Inverse design and implementation of a wavelength demultiplexing grating coupler

Results

• The design targets a 4.4 µm vertically incident Gaussian input and fundamental TE output modes, reversing the favored waveguide between 1300 and 1550 nm.Optimization first allows permittivity to vary continuously, then converts the design to a level-set representation for fine-tuning.Source: Results · p. 3
• The fabricated device uses a fully etched 220 nm silicon layer on 3 µm of buried oxide, with curved broadband output gratings placed away from waveguide ends.The output-grating geometry was chosen to provide broadband performance and minimize Fabry–Perot resonances from back-reflections.Source: Results · p. 3, Figure 1 · p. 4
• The authors suggest edge-coupling the output waveguides to optical fibers in future work to obtain a well-characterized out-coupling efficiency.Source: Results · p. 5
• At 1320 nm, light coupled only to the left output grating, while at 1540 nm it coupled only to the right, demonstrating the device’s routing function.The grating was excited with a focused laser beam in the experimental setup.Source: Results · p. 5, Figure 3 · p. 6
• At 1293 nm, simulated coupling efficiencies were 0.2937 into the left waveguide and 0.0008 into the right; at 1540 nm, they were 0.0027 and 0.4544, respectively.Measured signals broadly matched the simulated coupling spectra; measured output intensities were normalized to source brightness, not output-grating efficiency.Source: Results · pp. 5, 7
• 17±2 dB at 1310 nm and 12±2 dB at 1540 nm were measured, versus designed splitting ratios of 19.6 dB and 22.2 dB, respectively.The authors say the discrepancy is likely due to fabrication imperfections.Source: Results · p. 7

Optimization Algorithm and Electromagnetic Simulations

• The optimization uses a steepest-descent analogue with FDFD-calculated local gradients, then evaluates broadband performance using 2D FDTD simulations.Source: Optimization Algorithm and… · p. 7

Fabrication

• Fabrication used electron-beam lithography and HBr/Cl2 reactive-ion etching to pattern the nominal 220 nm silicon layer on a 3.0 µm BOX wafer.A 330 nm ZEP-52A resist mask was stripped after etching.Source: Fabrication · p. 7
• 17±2 dB at 1310 nm and 12±2 dB at 1540 nm were the experimentally measured splitting ratios.The figure also presents simulated coupling efficiencies and source-normalized measured output intensities.Source: Figure 4 · p. 8

1.1 Problem description

• Input and output modes are held fixed while their output-field amplitudes are bounded, and the permittivity must also satisfy fabrication constraints.The input modes are represented by current densities; each output mode is specified over an output surface with lower and upper amplitude bounds.Source: §1.1 · p. 12

1.2 Formulating the Optimization Problem

• The grating uses the steepest-descent-based inverse-design method, rather than the paper’s objective-first method.The authors report using only the steepest-descent-based method for this device.Source: §1.2 · p. 13

1.3 Optimization Algorithm

• Each iteration locally approximates the field-constraint penalty with a quadratic function, and the structure gradient is computed by brute force because the parameter count is small.Computing Q(z) is identified as the algorithm’s main computational cost.Source: §1.3 · p. 14

1.4 Choice of Q(z)

• Q(z) is chosen so its minimum gives a steepest-descent step of size κ from the previous permittivity value z0.The quadratic uses z0 from the preceding iteration.Source: §1.4 · p. 14
• The structure-objective gradient is obtained by differentiating the discretized Maxwell equations; rearrangement reduces the computation to one matrix solve, and complex absolute values use Wirtinger derivatives.The passages identify the derivative of the field-constraint penalty with respect to the fields as part of this calculation.Source: §1.4 · pp. 14–15

1.5 Parameterizing the structure

• The structure is first parameterized with two material permittivities, then converted to a thresholded level-set representation with anti-aliased borders.The initial parameterization is linear; fixed portions of the structure are represented separately.Source: §1.5 · p. 15

1.6 Solving Maxwell’s equations using FDFD

• FDFD solves are needed to evaluate fields at each iteration and solve the adjoint problem for the field-constraint gradient; the authors use MaxwellFDFD for MATLAB.The package is cited as.Source: §1.6 · p. 16

• Supplementary Table 1 lists parameters for the grating coupler.Source: Table 1 · p. 17
• The listed trench widths are denoted t_n.Source: Table 1 · p. 17
• The listed trench spacings are denoted w_n and are indicated in figure 2a.Source: Table 1 · p. 17