3.1. Inverse Design of As2S3 Waveguide
We generated the As2S3 waveguide data set with a scale of about 6000, and these data are divided into the train set and test set at a ratio of 80:20. Both the train and test set contain a number of complete pieces of data, which are used to train the network and check the performance of the network, respectively. Among the settings for the network, the ratio of 80:20 was one available option, and the algorithm automatically completes the division. The train set was used for training, including calculating the loss function and the optimizer to adjust the weight throughout the training process. Whereas, the test set was used for the network to check the values of loss function—it was not involved in the training process as it could only be used for testing. In this way, we ensured the reliability performance of the network evaluation via the test set. The results are displayed below.
For the inverse design of the As2S3 waveguide, we selected the target wavelength bands, ranging from 2000 nm to 2800 nm and from 1800 nm to 2900 nm, respectively (a wide and a narrow band), which means the input of network is on 9 [D1, D2, D3…, D9] dimensions or 11 [D1, D2, …, D11] dimensions, and the output was made up of 6-dimensional structural parameters [H1, H2, H3, Hs1, Hs2, W].
For the different target wavelength bands, where the structures of the network are the same, the different combinations of dispersion values were extracted from the data set to feed the network as input data. This also represents the efficiency and reusability of inverse design method. We did not use the entire dispersion values (1500 nm to 3500 nm for As2S3 waveguide and 1000 nm to 3000 nm for Si3N4 waveguide) due to the influence of the intrinsic zero dispersion of materials, which result in a rapid variation of waveguide’s dispersion inevitably. The reason for generating a data set with extra dispersion values was to retain the possibility of adjusting the target wavelength band freely by selecting different dispersion values as input of the network. This should not be time-consuming, and with different input data, the training process only lasts several tens to hundreds of seconds.
In
Figure 3, the prediction of two of the six structural parameters were used to display the training results for examples. The target wavelengths, in this study, are 2000 nm to 2800 nm. The X-axis is the actual parameter, and the Y-axis is the predicted values obtained by setting target dispersion as the corresponding dispersion values from test set. This scatter diagram does not display all the testing results, which are in a large amount. As shown in
Figure 3, most of points are close to the straight line of y = x with a small variation, which represents good performance of the network with the predicted values well close to actual values. We calculated the mean absolute error values of each structural parameters predicted by the trained network to check performance of the network more straightforward, as shown in
Table 3. We found that the MAE values of those structural parameters are all less than 1 nm, which is highly within the tolerance for practical fabrication.
In order to achieve the practical application of photonic integrated circuit design, we set three goals for dispersion engineering: Broadband low dispersion in different wavelength bands, broadband constant dispersion with positive or negative values, and slope-maintained linear dispersion, where the target dispersion values are all 0, a certain constant value, and a linear function respectively. By applying the trained network, predicted structures corresponding to the target dispersion can be obtained. We place these structural parameters into simulation, calculate the actual dispersion profiles, then compare the actual and target dispersion for verification.
The results of the inversely designed dispersion profiles of As2S3 waveguide obtained from trained Neural Network are shown in
Figure 4. The detailed structural parameters, mentioned in
Figure 4, is shown in
Table 4.
In
Figure 4a, the target wavebands are 1800–2900 nm and 2000–2800 nm respectively, which represents a wider band and a narrower band. Horizontal dotted line in the expanded Figure refers to the target dispersion values, which remained at zero, and represents the broadband low dispersion. The corresponding structures are A1 and A2, predicted by the network. Dispersion of A1 varies between 0–4.5 ps/(nm·km) for a 1154-nm bandwidth from 1885 nm to 3039 nm and dispersion of A2 varies between 0–1.5 ps/(nm·km) for an 861-nm bandwidth from 1945 nm to 2806 nm. The results of the predicted structures do not exactly follow the target bands due to the error of network. We found that, given that Structure A1 has a longer target band, its flatness of dispersion curve is not better than that of A2. However, it has a wider wavelength range with a relatively low dispersion. Compared with Reference [
6], a much lower dispersion in a slightly narrower bandwidth was obtained by the proposed network.
Figure 4b,c show the results of broadband constant dispersion and slope maintained linear dispersion. The target waveband is 2000–2800 nm.
Figure 4b shows five dispersion curves for structure A1, A3, A4, A5 and A6. The dispersion curve for A1 is for comparison. Broadband constant dispersion of 30, −30, 50, and −50 ps/(nm·km) can be predicted by the network. All five dispersion curves have wavebands of constant dispersion containing the target band, and the range of constant dispersion slightly exceeds the target band. As in
Figure 4a, the horizontal dotted lines in
Figure 4b refer to target values being fed into the network, corresponding to constant values mentioned above. Moreover,
Figure 4c shows the linear dispersion curves with four different slopes, corresponding to structures A7, A8, A9, and A10 predicted by the network. The vertical dotted lines, lying on 2000 nm and 2800 nm, indicate the linear area, whereby the four dispersion slopes are maintained to be 0.04, 0.02, −0.02, −0.04 ps/(nm
2·km), respectively. This kind of linear dispersion may be useful for compensation in fiber systems.
3.2. Inverse Design of Si3N4 Waveguide
Our proposed dispersion engineering method can also be applied to Si3N4 double slot waveguide, and it proves universality in photonic waveguide design. The process of network training is basically similar to that of As2S3 waveguide, but the new data set needs to be generated for theSi3N4 waveguide. The selected target wavelength band range from 1300 nm to 2200 nm and 1200 nm to 2400 nm, a wider band and a narrower band. Different material dispersion and modal refractive index should be expected for Si3N4 waveguide.
The scatter diagram containing the prediction of two parameters for Si
3N
4 waveguide is shown in
Figure 5, and the mean absolute error values of the results for Si
3N
4 waveguide are shown in
Table 5. The target band is 1300 nm to 2200 nm. The MAE values are also at a lower level, indicating good performance of prediction. However, the performance of the inverse design for Si3N4 waveguide is not as good as that of As
2S
3 waveguide, in terms of the predict accuracy in scatter diagram and mean absolute error value in
Table 5. In fact, there might be a best interval for scale of data set and variation range of the parameters in the data set, if deviated from this interval, the performance of network will decrease when other conditions remain unchanged. We attempted to find the best interval for Si
3N
4 waveguide, but a difference exists, maybe due to the sub-par interval for that of Si
3N
4 waveguide. Nonetheless, the results in
Figure 6 show that the network for Si
3N
4 waveguide also possess satisfying performance, thus the difference between
Table 3 and
Table 5 can be tolerated. The detailed structural parameters mentioned in
Figure 6 are shown in
Table 6.
As for the predicting results, in
Figure 6a, for structure B1, the broadband low dispersion varies from 0 to 5.8 ps/(nm·km) ranging from 1226 nm to 2368 nm for a 1142-nm bandwidth and for structure B2, dispersion varies from 0 to 1.4 ps/(nm·km) from 1306 nm to 2067 nm for a 761-nm bandwidth. Horizontal dot lines in zoom-in figure are the target dispersion values, all zero here, which represents broadband low dispersion. These well performed results in terms of much broader and lower dispersion compared with Reference [
7] would be highly useful for broadband nonlinear processing applications on Si3N4 platform.
Similar to As
2S
3, dispersion engineering, with maintained constant values or maintained dispersion slopes, is also carried out using the proposed method, as shown in
Figure 6b,c. Constant dispersions of 30, −30, 50, and −50 ps/(nm·km), ranging from 1300 nm to 2200 nm with small variations, are obtained by structure B3 to B6. While, the horizontal dot lines are target values, corresponding to the certain dispersion. The dispersion slopes of 0.04, 0.02, −0.02, −0.04 ps/(nm
2·km) from 1300 nm to 2200 nm are maintained, as indicated by the vertical dot lines, and obtained by Structure B7 to B10, as shown in
Figure 6c.
3.3. Influence of Sidewall Angle in Fabrication
Considering the fabrication of the horizontal double slot waveguide, there always exists a sidewall angle α, which means the wall of the waveguide is not perfectly vertical, as shown in the
Figure 7a. This may influence the light field distribution and then correspondingly change the waveguide dispersion. As shown in
Figure 7b, the waveguide in the left is perfectly vertical and the waveguide on the right has a sidewall angel of 5 degrees.
Using Structure A1 and B1 as examples, the dispersion of the waveguide, with different sidewall angles from 0 to 5 degrees, was investigated. The results are shown in
Figure 7c,d, respectively. We found that when the sidewall angle increases, the dispersion curves of the waveguide bend slightly. The dispersion of As
2S
3 waveguide changes more significantly than that of Si
3N
4 waveguide. A relatively high precision for As
2S
3 and Si
3N
4 waveguide fabrication needs to be maintained to keep the sidewall angle at a small level, in order to obtain the designed dispersion profile.
The fabrication of horizontal slot waveguide has already been studied in several works [
33,
34]. In view of the silicon nitride horizontal slot waveguide, the thin film of silicon nitride and silica can be deposited on silica wafer in the specific order via low pressure or plasma enhanced chemical vapor deposition (LPCVD/PECVD). The depositing thickness of film can be controlled by the duration time of deposition. After that, a dry etching method, such as reactive ion etching (RIE), can be applied to complete the processing of waveguide, which will result in a high performance of fabrication.
3.4. Generation of Frequency Combs via Double Slot Micro-Ring Resonator
It can be inferred from the study in [
29] that a well-designed flat and low anomalous dispersion is conducive for generating the Kerr frequency comb with board bandwidth and small power variation. We formulated micro-ring resonators with double-slot structure, with B1 and B2 specific structures. Then, we adopted their dispersion curve into a simulation of Kerr frequency comb, in order to verify the performance of practical application via inverse design process.
The dynamics of comb generation in micro-ring resonators is well-described by the Lugiato-Lefever equation, which can be described as Equation (2) below [
35,
36]:
where
and
means spectral and temporal envelopes of light field in resonators, respectively, and their relationship follows the Fourier transform. While,
is loss rate of the resonator and
is coupling rate between bus waveguide and resonator,
is Kerr frequency shift, and
represents the pump power.
means the integrated dispersion of
frequency component and this can be calculated through the dispersion curve obtained above.
is pump detuning. In our simulation, FSR of comb is set to be 200 GHz, corresponding to
and
. The loss of waveguide (Si3N4) is set to be 0.1 dB/cm and the resonator is assumed critical coupling, which means
. The Q value of resonator is
. The power and wavelength of pump is 1 W and 1800 nm. The pump detuning is 56 times of resonator FWHM, around 4.88 GHz.
Based on the conditions mentioned above, the flat and broad spectrums of Kerr frequency comb are obtained from Structure B1 and B2, as shown in
Figure 8, in order to compare influence of dispersion. The peak of spectrum in the longer wavelength area is generated because of the zero-value of integrated dispersion, according to Reference [
9]. The spectrum in
Figure 8a has a wide wavelength range and the power variation is relatively smooth. Quantificationally speaking, for a 10-dB power variation, the bandwidth of spectrum is measured to be 564 nm from 1515 nm to 2079 nm. For a 20-dB variation, the bandwidth of spectrum is measured to be 1068 nm from 1415 nm to 2483 nm. In
Figure 8b, the spectrum of B2 possesses smaller power variation around the narrower band due to the smaller dispersion. For a 10-dB power variation, the band width of spectrum is measured to be 777 nm from 1416 nm to 2193 nm. It lacks bandwidth due to the narrower range of low dispersion. In addition, comb spectrum of strip waveguide is an envelope of square of hyperbolic secant [
9]. In comparison, the spectrum of double slot waveguide has the potential to extend to low power variation. In practical terms, this could be adjusted according to the actual demands.