1. Introduction

Changing the gap width between the coupled waveguides is desired in situations when the power coupling coefficient needs to be tuned for purpose such as changing the power coupling efficiency, obtaining the critical coupling [1, 2]. There is a π/2 phase change for weak coupling and this phase change departs from π/2 as the coupling conditions deviates from weak coupling causing the coupling-induced resonance frequency shifts (CIFS) for a resonator[3–5]. The magnitude of the phase deviation from π/2 in the CIFS effect is small but for telecom-grade devices where precise wavelength control is required, the CIFS effect has to be avoided or compensated[6, 7]. In this paper, we report and analyze our findings on a large phase change, a π phase jump, for the coupled waveguides when the gap width between the coupled waveguides changes. We demonstrated and analyzed in Sec. 2.1 that the phenomenon occurs in the conventional straight co-directional coupler, and we showed in Sec. 2.2 by numerical example that the phenomenon also occurs in half-ring coupler. Basic characteristics and discussions of the phenomenon with respect to the gap width variation were given. Spectral shift, based on the phase jump phenomenon, by half free spectral range (FSR) in a single-arm and a double-arm microring resonator were demonstrated in Sec. 2.3.

For processing the dense channel spacing signal in optical communication, it is sometimes desired that the signals be interleaved such that the communication bandwidth or capacities can be increased in a cost-effective way. There are three approaches to achieve dense signals interleaving: lattice filter, Gires-Torunois-based Michelson interferometer and array-waveguide grating [8]. Bulk optics and integrated waveguide optics can be implemented for the interleaver such that a series of dense signal channels can be combined or separated. Waveguide-type inter-leaver has the advantage of compact size, moreover, compact waveguide-type interleaver with multi-stage design is more flexible than bulk one with multi-stage design. For the waveguide-type interleaver, Mach-Zehnder (MZ) interleaver [9, 10] was commonly used, and multi-stage MZ interleaver and microring-assisted MZ interleaver [11–14] were used to widen the 3 dB bandwidth and reduce the crosstalk. However, these devices still suffer from the large footprint, usually several millimeters to centimeters, to make the integration impractical. In this paper, we propose an all-microring interleaver based on employing the phase jump phenomenon, and chip size integration could be implemented for the all-microring device. A numerical example is given to demonstrate the feasibility of the idea in Sec. 3.

2. Theory

2.1. Symmetrical co-directional coupler

Figure 1 shows a symmetrical co-directional coupler. The input-output relationship derived from the rigorous coupled-mode theory (rCMT), as given by Ref. 15, can be recast into a simple form in terms of the scattering matrix or S-matrix:

 

Fig. 1. Symmetrical co-directional coupler

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where

where L is the coupling length, K_a = (K -CX)/(1- ∣C∣²), and Γ = (KC-X)/(1- ∣C∣²). K,C, and X are defined as:

E ₁ and E ₂ are the mode fields of the waveguide 1 and 2, respectively. L_c = π/(2K_a) is defined as the Transfer Distance. 2L_c is the coupling length for one complete cycle of power exchange. K is the coupling coefficient, C represents the cross-coupling coefficient, and X is the self-coupling coefficient of the coupled modes for the symmetrical co-directional coupler. When B ₀=0, the input-output relationship is:

For weak coupling (wCMT), i.e. C and X are equal to zero, Γ=0, K_a=K, and L_c=π/(2K), the matrix elements are given as:

When the gap width g is varied, K, C, X and hence K_a, Γ and L_c are varied accordingly, and up to a certain gap width the cosine and the sine function in Eq. (2a)–(2d) change sign, a π phase jump occur in s ₁₁, s ₂₁, s ₁₂ and s ₂₂. From the definition of L_c and Eq. (2a)–(2d), the phase jump occurs at the gap width for which

is satisfied, n= odd integer for s ₁₁ and s ₂₂, n= even integer for s ₁₂ and s ₂₁. The underlying physics of the π phase jump is fundamentally due to the phase accumulation of the back and the forth coupling each contributes π/2 phase, n + 1 is the number of the forth and the back coupling within L. Notice that for B ₀=0, the phase difference between A ₀ and A ₁ is ϕ ₁₁, and the phase difference between A ₀ and B ₁ is ϕ ₂₁, according to Eq. (6) and (7).

We demonstrate this phenomenon with a numerical example as follows: A co-directional coupler as shown in Fig. 1 is given with n_core=2.5, n_cladding=1.5, a=300 nm, L= 6 um, and λ=1550 nm. The single TE mode field distribution and the effective mode index for the single waveguide were obtained by using the beam propagation method (BPM) and they were then substituted into Eq. (3), (4), (5) and Eq. (2a)–(2d) to obtain all the matrix elements in Eq. (1b) and L_c for the co-directional coupler. Both rigorous coupled-mode theory (rCMT) and weak coupling assumption (wCMT) were calculated. The results for K_aL/π, L_c and the magnitude and phase for s ₁₁ and S ₂₁ vs. the gap width are shown in Figs. 2(a)–2(f). s ₂₂ and s ₁₂ are not shown but they are related to s ₁₁ and s ₂₁ by: s ₂₂ = s ₁₁, s ₁₂ = s ₂₁ as can be seen from Eq (2a)–(2d).

Figures 2(a)–2(f) reveal, firstly, L_c decreases with the gap width as expected, secondly, the π phase jump occurred for wCMT as well as for rCMT as predicted, there is a phase difference of ΓL between wCMT and rCMT consistent with Eq.(2a)–(2d), thirdly, the magnitude of the matrix element has a minimum at the gap width where there is phase jump, and fourthly, only limited number of phase jump occurred as the gap width is varying; the odd order phase jump occurs for ϕ ₁₁ and the even order phase jump occurs for ϕ ₂₁. According to Eq. (5) mathematically, the phase jump will keep on occurring with n >4 but at negative gap width that is not realistic and the CMT would not be valid, therefore, only limited number of phase jump occurs for a given coupling length L. When L increases, the first-order jump (n=1) occurs at larger gap width and more higher order jumps occur at smaller gap width.

 

Fig. 2. (a)Transfer distance (L_c). (b)K_aL/π. (c)ϕ ₁₁. (d)ϕ ₂₁. (e)∣s _ll∣. (f)∣s _2l∣. vs. gaP width g for the symmetrical co-directional coupler of Fig. 1. (n_core=2.5, n_cladding=1.5, a=300 nm, L=6 μm, and λ=1550 nm. - ○ -: Calculated with coupled mode theory (rCMT), -∆-: Calculated with weak coupled mode theory (wCMT).)

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2.2. Symmetrical half-ring coupler

In order to analyze the phase jump phenomenon for a half-ring coupler as shown in Fig. 3, we have developed an effective numerical method for numerically measuring the matrix elements of the S-matrix. Similar method was given by Ref. 3 for straight-half ring coupler. Our method is introduced as follows: Step1: the BPM is used to obtain the eigen-mode field distribution (M1) of a straight waveguide that has a cross-section identical to that of the ring waveguide. Step 2: M1 is taken as the launch field for a quarter ring waveguide in the FDTD simulation, the output filed of the quarter ring is then used as the input field for the second iteration, and the output field converged to the eigen-mode, M2, of the ring waveguide after several iterations. Step 3: M2 is then taken as the launch field at plane 1, referring to Fig. 3, with a specific pulse form and duration in the FDTD simulation, and the time evolution of the field at plane 1, 2 and 4 are monitored. Step 4: discrete Fourier transform is carried out on the time-domain fields to obtain the spectra for A ₀, A ₁ and B ₁. The matrix elements of S-matrix, s ₁₁ and s ₂₁, are then obtained through Eq. (6) and (7). Step 3 and 4 are repeated with launching M2 at plane 3, s ₂₂ and s ₁₂ are obtained accordingly. A numerical example, referring to Fig. 3, was carried out with the following parameters: n_core=2.5, n_cladding=1.5, waveguide width=300 nm, λ=1550 nm, material loss coefficient α_mat=0 dB/cm and outer radius R=12.15 μm. There was only one mode for the ring waveguide and it was a TE-like mode. The results are shown in Figs. 4(a) and 4(b). There is a noticeable phase protruding about 0.1π in ϕ ₁₁ right below the gap width of the phase jump, no satisfactory explanation could be given for the protrude phase. Nevertheless, it is clear that the phase jump indeed occurs in the half-ring coupler and its magnitude is close to π. The phase jump occurred at the minimum s ₁₁ and maximum s ₂₁. There is only one phase jump for 011 and no phase jump for 021 in this case. The criteria for the phase jump to occur in the half-ring coupler could be modified from Eq. (10) as L_eff=n L _c,eff, where L_eff is the effective coupling length and L _c,eff is the effective Transfer Distance, and n=1 in this particular example. For half-ring coupler with larger radius of curvature with the same gap width, we expect that L_eff would be larger and L_c,eff would be smaller such that the number of phase jump would be increased as the gap width is varying.

 

Fig. 3. Symmetrical half-ring coupler. (n_core=2.5, n_cladding=1.5, guide widths=300 nm, and α_mat=0 dB/cm).

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When the gap width and the radius of curvature for the half-ring coupler are fixed, the phase jump phenomenon will occur when the wavelength is varying. The simulation results are shown in Figs. 5(a) and 5(b) for the same half-ring coupler but fixed gap width at 85 nm. The phase jump occurs at 1481.14 nm for this case. We define this wavelength as λ_c. For longer wavelength, the evanescent field extends further out, a shorter L _c,eff is expected, and more phase jump would occur at longer wavelengths. The coefficient of K in Eq. (3) is inversely proportional to λ, the cosine and the sine function in Eq. (2a)–(2d) change sign whenever the wavelength changes by the order of magnitude of L, we therefore assert that the wavelength difference between two adjacent phase jump of the same order would be in the order of L_eff for half-ring coupler.

 

Fig. 4. (a)∣s ₁₁∣, and ∣s ₂₁∣ (b)ϕ ₁₁, and ϕ ₂₁ vs. gap width g for the symmetrical half-ring coupler of Fig. 3.

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Fig. 5. (a)∣s ₁₁∣, and ∣s ₂₁∣ (b)ϕ ₁₁, and ϕ ₂₁ vs. free-space wavelength for the symmetrical half-ring coupler of Fig. 3.

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We have calculated λ_c vs. gap width, g, and the result is shown in Fig. 6. Several important aspects with respect to the phase jump phenomenon, when the device is operated within a certain bandwidth, are implied: firstly, if one wants to avoid the phase jump, the gap width has to be larger than a certain value determined by the upper bound of the operating bandwidth, in the region of g_III as shown in Fig. 6 for C-band operation as an example, secondly, if one wants to utilize the phase jump for all operating wavelength, the gap width has to be smaller than a certain value determined by the lower bound of the operating bandwidth, in the region of g_I, and thirdly, if one wants to utilize the phase jump within the operating wavelength range, then the gap width has to be controlled and varied within a certain range g_II determined by the upper and the lower bound of the operating bandwidth.

 

Fig. 6. g vs. λ_c for the symmetrical half-ring coupler of Fig. 3.

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2.3. Single-arm and double-arm microring

Referring to Fig. 4(b), we designated the gap width as g_C at the phase jump, g_L when there is no phase jump, and g_S when there is phase jump. For a single-arm microring with g_S gap width, the through field undergoes π phase jump and the field that is coupled back to the bus after one round trip in the ring undergoes twice π/2 phase change from the coupling, the over-all through field at resonance is in constructive interference in contrast to that of the microring with g_L gap width where the over-all through field at resonance is in destructive interference, therefore, we expect that there will be a spectral shift between the microrings with g_L and g_S gap width, the amount of the spectral shift will be around half the free spectral range (FSR). For a double-arm microring, we define that the microring with both gap width is g_L or g_S as type I, and the microring with one gap is g_L and the other gap is g_S as type II. For type I, the field that couples back to the input bus after one round trip in the ring undergoes one phase jump (or no phase jump) same as the through field such that the over-all through field at resonance is in destructive interference due to twice π/2 phase change from the coupling, therefore, we expect that there is no spectral shift, neither in the through port nor in the drop port for type I, with respect to the single-arm microring with g_L. For type II, the field that couples back to the input bus after one round trip in the ring undergoes one phase jump (or no phase jump) that is opposite to the through field such that the over-all through field at resonance is in constructive interference, therefore, we expect that there is a half FSR spectral shift both in the through port and in the drop port for type II.

We prove the above-mentioned assertions by the following numerical examples. The microring is composed of half-ring bus and a ring with identical parameters, i.e β matching. The spectrum at the through port I_through and the spectrum at the drop port I_drop were calculated by the analytical expressions, Eq. (A.5), Eq. (A.14), and Eq. (A.15), as given in Appendix A by using the elements of the S-matrix for the half-ring coupler. Numerical measurement on the S-matrix elements of the half-ring coupler as introduced in Sec. 2.2 was performed first, and then the measured matrix elements were substituted into the analytical expressions in Appendix A to obtain the I_through and the I_drop spectra. We have also applied the FDTD method to obtain I_through and I_drop spectra in order to verify the validity of the analytical expressions. Same parameters for the half-ring coupler in Sec. 2.2 were used here except that the material power loss coefficient was chosen to be 50 dB/cm for the microring in order to, firstly, reduce the computational time for the FDTD method and secondly, profound spectra that is close to critical coupling could be obtained.

Figure 7(a) and 7(b) show the results for single-arm microring obtained by the FDTD method, red solid lines, and by the analytical expressions with the S-matrix elements, black solid lines. Both methods gave identical results except that the dip depths are slightly different due to the limited FDTD computational time. Fig. 7(a) shows I_through spectrum for a single-arm microring with g_L=350 nm, i.e. no phase jump. Fig. 7(b) shows I_through spectrum for the single-arm microring with g_S=20 nm, i.e. with phase jump. It is apparent that the spectra are shifted with each other by approximately half the FSR.

The reason that the shift is not exactly half FSR is that the resonance frequency is slightly different between these two cases due to that ϕ ₂₁ is not exactly π/2 for g_S=20 nm; weak coupling assumption is not exactly applied to g_S=20 nm. This assertion can be confirmed from Fig. 4(b) and from the non-zero ΓL in Fig. 2(c) for the co-directional coupler. This is basically the coupling-induced resonance frequency shift (CIFS) phenomenon when weak coupling assumption is not exactly applied [3].

Figure 8(a) and 8(b) show the through and the drop spectra of the type I and the type II double-arm microrings, respectively, with g_L=350 nm and g_S=20 nm. Again, the resonance frequencies are slightly different due to the CIFS effect. However, approximately half FSR spectral shift between the type I and the type II are clearly shown. Without the CIFS effect, the spectral shift should be exactly half FSR.

It is worth noticing that the phase jump in ϕ ₁₂ and ϕ ₂₁, if there is any, does not cause spectral shift, since the field undergoes both couple-in and couple-out that accumulates phase jump twice to yield 2π phase change.

For multi-ring add-drop device, where there are more than two gaps, the gap widths have to be arranged alternately between g_L and g_S in order to obtain the half FSR spectral shift. Multi-ring device of this type can be referred to as the generalized type II in contrast to the generalized type I in which all gap widths are g_L or g_S.

Notice that if there are more than one g_C, referring to Fig. 2(c), g_L and g_S are not restricted to the opposite side of a particular g_C, it is only required that one lies in the phase jump region and the other one lies in the region where there is no phase jump.

3. Application on wavelength interleaver

Based on the findings of the previous section, we propose an all-microring wavelength interleaver that consist a type I and a type II microring in series as shown in Fig. 9. An equal-spacing multi-channel input signal can be separated through the interleaver that has FSR twice the input channel spacing. We demonstrate the feasibility of the idea numerically through the following example: The spectra of the drop ports and the through port of the configuration in Fig. 9 can be calculated by the method as introduced in Appendix A.2; The elements of the S-matrix, including its magnitude and phase, for each half-ring coupler in Fig. 9 can be numerically measured as introduced in Sec. 2.2, and then substituting into Eq. (A.14) and Eq. (A.15), applying the method twice, all the output spectra can be obtained. The values for the relevant parameters of the waveguide in Fig. 9 were the same as those for the waveguides in Sec. 2.3 with g_L=350 nm and g_S=20 nm. The results are shown in Figs. 10(a)–10(c). It is apparent that the spectra of the drop ports were separated approximately half-way to each other.

 

Fig. 7. (a)I_through/I_i with g_L=350 nm. (b)I_through/I_i with g_S=20 nm for single-arm microring. (Black line: calculated from the S-matrix method, Red line: calculated from the full-domain FDTD method.)

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Fig. 8. I_through/I_i and I_drop/I_i for (a) Type I and (b) Type II for double-arm microring with g_L=350 nm and g_S=20 nm. (Black line: calculated from the S-matrix method, Red line: calculated from the full-domain FDTD method.)

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Fig. 9. All-microring wavelength interleaver.

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Fig. 10. (a)I _drop1/I_i. (b)I _drop2/I_i. (c)I_through/I_i. for all-microring wavelength interleaver with g_L=350 nm and g_S=20 nm.

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For a practical all-microring interleaver that is required to meet certain specifications, such as 3 dB bandwidth and cross-talk, the configuration needs to be optimized accordingly; multi-ring configuration, i.e. generalized type I and II in series, could be adopted and optimization of the structure could be carried out according to the methods provided in Ref. 16–19 for micro-ring devices to meet the specification.

The separation of the spectra in Figs. 10(a) and 10(b) was not exactly half the FSR due to the CIFS effect from the g_S that was 20 nm. The CIFS effect needs to be avoided or reduced as much as possible such that the type I and the type II has the FSR-matching, i.e. resonant frequencies located at exactly half FSR of each other, in order to interleave the equally spaced input channels. There are several solutions to the CIFS problem: firstly, it is feasible to optimize the 3 dB bandwidth of the multi-ring interleaver such that the FSR mismatch falls within the 3 dB bandwidth, secondly, type I with all g_L is preferred than that with all g_S, since the CIFS is negligible for large gap width microring, thirdly, adjusting the radius of curvature or the material refractive index for each type microring to match the FSR is also feasible, fourthly, using microrings with larger radius of curvature; since microring with larger R has longer effective coupling length (L_eff) that yields more phase jump and the first-order jump would occur at larger gap width with longer effective coupling length as was discussed in Sec. 2.1; we can select g_L and g_S that are around the first-order g_C that is large such that the CIFS effect could be minimized. The interleaver for dense channel spacing signal also calls for rings with large radius of curvature, for example: for rings with R= 12.15 μm as in the previous numerical examples, FSR= 13 nm; and for rings with R=115 μm, FSR=1.6 nm, 0.8 nm channel spacing signal could be interleaved. We would therefore expect that the CIFS effect could be greatly reduced for interleaver that targets DWDM application. We did not use R=115 μm ring for numerical example in this paper was due to the limited computational time available. Race-track microring with long coupling length and hence larger gap widths also serves the purpose for reducing the CIFS effect.

4. Conclusions

We showed that the phase jump phenomenon in the coupled-waveguide occurs in the co-directional coupler as well as the half-ring coupler. The spectra of the single-arm and the double-arm microring resonator could be shifted by approximately half FSR with properly selecting the gap widths. An all-microring wavelength interleaver was proposed and numerically demonstrated. Methods to reduce the possible CIFS effect in the interleaver were given. The advantages of the all-microring interleaver over the Mach-Zenhder type interleaver is smaller footprint and less sensitive to material loss that is inherent to microring. Either passive devices with fixed gap width or active devices with tunable gap width by MEMS technologies could be implemented.

Appendix A: Analytical expressions for the through and the drop spectra of a single-arm and a double-arm microring resonator based on the S-matrix method

A.1 Through spectrum of a single-arm microring resonator

From Eq. 1(a),

(A.1)$${\mathbf{A}}_1={\mathbf{s}}_{11}{\mathbf{A}}_0+{\mathbf{s}}_{12}{\mathbf{B}}_0$$

(A.2)$${\mathbf{B}}_1={\mathbf{s}}_{21}{\mathbf{A}}_0+{\mathbf{s}}_{22}{\mathbf{B}}_0$$

From the Fig. below,

(A.3)$${\mathbf{B}}_0={\mathbf{B}}_1\exp \left[-\pi R\left({\alpha }_{\mathrm{ring}}/2+j{\beta }_{\mathrm{ring}}\right)\right]$$

 

Fig. 11. Single-arm microring resonator.

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where α_ring is the power loss coefficient of the ring mode, and β_ring is the propagation constant of the ring mode. Substituted Eq. (A.3) into Eq. (A.2) then (A.1), we obtain:

(A.4)$${\mathbf{A}}_1=\left\{{\mathbf{s}}_{11}+{\mathbf{s}}_{12}{\mathbf{s}}_{21}\exp \left[\pi R\left({\alpha }_{\mathrm{ring}}/2+j{\beta }_{\mathrm{ring}}\right)\right]\right\}{\mathbf{A}}_0/\left\{1-{\mathbf{s}}_{22}\exp \left[\pi R\left({\alpha }_{\mathrm{ring}}/2+j{\beta }_{\mathrm{ring}}\right)\right]\right\}$$

Let I_through/I_i = ∣A ² ₁∣/∣A ² ₀∣. From Eq. (A.4), we obtain I_through/I_i, recast into the form of a standard Fabry-Perot cavity:

(A.5)$$I_{\mathrm{through}}/I_i=1-\genfrac{}{}{0.1ex}{}{I_{max}/I_i}{1+{\left(2\Im /\pi \right)}^2{\sin }^2(\theta /2)}$$

where ℑ = π√r/(1-r)

$$r=\mid {\mathbf{s}}_{22}\mid \exp (-{\alpha }_{\mathrm{ring}}\pi R)$$

$$\theta ={\varphi }_{22}+{\beta }_{\mathrm{ring}}\pi R$$

$$I_{max}/I_i=P_1+\left(P_2-P_3-P_4\right)/{\left(1-r\right)}^2$$

$$P_1=\left(1-{\mid {\mathbf{s}}_{11}\mid }^2\right)\left[1+{\left(2\Im /\pi \right)}^2{\sin }^2(\theta /2)\right]$$

$$P_2=2\exp \left(-{\alpha }_{\mathrm{ring}}\pi R\right)\mid {\mathbf{s}}_{11}\mid \mid {\mathbf{s}}_{12}\mid \mid {\mathbf{s}}_{21}\mid \mid {\mathbf{s}}_{22}\mid \cos \left({\varphi }_{11}-{\varphi }_{12}-{\varphi }_{21}+{\varphi }_{22}\right)$$

$$P_3=2\exp \left(-{\alpha }_{\mathrm{ring}}\mathrm{\pi R}/2\right)\mid {\mathbf{s}}_{11}\mid \mid {\mathbf{s}}_{12}\mid \mid {\mathbf{s}}_{21}\mid \cos \left({\varphi }_{11}-{\varphi }_{12}-{\varphi }_{21}-{\beta }_{\mathrm{ring}}\pi R\right)$$

$$P_4={\mid {\mathbf{s}}_{12}\mid }^2{\mid \mid {\mathbf{s}}_{21}\mid }^2\exp \left({-\alpha }_{\mathrm{ring}}\pi R\right)$$

A.2 Through and drop spectra of a double-arm microring resonator

Refering to the Fig. below, we define two S matrices, S ⁽¹⁾ and S ⁽²⁾, for the double-arm microring:

(A.6)$$\left[\begin{array}{c}\multicolumn{1}{c}{{\mathbf{A}}_1}\\ \multicolumn{1}{c}{{\mathbf{B}}_1}\end{array}\right]=S^{\left(1\right)}\left[\begin{array}{c}\multicolumn{1}{c}{{\mathbf{A}}_0}\\ \multicolumn{1}{c}{{\mathbf{B}}_0}\end{array}\right]=\left[\begin{array}{cc}\multicolumn{1}{c}{{\mathbf{s}}_{11}^{\left(1\right)}}& \multicolumn{1}{c}{{\mathbf{s}}_{12}^{\left(1\right)}}\\ \multicolumn{1}{c}{{\mathbf{s}}_{21}^{\left(1\right)}}& \multicolumn{1}{c}{{\mathbf{s}}_{22}^{\left(1\right)}}\end{array}\right]\left[\begin{array}{c}\multicolumn{1}{c}{{\mathbf{A}}_0}\\ \multicolumn{1}{c}{{\mathbf{B}}_0}\end{array}\right]=$$
$$\left[\begin{array}{cc}\multicolumn{1}{c}{\mid {\mathbf{s}}_{11}^{\left(1\right)}\mid \exp \left(-j{\varphi }_{11}^{\left(1\right)}\right)}& \multicolumn{1}{c}{\mid {\mathbf{s}}_{12}^{\left(1\right)}\mid \exp \left(-j{\varphi }_{12}^{\left(1\right)}\right)}\\ \multicolumn{1}{c}{\mid {\mathbf{s}}_{21}^{\left(1\right)}\mid \exp \left(-j{\varphi }_{21}^{\left(1\right)}\right)}& \multicolumn{1}{c}{\mid {\mathbf{s}}_{22}^{\left(1\right)}\mid \exp \left(-j{\varphi }_{22}^{\left(1\right)}\right)}\end{array}\right]\left[\begin{array}{c}\multicolumn{1}{c}{{\mathbf{A}}_0}\\ \multicolumn{1}{c}{{\mathbf{B}}_0}\end{array}\right]$$

 

Fig. 12. Double-arm microring resonator.

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(A.7)$$\left[\begin{array}{c}\multicolumn{1}{c}{{\mathbf{B}}_0}\\ \multicolumn{1}{c}{{\mathbf{C}}_0}\end{array}\right]=S^{\left(2\right)}\left[\begin{array}{c}\multicolumn{1}{c}{{\mathbf{B}}_1}\\ \multicolumn{1}{c}{{\mathbf{C}}_1}\end{array}\right]=\left[\begin{array}{cc}\multicolumn{1}{c}{{\mathbf{s}}_{11}^{\left(2\right)}}& \multicolumn{1}{c}{{\mathbf{s}}_{12}^{\left(2\right)}}\\ \multicolumn{1}{c}{{\mathbf{s}}_{21}^{\left(1\right)}}& \multicolumn{1}{c}{{\mathbf{s}}_{22}^{\left(1\right)}}\end{array}\right]\left[\begin{array}{c}\multicolumn{1}{c}{{\mathbf{B}}_1}\\ \multicolumn{1}{c}{{\mathbf{C}}_1}\end{array}\right]=$$
$$\left[\begin{array}{cc}\multicolumn{1}{c}{\mid {\mathbf{s}}_{11}^{\left(2\right)}\mid \exp \left(-j{\varphi }_{11}^{\left(2\right)}\right)}& \multicolumn{1}{c}{\mid {\mathbf{s}}_{12}^{\left(2\right)}\mid \exp \left(-j{\varphi }_{12}^{\left(2\right)}\right)}\\ \multicolumn{1}{c}{\mid {\mathbf{s}}_{21}^{\left(2\right)}\mid \exp \left(-j{\varphi }_{21}^{\left(2\right)}\right)}& \multicolumn{1}{c}{\mid {\mathbf{s}}_{22}^{\left(2\right)}\mid \exp \left(-j{\varphi }_{22}^{\left(2\right)}\right)}\end{array}\right]\left[\begin{array}{c}\multicolumn{1}{c}{{\mathbf{B}1}_{\mathrm{}}}\\ \multicolumn{1}{c}{{\mathbf{C}}_1}\end{array}\right]$$

From the Fig. above, the following relationships between B ₀, B ₁, A ₀, and C ₀ can be obtained :

(A.8)$${\mathbf{B}}_0={\mathbf{B}}_1{\mathbf{s}}_{11}^{\left(2\right)}$$

(A.9)$${\mathbf{B}}_1={\mathbf{B}}_0{\mathbf{s}}_{22}^{\left(1\right)}+{\mathbf{A}}_0{\mathbf{s}}_{21}^{\left(1\right)}$$

(A.10)$${\mathbf{C}}_0={\mathbf{B}}_1{\mathbf{s}}_{21}^{\left(2\right)}$$

Substituting Eq. (A.8) into Eq. (A.9) then Eq. (A.10), we obtain:

(A.11)$${\mathbf{C}}_0={\mathbf{s}}_{21}^{\left(1\right)}{\mathbf{s}}_{21}^{\left(2\right)}{\mathbf{A}}_0/\left(1-{\mathbf{s}}_{11}^{\left(2\right)}{\mathbf{s}}_{22}^{\left(1\right)}\right)$$

Similarly, from the Fig. above, we obtain:

(A.12)$${\mathbf{A}}_1={\mathbf{B}}_0{\mathbf{s}}_{12}^{\left(1\right)}+{\mathbf{A}}_0{\mathbf{s}}_{11}^{\left(1\right)}$$

Substituting Eq. (A.8) into Eq. (A.9) then Eq. (A.12), we obtain:

(A.13)$${\mathbf{A}}_1=[{\mathbf{s}}_{11}^{\left(1\right)}+{\mathbf{s}}_{11}^{\left(2\right)}{\mathbf{s}}_{21}^{\left(1\right)}{\mathbf{s}}_{12}^{\left(1\right)}/\left(1-{\mathbf{s}}_{11}^{\left(2\right)}{\mathbf{s}}_{22}^{\left(1\right)}\right)]{\mathbf{A}}_0$$

Let I_through/I_i = ∣A ₁∣²/∣A ₀∣² and I_drop/I_i = ∣C ₀∣²/∣A ₀∣² respectively. From Eq. (A.11), and Eq. (A.13) we obtain I_through/I_i and I_drop/I_i, recast into the form of a standard Fabry-Perot cavity:

(A.14)$$I_{\mathrm{drop}}/I_i=\genfrac{}{}{0.1ex}{}{I_{max,\mathrm{drop}}/I_i}{1+{\left(2\Im /\pi \right)}^2{\sin }^2(\theta /2)}$$

(A.15)$$I_{\mathrm{through}}/I_i=1-\genfrac{}{}{0.1ex}{}{I_{max,\mathrm{through}}/I_i}{1+{\left(2\Im /\pi \right)}^2{\sin }^2(\theta /2)}$$

where ℑ = π √r/(1-r)

$$r=\mid {\mathbf{s}}_{22}^{\left(1\right)}\mid \mid {\mathbf{s}}_{11}^{\left(2\right)}\mid$$

$$\theta ={\varphi }_{22}^{\left(1\right)}+{\varphi }_{11}^{\left(2\right)}$$

$$I_{max,\mathrm{drop}}/I_i={\mid {\mathbf{s}}_{21}^{\left(1\right)}\mid }^2{\mid {\mathbf{s}}_{21}^{\left(2\right)}\mid }^2/{\left(1-r\right)}^2$$

$$I_{max,\mathrm{through}}/I_i=Q_1+\left(Q_2-Q_3-Q_4\right)/{\left(1-r\right)}^2$$

$$Q_1=\left(1-{\mid {\mathbf{s}}_{11}^{\left(1\right)}\mid }^2\right)\left[1+{\left(2\Im /\pi \right)}^2{\sin }^2(\theta /2)\right]$$

$$Q_2=2{\mid {\mathbf{s}}_{11}^{\left(2\right)}\mid }^2\mid {\mathbf{s}}_{11}^{\left(1\right)}\mid \mid {\mathbf{s}}_{12}^{\left(1\right)}\mid \mid {\mathbf{s}}_{21}^{\left(1\right)}\mid \mid {\mathbf{s}}_{22}^{\left(1\right)}\mid \cos \left({\varphi }_{21}^{\left(1\right)}+{\varphi }_{12}^{\left(1\right)}-{\varphi }_{11}^{\left(1\right)}-{\varphi }_{22}^{\left(2\right)}\right)$$

$$Q_3=2\mid {\mathbf{s}}_{11}^{\left(2\right)}\mid \mid {\mathbf{s}}_{11}^{\left(1\right)}\mid \mid {\mathbf{s}}_{12}^{\left(1\right)}\mid \mid {\mathbf{s}}_{21}^{\left(1\right)}\mid \cos \left({\varphi }_{21}^{\left(1\right)}+{\varphi }_{11}^{\left(2\right)}+{\varphi }_{12}^{\left(1\right)}-{\varphi }_{11}^{\left(1\right)}\right)$$

(44)$$Q_4={\mid {\mathbf{s}}_{12}^{\left(1\right)}\mid }^2{\mid {\mathbf{s}}_{21}^{\left(1\right)}\mid }^2{\mid {\mathbf{s}}_{11}^{\left(2\right)}\mid }^2$$

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