The photonic integrated circuit (PIC) has developed rapidly in recent years and attracted a great deal of attention because of its high performance, tiny size, and low cost compared to conventional fiber systems.1^–3 Various on-chip devices of PICs have been studied with different functions to undertake the task of increasing demand for optical communication, sensing, source, and so on.4^–6 In those practical applications of PICs, multiwavelength light sources are usually applied, for example, in wavelength division multiplexing systems. Therefore, to maintain stability and reliability for different operating wavelengths, devices that have constant performance over an extensive wavelength range are essential. In addition, the optical power splitter is one of the indispensable building blocks in PICs, mainly serving for optical power distribution or as components of complex on-chip devices, such as microrings, wavelength division multiplexers, and Mach–Zehnder interferometers.7^–9 Obtaining splitters with an arbitrary splitting ratio (ASR), which provides flexibility and adaptability in forming photonic circuits for various purposes, has recently been a research focus.10^,11

Thus, as described above, achieving optical power splitters with an ASR over an extensive wavelength range proves to be essential and valuable, and multiple kinds of structures have been proposed and adopted by researchers. Directional couplers (DCs),12^–14 multimode interference (MMI),15 subwavelength gratings,16 and Y-branch waveguides17 are some of the corresponding structures that have been widely discussed among them. The MMI power splitter is one of the most common splitters due to its simplicity and fabrication tolerance, but it suffers from high wavelength dependency and excess insertion loss by operating on the self-image effect. The subwavelength grating-optimized splitter has merits, such as wideband stability. However, processing accuracy and insertion loss are hard problems to resolve. The Y-branch waveguide works like a road junction that separates the light field in two ways. It has the advantages of compact size and large bandwidth. Nevertheless, mode mismatch loss is annoying when the branching angle is not sufficiently small. The DC is the original power splitter, relying on the coupling effect of adjacent waveguides and owning superior compactness and low loss, but it also has the problem of wavelength sensitivity. Nowadays, asymmetric tapers,18 adiabatic optimization,19 rib structures,20^,21 or other topological structures22 have also been utilized to improve the wideband characteristics of DC-based power splitters.

In this paper, we optimize the DC-based power splitters by adopting an artificial gauge field (AGF) waveguide structure, assisted by a neural network-aided inverse design method. The design of power splitters with wideband constant ratios of 5%/95%, 10%/90%, 30%/70%, and 50%/50% has been completed in simulation over a 100-nm wavelength range, and the first two splitting ratios have been realized with only 0.9 dB power fluctuation experimentally on a silicon on insulator (SOI) platform. We exploit this optimization to the coupling region of the microring and obtain a wideband uniform extinction ratio (ER) with a variation <1.6dB experimentally. These demonstrations of DC-based power splitters indicate great potential and universality in wideband applications.

The AGF, as an important physics concept, draws many novel and valuable phenomena into diverse physical fields, for example, ultracold atoms23 and acoustics.24 Furthermore, some applications of photonics, such as Floquet topological insulators,25 also benefit from adopting AGF structures. Especially, for PICs, precise light guiding, controlling, and confining can be achieved through AGF-based waveguide structures. The initial design of wavy shapes for AGF waveguides was proposed on an SOI platform, which aimed to achieve on-chip superlens imaging.26 Then various integrated optical devices, such as high-density waveguide arrays, optical switches, and mode converters, have been demonstrated with the help of AGF waveguides.27^–32

In the field of PICs, these mentioned AGF-based structures are introduced by the curved trajectory of waveguides, which means modulating the waveguide edges with specific amplitude periodically; one typical trajectory for AGF is the sinusoidal curve, shown in Fig. 1(a). The schematic and simulated light field of one specific waveguide array, composed of sinusoidal AGF waveguides, is shown in Fig. 1(b). The schematic and light field of a conventional straight waveguide array are also shown in Fig. 1(b) for comparison. It can be observed that by introducing these AGF structures into waveguide shapes, the coupling conditions between two adjacent AGF waveguides turn out to be entirely different from those of straight waveguides. The figures indicate that the light field, propagating in the straight waveguide, has a large cross talk and couples into the adjacent waveguide easily, while that in the AGF waveguide is strongly confined with hardly any dissipation, expressing an extremely strong coupling suppression. This property is beneficial for the design of dense waveguide arrays, for example, in Ref. 33, which results in a reduction of light cross talk, occupied chip area, and further improvement of device performance. However, for waveguide arrays with AGF structures, the light field may not only suffer strong coupling suppression but also be stably split. By adjusting the AGF waveguide structures properly, wideband stable light power splitting might be achieved with a constant coupling length of the waveguide.26 The simulated light-field distributions of two straight and AGF waveguides are shown in Fig. 1(c). The white dotted lines in each field refer to the position where the light field is fully coupled to the adjacent waveguide. This indicates that the distribution of two AGF waveguides does not obviously change with the change of light wavelength, from 1500 to 1600 nm, while that of straight waveguides varies a great deal; the dotted lines reveal this. Theoretically speaking, the coupling dispersion of a straight waveguide is the key point that influences its coupling, and it is precisely because the dispersion changes with the wavelength that the coupling condition of straight waveguides changes with wavelength. For AGF waveguides, there exist two kinds of dispersion: straight coupling and AGF coupling dispersion; the latter can be an opposite number of the former by adjusting the waveguide structures. In this way, coupling dispersion can be fully compensated for by these AGF structures, achieving goals of flat coupling dispersion and further robust coupling.26 This property is undoubtedly beneficial for those on-chip devices that aim at wideband systems; by adopting AGF structures, wavelength insensitivity and wideband stability will be improved.

Accordingly, to achieve robust waveguide coupling and further stable power splitting, we apply AGF structures with the typical sinusoidal curve to our optimization of power splitters and microrings.

The main parameters of an AGF waveguide include the modulation amplitude A, period length P, waveguide width W, gap G, and the number of periods C, as shown in the right part of Fig. 2. We should adjust these structural parameters properly; then the sinusoidal directional coupler would have a specific constant splitting property that meets the demand. Considering this target, machine learning might be a valid method to achieve this multi-object optimization. The neural network applied in the machine-learning method can generalize the input–output relationship of a complex nonlinear function. For the design of an AGF waveguide, this can be used as a tool to help us fit the complex mapping relationship between the target splitting ratio and corresponding structural parameters, as shown in the middle part of Fig. 2. The splitting values [t1,t2,t3,…] are treated as input of the network, and the structural parameters are treated as output. After the two important steps of building a data set and a training network, precise structural parameters would be found automatically by inputting the target splitting ratio values into the properly trained neural network.

The first key point to achieve inverse design is building a data set. In order to ensure compactness, design, and fabrication feasibility, we let C=1 for the aim of a DC power splitter with wideband flat ASR. Then, we sweep the four structural parameters, A, P, W, and G, in simulation via FDTD to obtain the corresponding splitting ratio. Each piece of data in the data set contains the input and output parts that correspond to the input and output of the neural network. The input part includes splitting ratio values (transmission of bar and cross port), which are sampling values at different wavelengths λ. The output part includes the corresponding structural parameters.

The sweeping ranges of four structural parameters in the data set are shown in Table 1, and for each parameter, the sweeping points are at equal intervals in their range. These ranges are determined according to a pre-analysis. For the pre-analysis, we set larger ranges and intervals for every parameter, then simulate and obtain a pre-data set with a small amount (around 200). According to the pre-data set, we do analysis and remove a part of the ranges, in which the waveguide cannot propagate a light field, or the wavelength insensitivity is not presented. This is the process to determine the ranges mentioned in Table 1, and according to this range, the simulation is then completed to build the training data set. The total number of values in the training set is 4500, and for splitting ratio values, the sampling interval of wavelength is 11.1 nm, ranging from 1500 to 1600 nm, which means for each piece of data, there are 10 values in the input part.

For the second point of training, the neural network Toolbox based on MATLAB is applied to create, train, and test the network. The loss function is the mean square error, and the activation function is Tansig in hidden and output layers, which can effectively normalize and transfer the output data, unify the input range, and maximize the use of the active interval of the activation function. Mapping between input (splitting ratio) and output (structural parameter) coefficients is solved through feeding and training of the network, which can be used for prediction.

After completing these two points, the structure that possesses one specific splitting performance is found by inputting the ratio value; the prediction network gives the structural parameters. Through this process, the wavelength insensitivity of AGF waveguide power splitters and the feasibility of the inverse design method are proved. Such a kind of inverse design method has been applied to quite a few photonic structures and fiber designs, and it was found to be competent for supporting photonic design automation.34^–37

As for the results, we designed four kinds of DC power splitters via this method, which possess splitting ratios of 5%/95%, 10%/90%, 30%/70%, and 50%/50%, respectively. Structural parameters and actual transmission curves of the bar and cross port in the simulation are shown in Table 2 and Fig. 3. It should be noted that although the coupling region looks like an arc in the figures, it is a sinusoidal curve of a full period. As indicated in Figs. 3(a)–3(d), the transmission variations of these four kinds of power splitters are below 1%, 2%, 3%, and 8%, respectively, from 1500 to 1600 nm, which shows the feasibility of the inverse design method. In addition, we also analyze the propagation loss introduced by the AGF-DC device in Fig. 3(e). The total transmission, shown as the y axis, is the normalized sum of the power transmission on the cross and bar ports. It is obvious that the total transmissions of all four different splitters, ranging from 1500 to 1600 nm, are basically above 88% in simulation. This means more than 88% of the power has passed through the device and been split to cross and bar ports. In other words, the excess loss introduced by the device is lower than 0.6 dB, which is at a low and tolerable level. With this being done, DC power splitters with ASRs can be designed efficiently.

To verify the actual wideband performance of the power splitters mentioned above, we fabricated those with the structural parameters A and B in Table 2 on the SOI platform.

The image taken by microscope is shown in Fig. 4(a). The transmission of the cross port of the AGF power splitter is tested, along with a conventional straight waveguide power splitter as a comparison. A 3 dBm light is coupled to the chip through the lensed fiber; the whole edge coupling loss is ∼4.5dB. The transmission results are shown in Fig. 4(b), where the source power and coupling loss are subtracted. Splitters A and B in the figure correspond to structural parameters A and B, respectively, and splitter Strt corresponds to the straight waveguide splitter. It can be deduced from the figure that splitter A has a low 0.9-dB transmission variation, from −13.6 to −12.7dB, which refers to the splitting ratios from 4.4% to 5.4%. The splitter B has a 1.4-dB transmission variation from −10.3 to −8.9dB, which refers to the splitting ratios from 9.3% to 12.8%. Both are over a 100-nm wavelength range from 1500 to 1600 nm. The transmission curves of A and B show a noticeable flatness; however, the transmission of splitter Strt varies from −9 to −14.0dB, splitting the power with the ratio from 4.0% to 11.0% in the cross port and showing a much larger variation compared to A and B. The results of the splitting ratio are shown in Figs. 4(c) and 4(d), transferred from the transmission in Fig. 4(b), and we compare them with the simulated results mentioned in Figs. 3(a) and 3(b). This indicates that the splitting ratio lines fit well with the previous simulation results. Therefore, the inverse design method and fabricated splitters indicate great performance in obtaining arbitrary power splitters with noticeable wideband wavelength insensitivity. The power splitters are fabricated by AMF, Singapore: |p|2+|r|2=1,(1)T=1−α2+|p|2−2α|p|cos(θ+ϕp)1+α2|p|2−2α|p|cos(θ+ϕp).(2)

The ER is a key factor of the microring, and a uniform ER is beneficial for applications, such as microring modulators and the generation of soliton combs. Moreover, the ER has a strong connection with the coupling condition of the microrings. The transmission formulas of microrings are shown in Eqs. (1) and (2), based on coupled-mode theory.38T is the transmission ratio of the optical power, α is the loss ratio of the microring, p is the self-coupling coefficient for the bus waveguide, and r is the mutual coupling coefficient between the bus waveguide and the microring. θ is the total phase and ϕp is the phase mismatch caused by coupling. As described above, the mutual coupling coefficient r of the coupling region is seriously wavelength-sensitive, which affects the ER of the microring on a wide waveband. Figures 5(a) and 5(b) show the transmission curves of two rings and their corresponding coupling regions as examples, where r are different from each other, while other parameters remain the same. The coupling coefficient r in Fig. 5(a) varies linearly, and the corresponding ER of the microring suffers a fluctuation as a result, while the ER in Fig. 5(b) remains stable. It can be easily deduced that a uniform coupling region of microring results in a uniform ER.

For conventional microring, the coupling region is a couple of straight waveguides whose coupling property is sensitive to wavelength, just as described above, which degrades wideband ER performance. As a demonstration, we adopt the above AGF-based power splitters in the microring coupling region experimentally on the SiN platform instead of straight waveguide coupling, as shown in Fig. 6(a). The reason we fabricate these microrings on a SiN platform is to illustrate the broad application of power splitters in forming complex on-chip devices; moreover, the universality of AGF structures and the inverse design method are based on different materials. The microscope image of this microring is shown in Fig. 6(b); the right bottom part is the conventional coupling region of microrings.

For the microring with such a coupling region, similar parameter optimization via the inverse design method is completed, and the target of the power splitting ratio of the splitter’s cross port is set to be 2%, considering the intrinsic loss of the SiN waveguide. The parameters of this sinusoidal power splitter are C=1, W=1μm, A=1.52μm, P=13μm, G=0.34μm, and the microring is in a racetrack shape, with a 1500-μm perimeter. Figure 7(a) is the spectrum sweeping results from 1500 to 1600 nm for the optimized microring with a source power of 0 dBm; as a comparison, Fig. 7(b) is that of a conventional microring. It is obvious that the ER of the optimized microring is uniformly wideband uniform and possesses a variation of only 1.6 dB, from 3.9 to 5.5 dB, as shown in Fig. 7(c). However, conventional microring suffers a severe ER fluctuation. It can be indicated that this sinusoidal waveguide-optimized coupling region is greatly beneficial for maintaining the ER of the microring, keeping it constant in an extensive wavelength range. The microring is fabricated by LIGENTEC, Switzerland.

In this paper, we demonstrated the efficient and high-performance AGF-based DC power splitters and corresponding neural network-aided inverse design method. Wideband-flat DC power splitters with ASRs of 5%/95%, 10%/90%, 30%/70%, and 50%/50% have been obtained in simulation. The former two have been achieved experimentally, with a variation of less than 0.9 and 1.4 dB over a 100-nm wavelength range on an SOI platform. An AGF waveguide-optimized microring with a uniform ER whose variation is lower than 1.6 dB is also fabricated. These AGF waveguide-optimized structures and the relevant inverse design method can be valuable for most wideband-integrated applications by enhancing system performance.

Zhaonian Wang is a PhD student at Shanghai Jiang Tong University. He received his BS degree from the University of Electronic Science and Technology of China in 2019.

Biographies of the other authors are not available.