1.

Introduction

Programmable photonic integrated circuits (PICs), consisting of meshes of tunable beam splitters and/or optical phase shifters,¹ are extending their applications from widely used optical communication to optical computing,² microwave photonics,³ and quantum information processing.^4,5 To further scale up the programmable PICs, it is necessary to achieve low-loss, low-crosstalk, compact, low-power-consumption, multilevel programmable units. However, most programmable PICs utilize weak and volatile tuning mechanisms such as thermo-optic,^6,7 electro-optics,⁸ or free-carrier effects,⁹ providing very small and volatile refractive index change ($\mathrm{\Delta }n<0.01$). As a result, the devices usually have a large footprint ($>100\mu \mathrm{m}$¹⁰) and require a constant power supply ($\sim 10\mathrm{mW}$), incurring limited density and energy efficiency.

Nonvolatile chalcogenide phase-change materials (PCMs)^11–14 are a very promising platform. These materials have two stable microstructural phases in the ambient environment, i.e., the thermodynamically favorable crystalline (c) phase and metastable amorphous (a) phase. Due to the change in atomic arrangement, a and c phases differ significantly in their refractive index ($\mathrm{\Delta }n\sim 1$). Earlier experiments have shown that the phases can be controlled both optically using pulsed lasers^15–18 and electrically via on-chip microheaters.^19–23 It is worth noting that since both phases are stable, zero power is needed to maintain the state once the PCM phase is changed. In addition, this memory effect can reduce the complexity of control circuits as electronic memory blocks and time-division multiplexing are avoided. The large $\mathrm{\Delta }n$ and nonvolatility together offer compact devices ($\sim 10\mu \mathrm{m}$^23,24) with zero static energy consumption, hence a unique opportunity to achieve ultra-low-power programmable PICs.

Most technologically mature PCMs, such as ${\mathrm{Ge}}_2{\mathrm{Sb}}_2{\mathrm{Te}}_5$ (GST) and GeTe, are very lossy in the c phase particularly in the near-infrared regime ($\sim 1550\mathrm{nm}$), although they are transparent in the a phase. In spite of the emergence of wide bandgap PCMs, such as ${\mathrm{Sb}}_2{\mathrm{S}}_3$ and ${\mathrm{Sb}}_2{\mathrm{Se}}_3$,^{21–23,25–28} they are much less understood and therefore are less likely to be integrated in large-scale foundry processes. In this work, we consider the prototypical PCM GST, which has been widely used for electronic memory applications^29,30 and holds promise to integrate into the photonic foundries. As explained earlier, one major limitation of GST at 1550 nm is the absorption loss of c-GST, which prohibits it from phase-only modulation as needed for large-scale switching fabrics³¹ or interferometry-based optical signal processors.² To mitigate the loss, a phase-matched three-waveguide directional coupler (PM-TDC) design was reported in previous studies.^20,24,31 In addition to the loss, deterministic multilevel operation also remains challenging. Although pulse width or amplitude modulation^22,23 could produce mixed a and c phases, thereby having multiple operation levels, such a process is inherently stochastic.³² Furthermore, these schemes cannot be used in the PM-TDC design, which causes a large loss due to either material absorption or radiation loss at the end of the transition waveguide (WG).

In this work, we propose and numerically demonstrate low-loss optical programmable units based on a lossy PCM GST, including $1\times 1$, $1\times 2$, and $2\times 2$ programmable units. Our devices support deterministic multilevel operation with individually controlled GST segments, which are controlled by a series of interleaved p++-doped-intrinsic-n++-doped (PIN) silicon microheaters. Each microheater controls the corresponding segment in a binary way, and different states can be achieved by programming the phases of the GST segments. We emphasize that this scheme is inherently deterministic compared with engineering the electrical pulse amplitude,²² duration, or number of pulses,²¹ as it solely relies on the binary phase transition, which is more deterministic than achieving intermediate states. We show that $1\times 1$ and $1\times 2$ programmable units can achieve multiple operation levels using segmented PCM. We reported two methods to achieve $2^N$ and ($N+1$) states with $N$ segments. A preliminary experiment on $1\times 1$ WG programmable units with the segmented GST shows distinct multiple operation levels without significant thermal crosstalk. The rectified property of PIN diode heaters was exploited in this programmable unit to reduce the number of metal contacts to only one pair to give four operational levels. Numerical simulation shows that this method can be easily extended to $1\times 2$ programmable units. However, $2\times 2$ programmable units based on the same method show a very high loss in the intermediate levels. We model the PM-TDCs both analytically and numerically to unveil that it is the fundamental wave dynamics in coupled-WG systems that cause a combination of material absorption loss (type I) and radiation loss at the end of the transition WG (type II). In a PM-TDC design, one of the two types of loss presents intermediate states, incurring high optical loss. A modified structure, termed here as phase-detuned three-WG directional coupler (PD-TDC), is proposed to mitigate both types of loss, where the center transition WG is deliberately phase-detuned from the other two WGs for a-GST. By programming the GST patches, we numerically show a three-level $2\times 2$ programmable unit with low insertion loss (IL; $\sim 1.2\mathrm{dB}$). Our work provides a new design methodology for programmable photonic devices using lossy PCMs or any other modulation mechanism.

2.

Deterministic Multilevel Scheme Using Individually Controlled GST Segments

Figure 1 shows our scheme to achieve deterministic multilevel operation using the binary phase transition of PCM. Instead of engineering the electrical pulse conditions,^21,22 we propose to individually control multiple segments of GST thin films on silicon WGs to either fully a or c phase using multiple PIN diode heaters with opposite polarity.^13,33 By programming the state of each GST segment, multiple operation levels can be obtained.

Fig. 1

Our proposed deterministic multilevel programmable units, with multiple PIN diode heaters with reversed polarity to control the phase of each PCM segment in a binary manner: programmable units using (a) lossy and (b) transparent PCMs. Two PIN diodes are controlled by one pair of metals thanks to PIN diode’s rectified behavior. Depending on the sign of the applied voltage, a large electric current only occurs on the forward-biased diodes and switches the corresponding PCMs. The reversely biased diode will have limited current; hence, no PCM switching. An example of uneven segment design is shown at the bottom right. We omit the metals in the following pictures to simplify the schematics (colors: pink, p++ doped silicon; green, n++ doped silicon; gold, metal contact; blue, intrinsic silicon; white, lossy PCMs; gray, transparent PCMs).

3.

Experimental Demonstration of Multilevel 1 × 1 Programmable Units with Two Unequal Segments

A multilevel $1\times 1$ WG programmable unit can be implemented for either amplitude or phase modulation, depending on whether the PCM is lossy in the c phase. At the telecommunication wavelength of 1550 nm, GST,¹⁹ ${\text{Ge}}_2{\mathrm{Sb}}_2{\mathrm{Se}}_5$,³⁴ or ${\text{Ge}}_2{\mathrm{Sb}}_2{\mathrm{Se}}_4{\text{Te}}_1$³⁵ can be used for amplitude modulation, and ${\mathrm{Sb}}_2{\mathrm{Se}}_3$^22,23 or ${\mathrm{Sb}}_2{\mathrm{S}}_3$^21,26,36 can realize phase-only modulation. In the design in Fig. 1(a), each GST segment represents 1 bit, and a device with $N$ segments can present $2^N$ bits at most. This can be achieved by designing the lengths of GST segment $i$ twice that of segment $i-1$, see the bottom right sketch in Fig. 1. We note that, in general, unequal segments can generate $2^N$ levels, but the difference between levels is not the same unless the lengths satisfy the double-length condition.

The rectification behavior of PIN diode heaters can be leveraged to reduce the number of metal wires. Using a series of alternating PIN heaters [Fig. 1(a)], two PIN diode heaters can share the same metal pad because the reversely biased current is small below the breakdown voltage and not enough to trigger any phase transition. Therefore, the number of metal pads can be reduced by half, leading to simplified electrical wire routing. This is beyond the capability of traditional resistive heaters such as the tungsten heater in Ref. 34. Despite controlling two PIN diodes with a single pair of electrodes, one diode remains reverse-biased, resulting in minimal additional current. This current is typically less than $1\mu \mathrm{A}$, varying with the diode’s quality. Such a small extra current guarantees no electromigration on the contact metal. However, the reverse-biased diode and metal could be damaged if the amorphization voltage exceeds the PIN diode’s breakdown voltage usually between 10 and 20 V. For PCM-based directional coupler switches, a large voltage close to 10 V was used in previous works.^20,21,28 Therefore, the amorphization pulse must be carefully engineered to use this method.

One potential limitation is the thermal crosstalk between adjacent GST segments. We experimentally show that this thermal crosstalk does not trigger unintentional phase transition by fabricating and characterizing a $1\times 1$ programmable unit. The fabrication process is similar to our previous works.^20,21,28 We designed the programmable unit with two unequal GST segments (10 nm thick and 2 and $4\mu \mathrm{m}$ long) and demonstrated four distinct levels. Figure 2 shows the device layout, fabricated devices, and characterization results. The lengths of the PCM patches are engineered [Fig. 2(b)] to provide a maximum number of $2^N$ states ($N=2$, therefore four levels). The silicon WGs are 500 nm wide and 220 nm thick without ${\mathrm{SiO}}_2$ top cladding. The boron- and phosphorus-doping regions have a doping concentration of $\sim {10}^{20}{\mathrm{cm}}^{-3}$, which are 200 nm away from the WG edge to avoid free-carrier absorption loss. The dimensions of the doping areas are annotated in Fig. 2(b). Metal contacts are formed by 5 nm/180 nm Ti/Pd, which are placed $1\mu \mathrm{m}$ away from the WG to avoid metal absorption loss. The fabricated device is shown in Fig. 2(c). Time trace sequences in Fig. 2(d) were obtained by recording the transmission at 1550 nm while alternatively applying amorphization (4 V, 100 ns) and crystallization pulses (1.4 V, $50\mu \mathrm{s}$) on segment 2. The on–off extinction ratio (ER) is 0.6 dB, working as a high-precision tuning knob. Low-precision tuning ($T0$ to $T1$ and $T1$ to $T2$, on–off ER of $\sim 3\mathrm{dB}$) was realized by sending in pulses in a reversed bias to control the longer GST segment 1, with $-4.5\mathrm{V}$ for amorphization and $-1.5\mathrm{V}$ for crystallization. We note the polarity is reversed compared with the longer segment. This result shows that the short and long segments are separately tuned without noticeable thermal crosstalk.

Fig. 2

Deterministic four-level $1\times 1$ programmable unit using two PIN heaters with reversed polarity: (a) schematic of the device; (b) zoomed-in schematic; (c) optical microscope image of the fabricated device, scalebar: $25\mu \mathrm{m}$; and (d) time-trace measurement, showing four distinct and reversible transmission levels. Measured results for time traces $T0$, $T1$, and $T2$ are in blue, orange, and green, respectively, where fine-tuning is achieved in each time trace by switching segment 2 and coarse-tuning is shown between two adjacent time traces by switching segment 1.

4.

Optical Design for Deterministic Multilevel 1 × 2 Programmable Unit

The experimental result in Sec. 3 validates our deterministic multilevel scheme in the $1\times 1$ programmable unit. In this section, we extend the idea to $1\times 2$ and $2\times 2$ programmable units, which require phase-only modulation and are building blocks for large-scale programmable PICs. To achieve a low-loss operation, a three-WG directional coupler consisting of GST-loaded silicon (hybrid) WG and bare silicon WG was previously reported.^20,24,31 When GST is in the a phase, the hybrid WG is designed to phase-match with the bare silicon WGs. Therefore, the input light couples to the cross port. When the GST changes to the c state, due to the large effective index difference, the WGs are phase mismatched. As a result, the light barely couples and stays in the bar port, thus bypassing the high loss of c-GST. We emphasize that achieving multilevel operation using this idea requires extra design considerations since the absorption loss of c-GST limits the available phase sequence of the GST segments.

Figure 3 shows the simulation result for a $1\times 2$ optical programmable unit, for which the design is the same as Ref. 20. One change is that two individually controlled GST segments, instead of one, are used to achieve multilevel operation. The achievable low-loss states are, however, not $2^N$ as in the $1\times 1$ programmable unit case, but $N+1$ since any a-GST segment before the c-GST segment results in light coupling to the GST-loaded WG, which then gets completely absorbed by the highly lossy c-GST segments. Our simulation results agree with this intuition. We denote the phase sequence from left to right as {phase of segment 1, phase of segment 2} and show the electric field distribution and the transmission spectrum for all four possible phase sequences. Figure 3(b) shows that sequence {a, c} exhibits a high IL of $\sim 3\mathrm{dB}$ due to high absorption of the c-GST segment 2. This is verified by the electric field distribution image, where the light first couples into the cross port along the a-GST segment 1 and then gets completely absorbed by c-GST segment 2. Since the length of the first segment is half of the coupling length, exactly half of the light is coupled and absorbed, agreeing well with the simulation.

Fig. 3

FDTD simulation for a deterministic three-level $1\times 2$ programmable unit using GST: (a) schematic, (b) intensity distribution, and (c) the corresponding transmission spectra for different GST phase combinations. GST sequence {a, c} introduces an IL of $\sim 3\mathrm{dB}$ due to c-GST absorption loss, pointed by a white arrow in the field distribution map.

5.

Optical Design for Deterministic Multilevel 2 × 2 Programmable Unit

Previous GST-based low-loss $2\times 2$ programmable units leveraged a PM-TDC structure to circumvent the loss of c-GST.^20,24,31 In these works, an extra transition WG is placed between two silicon WGs with 20-nm-thick GST deposited on top for active switching. When the GST is in the a phase, the transition WG is designed to phase-match with the other two WGs. Therefore, the input light can be coupled into the transition WG and then to the cross port. On the contrary, once switching the GST to the c phase, a large effective index difference is introduced, and the light can barely couple to the high loss transition WG. This bypasses the high loss of c-GST. Unfortunately, a straightforward extension of this PM-TDC method to multilevel operation does not work as we will show in the following analysis.

5.1.

Challenges of Multiple Segments Switching Using the PM-TDC Approach

Figure 4 shows a simulation using a similar design to Ref. 20 but with two GST segments. It shows that both {a, c} and {c, a} phase sequences have a high loss of $\sim 3\mathrm{dB}$. The former can be explained based on the same concept as in the $1\times 2$ programmable unit case: no c-GST should be placed after an a-GST segment when finite light is still in the transition WG, defined as the type I absorption loss. Besides, the transition WG, which is open at the end, adds another constraint for low-loss operation: no light can remain in the transition WG at the end of this transition WG. Otherwise, that portion of light will radiate and get lost, defined as type II radiation loss.

Fig. 4

Phase-matched three-WG $2\times 2$ directional couplers have a high loss for intermediate levels: (a) schematic, (b) the transmission spectra for different GST phase sequences, and (c) the corresponding intensity distribution. Both sequences {a, c} and {c, a} introduce an IL of $\sim 3\mathrm{dB}$. The former is due to c-GST absorption loss (type I loss), while the latter is due to the radiation loss at the end of the transition WG (type II loss), both indicated by white arrows.

5.2.

Analytical and Numerical Study of the Three-WG Directional Coupler Systems

To tackle this loss issue, it is crucial to understand the intensity dynamics inside of each WG in this coupled WG system. We describe the coupled three-WG system using the following coupled partial differential equation:³⁷

Eq. (1)

$$\{\begin{array}{c}\frac{d{a}_0}{dz}=i{\kappa }^{*}(a_1+a_2)\\ \frac{d{a}_1}{dz}=i\kappa a_0-i\delta a_1\\ \frac{d{a}_2}{dz}=i\kappa a_0-i\delta a_2,\end{array}$$

where $z$ is the distance in the light propagation direction, $a_{i,i=0,1,2}$ is the amplitude in three WGs in Fig. 5(a), $\kappa =\mathrm{\Delta }{n}_{\mathrm{eff},c}·\frac{2\pi }{{\lambda }_0}$ and $\delta =\mathrm{\Delta }{n}_{\mathrm{eff},i}·\frac{2\pi }{{\lambda }_0}$ are the coupling coefficient and detuning between the center and other WGs, $\mathrm{\Delta }{n}_{\mathrm{eff},c}$ is the effective index difference between the super-modes formed by coupled WG0 and WG1 (or WG2), $\mathrm{\Delta }{n}_{\mathrm{eff},i}$ is the effective index difference between the modes in WG0 and WG1 (or WG2) assuming they are isolated, and ${\lambda }_0$ is the vacuum wavelength of light. We note that only the nearest coupling is considered in this model, which is usually a valid assumption in a two-dimensional photonic WG array. We also assume that WG1 and WG2 have the same geometry to simplify the model and neglect the loss term for simplicity. We note that this coupled equation is analytically solvable, see Appendix A. To quickly capture the physics, we show the numerical analysis below.

Fig. 5

Analytical model reveals that phase-matched three-WG couplers can never have low-loss intermediate levels: (a) schematic for a generic GST three-WG coupler. The GST patches are denoted in white. In all the discussions, we assume that the WG1 is the input port. (b) Numerically calculated light intensity in three WGs. Blue, orange, and green colors denote WG0, WG1, and WG2, respectively. In the case of two GST segments, phase sequence {c, a} would lead to a propagation distance of $L_c/2$, indicated by the gray dotted line. It is clear that finite energy remains in WG0, causing type II radiative loss.

First, to analyze the case where WG0 is phase-matched with WG1 and WG2, we set $\delta$ to zero and solve this equation numerically. We found that the middle WG intensity is zero only when the light is completely in the cross or bar port [Fig. 5(b)]. Therefore, type I loss occurs in the {a, c} state, while type II loss occurs in the {c, a} state. Therefore, we summarize the requirements for low-loss multilevel operation in a three-WG directional coupler as (1) no light can be in the middle WG at a c-GST segment to avoid type-I loss, (2) no light can be at the end of the middle WG to avoid type-II loss, and (3) there must exist some intermediate splitting ratios (except from complete bar or cross) when the intensity of light in WG0 is zero. Obviously, the last requirement can never be fulfilled in a PM-TDC.

5.3.

New Degree of Freedom: PD-TDCs

To solve this issue, a new degree of freedom must be introduced. We deliberately designed WG0 to have a phase detuning $\delta$ with WG1 and WG2, i.e., a phase-detuned TDC scheme. The phase detuning between WGs can be quantitatively represented by the detuning rate $\delta$, where $\beta =2\pi /\lambda$ is the vacuum wavevector and $\mathrm{\Delta }{n}_{\mathrm{eff}}$ is the effective index difference between WGs. To intuitively understand the wave dynamics in this system, we performed a numerical simulation as shown in Fig. 6, where different plots show the intensity in the three WGs versus propagation length under different detuning rates $\delta$. It can be clearly seen that different splitting ratios between WG1 and WG2 can be obtained when zero light is in WG0. We note that Eq. (1) is analytically solvable, and the solution is presented in Appendix A. Another possible solution emerging from this wave dynamics is adding a tapered structure at the end of WG0, which is discussed in more detail in Appendix B.

Fig. 6

Detuning the middle WG provides rich optical intensity dynamics, enabling low-loss intermediate levels: (a)–(i) intensity dynamics in the three WGs as the detuning $\delta$ varies from 0 to $4\kappa$. As the detuning gets larger, the intensity in WG0 becomes smaller, and that in the other two is smoother. As $\delta$ increases from zero, multiple zeros in the blue curve emerge (e.g., pointed by red arrows in panel (b), overcoming type II loss. Note that this simulation assumes a coupling length $L_c$ of $10\mu \mathrm{m}$ between WG0 and WG1 (or WG2) when $\delta =0$.

A tradeoff must be considered when choosing detuning rate $\delta$. A large detuning $\delta$ generally creates more zeros in the middle WG within the same coupling region length, hence more achievable intermediate levels. However, a large $\delta$ also leads to a longer device (for complete power transfer to the cross port) and hence larger overall loss when GST is in the c phase. To compensate for this, a thicker GST can be used to provide a larger $\mathrm{\Delta }n$. However, the loss due to a-GST also increases. Furthermore, this makes reversible electrical switching more difficult.

5.4.

Design Example of Nonvolatile Multilevel 2 × 2 Programmable Units Based on GST

Based on the above discussion, we choose $\delta =1.5\kappa$ and design a multilevel $2\times 2$ programmable unit with an ellipsometry measured GST refractive index (see Appendix B for detailed design procedures). Figure 7 shows the finite-difference time-domain (FDTD) simulation results with a low IL ($\sim 1.2\mathrm{dB}$) and three achievable splitting ratios of 100:0, 50:50, and 0:100. The designed parameters are summarized in Table 1.

Fig. 7

Optimized deterministic three-level $2\times 2$ programmable units with phase-detuned TDC design ($\delta =1.5\kappa$): (a) schematic (left) and the cross-sectional view (right). Designed geometry parameters are listed in Table 1. (b) Simulated intensity distribution and (c) the corresponding transmission spectra for different GST phase sequences. The IL is reduced to $\sim 1.2\mathrm{dB}$ compared with $\sim 3\mathrm{dB}$ in traditional PM-TDC designs.

Table 1

Designed parameters for the multilevel 2 × 2 GST programmable unit with δ=1.5κ.

h (nm)	hslab (nm)	gd (nm)	wb (nm)	wh (nm)	g (nm)	hGST (nm)	wGST (nm)	lc (μm)
220	100	200	630	500	315	30	450	115

Figure 8 shows the IL and ER of the optimized $2\times 2$ programmable unit. The IL is low ($\sim 1.2\mathrm{dB}$) across a broad wavelength (60 nm) for all three levels (four configurations). We emphasize that this IL is mainly due to the relatively thick GST film used in the design, which introduces absorption loss in its a state. By replacing GST with lower loss PCMs such as GSST, we can potentially further reduce the loss. However, the index contrast $\mathrm{\Delta }n$ of GSST is much smaller than GST, which may result in a thicker PCM and longer devices. The ERs of the complete cross and bar states remain high across 60 nm, but the intermediate states are more wavelength-sensitive with a 3 dB variation bandwidth of $\sim 22.6\mathrm{nm}$.

Fig. 8

Simulated transmission spectrum of the designed three-level $2\times 2$ TDC programmable unit ($\delta =1.5\kappa$): The device shows (a) a low loss ($<1.2\mathrm{dB}$) for all four GST sequences and (b) a bandwidth of $\sim 22.6\mathrm{nm}$ with less than 3 dB ER variation.

6.

Discussion

The number of operation levels of our device is determined by the phase detuning $\delta$ and by the wave dynamics of the three-WG system. From Fig. 6, an arbitrary number of levels can be achieved if the detuning $\delta$ is large enough, which, however, incurs a long device length. Moreover, a-GST has finite absorption loss at 1550 nm, and a longer device leads to not only limited device density but also a larger IL. In the future, the number of operation levels can be improved using other PCMs with zero loss in the a phase, such as GSST. Another approach is to add a tapered structure at the end of the middle WG, which forces light to transfer completely to the other two WGs, see Appendix B. However, both approaches would cause a longer device, limiting the integration density of the system.

Using transparent PCMs, such as ${\mathrm{Sb}}_2{\mathrm{Se}}_3$ and ${\mathrm{Sb}}_2{\mathrm{S}}_3$, we can easily extend the multilevel design to a $2\times 2$ programmable unit.^21,38 The design principle is similar, and the switching of output ports relies on the effective index change of the hybrid WG when the PCM is switched. Importantly, since transparent PCMs introduce negligible excess loss, $2^N$ distinguished splitting ratios can be easily achieved by varying PCM segment lengths.³³ This is a result of the reduced design complexity due to the relaxed loss requirements. Table 2 summarizes some recent results of broadband $2\times 2$ multilevel switches using transparent PCMs in terms of IL, PCM length, number of levels, number of electrodes, and whether the devices support deterministic multilevel operation. Notably, Ref. 39 showed that c-GST exhibits an unacceptable absorption loss when directly used for phase modulation in a Mach–Zehnder interferometer. Unsurprisingly, transparent PCMs show lower IL, a higher number of levels, and a smaller number of electrodes. However, these transparent PCMs are more challenging to be integrated into photonic foundries for high-volume production.

Table 2

Comparing multilevel 2 × 2 switches based on lossy and transparent PCMs.

In summary, we have numerically demonstrated GST-based silicon photonic programmable units with low IL and deterministic multilevel. The multilevel operation is achieved via a novel design where interleaved PIN silicon heaters separately control each GST segment. By programming each GST segment’s phase, deterministic multilevel operations are obtained. To demonstrate that the thermal crosstalk does not introduce unwanted PCM phase transition, we fabricated and characterized a two-GST-segment, four-level $1\times 1$ programmable unit. The experimental result shows four distinct transmission levels, and each level is accessible, reliable, and reversible. Based on this idea, we also designed $1\times 2$ and $2\times 2$ programmable units with low IL. A phase-detuned TDC geometry was proposed to achieve a low-loss $2\times 2$ programmable unit, and we numerically demonstrated a design with a low IL ($\sim 1.2\mathrm{dB}$) and three operation levels. This novel design idea provides a way to achieve low-loss photonic switches with multiple operation levels using lossy tuning media.

7.

Appendix A: Low-Loss Multilevel 2 × 2 GST Programmable Unit Design Procedure

7.2.

Design Procedures for PD-TWDs

The design procedure can be described as the following steps:

• 1. Assume an operation wavelength of ${\lambda }_0=1.55\mu \mathrm{m}$ and standard 220 nm silicon on insulator wafer. The WG is formed by partially etching the silicon by 120 nm. The remaining 100-nm slab is used for doping. Determine suitable constants $M$ and $N$ to achieve reasonable numbers of levels, phase mismatch between a- and c-GST, and device length. Here, we used $M=1.5$ and $N=30$ to achieve three operation levels and reasonable ER between a- and c-GST.

• 2. Fix the width of the WG0 ($w_h$) and PCM thickness ($h_{\mathrm{PCM}}$), such as to $w_h=500\mathrm{nm}$ and $h_{\mathrm{PCM}}=20\mathrm{nm}$. Then, calculate the effective indices for both a and c phases to determine $d{n}_{\mathrm{ca}}$. Calculate the desired detuning rate ${\delta }_{\mathrm{a}}$, coupling coefficient $\kappa$ from $M$, $N$, $d{n}_{\mathrm{ca}}$, and ${\lambda }_0$. Estimate the device length $L_{\mathrm{dev}}$. If the device length is too long, increase PCM thickness ($h_{\mathrm{PCM}}$) and repeat this step. Here, we used $h_{\mathrm{PCM}}=30\mathrm{nm}$ to achieve a high refractive index contrast and obtained the following parameters: ${\delta }_{\mathrm{a}}=0.04959\mu {\mathrm{m}}^{-1}$, $\kappa =0.03306\mu {\mathrm{m}}^{-1}$, and $L_{\mathrm{dev}}=115\mu \mathrm{m}$.

• 3. Sweep the width of the bare silicon WG ($w_b$) to optimize the detuning rate ${\delta }_{\mathrm{a}}$. We calculated the desired effective index as 2.67177 and optimized $w_b=630\mathrm{nm}$.

• 4. Sweep the gap to optimize the coupling coefficient $\kappa$. We sweep the gap between 300 and 400 nm and obtained the desired $\kappa$ when it is 315 nm.

• 5. Run an FDTD simulation to verify the field distribution in the WG. Our FDTD results are shown in Figs. 7 and 8.

Fig. 9

An alternative design with phase-matched middle WG and a tapered structure at the end of the middle WG. It can potentially enable arbitrary multilevel switching but at the cost of a much longer device.

8.

Appendix B: Discussion on Using a Tapered Structure in the Middle WG for Multilevel Switching

The tapered WG structure can usually reduce the loss from mode mismatches when the WG structure changes,²³ which is rarely explored in three-WG systems. In this section, we briefly discuss the potential of using a tapered middle WG in phase-matched three-WG systems to achieve multilevel switching. Figure 9 shows our proposed device, where the middle WG has a taper at the end. The taper can adiabatically guide all the light to the other two WGs to ensure no light remains in the middle WG at the end and hence no type II loss.

The rationale of this design can be understood from Fig. 6. The three-WG system with a tapered middle WG has the detuning $\delta$ increasing with the propagation length, causing the amplitude in the middle WG to decrease. One can see from Fig. 6 that the wave dynamics in the three-WG system change drastically with $\delta$. To avoid extra mode mismatch loss, detuning $\delta$ should vary as slowly as possible, incurring a long coupling region. Such a long coupling distance is quite common for adiabatic directional couplers unless adiabatic mode evolution is carefully engineered.^40,41 A more detailed study of this structure is already beyond the scope of this paper and may constitute future research directions.