Type of the Paper (Article

The 2 × 2 3-db couplers are one of the most widely used and important components in silicon photonics. We propose an ultra-broadband and compact 2 × 2 3-dB adiabatic coupler defined by b-splines and optimized with an efficient supermode-injected adjoint shape optimization. By combining both the mode adiabatic evolution and mode coupling at two different wavelength ranges, we achieve an ultrabroad bandwidth of 530 nm from 1150nm to1680nm with a power imbalance under 0.76db and a low loss in a compact coupling length of 30 μm according to our simulation results. The supermode-injected adjoint shape optimization can also be applied to the design of other photonic devices based on supermode manipulation.


Introduction
With the development of artificial intelligence (AI) technology, high-capacity, and low-latency interconnects are required for the training and inference of AI large models in high-performance computing (HPC) clusters.Silicon photonics is a promising solution for its high integration density, low cost, and CMOS process compatibility [1].The 2 × 2 3db couplers are one of the most widely used and important components in silicon photonics.Conventional 3-db couplers based on directional couplers (DCs) have limited bandwidth [2].Couplers based on Multimode interference (MMI) offer a broad bandwidth but at the cost of considerable loss and a large footprint [3].There have been several approaches to enhance the performance of the 3-db couplers, including bent DCs [4,5], subwavelength grating (SWG) assisted DCs [6,7], and adiabatic couplers [8][9][10].Bent DCs can operate at wide bandwidths by achieving phase matching between two waveguides with different propagation constants [4,5].This requires precise control of the waveguide and gap widths.Recently, SWG structure has been used to tailor the dispersion of waveguides, achieving broad bandwidths in compact length.An ultrabroad bandwidth of 270nm from 1400nm to 1670nm in a total length of 24.4um in SWG-assisted DCs is demonstrated, but it is designed in a minimum feature size of 64nm, which may be challenging to fabricate [6].Adiabatic couplers often require a long length to achieve a broad bandwidth.A variety of adiabatic curves have been proposed to realize the fast adiabaticity and reduce the length of the device [9][10][11][12].A short coupling length of 11.7um can be achieved in an adiabatic coupler optimized by the FAQUAD protocol.However, this comes at a price of a limited bandwidth of 75nm [10].Therefore, achieving broadband 2x2 adiabatic couplers with compact footprints is still demanding.
In recent years, there has been a growing interest in the inverse design, which has resulted in the design of various devices, such as mode manipulation devices [13,14], wavelength (de)multiplexers [15,16], grating couplers [17], and micro resonators [18]. 2 × 2 3-dB bent couplers with low loss and compact footprint can also be realized by inverse design [19,20].Adjoint shape optimization is a gradient-based inverse design that is suitable for design with an initial structure and easy to apply shape constraints, lead to fabrication-friendly devices such as 1×2 splitters [21], waveguide crossing [22], polarization rotator [23].However, there have been limited investigations into the utilization of adjoint shape optimization for the design of adiabatic couplers.
In this work, we propose an ultrabroadband and compact 2 × 2 3-dB adiabatic coupler based on silicon strip waveguides of the SOI platform.The adiabatic coupler operates in mode evolution at longer wavelength and mode coupling in shorter wavelength.The coupling region defined by b-splines is optimized efficiently by a novel supermode-injected adjoint shape optimization paradigm, which is using supermode as the forward and adjoint source in the optimization algorithm.The simulation result indicates that we achieve an ultrabroad bandwidth of 530 nm from 1150nm to1680nm with a power imbalance under 0.76db and a low loss in a compact coupling length of 30 µ m.

Design and principle
Figure 1(a) shows a three-dimensional view of the device, which is based on SOI platform with a 220 nm top layer of silicon, a 3 µ m buried oxide layer, and a 2 µ m silicon dioxide cladding.The device consists of three regions as shown in figure 1(a).
In Region I, WG1 and WG2 of widths W1=380nm and W2=500nm, respectively, are brought closer via two S-bends of length L1, resulting in a decrease of the gap between them from G0 to G1.In Region II, the supermode achieves adiabatic evolution as the widths of WG1 and WG2 slowly vary to reach the same W3=410nm, resulting in an equal distribution of optical power in the two waveguides.In Region III, the gap between them is increased from G1 to G2 by using two S-bends of length L3.G0 and G2 should be large to prevent excitation of any modes in the other waveguide.Therefore, we choose G0=1.6um and G2=1.65um.L1 and L2 should have sufficient length to deteriorate/avoid the coupling among supermodes.Thus, we choose L1=30µ m and L2=12µ m.
In Region II, WG1's upper edge and WG2's lower edge are straight and aligned with the x-axis, which is also the propagation direction, and the gap between them is G1=100 nm to satisfy the actual fabrication process requirements.Meanwhile, the optimization of the lower edge shape in WG1 (B1) and upper edge shape in WG2 (B2) is to be carried out using two fourth-order clamped B-splines, defined by their control points as follows: where    represents the ( + 1)ℎ control point and _(, 3) () denotes B-spline basic functions of degree three.This indicates each point on the curve is a linearly weighted sum of the four closest control points.The curve's control points are distributed evenly along the x-axis with a large distance of D0=2μm between adjacent control points ensuring a smooth and flat curve, resulting in a length of L2=30μm containing 16 control points.To reduce the number of optimization parameters, the x coordinates of each control point are fixed, while the y coordinates of B1 are proportional to those of B2, as specified by: To guarantee a seamless connection between the regions, both endpoints of B1 and B2 will be firmly fixed to eliminate any discontinuities.Thus, the total number of optimization parameters is 14.The aim of Region II is to ensure adiabatic supermode evolution while avoiding any coupling with other supermodes.This is similar to the principle of multimode waveguide bend, which prevents inter-mode crosstalk while the geometry undergoes bending.Therefore, adapting shape optimization to an adiabatic coupler only requires extending the optimization object from a single waveguide system's mode to a dual waveguide system's supermode.We defined the FOM (figure of merit) as follows: where _( ) is the transmission of the odd supermode.In Region II, the main physical process involves the evolution of the supermodes, resulting in a slower change in the shape of the coupler.As a result, the system exhibits almost lossless characteristics, and the main factor that deteriorate the FOM is the inter-supermode coupling rather than the total loss.Since there are only two supermodes in the dual waveguide system, if one supermode's transmittance is high, then the transmittance of the other supermode should also be high.Actually, according to our simulation during iterations, the transmittance of both the odd and even supermodes are always nearly identical.However, the odd supermode performs slightly worse than the even supermode due to the fact that WG1, which has a narrower width, is more sensitive to geometry variations.Therefore, we simply select the transmittance of the odd supermode as the FOM and thus cut the simulation time in half.We use 3D-FDTD to obtain the FOM.In every iteration, we employ both a forward simulation and an adjoint simulation to compute the FOM gradient with respect to the optimized parameters.The forward source is set to be the odd supermode at the begin of Region II as shown is fig 4(a), while the adjoint source is set to be the same odd supermode at the end of Region II.The L-BFGS-B algorithm is subsequently utilized to update the parameters based on the acquired FOM and gradients.

Simulation
After 28 iterations, the figure of merit (FOM) reached 0.9966, as depicted in figure 2b.The iteration process takes 10h on a server equipped with dual 2.90-GHz Intel Xeon Platinum 8268 CPU and 256-GB RAM.The optimized B2 shape is depicted as the red solid line in Figure 2a, while the conventional linear taper shape is presented for comparison purposes in the same figure.The device's transmittance from the input port in Region I to the bar and cross output port in Region III is illustrated in Figure 2c across the wavelength range of 1150nm-1680nm.In the linear case, there is a power imbalance of approximately 3+-1db within the 1400nm-1650nm range, with an obvious degradation at shorter wavelengths.Conversely, the optimized case, the power imbalance stands at around 3+-0.76db over the entire range of 1150nm-1680nm.Fig. 2d illustrates the odd supermode's transmittance through Region II.The data clearly indicates that the optimized case performs better than the linear case, particularly in the shorter wavelength.However, the transmission still remains low, especially when compared to longer wavelengths.Nonetheless, the power splitting ratio does not decrease much.This is because the splitting ratio depends not only on the magnitude of supermode transmission, but also on the phase difference between odd and even supermode, which can be expressed as follows: Here, T_odd is the transmission of the odd supermode, while T_even refers to the transmission of the even supermode.Δφ represents the phase difference between the odd and even supermodes.Additionally, T_bar denotes the transmission at the bar port, and T_cross the transmission at the cross port.When either T_even or T_odd is sufficiently small, the power splitting ratio T_bar/T_cross approaches 1, indicating the adiabatic coupler's operational status.When the phase difference Δφ equals (2n+1)π/2, the power splitting ratio T_bar/T_cross is also equal to 1, indicating that the device is operating as a 3-db directional coupler.Thus, at longer wavelengths, the device is optimized for realizing supermode adiabatic evolution, while at shorter wavelengths, the device operates as a directional coupler by selecting an appropriate length L2 = 30um to achieve the (2n+1)π/2 phase difference between odd and even supermode.Fig. 4 (a)-(f) show the simulated light propagation profiles at 1.2μm, 1.4um and 1.6um when the light is launched from the WG1 and WG2 input ports, respectively.At all three wavelengths, it can be seen that the light from the two input ports is evenly split into the two output ports.

Conclusions
In conclusion, we have proposed an Ultra-broadband and compact 2×2 3-dB adiabatic coupler on SOI.We adapted the adjoint shape optimization for the design of adiabatic coupler by using supermode as the forward and adjoint source.Ultra-broadband is achieved in a short length by combining the mode adiabatic evolution at longer wavelength and mode coupling at shorter wavelength.The simulation result shows the power imbalance of the 3-db coupler is under +-0.76 dB over the 530nm bandwidth of 1150nm-1680nm with coupling length being 30um and total length being 62um.We believe our device can find numerous applications for MZI-base system in integrated photonics.The supermode-injected adjoint shape optimization can also be applied to other photonic device based on supermode manipulation, resulting in the more compact device footprint.

Figure 1 .
Figure 1.Schematic diagram of the adiabatic coupler.

Figure 2 .
Figure 2. Schematic diagram of the forward and adjoint source.

Figure 3 .
Figure 3. Schematic diagram of the shape of B2 and FOM v.s.iteration.

Figure 4 .
Figure 4. Schematic diagram of the power splitting ration and transmission of supermode.