Inverse Design of Ultra-Compact and Low-Loss Optical Phase Shifters

: In previous works, inverse design methods have primarily focused on manipulating the optical power to achieve speciﬁc design targets. In this paper, we use the inverse design method to enable the precise engineering of the optical phase. As a proof of concept, we present a series of phase shifters (PSs) with varying phase shifts, which are inversely designed and theoretically validated on the silicon-on-insulator (SOI) platform. The designed PSs exhibit remarkable accuracy, with deviations of below 1 ◦ . These proposed PSs feature an ultra-compact footprint of 3 µm in length and offer superior fabrication tolerances compared to conventional structures. Over the 1535–1565 nm bandwidth, the discrepancy between the actual and target phase shifts remains below ±1 ◦ for all phase shifters, while the insertion loss is consistently below 0.035 dB. Moreover, the feasibility of the designed ﬁve PSs is veriﬁed using 2 × 2 multimode interference couplers (MMI).


Introduction
Photonic Integrated Circuits (PICs) based on the Silicon-on-Insulator (SOI) platform have attracted much attention due to their high integration density and compatibility with CMOS processes [1][2][3].Among the various integrated optical devices on the SOI platform, optical phase shifters (PSs) can be used not only for sensors [4], telecom and datacom applications [5,6], but also for quantum computing [7,8].Of the reported PSs, there are two main types of structure, active and passive, which produce phase shifts by adjusting the corresponding optical path lengths or changing the propagation constants of the waveguide modes.Active PSs mainly use the thermo-optical effect of silicon photonic waveguides to modify the effective refractive index of the waveguide by means of on-chip metal microheaters to achieve a corresponding phase shift [9,10].In contrast, passive PSs have no requirement for control elements or driving power, eliminating the constraint of power consumption and enabling applications in novel optical 90 • hybrids [11], arbitrary ratio power splitters [12,13], and mode-division multiplexing (MDM) systems [14,15].
To date, only a few studies have been reported on passive PSs on SOI and the primary design approach is to induce a phase shift by controlling the waveguide length or width.Chaen et al. achieved a phase shift of 180 • between two similar waveguides by simply increasing the length of one waveguide relative to another waveguide [16].By changing the width of one waveguide, a phase shift of 90 • or 180 • between the two waveguides can be achieved, enabling butterfly-shaped PSs that can be used for TE 0 to TE 1 [17], TE 0 to TE 2 mode conversion [18] and demultiplexing of TE 0 and TE 1 modes [19].Morrissey et al. implemented a PS with a phase shift of 180 • by optimizing the width of a 1 × 1 multimode interference (MMI) coupler [20].Moreover, using anisotropy and dispersion engineering in subwavelength metamaterial waveguides, a 90 • PS with an ultra-broadband of 400 nm was realized by varying the longitudinal width of the subwavelength grating (SWG) [21].
However, the phase shift of the PS for these designs is typically 90 • or 180 • to apply to the higher-order modes required for processing in MDM systems.There is little research on PSs for special phase shifts such as 30 • , 60 • , etc.The flexibility of PICs can be effectively improved if PSs with arbitrary phase shifts can be implemented.
More recently, inverse design methods for photonic devices have been extensively reported, which can implement high-performance devices at a much smaller size [22][23][24].However, most of the inverse design methods concentrate on the design of the mode power distribution in the waveguide, while very little inverse design research has been carried out for phase design.In this paper, we inversely design the passive PSs based on shape optimization.By optimizing the multi-segment width of the PS using particle swarm optimization (PSO), arbitrary phase shifts can be achieved.This paper validates a series of PSs with phase shifts of 1 • , 30 • , 60 • , 90 • , and 180 • , respectively.The length of each PS is only 3 µm.Simulations show that the maximum phase shift error (PSE) for five PSs is below ±1 • over the wavelength range 1535 to 1565 nm and all PSs exhibit extremely low insertion loss (IL) (<0.035 dB).In addition, the designed PSs are verified using 2 × 2 MMI.The verification results match with the simulation calculation results and prove the feasibility of the designed PSs.To the best of our knowledge, the proposed 90 • and 180 • PSs exhibit the smallest size and lowest losses.

Design Principle
Figure 1a depicts the schematic diagram of the proposed PS, which is designed on an SOI platform with a 220 nm core silicon layer and oxide cladding.The phase shift is achieved by varying the shape of the waveguide in the phase shift section (indicated by the dashed box).A straight waveguide with the same length is used as a reference to determine the resulting accumulated phase shift, denoted as ∆Φ.All designed PSs have a fixed length L of 3 µm and are connected to an input/output waveguide with a width W of 0.5 µm.The shape optimization principle for the PS is illustrated in Figure 1b.The shape of the PS is defined using three different widths (W 1 , W 2 , and W 3 ) equally distributed along the x-axis.For easier fabrication, the PS has a smooth and continuous boundary constructed by the cubic spline interpolation method.By optimizing the values of W 1 , W 2 , and W 3 , the shape of PS can be adjusted to achieve the desired phase shift.Here, we use the PSO algorithm to optimize the three parameters.The PSO is a population-based stochastic optimization technique inspired by the social behavior of flocks of birds or schools of fish, which has been widely used to solve multi-parameter optimization problems due to its fast convergence [25,26].The three-dimensional finitedifference time-domain (3D FDTD) method is used for the simulation of PS, phase calculation, and optimization objective calculation.The optimization objective is quantified by a figure of merit (FOM) and is expressed by the following equation:

Silicon Silica
where ∆Φ i and ∆Φ * i are the actual and target phase shifts at the i-th wavelength channel within the wavelength range of 1535-1565 nm.T i is the normalized transmittance of the input TE 0 mode at the i-th wavelength channel.Most of the phase shifters are designed by considering only the device parameters as a function of phase shift and ignoring the possible variations in insertion loss.In Equation (1), we design the combination of phase shift and normalized transmittance of the device as FOM.The FOM is minimized by PSO, i.e., the FOM converges to 0. In this way, ∆Φ converges to the target value ∆Φ * and T converges to 1, which ensures that the designed PS achieves the required phase shift with minimum losses.Considering the numerical balance between the two parts in Equation ( 1), the units of phase shift are converted from degree to radian to facilitate the calculation.
The detailed flow of optimizing W 1 , W 2 , and W 3 for phase design using the PSO algorithm is shown in Figure 2. The positions of the particles represent the values of W 1 , W 2 , and W 3 , while the velocities of the particles correspond to the variation range for these parameters.The positions and velocities of the particles are updated iteratively based on the following Equation [27]: where z and v represent the position and velocity of the particle, respectively.n denotes the iteration.p best represents the best position achieved by the particle so far, and g best represents the best position achieved by any particle in the entire swarm.The coefficients ω, r 1 , and r 2 control the impact of the previous velocity, the particle's personal best, and the swarm's global best on the update.η 1 and η 2 are random coefficients between 0 and 1.
To ensure better search results, a wide range of parameters is considered for optimization, where W 1 , W 2 , and W 3 ∈ [0.2 µm, 0.8 µm].At each iteration, the FOM obtained from the 3D FDTD is used as an evaluation metric to find the appropriate p best and g best to minimize the FOM.Afterward, the position and velocity of the particles are updated according to Equations ( 2) and (3), and the optimization is stopped to obtain the final result when the set number of iterations is reached.
A common approach to designing phase shifters usually starts with parameters such as the length or overall width of the device.By scanning these basic parameters of the device, it is possible to obtain them as a function of phase shift.However, this approach may face the problem of unconstrained device dimensions and the fact that the phase shift of a device is relatively sensitive to variations in its parameters.In contrast, the inverse design method in this paper starts with the target phase shift and allows the device shape to be designed for arbitrary phase shifts by optimizing the multi-segment width of the device in an ultra-compact footprint of 3 µm × 0.8 µm.As a result, the designed devices are smaller in size and robust to fabrication errors.We validated the design of five PSs with different phase shifts by setting ∆Φ * = 1 • , 30 • , 60 • , 90 • , and 180 • .After 80 iterations of PSO, the W 1 , W 2 , and W 3 values corresponding to each PS of the design are listed in Table 1.The final structures with smooth boundaries are shown in Figure 3.All PSs have a length of 3 µm and a width between 0.3 µm and 0.8 µm.The inset of Figure 3 shows the electric field distribution at 1550 nm for the corresponding PSs and it can be clearly observed that the light propagates directly along the PSs with negligible loss.

Results and Discussion
The performance of the PSs is evaluated by examining two key metrics: the phase shift error (PSE) and the insertion loss (IL).The PSE is defined as the deviation between the achieved phase shift and the target phase shift, which can be written as: and the IL can be written as: where T output is the transmittance at the output port of the PS.The calculated phase shifts and corresponding PSEs for the five PSs are shown in Figure 4a-e The tolerance to fabrication errors of the five PSs is investigated by varying the waveguide width (∆W) within a range of −20 nm to 20 nm, as illustrated in Figure 5a.The maximum absolute value of PSE at each wavelength is calculated for each wavelength within the range of 1535 nm to 1565 nm and is shown in Figure 5b.It is observed that the maximum absolute value of PSE for all PSs in the 30 nm wavelength range is 7.6 • .In comparison, the maximum absolute value of PSE reaches 10 • with the same variation for both the conventional butterfly PS and the MMI-based PS [20,21].Figure 5c presents the maximum variation in IL due to the width changes for all PSs within the wavelength range of 1535 nm to 1565 nm.It can be observed that the IL is minimally affected by the width variation, with the maximum IL remaining below 0.035 dB in the entire 30 nm wavelength range.This demonstrates that our designed PSs exhibit favorable fabrication tolerances, as the IL remains consistently low even with variations in the waveguide width.To validate the feasibility of the designed PSs, the characterization setup of 2 × 2 MMI was used to further the simulation verification of the PSs, as depicted in Figure 6a.In this validation structure, TE 0 lights with equal power are injected at input port 1 and input port 2.Then, the PS on the lower arm introduces the corresponding phase shift to the transmitted light.Finally, the two TE 0 lights with phase difference ∆Φ are coupled into the 2 × 2 MMI.The dimensions for 2 × 2 MMI are adopted from Guan et al. [28].Figure 6b- Additionally, Table 2 provides a comparison between our work and previously reported passive optical PSs.It is evident that the PSs designed in this paper exhibit significant advantages in terms of footprint and losses, representing the smallest passive PSs reported to date.

Conclusions
In conclusion, our study presents a novel PS with arbitrary phase shift designed using a shape optimization method.Five PSs with phase shifts of 1 • , 30 • , 60 • , 90 • and 180 • are demonstrated.All five PSs are designed in an ultra-compact size of only 3 µm in length.Simulation results show that the maximum PSE for all PSs remains below ±1 • over a wavelength range of 30 nm (1535-1565 nm) and the maximum IL for all phase shifters is only 0.035 dB.The designed PSs are verified using 2 × 2 MMI.The verification results match the simulation results with an error of less than ±0.25 • between the results.We believe that the arbitrary phase shift PS demonstrated in this paper could find potential applications in coherent communications, quantum photonics, and MDM circuits.

Figure 1 .
Figure 1.(a) The schematic diagram of the proposed PS.(b) The schematic diagram of the shape optimization principle for the PS.

Figure 5 .
Figure 5.As ∆W varies from −20 nm to 20 nm, (a) the schematic of the variation in the width of the designed PS, (b) the maximum absolute value of PSE, and (c) maximum IL within the wavelength range of 1535 to 1565 nm.
f illustrate the electric field distribution at 1550 nm for the validation structures of five PSs.It is evident that the power distribution at output port 1 and port 2 varies for different phase shifters.The transmission spectra of employed 2 × 2 MMI as a function of phase difference ∆Φ are shown in Figure 6g.By analyzing the output power distribution, we can determine the phase shift induced by the PSs.The power distribution at the outputs for all validation structures is also marked in the figure, with circles indicating the power from the out port 1 and forks indicating the power from the out port 2. The phase shifts for the proposed PSs at 1550 nm are estimated as 1.04 • , 29.96 • , 59.94 • , 89.91 • , and 179.98 • , respectively.Consequently, the corresponding PSEs are 0.04 • , −0.04 • , −0.06 • , −0.09 • , and −0.02 • , respectively.The phase shifts of all PSs are also verified within the wavelength range from 1535 to 1565 nm, and the calculated PSE curves are shown in Figure 6h.It can be observed that the PSEs for 1 • PS, 30 • PS, 60 • PS, 90 • PS and 180 • PS are below ±0.16 • , ±0.3 • , ±0.63 • , ±0.52 • and ±0.99 • over the 30 nm bandwidth, respectively.Compared to the theoretical analysis, There is a slight mismatch (<±0.25 • ) with the results obtained from 2 × 2 MMI structures.The slight discrepancy may be attributed to differences in the overall simulation area, leading to variations in the grid accuracy for the 3D FDTD simulations employed in the characterization setup.

Figure 6 .
Figure 6.(a) The Schematic diagram of the test structure used to characterize the PSs.The electric field distribution of the tested structures at 1550 nm for (b) 1 • PS, (c) 30 • PS, (d) 60 • PS, (e) 90 • PS, and (f) 180 • PS.(g) Transmission spectra of 2 × 2 MMI at 1550 nm when two signals with phase difference ∆Φ are input.Circles indicate power from test structure output port 1 and forks indicate power from test structure output port 2. (h) PSE curves for all PSs validated in the wavelength range of 1535 to 1565 nm.

Table 1 .
Optimization results for parameters W 1 , W 2 and W 3 .

Table 2 .
Comparison of Reported Phase shifters.