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],

 

Fig. 1 Two typical coupler architectures with different coupling geometries. a ₁, a ₂ are the input signals and b ₁, b ₂ are the corresponding output signals; L is the length of the middle straight part; R is the outer bend radius; gap is the spacing between edges of the two waveguides, and w ₁, w ₂ are the waveguide widths as shown. The brown region is the waveguide core at the resonator side, and the yellow region, which shows the access waveguide or part of the adjacent resonator, can be changed to be either waveguide core or cladding to obtain the net phase shift from the coupling effect. (a) Coupler CA with vertical symmetry axis V and horizontal symmetry axis H. (b) Coupler CB with vertical symmetry axis V.

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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],

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],

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 κ² (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).

 

Fig. 2 (a) Simulation parameters for the coupler CA shown in Fig. 1(a). The bend radius R is 6 μm; the waveguide width w ₁ is 400 nm, and the gap is 100 nm. n _WG and n _Cladding are the refractive indices of the waveguide core and cladding, respectively, as obtained from the effective index method. Simulation (marker) and modeling (dotted line) results of the coupler CA as a function of the straight part length L are shown for (b) phase responses (for the sake of clarity, ϕ ₁₂ (ϕ ₂₁) is plotted relative to its value at L = 0), and (c) power coupling coefficient κ².

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Fig. 3 (a) Simulation parameters for the coupler CB shown in Fig. 1(b). The bend radius R is 6 μm; the waveguide widths w ₁ and w ₂ are both 400 nm, and the gap is 100 nm. Simulation (marker) and modeling (dotted line) results of the coupler CB as a function of the straight part length L are shown for (b) phase responses ϕ ₁₁, ϕ ₂₂, (c) phase responses ϕ ₁₂ ,ϕ ₂₁, and (ϕ ₁₁ + ϕ ₂₂)/2 (for the sake of clarity, ϕ ₁₂ (ϕ ₂₁) is plotted relative to its value at L = 0), and (d) power coupling coefficient κ².

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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]^T while the transfer matrix T for the middle straight part relates [d ₁, d ₂]^T to [c ₁, c ₂]^T. Here [c ₁, c ₂]^T and [d ₁, d ₂]^T are the input and output optical signals for the middle straight part. Power conservation requires that both matrices 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^T, which relates [b ₁, b ₂]^T to [d ₁, d ₂]^T. The total transfer matrix U, as a result, is given by [10]

 

Fig. 4 Three components comprising the coupler CA (lower part of input and output bends by dashed line) and CB (lower part of input and output bends by solid line) originally defined in Fig. 1: input bend, middle straight part, and output bend, which are modeled by transfer matrices B, T and B^T, respectively. The two fundamental system modes of the middle parallel waveguide coupler are schematically shown on the middle part of the coupler.

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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_as, respectively, are schematically shown in Fig. 4. Although in the case w ₁ ≠w ₂, the two system modes do not have simple symmetry relating to the individual waveguide modes, we shall still use β_s and β_as 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 T for the middle parallel waveguide coupler can be expressed as [16],

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

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_as 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 L observed in Figs. 2(b) and 3(c) can be identified with this second-order phase term exp(iΔβL), with the slope Δβ provided in Table 1 for the middle parallel waveguide coupler simulated in Figs. 2 and 3.

Table 1. Modeling parameters and fitting results for the coupled structures shown in Figs. 2, 3 and 6

View Table

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

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 + Δψ)) for the upper arm, and exp(i(α ₀-Δψ)) for the lower arm, respectively (we put the details in the Appendix B). For the output bend, the transfer matrix is B^T, and the free propagation phases will be the same as those of the input bend. Finally, we obtain N as,

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 Δψ = 0 due to the symmetry about the horizontal axis. After a straightforward calculation for the transfer matrix U, we obtain,

For the coupler CB Δψ≠0, and the results are more complicated. For small |Δψ| (<0.5 rad), exp(iΔψ)≈1 + iΔψ 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,

The negative sign of Δψ 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 Δψ when w ₁ = w ₂ (from Eq. (14) exp(i2Δψ) is the phase difference between the upper and the lower arms). This explanation could be qualitatively understood from Fig. 5(a) .

 

Fig. 5 (a) Input bend of the coupler CB shown in Fig. 1(b), which consists of the upper bent waveguide with width w ₁ and the lower straight waveguide with width w ₂. β ₁, β ₂ are the propagation constants of individual straight waveguides with widths w ₁ and w ₂, respectively. The reference planes of [a ₁, a ₂]^T and [c ₁, c ₂]^T are shown by the dashed line. From the simulation and modeling results of Fig. 3 we conclude that the length of propagation path from a ₁ to c ₁ is smaller than that from a ₂ to c ₂, resulting a negative Δψ when β ₁ = β ₂. (b) A schematic drawing Δψ of as a function of (β ₁-β ₂). The numbers are just shown for illustrative purpose and not exact.

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Based on this understanding, the variations of Δψ 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, Δψ 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, Δψ will be larger than that in the w ₁ = w ₂ case. If we keep reducing w ₂, Δψ will increase from a negative value to an eventual positive value. This behavior of Δψ 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 Δψ, 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 |Δψ| (<0.5 rad), exp(iΔψ)≈1 + iΔψ, 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, Δψ 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, Δψ 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.

 

Fig. 6 (a) Simulation parameters for the coupler CB shown in Fig. 1(b). The bend radius R is 6 μm; the gap is 100 nm; the upper waveguide width w ₁ is fixed at 400 nm, and the simulations are performed for three different values of the lower waveguide width w ₂: 390 nm, 400 nm, and 410 nm. Simulation (marker) and modeling (dashed line) results of the coupler CB as a function of the straight part length L are plotted in (b), (c) and (d): (b) phase response ϕ ₁₁, (c) phase response ϕ ₂₂, and (d) power coupling coefficient κ².

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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:

The second term in Eqs. (23) and (24), which is responsible for the difference between ϕ ₁₁ and ϕ ₂₂, is proportional to Δψ. Contrary to Δβ whose property is difficult to predict without specific numerical calculations, Δψ 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 Δψ 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, Δψ 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 |Δβ| 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 Δψ is nonzero), an interesting feature mentioned in the beginning of this paper [9,19].

 

Fig. 7 (a) Simulation parameters for the coupler CB shown in Fig. 1(b) with L = 0. The bend radius R is 6 μm; the gap is 100 nm; the upper waveguide width w ₁ is fixed to be 400 nm, and the lower waveguide width w ₂ varies. Simulation results as a function of w ₂ are plotted in (b), (c): (b) phase responses ϕ ₁₁ and ϕ ₂₂ (the red dotted line denotes the waveguide width that Δψ = 0), and (c) power coupling coefficient κ² (the two red dotted lines denote the waveguide widths corresponding to maximum power transfer and Δψ = 0, respectively).

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Δψ also affects the power coupling coefficient κ² as shown in Eq. (25), and we notice that the maximum power transfer will be less than 100% if Δψ≠0, a characteristic feature in a phase mismatched coupler. Thus, we conclude that Δψ is a measure of the phase mismatch of the input bend. Figure 7(c) plots the power coupling coefficient κ² for the coupler CB by fixing 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 Δψ but also z_a in Eq. (25)). The maximum power transfer will occur when the upper and the lower arms are close to be phase matched, i.e., Δψ≈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 κ² is presented.

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 Δβ could be either positive or negative, depending on specific material systems and coupling structures. The geometric parameter Δψ, 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 Δβ 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 Δψ, 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.

 

Fig. 8 (a) Structure used for CIFS simulation using a complex-frequency eigenmode solver: a microring with an outer radius R coupled to a straight waveguide. The brown region is the waveguide core with refractive index n _WG, and the yellow region can be either waveguide core or cladding to obtain the coupling effect on the resonance frequency shift of the resonator. (b) Simulation parameters for the structure shown in (a). (c) CIFS obtained via the eigenvalue solution (dashed line) and the phase response method (cross), respectively (note that CIFS is shown in shift in wavelength, and a positive shift in wavelength means a negative CIFS).

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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, Δβ~0) the free propagation phases are not necessarily exp(i(α ₀+Δψ)) for the upper arm, and exp(i(α ₀-Δψ)) for the lower arm for the fixed reference planes of [a ₁, a ₂]^T and [c ₁, c ₂]^T. Suppose Δψ needs to be replaced by Δψ’ in Eq. (15) for the normalization matrix N, then Eqs. (19) and (20) become,

(26)$${\varphi }_{11}\approx \Delta \beta (L+2z_p)+2\Delta \psi \sin ({\beta }_0{z}_a)\frac{\sin ({\beta }_0(L+z_a))}{\cos ({\beta }_0(L+2z_a))}+2(\Delta \psi -\Delta \psi \text{'})$$

(27)$${\varphi }_{22}\approx \Delta \beta (L+2z_p)-2\Delta \psi \sin ({\beta }_0{z}_a)\frac{\sin ({\beta }_0(L+z_a))}{\cos ({\beta }_0(L+2z_a))}-2(\Delta \psi -\Delta \psi \text{'})$$

From Eqs. (26) and (27) we see that there will be a constant proportional to (Δψ−Δψ’) (a factor of 2) added to ϕ ₁₁, and the same constant subtracted from ϕ ₂₂. Clearly when the gap is large, Δψ−Δψ’ is negligible. We can estimate the effect of the coupling on Δψ−Δψ’ 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

(28)$$(\Delta \psi -\Delta \psi \text{'})\approx \delta L(\frac{\tan ({\beta }_{0}L)}{{\beta }_{0}L}-1)$$

For the input bend, the effective length 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 be

(29)$$(\Delta \psi -\Delta \psi \text{'})\approx \delta z_a{({\beta }_0{z}_a)}^2/3$$

For 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, Δψ−Δψ’ 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.

References and links

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19. For example, for the material system used in Ref. [9], Δβ is numerically found to be negative and small (~-0.015 μm⁻¹ at 1578nm for TE polarization, w₁/w₂ = 400/400 nm, and gap = 100 nm). The corresponding simulation results are shown in Fig. 6 in Ref. [9]. We already know Δψ 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 ϕ₁₁ (ring-ring phase) is negative (corresponding CIFS positive) and ϕ₁₁ and ϕ₂₂ are of opposite signs are readily understood from Eqs. (23) and (24).

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