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In modern power electronics, materials like GaN play a crucial role in numerous industrial applications. When it comes to photonic applications, subwavelength gratings have many useful properties like tunable reflectivity and polarization selectivity which makes them very versatile. Due to the numerous benefits it offers, this SWG technology is employed in a variety of devices, including light-emitting diodes, thermal photovoltaics, resonators, optical filters, and others. III-Nitride VCSELs is one such similar device where subwavelength grating reflectors have been experimented with, as it has been successfully commercialized. Unfortunately, developing high-quality Distributed Bragg Reflectors (DBRs) is challenging due to issues such as high lattice mismatch ratio, low index contrast, and low p-type doping.

In designing subwavelength gratings for targeted applications, photonic designs can be used by forming complex non-closed maps between design parameters and reflectivity characteristics. There are also many other designs like several part gratings, and spherical and triangular arrays that have certain limitations in the fabrication process or issues caused during lithographic technology. To achieve and optimize subwavelength grating, machine learning-based methods were tested where 500 nm and 1.55 μm regimes of wavelength were taken into consideration.

**Experiment methodology**

As shown in Figure 1, two types of gratings were tested, where both the gratings were 2-part periodic and dimensional gratings. Since the gratings are zero contrast, they exhibit a superior reflectivity to high contrast gratings, and hence zero contrast and high contrast gratings were taken into consideration (H_{c}=0). There are four variable parameters in rectangular grating (Figure 1a), i.e. duty cycle (f), grating thickness (Hg), homogeneous cladding thickness (Hc) and period (Δ). Contrastingly, as shown in Figure 1(b), the complex-shaped grating has a curved grating profile. In the x-y plane, the polynomial grating is truncated at x = fΔ as the region from x = fΔ to Δ is set to 0. From the equation, x is in microns and all the coefficients c0, c1, c2, c3 and c4 have the units as μm^{−1}, μm^{−2} and μm^{−3}, respectively.

From the above equation, g(x ) = c4x4 + c3x3 + c2x2 + c1x + c0, which means that there are eight grating parameters for the polynomial-shaped grating and eight dimensional search space.

**Optimization methods and Algorithms used**

An experiment was done by applying a differential evolution algorithm for cost function minimization as two goals were set: stopband width maximization, and fabrication tolerance maximization. In stopband width maximization, the stopband width was maximized without consideration for the fabrication tolerance and then tolerance was maximized while a 20 nm stopband width constraint was imposed.

Figure 2 shows the complete optimization process. Initially, the cost function evaluations need to be done for vectors that are in the search space. This measures the deviations in the reflectivity characteristics in a grating design vector that is within the search space from the desired reflectivity characteristics, which can be calculated by the RCWA method. In stopband width optimization, the cost function can be defined as:

Where λ_{i} are the discrete data points along the desired reflectivity characteristic spectrum taken about the centre wavelength under consideration. R_{s,λi} are the calculated reflectivity values of a specific vector in the search space and R_{d,λi} represents the desired reflectivity value for λ_{i}. When it comes to fabrication tolerance maximization,

Where M is the number of discrete gathering parameter values spanning the fabrication tolerance and N is the corresponding wavelength positions, therefore, M x N is an array of evaluation points for the reflectivity values.

During the experiment, a differential evolution algorithm was also used to minimize the cost function. By doing so, mutant vectors were created by combining random population vectors for each vector in the population. In the end, wherever a low-cost function value is obtained, the trial vectors replaced the population vectors. The search vectors were designated with X _{i}. X_{i} = {f, Δ, Hg, Hc} for rectangular grating and X _{i} = {f, Δ, Hc , c0, c1, c2, c3, c4} for polynomial grating.

**Results Obtained after Simulation**

**Stopband width maximization**

As shown in Figure 4, the reflectivity characteristics of the optimized rectangular grating are depicted. For 500 nm center wavelength, f = 0.434, Δ = 415 nm, H_{g} = 157 nm and H_{c} = 31 nm. It was noted that a stopband width of 170 nm (λ/λcenter = 34%) was the outcome.

The above table shows the stopband width maximization results. Both rectangular and polynomial had shown comparable stopband widths and tolerances for all wavelength regimes, but for polynomial grating, results were comparatively poor.

**Fabrication tolerance maximization**

As shown in Figure 5, when it comes to polynomial-shaped grating where f = 0.683, H_{g }= 1.73 *μ*m, H_{c }= 29 nm, g(x) = 6.19x^{4 }– 7.68x^{3 }– 3.71x^{3} – 0.124x^{2 }+ 0.185. For this case, a stopband width constraint of 20 nm is imposed while maximizing the fabrication tolerance.

The above table shows the results for fabrication tolerance maximization.

**Conclusion**

An experiment was conducted in developing GaN subwavelength gratings by using evolutionary algorithms in the design and fabrication process. Results showed that polynomial gratings of order 4 or less can be used in achieving comparable broadband reflectivity features. This is an efficient way to explore the search space for the best design.

**References**

[1] O. N. Ogidi-Ekoko, W. Liang, H. Xue and N. Tansu, “Machine Learning Inspired Design of Complex-Shaped GaN Subwavelength Grating Reflectors,” in IEEE Photonics Journal, vol. 13, no. 1, pp. 1-13, Feb. 2021, Art no. 2700213, doi: 10.1109/JPHOT.2020.3048182.

[2] C. J. Chang-Hasnain and W. Yang, “High-contrast gratings for integrated optoelectronics,” Adv. Opt. Photon., vol. 4, no. 3, pp. 379–440, 2012.

[3] Y. Kanamori, M. Ishimori, and K. Hane, “High efficient light-emitting diodes with antireflection subwavelength gratings,” IEEE Photon. Technol. Lett., vol. 14, no. 8, pp. 1064–1066, Aug. 2002.

*This article was originally published on **Power Electronics News**.*