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Abstract
Optical power splitters with ultra-broadband and ultra-low insertion loss are desired in the field of photonic integration. Combining two inverse design algorithms for staged optimization, we present the design of a Y-junction photonic power splitter with 700 nm wavelength bandwidth (from 1200 nm to 1900 nm) within a 0.2 dB insertion loss, corresponding to a 93 THz frequency bandwidth. The average insertion loss is approximately −0.057 dB in the valuable C-band. Moreover, we comprehensively compared the insertion loss performance of different types and sizes of curved waveguides, and also give the cases of 1:4 and 1:6 cascaded power splitters. These scalable Y-junction splitters provide new alternatives for high-performance photonic integration.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
In the era of big data, the requirements for broad bandwidth photonic integrated circuits (PICs) are evident. Compact optical power splitters with high bandwidth and low insertion loss are important components of PICs. Various optical power splitter device designs have been proposed. The multimode interference (MMI) coupler structure can be applied to diverse photonic devices [1–5] because of its compactness and good manufacturing tolerances. Another common optical power splitter scheme is the directional coupler (DC) [6–9], which has a simple fabrication process and can realize any power split ratio. However, traditionally, MMI and DC devices are sensitive to wavelength changes, and their operating bandwidths generally do not exceed 200 nm. Y-junction waveguides are widely used in optical power splitters because of their wavelength independence and low insertion loss characteristics; further, they have various design schemes, such as the antenna coupled Y-junction [10,11], wide-angle Y-junction [12,13], adiabatic tapers [14–16], and subwavelength grating (SWG) Y-junction [17–19]. These Y-junction schemes are effective in principle, but most of them are designed based on manual parameter selection or semi-analytical models. Limited by the small parameter search space, they are hindered from realizing optimal multiple performance indicators.
Driven by the demands for PIC development, inverse design offers a novel approach to reconcile high performance requirements and small size constraints through advanced iterative algorithms [20,21]. These algorithms include direct binary search (DBS) [22–25], the adjoint method [26–28], objective-first method [29,30], and deep learning [31–33]. In recent years, various inverse-design photonic integration devices have been reported, such as the mode-division multiplexer [34,35], wavelength demultiplexer [36,37], and polarization beam splitter [38,39]. The implementation schemes of various inverse design power splitters [40–44] include the following: a dual-mode 3-dB power splitter [42], with a footprint of 2.8 × 2.8 µm, an excess loss of −1.5 dB, and a bandwidth of 80 nm; a 1 × 3 power splitter with an insertion loss of −0.642 dB in the wavelength range of 1400–1700 nm [43]; and an asynchronous double-deep Q-learning algorithm to realize an MMI power splitter with a broad bandwidth of 1200–1650 nm, breaking through the bandwidth limitation of traditional MMI [44]. Reference [45] proposed a unified density-based topology optimization framework, and on this basis designed a low insertion-loss T-branch device in the bandwidth range of 1500–1600 nm. These research progresses motivated us to obtain a Y-junction power splitter with ultra-broadband and ultra-low insertion loss through inverse design.
Here, we combined the shape-adjoint optimization algorithm with DBS in the inverse design, fully combining the advantages of the two algorithms. The Y-junction waveguide designed in this way has not only extremely low insertion loss, averaging −0.057 dB in the C-band, but also an ultra-broad bandwidth, with a frequency bandwidth of 93 THz and a wavelength bandwidth of 700 nm in the transmission variation range of 0.2 dB. At present, the research of silicon photonics technology focuses on the traditional wavelength range of 1200–1600 nm [46]. The proposed Y-junction waveguide device completely covers the wavelength range of 1200–1900 nm, which, to the best of our knowledge, is the broadest bandwidth reported thus far for a Y-junction power splitter. Our device could satisfy the requirements of the next generation of high-performance photonic integration.
2. Design and optimization
Figure 1(a) shows a schematic of the proposed Y-junction optical power splitter, which is based on a standard silicon-on-insulator platform and consists of an input waveguide, a Y-junction waveguide, several pixel points, and two output waveguides. The silicon core layer on the top is 220 nm, and the thickness h of the buried silicon dioxide layer is 3 µm. The width d of the input and output waveguides is set to 590 nm, the length L1 of the Y-junction coupling region connected to the input waveguide is 2 µm, the width W1 is 1.2 µm, the length L2 of the curved waveguide region is only 3 µm, and the width W2 is only 2.68 µm. The gap g between the two output waveguides is set to 50 nm, the size of the pixel is set to 100 × 100 nm, and the depth of each pixel is 220 nm. Figure 1(b) shows that the insertion loss of the proposed device has a bandwidth of 700 nm within a 0.2-dB variation range, covering the wavelength range from 1200 nm to 1900 nm, and its frequency bandwidth is 93 THz, covering the frequency range from 157 to 250 THz. The wavelength range of the traditional low-loss optical communication band from the O-band to the L-band is 1260 nm to 1625 nm, which is completely covered by the working wavelength of the proposed device. The insertion loss variation range from the O-band to the L-band is only −0.08 dB, which greatly improves the applicability and scalability of the device.
Fig. 1. (a) Schematic of the designed Y-junction optical power splitter; the inset on the right shows its top view and the size of the output waveguide gap. (b) Simulated insertion loss curves of the device. (c) Transmission contrast curve.
The structure was simulated using the three-dimensional finite difference time domain (3D FDTD) method; the overall algorithm flow is shown in Fig. 2(a). The algorithm flow is divided into three stages. In the first stage, the shape-adjoint optimization algorithm is used to design the Y-junction coupling region. In the second stage, the optimized coupling structure is used to design the curved waveguide, which also uses the shape-adjoint optimization algorithm. In the third stage, the DBS algorithm is used to continue to optimize the structure after the second-stage optimization, and we apply DBS algorithm outside the branch area. The purpose of this method is to recycle some of the light field energy leaked outside the waveguide into the waveguide, so as to achieve low insertion-loss and ultra-broadband. If these discrete mosaic patterns are attached to the branches, it will cause irregular changes in part of the outer boundary of the curved waveguide, which will further lead to increased insertion loss.
Fig. 2. (a) Flowchart of the overall algorithm. (b) Schematic of the specific optimization process. The blue, green, and purple boxes and numbers represent the coupled region, the curved waveguide, and the DBS algorithm optimization process, respectively.
In the shape adjoint optimization stage, we need to maximize the absolute value of the electric field of point ${x_0}$ in the given optimization region Ω, so figure-of-merit (FOM) can be expressed in the form of Eq. (1).
When ${x_0}$ changes slightly, that is, when the optimized shape changes, the corresponding FOM changes can be expressed as:
In Eq. (2), ${E^{old}}({{x_0}} )$ represents the electric field before ${x_0}$ changes. The change of ${x_0}$ will cause the change of dielectric value, and ΔE(x_0) represents the change of electric field caused by dielectric modification. Introduce the variation of the dielectric permittivity Δ${\varepsilon _r}$ into Eq. (2) and derive Eq. (3) accordingly. Please refer to Ref. [26] for the detailed derivation process.
Here, ${E^{adj}}(x )$ represents the adjoint simulation, while ${E^{old}}(x )$ represents the electric field distribution obtained from a forward simulation. Therefore, in the shape adjoint optimization algorithm, each time the shape changes, only one forward simulation and one adjoint simulation are required to calculate the gradient information of the shape derivative and quality factor of the optimization region.
The above equations are the mathematical explanation of the algorithm. In the actual application process, we define FOM as Eq. (4):
${T_0}$ represents the set target transmittance value, and T represents the actual transmittance value obtained at the output waveguide. ${\lambda _1}$ and ${\lambda _2}$ represent the start and end of the wavelength, respectively. ${\lambda _i}$ represents the wavelength within the range of ${\lambda _1}$ and ${\lambda _2}$, and its value is related to N. N is the number of wavelength points we select within the range of ${\lambda _1}$ and ${\lambda _2}$, the larger N is, the finer the division of ${\lambda _i}$ is. We set ${T_0}$=1, ${\lambda _1}$=1200 nm, ${\lambda _2}$=2000nm, N = 81.
However, we think that different algorithms are suitable for the design of different devices. We slightly improved the traditional shape adjoint optimization algorithm to make it more suitable for the Y-junction waveguide we designed. For the shape adjoint optimization algorithm, it is necessary to establish the functional relationship between the x-coordinates and y-coordinates of the outer boundary points in advance, so that the corresponding boundary points can be found according to the functional relationship when the shape changes next time. In the initial configuration, we constrained the y-coordinate optimization range for both the outer and inner boundaries of the upper half of the curved waveguide to a range of 0-1.8µm. The lower half was designed symmetrically. However, in the actual design process, it is found that because the limit range of y coordinate changes is not enough. It may produce some abrupt points in the optimization process, and for the curved waveguide, there are two boundaries that need to be optimized at the same time, if not restricted, it may produce the phenomenon that the two boundaries have intersection. In order to make our optimization structure smoother, we incorporated the min-max normalization method into the shape adjoint optimization. However, since this method can only normalize data values to [0,1] or [-1,1], certain modifications were necessary to normalize the data into the specified interval [a, b], as indicated in Eq. (1).
Taking the optimization of the upper part of a curved waveguide as an example, X represents the y coordinate value of the outer or inner boundary of the curved waveguide, which varies from 0 to 1.8µm. ${X_{min}}$ and ${X_{max}}$ respectively represent the minimum and maximum values in the set of data X. a, b represents the normalized interval, and ${X_{nom}}$ represents the y coordinate value after the normalization of the outer boundary or the inner boundary. The outer boundary of the curved waveguide corresponds to 0.6µm and 1.34µm, and the inner boundary is 0.025µm and 0.75µm, respectively.
In the DBS algorithm optimization stage, each pixel is traversed in turn, and the state of each pixel is determined according to the change of the FOM value (“0” is air, “1” is silicon). The influence of a single pixel on the transmittance of different wavelengths is not the same. Simulation experiments demonstrated that the influence of a single pixel on wavelengths with higher transmittance can be neglected. In this case, taking the average transmittance at each wavelength as a measure of FOM will not improve the overall performance of the device. Therefore, we introduced a weight factor into the FOM calculation expression:
Here, N represents the number of wavelength points without using weight factors, i.e., the number of wavelength points between ${\lambda _a}$ and ${\lambda _b}$. M represents the number of wavelength points using weight factors, i.e., the number of wavelength points between ${\lambda _b}$ and ${\lambda _c}$. w is the weight factor. $\mathop \sum \limits_{{\lambda _a}}^{{\lambda _b}} T_{{\lambda _i}}^{up + down}$ represents the sum of the transmittances of the upper channel and the lower channel of N wavelengths in the wavelength range of ${\lambda _a}$ and ${\lambda _b}$, and its main function is to improve the performance of ${\lambda _b}$ to ${\lambda _c}$ band without causing too much impact on the performance of ${\lambda _a}$ to ${\lambda _b}$ band. $\mathop \sum \limits_{{\lambda _b}}^{{\lambda _c}} T_{{\lambda _j}}^{up + down}$ represents the sum of the transmittances of the upper channel and the lower channel of M wavelengths in the range of ${\lambda _b}$ and ${\lambda _c}$. Here, N = 15, M = 5, $w$=5, ${\lambda _a}$=1200 nm, ${\lambda _b}$=1800 nm, ${\lambda _c}$=2000 nm. The final structure is obtained after several iterations of optimization.
Figure 2(b) shows the specific optimization process of the entire Y-junction waveguide. Each optimization process corresponds to the different colored numbers in Fig. 2(a). The blue “1” represents the initial structure of the coupling region; the size of the optimized region is 2 × 2 µm, symmetric about the x-axis. The outer boundary of the optimized structure is composed of discrete points. In the coupling area, we set the x-coordinate of the outer boundary to 30 equally spaced nodes. Their corresponding y-coordinates are the targets to be optimized, and the optimization range of the y-coordinates is set to 0–2 µm. The shape of the first stage, along with the coupling region structure obtained after optimization, is shown at blue “5.” Blue “4” represents the different shapes obtained during the optimization process, and blue “3” represents the change curve of the FOM during the optimization process.
Optimizing the curved waveguide according to the obtained coupling region structure is the second stage of optimization. The initial optimized structure of the curved waveguide is shown at green “1.” The size of the optimized area is 3 × 3 µm, and the number of nodes is set to 50. The optimized structure in the second stage is shown at green “5.” The final structure formed by DBS algorithm optimization is shown at purple “2.”
3. Simulation and results
Different gap sizes not only have different effects on the simulation results, but also have different degrees of manufacturing difficulty. Figure 3(a) shows the insertion loss comparison for six different gap sizes. The results show that a smaller gap corresponds to lower insertion loss of the device, but the gap between 0 and 50 nm is not significant. But the smaller gap, the greater difficulties during manufacturing process. for example, the reactive ion etching (RIE) lag effect is a typical manufacturing error in etching process. The RIE lag effect causes under-etching occur in regions with small feature sizes [47,48]. On the other hand, when the gap is larger than 100 nm, the performance of the Y-junction waveguide decreases significantly, which is also reflected in the distribution of the light field. Considering the balance of fabrication and device performance, we chose a gap size of 50 nm for the discussion of the subsequent experimental results.
Fig. 3. Effect of different gap sizes on the performance of the device. (a) Insertion loss curves of different gaps. (b)–(g) Light field distributions of different gaps at 1550 nm.
To verify the advantages of phased optimization method, we compared the performance of three different states in the optimization process. Figure 4(a) shows the initial state structure, which consists of a tapered waveguide and two pairs of curved waveguides spliced into 90° arcs with a radius of 1.5 µm. Figure 4(b) shows the structure after shape-adjoint optimization of the coupling region. Figure 4(c) shows the result of shape-adjoint optimization of the curved waveguide using the structure optimized in the coupling region. Figures 4(d)–(f) show the light field distributions of these three different states at 1900-nm wavelength. Figure 4(g) shows the insertion loss contrast curves of the three different states in the wavelength range of 1200–2000 nm. The simulation results show that the insertion loss of the initial shape is higher than 0.2 dB, and the bandwidth of the middle state obtained after one-time shape-adjoint optimization is approximately 400 nm within the range of 0.2 dB change. However, the Y-junction waveguide obtained after two shape-adjoint optimizations has a bandwidth of up to 650 nm.
Fig. 4. (a)–(c) Three different state structures during two shape-adjoint optimizations and their (d)–(f) light field distributions. (g) Insertion loss curves of the three states.
Since our Y-branch waveguide has reached a bandwidth of 650 nm after two shape adjoint optimizations, we want to further reduce the insertion loss near the wavelength of 1900 nm without affecting the performance of the existing device. The DBS algorithm is crucial in the inverse design method because of its good scalability and simple manufacturing process [39]. It can further improve the overall bandwidth of the device by changing the weight ratio of different wavelengths in the Eq. (6). Through introducing the DBS algorithm, some of the light field energy leaked outside the waveguide can be recycled back into the waveguide, so as to achieve low insertion-loss and ultra-broadband.
Considering that each pixel in the DBS algorithm has different effects on the insertion loss at different wavelengths, and the Y-branch waveguide we designed has ultra-broadband. Therefore, we adopt an asymmetrical DBS scheme, and use the difference in transmittance of the two output ports as one of the indicators to determine the state of each pixel point, so as to avoid the power imbalance of the two output ports.
In Eq. (6), we assigned the weight factor to the wavelength range of 1800–2000 nm and compared the performance of DBS before and after optimization, as shown in Fig. 5. Visually, this is not obvious in Fig. 5 where the width of the abscissa is 800 nm. But we think that the improvement of 50 nm is already equivalent to the bandwidth of a communication band (for example, the range of C-band is 1530–1565 nm). In addition, the introduction of the DBS algorithm allows us to selectively improve the performance of a specific band through different weight distribution ratios. In practical applications, the performance of devices can be selectively improved according to different application bands.
Fig. 5. Simulation of device performance (a), (c) before and (b), (d) after adding the DBS algorithm. (e) Insertion loss contrast curve.
Finally, after DBS optimization, an ultra-low loss Y-junction waveguide device design with a wavelength bandwidth of 700 nm was obtained. Table 1 compares the various performance parameters between our device and existing Y-junction optical power splitters.
Table 1. Performance Parameters of Various Y-Junction Optical Power Splitters
To verify the scalability of the phased shape optimization algorithm, we compared the performance of four different curved waveguides, as shown in Fig. 6. Considering that the region of DBS algorithm applied to different curved waveguides is also different, it may cause a certain deviation to the experimental results. Figure 5(e) has verified that adding DBS algorithm can further improve the device bandwidth. In addition, the optimization of the DBS algorithm needs a lot of iterative calculations, which also takes a lot of computing time. With the expansion of the curved waveguide region, the calculation time will increase linearly. Taking these factors into consideration, we did not add the DBS algorithm in the comparison of these four different curved waveguides.
Fig. 6. Light field diagram and performance comparison of curved waveguides with different sizes. (a), (e) 2 × 1.79 µm; (b), (f) 3 × 2.68 µm; (c), (g) 4 × 3.58 µm; (d), (h) 5 × 4.48 µm. (i) Insertion loss contrast curve.
The dimensions of the three curved waveguides were 2 × 1.79 µm, 3 × 2.68 µm, 4 × 3.58 µm, and 5 × 4.48 µm. According to the curve comparison results shown in Fig. 6(i), the four curves closely overlap in the wavelength range of 1200–1800 nm, which verifies the strong scalability of our phased shape-adjoint optimization algorithm.
To achieve different splitting ratios, we cascaded simulations of 1:4 and 1:6 optical power splitting devices, as shown in Fig. 7. For cascade schemes, simplicity and efficiency of the basic elements are important. However, adding the DBS algorithm will make our cascading scheme very complicated and weak in scalability, so we did not add the DBS algorithm. We changed the size of the curved waveguides in region 1 to 4 × 3.58 µm to prevent the output waveguides from being too close; the size of the curved waveguides in the other regions was 3 × 2.68 µm. The transmittance curves of the four output ports in the 1:4 cascade simulation are shown in Fig. 7(c). The blue curve represents the total transmittance of the four ports. In the wavelength range of 1400–1900 nm, the total transmittance was above 0.95, and the transmittance of each port was approximately 0.25. The transmittance curves of port 1 and port 4 closely overlap, and the transmittance curves of port 2 and port 3 closely overlap. The transmittance and total transmittance curves of the six output ports in the 1:6 cascade simulation are shown in Fig. 7(f). The transmittances of ports 1-4 are approximately 0.15, and the transmittances of ports 5 and 6 are approximately 0.25.
Fig. 7. (a), (d) Schematic of 1:4 and 1:6 cascade simulations. (b), (e) Light field distribution at 1550 nm. (c), (f) Transmittance curve of each port.
4. Conclusion
In summary, we combined the shape-adjoint optimization algorithm and DBS algorithm to obtain an ultra-broadband Y-junction optical power splitter with ultra-low insertion loss. In the broad range of 1200–1900 nm, the variation range of insertion loss of the entire device is only 0.2 dB, the frequency bandwidth is as high as 93 THz, and the wavelength bandwidth reaches 700 nm. We simulated the performance of four different sizes of curved waveguides; the simulation results show that the applicability of our design method is not limited to one specific size. The 1:4 and 1:6 cascade simulation results show that our cascaded device still has good performance in the wavelength range of 1400–1900 nm, and the overall insertion loss is lower than −0.2 dB. Overall, Y-junction devices designed with the proposed method exhibit considerable potential for application in high-density photon integration and 6 G optical communication networks.
Funding
Innovation Support Program for Overseas Students in Chongqing (cx2021008); China Postdoctoral Science Foundation (2018M633704); Innovation Research 2035 Pilot Plan of Southwest University (SWU-XDPY22012); Chongqing Science Funds for Distinguished Young Scientists (cstc2021jcyj-jqX0027); National Natural Science Foundation (60907003, 61805278, 62275271).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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