## Abstract

This paper presents 1550-nm simulation results on the waveguided silicon-on-insulator four-port optical filtering and switching devices known as “SCISSOR” (an in-line array of microring resonators wherein each ring is coupled to two bus waveguides). We optimized the array number, the ring-bus coupling and the inter-ring spacing in order to obtain “rectangular” filter-passband shapes that have not heretofore been reported in the resonant-optics literature. We were able to engineer a box-like bandpass whose wavelength width could be anywhere from 5 to 50 % of the free spectral range (FSR). We then performed ring-bus apodization of the array that increased side-lobe suppression on the main filter band and widened the band. By reducing the FSR to 2.51 nm with increased ring diameter, we also showed that complete, high-extinction 2×2 optical switching is attained when the effective index of each ring in the group is changed by 2×10^{-3}, giving 1.02-nm shift of the 0.77-nm passband. Tunable filtering, sensing, reconfigurable add/drop and wavelength-division de-multiplexing is offered in addition to switching.

©2008 Optical Society of America

## 1. Introduction

The optics literature contains a number of theoretical and experimental papers on the so-called *coupled-resonator optical waveguides* (CROWs) and *side-coupled integrated spaced sequence of resonators* (SCISSORs) structures comprised of waveguided microring-resonator arrays [1–14]. In the latter device, all rings can be side-coupled to one or two bus waveguides. The focus in this paper is on the two-bus SCISSORs which we call the “dual channel” SCISSORs. The purpose of this theoretical paper is to perform optical engineering on the device in order to demonstrate its unique filtering and switching properties. We have engineered the ring separations to attain a special Bragg resonance and we adjusted the ring couplings to produce rectangular filter passbands that are a desired fraction of the free spectral range (FSR).

The “unapodized” SCISSOR has ring-bus coupling coefficients that are uniform across the device. Apodization refers to optimization of the filter characteristic attained by a spatial variation of the coupling coefficients across the device according to a “profile” such as Hamming or Bartlett [15]. We present apodizations that create improved side-lobe rejection and steeper side walls on the main passband.

The SCISSOR optical filter, often used as a “passive” device, can be transformed into an “active” device by perturbing the effective refractive index. A bus perturbation will modify the shape and multiplicity of the Bragg resonances whereas perturbation of the rings and the buses together shifts the overall wavelength-transmission profile while preserving its shape. In this paper, we choose the latter perturbation method to achieve complete 2×2 switching for suitably narrow passbands. We report the 2×2 switching response of 11-ring and 5-ring apodized SCISSORs by index perturbation. The specific layouts of 1×4 routing switches, 2×2 reversing switches, 4×4 matrix switches and 1×8 wavelength-division de-multiplexers for a variety of applications are illustrated in section 7.

## 2. Dual-channel SCISSORs

A schematic diagram of a dual channel *side-coupled integrated spaced sequence of resonators* (DC-SCISSOR) is shown in Fig. 1. The DC-SCISSOR is analyzed using the powerful optical transfer matrix method [16].

For example, the internal fields (*e*
_{x,N}) and the external fields (*E*
_{x,N}) are related by the coupling matrices(**K**) and the propagation matrices (**P**) of the *N*-th microring resonator,

$$\left[\begin{array}{c}{E}_{3,N}\\ {E}_{4,N}\end{array}\right]={\mathbf{K}}_{N}\left[\begin{array}{c}{e}_{3,N}\\ {e}_{4,N}\end{array}\right],$$

$${\mid {t}_{i}\mid}^{2}+{\mid {c}_{i}\mid}^{2}=1,i=1,\dots ,N,$$

where *j*=√-1, *t* and *c* are normalized transmission and cross-coupling coefficients, respectively. The subscripts represent each microring. In this paper, we assume lossless and symmetric evanescent coupling for each microring. The internal propagation matrix relates internal fields of the two evanescent coupling regions in each resonator.

where *β*=2*πn*
* _{eff}*/

*λ*

_{0}is the propagation constant,

*λ*

*is the free space wavelength,*

_{0}*n*

*is the effective refractive index of a guided mode, and*

_{eff}*R*is the ring radius. The transfer matrix (

**M**) of the

*N*-th resonator is

After rearranging the transfer matrix in Eq. (3),

The external fields of the *N*-th resonator and the *N*-*1*-th resonator are related by the external propagation matrix (**L**),

where, *L*
* _{s}* is the separation between the

*N*-th and

*N*-

*1*-th resonator.

The overall transfer matrix of the DC-SCISSOR with *N* microring resonators is obtained by a simple matrix multiplication,

After applying a boundary condition (*E*
_{3},* _{N}*=0: single input at

*E*

_{1,1}) to Eq. (6), the normalized output intensities at the “drop” (

*E*

_{4,1}) port and “through” (

*E*

_{2,N}) port of the DC-SCISSOR are,

Note that light exits the through port in the forward direction while light emerges from the drop port in the backward direction.

## 3. DC-SCISSORs with different ring separations

The spectral responses at the drop port of a DC-SCISSOR with ring separations are calculated for the case of silicon-on-insulator (SOI) strip waveguides as in our earlier paper [16] with the assumption of 3.05 for the effective waveguide index at 1550 nm. A propagation loss of 0.5 dB/cm was assumed for the waveguides’ silicon core, and this loss was taken into account using the analytic procedures presented in Ref 16. The output characteristics of a DC-SCISSOR are determined by two types of resonance: (1) resonance from an array of resonators and (2) Bragg resonance from the periodic backward reflections guided by one of the bus waveguides. Figure 2 shows the output spectral responses of DC-SCISSORs with different *L*
* _{s}*. In the calculation, each DC-SCISSOR has 30 identical microring resonators: the ring radius is 10 µm and the cross-coupling efficiency is 0.31. The middle-passbands can be engineered for single or multiple sub-bands. In Fig. 2(b), the spectral response of the DC-SCISSOR with

*L*

*=*

_{s}*πR*shows a wide middle-passband whose shape is identical to the “main bands” on either side since the reflected fields and the internal fields at each output coupling section are in phase. This spectral response is useful for filtering and switching applications since all the passbands are identical and wide. The special case

*L*

*=*

_{s}*πR*is important because the overall characteristic has a simple periodic response without “complicated” Bragg peaks. Now the SCISSOR response is truly symmetric and the FSR is half of what it was. We proceed to use this

*L*

*condition to fashion unique filters and switches.*

_{s}## 4. Effects of the cross-coupling efficiency

Selection of the coupling strength is a powerful technique because the choice of c will govern the shape of the main passband including the side-lobe strengths. Using the above matrices, we have investigated an 11-resonator DC SCISSORs for the cross-couplings of 0.60, 0.53, 0.31 and 0.14 during the *L*
* _{s}*=

*πR*condition. Figure 3 presents the output spectra at the drop port of a DC-SCISSOR for the four cases. Looking at the results, we see that the passbands of the strong-coupling structure are wide and box-like as desired: however, the unwanted ripples on the sides are fairly significant. For the weak coupling cases, the ripples become insignificant as desired but the passbands become more triangular in shape. Evidently, there is a tradeoff between the passband width and ripple height in the design of a DC-SCISSOR. In the next section, we optimize these spectral response shapes by apodizing the ensemble of transmission coefficients

*t*

*.*

_{i}## 5. Apodized DC-SCISSORs

Apodization of the SCISSOR filter is an important approach because it offers a way to obtain greater out-of-band rejection and better band shapes. A DC-SCISSOR is apodized by progressively changing the transmission coefficient of each resonator. In recent literature, several windowing functions are evaluated for coupled resonator waveguides [17, 18]. In this paper, we apply a simple linear apodization function to the DC-SCISSOR. Figure 4 is a schematic diagram of an apodized DC-SCISSOR.

We performed this apodization upon the Fig. 4 filters by linearly decreasing the transmission coefficient strength for the first six rings, and linearly increasing the coefficients for the following five rings. The apodization function applied to the transmission coefficient is

$$\Delta t=\left({t}_{1}-{t}_{\frac{\left(N+1\right)}{2}}\right),$$

where *i* is an integer, *N* is the total number of the resonators, *t*
_{1} is an initial transmission coefficient, and Δ*t* is the apodization depth. Figure 5 shows the calculated spectral responses of apodized DC-SCISSORs having the four different initial transmission coefficients that were studied in Fig. 4. In the calculation, the apodization depth (Δ*t*) is fixed and assumed *t*
_{1} be 0.14 with *N*=11 as before. The solid-line graphs are the spectral responses of unapodized DC-SCISSORs and the dashed-line graphs are of apodized. On these logarithmic-scale graphs, we can see that the unapodized (Δ*t*=0) weak-coupling DC-SCISSORs exhibit stronger side-lobe extinction than the strong-couping devices. After applying the apodization function (Δ*t*=0.14) in Eq. (8), the passbands of the DC-SCISSORs were significantly widened in all cases. Apodization improved the out-of-band extinction in Figs. 5(a) and 5(b), but worsened it in Figs. 5(c) and 5(d).

## 6. The prior art of apodized DC-SCISSORs

Apodized DC-SCISSORs have previously been studied by Darmawan *et al* [13] and Ito *et al* [14]. Darmanwan assumed lossless rings and presented the normalized responses of an unspecified material around the normalized “unity” center wavelength. By contrast, we make specific numerical predictions for the silicon-on-insulator system in the 1550-nm telecomm band and we include ring loss. For N rings, Darmawan proposed active apodization by tuning each of the 2N bus-ring couplings using 2N active multimode-interference couplers. He did not consider switching as we do in Section 7 below. We do not perturb the ring-bus couplings and instead change the refractive index of each entire ring.

Ito examined apodized 4-ring DC-SCISSORs in the lossless Ta_{2}O_{5}-SiO_{2} system unlike our SOI analysis. He did not examine switching. He proposed an 1×2 interleaver which, in the 8-wavelength case, routes the input wavelengths λ_{1}, λ_{3}, λ_{5}, λ_{7} to the though port, while wavelengths λ_{2}, λ_{4}, λ_{6}, λ_{8} travel to the drop port. He then uses 1:4 demultiplexers such as arrayed waveguide gratings to separate the two quartets into individual wavelengths. We take a different all-SCISSORS approach to wavelength division demultiplexing. In our first SCISSORs stage, we route λ_{1}, λ_{2}, λ_{3}, λ_{4} to the through port and λ_{5}, λ_{6}, λ_{7}, λ_{8} to the drop port. As described in Section 8-4 below, the stage-1 FSR is divided by two in the second stage and divided then again in the third stage-- producing eight individual wavelength outputs.

## 7. Apodized DC-SCISSOR switches

There is considerable literature on electrooptical (EO) single-microring switches. Inherently, the optimized single-ring switch has a triangle-shaped narrow passband. For example, the 3-dB operation band of the Watts *et al* 4-µm-diameter SOI device is only 0.17 nm wide at the 1578 nm resonance [19]. By contrast, the passband in the SCISSORs device is generally 4x to 5x wider than that of the single ring. There are two reasons why these wideband DC SCISSORs will perform as better-than-single switches: (1) the SCISSORs can handle a larger information bandwidth—a higher data rate on the modulated laser signal and (2) the SCISSORs performance is less affected by a change in the temperature of the SOI chip.

Let us consider briefly the expected “dynamics” of the EO SCISSORs switch. Here, the experimental performance of single-ring EO switches reported in the literature can offer important guidelines. Because the rings within the SCISSORs are driven simultaneously and electrically in parallel, their EO response is very similar to that of a single ring, including the rise-and-fall times and the voltage required to reach the ON state of the two-state device—but the SCISSORs requires higher current than the single ring.

As Fig. 5 demonstrates, apodized DC-SCISSORs exhibit box-like spectral responses that are useful for temperature-insensitive optical switching. For high performance optical switching, the passband side walls must be steep and the top fairly flat. There is also a requirement on the filter’s full-width at half maximum B. During index perturbation, the passband shifts by an amount δλ, therefore it is essential that B<δλ for complete switching.

The effective index of a DC-SCISSOR can be disturbed by electrooptical, thermooptical, opto-optical and mechano-optical means. The EO effect offers “push pull” operation in which the index of some part of the DC-SCISSOR is increased while the remaining part experience decreased index. Here we consider the “push only” case which is easier to implement. For quantitative analysis of switching, we substitute the expression *n*
* _{eff}*+

*Δn*for

*n*

*in*

_{eff}*β*as in Ref. 16, where

*Δn*represents the induced index. We emphasize the “spectrally enhanced” apodized SCISSORs.

## 7.1. Eleven-ring simulations

In practice, there is an upper limit of approximately 2×10^{-3} for the real index shift *Δn*. Our preliminary calculations showed that this *Δn* causes δλ~1 nm at the telecomm wavelengths. Therefore, we must engineer B<1 nm in order to make practical switches. This is indeed feasible. We found in Fig. 5(c) a square profile with low lobes and here B was about 30 % of FSR defined by the separation from peak to peak in the output spectrum. If we write B/FSR=*γ*, we note that *γ* remains constant (as does the B shape) if the FSR is decreased by increasing the diameter of each ring. We can use this scaling law to get B<1 nm by enlarging the 10 µm ring radius to 50 µm or more. We examined the *R* cases of 50 µm and 60 µm for *N*=11. In addition, we considered the case of *N*=5 because it is beneficial to reduce the array size in order to minimize the on-chip footprint of the DC SCISSORs.

We considered two 11-ring apodized DC-SCISSORs with *R*=50 µm and the same Δ*t*=0.08 apodization depth. Optical switching was achieved by changing the refractive index of the each ring in the same way. Figures 6(a) and 6(c) show the spectral response of the indexmodulated 2×2 switch for the *t*
* _{1}* choices of 0.95 and 0.98. For both switches, the entire passband is completely shifted with an index perturbation of 2×10

^{-3}. The semi-log plots of Figs. 6(b) and 6(d) show more than 25 dB of ON/OFF extinction at

*Δn*=2×10

^{-3}

## 7.2. The choice of the apodization function

Several window functions are given in Refs. [17] and [18], and we think it will be illuminating to determine how the linear function performance differs from that of representative functions like Gaussian and Hamming. To reveal those differences, we use the example of the 11-ring DC-SCISSORs with *R*=50 *µm*. For fair comparison, each apodization function has same initial value (*t*
_{1}) and same apodization depth (*Δt*), as shown in Fig. 7(a). The results are shown in Fig. 7(b) where we see that the band shapes did not change much from one apodization function to the next. In summary, there are three reasons why the linear apodization has been chosen in this paper: (1) the device performance here is just as good as that of other window functions, (2) in practice, it is relatively easy to fabricate the linear progression, (3) for devices containing a small number of rings, such as five, the piece-wise fitting of the “complex” functions is not a good approximation, whereas the linear fit is exact.

## 7.3. Five-ring simulations

Turning now to the 5-resonator apodized 2×2 switches, we investigated the case of *R*=60 µm (where B=0.72 nm for Figs. 8(a) and 8(b); B=0.48 nm for Figs. 8(c) and 8(d)) and *Δt*=0.08. The predicted spectral responses of two DC-SCISSOR switches with transmission coefficients of 0.90 and 0.95 are shown in Fig. 8.

Once again, for an index perturbation of 2×10^{-3}, the passband of the DC-SCISSOR switch was completely shifted out of its initial position and the extinction ratio of the ON-to-OFF states exceeds 25 dB.

## 8. Applications

The variety of potential applications is illustrated here schematically by four examples. The drawings of the proposed devices show a top view of the SOI integrated optical network-on-a-chip, and the two ring colors show the ON and OFF states, respectively.

## 8.1. The 4 by 4 matrix switch

Figure 9 shows a 4×4 DC-SCISSOR matrix switch. There are 16 elemental 2×2 switches within this 4×4 non-blocking crossconnect switch (along with 16 waveguide crossovers); however only 4 of the elemental devices need to be addressed ON at any time. The remaining 12 devices are OFF.

## 8.2. The 1 by 4 matrix switch

Figure 10 is an example of a 1-to-M routing switch (the 1 to 4 can be extended in a tree-like fashion to 1-to-8, etc), and here we expect low crosstalk at any output port as well as low insertion loss.

## 8.3. The 2 by 2 optical switch

Figure 11 is a reversing optical switch that has a compact structure and one 90° waveguide intersection. Again the loss and crosstalk are anticipated to be low.

## 8.4. The 1 by 8 wavelength-division DEMUX

Eight different wavelength signals are launched into the single input waveguide, and those wavelengths are “completely” separated at the eight outputs. Figure 12 shows a 1 by 8 wavelength-division de-multiplexer (DEMUX) using DC-SCISSORs. The branching geometry utilizes a unique spectral response feature of the DC-SCISSORs suggested in Fig. 5. There we found that the fraction *γ*=B/FSR is governed by the coupling strength t_{int} of the rings, and it is clear that *γ*=0.50 can be engineered. Therefore we can obtain a “square wave” filter response; a box-like passband with a 50 % duty cycle. That response is used here in the DEMUX where the first four wavelengths are routed to the first output, while the second four go to the second output. The first filter has a “large” FSR governed by the choice of R. Then in the next DEMUX stage, the FSR is only half as large as in the previous stage, which means that λ1…λ4 is divided into “sub bands” λ1,λ2 plus λ3,λ4, while λ5…λ8 divides into λ5,λ6 and λ7,λ8. Finally, the third stage has an FSR one half that of the second stage, which completes the desired wavelength division.

## 9. Conclusion

We have demonstrated that the two-bus SCISSORs has main passbands and multi-peaked Bragg resonance passbands between the mains. We presented the detailed transfer matrix theory for N-fold SOI microring (or microdisk) resonator devices and showed via simulations that the Bragg bands can be transformed into a single shape identical to that of the main band by constraining the bus separation between microrings to be *πR* where *R* is the ring radius. This produces a “square wave like” response. We predict that the ring-bus cross-coupling coefficient can be chosen to attain rectangular box-like 1550-nm passbands whose bandwidth B can be engineering to be 5 to 50 % of the free spectral range.

Apodization studies were then performed upon the DC-SCISSORs utilizing a linear profile for the change in transmission coefficients. We found widened rectangular bands with improved side-lobe rejection when the initial coupling is strong. The apodized filters were then employed in a simulation of 2×2 optical switching induced (for example) by carrier injection into the SOI SCISSOR which modifies the effective index by an amount *Δn*.

We find that the filter shape is translated along the wavelength scale (without shape change) in proportion to Δn. Since the upper limit on *Δn* is about 2×10^{-3} in practice, the filter shift is about 1 nm. Therefore, to attain complete 2×2 switching of an optical signal within B, we must engineer B to be narrower than 1 nm. We accomplished this by increasing the ring radius from 10 µm into the 50-to-60 µm range, thereby reducing the FSR. Since B is a fixed fraction of FSR, we were able to simulate filter passbands in the 0.48 to 0.72 nm range, yielding the desired low-crosstalk switching.

We have proposed apodized-SCISSOR designs for 1×N switches, 2×2 reversing switches, 4×4 matrix switches, and 1 to 8 wavelength-division DEMUXs, all of which are amenable to CMOS integration and electrooptical control.

Generally, the SCISSORs can be applied as modulators, switches, tunable filters, reconfigurable add/drop multiplexers, de-multiplexers and sensors. Regarding sensing, chemical and biological agents can be detected by having the target molecules alter the refractive index of “exposed” rings [20]. Another technique is to modify the filter response into a sinusoidal dependence upon wavelength (cosine-squared transmission). Then an array of sinusoidal SCISSORs with progressively larger FSRs can be arrayed side-by-side on a silicon chip in the manner described by Florjanczyk *et al* [21] to create a Fourier Transform Infrared Spectrometer-on-chip that has no moving parts, and the SCISSOR array will have a smaller footprint than the unbalanced Mach-Zehnder interferometers.

## Acknowledgements

This work is supported in part by the Air Force Office of Scientific Research, Dr. Gernot Pomrenke, Program Manager.

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