Optical quantizer is one of the key components in photonic analog-to-digital converters, which is promising for advanced signal processing applications, such as radar, medical imaging, and high-speed communication^[1]. In recent years, phase-shifted optical quantization (PSOQ) schemes have attracted much attention due to their advantages in on-chip integration, stability, and low input power requirement^[2]. So far, extensive PSOQ schemes have been proposed such as an array of Mach–Zehnder-modulators (MZMs) with identical half-wave voltages but with different bias voltages^[3], a cascaded-unbalanced-MZM with full use of phase space^[4], and a cascaded step-size MMI with one modulator^[5]. Recently, a PSOQ scheme based on the cross-polarization modulation effect in a double-ridge a-Si:H-Si3N4 waveguide has been reported^[6]. Among these schemes, the structure of the MMI-based optical quantizer is simple and compact, which meets the requirement of on-chip integration. However, its performance is limited by the imaging quality of the MMI. When MMIs are designed based on a high index contrast platform such as silicon-on-insulator (SOI), phase errors between different modes arise, especially for MMIs with more than 2 input/output ports^[7]. The phase errors result in low-quality imaging, thus leading to device performance degradation such as excess insertion loss, nonuniformity, and narrow operation bandwidth.

Therefore, essentially, enhancing the performance of the MMI-based optical quantizer suppresses the phase errors of the MMI. So far, there are three typical methods. The first one is using a shallowly etched multimode region to reduce the index contrast^[8], thus alleviating the phase errors. Since the etching depth cannot always be available due to fabrication limits, it is difficult to fully suppress the phase errors. The second one is using a subwavelength grating (SWG) structure in the lateral cladding regions of the MMI to control the index^[9]. However, it still faces fabrication challenges due to the small feature size of the SWG (typically<100nm). In contrast, the third method, the inverse design method, can explore the entire design space of devices, rather than being limited to a few parameters^[10,11]. Nevertheless, most inverse design algorithms such as direct-binary search (DBS)^[12], genetic optimization^[13], and particle swarm optimization (PSO)^[14,15], rely on random testing of a large number of different parameter sets, which require high computational costs. In addition to these algorithms, the adjoint shape optimization method is more efficient^[16], as it requires only two electromagnetic simulations per iteration to calculate the shape derivatives, thus significantly reducing computational costs. Furthermore, the designed devices based on shape optimization usually have continuous smooth structures, which can be easily fabricated using DUV photolithography^[17,18].

In this paper, in order to improve the performance of an MMI-based optical quantizer, a co-optimization of single and double inputs is proposed in an adjoint shape optimization algorithm of inverse design to optimize the shape of the MMI. As a result, the imaging quality of the MMI associated with the phase errors improves significantly, leading to performance [insertion loss (IL), uniformity, and effective number of bits (ENOB) of the optical quantizer] improvement. Simulation results show that the IL is optimized from 0.24 to 0.09 dB, and the uniformity is optimized from 1.27 to 0.16 dB at 1550 nm. The ENOB is much closer to the ideal value of 3.32 bit. The optimized optical quantizer has a continuous smooth profile that is fabrication-friendly. We fabricated the device on a 220-nm-thick SOI platform. Measurement results show that the IL is optimized from 0.85 to 0.35 dB and the uniformity is optimized from 1.21 to 0.24 dB at 1550 nm, respectively. The maximum ENOB is 3.31 bit, and the bandwidth of ENOB>3 bit is expanded to more than 50 nm.

Figure 1(a) is a schematic diagram of the sampling and quantization process based on our proposed phase-shifted optical quantizer. An optical pulse train is divided into two through a 1×2 MMI, one of which passes through a phase modulator (PM) as Es and the other is Er. The PM is driven by the analog electrical signal VRF(t) to be quantized, so the amplitude information of the input analog signal is converted into the phase difference. Next, Es and Er with a phase difference of Δφ(t) enter the two input ports of a 2×5 MMI [which acts as an optical quantizer, as shown in Figs. 1(b) and 1(c)]. The outputs of the 2×5 MMI are labeled as In (n=1, 2, 3, 4, 5). Based on the self-imaging properties of the MMI^[19], the output optical intensities can be expressed as^[20]In=P5[1+cos(Δφ+6−2n5π)],(1)where P is the power of the input optical pulse train. Equation (1) shows that the output power of each port is in a sinusoidal relationship with Δφ(t), and there is a phase shift of 2π/5 between the adjacent output ports. The curves of In, so-called quantization curves, are shown in Fig. 1(d). Taking the middle value of transmittance as the decision threshold, we can obtain 10 quantization levels [as shown in Fig. 1(e)], corresponding to a theoretical quantization bit of 3.32. However, due to the imperfect imaging of the MMI, especially when the number of output ports is larger than 2, there would be additional loss and nonuniformity in the outputs, leading to performance degradation of the optical quantizer. For example, as previously reported^[5], the MMI-based device has an insertion loss of 1.26 dB and a small bandwidth of 12 nm for the ENOB>3 bit.

To solve this issue, we utilize a shape optimization method to improve the performance of the optical quantizer, which is evaluated by the IL, uniformity (U), and ENOB, wherein U and ENOB are defined as U=−10lgmin(P1,P2,…,P5)max(P1,P2,…,P5),(2)ENOB=206.02lgPFS/12112(PFS2K)2+12K∑i=12K−1Δstep−i2,(3)where Pn (n=1,2,…,5) is the maximum value of each quantization curve. PFS is the maximum phase difference induced by the analog signal. K is the number of output ports, and Δstep−i is the phase error of each quantization level^[21]. The IL and U depend on the power of the outputs, while the ENOB mainly depends on the phase relationship of the outputs.

The quality of the optimization result is characterized by a figure of merit (FOM). For power optimization, the FOM can be expressed by the normalized output power, which is given as FOMi=ti4|∫(E×Hm¯+Em¯×H)·dS|2∫Re(Em×Hm¯)·dS,(4)where ti is the target value, and Em and Hm are the field profiles of the fundamental mode at the surface S, while E and H are the actual fields, respectively^[16].

To optimize IL and U, a single-light injection is considered. When a light is injected from one of the input ports, the power is theoretically divided evenly at the five outputs. Therefore, the target values for FOMi (i=1,2,…,5) are 0.2, which means the lowest IL and the best uniformity. To optimize the ENOB, two-light simultaneous injection should be considered. When two lights with the same power and a phase difference of Δφ are injected into two ports simultaneously and symmetrically, the output power distribution changes with the variation of Δφ. Theoretically, the phase differences between adjacent outputs are equal to 2π/5. However, due to the imperfect imaging of the MMI, the phase differences are not always 2π/5. On the other hand, Eq. (4) is only suitable for power optimization, while the phase derivative of the electromagnetic field is difficult for analytic expression and calculation. However, we notice from the quantization curves in Fig. 1(d) that when Δφ=0, the transmission values are 0.262, 0.038, 0.4, 0.038, and 0.262. Based on the inherent interference characteristic of the MMI, if the starting points of all curves are fixed to the ideal values, then the phase errors can be minimized. In other words, instead of optimizing the phase relationship directly, we transform the phase differences between five output ports into a power relationship based on the quantization curves in Fig. 1(d). Therefore, to define the FOM with two injections and optimize quantization curves when two lights with Δφ inject, FOM6 is defined as the powers of port 1 and port 5, the target of which is 0.262. Similarly, the target of FOM7 is 0.038, as the powers for port 2 and port 4, and the target of FOM8 is 0.4, as the power for port 3. A total FOM is defined as the sum of FOMi (i=1,2,…,8) to perform the co-optimization.

The initial structure of the 2×5 MMI is shown in Fig. 1(b). This structure is defined in a fully etched silicon layer with a thickness of 220 nm. The widths of the input and output waveguides Wwg are 450 nm. The length and width of the tapered waveguide are set to be Lt=10μm and Wt=1.2μm, respectively. The width WMMI is set to be 8 µm for a compact size, and the length LMMI is optimized to be 90 µm for minimum loss at 1550 nm. As shown in Fig. 1(c), the objects of the shape optimization are the upper and lower boundaries of the MMI, which are optimized by modifying the positions of 46 points on each boundary. The x-coordinates of 46 points are evenly distributed and fixed, and the y-coordinates are varied within ±0.4μm. During the optimization and considering the fabrication level, cubic spline interpolation is performed between these points to avoid sharp corners. An adjoint method is used to calculate the gradient of FOMi with respect to the parameter points. This method requires only two electromagnetic simulations per iteration, which are forward simulation and adjoint simulation, significantly improving optimization efficiency. The electromagnetic simulations are performed by the 2.5-D finite-difference time-domain (FDTD) solver, and then the gradient descent algorithm is used to find the optimal values of parameter points. The whole process took about 20 h when ran on 48 cores of a server that had two AMD EPYC 74F3 processors. As shown in Fig. 1(f), the value of total FOM/target reaches maximum and tends to stabilize after 24 iterations. The inset in Fig. 1(f) shows the boundary curve of the final optimized MMI.

Figure 2 shows the initial and optimized electric field profiles of the MMI at 1550 nm. We can see from the outputs in Fig. 2 that the output power of the optimized MMI is more stable than that of the initial one, in both individual-injection and simultaneous-injection cases.

To evaluate the performance of the MMI-based optimized optical quantizer, we swept the phase difference of two input lights and drew the transmittance curves, which are shown in Fig. 3(a). The dashed lines represent the initial results, while the solid lines represent the optimized results. It can be observed that the optimized transmittance curves have better uniformity. Especially, the optimized transmittance values at Δφ=0 are 0.261, 0.038, and 0.398, which are better than the initial results of 0.226, 0.079, and 0.353 and are very close to the theoretical target values of 0.262, 0.038, and 0.4. We then simulated and calculated the wavelength dependence of the IL, the U, and the ENOB for the optical quantizer in a range of 1530–1580 nm. As shown in Figs. 3(b) and 3(c), the IL is optimized from 0.24 to 0.09 dB, and the U is optimized from 1.27 to 0.16 dB at 1550 nm. The loss decreases by 0.15 dB, and uniformity improves by 1.11 dB. From Fig. 3(d), we can see that in a wavelength range of 1530–1580 nm, the ENOB of the initial optical quantizer is less than 3.2 bit, while the ENOB of the optimized optical quantizer is always larger than 3.2 bit. Especially, the ENOB is improved from 2.91 to 3.3 bit at 1550 nm. Figure 3 shows that the overall performance of the optimized optical quantizer is improved compared to that of the initial design.

Next, to investigate the fabrication tolerance of the proposed device, we changed the MMI width to simulate the over-etching and under-etching during fabrication. Figures 4(a)–4(c) show the spectra of the IL, U, and ENOB of the optical quantizer as a function of width variation (ΔW=−40, 0, and 40 nm, respectively). The changes in width cause small shifts of the center wavelength, but the devices still maintain excellent performance in the same wavelength band, indicating a large fabrication tolerance characteristic of the device.

We fabricated the proposed device on a commercial SOI platform with a 220-nm-thick top silicon layer, a 3-µm-thick buried oxide layer, and a 1-µm-thick top oxide cladding. Using deep ultraviolet photolithography and inductively coupled plasma etching, the optimized structure was patterned and fully etched. The initial structure was also fabricated for comparison. To measure the performance of the optical quantizer, we designed a system structure for the device according to Fig. 1(a). The microscopic images of the fabricated devices are shown in Fig. 5(a). Figures 5(b) and 5(c) are the locally enlarged images of the initial and optimized MMI, respectively.

Light from a tunable laser source with the TE mode controlled by a polarization controller was fed into the chip through a grating coupler. The phase difference Δφ was obtained by a heater, which was driven by a voltage source. A multi-port optical power meter was used to measure the power of the output ports. We gradually increased the voltage of the heater and recorded the output powers to draw the transmittance curves, as shown in Fig. 6(a), of which the blue dashed and orange solid lines represent the transmittance curves for the initial and optimized optical quantizer, respectively. We can see from Fig. 6(a) that the power intensity and uniformity of the optimized device are better than those of the initial one.

By calculating the crossing points between the normalized power and the decision threshold of 0.5 from Fig. 6(a), the transfer functions at 1550 nm are obtained, as shown in Figs. 6(b) and 6(c), where the orange solid lines are the measured results, while the green dashed lines are the ideal ones. As can be seen from Fig. 6(b), there is a considerable deviation between the initial result and the ideal value, corresponding to the phase error in Eq. (3) and causing a decrease in the ENOB. In contrast, the transfer function of the optimized device in Fig. 6(c) exhibits more uniform quantization levels and a smaller deviation from the ideal value.

Lastly, we changed the wavelength of the laser from 1530 to 1580 nm with a step of 2 nm, and measured the IL, U, and ENOB, as shown in Figs. 6(d)–6(f), respectively. We can see from Fig. 6(d) that the IL of the optimized optical quantizer is reduced compared to the initial one in the whole wavelength range from 1530 to 1580 nm. In particular, the IL is reduced from 0.85 to 0.35 dB at 1550 nm. The IL is much better than that of the previously reported MMI-based quantizer (the IL of which is 1.26 dB). From Fig. 6(e), we can see that the uniformity is optimized from 1.21 to 0.24 dB at 1550 nm. As shown in Fig. 6(f), the ENOB has an improvement of 0.45 bit at 1550 nm and is larger than 3.06 bit in a range of 1530–1580 nm. It can be noticed that the uniformity and the ENOB of the initial quantizer are better than those of the optimized one at the wavelength (such as 1532 nm) far from 1550 nm, despite the larger IL. This is because the phase errors result in inconsistent center wavelengths of performance indicators of the initial MMI. The overall optimization result is that the IL, the U, and the EONB all exhibit good values in the range of 1530–1580 nm and their center wavelengths all appear around 1550 nm. The experimental result of the ENOB is also better than that of the previously reported devices (12 nm bandwidth for the ENOB>3bit^[5], or a maximum value of 3.17 bit on the thin-film lithium niobate platform^[22]). To confirm the fabrication robustness of the optimized device, we measured two optimized devices fabricated at different positions on the same wafer. The performance of the two optimized devices is similar (more details can be found in the Supplementary Material). The co-optimization approach is also applicable to the optimization of other optical quantizers with multiple input/output ports and MMI devices with specific phase relationships, such as 90-deg hybrid and 7-port quantizer (see the Supplementary Material).

In conclusion, we have proposed and experimentally demonstrated a performance-enhanced optical quantizer. The initial structure of the optical quantizer is a 2×5 MMI, and the boundaries are optimized using an efficient adjoint shape optimization method. We propose the co-optimization for one input and two inputs to simultaneously improve the IL, the uniformity, and the ENOB. We fabricate the initial and optimized devices on an SOI platform using deep ultraviolet photolithography. Experimental results show that the IL is optimized from 0.85 to 0.35 dB, and the uniformity is optimized from 1.21 to 0.24 dB at 1550 nm. The optimized ENOB is larger than 3.06 bit with a maximum of 3.31 bit in the range of 1530–1580 nm. The proposed high-performance optical quantizer is expected to have significant applications in advanced signal processing systems.