## Abstract

New designs are proposed for $2\times 2$ electro-optical switching in the 1.3–12 μm wavelength range. Directional couplers are analyzed using a two-dimensional effective-index approximation. It is shown that three or four side-coupled Si or Ge channel waveguides can provide complete crossbar broad-spectrum switching when the central waveguides are injected with electrons and holes to modify the waveguides’ core index by an amount $\mathrm{\Delta}n+i\mathrm{\Delta}k$. The four-waveguide device is found to have a required active length $L$ that is 50% shorter than $L$ for the three-waveguide switch. A rule of $\mathrm{\Delta}\beta L>28$ for 3w and $\mathrm{\Delta}\beta L>14$ for 4w is deduced to promise insertion loss $<1.5\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{dB}$ and crosstalk $<-15\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{dB}$ at the bar state. At an injection of $\mathrm{\Delta}{N}_{e}=\mathrm{\Delta}{N}_{h}=5\times {10}^{17}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\mathrm{cm}}^{-3}$, the predicted $L$ decreased from $\sim 2$ to $\sim 0.5\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{mm}$ as $\lambda $ increased from 1.32 to 12 μm. Because of Ge’s large $\mathrm{\Delta}k$, the Ge bar loss is high in 4w but is acceptable in 3w.

© 2014 Chinese Laser Press

## 1. INTRODUCTION

Techniques for electro-optical (EO) switching have been an ongoing concern throughout the history of integrated photonics, with the $\mathrm{\Delta}\beta $ coupler being an early choice and the Mach–Zehnder interferometer (MZI) later becoming a favorite. The EO coupler approach is “revived” here and is expanded to multiple waveguides. This paper focuses on the design and numerical simulation of active broadband directional couplers consisting of three or four Si or Ge side-coupled channel waveguides. The outer two waveguides are the switching channels, and the central waveguide “island(s)” have a PIN diode structure for injecting electrons and holes into the intrinsic waveguide core(s). The active length $L$ is chosen equal to the coupling length ${L}_{c}$ so that input light crosses over to the second outer waveguide in the zero-bias PIN state, yielding the optical cross state. With sufficient bias applied (as determined here), the launched lightwave “decouples” and remains principally in the initial waveguide, forming the optical bar state and providing the desired $2\times 2$ crossbar switching.

Using a two-dimensional (2D) effective-index approximation of in-plane three-dimensional (3D) structures, an analytic formalism [1] and a mode-simulation software approach [2] were employed here to determine the infrared power in each waveguide as a function of propagation distance $z$ for both the voltage-off and voltage-on conditions. That modeling allowed determination of the infrared insertion loss (IL) and crosstalk (CT) at any length $z$. The electron-and-hole concentrations $\mathrm{\Delta}{N}_{e}$ and $\mathrm{\Delta}{N}_{h}$ injected into an active waveguide produce a modification of the initial $n+i$ 0.0 bulk core index by an amount $\mathrm{\Delta}n+i\mathrm{\Delta}k$. Thus free-carrier physics comes into play to determine how the wavelength of operation $\lambda $ and the carrier concentrations $\mathrm{\Delta}N$ influence $\mathrm{\Delta}n$ and $\mathrm{\Delta}k$. These free-carrier-effect (FCE) relationships [3] are specified here for both Si and Ge over their transparency $\lambda $ ranges: 1.32 to 12 μm for Si and 1.8 to 12 μm for Ge. (The Ge transparency continues beyond 12 μm, but we arbitrarily stopped there.)

A level of electron-and-hole injection $\mathrm{\Delta}{N}_{e}=\mathrm{\Delta}{N}_{h}=5\times {10}^{17}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\mathrm{cm}}^{-3}$ was selected as being an induced $\mathrm{\Delta}N$ that is effective for switching. Although this injection level might seem “high,” it is readily reachable in practice. The EO bulk-waveguide perturbations $\mathrm{\Delta}n$ and $\mathrm{\Delta}k$ were then calculated from free-carrier theory at this level of injection. These perturbations were then entered into the simulation to show the $2\times 2$ switching response.

The inter-waveguide spacing $s$ enables evanescent-wave coupling of the various air-clad waveguides, and $s$ can be adjusted to determine the distance $z$ at which the first cross state occurs, which means that “any” value of ${L}_{c}$ can be obtained for the device such as those in the 10–4000 μm range. The change in propagation coefficient of a carrier-injected waveguide is $\mathrm{\Delta}\beta $, and we determine the two channel output powers as a function of $\mathrm{\Delta}\beta L$, assuming an initial cross state. The component $\mathrm{\Delta}k$ was fully taken into account in our work. The obtained graph is the generic $2\times 2$ switching characteristic. We found the generic characteristic of directional coupler switches composed of two, three, or four waveguides. In this paper, the abbreviations 2w, 3w, and 4w are used to denote the two-waveguide, three-waveguide, and four-waveguide directional coupler switches, respectively. The generic $2\times 2$ plots just mentioned gave us the value of $\mathrm{\Delta}\beta L$ required in the 2w, 3w, and 4w cases to get a reasonably low-loss bar state such as $<1\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{dB}$ of loss. The results for 2w, 3w, and 4w were $\mathrm{\Delta}\beta L\sim 6$, 28, and 14, respectively. Then, starting with the $\lambda $-dependent FCE theory cited above, we determined a specific $\mathrm{\Delta}\beta $ with which we were able to find the device length $L$ required for high-performance crossbar switching for both the Si and Ge cases. The results presented here show $L$ in the range of 0.24–0.80 mm (at $\lambda =12\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{\mu m}$) to 1.6–3.6 mm (at $\lambda =1.3\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{\mu m}$).

## 2. BACKGROUND DISCUSSION

The MZI may be operated as a high-performance $2\times 2$ EO switch when a pair of $2\times 2$ 3 dB couplers is employed—one at input, the other at output. The EO perturbation of active length $L$ is utilized in one arm of the MZI. This switch has the desired “cross” and “bar” states, with the cross usually occurring at perturbation-off, and with the bar state found at perturbation-on, as shown in Figs. 1(a) and 1(b). In $\lambda =1.32/1.55\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{\mu m}$ telecoms practice, the MZI is constructed of single-mode semiconductor channel waveguides, and the FCE is typically used as the EO switching mechanism—for example, by accumulating or injecting electrons and holes into one of two $L$-length arms of the interferometer.

For the MZI and the present 2,3,4w’s, electron–hole injection will produce a change in the complex bulk index of the intrinsic waveguide core by an amount $\mathrm{\Delta}n+i\mathrm{\Delta}k$, where $\mathrm{\Delta}n$ is the real-index change, and $\mathrm{\Delta}k$ is the change in extinction coefficient. The loss component $\mathrm{\Delta}k$ is always present and is “inevitable.” If we define the ratio $\rho =\mathrm{\Delta}n/\mathrm{\Delta}k$, we find that $\rho $ depends upon $\lambda $, $\mathrm{\Delta}N$, and the spectral absorption curve of Si, Ge, or SiGeSn.

The MZI $2\times 2$ has high performance and can act as a standard of comparison. The induced shift $\mathrm{\Delta}\beta $ in the propagation coefficient $\beta $ of the MZI active waveguide is proportional to the real-index shift $\mathrm{\Delta}\beta =2\pi \mathrm{\Delta}n/\lambda $, and the $\mathrm{\Delta}n$ swing in the MZI required for complete switching to the bar state is $\mathrm{\Delta}\beta L=\pi $ rad. During that switching, because of $\mathrm{\Delta}k$ there will be optical propagation loss in the active arm at the bar state: $\alpha =4\pi \mathrm{\Delta}kL/\lambda $, inducing IL and CT at the outputs. The normalized output power of the MZI in the presence of $\mathrm{\Delta}k$ and $\mathrm{\Delta}\beta L=\pi $ phase shift is shown in Fig. 1(c), which can be derived from coupled-mode theory.

The loss parameter $A=\mathrm{exp}(-2\pi \mathrm{\Delta}kL/\lambda )$ in Fig. 1(c) is an amplitude-attenuation factor, which can be further simplified to $A=\mathrm{exp}(-2\pi \mathrm{\Delta}nL/\lambda \rho )=\mathrm{exp}(-\mathrm{\Delta}\beta L/\rho )=\mathrm{exp}(-\pi /\rho )$. Therefore the IL and CT of the MZI $2\times 2$ switch at the bar state are reduced to a function of $\rho $ as plotted in Fig. 2. It is seen that the IL of the MZI $2\times 2$ will exceed 2.3 dB and the CT will exceed $-12\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{dB}$ when $\rho <10$, a regime of $\rho $ that occurs at “longer wavelengths” in both Si and Ge as is shown in $\rho $-vs-$\lambda $ tables given later in this paper.

This paper presents the new EO 3w and 4w $2\times 2s$ as useful alternatives to the MZI. These switches are nonresonant (broad spectrum) noninterferometric devices. As shown here, complete switching appears feasible. The MZI, with its two 3 dB couplers and straight-arms section, has a “three-piece” construction, whereas our couplers offer one-piece construction. Our work here reveals a disadvantage—that the length of the 2,3,4w’s active region is always greater than that of the MZI $2\times 2$; however, the 2,3,4w’s overall switch length ${L}_{\text{tot}}$ may comparable to the MZI total length. An advantage found here is that the 2,3,4w’s offer greater immunity against $\mathrm{\Delta}k$-induced bar-state propagation loss than do the MZIs.

The Fig. 3 top view of the 3w illustrates the coupling length ${L}_{c}$, the active length $L$, and the total device length ${L}_{\text{tot}}$. Shown here are connecting waveguides that “fan into” and “fan out of” the coupling zone at the input and the output to the active length. The fan in/out region for 3w may be more compact than that for 2w. An advantage of the 4w is that it often does not require any flared-out connections, and the straight-line nature of the outer guides is preserved. These properties can be observed when individual $2\times 2s$ are joined and interconnected into an $N\times N$ EO matrix switch.

What are the $N\times N$ possibilities? The 2w is unfortunately an asymmetric switch, because, for a given input, the switching characteristic changes when the EO zone changes from the second to the first waveguide. By contrast, the 3w and 4w have the property of being completely symmetric. Because of asymmetry, the 2w would be used functionally as a $1\times 2$ switch, and that operation works well when a group of 2w’s are “cascaded” into the crossbar matrix shown in Fig. 4. In comparison to the crossbar, the 3w’s and 4w’s would be cascaded into the nonblocking $N\times N$ permutation matrix switches illustrated in Figs. 5 and 6, which rely on switch symmetry. The “nonfan” configuration for the 4w’s has been selected in Fig. 6, and this reveals the compact arrangements that are possible.

## 3. APPROACH AND ANALYSIS

Simplifying assumptions were made in the analysis. Although the actual couplers are 3D, our 2D effective-index simulations give guidelines on what to expect for the 3D case. Also, all waveguides are assumed identical—yielding synchrony or phase matching. Actually, of course, $P$ and $N$ contacts would introduce a small amount of loss ${k}_{o}$ in the central waveguides, thereby changing the core’s real index slightly. Hence a small IL and phase mismatch would really exist at the zero-bias cross state, violating our assumption of a perfect cross state existing at $L={L}_{c}$. From knowledge of ${k}_{o}$ and ${L}_{c}$ we estimate the real cross-state IL as ${\alpha}_{o}{L}_{c}=4\pi {k}_{o}{L}_{c}/\lambda $. Here we also equate the material index change and the effective modal index change to simplify the analysis, i.e., $\mathrm{\Delta}\beta =2\pi \mathrm{\Delta}{n}_{\text{eff}}/\lambda =2\pi \mathrm{\Delta}n/\lambda $, which is generally valid for the strongly guiding waveguide.

Our approach is divided into two parts. The analytical coupled-mode theory of Chen *et al.* [1] was used for the 3w, whereas the 4w research employed beam-propagation software [2] to find the infrared output power emanating from WGs 1 and 4, under varying conditions. For both devices, free-carrier theory gives effective indices in the central waveguides: $\mathrm{\Delta}n+i\mathrm{\Delta}k$ at “voltage-on” and $\mathrm{\Delta}n=0$, $\mathrm{\Delta}k=0$ for voltage-off.

#### A. Analytical Formulation for the 3w Switch

In the 3w, as shown in Fig. 7, the passive WG1 and WG3 are identical with propagation constant ${\beta}_{1}$. The central (active) waveguide, WG2, is absorptive, with propagation constant ${\beta}_{2}$, where ${\beta}_{2}={\beta}_{2r}+i{\alpha}_{am}$. The imaginary part ${\alpha}_{am}$ is the absorption coefficient for the optical field amplitude ($\alpha =2{\alpha}_{am}$ denotes the absorption coefficient of optical power). Here ${\beta}_{2}$ is the quantity that is modulated or “driven.” CW light is launched into WG1 at $z=0$.

The direct coupling between WG1 and WG3 is neglected. (This is usually true for high-index-contrast waveguides.) The optical field is launched into WG1. We are going to investigate the optical amplitude-squared ${|{a}_{1}(z)|}^{2}$, ${|{a}_{2}(z)|}^{2}$, and ${|{a}_{3}(z)|}^{2}$, the power that comes out of WGs 1, 2, and 3 respectively. We can write the following three equations to describe the light propagation within 3w:

where ${\kappa}_{12}={\kappa}_{32}={\kappa}^{*}$ and ${\kappa}_{21}={\kappa}_{23}=\kappa $. The solutions ${|{a}_{1}(z)|}^{2}$, ${|{a}_{2}(z)|}^{2}$, and ${|{a}_{3}(z)|}^{2}$ for the initial condition of ${a}_{1}(0)=1$ and ${a}_{2}(0)={a}_{3}(0)=0$ are given byIn the present context, the zero-bias state corresponds to the case of ${\alpha}_{am}=0$ and $\mathrm{\Delta}\beta =0$. Therefore, Eqs. (2) are reduced to

Thus, by substituting $z={L}_{c}$ into Eqs. (2), we are able to calculate power transmission with respect to the complex propagation-value changes in WG2, $\mathrm{\Delta}\beta +i\mathrm{\Delta}{\alpha}_{am}$, values that are related to the carrier-induced index shifts during voltage-on carrier injection: $\mathrm{\Delta}\beta +i\mathrm{\Delta}{\alpha}_{am}=(2\pi /\lambda )(\mathrm{\Delta}n+i\mathrm{\Delta}k)$. Then, from close inspection of Eqs. (2) at $z={L}_{c}$, it is found that the switching characteristics of 3w are essentially determined by two main parameters, $\mathrm{\Delta}\beta {L}_{c}$ and $\rho $. This also applies to the 2w and 4w cases.

#### B. Assumption of Weak Coupling

In arrays of parallel 3D strip waveguides, the strength of evanescent-wave coupling between adjacent waveguides is determined by the width and height of the waveguides as well as the inter-guide gap. This strength can be weak, moderate, or strong. Weak coupling is assumed here. Such coupling can be described in physical and/or mathematical language. An approximate way to talk about such coupling is to say that coupling is weak if the evanescent mode field from one waveguide does not extend into a nonadjacent waveguide.

It is difficult to state an “exact criterion” for weak coupling. A realistic way to characterize “weakness” in the present 3w and 4w cases is to say that the desired weak coupling is attained when ${L}_{c}$ is greater than $40\lambda $ at the wavelength of operation. Here ${L}_{c}$ refers to the device length needed to obtain complete cross coupling of light from one outer waveguide to the other outer waveguide when the voltage is off.

In situations in which the coupling is moderate to strong, for example, when ${L}_{c}\sim 20\lambda $, the same cross-to-bar switching is expected, but the IL and CT predicted by the “weak assumption” will not be accurate. Weak coupling is sometimes violated in the 7–12 μm wavelength region because the free-carrier EO effects are strong there, leading to a large perturbation $\mathrm{\Delta}\beta $ that shrinks ${L}_{c}$ in the $\mathrm{\Delta}\beta {L}_{c}$ switching requirement. Strong coupling is not necessarily bad. Strong coupling could lead to some very good switching performance, but that analysis is beyond the scope of this paper and is recommended for future study.

#### C. Free-Carrier Theory

For crystalline silicon, the FCE has been worked out over the 1.3–14 μm wavelength range to yield detailed predictions of $\mathrm{\Delta}n$ and $\mathrm{\Delta}k$ for injected or depleted concentrations of electrons and/or holes [4]. That theory is employed here, specifically in Table 1 and Eqs. (4) and (5) found in [4], to produce estimates of $\mathrm{\Delta}{n}_{e}$, $\mathrm{\Delta}{n}_{h}$, $\mathrm{\Delta}{k}_{e}$, and $\mathrm{\Delta}{k}_{h}$ over 1.32–12 μm for the aforementioned injection of $\mathrm{\Delta}{N}_{e}=\mathrm{\Delta}{N}_{h}=5\times {10}^{17}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\mathrm{cm}}^{-3}$, which we feel is a feasible or “conservative” level. The corresponding FCE in crystalline germanium has been investigated by Nedeljkovic *et al.* [5]. The empirically based Kramers–Kronig theoretical predictions found in [5] are presented here. Assuming the dual $e+h$ injection just mentioned, specific estimates of the resulting combined real-index change and the combined extinction-coefficient change are given here along with the phase/amplitude ratio $\rho =(\mathrm{\Delta}{n}_{e}+\mathrm{\Delta}{n}_{h})/(\mathrm{\Delta}{k}_{e}+\mathrm{\Delta}{k}_{h})$. These Si and Ge FCE predictions are listed as a function of wavelength in Tables 1 and 2, where we present the $\mathrm{\Delta}\beta $ that corresponds to $(2\pi /\lambda )(\mathrm{\Delta}{n}_{e}+\mathrm{\Delta}{n}_{h})$. These are induced changes in the bulk semiconductor index, and in a real waveguide the mode overlap factor must be applied to find the change in effective index.

## 4. SWITCHING-CHARACTERISTIC RESULTS

#### A. Comparison of 2w and 3w based on Coupled-Mode Analysis

Our first calculation is a comparison of 3w with the prior-art 2w. In the 2w, as indicated above in Fig. 4, it is advantageous to locate the EO injector in WG2 (the cross-state waveguide) when WG1 is the input. The theory and equations for the 2w are presented in the 1988 paper of Soref *et al.* [6], and the resulting switching curves (Fig. 3b of [6]) are reproduced here. Turning to the 3w predictions, our Eqs. (2a) and (2c) above are used to reveal the powers ${|{a}_{1}(z)|}^{2}$ and ${|{a}_{3}(z)|}^{2}$ exiting WG1 and WG3. In those relations, the voltage-off condition is ${\beta}_{2}(\text{off})={\beta}_{2r}(\text{off})+j{\alpha}_{am}(\text{off})={\beta}_{1}+j0.0$, and with voltage on, ${\beta}_{2}(\text{on})={\beta}_{2r}(\text{on})+j{\alpha}_{am}(\text{on})={\beta}_{1}+\mathrm{\Delta}\beta +j\mathrm{\Delta}{\alpha}_{am}$. Switching results are presented in Fig. 8, where the propagation phase shift $\mathrm{\Delta}\beta L$ is taken as the independent variable and the ratio $\rho =\mathrm{\Delta}n/\mathrm{\Delta}k$ is used as a parameter running from 5 to 5000. As mentioned above, $\mathrm{\Delta}\beta +i\mathrm{\Delta}{\alpha}_{am}=(2\pi /\lambda )(\mathrm{\Delta}n+i\mathrm{\Delta}n/\rho )$. At $\mathrm{\Delta}\beta =0$, a cross state is assumed ($L={L}_{c}$), and the plot shows the phase-shift-driven evolution of cross-into-bar, and bar-into-cross. A general result is a low-loss induced bar state for both devices. Specifically, for 3w, even at $\rho =5$, $\mathrm{\Delta}\beta L\ge 28$ gives bar-state IL $<0.6\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{dB}$ with CT below $-15\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{dB}$. For 2w, $\rho =15$ and $\mathrm{\Delta}\beta L=6$ gives 0.4 dB IL, while $\mathrm{\Delta}\beta L=12$ offers 0.4 dB IL for $\rho =5$ together with CT below $-18\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{dB}$.

#### B. Beam-Propagation Simulation for 3w and 4w

Next, beam-propagation simulation is adopted to characterize the switching performance in 3w and 4w couplers. Simulation is carried out at the TE polarization (parallel to the device plane), since the TM beam-propagation simulation is often erroneous in high-index-contrast waveguides. However, the rules deduced here apply to both polarizations. Silicon at $\lambda =1.32\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{\mu m}$ and germanium at $\lambda =12\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{\mu m}$ were chosen to examine the effects of spectrum change and materials change. Metaphorically, these choices are “end points” that “bracket” the behavior of 3w and 4w. At $\lambda =1.32\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{\mu m}$, a waveguide width of 0.4 μm is selected for the single-mode Si channels by scaling down the well-known width for 1.55 μm. At the $\lambda =12\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{\mu m}$ upper end, the Ge width of 3.6 μm is adopted because GeSn waveguide work [7] shows widths in the range $\lambda /n<w<2\lambda /n$, depending upon the core/cladding index contrast.

Figure 9(a) shows a top view of the Si 3w with index 3.5 at 1.32 μm, along with its 4w counterpart in Fig. 9(b). Generally, the procedure for 3w and 4w was as follows: (1) select the gap $s$ to give ${L}_{c}$ of 200–1000 μm; (2) find both output powers as a function of $z$ with voltage-off for an initial cross; (3) determine those powers with voltage-on using trial values of $\mathrm{\Delta}n$ and $\mathrm{\Delta}k$. Because ${L}_{c}<1\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{mm}$ is chosen, the level of injection $\mathrm{\Delta}n$ is high, but a smaller $\mathrm{\Delta}n$ can be traded off against a larger ${L}_{c}$.

As in the 2w and 3w cases, the bar/cross states in 4w are associated with $\mathrm{\Delta}\beta L$ and $\rho $. Thus at fixed $\lambda $, the switching characteristics are determined by $\mathrm{\Delta}nL$ and $\mathrm{\Delta}n/\mathrm{\Delta}k$. Considering the Si 4w at 1.32 μm, in order to verify that the product $\mathrm{\Delta}n{L}_{c}$ governs switching behavior, the responses of two devices were compared. One had ${L}_{c}=370\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{\mu m}$, and the other had ${L}_{c}=750\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{\mu m}$. Those lengths were attained by adjusting $s$ in each case. The cross state at $\mathrm{\Delta}n=\mathrm{\Delta}k=0$ for the two cases is shown by the solid lines in Fig. 10. The index perturbation $\mathrm{\Delta}n=0.004$ is introduced to the ${L}_{c}=750\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{\mu m}$ 4w, while $\mathrm{\Delta}n=0.008$ is introduced to the ${L}_{c}=370\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{\mu m}$ 4w to give the same $\mathrm{\Delta}\beta L=14.3$ (dashed lines). Meanwhile, the presence of $\mathrm{\Delta}k$ at the same $\rho =10$, $\mathrm{\Delta}k=0.0004$ for ${L}_{c}=750\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{\mu m}$ and $\mathrm{\Delta}k=0.0008$ for ${L}_{c}=370\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{\mu m}$, induces the same amount of IL as shown by the dotted lines. In addition, we can see that the bar-state IL in 4w is as low as 0.7 dB, which is much lower than the 2.3 dB IL in the MZI $2\times 2$ switch at $\rho =10$ as shown in Fig. 2.

We next compare the Si 3w and 4w at 1.32 μm by considering the same ${L}_{c}=750\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{\mu m}$ with $L={L}_{c}$. The dependence of the power output upon $\mathrm{\Delta}\beta $ is determined through beam-propagation simulation for 4w and through analytical calculation for 3w. In the lossless injection case ($\mathrm{\Delta}k=0$), $\mathrm{\Delta}n$ ranging from 0.001 to 0.006 is introduced to give $\mathrm{\Delta}\beta L$ going from 4 to 25. The resulting IL and CT are plotted versus $\mathrm{\Delta}\beta L$ in Fig. 11. We find in Fig. 11(a) that 4w is superior to 3w in achieving low CT at a shorter device length. Figure 11(b) indicates CT $<-20\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{dB}$ is achieved at $\mathrm{\Delta}\beta L>14$ in 4w together with an IL $<0.4\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{dB}$. For 3w, $\mathrm{\Delta}\beta L>28$ is required to achieve CT $<-15\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{dB}$, and IL $<0.4\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{dB}$ is promised when $\mathrm{\Delta}\beta L>18$.

For the remainder of this paper, the effect of finite $\mathrm{\Delta}k$ is examined for 4w at $\mathrm{\Delta}\beta L=14$ and for 3w at $\mathrm{\Delta}\beta L=28$. Thus, at $\lambda =1.32\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{\mu m}$, keeping ${L}_{c}=750\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{\mu m}$ for 4w, the ${L}_{c}$ for 3w is increased in Fig. 12 to 1500 μm by adjusting the gap $s$. Then the normalized propagation power at the bias-off state is shown by the solid lines in Fig. 12. Next, $\mathrm{\Delta}n=0.004$ and $\mathrm{\Delta}k=0$ are introduced to give $\mathrm{\Delta}\beta L=14$ in 4w and $\mathrm{\Delta}\beta L=28$ in 3w (dashed lines), resulting in excellent bar-state response as predicted in Fig. 11. Finally, we consider the high-loss case of $\rho =4$ ($\mathrm{\Delta}k=0.001$) for which the bar states are indicated as dotted lines. The finding is that this high $\mathrm{\Delta}k$ induces a higher IL in 4w than in 3w (1.1 dB versus 0.4 dB), but both ILs are not large.

Before turning to the Ge devices, let us update the lossless Fig. 11 studies by putting in the effect of actual $\mathrm{\Delta}k$ in the active waveguides. The ratio $\rho $ is a figure of merit for these waveguides, and the influence of $\rho $ upon the bar-state IL and CT was simulated, taking the $\mathrm{\Delta}\beta L=28$ and $\mathrm{\Delta}\beta L=14$ required for high-performance switching in 3w and 4w, respectively. The simulation results for IL are presented in Fig. 13, and it is found that the 3w IL is quite immune to the high loss in the $1<\rho <5$ range. However, in that same range, the 4w does suffer a significant IL penalty as shown. The good news is that both 3w and 4w maintain excellent CT suppression over the full $\rho $ range.

Next, as illustrated in Fig. 14, we examine Ge 3w and Ge 4w behavior at 12 μm using the $\mathrm{\Delta}\beta L=28$ and $\mathrm{\Delta}\beta L=14$ conditions. From Table 2, $\mathrm{\Delta}n\sim 0.07$ and $\mathrm{\Delta}k\sim 0.04$ are assumed, which leads to device lengths $L$ of 780 and 390 μm for Ge 3w and Ge 4w, respectively, as discussed below. The results are presented in Fig. 15. Solid lines show the cross state at zero bias, dashed lines show lossless injection, and the dotted lines indicate the $\mathrm{\Delta}k\sim 0.04$ level of Table 2. Figure 15 reveals very low CT during injection in all cases. As seen in Fig. 13, the $\rho =1.75$ in Fig. 15 does impact the bar-state IL; thus 2.2 dB IL is expected for 4w, while 3w does much better with 0.86 dB IL. To achieve the ${L}_{c}$’s of Fig. 15, the gap size $s$ is 1470 and 980 nm for 3w and 4w, respectively, according to the beam propagation method.

Based on the criteria $\mathrm{\Delta}\beta L>28$ for 3w and $\mathrm{\Delta}\beta L>14$ for 4w, it is straightforward to predict the minimum device length $L$ required for high-performance Si and Ge 3,4w switches at different wavelengths and at various carrier injection levels. Taking the $\mathrm{\Delta}n$ presented in Tables 1 and 2, those $\mathrm{\Delta}n$ are substituted into the $\mathrm{\Delta}\beta L$ constraints to find the $L$ given in Table 3. Looking at the predictions in Table 3, the lengths of 1.5–3.7 mm needed for Si and Ge at $1.32<\lambda <2.00\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{\mu m}$ are somewhat long, but those lengths can be shortened via higher injection such as ${10}^{18}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\mathrm{cm}}^{-3}$. However, that strategy increases the required switching energy. It is a trade-off situation. The $L$ values in Table 3 decrease with increasing $\lambda $; thus operation at the longer wavelengths is more favorable from an energy standpoint, a longwave advantage. Alternatively, instead of injection, the EO depletion of carriers in doped central waveguides is an approach that goes from an initial bar state to a depleted cross state. The issue there is to deplete 80% or more of the active waveguides so as to attain a low-loss cross.

## 5. ALTERNATIVE ELECTRO-MODULATION TECHNIQUES

As viable alternatives to the FCE, there are several EO mechanisms available for altering the effective index of a semiconductor waveguide. Touching briefly on these in the present switching context, the first is the index shift $\mathrm{\Delta}n$ induced by the Pockels effect of a DC-poled second-order nonlinear optical polymer that is embedded in a central slot of the Si or the Ge strip channel waveguide(s) in the mid-region of the 3w or 4w switch. Both “half strips” of the Si or Ge waveguide are doped for Ohmic contacting so that voltage can be applied across the EO polymer. An example is the ${r}_{33}=250\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{pm}/\mathrm{V}$ polymer developed by the University of Arizona group [8] for the 1.55 μm wavelength (and for part of the mid-infrared). In addition, a recent paper suggests that an ${r}_{33}$ effective value of $1000\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{pm}/\mathrm{V}$ is feasible [9]. Let us assume that the slot contact spacing is just a few hundred nanometers. This assumption then leads to the estimate that an RF electric field of $E=10\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{V}/\mathrm{\mu m}$ can be applied to the polymer. Then we turn to the traditional formulation: $\mathrm{\Delta}n=(1/2){n}^{3}{r}_{33}E$. Taking ${r}_{33}(\mathrm{max})=250\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{pm}/\mathrm{V}$ under the trade name “soluxra” with its index $n=1.80$, we find a change $\mathrm{\Delta}n$ of 0.0073 in the bulk polymer index. Conservatively, this should lead to an effective $\mathrm{\Delta}n$ of 0.003 or more in the waveguide. The conclusion here is that the 4w organic-semiconductor hybrid looks quite feasible for $L=1\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{mm}$ at $\lambda =1.31\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{\mu m}$ and for $L=3\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{mm}$ at $\lambda =4\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{\mu m}$.

The thermo-optic effect is also a good choice, mainly in Ge devices at $\lambda =1.8\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{\mu m}$ [10] and at mid-infrared, where $\mathrm{\Delta}n/\mathrm{\Delta}T$ is about $5.8\times {10}^{-4}/\xb0\mathrm{C}$. Taking $\mathrm{\Delta}T=10\xb0\mathrm{C}$, an effective index shift around $\mathrm{\Delta}n=0.005$ looks feasible, which could actuate a millimeter-scale 4w at $\lambda =1.8\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{\mu m}$. Looking at quadratic EO effects in Ge at $\lambda \sim 1.6\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{\mu m}$ (and in GeSn for longer wavelengths), we have the Franz–Keldysh and quantum-confined Stark effects, both of which are judged to be unlikely candidates for directional coupler switches because these effects are $\rho \sim 1$ electro-absorption effects that have a large background absorption ${k}_{o}$ that would decrease the cross-state transmission.

## 6. PLASMONIC WAVEGUIDE COUPLERS

A new kind of channel waveguide is coming on-stream in the mid-infrared region, a strip structure that has “hybrid” waveguiding wherein a surface-plasmon-polariton mode trapped at a conductor interface [11] is combined with internal infrared reflection within a dielectric strip, forming the hybrid plasmon-polariton (HPP) mode. Experiments on this HPP strip structure at $\lambda =1.32\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{\mu m}$ showed a large change in effective index of $0.032+i0.025$. However, the issue here is an initial background loss of 0.42 dB per μm. The solution to this problem is to go into the strong-coupling regime, where the switch length shrinks down to just a few wavelengths so as to minimize the initial cross-state loss. Sorger and co-workers [12,13] have presented evidence that this can be done in a 3w.

## 7. ACTIVE WAVEGUIDED RESONATORS

Looking to the future, the topic of EO waveguided resonators used as the central waveguide(s) in 3w and 4w is ripe for exploration. The Si or Ge strip waveguide can be made resonant by forming an inline array of cylindrical holes or rectangular slots in the waveguide so as to create a one-dimensional photonic crystal (PhC) cavity having reasonably high $Q$. Such a resonator is known as a nanobeam (NB), and this PhC device is quite scaleable with wavelength. The FCEs are quite applicable to NBs and can cause considerable shift and damping of a particular resonant mode by the depletion, injection, or accumulation of charge. For example, a recent paper [14] pointed out the advantages of using a lateral PN junction to deplete a 1.55 μm group IV NB. The technique of resonance shifting within 3w and 4w is expected to yield switch lengths that are much smaller than those derived here for broadband structures. The much-reduced device size in the cavity cases comes at the expense of a narrow spectral window.

## 8. CONCLUSION

The 2D effective-index approximation allows us to sketch out the main features of actual 3D directional coupler switches that are integrated in plane with side-coupled parallel channels. An analytic coupled-mode formalism was taken for 3w, and a beam-propagation software numerical-simulator approach was taken for 4w. All three devices, 2w, 3w, and 4w, are readily cascadeable into the $N\times N$ crossbar matrix switch. Unlike the asymmetric 2w, the symmetric 3ws and 4ws are easily cascaded into the nonblocking $N\times N$ permutation matrix switch. The 4w is generally more compact than either 3w or 2w since it does not always require spatial “fan out” of the outer waveguides.

In the 3w,4w analysis, phase-matched synchrony of WGs in the weak-coupling regime is assumed, resulting in a perfect cross state with injection off. (In reality, this cross state has a small IL due to the ${k}_{o}$ background loss of the WGs.) Our simulations for silicon show the new result that the bar state with injection on is well shielded from IL induced by the $\mathrm{\Delta}k$ component. Silicon 3w’s and 4w’s are projected to give bar-state IL $<1\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{dB}$ and CT $<-15\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{dB}$ over the full 1.3–12 μm wavelength range when the PIN injection level is chosen to yield an index change of $\mathrm{\Delta}\beta L>28$ for the 3w and $\mathrm{\Delta}\beta L>14$ for the 4w. This simulation takes into account the actual loss $\mathrm{\Delta}{k}_{e}+\mathrm{\Delta}{k}_{h}$ that accompanies the $\mathrm{\Delta}{n}_{e}+\mathrm{\Delta}{n}_{h}$. Generally, the 4w can achieve switching performance equivalent to 3w in half of the 3w’s length, but the bar-3w is less sensitive to $\mathrm{\Delta}k$ as compared to bar-4w. It is clear that 3w, 4w have a significantly larger active length when compared to the reference $2\times 2$ MZI active length whose switching requirement is $\mathrm{\Delta}\beta L=3.14$. However, the MZI is not protected against bar-state IL loss linked to $\mathrm{\Delta}k$, which would be $>2.3\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{dB}$ in the MZI during the presence of $\mathrm{\Delta}k>0.1\mathrm{\Delta}n$. At that same $\mathrm{\Delta}k$ level, the IL for 4w is only 0.8 dB. As mentioned, the cross state at zero bias would experience contact loss ${k}_{o}$ from P and N doping, but that loss is expected to be small in practice.

For the free-carrier scenario, switching speed and energy consumption were not addressed here. Instead, theoretical estimates of CT and IL were targeted. The projected performance is “good” for the following reasons: the bar state CT was less than $-15\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{dB}$ for Si and Ge, in both 3w and 4w. The bar state IL depended upon ρ, which ranged from 5.3 to 17.8 for Si and from 1.5 to 4.8 for Ge. Analysis of Si over the 1.32–12 μm $\lambda $ range, and of Ge over the 1.8–12 μm $\lambda $ range, gave IL of 0.30 to 0.55 dB in Si 3w, 0.65 to 1.0 dB in Si 4w, 0.7 to 1.6 dB in Ge 3w, and 1.2 to 2.2 dB in Ge 4w. Those losses are low except for the 2.2 dB longwave loss in Ge 4w.

A benefit of PIN-waveguide free-carrier injection is the decreased switch length in the mid-wave and long-wave infrared regions. That happens because FCE grows stronger with increasing wavelength. If a representative injection is assumed, and if $\lambda $ is raised from 1.32 to 12 μm, then $L$ decreases from 3.6 to 0.49 mm (Si 3w), from 1.8 to 0.24 mm (Si 4w), from 3.2 to 0.8 mm (Ge 3w), and from 1.6 to 0.4 mm (Ge 4w).

It is likely that improvements in switching performance for these Si and Ge devices can be attained by selecting a different EO mechanism, by choosing plasmonic EO waveguides (instead of photonic waveguides), or by inserting cavity resonators within active photonic waveguides. For example, analysis indicates that a fast, low-loss EO polymer “slot layer” embedded in a Si or Ge strip waveguide would yield low-loss switching in a 1 mm length at $\lambda =1.32\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{\mu m}$ and in a 3 mm length at $\lambda =4\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{\mu m}$.

## ACKNOWLEDGMENTS

The author appreciates the valuable help of Dr. Yijing Chen (Data Storage Institute of Singapore and the ECE Dept., National University of Singapore), who performed all of the Beam Prop simulations. Dr. Chen’s wisdom and technical advice were essential to this paper. Support of the Air Force Office of Scientific Research (Dr. Gernot Pomrenke, Program Manager) and the UK Engineering and Physical Sciences Research Council (Project MIGRATION) is also gratefully acknowledged.

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