Due to the ability to ensure ultracompact photonic integrated circuits (PICs) and the advantage of CMOS-compatible processing, the silicon-on-insulator (SOI) platform has drawn appreciable attention in the past few years. However, nanostructures based on the SOI platform usually show strong birefringence, which can cause undesired polarization-dependent mode dispersion. Hence, the devices that can manipulate the polarization state of light become a vital part of PICs. Many functional devices have been proposed for this reason, including polarization beam splitters (PBSs) ^[1–15], polarization rotators (PRs) ^[16,17], and polarizers ^[18,19]. As a typical polarization-handling device, structures like multimode interferometers (MMIs) ^[1,2], Mach–Zehnder interferometers (MZIs) ^[3], photonic crystal structures ^[4], hybrid plasmonic silicon waveguides (HPWs) ^[5,6], grating structures ^[7–10], and directional couplers (DCs) ^[11–14] have been employed to realize polarization splitting. Among those structures, DC-based PBSs are preferred because they are easy to design and usually provide a smaller footprint ^[11–14]. However, due to the underlying phase-matching principle of DCs, the performance of these devices is very sensitive to fabrication errors. To relax the constraint of the phase-matching condition, grating-based PBSs have been extensively studied ^[7–10]. For example, Zhang et al. experimentally demonstrated a PBS with an extinction ratio (ER) of more than 30 dB and an insertion loss (IL) less than 1 dB ^[9]. More importantly, the proposed PBS exhibits a high tolerance for manufacturing errors and a low sensitivity to the coupling length. Unfortunately, due to the lack of a full analytic theory, the design of grating-based PBSs is very complicated. Recently, based on the concept of free-form metamaterials, Shen et al. proposed a PBS with a footprint of only 2.4μm×2.4μm ^[19]. This kind of device is extremely robust to fabrication errors, but under the limitation of scattering, the bandwidth of ER greater than 10 dB is only 32 nm.

In this paper, based on the simple DC structure and the concept of “objective first” ^[20–23], we use the inverse design method ^[24–28] to design PBSs numerically. Compared to traditional design methods, which require a large amount of manual tuning, we can easily design PBSs according to specific demands. The only thing we need to do is to give proper initial conditions for the design process. Due to the ability to search for the full design space of target devices with arbitrary topologies ^[25], the designed devices can work in C-band, L-band, and even O-band. Here, we have designed a series of PBSs with a coupling area of only 0.48μm×6.4μm. The insertion losses of these devices can be less than 0.4 dB (0.35 dB) for the TE (TM) mode at three far apart isolated wavelengths (1310, 1550, and 1600 nm), and the extinction ratios are bigger than 19.9 dB for the TE and TM modes at all three wavelengths. When it comes to broadband situations, the ERs of the two polarization states are more than 14.50 dB within the wavelength range of 1286–1364 nm, 1497–1568 nm, and 1553–1634 nm. As we know, the full coupling length for the fundamental TE (TM) mode for a traditional DC varies with the wavelength due to modal dispersion ^[10]. Generally speaking, the DC-based PBSs that work in a shorter band have a bigger footprint, and the design process of these devices could be more complicated ^[10,29]. Here, benefiting from using the full design space, the ERs of the TE (TM) mode can still be kept higher than 15.48 dB (18.87 dB) and the ILs are lower than 0.46 dB (0.31 dB) when the operation wavelength is located at the O-band (1286–1364 nm). More than that, as long as it does not violate the principles of physics (the coupled-mode theory), we can theoretically design a PBS in any communication band without changing the size of the device.

Similar to traditional DCs, the presented PBSs are composed of two parallel straight waveguides and a rectangular coupling region, which is divided into 160pixel×12 pixel. The sides of these square pixels are 40 nm, giving a coupling area of 0.48μm×6.4μm [Fig. 1(b)]. To minimize the device size, as depicted in Fig. 1, we choose the widths of the straight waveguides to be 360 nm (9 pixels). To ensure our devices can be fabricated in a single lithography step, the heights of the waveguides also are set to be exactly like the coupling region; that is, 220 nm. And the 220-nm-thick Si (nSi=3.49) layer is placed in the middle of the silica (nSiO2=1.45) cladding layer, which has a height of 40 pixels (1.6 μm). By changing the refractive index of each pixel in the coupling region (to be silicon or silica), the devices can work in different target bands. In fact, in addition to the refractive index distribution, the material dispersion also has an impact on the characteristics of those devices ^[30,31]. In this paper, however, to simplify the optimization model and improve the optimization efficiency, we ignored the effect of material dispersion.

Considering that the convergence of the inverse design method is determined by the “physics residuals” ^[20–23] of design objects, we start by giving proper initial conditions for the design process. For the structure described in Fig. 1, when the wavelength is specified, the effective indices of supported modes are uniquely determined by refractive indices of pixels in the cross section [yellow region in Fig. 1(b)]. The refractive indices can take the maximum value and the minimum value when all of the pixels in the cross section are occupied by silicon and silica, respectively [Fig. 2(a)]. According to the description of Section 3.2.1 in Ref. ^[30] and Section 4.3 in Ref. ^[31], the full coupling length (the length necessary for full power to transfer from one waveguide to the other, which is one-half of the beat length) of the fundamental mode is determined by the effective indices of the two first-order supermodes. Hence, the range of the full coupling length is determined by where neff,1 and neff,2 are the effective indices of two first-order supermodes (neff,1−neff,2≥0). Figure 2(a) shows that there is an overlap between the value range of the effective indices of the TE0 (TM0) mode and the TE1 (TM1) mode [shaded parts of Fig. 2(a)], which means the extended maximum value of the full coupling length of the fundamental TE (TM) mode is infinity (neff,1−neff,2=0). Therefore, we cannot effectively estimate the maximum value of the full coupling length from Eq. (1). Fortunately, for the structure described in Fig. 1, with the increases of silicon composition in the cross section, the difference of the effective indices between the fundamental mode and the first-order mode will increase. Hence, the full coupling length will take the maximum value when the cross section of the coupling region is totally occupied by silica. In contrast to the maximum value, Fig. 2(a) clearly shows that the minimum value of the full coupling length can be calculated using λ/2[(neff,1)max−(neff,2)min]. Figure 2(b) shows that the full coupling length of the TE mode can vary from 1.04 μm (black star) to 1539 μm (yellow star), but the full coupling length of the TM mode is limited in the scope of 1.98 μm (black triangle) to 206.5 μm (yellow triangle). This phenomenon can be simply explained. For the structure described in Fig. 1, the TE mode is mainly resonant in the x–y plane ^[10]; hence, changing the refractive index distribution will mainly affect the TE mode. Besides, we can see that both the maximum values of the TE mode and the TM mode rapidly increase when the wavelength decreases. In other words, it would be much more difficult to design a PBS in the O-band. Now the question is: how much of the coupling length should be taken? For a general inverse design problem, the design process is relatively simple. First, you specify the performance of the device by defining the conversion efficiency between the sets of input modes and output modes at several discrete frequencies. Second, use the “objective first” method and the steepest descent method to alternately update the field distribution and refractive index distribution until the “physics residuals” are lower than 10−6, or the upper limit of the iteration times has been satisfied ^[20–28].

To determine the size of the coupling region, we perform some tests at the wavelengths of 1310 nm and 1550 nm. As depicted in Fig. 1, the fundamental TE mode and the fundamental TM mode are coupled into the PBS devices from port 1, and, after passing through the PBS, TE mode, and TM mode, are coupled out from port 3 and port 4, respectively. During this design process, we specified that >95% input power of the TE (TM) mode should be coupled out of port 3 (4) and less than 1% should be coupled out of port 4 (3). Figure 2(b) shows that the full coupling lengths of the TE and TM modes are in the range of 1.13 μm to 63.33 μm, and 2.08 μm to 11.49 μm (at the wavelength of 1550 nm). The information tells us that the minimum coupling length of the PBS is max{2×1.13μm;2.08μm}=2.26μm. Similarly, Fig. 2(b) also shows that the minimum coupling length is max{2×1.36μm;2.01μm}=2.72μm at the wavelength of 1310 nm. Therefore, if we want the device to achieve high performances at those two target wavelengths, the length of coupling region should be larger than 2.72 μm. Figure 3 shows the design results at the wavelengths of 1310 nm (solid line) and 1550 nm (dashed line), where the coupling lengths are chosen to be 1.4, 3.4, and 6.4 μm. As illustrated in Fig. 3, the performance (ERs and ILs) of the designed devices increases with the increasing of coupling length. The PBSs have poor performance when the coupling length is 1.4 μm (<2.72μm), but when the coupling length reaches 6.4 μm, the ERs of two polarization states can approach 20 dB and the ILs can be less than 0.4 dB at both wavelengths. Besides, Fig. 3 shows that the requested coupling length is inversely proportional to the wavelength. The coupling length of 3.4 μm is enough for a wavelength of 1550 nm, but the PBS requires a bigger coupling length (6.4 μm) at a wavelength of 1310 nm. The reason for this phenomenon is that the full-coupling length of the TE and TM modes is more sensitive to effective indices disturbance in a smaller wavelength range. As a result, a longer coupling length, which can provide more design freedom, is needed for a shorter wavelength. Here, to make sure the working wavelength can be expanded to C-band, L-band, and O-band, we choose the coupling length to be 6.4 μm.

To illustrate the scalability of the proposed PBSs, except the two target wavelengths (1310 nm and 1550 nm) that have been described in Fig. 3, we choose another target wavelength (1600 nm), which belongs to the L-band. The geometry of the designed structure, the electric field amplitude distribution at the center of the devices, and the transmittance spectra of the three devices are shown in Fig. 4. As shown in Fig. 4, when the coupling length is chosen to be 6.4 μm, the inverse design method can realize polarization splitting at those three far apart target wavelengths, the ERs of the two polarization states can approach 20 dB (1%), and the ILs can be less than 0.4 dB (91.2%). Naturally, based on the conclusions of Figs. 3 and 4, we can predict that we can easily design PBS devices at any target wavelength (within the range of 1310–1600 nm).

The transmittance spectra in Figs. 4(i)–4(k) reveal that the device performance quickly drops off as one moves away from the target wavelength. To design broadband PBSs, we specified multiple target wavelengths in our design process that have identical desired performances (95% versus 1%). The geometry of the designed structures and the corresponding broadband operation spectra are shown in Figs. 5(a)–5(c) and Figs. 5(d)–5(f), respectively. Figure 5 reveals that the broadband operation has been achieved. As shown in Figs. 5(g)–5(i), under the limitation of IL<0.46dB (90%), the bandwidths of both polarization states are larger than 72 nm (1286–1364 nm, 1497–1568 nm, and 1553–1634 nm). At the same time, the ER can be larger than 14.5 dB (3.55%).

In conclusion, we have numerically proposed a scheme to design on-chip polarization beam splitters. In our scheme, the footprints of the coupling regions are only 0.48μm×6.4μm. By manipulating the refractive index of a coupling region, the PBS devices can work in C-band, L-band, O-band, or any other communication band. The simulation results show that the insertion losses of the designed polarization beam splitters are less than 0.4 dB (0.35 dB) for TE (TM) mode at three wavelengths, and the extinction ratios are larger than 19.9 dB for the TE and TM modes. Besides, the extinction ratios of both polarization states are more than 14.5 dB within the wavelength ranges of 1286–1364 nm, 1497–1568 nm, and 1553–1634 nm. At the same time, the ILs are smaller than 0.46 dB. Our results indicate that the inverse design method can efficiently design high-performance PBS under the guidance of basic physics theory (the coupled-mode theory). In addition, this method can easily be applied to design most types of photonic devices. Unlike design methods such as the direct binary search (DBS) algorithm ^[19,32], the genetic algorithm ^[33,34], and the topology optimization method ^[35], which empirically give a design region (including shape and size), we proposed a semi-quantitative analysis method based on the coupled-mode theory to determine the size of the design region.