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Abstract
Ultracompact mode multiplexers based on mosaic structure for various wavelength bands designed by Bayesian technique are investigated. C-, O-, and C + O band, TE0-TE1 2-mode multiplexers can be designed with the same footprint, by only changing the mosaic-pattern, showing the great flexibility of mosaic-based devices. Bayesian direct binary search method is used for the design, and it is demonstrated that the Bayesian technique is superior to conventional design method in terms of the best-structure search for the same number of iterations. The designed devices are fabricated for Si-waveguide platform, and the proof-of-concept results are obtained. These results indicate that the mosaic-based devices are promising candidates for future compact optical transceivers.
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1. Introduction
Recent advances in data center network requires a lot of developments in optics. The transceivers should be as small as possible to increase the spatial density, leading to the necessity of various small-form-factor transceivers. Since the area of the transceiver is limited, the optical chip is also as small as possible, and some kinds of integration have to be done, namely, light sources, detectors, and various passive components.
Mosaic-based devices are one of the promising candidates for such ultrasmall photonic devices. Mosaic-based devices are usually fabricated on Si-photonics platform and composed of input and output Si-wire waveguides and Si-plate placed between them. The Si-plate is divided into square pixels, whose typical side length is about 100 to 150 nm. The mosaic pattern is composed by etching selected pixels, and various components have been demonstrated, such as, ultrasmall polarization beam splitter [1], single(multi)-mode power splitters [2–6], waveguide crossings [7,8], and mode multiplexers (MUXs) [9,10].
As well as sizes, the data capacity should also be increased. Since wavelength-division-multiplexing (WDM) and digital-coherent technology have already introduced, different technique to increase the capacity is highly desired. A mode-division-multiplexing (MDM) technique may be one possible candidate. Although the mosaic-based mode MUXs for C-band use were demonstrated [9–11], the wavelength used in data centers depend on the system (C- or O-band are typically used). Therefore, mode MUXs for various wavelength bands are attractive, especially for data center networks.
In most of studies, the target mosaic pattern is designed by so-called direct binary search (DBS) algorithm. In the DBS algorithm, a pixel is randomly chosen and inverted, then, the device characteristic is calculated based on electromagnetic (EM) solver. If the characteristic is improved, the inversion is retained, if not, the inversion is cancelled. Although the DBS optimization is powerful, huge computational time is necessary and the results are changed with each trial [12]. Therefore, multiple trials should be done to obtain the (nearly) best structures. Recently, to overcome this problem, Bayesian DBS method was proposed in [13]. By using Bayesian inference, it was demonstrated that the better solutions can be obtained for asymmetric power splitter case. In [13], a lot of training data was gathered for specific splitting ratio before starting Bayesian design, and they are used as training data for other splitting ratio design. For the machine-learning based design, if there are a lot of training data, the effective design is possible. However, for the general device design, there are not always enough training data. Therefore, the effectiveness of the Bayesian DBS for just one device design and zero training data at the beginning was not demonstrated.
In this paper, ultracompact mode MUXs based on mosaic structure for various wavelength bands designed by Bayesian technique are investigated. C-, O-, and C + O band, TE0-TE1 2-mode MUXs can be designed with the same footprint, by only changing the mosaic-pattern. Bayesian DBS method [13] is used for the design. Here, the number of training data is zero at the beginning, and Bayesian approach is used after some DBS iterations. With the same condition, it is demonstrated that the Bayesian design is superior to conventional DBS method in terms of the best-structure search for the same number of iterations. The designed devices are fabricated for Si-waveguide platform, and the proof-of-concept results are obtained. Our work reveals that the great flexibility of the mosaic-based devices, which will be beneficial for future tunable mosaic-based devices. They are also promising candidates for future compact optical integration components.
2. Device structure and Bayesian direct binary search method
2.1 Device structure
Figure 1 shows the mosaic-based mode MUX considered here. The device is based on Si waveguide with the thickness of 220 nm. It consists of two inputs and one output waveguides. Between input and output waveguides, there is a Si-plate, in which mosaic-pattern is placed. The size of mosaic region is rectangle with the size of Lx = 3 µm, Lz = 3.75 µm. The region is divided into 20 × 25 = 500 square pixels, whose side length is 150 nm. Here, photonic crystal type mosaic structure is employed. In each pixel, there is a circular hole or not. If a hole is placed in the pixel, the diameter of the hole is 120 nm. The hole is embedded with the cladding material, SiO2. The widths of the waveguides are win = 0.6 µm, wout = 1.08 µm. 30-nm margin is added outside the mosaic region to protect hole shape [2]. The output waveguide is placed at the edges of Si-plate, as shown in Fig. 1. The target operation of the device is as follows. If the TE0 mode is launched from upper (lower) input waveguide, TE0 (TE1) mode is outputted from the output waveguide. The performance of the device is defined by fitness function, shown in section 3.
2.2 Bayesian DBS
Bayesian DBS is divided into two parts. One is the regression of the performance of mosaic-based devices based on known data. For the regression, we use a Gaussian process (GP) [14], which is a kind of regression technique based on Bayesian inference. The purpose of using GP is to learn output vector y from input vector x. In terms of mosaic-based device design, x and y correspond to the mosaic-pattern of the device and the device characteristics, respectively. In this paper, the GP is used to predict the performance of the device for unknown mosaic-pattern.
Once the regression can be done, the next task is to select next mosaic-pattern to be investigated in the DBS algorithm. For this purpose, we used Bayesian optimization [13,15,16]. In the Bayesian optimization, the input vector x for the maximum (or minimum) value of f(x) (fitness of the device with the mosaic pattern x) is searched as following. The next x* (the next mosaic pattern) to be searched, is determined by evaluating an acquisition function based on expected improvement (EI) as [13],
When selecting the next pixel in conventional DBS, there are 500 possibilities. The acquisition function, (1), is evaluated for all possible structure and select the structure with maximum aEI as the next structure. Then, the characteristic of the selected structure is calculated by three-dimensional finite-element method (FEM) [17,18], and if the performance is improved, the changed pixel is retained. If not, the changed pixel is returned to the original state. For both cases, the result is added to known data to increase the accuracy of the regression.
3. Design of mosaic-based mode multiplexers
Here, the design of mosaic-based mode MUXs for various wavelength bands are demonstrated without changing the footprint (the area of Si-plate).
3.1 Mode MUX for C-band
The device for C-band is designed here. The considered wavelengths are 1530, 1550, and 1570 nm. The fitness function used here is
For optimizing problem, the selection of the initial structure is very important. Here, we employ two initial structures. One is just a Si plate, where all pixels are Si. The other is the mosaic-based mode MUX designed for two-dimensional (2-D) equivalent structure [13]. For the optimization, a conventional DBS method is used for the first round of iteration period (500 times). If the conventional DBS is used, two more rounds of iteration periods are done. For the Bayesian DBS, after the first round of iteration period, the accumulated data are used as the input of GP, and next structure is selected by the algorithm shown in 2.2. After each calculation, the results of EM analysis is added to the known data. For the Bayesian DBS, 1000 times iterations (two iteration periods) are done after first DBS iteration period. Therefore, for both conventional and Bayesian DBSs, 1500 iterations were done.
Solid and dashed lines in Fig. 2 show the fitness as a function of iterations for C-band mode MUX. Two sets of lines correspond to different initial structures as shown in the inset. Up to 500 iterations, the results are the same for both methods. From 501 iteration, the behavior is changed, and the Bayesian DBS gives slightly faster convergences and better results, especially for Si-plate initial structure. For 2-D designed initial structure, the final fitness for DBS and Bayesian DBS are the same, and the same patterns are obtained accidentally. The designed pattern is shown in Fig. 3(a). Figure 3(b) and (c) show the electric field distributions of the device with 2-D designed initial structure for the Ports 1 and 2 inputs at 1550 nm. TE1 mode is successfully generated for Port 2 input. Figures 4(a) and (b) show calculated transmission spectra of the device for Ports 1 and 2 inputs. Rij is the reflected power for input port i to input port j. Left (right) panels correspond to Port 1 (2) input. Wavelength dependence is very small, and T1,TE0 > −0.25 dB, T2,TE1 > −0.84 dB for C-band. Figures 4(c) and (d) show the O-band spectra of C-band design. The losses are, of course, increased, and T1,TE0 = −2.35 to −2 dB, T2,TE1 > −6.4 to −5 dB from 1290 to 1310 nm. Compared with previous studies of C-band mode MUX, the loss is similar ( [9], T1,TE0, T2,TE1 > −0.47 dB, [10] T1,TE0 > −0.39, T2,TE1 > −0.49 dB). In this work, T1,TE0 is smaller and T2,TE1 is larger compared with [9,10]. This is probably due to the value of α in the fitness function (5). If we use smaller α, the balance between T1,TE0 and T2,TE1 will be improved.
Fig. 3. (a) Mosaic patterns of C-band mode MUXs designed by conventional and Bayesian DBS. (b) and (c) Field distributions of mode MUX designed by Bayesian DBS at 1550 nm. 2-D designed initial structure is used.
Fig. 4. Calculated transmission spectra of C-band mode MUX. (a) and (b) C-band characteristics. (c) and (d) O-band characteristics. The left (right) panels are for Port1 (2) input.
3.2 Mode MUX for O-band
Here, the device for O-band is designed. The considered wavelengths are 1290, 1300, and 1310 nm. These wavelength range is used in 100-Gbit/s Ethernet [19]. As in the C-band case, two initial structures are considered, namely, Si-plate and 2-D designed pattern. Solid and dashed lines in Fig. 5 shows the fitness as a function of iterations for O-band mode MUX. Two initial structures are also shown as the inset. For both initial structures, the final fitness for DBS and Bayesian DBS are almost the same. For Si-plate, the final fitness for conventional DBS design is slightly better. However, this does not mean conventional DBS is superior to Bayesian DBS. As explained in the Introduction, the results of the conventional DBS changes with each trial. As shown later in the next section, the Bayesian DBS gives better results stably than conventional DBS. Figure 6 shows the designed mosaic pattern and electric field distributions at 1300 nm for the device designed by Bayesian DBS with 2-D designed initial structure. Figures 7(a) and (b) show the C-band spectra of the O-band design. The losses are, of course, increased from C-band design, and T1,TE0 = −2.1 to −1.5 dB, T2,TE1 > −2.9 to −2.6 dB from 1530 to 1570 nm. Figures 7(c) and (d) show calculated transmission spectra of the device for Ports 1 and 2 inputs. Left (right) panels correspond to Port 1 (2) input. Wavelength dependence for the target wavelength range is very small, and T1,TE0 > −0.7 dB, T2,TE1 > −0.9 dB from 1290 to 1310 nm.
Fig. 6. (a) Mosaic pattern of O-band mode MUXs designed by Bayesian DBS. (b) and (c) Field distributions of the mode MUX at 1300 nm. 2-D designed initial structure is used.
Fig. 7. Calculated transmission spectra of O-band mode MUX. (a) and (b) C-band characteristics. (c) and (d) O-band characteristics. The left (right) panels are for Port1 (2) input.
3.3 Mode MUX for C + O-band
The results in 3.1 and 3.2 show that the mosaic-based devices can be applied to various wavelength bands with the same footprint, just by changing its mosaic pattern. Here, we try to design the mode MUX for C + O bands. The considered wavelengths are 1300 and 1550 nm.
Figure 8 shows the fitness as a function of iterations for C + O-band mode MUX with each initial structure. For both initial structures, the difference in the final fitness is clear and the Bayesian DBS gives better results and faster convergences compared with the conventional DBS. Figure 9(a) shows the mosaic pattern of C + O-band mode MUX with the 2-D designed initial structure and Fig. 9(b) shows the electric field distributions for 1300 and 1550 nm. Intended characteristics are obtained for two wavelength bands. Figure 10 shows the calculated transmission spectra of the device for Ports 1 and 2 inputs for C- and O-bands. Left (right) panels correspond to Port 1 (2) input. Again, wavelength dependence for the target wavelength range is very small, and T1,TE0 > −0.4 dB, T2,TE1 > −1.3 dB from 1290 to 1310 nm, and T1,TE0 > −0.16 dB, T2,TE1 > −1.1 dB from 1530 to 1570 nm.
Fig. 9. (a) Mosaic pattern of C + O-band mode MUXs designed by Bayesian DBS. (b) Field distributions of the mode MUX at 1300 and 1550 nm. 2-D designed initial structure is used.
Fig. 10. Calculated transmission spectra of C + O-band mode MUX. (a) and (b) C-band characteristics. (c) and (d) O-band characteristics. The left (right) panels are for Port1 (2) input.
To show the fabrication tolerance, transmission characteristics at 1300 and 1550 nm are investigated for hole diameter change, Δd (difference from 120 nm). Figure 11(a) shows transmission of C + O band mode MUX for Port1 input. Red and blue lines correspond to 1550 and 1300 nm. For Port1 input (TE0 to TE0 mode), there are almost no transmission change for 20-nm change in the diameter. Figure 11(b) shows the same thing for Port2 input (TE0 to TE1 mode). For Port2 input, the hole diameter change affects much compared with Port1 input. For Δd = + 20 nm, the transmissions are degraded by 1.4 and 2.4 dB for 1550 and 1300 nm. Although the losses are increased, the mode MUX operation is still possible due to small crosstalk components.
Fig. 11. Transmission as a function of Δd for (a) Port1 and (b) Port2 inputs of C + O band mode MUX.
As explained in the Introduction, the final results of DBS design are changed with each trial. Therefore, 10 different trials were done for C + O band mode MUX design. The initial structures are Si-plate. As previous examples, first iteration period (500 times) is done with conventional DBS, and second and third iteration periods (1000 times) are done with conventional or Bayesian DBS. Black line in the Fig. 12 shows the fitness at the first iteration. Since conventional DBS algorithm is used, the values are fluctuated. Blue and red solid lines are the final fitness obtained by conventional and Bayesian DBSs. Bayesian DBS gives better results in 9 trials (except for No.7, but the difference is very small), showing the usefulness of the Bayesian DBS for the stable design of the mosaic-based devices.
The results in this section show that the great flexibility of the mosaic-based devices. The device for various wavelength bands can be designed only by changing the mosaic-pattern without changing their footprint. If a tunability can be added by using, for example, phase change material [20], flexible and reconfigurable operation of the device is possible, which is useful for all-optical network and photonic computing with ultrasmall footprint.
4. Experimental results
The C + O band mode MUX shown in 3.3 is fabricated for the proof-of-concept devices. The 220-nm SOI wafer is patterned with electron beam lithography. The pattern is etched with reactive ion etching. Finally, SiO2 cladding is deposited by using chemical vapor deposition. The wafer is diced to the chip and the light from the amplified spontaneous emission source is used as the light source and an optical spectrum analyzer is used to receive the light from the chip. For the measurement, we fabricated the same device on the same chip as shown in the left panel of Fig. 13, and the transmission is measured for the four optical paths. The distance between two devices is Lmid = 50 µm. The SEM picture of the fabricated device is shown in the right panel of Fig. 13. The picture was taken by removing the SiO2 cladding of the fabricated device by using HF. The diameters of the fabricated holes are around 140 to 145 nm, which is significantly larger than the design value. Also, the etching depth in the hole is about 150 nm. These values can be improved by optimizing the process conditions.
Fig. 13. (Left) The measurement schematic and (right) the SEM picture of the fabricated C + O-band mode MUX.
Solid lines in Fig. 14 show measured spectra of the device. For the measured data, the loss of the reference straight waveguide is subtracted, and therefore, the spectra are transmissions for two mode MUXs. Upper (a) and (b) (bottom (c) and (d)) panels are for C- (O-) bands. Left (right) panels correspond to Port 1 (2) input. To compare with the measured data, we calculated the transmission spectra of two mode MUX by using following equation:
Fig. 14. Calculated and measured transmission spectra of C + O-band mode MUX. (a) and (b) C-band characteristics. (c) and (d) O-band characteristics. The left (right) panels are for Port1 (2) input.
5. Conclusion
Mosaic-based ultracompact mode MUXs for various wavelength bands are demonstrated. C-, O-, and C + O band, TE0-TE1 2-mode MUXs can be designed with the same footprint, by only changing the mosaic-pattern, showing the great flexibility of the mosaic-based devices. Bayesian DBS method is used for the design, and it is demonstrated that the Bayesian design is superior to the conventional design method in terms of the best-structure search and stability. The proof-of-concept device was fabricated and the intended device operation was successfully demonstrated. Although we treat only TE1 mode MUX, the design of other modes is straight forward. These results indicate that the mosaic-based devices are promising candidates for future optical applications, requiring ultrasmall photonic integration.
Funding
Japan Society for the Promotion of Science (21H01378).
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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