## Abstract

We present modeling and simulation results on a new family of waveguided interferometric 2×2 optical routing switches actuated by electro-optic or thermo-optic or all-optical control. Two pairs of coupled microring resonators provide two 3dB coupling regions within a compact Mach-Zehnder geometry. An index perturbation *Δn* of 2×10^{-3} is sufficient to produce 100% 2×2 switching. This perturbation can be applied to one arm of the MZI or to the four rings in the device or to an additional ring that is coupled to one arm. We find that push-pull control is effective for switching: for example, when carriers are injected in one region and depleted in a corresponding second region. An optical transfer-matrix technique is employed to determine the electromagnetic response (the 1550-nm switching characteristics) of the three device-types. Microdisks can be employed instead of microrings, if desired.

© 2008 Optical Society of America

## 1. Introduction

Tandem or double-microring switches were proposed in several articles [1–3] and were realized quite recently by Watts and co-workers [4]. The innovative device of [4] was realized in silicon-on-insulator (SOI) waveguides where two coupled 6-µm-diameter silicon disks comprised the active region. A vertical PN junction within each disk was employed to inject (or deplete) carriers into the disks’ whispering-gallery mode region thereby blue-shifting (or red-shifting) the mode resonance. Their 2×2 routing device had excellent performance in the areas of speed, insertion loss, free spectral range, and extinction ratio.

The prior-art microring devices employ resonance or phase response to implement optical switches and modulators [4–12]. In this paper, we combine interference and resonance to create active devices possessing a micro-scale footprint suitable for optoelectronic on-chip network integration. Here, we propose and analyze three active devices based upon the waveguided Mach-Zehnder interferometer (MZI) geometry. Each device includes two 3 dB coupling regions and each 3 dB coupler consists of two microring resonators with bus-ring, ring-ring and ring-bus coupling zones (four rings per MZI). The balanced interferometer devices are: the MZI with a variable-phase arm, the MZI with variable-resonance couplers and the MZI with an added single-ring-to-arm coupling (utilizing resonant mode shifting of the single ring). These devices have much smaller area than the conventional directional-coupler MZI. In addition to being switches and modulators, the latter two devices can be operated as tunable filters or as reconfigurable add-drop wavelength-division multiplexers. The devices simulated here require a relatively small, low-power, external control signal (such as electro-optic or thermo-optic) because only a small localized change in the effective index of some waveguide structures induces complete 2×2 switching.

The organization of this paper is as follows. We begin with a detailed explanation of the optical transfer-matrix method, and we apply it to the double ring coupler (DRC). Next we analyze the filter characteristics of our balanced MZIs in the voltage-off state. The filter passband shape is optimized. Next we examine the switching characteristics of three devices as a function of effective-index perturbation and show the conditions for complete 2×2 switching.

## 2. Double ring coupler

The optical transfer matrix approach is a powerful technique for analyzing “cascaded” optical elements. The transfer matrix method has been successfully applied to optical microresonators and shows excellent agreement with experimental results [13–18]. In the transfer matrix analysis, each optical component of a microresonator device is represented by a matrix which relates input and output of the resonator. The overall transfer matrix (**W**) is a matrix multiplication of each transfer matrix

where *N* is the total number of optical components and **Z**
_{i} is a transfer matrix, representing each component.

A DRC consists of three evanescent coupling regions and two microring sections as shown in Fig. 1. The internal and external fields of the DRC are related via three coupling matrices and two propagation matrices describing phase and amplitude relations inside each microring.

First, the coupling matrices (**K**
_{i}) of each evanescent coupling region are

$$\left[\begin{array}{c}{e}_{\mathrm{4,2}}\\ {e}_{\mathrm{3,2}}\end{array}\right]=\frac{j}{{c}_{2}}\left[\begin{array}{cc}{t}_{2}& -1\\ 1& {-t}_{2}\end{array}\right]\left[\begin{array}{c}{e}_{\mathrm{4,1}}\\ {e}_{\mathrm{3,1}}\end{array}\right]\equiv {\mathbf{K}}_{2}\left[\begin{array}{c}{e}_{\mathrm{4,1}}\\ {e}_{\mathrm{3,1}}\end{array}\right],\left[\begin{array}{c}{E}_{3}\\ {E}_{4}\end{array}\right]={\mathbf{K}}_{1}\left[\begin{array}{c}{e}_{\mathrm{1,2}}\\ {e}_{\mathrm{2,2}}\end{array}\right],$$

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

where
$j=\sqrt{-1}$
, *t* and *c* are normalized transmission and cross-coupling coefficients, respectively. The subscripts represent each coupling region. In this paper, we assume lossless evanescent coupling. Second, the internal fields of each microring are related by a propagation matrix, **P**
_{i},

$$\left[\begin{array}{c}{e}_{\mathrm{1,2}}\\ {e}_{\mathrm{2,2}}\end{array}\right]={\mathbf{P}}_{2}\left[\begin{array}{c}{e}_{\mathrm{4,2}}\\ {e}_{\mathrm{3,2}}\end{array}\right],$$

where *β*=2*πn _{eff}*/

*λ*

_{0}is the propagation constant,

*λ*

_{0}is the free space wavelength,

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

_{eff}*R*is the ring radius. By combining the coupling matrices in Eq. (2) and the propagation matrices in Eq. (3), the overall transfer matrix (

**M**) of the DRC is

After applying a boundary condition (*E*
_{3}=0: single input at *E*
_{1}) to Eq. (4), the normalized output intensities at the “drop” (*E*
_{4}) port and “through” (*E*
_{2}) port of the DRC are,

In this paper, we investigate single-mode strip-like (nanowire) SOI waveguides whose bulk crystal index is 3.48 at the 1550 nm operation wavelength. Throughout this paper, we assume the effective index of those guides to be 3.05 and the ring radius is assumed to be 20 µm. We can obtain a realistic picture of the general SOI-device switching behavior even if we make the assumption that the active regions do not have any material loss or carrier-induced propagation loss (the lossless assumption is made here). There are several options for the cross and transmission coupling coefficients. For example, we chose *c*
_{1}=0.707, *c*
_{2}=0.14 and obtained the desired 3dB transfer on resonance (Fig. 2(a)) rather than the 100% drop in [4]. The spectral response of the 2×2 DRC shows complete “channel drop” by engineering the ring-to-ring cross coupling efficiency (*c*
_{2}=0.34) shown in Fig. 2(b). The cross and transmission coupling efficiencies can be adjusted by controlling the gap of the two rings and the separation between the ring and the bus waveguide. However, when two of those Fig. 2(b) DRCs are connected by buses to make an MZI, the resulting filter characteristic of this MZI had a complicated double-dip spectral shape which had previously reported by one of the present authors [2], and we rejected that choice as giving an unreasonably complex spectral response. Instead, we construct novel interferometric optical filters and switches by utilizing the DRCs with single-dip Fig. 2(a) filter response as the building blocks.

## 3. Microring resonator Mach-Zehnder interferometer

The microring resonator Mach-Zehnder interferometer (MRMZI) is constructed by connecting two DRCs with the waveguide arms shown in Fig. 3. The first DRC splits the input into two waveguide arms and the second DRC combines them. Depending on the relative difference in phase between the arms, the combined signals produce constructive or destructive interference at each output port. The MRMZI produces resonant interference, offering high-Q spectral response on resonance.

The transfer matrix of the MRMZI is obtained by cascading three matrices: two matrices representing the input and the output DRCs and a third matrix describing the waveguide arms. First, the matrix components of the DRC in Eq. (4) are rearranged

Then, the outputs of the first DRC are connected to the inputs of the second DRC through the matrix which is given by

where Δ*ϕ* is the relative phase difference of the propagating fields in the two arms and *L _{arm}* is the identical length of each arm. The transfer matrix (

**S**) of the MRMZI is

The normalized output intensities at each port are

The spectral responses of the MRMZI with different applied Δ*ϕ* are plotted in Fig. 4. The cross coupling efficiencies of waveguide-to-ring and ring-to-ring are assumed to be 0.707 and 0.14, respectively. The filter characteristic of the MRMZI is engineered to obtain single-valleys passband by optimizing the cross coupling coefficients. As expected, complete 2×2 optical switching from the drop port to the through port is achieved with π-radian phase shift of one of the arms.

As mentioned earlier, this Δ*ϕ* is induced electro-optically (EO), thermo-optically (TO) or opto-optically (OO). It is interesting to note that the Fig. 3 MRMZI changes from a *narrowpass* filter at zero bias to an *all-pass* filter at full bias.

Let us consider in more detail the EO, TO and OO switching mechanisms for push-pull operation. The TO effect uses a temperature rise to increase the waveguide index. The electron-hole pair generation in the OO effect, whether resulting from two-photon absorption of waveguided sub-bandgap light or one-photon absorption of free-space incident visible light, serves to decrease the index. Free-carrier injection gives a decreased index, while carrier depletion increases the index. Depletion gives a strong “pull”, and injection a strong “push”, thus the EO carrier mechanism is suitable for push-pull as well as pull-only or push-only operation.

For the TO mechanism, push-only operation may be the only practical approach because, when push-pull is wanted, heating in one DRC and cooling in an adjacent DRC may not be feasible simultaneously if the two DRC’s are close to each other. (A temperature rise of 11 °C is needed to attain Δn=2×10^{-3}). For the OO mechanism, push pull would require going from no illumination to full illumination in one DRC when changing from full illumination to no illumination in the second DRC, an arrangement that is probably clumsy. And so the OO push-pull seems impractical, although the push-only OO is quite feasible.

The best approach to push pull is carrier depletion in the first region and carrier injection in the second region. In fact, Watts *et al* have demonstrated push and pull in the same DRC. They turned depletion into injection simply by changing the polarity of the bias from negative to positive. However, it is probably not necessary to use a given DRC for both depletion and injection. Rather, the best way would be to optimize one DRC (or arm) for depletion, and to optimize the other DRC (or arm) for injection. It takes strong injection to reach Δn=2×10^{-3}, an index shift that is probably the upper limit for SOI injection at 1550 nm. For depletion, the upper limit is probably Δn=1×10^{-3} in practice. However, both carrier effects increase strongly with the wavelength of operation, rising approximately as λ^{2}. Thus, the push-pull Δn requirements are met easily at wavelengths longer than 1550 nm

## 4. MRMZI optical switches

We now examine 2×2 switching induced by perturbing the microring pairs via the EO or TO or OO effect. The first case is perturbing the rings in the balanced Δϕ=0 version of Fig. 3. The second case is a modified MRMZI in which a single microring is coupled to each arm of the Fig. 3 Δϕ=0 MRMZI. This is a “resonant arm” device. We propose and analyze three perturbation methods: “double push”, “single push”, and “push-pull” using the transfer matrix analysis method.

#### 4.1 MRMZI switches with non-resonant arms

Two perturbation methods are applied to the MRMZI by directly perturbing the DRCs. Figure 5 shows “double push” and “push-pull” MRMZI switches. The term “push” refers to an increase in n_{eff}, with +Δn giving a red shift of the DRC’s resonance wavelengths, while the term “pull” means an induced decrease in n_{eff} with -Δn causing a blue shift. Independent control of each DRC is feasible; thus the first DRC can be pushed while the second is pulled.

Looking now at Fig. 5, we see the double-push and push-pull switches. To simulate perturbation of the DRCs, we modify the transfer matrix (**T**) of the each DRC in Eq. (6) using n_{eff} ±Δn to replace n_{eff} within β of Eq. (3), Then Eq. (8) yields the switching characteristics. Figures 6(a) and 6(b) are the optical responses at the through port of the “double push” and the “push-pull” MRMZI switches, respectively. Excellent ON-OFF switching at resonance is achieved with 2×10^{-3} index perturbation (*Δn*) for both cases. The “push-pull” perturbation of the MRMZI creates splitting in resonance quite different from the Fig. 6(a) simple shift. This resonance splitting induces a fairly wide symmetric transparent region within the operating bandwidth as shown in Fig. 6(b). The phase responses of the double push and the push-pull MRMZI for different index perturbations are shown in Fig. 6(c) and Fig. 6(d), respectively. The phase responses of the two MRMZI switches show abrupt transitions near the operating wavelengths. The spectral width of the resonance splitting extends to the entire operating bandwidth of the MRMZI with 3.4×10^{-3} index perturbation. An asymmetric induced transparency in two single-microring resonators has been recently demonstrated by Lipson’s group [19].

#### 4.2 MRMZI switches with arm-coupled microring resonators

The nonlinear phase response of individual optical microring resonators coupled to the arms of a conventional MZI, has been utilized to create linear and nonlinear device applications by several researchers [20–25]. We propose new optical switches by utilizing the nonlinear phase response of a single microring resonator added to each arm of our MRMZI. Figure 7 shows schematic diagrams of “single push” and “push-pull” MRMZIs obtained with two arm-coupled microring resonators.

To derive the transfer matrix of MRMZI with arm-coupled MRs, we include the field transfer function of a microring resonator in the propagation matrix (**L**) of Eq. (7). The field transfer function (*T _{MR}*) of a microring resonator is described by,

where ${c}_{3}\left(=\sqrt{1-{t}_{3}^{2}}\right)$ is the cross coupling coefficient between the waveguide arm and the microring resonator. The optical transfer matrix of the MRMZI with two coupled microring resonators is

where **I** is the 2×2 identity matrix.

The spectral responses of the Fig. 8(a) “single push” and the Fig. 8(b) “push-pull” MRMZI with coupled microring resonators are plotted using Eq. (11). Figure 8(a) shows a set of asymmetric spectral responses of the “single push” MRMZI for different refractive index perturbations. The asymmetric responses originate from the nonlinear phase response of the microring resonator coupled to the upper arm of the MRMZI. The “push-pull” MRMZI exhibits symmetric spectral response, resulting from the resonance splitting effect which is discussed in section *4-1*. The “push-pull” MRMZI requires one-half of the refractive index perturbation that is needed in the “single push” for complete ON-OFF switching. Returning for a moment to Fig. 3, we again see the effectiveness of push pull because Δϕ of +π/2 rad in the upper arm together with -π/2 rad in the lower arm gives the same result as +π rad in one arm only.

## 5. Effects of optical loss upon device performance

We have assumed ideal lossless resonators and a lossless phase shifting mechanism, but it is clear that loss will be present in practice in both the resonators and the mechanism. Those losses will affect the device’s optical extinction ratio and its optical insertion loss. Since our devices are based upon the DRC, we can make a rough estimate of the 2×2 loss effects by drawing upon the experimental DRC work of Watts and the one-ring results of Dong. At 1533 nm, Watts [4] observed 4 dB insertion loss in the OFF state and 8 dB in the ON state. We would expect somewhat lower losses in the MZMZI since Watt’s device was not optimized. Dong [6] observed less than 3 dB loss and projects 0.5 dB loss in an optimized structure. The OFF state loss is comprised of waveguide material loss, waveguide scattering loss linked to wall roughness, unwanted coupling loss to the substrate, and bus-to-ring coupling loss. The ON state loss has an added component due to intraband absorption from free carrier concentrations. The added loss due to electron and hole injection has been analyzed by Emelett and Soref [1] for an SOI DRC at 1550 nm. They showed that loss is governed by the imaginary part of the ring’s index perturbation Δn+jΔκ, and that Δk=0.053 Δn.

We investigated the effects of optical loss upon device performance by calculating the spectral responses at the through port of the double push MRMZI switch using an OFF-state propagation loss of 0.5 dB/cm and the added ON-state loss of Δk=0.053 Δn.. These losses are introduced into the calculation by modifying the effective refractive index of the doublepush MRMZI as n_{eff}=n_{r}-j(κ+Δκ). The first term of the two extinction coefficients, κ, describing the propagation loss, is assumed to be 1.42×10^{-6}. The second extinction coefficient, Δκ, represents the added loss at the ON state of the switch. The dimensions and device parameters of the MRMZI are same as in Fig. 6(a). Figure 9 shows the spectral responses of the “lossy” double push MRMZI switch for different index perturbations. The ON state loss induced by the index perturbation decreases the drop/thru extinction ratio but increases the operating bandwidth of the switch. The operating bandwidth of the MRMZI switch defined by the full width at half maximum (FWHM) of the spectral response of the switch is 0.76 nm. Using the relation, Δλ/λ=Δf/f, we find that the MRMZI information bandwidth is 95 GHz which is acceptable for telecomm applications. At the 1551.7-nm center wavelength, the extinction ratio of the Fig. 9 double push MRMZI is 46.8 dB.

## 6. Conclusion

We have predicted the performance of lossless SOI-waveguide-based resonant interferometric microring switches. (It was assumed that the waveguides did not have any material-induced loss or carrier-induced propagation loss). The 2×2 optical routing characteristics of three different balanced MZI devices were investigated, each of which has two tandem-ring 3 dB couplers between the parallel bus waveguides. A localized perturbation of the waveguided “rings” or “arms” index, *Δn*=2×10^{-3}, was enough to produce total switching and the pushpull versions of the MRMZIs presented here required only one-half of the control signal needed for push-only switching. We can easily employ the 6-µm-diameter disks of Watts [4] in which case the area of our mode-shifted 2×2s will always be less than 30 µm×40 µm. The “push-pull” MRMZI exhibits optical transparency, resulting from resonance splitting effects. In all of our devices, the drop and through outputs point in the forward direction, unlike the backward-and-forward outputs of previous single-ring devices. Therefore, it is easy to cascade-interconnect several of our 2×2 MRMZIs to create an N×N optical matrix switch. In summary, our proposed MRMZI switches offer great potential in a number of applications including on-chip signal routing, modulation, tunable filtering and reconfigurable add-drop WDM.

## Acknowledgments

The authors wish to thank AFOSR/NE (Dr. Gernot Pomrenke, Program Manager) for ongoing support of this in-house research. The authors also wish to thank Dr. Walter Buchwald of AFRL for helpful technical discussions. Sang-Yeon Cho is an AFOSR Summer Faculty scientist at AFRL Hanscom.

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