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Abstract
Reconfigurable photonic devices integrated with silicon waveguides are important building blocks for future on-chip photonic circuits. In this paper, we focus on the mode order conversion in silicon waveguides with non-volatile reconfigurable capability. Deformed phase change material Sb2Se3 (antimony triselenide) stripes are introduced at the edges of the functional region to provide the refractive index difference required by mode conversions. The shapes of stripes are inversely designed by a gradient-based iterative optimization strategy with 57 (19) iterations for TE0-to-TE1 (TE0-to-TE2) mode converter. The footprint of the functional region is as compact as square center wavelength. In the crystalline phase, TE0-to-TE1 and TE0-to-TE2 mode conversions are realized with conversion efficiencies of 98.5% and 96.3% at a center wavelength of 1550 nm, respectively. While in the amorphous phase, the input TE0 mode directly passes through the functional region with efficiencies of 93.0% and 92.4%, respectively. The output mode can be reconfigured by changing the phase of Sb2Se3 stripes. Moreover, after introducing ±10 nm geometrical deviations to the perfect Sb2Se3 stripe design, corresponding red and blue shifts of conversion efficiency spectra can be observed, and the simulation results reflect the reasonable robustness of the proposed mode converters.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
All-optical signal processing is of great importance to ultra-high-speed optical communication networks for their fast response time, high communication capacity, flexibility, reliability, and low power consumption [1–5]. Mode converters are devices that can change the spatial distribution of optical waves propagating in waveguides. They enable modulation [6,7], wavelength-division multiplexing (WDM) [8–13], mode-division multiplexing (MDM) [14–17], and mode-locked lasers [18,19], which are essential for all-optical signal processing and integrated photonic systems [20,21]. Additionally, mode converters play a crucial role in integrated photonic systems due to their ability to manipulate optical signals [22–24]. However, there are still significant challenges in the development of mode converters, including issues such as excessive size and high transmission loss. To address these challenges, many designing methods for mode converters have been proposed, including genetic algorithms [25,26], particle swarm optimization [27,28], gradient-based binary searches [29,30], deep learning methods [31–33], and adjoint methods [34–36].
In recent years, a variety of mode converters have been proposed to further reduce footprint and insertion loss, and develop new functionalities. Jia et al. [37] proposed a multichannel parallel mode converter based on an inverse design optimization method with a footprint of 4 × 3 µm2 to achieve TE0-to-TE2 mode and TE1-to-TE3 mode with insertion losses of 0.8 to 1.0 dB and 0.9 to 1.2 dB in the wavelength range of 1525 to 1565 nm. Ye et al. [38] proposed ultra-compact integrated photonic devices based on mode manipulation. Using a digital metamaterial composed of silicon and air pixels, they achieved six high-quality and efficient transitions between four TE polarization modes in a functional region of only 1.0 × 1.55 µm2, with a topology-optimized design. In 2021, Chen et al. [39] proposed an ultra-compact broadband inline mode converter based on quasi-TE00 and quasi-TE10 silicon on an insulator. The mode conversion region consisted of a continuously width-modulated waveguide with a small footprint of 1.32 × 4.52 µm2. The design was carried out by a particle swarm optimization algorithm and achieved an analog conversion efficiency of about -0.174 dB and an insertion loss of less than 0.153 dB over a bandwidth of 100 nm. In 2023, Dou et al. [40] used an intelligent inverse design algorithm combined with an adaptive genetic algorithm (AGA) with finite element simulation to design a set of arbitrary order mode converters with low excess losses (ELs) and low crosstalk (CT). The designed TE0-n (n = 1,2,3,4) and TE2-n (n = 0,1,3,4) mode converters had a footprint of only 1.8 × 2.2 µm2 at the communication wavelength of 1550 nm. The maximum and minimum conversion efficiencies were 94.5% and 64.2%, respectively. Recent studies have highlighted the potential of mode converters with intelligent algorithm design to enhance all-optical processing integration and the development of efficient integrated photonics systems. To this end, we establish a shape description scheme for the internal part of the functional region for iterative optimization algorithms of gradients for intelligent inverse design of mode converters. To further develop the potential of photonic integrated circuits (PICs) in applications of all-optic routing, neuromorphic computing, or to realize highly reconfigurable and general-purpose platforms, programmable/reconfigurable optical devices are indispensable [41,42].
Adjustable optical properties are needed to achieve full manipulation of features of photons, i.e., to realize reconfigurable/reprogrammable photonic devices. Phase-change materials (PCMs) are those materials with optical properties that can be changed upon applying an external stimulus (optical, electrical, or thermal) [43]. PCMs are promising for a wide range of applications in tunable optics because they enable non-volatile, low-power, high-performance and highly integrated optical functions [44]. These materials have been used in various situations. Most commonly used PCMs include ITO [45], VO2 [46], GST [47], GSST [48,49], Sb2Se3 [50,51], Sb2S3 [52–54], etc. Among the mentioned materials, Sb2S3 and Sb2Se3 are proven to have relatively small absorption in the communication c-waveband with a decent refractive index difference relative to Si [55]. It is also proven to be relatively durable, which is able to withstand 4000 phase-change cycles without noticeable performance degradation [51]. The application of PCMs in reconfigurable optical devices is becoming more popular in recent years. Chen et al. put forward designs of chip-scale nanophotonic switches using ITO and VO2, respectively [45,56]. In 2017 Shi et al. proposed a reconfigurable optical power splitter using Sb2Se3-based digital metamaterials, whose power splitting ratio can be 9:1, 8:2, 7:3, 6:4 in “on” state, while remaining 5:5 in “off” state [57]. In 2020, Wu et al. proposed a TE0-to-TE1 mode converter with Ge2Sb2Te5-based phase gradient metasurface integrated on a waveguide [58]. In the same year, Chen et al. proposed a Tunable TE0-to-TE1 converter using GST, with a footprint of 22 × 3.5 µm2 [59]. Additionally, Chen et al. demonstrated a compact switchable mode converter using GeSe with a footprint of 0.9 × 5 µm2 [60]. In 2022, Abdollahramezani et al. demonstrated an optical modulator using GST, with multiple intermediate states and a large tuning range [47]. In 2022, Fei et al. illustrated a reconfigurable silicon waveguide mode converter using Sb2Se3, with a footprint of 2.3 × 1.1 µm2 [61]. As a result of introducing digital metasurface into designing reconfigurable optical devices, functional areas are getting increasingly complicated, while physical footprint of these areas is getting smaller. To achieve a better optical performance while reducing design cost, we plan to introduce sensible understandings into our design.
In this paper, we demonstrate reconfigurable TE0-to-TE1 and TE0-to-TE2 mode converters based on Sb2Se3 phase change materials for waveguide integration. The Sb2Se3 stripes in the functional region are inversely designed by combining the finite element method (FEM) and the method of moving asymptote (MMA), which allows high conversion efficiency shapes to be obtained in less than 60 optimization iterations. The footprint of TE0-to-TE1 and TE0-to-TE2 designs is 2 × 1 µm2. The robustness of the designs is verified by 3D simulation involving deviations of ±10 nm.
2. Methodology
To achieve the desired configurability, we employ an innovative approach to waveguide optimization making use of inverse design techniques. The optimized design schematics of the TE0-to-TE1 and TE0-to-TE2 mode converters are represented in Fig. 1(a) and Fig. 1(b), respectively. We choose Sb2Se3 as the phase change material for waveguides because of its advantages over conventional phase change materials in terms of high optical tolerance, stability, and reparability [61,62]. In addition, Sb2Se3 has a high refractive index in both the crystalline and amorphous phases, which provides flexibility and versatility for the integration of waveguides, allowing the Sb2Se3 material to enable the proposed reconfigurable mode converter to operate with low loss and low power consumption. Specifically, Sb2Se3 in the crystalline phase has a refractive index (4.05 + 0i) significantly different from air, which ensures a dielectric contrast for scattering and interference and facilitating TE0-to-TE1 or TE0-to-TE2 mode conversion. Sb2Se3 in the amorphous phase has a refractive index close to silicon (3.285 + 0i), enabling direct transmission of TE0, TE1 and TE2 modes through the functional region [55]. Both TE0-to-TE1 and TE0-to-TE2 mode converters have a functional region with a width of 1 µm and a length of 2 µm. The functional region consists of silicon and crystalline Sb2Se3 strips, with SiO2 as a substrate below the waveguide. To achieve beam splitting and phase separation, the TE0-to-TE1 converter has one Sb2Se3 strip on one side of the functional region, while the TE0-to-TE2 mode converter has two Sb2Se3 strips on both sides. The boundary of the Sb2Se3 strip coincides with the boundary of the adjacent silicon waveguide. In experiments, the combination of etching and thermal evaporation techniques is necessary for fabricating a reconfigurable mode converter [63,64]. Waveguides are fabricated by first etching silicon substrates to create parallel waveguide structures and then etching the optimized shapes of waveguides into functional regions. A thin film of Sb2Se3 is deposited on the substrate by thermal evaporation and patterned into the desired shape. To achieve phase change of the PCM, we refer to different methods proposed in current researches. One method is to use a laser pulse to switch the PCM between the amorphous and crystalline phases [51]. Another method is to use a bridge-type heater structure, which realizes partial crystallization of the PCM by the thermal distribution difference of different bridges [65]. The third method is to use a microheater based on graphene and alumina (Al2O3) layers to achieve reversible optical switching of the PCM [61]. These methods can achieve fast and low-power phase change control, and are comparable or larger than the size of our device.
Fig. 1. Three-dimensional schematic diagram of the proposed reconfigurable silicon waveguide (a) TE0-to-TE1 mode converter. (b) TE0-to-TE2 mode converter.
We employ a two-dimensional finite element method (FEM) simulation implemented in COMSOL Multiphysics to evaluate the performance of the reconfigurable mode converter designed by our inverse design approach. To ensure stable simulation results, a grid size convergence test is adopted. Additionally, 3 µm perfectly matched layers (PML) are used on all boundaries to prevent reflection. As is known to all, the working range of optical devices, such as waveguides, is dependent on the thickness of the waveguide [66,67]. However, since the 2D simulation only deals with a planar surface, it cannot directly model the finite thickness of the waveguide. To overcome this limitation, an equivalent refractive index is introduced in the simulation, which takes into account the waveguide thickness and allows for an accurate representation of the waveguide behavior in a 2D simulation. We assign equivalent refractive indices of 3.0, 3.62, and 2.81 to silicon, crystalline Sb2Se3, and amorphous Sb2Se3, respectively, for a 340 nm-thick device [57]. Similarly, for a 220 nm-thick device, we assign equivalent refractive indices of 2.7, 3.32, and 2.71 to silicon, crystalline Sb2Se3, and amorphous Sb2Se3, respectively.
We utilize the gradient-based MMA algorithm to optimize the shape of the deformed Sb2Se3 strips in the functional region [66]. In the initial design, one and two deformed Sb2Se3 stripes of size 0.3 × 2 µm2 are used in the functional region to implement TE0-to-TE1 and TE0-to-TE2 mode conversions, respectively. The MMA algorithm can find the optimal shape by iteratively minimizing the objective function, subject to the constraints imposed by the fixed points and maximum deformation. We define the objective function ${F_{MC}}\left( \lambda \right) = {\sum _i}{w_i}{\smallint _{{A_i}}}{P_x}\left( \lambda \right)ds$ by integrating the forward power flux near the output port. For each region, the integral quantile function is the weight of the integral is given by ${w_i}$, while the integral region is labeled by ${A_i}$ [38]. Besides, we set the maximum deformation allowance to 150 nm for the inner boundary, and 200 nm for the outer boundary, respectively. To avoid a noticeable deviation in the shape of the Sb2Se3 strip, fixed points are set at the vertices of the Sb2Se3 strips near the boundary. The boundary deformations of the Sb2Se3 strip is described by Bernstein polynomials, which is a type of mathematical function that can describe smooth curves: ${\Delta } = {\sum _{\textrm{d} = 1,2}}\left[ {{{\mathop {\textrm e}\limits^{\hat{} } }_{\textrm d}}\sum _{{\textrm j} = 1}^{\textrm{n} - 1}{{\zeta }_{\textrm{d},\textrm{j}}}{{\textrm s}^{\textrm j}}{{(1 - {\textrm s})}^{{\textrm n} - {\textrm j}}}} \right]$, where Δ is the shape function, d denotes the dimensionality of the model, ed is the unit vector in the d-direction, s serves as the boundary parameter, $\zeta_{d,j}$ represents the Bernstein coefficients, the values of n represent the order of Bernstein polynomial(the higher the order, the more complex the shape). A three-dimensional Finite Difference Time Domain (3D FDTD) is applied to simulate a 3D SOI-based (Silicon-on-insulator) model and to calculate the conversion efficiency, transmission efficiency and optical field distribution of the inversely-designed reconfigurable mode converter. The transmission efficiency is evaluated by T = Pout/Pin while the conversion efficiency is CE = PTEi/Pin. PTEi shows the output power for the ith TE mode, which is obtained by FDTD mode decomposition. Pout represents the total output power. And Pin is the input power. The mode purity on the output port is calculated by CE/ T.
3. Results and discussion
Our approach aims to establish a shape description scheme that can facilitate the development of reconfigurable mode converters. We conduct a shape optimization on the Sb2Se3 stripes for mode conversion. Since the thickness of silicon waveguide is set to 340 nm, therefore equivalent refractive index of Si and Sb2Se3 is set to 3 and 3.62/2.81 (crystalline/amorphous), respectively. Figure 2 shows the objective functions of our optimization and snapshots of Sb2Se3 stripes and power flux distribution in the functional region at various optimization steps.
Fig. 2. Optimization process of the proposed design. Objective function convergence diagram, power flux and Sb2Se3 distribution in the optimization process for (a) TE0-to-TE1 and (b) TE0-to-TE2 mode converter. Waveguide thickness is 340 nm.
It takes 57 iterations steps for the TE0-to-TE1 converter to reach convergence, as shown in Fig. 2(a), the power flux bifurcates into two branches after Step1. Then objective function decreases dramatically in the first 15 steps, as the Sb2Se3 stripe deforms substantially. After 20 steps, change in objective function and deformation of Sb2Se3 stripe become subtle, then reach convergence in step 57. Similarly, in Fig. 2(b), the power flux splits into three branches after Step1. Objective function decreases significantly in the first 5 steps, then decreases slightly until it reaches convergence. Noticeably, two Sb2Se3 stripes in TE0-to-TE2 converter exhibit symmetry in every step, without us giving any constraint on symmetry. Through the introduction of MMA, our optimization efficiency is greatly improved.
In addition, we carry out a similar optimization search for a 220 nm thick silicon waveguide based on the previously obtained design. As shown in Figure S1, it takes 12 iterations and 24 iterations, respectively, for TE0-to-TE1 converter and TE0-to-TE2 converter. The obtained optimal design for two different silicon thicknesses is proven to be relatively similar. The functional regions of the mode converters have a length of only 2 µm, and the initial width of each Sb2Se3 stripe is set to 300 nm. Figure 3 shows the design shapes of the TE0-to-TE1 and TE0-to-TE2 converters with thicknesses of 340 nm and 220 nm for the silicon waveguide, respectively. The optimal designs for the two different silicon thicknesses are remarkably similar; however, the deformation of the optimized Sb2Se3 stripes is more significant.
Fig. 3. Shape of Sb2Se3 stripes from the functional region in (a) TE0-to-TE1 mode converter and (b) TE0-to-TE2 mode converter for 340 nm and 220 nm thick silicon waveguides, respectively.
Based on the above inverse designs, we conduct 3D FDTD simulations with SOI configuration to validate the functionality and performances. The thickness of the silicon waveguide is set to 340 nm according to common industry practice, its width is set to 1 µm. The simulated magnetic fields (Hz) at the center wavelength 1550 nm are illustrated in Fig. 4. We measure transmission efficiency (T) and conversion efficiency (CE) of both forward and backward transmission. When Sb2Se3 is in the crystalline phase, the electromagnetic wave going through the given functional region from the input waveguide is split into two/three separate beams of wave. As a result of the refractive index difference between Si and Sb2Se3 in the functional region, different beams going through the functional region gradually form a phase deviation of 180°. As a consequence, stable higher mode propagations are remained in the output waveguide. The results show that for TE0-to-TE1 converter, T reaches 99.1% for both forward and backward, while CE reaches 98.5% and 98.6%, respectively. For TE0-to-TE2 converter, T reaches 96.7% and 97.4% for forward and backward transmission, while CE reaches 96.3% and 96.7%, respectively. When Sb2Se3 is in the amorphous phase, the input light in TE0 mode is expected to pass through the functional region without any mode change. Result shows that T reaches 99.9% and 99.8% while CE reaches 93.0% and 92.4%, respectively, for both functional regions. Both mode converters have shown outstanding conversion efficiency and transmission efficiency when configured to conversion mode, i.e., Sb2Se3 is set to crystalline phase. Besides, our designs are proven reciprocal and follows time reversal symmetry, as T and CE show little difference between forward and backward conversion. It should be recognized that after the insertion of Sb2Se3, the transmission efficiency of the direct transmission TE0 mode is high but the conversion efficiency decreases. This is caused by the small difference in refractive index between Sb2Se3 and silicon. Moreover, we carry out the similar simulations at a thickness of 220 nm in the silicon waveguide. As shown in Figure S2, the results illustrate that both satisfactory conversion efficiency and transmission efficiency can be achieved in a silicon waveguide of 220 nm thickness as well.
Fig. 4. Magnetic field (Hz) distribution of (a) TE0-to-TE1 (b) TE0-to-TE2 converters at the central wavelength of 1550 nm. Simulated by 3D FDTD with SOI configuration. Waveguide thickness is 340 nm.
We conduct some frequency-domain analyses in 3D FDTD simulation in order to validate the operational frequency range of our converters. For a silicon waveguide with a thickness of 340 nm, Fig. 5(a) depicts the variation of transmission efficiency and conversion efficiency of the proposed TE0-to-TE1 mode converter at wavelengths between 1520 nm and 1580 nm. The transmission efficiency remains around 99% for the amorphous and crystalline phases, meanwhile, the conversion efficiency remains above 97% and 93%, respectively. For the TE0-to-TE2 converter, as shown in Fig. 5(b), the transmission efficiencies of the amorphous and crystalline phases remain above 99% and 94%, respectively, and the conversion efficiencies are higher than 92% in the wavelength range from 1520 nm to 1580 nm. It can be seen that the mode converter with the addition of the phase change material Sb2Se3 undergoes some degradation in the conversion efficiency in the amorphous phase, while the conversion efficiency and transmission efficiency remain good in the crystalline phase. Notably, both T and CE exhibit a linear relationship with changes in the input wavelength.
Fig. 5. Transmission efficiency and conversion efficiency of (a)TE0-to-TE1 and (b)TE0-to-TE2 mode converter, simulated by 3D FDTD with SOI configuration.
In general, for both mode converters demonstrate the potential to achieve a broad operating bandwidth, which is one of the desired functionalities for a general mode converter. As shown in Figure S3, we similarly perform a parametric scan of the silicon waveguide with 220 nm thickness. The T and CE of the TE0-to-TE1 mode converter remain above 94% for both modes. Although the transmission efficiency and conversion efficiency of the TE0-to-TE2 mode converter crystalline phase decrease significantly after 1550 nm, the performance is still acceptable. Additionally, we include SiO2 cladding in our 3D FDTD simulations to account for its possible effects on the mode conversion. Figure S4(a) shows the distribution of magnetic field (Hz) for both crystalline and amorphous phases with cladding. Figure S4(b) shows the conversion efficiency versus wavelength for the crystalline and amorphous phases with and without cladding. The results show that the SiO2 cladding has negligible impact on the conversion efficiency of mode converters.
In the actual manufacturing process, errors are inevitable, leading to discrepancies between the fabricated samples and the originally designed structure. Such discrepancies may result in degraded device performance. To validate the robustness and manufacturing tolerance of our designs, we introduce a geometrical deviation of expanding or shrinking 10 nm from the target shape and run FDTD simulations with the same settings as above. As shown in Fig. 6, the results show: when Sb2Se3 stripes are in the crystalline phase, the parabolic curves of CE red shift when Sb2Se3 stripes shrink, blue shift when Sb2Se3 stripes expand. If we focus on central wavelength 1550 nm, CE show slightly degradation. When Sb2Se3 stripes are in the amorphous phase, CE show decrease by a small margin when Sb2Se3 stripes shrink, increase when Sb2Se3 stripes expand, while remaining the linear relationship to input wavelength. This feature makes it suitable for practical application in optical communications and photonic integrated circuits.
Fig. 6. Conversion efficiencies of (a) TE0-to-TE1 and (b) TE0-to-TE2 mode converter introducing size deviation of expanding or shrinking 10 nm in 340 nm thick silicon waveguide. 3D FDTD simulations are carried out with SOI configuration.
Moreover, we examine the impact of two other imperfections on the proposed reconfigurable mode converters using 3D FDTD simulations. As shown in Figure S5, we introduce a height difference of up to 20 nm between the PCM stripes and the Si waveguide. The results show that the transmission and conversion efficiencies of the mode converters remain high and stable at different heights. As shown in Figure S6, we simulate the impact of incomplete crystallization of Sb2Se3 by varying the refractive index difference (Δn) between the crystalline and amorphous phases. We set the refractive index of Sb2Se3 to 4.05, 4.01175, 3.9735 at crystalline state; 3.285, 3.32325, 3.3615 at amorphous state, corresponding to Δn decrease by 0%, 10%, 20%. The results show that the conversion efficiency decreases when Sb2Se3 is in the crystalline phase, but increases when Sb2Se3 is in the amorphous phase. This is because the refractive index of Sb2Se3 gets closer to that of Si, which affects the phase modulation ability. However, the conversion efficiencies of the mode converters remain high and stable, indicating the robustness and fabrication error tolerance of our design.
4. Conclusions
In this paper, deformed Sb2Se3 stripes integrated with silicon waveguide are proposed for reconfigurable mode converters. Our inverse design strategy is built based on polynomial description of boundary deformation and the gradient-based MMA. The optimal designs are obtained with only 57 and 19 optimization iterations for TE0-to-TE1 and TE0-to-TE2 mode converters, respectively. The results demonstrate high transmission efficiency and conversion efficiency when Sb2Se3 is in the crystalline phase (T = 99.1% and CE = 98.5% for TE0-to-TE1 conversion, T = 96.7% and CE = 96.3% for TE0-to-TE2 conversion), while amorphous Sb2Se3 allows direct mode transfer with little loss. The designed mode converters also guarantee the same high performance in backward transmission, following time reversal symmetry. When introducing ±10 nm geometrical deviations to the perfect Sb2Se3 stripe, the designs show good robustness as well. The proposed reconfigurable mode converters provide a compact and efficient solution for on-chip mode conversion and have great potential for applications in optical communications and photonic integrated circuits.
Funding
State Key Laboratory of Information Photonics and Optical Communications (IPOC2020ZT01).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
Supplemental document
See Supplement 1 for supporting content.
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