Harnessing self-heating effect for ultralow-crosstalk electro-optic Mach–Zehnder switches

Peng Bao; Qixiang Cheng; Jinlong Wei; Giuseppe Talli; Maxim Kuschnerov; Richard V. Penty

doi:10.1364/PRJ.492807

Abstract

This paper presents a novel approach to counterbalance free-carrier-absorption (FCA) in electro-optic (E-O) Mach–Zehnder interferometer (MZI) cells by harnessing the self-heating effect. We show insights on crosstalk limitations in MZIs with direct carrier-injection and provide a detailed design methodology on a differential phase shifter pair. Leveraging both free-carrier dispersion (FCD) and self-heating effects, our design enables arbitrary phase tuning with balanced FCA loss in the pair of arms, eliminating the need for additional phase corrections and creating ultralow crosstalk MZI elements. This neat design disengages from the commonly used nested structure, thus providing an opportunity of embedding tunable couplers for correcting imperfect splitting ratios given that only two are needed. We show that with the use of tunable directional couplers, a standard

\pm 10 nm

process variation is tolerated, while achieving a crosstalk ratio below

- 40 dB

. By direct carrier injection in both arms, the proposed device operates at nanosecond scales and can bring about a breakthrough in the scalability of E-O switch fabrics, as well as other silicon integrated circuits that have stringent requirements for crosstalk leakage.

1. INTRODUCTION

The growing demand for cloud services due to emerging applications such as data analytics, the Internet of Things (IoT), virtual reality (VR), and artificial intelligence (AI) has resulted in a significant increase in data traffic within data centres [1]. The current trend towards mega-data centres with hundreds of thousands of servers has been driven by economy of scale cost reduction [2]. In addition, the traffic flow of the data centre networks has shifted from north–south to east–west, i.e., the internal data traffic between servers and storage now surpasses that of inbound or outbound data traffic [3], posing unprecedented challenges to the intra-data centre interconnects in terms of bandwidth and connectivity. Optical switched networks could offer many advantages, including adaptive resource provisioning, lower latency, greater bandwidth per port, and higher efficiency. In particular, it has been well recognized that the combination of optical circuit switching along with electrical packet switching is one of the most attractive solutions for the future data centre network architecture [4,5].

The silicon-on-insulator (SOI) platform takes advantage of well-established CMOS technologies, positioning itself as a highly promising platform for optical integrated switch fabrics. By embedding phase shifters in interferometric structures, such as Mach–Zehnder interferometers (MZIs) and ring resonators, planar optical switching can be achieved, and higher radix switch fabrics can be assembled by logically wiring the switch cells together [6,7]. Layered switching structures are also being explored to facilitate the development of vertical microelectromechanical systems (MEMS) couplers [8 –10]. While optical switching with microsecond-scale reconfiguration time would already benefit the system, faster switching speeds better suit latency-sensitive applications [11]. The free-carrier dispersion (FCD) effect offers nanosecond-scale switching time although it indeed comes at a price of higher loss and crosstalk due to the induced free-carrier-absorption (FCA) [12]. While gain integration has been widely demonstrated as a viable solution to tackle the switch insertion loss [13 –15], coherent crosstalk remains trickier to handle, and it adds penalties to the optical power budget creating a major signal integrity challenge. Efforts have been made in this respect by using nested switch elements [16 –18] or dilated switch topologies [19], both of which trade off device complexity for improved crosstalk performance.

This paper introduces a novel design method to achieve ultralow-crosstalk electro-optic (E-O) MZI cells with direct carrier injection. Here, the unwanted self-heating in a doped E-O phase shifter is enhanced instead and is manipulated to offset the index modulation by FCD, enabling us to engineer a pair of differential E-O phase shifters in an MZI that always maintain the same insertion loss, while being capable of providing an arbitrary overall phase difference. This effectively corrects any phase errors, balances optical power in the MZI arms, and thus cancels FCA-induced impairments. Furthermore, a curved tunable directional coupler (CTDC) is developed to mitigate fabrication imperfections in power splitting and to widen its operation bandwidth. We also examined the proposed switch cell in various topologies because the characteristics of elementary cells play a decisive role in topology selection.

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The remainder of this paper is structured as follows. Section 2 reviews state-of-the-art silicon switches by carrier injection and switch cells. Section 3 shows insights on crosstalk limitations in MZIs and details design considerations of the ultralow-crosstalk MZI cell. Section 4 investigates the practicability of implementing the proposed switch cell in various topologies. Finally, Section 5 summarizes this work.

2. SILICON MZI SWITCHES BY CARRIER INJECTION

Silicon does not possess linear electro-optic effects, and its quadratic effects are very weak. The plasma dispersion effect (PDE) through carrier injection offers the best all-silicon solution for E-O switch fabrics, and to date, silicon E-O MZI switches have been demonstrated scaling from 4 to 32 ports [6,14,19 –22]. The non-blocking Beneš topology is undoubtedly the most popular selection when constructing large-scale switch fabrics because it requires the minimum number of switch cells. While they are excellent demonstrations of high-level integrations, crosstalk arising from FCA, exacerbated by the single-ended drive scheme, poses a strong limitation on the performance [22]. First-order crosstalk gets effectively suppressed in Ref. [19], but the applied dilated topology pays the price of a much larger footprint. The chip insertion loss can be viably addressed by having integrated gain elements, as demonstrated by the semiconductor optical amplifier (SOA)-integrated E-O switch fabric reported in Ref. [14]. This fabric uses a 4-channel SOA array bonded to an on-chip etched cavity with butt-coupled waveguide interfaces, resulting in a net neutral insertion loss.

The design of elementary cells plays a critical role, being decisive for the circuit-level performance of a switch fabric. Table 1 summarizes representative elementary MZI cells actuated by direct carrier injection with nanosecond-scale reconfiguration times. Bounded by FCA, typical E-O MZI cells exhibit an insertion loss of approximately 1 dB and crosstalk between $- 20$ and $- 27 dB$ , depending on whether the switch is driven single ended [22] or push-pull [23]. The use of augmented designs can improve crosstalk performance. The Chinese Academy of Sciences (CAS) presented a $2 \times 2$ dilated MZ switch (DMZS) consisting of four MZI elements [24] that can suppress the crosstalk to less than $- 31 dB$ over a 40 nm wavelength range, though individual MZIs exhibiting only a crosstalk ratio of $- 14 dB$ . IBM reported a nested MZS (NMZS) design with a variable optical attenuator [16]. The switch cell could ideally remove crosstalk leakage by setting matched power attenuation but with wavelength dependence. The University of British Columbia subsequently proposed a balanced nested MZS (BNMZS) [17] providing broadband tri-state operation and excellent crosstalk suppression in the extra blocking state. IBM later further improved their work with a shift-and-dump MZS (SDMZS) drawing from modulator designs [18]. The switch cell achieves minimum crosstalk of $- 33 dB$ but consumes rather higher power at 99 mW due to the use of six additional heaters. These augmented designs very much advance the performance of MZI switch fabrics but inevitably trade off device complexity and more importantly, the fabrication variation in power coupling coefficient remains as a thread. Variable splitters have been proposed to address the imperfect power splitting ratio [27], which, however, are prohibitive in the augmented structures because multiple pairs are needed to make the device intricate.Table 1.

Example Elementary Silicon MZI Cells by Direct Carrier Injection

Reference	3 dB Splitter	Loss (dB)	Crosstalk (dB)/Bandwidth (nm)	Switching Time (ns)	Power (mW)	Description
[22]	MMI	1	$- 18 dB$ at 1.55 μm^a	3.2	20.9	E-O MZS (single-ended)
[23]	DC	1.2	$- 20 dB / 12 nm$	4	26	E-O MZS (push-pull)
[24]	MMI	8	$- 30 dB / 40 nm$	N.A.	40.8	E-O DMZS
[16]	DC	2	$- 20 dB / 1 nm$	4	34	E-O NMZS
[17]	DC	0.65^b	$- 28 dB$ ^b at 1.55 μm^a	N.A.	N.A.	E-O BNMZS
[18]	DC	1.2	$- 30 dB / 5 nm$	6	99	E-O SDMZS
[25]	Y-splitter	3	$- 20 dB$ at 1.5 μm^a	50	6	T-O MZS (strip waveguide)
[26]	Y-splitter	1.9	$- 18 dB$ at 1.55 μm^a	36	20.7	T-O MZS (MMI-PS)
This work	Curved DC	$< 2$	$- 30 dB / 24 nm$	20	18	Differential PS pair
This work	Curved DC & CTDC	$< 2$	$- 30 dB / 27 nm$	20	23	Differential PS pair + one CTDC
This work	CTDC	$< 2$	$- 30 dB / 30 nm$	20	28	Differential PS pair + two CTDCs

Crosstalk reported at a certain wavelength.

Obtained when 3 dB couplers have an identical splitting ratio of 0.52:0.48.

In addition, silicon exhibits a strong thermo-optic (T-O) coefficient ( $1.8 \times 10^{- 4} K^{- 1}$ ) [28]. Metal and doped heaters have been widely applied in silicon photonic circuits to provide an almost ideal way to manipulate phase at microsecond scales. However, by direct current injection that heats up the waveguide, Joule effect has been successfully engineered to actuate optical switching in nanosecond scales [25,26]. The underlying reason is that it dramatically reduces heat capacity. Massachusetts Institute of Technology (MIT) first reported an MZI switch with strip waveguides achieving 50 ns switching time by pulsing the heating power [25], and modelling indicates a 10-ns switching time with increased pulsed power. Direct carrier injection inevitably incurs loss due to FCA and thus Waseda University recently demonstrated an MZI element with direct-heating phase shifters in a multimode interference (MMI) structure [26]. A switching time of 36 ns is obtained, and further speedup is expected as the width of MMI poses a limitation. Nevertheless, like E-O MZIs, the crosstalk ratio of T-O cells by direct current injection is also bounded by FCA-induced loss to around $- 20 dB$ , which potentially inhibits its wide adoption.

Fortunately, direct carrier injection can be exploited to trigger both FCD and self-heating effects at nanosecond scales. It should be noted that the former incurs blue shift, while the latter incurs red shift, providing an interesting design space.

3. ELEMENTARY MACH–ZEHNDER SWITCH CELL

In Fig. 1, a schematic representation of an MZI, serving as a $2 \times 2$ switch cell, is presented. The schematic includes an input 3 dB coupler with a splitting ratio of $t_{L}^{2} : k_{L}^{2}$ , two arms equipped with phase shifters that modulate the optical phases $ϕ_{1}$ and $ϕ_{2}$ with field transmission coefficients $α_{1}$ and $α_{2}$ , and an output 3 dB coupler with a splitting ratio of $t_{R}^{2} : k_{R}^{2}$ .

Schematic of a 2 × 2 switch cell.

Figure 1.Schematic of a $2 \times 2$ switch cell.

The relationship between the input and output electric fields of the switch cell can then be expressed as $[\begin{matrix} E_{o 1} \\ E_{o 2} \end{matrix}] = [\begin{matrix} t_{R} & k_{R} e^{- j \frac{π}{2}} \\ k_{R} e^{- j \frac{π}{2}} & t_{R} \end{matrix}] [\begin{matrix} α_{1} e^{- j ϕ_{1}} & 0 \\ 0 & α_{2} e^{- j ϕ_{2}} \end{matrix}] [\begin{matrix} t_{L} & k_{L} e^{- j \frac{π}{2}} \\ k_{L} e^{- j \frac{π}{2}} & t_{L} \end{matrix}] [\begin{matrix} E_{i 1} \\ E_{i 2} \end{matrix}],$ (1)where $E$ is the electric field.

Under the assumption that light is introduced into port $I_{1}$ (i.e., $E_{i 1} = 1$ and $E_{i 2} = 0$ ), the power present at the two outputs can be calculated as follows: $P_{o 1} = {| E_{o 1} |}^{2} = t_{L}^{2} t_{R}^{2} α_{1}^{2} + k_{L}^{2} k_{R}^{2} α_{2}^{2} - 2 t_{L} k_{L} t_{R} k_{R} α_{1} α_{2} \cos (ϕ_{1} - ϕ_{2}),$ (2) $P_{o 2} = {| E_{o 2} |}^{2} = t_{L}^{2} k_{R}^{2} α_{1}^{2} + k_{L}^{2} t_{R}^{2} α_{2}^{2} + 2 t_{L} k_{L} t_{R} k_{R} α_{1} α_{2} \cos (ϕ_{1} - ϕ_{2}) .$ (3)

The switch cell is in the cross state when both phase shifters are turned off, i.e., $ϕ_{1} - ϕ_{2} = 0$ . Under this condition, the light comes out of port $O_{2}$ , and the crosstalk can be expressed as the power ratio of the leakage at port $O_{1}$ to the output at port $O_{2}$ : $CT (cross) = \frac{P_{o 1}}{P_{o 2}} = \frac{{| E_{o 1} |}^{2}}{{| E_{o 2} |}^{2}} = \frac{t_{L}^{2} t_{R}^{2} α_{1}^{2} + k_{L}^{2} k_{R}^{2} α_{2}^{2} - 2 t_{L} k_{L} t_{R} k_{R} α_{1} α_{2}}{t_{L}^{2} k_{R}^{2} α_{1}^{2} + k_{L}^{2} t_{R}^{2} α_{2}^{2} + 2 t_{L} k_{L} t_{R} k_{R} α_{1} α_{2}} .$ (4)

Furthermore, the switch cell is in the bar state when one of the phase shifters is turned on, i.e., $ϕ_{1} - ϕ_{2} = π$ . In such a state, the light goes to port $O_{1}$ , and the crosstalk ratio is given by $CT (bar) = \frac{P_{o 2}}{P_{o 1}} = \frac{{| E_{o 2} |}^{2}}{{| E_{o 1} |}^{2}} = \frac{t_{L}^{2} k_{R}^{2} α_{1}^{2} + k_{L}^{2} t_{R}^{2} α_{2}^{2} - 2 t_{L} k_{L} t_{R} k_{R} α_{1} α_{2}}{t_{L}^{2} t_{R}^{2} α_{1}^{2} + k_{L}^{2} k_{R}^{2} α_{2}^{2} + 2 t_{L} k_{L} t_{R} k_{R} α_{1} α_{2}} .$ (5)

To eliminate crosstalk in both states, the power in the two arms of the interferometer must be equal, i.e., $α_{1} = α_{2}$ , and the splitting ratios of the input and output 3 dB couplers should be precisely 50:50, i.e., $t_{L}^{2} = k_{L}^{2} = t_{R}^{2} = k_{R}^{2}$ . The same holds true for launching light into port $I_{2}$ . However, for E-O switch cells, the PDE-based phase-shifting mechanism causes unbalanced FCA loss in the two MZI arms. Additionally, fabrication imperfections often result in power splitting ratio deviations from the ideal 50:50. Both deteriorate crosstalk.

A. Free Carrier Absorption Loss

PIN junctions, which consist of an intrinsic region sandwiched by $p$ - and $n$ -doped regions, are dominantly utilized in E-O switches for fast phase shifting. When a forward bias is applied to the PIN junction, free carriers are injected from the doped regions into the intrinsic region, i.e., the waveguide, altering its refractive index and producing a phase shift. However, the injected carriers not only modify the waveguide’s refractive index but also induce undesirable loss due to FCA. Soref’s model [29] can be used to tie the change of refractive index $Δ n$ to that of the absorption coefficient $Δ α$ , $Δ α (λ) = a (λ) Δ N_{e}^{b (λ)} + c (λ) Δ N_{h}^{d (λ)},$ (6) $- Δ n (λ) = p (λ) Δ N_{e}^{q (λ)} + r (λ) Δ N_{h}^{s (λ)},$ (7)where $Δ N$ is the change in carrier concentration, and $a$ , $b$ , $c$ , $d$ , $p$ , $q$ , $r$ , and $s$ are measured wavelength-dependent coefficients.

The FCA-induced loss fundamentally bounds the crosstalk ratio of an E-O switch cell. Such loss creates a power imbalance within the interferometer and thus incomplete interference occurs, leading to power leakage, i.e., crosstalk, as per Eqs. (4) and (5). For single-ended driven E-O switches, their phase shifters are inactive in the cross state, leading to no power imbalance and hence no crosstalk. While in the bar state, due to the FCA-induced loss, a power imbalance of roughly 1.5 dB [23] arises, causing around $- 22 dB$ crosstalk. Push–pull driven switches have an approximately 0.6 dB power imbalance in both states [23], translating to about $- 22 dB$ of crosstalk.

B. Self-heating Effect

Soref’s equations also reveal that the induced phase change is always tied with a fixed amount of loss, leaving little room to engineer the E-O phase shifter to manipulate its phase change and FCA loss individually. To break this bound, we propose introducing the self-heating effect that offsets the phase change but leaves the FCA loss [30]. The heat $Q$ is caused by both Joule heating from the carrier currents ( $Q_{n, p}$ ) and by carrier recombination ( $Q_{R}$ ): $Q = Q_{n} + Q_{p} + Q_{R},$ (8) $Q_{n, p} = J_{n, p} \cdot E_{n, p},$ (9) $Q_{R} = q (E_{g} + 3 k T) R,$ (10)where $J_{n, p}$ is the current density, $E_{n, p}$ is the electric field, $q$ is the electron charge, $E_{g}$ is the bandgap energy, $k$ is the Boltzmann constant, $T$ is the temperature, and $R$ is the net recombination rate [31].

We particularly look at phase shifters with two doping structures, as illustrated in Fig. 2. The first one only contains heavily doped sectors ( $P +$ and $N +$ ) that are separated by an intrinsic region, while the second has additional lightly doped sectors ( $P$ and $N$ ) that are respectively sandwiched by the heavily doped sectors and the intrinsic region. In both cases, the width of the intrinsic region is set at 2 μm to offer lower driving voltage, while maintaining the separation of the optical mode from doping regions. Concurrently, a 600-nm-wide ridge waveguide is employed to ensure a single mode and its maximal overlap with injected carriers. The waveguide and slab are 220 nm and 90 nm in height, respectively. The doping levels for the lightly and heavily doped regions are $1 \times 10^{18} {cm}^{- 3}$ and $1 \times 10^{20} {cm}^{- 3}$ , correspondingly.

Figure 2.Cross-section schematic of the differential E-O phase shifter pair.

A coupled heat and charge transport solver in Lumerical [32] is set up to model the heat propagation and carrier distribution within the phase shifter at different bias voltages. The solver self-consistently solves the drift-diffusion equations with Poisson’s equation and the heat transport equation using the finite-element method (FEM). The generated profiles are subsequently imported into a finite-difference eigenmode (FDE) solver to calculate the resultant effective refractive index change $Δ n_{eff}$ and FCA loss. Finally, $Δ n_{eff}$ is translated into phase shift $Δ ϕ$ using the following equation: $Δ ϕ = \frac{2 π}{λ} Δ n_{eff} Γ L,$ (11)where $λ$ is the wavelength of the signal, $Γ$ is the confinement factor, and $L$ is the length of the phase shifter.

Figure 3(a) shows that self-heating is more prominent in shorter phase shifters, as a larger increase in carrier concentration is needed for the same amount of phase change, generating more heat per unit volume, thus resulting in a higher temperature. The rise of temperature increases the refractive index with no additional loss but counteracting the phase change. The waveguide doping profile also impacts self-heating because the carrier recombination rates are higher in lightly doped regions, leading to a higher temperature rise. Figures 3(b) and 3(c) illustrate the overall phase shift as a function of bias voltage, with the phase change broken down into that due to the self-heating effect (red shift) and the FCD effect (blue shift) in a 50 μm and a 1000 μm E-O phase shifter, as an example. In the shorter phase shifter, the overall effect first blue shifts the phase since the self-heating effect elevates the refractive index gently in the beginning; however, its impact grows more sharply as the voltage increases, and quickly offsets the FCD effect. The phase, therefore, will then get red shifted. Conversely, the longer phase shifter experiences minimal self-heating, resulting in consistent blue shift. Figures 3(d)–3(f) show the impact of doping concentrations on the self-heating effect. In lightly doped regions, a higher doping density results in more significant self-heating due to increased majority carriers and decreased minority carriers, raising recombination rates. In heavily doped regions, increased doping, however, reduces overall recombination rates by promoting electron-hole pair separation with a stronger built-in electric field. It is worth noting that in long E-O phase shifters (generally over 500 μm length), variations in doping concentration can hardly impact the relationship between the insertion loss and phase change, since the self-heating effect is insignificant.

Figure 3.(a) Insertion loss for the E-O phase shifter with varying lightly doped region widths (0–2 μm) plotted against absolute phase shift for different device lengths. (b) and (c) Insertion loss and phase shift for 50-μm- and 1000-μm-long E-O phase shifters against bias voltage. Both phase shifters share a common lightly doped region width of 2 µm. (d) and (e) Insertion loss for the E-O phase shifter with a 50 μm length and 2 μm lightly doped region width, plotted against absolute phase shift for different doping concentrations. (f) Insertion loss for the E-O phase shifter with a 1000 μm length and no lightly doped region, plotted against phase shift for different doping concentrations. Note that the dark curves in (d)–(f) represent the projection of the original 3D curves onto the loss–bias plane.

C. Differential Phase Shifter Pair

Drawing on these results, a differential E-O phase shifter pair is designed to be implemented in an MZI cell, consisting of a 500 μm phase shifter with heavily doped regions only (PS1) and a 50 μm phase shifter with additional 2-μm-wide lightly doped regions (PS2). The relatively large length of the first phase shifter ensures minimal impact from the self-heating effect without significantly increasing the device footprint. It exploits the FCD effect only and obeys Soref’s equations presented in Eqs. (6) and (7), providing fast phase shifting with FCA-induced loss. By contrast, the second phase shifter experiences both FCD and self-heating effects that offer a nearly counterbalanced phase change but also FCA-induced loss. Its length could be further shortened to enhance the self-heating effect, thereby increasing the phase difference between the two phase shifters [as shown in Figs. 4(a) and 4(b)]. However, such a modification would require a higher current density (from $9 \times 10^{8}$ to $15 \times 10^{8} A / m^{2}$ when the length is reduced from 50 to 30 μm) within the PIN junction. Such a phase shifter pair thus can operate differentially to simultaneously achieve a balanced loss and an arbitrary overall phase difference to trigger switching, with minimal crosstalk. The insertion loss versus phase shift is plotted for both longer and shorter phase shifters in Fig. 4(a), and their overall phase difference versus loss is plotted in Fig. 4(b). At $π$ phase difference in the bar state, the longer and shorter phase shifters are biased at 0.89 V and 1.25 V, respectively, resulting in an overall insertion loss of 2 dB. The estimated power consumption is about 18 mW. Figure 4(c) details the I-V characteristics for both phase shifters individually. Shortening PS2 can lead to a lower insertion loss [as shown by Fig. 4(b)], i.e., $< 2 dB$ , with an increased current density but may require further investigation.

Figure 4.(a) Insertion loss of E-O phase shifters plotted against the provided phase shift, with the yellow dashed line representing PS2 at 30 μm length. (b) Insertion loss of the differential E-O phase shifter pair plotted against the provided phase shift, featuring a red dashed line for PS2 at the 30 μm length. (c) Current in the two E-O phase shifters plotted against the applied bias voltage.

The proposed design can also compensate for potential phase errors as it is capable of providing arbitrary phase shifts, as stated above. Consequently, additional heaters are not needed for phase corrections, reducing the control complexity.

D. Switching Speed Enhancement

In the proposed switch cell, the self-heating effect plays a crucial role in determining the switching speed, especially in the shorter phase shifter (PS2). To estimate its switching time, a transient analysis is performed. Initially, the steady-state current density and recombination rate of the phase shifter with the cell in the bar state are determined using a charge transport solver. These parameters are subsequently translated into heat generation using Eqs. (8)–(10). To simulate the switching process, the generated heat is treated as a source, and its activation and deactivation are controlled by a step signal within the heat transport solver operating in a transient mode. This allows for the extraction of temperature variations in the waveguide core over time, which in turn enables the determination of the switching time. A 5 ns rise time is incorporated into the step signal, following a logarithmic profile to represent the exponential change in carrier concentration during actuation. Throughout the simulation, the ambient temperature around the simulation region (i.e., the silicon substrate) is maintained at 300 K. Initial results show that both the rise and fall times triggered by the self-heating effect, $τ_{rise}$ and $τ_{fall}$ , are approximately 12 μs.

In addition to transient simulation, the temperature change with time can also be approximated by the following equation: $Δ T (t) = Δ T_{steady} (1 - e^{- \frac{t}{τ}}),$ (12)where $Δ T_{steady}$ is the steady-state temperature change, and $τ$ is the time constant. The two terms can be expressed as $Δ T_{steady} = \frac{P}{GA},$ (13)and $τ = \frac{H}{GA},$ (14)where $P$ is the applied power, $G$ is the thermal contact conductance between the heated waveguide and the heat sink, $A$ is the area traversed by the heat flow, and $H$ is the heat capacity of the heated arm [33].

Therefore, the switching speed of the device can be greatly enhanced either by reducing the time constant or by increasing the applied power. An approach to lower the time constant is to decrease the heat capacity, which can be achieved by positioning the $p − i$ and $n − i$ junctions closer to the waveguide, given that they are the primary sources of heat generation. Nonetheless, this approach also presents a trade-off, as it may lead to increased insertion loss due to the overlap of the optical mode and doped regions.

On the other hand, the rate of temperature change can be enhanced by increasing the bias voltage. A pulse excitation technique [34] can be implemented to effectively decrease $τ_{rise}$ . This technique utilizes an excitation signal comprising a high-energy pulse to overdrive the phase shifter, thereby accelerating the temperature rise. Simulation results depicted in Fig. 5(a) confirm that $τ_{rise}$ of less than 20 ns can be achieved when the bias voltage exceeds 3.5 V. We thus implement an excitation pulse with a voltage of 3.5 V and a duration of 20 ns to the PS2, as shown in Fig. 5(b). Heat generation occurs throughout the entire slab, with the $p − i$ and $n − i$ junctions contributing the most heat due to Shockley–Read–Hall (SRH) recombination. Because of the relatively high thermal conductivity of silicon compared to that of silicon oxide, this heat rapidly elevates the temperature of both the silicon waveguide and slab before dispersing into the surrounding oxide claddings. The majority of the heat diffuses in the upward direction, as other directions are blocked by the presence of the electrodes as well as the silicon substrate, both of which serve as heat sinks. The heat propagation process is visualized in Fig. 5(c). It can be seen that the temperature at the waveguide core rises rapidly, reaching 90% of its steady-state value within 20 ns. It then stays almost unchanged, but the generated heat gradually diffuses into its surroundings leading to a temperate increase in this area. Notably, the heat distribution is slightly asymmetric, and this is attributed to a higher recombination rate in the p-doped region [35]. The analytical model proposed in Ref. [34] can be used to derive an optimal excitation pulse to avoid overshoot and thus eliminate the need for a feedback loop.

Figure 5.(a) $τ_{rise}$ for PS2 under pulse excitation technique with varying overdrive voltages. Inset shows a close-up of $τ_{rise}$ for bias voltage over 3 V. (b) Control scheme for reducing $τ_{rise}$ : bias voltage applied to PS2 (top) and corresponding waveguide core temperature change (bottom); shadowed region indicates 90%–100% of steady-state temperature. (c) Temperature distribution for PS2 at 20 ns (left) and 20 μs (right) after an excitation pulse followed by a step signal. Insets illustrate waveguide core temperature. (d) Control scheme for reducing $τ_{fall}$ : bias voltage applied to PS2 and PS1 (top), corresponding waveguide core temperature change (middle), and resulting phase shift (bottom); shadowed region indicates 0%–10% of steady-state phase difference between the two phase shifters.

Additionally, a differential control scheme [26] can be adopted to reduce $τ_{fall}$ , since it is possible to heat PS1 to swiftly decrease the phase difference between the two arms in an MZI cell, as illustrated by Fig. 5(d). The phase shift is obtained by converting the temperature change into the change of refractive index $Δ n_{eff} = \frac{Δ n}{Δ T} \cdot Δ T$ , where $\frac{Δ n}{Δ T}$ is $1.84 \times 10^{- 4} K^{- 1}$ [28]. Both phase shifters thus cool down at similar rates, maintaining minimal phase difference. In this way, $τ_{fall}$ is diminished to approximately $τ_{rise}$ of PS1, which is $< 20 ns$ .

Therefore, the switching speed of the proposed device can be effectively engineered to the nanosecond regime and is ultimately bounded by the maximum forward surge current of the PIN junction. The proposed device draws about 70 mA current with a 3.5 V bias, corresponding to a current density of $\sim 6.4 \times 10^{9} A / m^{2}$ . This is comparable to the device reported in Ref. [26], showing its viability.

E. Splitting-ratio Correction

In practice, the power splitting ratio of the 3-dB couplers in an MZI cell is likely non-ideal due to manufacturing imperfections. According to Eq. (1), a 1% deviation from the perfect 50% coupling ratio of couplers can degrade the crosstalk ratio to approximately $- 35 dB$ . This value deteriorates to $- 28 dB$ and $- 20 dB$ when the deviation increases to 2% and 5%, respectively. A CTDC is proposed to correct any manufacturing imperfections in this work, with the additional advantages of high bandwidth and large fabrication tolerance compared to conventional directional couplers (DCs) [36]. Their asymmetric nature is also favoured for effective splitting-ratio-tuning at low heating powers [37].

Figure 6(a) shows the schematic of the CTDC, where $w$ is the waveguide width, $g$ is the waveguide separation in the coupling region, and $R$ and $α$ are the bending radius and bending angle of the coupling region, respectively. The bottom waveguide is bent at a larger angle $β$ to increase its separation from the top waveguide. Compensations are made at both ends to make them align horizontally. Two TiN heaters (H1 and H2) are positioned 1 µm above the silicon layer and at a lateral distance $d$ from the centre of the two waveguides, respectively, enabling us to manipulate the temperature gradient between the coupled waveguides and thus the tuning of the splitting ratio [37]. We select TiN as the material of the heater, given its widespread accessibility in foundries. One could employ alternate metals or alloys instead.

Figure 6.(a) Schematic of the CTDC. (b) Wavelength response for the CTDC under different width variation values. (d) Cross-coupling ratio at 1.55 μm versus the power dissipated for the CTDC.

Here $g$ is set to 300 nm ensuring sufficient coupling length for thermal tuning, and $β$ is set to twice of $α$ . The rest parameters are optimized by employing the particle swarm optimization (PSO) method [38] to primarily maximize the device bandwidth, using a finite-difference time-domain (FDTD) solver. The optimal parameters obtained are $w = 385 nm$ , $R = 60$ , and $α = 9.5 °$ . A thermal simulation is then performed to determine the lateral distance $d$ at 2 µm for a maximum temperature gradient. Finally, the obtained temperature distribution is imported into the FDTD solver to investigate the splitting ratio of the coupler at different heating powers.

The impact of fabrication variation on the curved DC is depicted in Fig. 6(b), accounting for a standard $\pm 10 nm$ process variation in waveguide width, which we regard as the most significant impact due to the fabrication imperfections [39]. We assume that a deviation $Δ w$ in the waveguide width corresponds to an inverse gap deviation $- Δ w$ in the DC, thereby preserving the distance between the centres of the waveguides. In the absence of fabrication variation, the device exhibits a nominal $50 % \pm 2 %$ cross-coupling ratio between 1.525 and 1.575 μm. When variations are present, the centre wavelength of the coupling-ratio curve shifts slightly from 1.55 μm, resulting in a $\sim 2 %$ deviation in the coupling ratio at this wavelength. Our model may underestimate actual deviations, as it disregards other fabrication variations, including alternations in waveguide thickness or sidewall angle. However, it shows a clear path towards after-fabrication correction on the MZIs for performance enhancement. Figure 6(c) displays the cross-coupling ratio both at 1.55 μm and across a 60 nm wavelength span as a function of bias power. A tuning range of 40% to 60% is observed when the power remains below 20 mW. This $\pm 10 %$ correction range is sufficient to address any fabrication variations. Further analysis on fabrication tolerance at the cell-level is provided in the following section.

F. Performance Evaluation

A transfer matrix analysis is performed to evaluate the performance of the proposed MZI cell. The matrix is obtained by substituting the rigorously simulated results for both the E-O phase shifter pair and the curved DC into Eq. (1) at different wavelengths.

The transfer matrix of the curved DC is expressed as $C_{curved} = [\begin{matrix} t e^{- j (\frac{π}{2} + Δ φ^{'})} & k e^{- j \frac{π}{2}} \\ k e^{- j \frac{π}{2}} & t e^{j (\frac{π}{2} + Δ φ^{'})} \end{matrix}],$ (15)to account for wavelength-dependent phase difference $Δ φ^{'}$ between the two output ports due to its asymmetric structure. Such phase difference limits the bandwidth of the device but can be eliminated by utilizing a point symmetry configuration [40] to balance the two optical arms. We evaluate its performance with three cases: two passive DCs, one passive DC and one active DC, and two active DCs, with the splitting ratio of the active DCs being optimized in each scenario, to investigate the trade-off between performance and control complexity. Figure 7 shows transmission spectra for the three MZI configurations including cases of both ideal fabrication and the extreme width variations ( $\pm 10 nm$ ). All exhibit an insertion loss of $< 0.1 dB$ in the cross state and $< 2 dB$ in the bar state, owing to the FCA loss associated with the $π$ phase shift.

Figure 7.(a)–(c) Schematics of the three configurations. (d)–(f) Transmission spectra for each configuration in the cross and bar states under different width variations.

We further perform Monte Carlo (MC) simulation that incorporates fabrication variations in curved DCs. In each trial, we randomly pick waveguide widths for the pair of DCs in the MZI from a uniform distribution ranging between 375 and 395 nm, representing a $\pm 10 nm$ variation. The transmission spectra of these DCs are computed through FDTD simulation, which are subsequently substituted into Eq. (1) for calculating the MZI’s transmission spectrum, allowing us to retrieve its crosstalk ratio and operation bandwidth. The applied power to the active DCs is optimized to achieve a 50:50 splitting ratio at the centre wavelength. Figure 8 details the distribution of crosstalk ratio at 1.55 μm and the distribution of operation bandwidths at crosstalk of $- 30 dB$ for 400 MC trials. The results indicate that the proposed MZI cell with even two passive DCs can achieve a crosstalk ratio below $- 30 dB$ with a bandwidth over 20 nm in the cross state as the worst-case and extending over 30 nm in the bar state. Incorporating one and two active DCs respectively suppresses the worst-case crosstalk at the centre wavelength in both states to below $- 35 dB$ and $- 50 dB$ , respectively. The two-CTDC design can always correct fabrication errors achieving a crosstalk ratio below $- 40 dB$ throughout a wavelength range of 10 nm in both states as the worst case. The narrow-down of operational bandwidth stays as the remaining impact of imperfect coupling coefficients. This can be addressed by further improving the intrinsic bandwidth of DCs, such as the asymmetric curved directional coupler that achieves a wavelength bandwidth of 100 nm [41].

Figure 8.Distribution of crosstalk ratio at 1.55 μm (left) and bandwidth at crosstalk of $- 30 dB$ (right) for the three configurations in cross (top) and bar (bottom) states across 400 trials assuming uniformly distributed fabrication variations. Note that the crosstalk ratio for the two-CTDC case is not visible on the left due to its complete suppression at 1.55 μm.

4. TOPOLOGY EXPLORATION

The choice of switch topology considerably influences its circuit-level performance [11]. A key consideration is to make the proposed MZI cell best complement the switch topology. As a binary butterfly network derivative, Beneš optimizes the number of switch cells necessary for a non-blocking $N \times N$ network. Each MZI cell concurrently transverses two signals, making the ultralow crosstalk design valuable. We do not include Beneš in this article as its path diversity could get exacerbated by the unequal loss states in the cross and bar of the differential cell. On the contrary, switch-and-select and dilated Banyan offer a dedicated pathway for each input–output pair, while ensuring each cell carries one signal that cancels first-order crosstalk. Both scale poorly in overall switch count and do not fall within the scope of this work. We thus focus on path-independent loss (PILOSS) [42] and double-layer network (DLN) [43] architectures that are not fully immune to first-order crosstalk but both scale nicely in waveguide crossing count. PILOSS network only sets one switch cell in the bar state in any path, while only the middle stage in DLN that suffers first-order crosstalk employs the proposed ultralow crosstalk MZI cell.

A. Methodology

To assess the performance of the switch fabrics, we follow the methodology described in Ref. [44]. This approach models each switch stage and the interconnecting shuffle networks using transfer matrices, enabling the entire switch fabric to be represented as a multiplication product of these matrices. Furthermore, we incorporate the insertion loss of the shuffle waveguides into the model, as this factor cannot be neglected for large-scale switches. The transmission amplitude from input port $i$ to output port $j$ , $T_{i j}$ , is subsequently computed as $T_{i j} = {(M_{t}^{i j} \cdot e_{i})}_{j},$ (16)and the crosstalk amplitude $X_{i j}$ , $X_{i j} = {(M_{t}^{i j} \cdot (1 - e_{i}))}_{j},$ (17)where $M_{t}^{i j}$ is the transfer matrix of the whole switch fabric, setting in a configuration that connects input port $i$ to output port $j$ , $1$ is an all-ones column vector, and $e_{i}$ is a column vector with component $i$ equal to one and the rest equal to 0.

We perform a statistical performance evaluation of each switch fabric by randomly selecting $10^{6}$ optical paths and computing the aggregated insertion loss and crosstalk penalty for each. In this process, the switch is programmed to be configured in a valid state, guaranteeing that each input port is connected to a distinct output port, and the aggressor input signals $(1 - e_{i})$ are considered coherent and in phase for calculating the worst-case crosstalk. The insertion loss and crosstalk for each component utilized in the simulations are detailed in Table 2. Some are extracted from simulations, assuming a bending radius of 10 μm for the 90° bend and a transition length of 10 μm between strip and slab waveguides for phase shifters.Table 2.

Loss and Crosstalk for Key Building Blocks

Component	Loss (dB)	Crosstalk
Si waveguide	1.5 per cm	N.A.
90° bend	0.014^a	N.A.
Waveguide crossing	0.05 [22]	–40 [45]
Transition waveguide	0.02^a	N.A.
Curved DC	0.01^a	N.A.
Regular E-O phase shifter	0.7^a	N.A.
Differential E-O phase shifter pair	0 (cross)/2 (bar)^a	N.A.
Edge coupler to fiber	1.5 [46]	N.A.

Simulation values.

B. Results and Discussion

The overall power penalty histograms for PILOSS and DLN switches, with scales of $4 \times 4$ , $8 \times 8$ , and $16 \times 16$ , are presented in Figs. 9(a). Its breakdowns of loss- and crosstalk-induced penalties are shown in Figs. 9(b) and 9(c), respectively. The aggregated crosstalk $ϵ$ is converted into power penalty $δ$ as [47] $δ = - 10 \log (1 - 2 \sqrt{ϵ}) .$ (18)

PILOSS switches exhibit a highly uniform distribution of insertion loss as each path includes exactly $N - 1$ MZI cells in cross but one in bar, and exactly $N - 1$ waveguide crossings for an $N \times N$ port count. First-order crosstalk can occur in up to $N - 2$ stages in the worst case, but the use of proposed differential cells performs nicely in suppressing the crosstalk leakage, while having moderate loss. DLN switches display increased loss diversity due to the differentiation of loss in the cross and bar states. Again, the single stage of first-order crosstalk in the centre of DLN is largely suppressed, with the benefit of reducing the switch hop count and the waveguide crossings per path. It can be seen that, with the use of proposed MZI cells, the switch crosstalk is well managed, and the induced penalty is marginal compared with that of the insertion loss. However, as the port count continues to scale up, the accumulated crosstalk from both switch cells and waveguide crossings is becoming likely a major source of degradation. When the port count gets to 16 and above, the total incurred penalty could well make the switch incompatible with current short-reach unamplified link specifications. Therefore, optical gain is needed and SOAs can be integrated via either flip-chip bonding [13,14] or micro-transfer printing [15].

Figure 9.(a) Overall power penalty histograms for PILOSS (top) and DLN (bottom) switches at different scales, featuring breakdowns of (b) insertion loss and (c) crosstalk-induced power penalty.

5. CONCLUSION

In conclusion, this paper introduces an innovative approach to mitigate FCA-induced crosstalk in E-O MZI switch cells by leveraging the self-heating effect. The design features a pair of differential E-O phase shifters, with one exploiting only the FCD effect for fast phase tuning, while the other with both FCD and self-heating effects facilitating a nearly counterbalanced phase change. This scheme delivers an arbitrary differential phase shift with strictly balanced FCA-loss, minimizing crosstalk. This method lowers design complexity with no need for extra phase corrections. By introducing the pulse excitation technique and differential control method, nanosecond-scale switching can be achieved. With the aid of CTDC, fabrication errors can be significantly tolerated, achieving a crosstalk ratio below $- 40 dB$ . This design proves to be particularly beneficial to complement both PILOSS and DLN switches. Furthermore, the design can bring about a breakthrough in the scalability of E-O switch fabrics, with great potential for high-performance switching applications in data centres.

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