through the coupling region, which is *ϕ*
_{11} as we have defined. Using the fact that a round-trip net phase shift of 2π will shift the resonance frequency by one free spectral range (FSR), CIFS can be approximated as [9,10],

*f*

_{FSR}is the FSR,

*c*is the speed of light in vacuum,

*n*

_{g}is the group index at the resonance frequency, and

*S*is the perimeter of the resonator. The negative sign in Eq. (2) is due to the convention used throughout this paper for the propagation phase term as exp(

*ißz*) with

*β*being the propagation constant and

*z*being the coordinate in the direction of propagation. Because of Eq. (2), calculation of CIFS is equivalent to the calculation of the phase term

*ϕ*

_{11}. In addition, if

*a*

_{2}and

*b*

_{2}represent the travelling signals inside an adjacent coupled resonator, CIFS of that adjacent resonator will be given by a similar expression as Eq. (2), with phase term

*ϕ*

_{11}replaced by

*ϕ*

_{22}. Therefore, in the rest of the paper, we will focus on the study of the phase responses of the coupling region instead of direct CIFS calculations. Once we have a thorough understanding of the phase behaviors of the transfer matrix

*U*, CIFS calculation using Eq. (2) is straightforward (see Appendix A for more discussions).

Before going into detailed calculation of these phase terms, it is worthwhile to mention some general properties of the transfer matrix *U* imposed by power conservation and symmetry considerations. Power conservation requires *U* to be unitary, and this imposes that *U* can be written as [9],

*θ*

_{0},

*θ*

_{1},

*θ*

_{2}are phase parameters related to

*ϕ*(

_{ij}*i, j*= 1, 2), and

*κ*is the magnitude of the field coupling coefficient between the two coupled structures shown in Fig. 1 (

*κ*is the power coupling coefficient). From Eq. (3), it is easy to observe that When the couplers have the symmetry about the middle vertical axis (i.e., axis

^{2}*V*shown in Fig. 1), reciprocity and symmetry considerations give [9,10]

*H*in Fig. 1(a), symmetry requires that [10]Note that for the coupler shown in Fig. 1(b) no such horizontal axis exists, and hence, in general

*ϕ*

_{11}≠

*ϕ*

_{22}.

To explore the phase properties of the transfer matrix *U*, we carry out a detailed analysis for the two typical coupling structures shown in Fig. 1. Waveguides are terminated and resonators are broken to implement the signal source and PMLs. The resulting couplers, named CA and CB, respectively, are then analyzed with the finite element method (details in Appendix A). The first coupler CA shown in Fig. 1(a) has two symmetry axes, the vertical axis *V* and the horizontal axis *H*. As a result, the phase terms have the properties given by Eq. (7). The second coupler CB shown in Fig. 1(b) only has the vertical symmetry axis *V*, and its phase properties are given by Eq. (6). Simulations for these two couplers are performed at a fixed frequency and for varying coupling parameters (e.g., the length of the middle straight part *L* and the waveguide widths *w*
_{1}, *w*
_{2} as shown in Fig. 1). The phase term *ϕ*
_{11} (*ϕ*
_{21}) can be obtained by comparing the phase difference of *b*
_{1} (*b*
_{2}) with and without the presence of the coupling waveguide (shown by the yellow region in Fig. 1) with the source implemented at the *a*
_{1} side, as the case shown in Fig. 1. Similarly, the phase terms *ϕ*
_{22} and *ϕ*
_{12} can be obtained with the source implemented at the *a*
_{2} side.

## 3. Simulation results and discussions

We choose the silicon-on-insulator (SOI) material system in our simulations, and we simulate the fundamental TE-like polarization (electric field predominantly in the plane of the structure). Using the effective index method to simplify the 3-dimensional (3-D) structures to 2-dimensional (2-D), one obtain an effective index of 2.829 for the silicon layer at wavelength 1578nm, assuming a silicon thickness of 220 nm on top of a thick enough (at least 1μm) buried oxide (BOX) layer and an air cladding.

#### 3.1 Simulations: couplers with identical waveguide widths

We first simulate couplers with identical waveguide widths, i.e., *w*
_{1} = *w*
_{2} as in Fig. 1. Phase responses are obtained for various straight part lengths *L*, with the bend radius *R* and the gap between waveguides fixed at 6 μm and 100 nm, respectively. Moreover, the power coupling coefficient *κ ^{2}* (defined in Eq. (3)), which reflects the magnitude of coupling strength, is also obtained. Figure 2
shows the simulation results (shown by marker) for the coupler CA with the simulation parameters listed in Fig. 2(a), while Fig. 3
shows the simulation results for the coupler CB in a similar way. In addition, we have plotted the modeling results (shown by dotted line) for both couplers (the details of the model will be discussed in the next subsection).

From Fig. 2(b), we observe that for the coupler CA: 1) the requirements from power conservation and symmetry considerations given by Eq. (7) are satisfied; 2) the phase terms *ϕ*
_{11} (*ϕ*
_{22}), *ϕ*
_{12} (*ϕ*
_{21}) are linear with *L* with the same slope. From Fig. 3, we observe that for the coupler CB: 1) the requirements from power conservation and symmetry considerations given by Eq. (6) are satisfied, as shown by Fig. 3(c); 2) *ϕ*
_{11}≠*ϕ*
_{22}, and both are nonlinear with *L,* as shown by Fig. 3(b). It is especially interesting to note that, *ϕ*
_{11} is positive at *L* = 0 and slightly increases for small values of *L*. When *L* is large enough, the coupling becomes strong, and *ϕ*
_{11} starts to decrease with *L*, eventually to a negative value. That means the corresponding CIFS, as easily seen from Eq. (2), is negative when *L* is small and becomes positive when *L* is large enough. In contrast, for the coupler CA, CIFS is always negative and doesn’t change sign as *L* increases; 3) *ϕ*
_{12}, *ϕ*
_{21} and (*ϕ*
_{11} + *ϕ*
_{22})/2 are linear with *L* with the same slope, and this slope is the same as that shown in Fig. 2(b).

#### 3.2 Modeling of transfer matrix U

To understand the simulation results presented in the previous subsection 3.1, we analyze the coupler structures shown in Fig. 1 in more detail in this subsection. Both the couplers CA and CB are composed of three parts: input bend, middle straight part, and output bend, as shown in Fig. 4
. Each component can be described by a transfer matrix. For example, the transfer matrix *B* for the input bend relates [*c*
_{1}, *c*
_{2}] * ^{T}* to [

*a*

_{1},

*a*

_{2}]

*while the transfer matrix*

^{T}*T*for the middle straight part relates [

*d*

_{1},

*d*

_{2}]

*to [*

^{T}*c*

_{1},

*c*

_{2}]

*. Here [*

^{T}*c*

_{1},

*c*

_{2}]

*and [*

^{T}*d*

_{1},

*d*

_{2}]

*are the input and output optical signals for the middle straight part. Power conservation requires that both matrices*

^{T}*B*and

*T*to be unitary. In addition, because of the symmetry about the vertical axis and the reciprocity, the transfer matrix for the output bend can be shown to be

*B*, which relates [

^{T}*b*

_{1},

*b*

_{2}]

*to [*

^{T}*d*

_{1},

*d*

_{2}]

*. The total transfer matrix*

^{T}*U*, as a result, is given by [10]

*N*is a normalization matrix to remove the free propagation phase contribution because of the convention made in defining the transfer matrix

*U*in Eq. (1).

The middle straight part is a parallel waveguide coupler and has been extensively studied using the CMT [13–15]. In principle, this coupled system can be exactly analyzed by expanding the individual waveguide modes by the complete set of system modes (or supermodes) [13]. If the waveguides are single mode, and the gap between them is not too small, limiting the expansion to the lowest two fundamental system modes usually suffices [14,15]. For the particular case of *w*
_{1}
*= w*
_{2}, the field profiles of the two fundamental system modes, i.e., the symmetric mode and the antisymmetric mode, with different propagation constants *β _{s}* and

*β*, respectively, are schematically shown in Fig. 4. Although in the case

_{as}*w*

_{1}

*≠w*

_{2}, the two system modes do not have simple symmetry relating to the individual waveguide modes, we shall still use

*β*and

_{s}*β*to denote the propagation constants for these two system modes. In this paper, we will focus on the cases that the two waveguides are either exactly or close to be phase matched. As a result, the transfer matrix

_{as}*T*for the middle parallel waveguide coupler can be expressed as [16],

*δ*is the phase mismatch between the two waveguides defined as

*δ*≡(

*β*

_{1}

*−β*

_{2})/2, with

*β*

_{1},

*β*

_{2}corresponding to the propagation constants of individual waveguides with widths

*w*

_{1},

*w*

_{2}respectively; $\overline{\kappa}$ is the field coupling coefficient between two waveguides, and

*β*

_{0}is obtained from the knowledge of $\overline{\kappa}$and

*δ*[9,13]. In the system modes approach outlined here,

*β*and

_{s}*β*are numerically calculated with desired accuracy, and

_{as}*β*

_{0}is obtained via Eq. (10), in which $\overline{\kappa}$ can also be obtained from the knowledge of

*β*

_{0}and

*δ*. Since

*β*and

_{s}*β*can be calculated accurately from numerical simulations, the system modes approach is in general more accurate than the CMT approach.

_{as}If *w*
_{1}
*= w*
_{2}, the parallel waveguide coupler is synchronous and hence *δ* = 0 (*β*
_{1} = *β*
_{2}) and *β*
_{0} = $\overline{\kappa}$ (Eq. (10)). In this case, Eq. (9) can be simplified as

*β*≡(

*β*+

_{s}*β*−

_{as}*β*

_{1}−

*β*

_{2})/2. Equation (11) shows that besides the free propagation phase contribution exp(

*iβ*

_{1}

*L)*, additional phase term exp(

*i*Δ

*βL*) appears. By combining the definitions of

*β*

_{0}and Δ

*β*, we can rewrite

*β*and

_{s}*β*

_{as}as

Equations (12) and (13) can be regarded as Taylor expansions of the propagation constants of the coupled system modes as a function of coupling coefficients. When the two identical waveguides are well separated (i.e., *β*
_{0}≈0), the system has two degenerate modes with propagation constants *β*
_{1} = *β*
_{2}. As the two waveguides are brought closer, the degeneracy is removed, and the new propagation constants *β _{s}* and

*β*can be expanded based on the different orders of the coupling perturbation. In Eqs. (12) and (13), we actually expand the propagation constants of the two fundamental system modes up to the second order of the coupling perturbation, while in CMT the expansion is only up to the first order. The linear phase behavior with

_{as}*L*observed in Figs. 2(b) and 3(c) can be identified with this second-order phase term exp(

*i*Δ

*βL*), with the slope Δ

*β*provided in Table 1 for the middle parallel waveguide coupler simulated in Figs. 2 and 3.

However, we observe that the CA and CB couplers show different results, as seen in Figs. 2 and 3. For the coupler CA, the phase responses shown in Fig. 2(b) are all linear with *L*, which is similar to that given by Eq. (11) for the synchronous parallel waveguide coupler except a constant difference. For the coupler CB, the phase responses shown in Figs. 3(b) and 3(c) illustrate some similarities with a synchronous parallel waveguide coupler, and also some significant differences, especially for the phase terms *ϕ*
_{11} and *ϕ*
_{22}. This suggests that the bent parts are important to the final phase responses, and the corresponding transfer matrix *B* has to be included in the model.

Since the transfer matrix *B* relating [*c*
_{1}, *c*
_{2}]* ^{T}* to [

*a*

_{1},

*a*

_{2}]

*as shown in Fig. 4 is assumed to be unitary, it has a similar expression as Eq. (3) and is expressed by*

^{T}*β*

_{0}, Δ

*β*} correspond to those of the middle parallel waveguide coupler with the specified gap. Unlike in Eq. (11) where the only length parameter is the physical length

*L*,

*z*and

_{p}*z*are introduced as equivalent lengths to lump the phase and amplitude responses of the bent part, respectively [17]. The reference planes of [

_{a}*a*

_{1},

*a*

_{2}]

*are chosen such that the phase difference between*

^{T}*c*

_{1}and

*a*

_{2}is equal to that between

*c*

_{2}and

*a*

_{1}, which is reflected by the fact that the off-diagonal elements of transfer matrix

*B*are equal in Eq. (14) (Appendix B). With this choice of the reference planes of [

*a*

_{1},

*a*

_{2}]

*, generally the phase difference between*

^{T}*c*

_{1}and

*a*

_{1}is not equal to that between

*c*

_{2}and

*a*

_{2}, and Δ

*ψ*is introduced to account for this difference.

*α*

_{0}is a constant and its meaning can be better understood in the following paragraph.

The normalization matrix *N* can be calculated based on its definition, i.e., to remove the free propagation phase contributions in the absence of coupling. In this sense, *N* has two diagonal elements, each corresponding to the phase difference between *a _{i}* (

*i*= 1, 2) and

*b*(

_{i}*i*= 1, 2) when the coupling effect is negligible. Because of the nature of free propagation, the phase contributions from different components in Fig. 4 can be added to obtain the overall propagation phases. For the middle parallel waveguide coupler, the calculation is straightforward: exp(

*iβ*

_{1}

*L*) for the upper waveguide, and exp(

*iβ*

_{2}

*L*) for the lower waveguide. For the input bend, it is a little complicated. Based on the definition of the transfer matrix

*B*(Eq. (14)), the free propagation phases can be approximated as exp(

*i*(

*α*

_{0}+ Δ

*ψ*)) for the upper arm, and exp(

*i*(

*α*

_{0}-Δ

*ψ*)) for the lower arm, respectively (we put the details in the Appendix B). For the output bend, the transfer matrix is

*B*, and the free propagation phases will be the same as those of the input bend. Finally, we obtain

^{T}*N*as,

*U*can thus be obtained from Eq. (8), with matrices

*T*,

*B*, and

*N*given by Eqs. (9), (14), and (15), respectively. The parameters {$\overline{\kappa}$,

*δ*,

*β*

_{0}, Δ

*β*} are numerically obtained from the middle parallel waveguide coupler by solving the coupler’s lowest two fundamental system modes, and the unknowns {Δ

*ψ*,

*z*,

_{p}*z*} are fitted based on the simulation results of the transfer matrix

_{a}*U*.

#### 3.3 Modeling results: couplers with identical waveguide widths

For couplers with identical waveguide widths *w*
_{1} = *w*
_{2}, simulation and modeling results are shown in Figs. 2 and 3 for the couplers CA and CB, respectively, where good agreement is observed. The corresponding modeling parameters and fitting results are provided in Table 1.

For the coupler CA Δ*ψ* = 0 due to the symmetry about the horizontal axis. After a straightforward calculation for the transfer matrix *U*, we obtain,

*z*and 2

_{p}*z*for the phase and amplitude responses, respectively. This is exactly what we observe from Fig. 2.

_{a}For the coupler CB Δ*ψ*≠0, and the results are more complicated. For small |Δ*ψ*| (<0.5 rad), exp(iΔ*ψ*)≈1 + iΔ*ψ* in Eq. (14), and we obtain for small |*ϕ*
_{11}| and |*ϕ*
_{22}| (<0.5 rad) after a simple though tedious computation for the transfer matrix *U*,

*ϕ*

_{12},

*ϕ*

_{21}, and (

*ϕ*

_{11}+

*ϕ*

_{22})/2 are linear with

*L*with a slope of Δ

*β*. Since Δ

*β*is defined for the middle straight part which is shared by CA and CB with the same parameters in Figs. 2 and 3, it is expected and verified that the slope observed in Fig. 2(b) is equal to that observed in Fig. 3(c). Moreover, both

*ϕ*

_{11}and

*ϕ*

_{22}have two terms, one is linear with

*L*and the other is nonlinear with

*L*. It is clear that the magnitude of the second term in Eqs. (19)-(20) is small at small values of

*L*, but at large

*L*, this term starts to dominate the first term. Thus, we expect

*ϕ*

_{11}and

*ϕ*

_{22}to vary linearly with

*L*only for small

*L*. At large

*L*, we expect

*ϕ*

_{11}to increase (if Δ

*ψ*is positive) or decrease (if Δ

*ψ*is negative) nonlinearly with

*L*. Results shown in Figs. 3(b) and 3(c) indicate that Δ

*β*is positive and Δ

*ψ*is negative, since

*ϕ*

_{11}eventually decreases to a negative value. On the contrary, in

*ϕ*

_{22}both terms are of positive sign, therefore it increases with

*L*monotonically as shown in Fig. 3(b).

The negative sign of Δ*ψ* is believed to be intrinsic to the geometry of the coupler CB, which has been explained using the geometry conformal mapping argument in Ref [10]. For the upper bend, the propagation constant of the travelling signal is roughly equal to that of a straight waveguide with the same width when the bend radius is not too small [18]. In writing Eq. (14), we have chosen the reference planes of [*a*
_{1}, *a*
_{2}]* ^{T}* such that the off-diagonal elements of the transfer matrix

*B*are equal. It seems that with this choice of reference planes, the propagation path from the reference plane of

*a*

_{1}to that of

*c*

_{1}is shorter than the propagation path from the reference plane of

*a*

_{2}to that of

*c*

_{2}, resulting a negative Δ

*ψ*when

*w*

_{1}

*= w*

_{2}(from Eq. (14) exp(

*i*2Δ

*ψ*) is the phase difference between the upper and the lower arms). This explanation could be qualitatively understood from Fig. 5(a) .

Based on this understanding, the variations of Δ*ψ* for the coupler CB with the geometric parameters can be predicted. For example, we fix the upper waveguide width *w*
_{1} and change the lower waveguide width *w*
_{2}. When *w*
_{2} is larger than *w*
_{1}, the propagation constant of the lower arm (*a*
_{2} to *c*
_{2}) is larger than that of the upper arm (*a*
_{1} to *c*
_{1}). Also, we already know that propagation length of the lower arm (*a*
_{2} to *c*
_{2}) is larger than that of the upper arm (*a*
_{1} to *c*
_{1}). As a result, Δ*ψ* will be smaller and more negative than that in the *w*
_{1}
*= w*
_{2} case. On the contrary, if *w*
_{2} is less than *w*
_{1}, the propagation constant of the lower arm (*a*
_{2} to *c*
_{2}) is smaller than that of the upper arm (*a*
_{1} to *c*
_{1}). Thus, Δ*ψ* will be larger than that in the *w*
_{1}
*= w*
_{2} case. If we keep reducing *w*
_{2}, Δ*ψ* will increase from a negative value to an eventual positive value. This behavior of Δ*ψ* is schematically shown in Fig. 5(b), and we will investigate these predicted properties in the following subsections.

#### 3.4 Modeling results: couplers with different waveguide widths

To verify our model and also explore the properties of Δ*ψ*, we proceed to survey more examples by simulating the coupler CB with different waveguide widths, i.e., *w*
_{1}
*≠w*
_{2}, and performing the modeling accordingly. For small |Δ*ψ*| (<0.5 rad), exp(iΔ*ψ*)≈1 + iΔ*ψ*, and if the phase mismatch |*δ*| of the middle parallel waveguide coupler is also small so *β*
_{0}≈*κ* (Eq. (10)), we obtain for small |*ϕ*
_{11}| and |*ϕ*
_{22}| (<0.5)

Comparing with Eqs. (19) and (20), an additional phase term which is proportional to the phase mismatch *δ* of the middle parallel waveguide coupler appears in the phase responses *ϕ*
_{11} and *ϕ*
_{22} in Eqs. (21) and (22). Figure 6
shows the simulation and modeling results for the coupler CB by fixing the upper waveguide width *w*
_{1} at 400nm and changing the lower waveguide width *w*
_{2}, with other simulation parameters listed in Fig. 6(a) and modeling results provided in Table 1. When *w*
_{2} = 410 nm, Δ*ψ* is smaller (−0.285, see Table 1) than that in the *w*
_{2} = 400 nm case (−0.261), and *δ* is also negative. As a result, *ϕ*
_{11} first slightly increases with *L* and then quickly decreases with *L*, as shown in Fig. 6(b). When *w*
_{2} = 390 nm, Δ*ψ* is larger (−0.22, see Table 1) than that in the *w*
_{2} = 400 nm case (−0.261), and more importantly, *δ* is positive. From Fig. 6(b) we can deduce that the magnitude of the third term in Eq. (21) is larger than that of the second term, so *ϕ*
_{11} keeps increasing with *L* instead of changing direction as the case of *w*
_{2} = 400 nm. The behavior of *ϕ*
_{22} for different *w*
_{2} shown in Fig. 6(c) can be analyzed in a similar way and will not be discussed here.

#### 3.5 A special case: L = 0 μm

When *L* = 0 μm, the resonators shown in Fig. 1 will be reduced to the circular structure, and the phase responses of the resulting coupler can be obtained from Eqs. (21) and (22) by setting *L* = 0 μm:

*β*

_{0}, Δ

*β*} correspond to the coupling parameters of a parallel waveguide coupler with the specified gap, though in the

*L*= 0 μm case the middle straight part does not exist. In fact, by inserting such an interferometric structure (i.e., the parallel waveguide coupler), we are able to extract the unknowns {Δ

*ψ*,

*z*,

_{p}*z*} for the input and output bends, as in the previous examples shown in Figs. 2, 3 and 6. Equations (23) and (24) show that even for the

_{a}*L*= 0 μm case, the phase responses

*ϕ*

_{11}and

*ϕ*

_{22}still have two terms. The first one is proportional to Δ

*β*(Δ

*β*≡(

*β*+

_{s}*β*−

_{as}*β*

_{1}−

*β*

_{2})/2), which has been shown by Eqs. (12) and (13) to be a second-order coupling parameter and will not appear in the first-order CMT. Thus, Δ

*β*has to be determined numerically based on its definition, and can be either positive or negative depending on the specific material systems, in which the sign and magnitude of Δ

*β*can vary dramatically [19].

The second term in Eqs. (23) and (24), which is responsible for the difference between *ϕ*
_{11} and *ϕ*
_{22}, is proportional to Δ*ψ*. Contrary to Δ*β* whose property is difficult to predict without specific numerical calculations, Δ*ψ* is mainly determined by the geometric properties of the input bend (the bend radius *R*, the waveguide widths *w*
_{1}, *w*
_{2}, and the gap), and its sign and magnitude are in principle predictable, as we have shown for the coupler CB in Fig. 5(b) (a quantitative model for estimating Δ*ψ* is out of scope of this paper and will be provided elsewhere). Figure 7(b)
plots *ϕ*
_{11} and *ϕ*
_{22} for the coupler CB by fixing *w*
_{1} and changing *w*
_{2} in the *L* = 0 case. When *w*
_{2} is reduced from a value larger than *w*
_{1}, Δ*ψ* increases from a negative value to an eventual positive value, reflected by the fact that when *w*
_{2}> 360 nm, *ϕ*
_{22}>*ϕ*
_{11}, and when *w*
_{2}<360 nm, *ϕ*
_{22}<*ϕ*
_{11}. This is exactly expected from Fig. 5(b). In Fig. 7(b), both *ϕ*
_{11} and *ϕ*
_{22} are positive, indicating a negative CIFS. However, in the material systems that |Δ*β*| is small, the second term in Eqs. (23) and (24) may dominate. As a result, *ϕ*
_{11} and *ϕ*
_{22} will be of opposite signs, which explains the possibility to observe opposite CIFS in two coupled microrings with different radii (which is necessary to ensure that Δ*ψ* is nonzero), an interesting feature mentioned in the beginning of this paper [9,19].

Δ*ψ* also affects the power coupling coefficient *κ ^{2}* as shown in Eq. (25), and we notice that the maximum power transfer will be less than 100% if Δ

*ψ*≠0, a characteristic feature in a phase mismatched coupler. Thus, we conclude that Δ

*ψ*is a measure of the phase mismatch of the input bend. Figure 7(c) plots the power coupling coefficient

*κ*for the coupler CB by fixing

^{2}*w*

_{1}and changing

*w*

_{2}in the

*L*= 0 case. When

*w*

_{2}reduces, the field overlap between the upper and the lower arms increases. On the other hand, the phase mismatch condition varies more dramatically as

*w*

_{2}changes and thus is a more decisive factor (the phase mismatch not only affects Δ

*ψ*but also

*z*in Eq. (25)). The maximum power transfer will occur when the upper and the lower arms are close to be phase matched, i.e., Δ

_{a}*ψ*≈0, which has been confirmed by Fig. 7(c) and also Ref [20], where a detailed discussion of the impact of phase mismatch on the power coupling coefficient

*κ*is presented.

^{2}## 4. Conclusion

We have shown that for microring resonators, the phase responses of the coupling region of the resonator coupling structure and therefore the CIFS of the corresponding resonators can be quantitatively analyzed by incorporation of the second-order coupling effect and also the geometric effects contributed by the input and output bent parts. The second-order coupling parameter Δ*β* could be either positive or negative, depending on specific material systems and coupling structures. The geometric parameter Δ*ψ*, on the other hand, is deterministic and is a measure of the phase mismatch of the input and output bends. With the help of the developed model, qualitative and in some cases quantitative analysis of CIFS for practical 3-D resonator coupling structures without full numerical simulations become feasible. As a particular example, for coupling scheme such as the CA coupler shown in Fig. 1(a), only the second-order coupling effect is present, and CIFS can be easily obtained by calculation of Δ*β* for a 3-D parallel waveguide coupler with the specified gap, whose calculation only involves an essential 2-D simulation [11]. We believe with the ongoing efforts on quantitative modeling of the geometric parameter Δ*ψ*, a fast and efficient estimation of CIFS for common coupling structures such as the CB coupler shown in Fig. 1(b) is possible in the near future.

## Appendix A. Simulation details and CIFS calculation

The simulations in this paper are performed with an in-house FEM code by implementing the 2-D Helmholtz equation in the Comsol environment. More than 200,000 cubic elements with grid sizes less than 20 nm are used to resolve the structure.

This paper has assumed that the CIFS calculation is equivalent to the calculation of the phase responses of the coupling region through the connection established by Eq. (2). We will provide an example here to verify this equivalence, which is critical since the most of discussions of CIFS in this paper are based on the phase responses of the coupling region. Figure 8(a)
shows the simulated structure, which is a microring side coupled to a straight waveguide. By solving the eigenvalue problem of the system, the resonance frequency of the resonator is obtained [21]. According to its definition, CIFS is the difference of the resonance frequencies of the same mode with and without the coupled waveguide. Figure 8(c) compares the simulation results of CIFS via this eigenvalue solution to the phase response method by simulating the coupler CB at *L*=0 case and converting the phase results to CIFS through Eq. (2) for different gaps. Good agreement is obtained, and our assumption that the phase response study is equivalent to the direct CIFS study is justified.

## Appendix B. Calculation of normalization matrix *N*

Strictly speaking, when the coupling is negligible (for example, by increasing the gap to a large enough value so that *β*
_{0}~0, Δ*β*~0) the free propagation phases are not necessarily exp(*i*(*α*
_{0}+Δ*ψ*)) for the upper arm, and exp(*i*(*α*
_{0}-Δ*ψ*)) for the lower arm for the fixed reference planes of [*a*
_{1}, *a*
_{2}]* ^{T}* and [

*c*

_{1},

*c*

_{2}]

*. Suppose Δ*

^{T}*ψ*needs to be replaced by Δ

*ψ’*in Eq. (15) for the normalization matrix

*N*, then Eqs. (19) and (20) become,

*ψ−*Δ

*ψ’*) (a factor of 2) added to

*ϕ*

_{11}, and the same constant subtracted from

*ϕ*

_{22}. Clearly when the gap is large, Δ

*ψ−*Δ

*ψ’*is negligible. We can estimate the effect of the coupling on Δ

*ψ−*Δ

*ψ’*by reducing the gap to the specified number from a general asymmetric parallel waveguide coupler, whose transfer matrix is given by Eq. (9). Note that the reference planes for the parallel waveguide coupler are chosen such that the off-diagonal elements are equal in Eq. (9) (that’s the reason we make the same choice of reference planes for the input bend in Eq. (14)). A simple calculation shows that

*L*is approximately

*z*

_{a}, and tan(

*β*

_{0}

*L*)≈

*β*

_{0}

*L*+ (

*β*

_{0}

*L*)

^{3}/3. Thus, we can simplify Eq. (28) to beFor the examples shown in Table 1, by substituting the corresponding values to Eq. (29) this amount can be calculated to be very small (<0.002 rad) and thus is negligible (the fitting error is on the order of 0.005 rad).

When the phase mismatch parameter |*δ*≡(*β*
_{1}
*−β*
_{2})/2| is very large so that |*δ*|~*κ*, Δ*ψ−*Δ*ψ’* can be comparable to other terms in Eqs. (26) and (27), and the presented model will have appreciable errors. However, in this case, expansion of the individual waveguide modes to the lowest two system modes usually is not sufficient, and the expansion basis has to be enlarged in Eq. (9). These complex situations are out of scope of this paper and will not be discussed.

## Acknowledgments

This work was supported by Air Force Office of Scientific Research under Contract No. FA9550-06-01-2003 (G. Pomrenke). The authors are also grateful to A. Eftekhar for helpful discussions.

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**19. **For example, for the material system used in Ref. [9], Δ*β* is numerically found to be negative and small (~-0.015 μm^{−1} at 1578nm for TE polarization, *w*_{1}/*w*_{2} = 400/400 nm, and *gap* = 100 nm). The corresponding simulation results are shown in Fig. 6 in Ref. [9]. We already know Δ*ψ* is negative for this coupling geometry (“CB” coupler), and therefore both terms in Eqs. (23) are negative and the second term is dominant. The two features that *ϕ*_{11} (ring-ring phase) is negative (corresponding CIFS positive) and *ϕ*_{11} and *ϕ*_{22} are of opposite signs are readily understood from Eqs. (23) and (24).

**20. **M. Soltani, *Novel integrated silicon nanophotonic structures using ultra-high Q resonator,* Ph.D. dissertation, Georgia Institute of Technology (2009)

**21. **M. Popovic, “Complex-frequency leaky mode computations using PML boundary layers for dielectric resonant structures,” in Proceedings of Integrated Photonics Research (Washington, DC, June 17, 2003).