From e58e00f5331b3235e0378a956b0caefbb538714a Mon Sep 17 00:00:00 2001 From: Ardavan Oskooi Date: Tue, 2 Jul 2024 22:08:49 -0700 Subject: [PATCH 01/13] simulation script for adjoint gradient of a multilayer stack with respect to layer thicknesses --- .../adjoint_optimization/multilayer_opt.py | 328 ++++++++++++++++++ 1 file changed, 328 insertions(+) create mode 100644 python/examples/adjoint_optimization/multilayer_opt.py diff --git a/python/examples/adjoint_optimization/multilayer_opt.py b/python/examples/adjoint_optimization/multilayer_opt.py new file mode 100644 index 000000000..4d4bf420b --- /dev/null +++ b/python/examples/adjoint_optimization/multilayer_opt.py @@ -0,0 +1,328 @@ +"""Shape optimization of a multilayer stack. + +The 1D stack consists of alternating materials of index N_LAYER_1 and N_LAYER_2 +in the arrangement: N_LAYER_1, N_LAYER_2, N_LAYER_1, N_LAYER_2, ..., N_LAYER_1. + +The design parameters are the N layer thicknesses: [t_1, t_2, ..., t_N]. +""" + +from typing import Callable + +from autograd.extend import primitive, defvjp +from autograd import numpy as npa +from autograd import tensor_jacobian_product +from mpl_toolkits.axes_grid1 import make_axes_locatable +import matplotlib.pyplot as plt +import meep as mp +import meep.adjoint as mpa +import numpy as np + + +RESOLUTION_UM = 200 +WAVELENGTH_UM = 1.0 +AIR_UM = 1.0 +PML_UM = 1.0 +NUM_LAYERS = 11 +MIN_LAYER_UM = 0.1 +MAX_LAYER_UM = 0.9 +N_LAYER_1 = 3.5 +N_LAYER_2 = 1.0 # TODO (oskooi): support arbitrary use case (not just air). +LAYER_PERTURBATION_UM = 1.0 / RESOLUTION_UM +DESIGN_REGION_RESOLUTION_UM = 10 * RESOLUTION_UM + +DEBUG_OUTPUT = True + +design_region_size = mp.Vector3(0, 0, NUM_LAYERS * MAX_LAYER_UM) +nz_design_grid = int(design_region_size.z * DESIGN_REGION_RESOLUTION_UM) + 1 +nz_sim_grid = int(design_region_size.z * RESOLUTION_UM) + 1 +num_unit_cells = int(0.5 * (NUM_LAYERS - 1)) + + +def design_region_to_grid(nz: int) -> np.ndarray: + """Returns the 1D coordinates of the grid for the design region. + + Args: + nz: number of grid points. + + Returns: + The coordinates of the grid points. + """ + + z_grid = np.linspace(-0.5 * design_region_size.z, +0.5 * design_region_size.z, nz) + + return z_grid + + +@primitive +def levelset_and_smoothing(layer_thickness_um: np.ndarray) -> np.ndarray: + """Returns the density weights for a multilayer stack as a levelset. + + Args: + layer_thickness_um: thickness of each layer in the stack. + + Returns: + The density weights as a flattened (1D) array. + """ + + air_padding_um = 0.5 * (design_region_size.z - np.sum(layer_thickness_um)) + + weights = np.zeros(nz_design_grid) + z_grid = design_region_to_grid(nz_design_grid) + + # Air padding at left edge. + z_start = 0 + z_end = int(air_padding_um * DESIGN_REGION_RESOLUTION_UM) + weights[z_start:z_end] = 0 + + z_start = z_end + for j in range(NUM_LAYERS): + z_end = z_start + int(layer_thickness_um[j] * DESIGN_REGION_RESOLUTION_UM) + weights[z_start:z_end] = 1 if (j % 2 == 0) else 0 + z_start = z_end + + # Air padding at right edge. + weights[z_start:] = 0 + + # Smooth the design weights by downsampling from the design grid + # to the simulation grid using bilinear interpolation. + z_sim_grid = design_region_to_grid(nz_sim_grid) + smoothed_weights = np.interp(z_sim_grid, z_grid, weights) + + if DEBUG_OUTPUT: + fig, ax = plt.subplots() + ax.plot(z_sim_grid, smoothed_weights, "b-") + ax.plot(z_grid, weights, "r-") + ax.set_xlabel(r"z ($\mu$m)") + ax.set_ylabel("design weights (smoothed)") + if mp.am_master(): + fig.savefig("multilayer_stack_levelset.png", dpi=150, bbox_inches="tight") + + return smoothed_weights.flatten() + + +def levelset_and_smoothing_vjp( + ans: np.ndarray, layer_thickness_um: np.ndarray +) -> Callable[[np.ndarray], np.ndarray]: + """Returns a function for computing the vector-Jacobian product.""" + + air_padding_um = 0.5 * (design_region_size.z - np.sum(layer_thickness_um)) + + jacobian = np.zeros((nz_sim_grid, NUM_LAYERS)) + + z_grid = design_region_to_grid(nz_design_grid) + z_sim_grid = design_region_to_grid(nz_sim_grid) + + for i in range(NUM_LAYERS): + weights = np.zeros(nz_design_grid) + + # Air padding at left edge. + z_start = 0 + z_end = int(air_padding_um * DESIGN_REGION_RESOLUTION_UM) + weights[z_start:z_end] = 0 + + z_start = z_end + for j in range(NUM_LAYERS): + layer_um = layer_thickness_um[j] + if j == i: + layer_um += LAYER_PERTURBATION_UM + z_end = z_start + int(layer_um * DESIGN_REGION_RESOLUTION_UM) + weights[z_start:z_end] = 1 if (j % 2 == 0) else 0 + z_start = z_end + + # Air padding at right edge. + weights[z_start:] = 0 + + # Smooth the design weights by downsampling from the design grid + # to the simulation grid using bilinear interpolation. + smoothed_weights = np.interp(z_sim_grid, z_grid, weights) + + jacobian[:, i] = (smoothed_weights - ans) / LAYER_PERTURBATION_UM + + if DEBUG_OUTPUT: + fig, ax = plt.subplots() + im = ax.imshow( + np.transpose(jacobian), + cmap="inferno", + interpolation="none", + aspect="equal", + ) + ax.set_title(r"$\partial \rho_{smooth-levelset} / \partial t$") + divider = make_axes_locatable(ax) + cax = divider.append_axes("right", size="5%", pad=0.05) + fig.colorbar(im, cax=cax) + if mp.am_master(): + fig.savefig("multilayer_stack_gradient.png", dpi=150, bbox_inches="tight") + + return lambda g: np.tensordot(g, jacobian, axes=1) + + +def input_flux() -> float: + """Returns the flux generated by the source.""" + + frequency = 1 / WAVELENGTH_UM + pml_layers = [mp.PML(direction=mp.Z, thickness=PML_UM)] + + size_z_um = PML_UM + AIR_UM + design_region_size.z + AIR_UM + PML_UM + cell_size = mp.Vector3(0, 0, size_z_um) + + src_cmpt = mp.Ex + src_pt = mp.Vector3(0, 0, -0.5 * size_z_um + PML_UM) + sources = [ + mp.Source( + mp.GaussianSource(frequency, fwidth=0.1 * frequency), + component=src_cmpt, + center=src_pt, + ) + ] + + sim = mp.Simulation( + resolution=RESOLUTION_UM, + cell_size=cell_size, + dimensions=1, + boundary_layers=pml_layers, + sources=sources, + ) + + tran_pt = mp.Vector3(0, 0, 0.5 * size_z_um - PML_UM) + + flux_mon = sim.add_flux( + frequency, 0, 1, mp.FluxRegion(center=tran_pt, size=mp.Vector3()) + ) + + sim.run( + until_after_sources=mp.stop_when_fields_decayed(25.0, src_cmpt, tran_pt, 1e-6) + ) + + flux = mp.get_fluxes(flux_mon)[0] + + return flux + + +def multilayer_stack(norm_flux: float) -> mpa.OptimizationProblem: + """Sets up the adjoint optimization of a multilayer stack. + + Args: + norm_flux: input flux to use for normalizing the objective function. + + Returns: + A meep.adjoint.Optimization object for the simulation. + """ + + frequency = 1 / WAVELENGTH_UM + pml_layers = [mp.PML(direction=mp.Z, thickness=PML_UM)] + + size_z_um = PML_UM + AIR_UM + design_region_size.z + AIR_UM + PML_UM + cell_size = mp.Vector3(0, 0, size_z_um) + + src_cmpt = mp.Ex + src_pt = mp.Vector3(0, 0, -0.5 * size_z_um + PML_UM) + sources = [ + mp.Source( + mp.GaussianSource(frequency, fwidth=0.1 * frequency), + component=src_cmpt, + center=src_pt, + ) + ] + + mat_1 = mp.Medium(index=N_LAYER_1) + mat_2 = mp.Medium(index=N_LAYER_2) + + matgrid = mp.MaterialGrid( + mp.Vector3(0, 0, nz_sim_grid), + mat_1, + mat_2, + weights=np.ones(nz_sim_grid), + do_averaging=False, + ) + + matgrid_region = mpa.DesignRegion( + matgrid, + volume=mp.Volume(center=mp.Vector3(), size=design_region_size), + ) + + geometry = [ + mp.Block( + material=matgrid, size=matgrid_region.size, center=matgrid_region.center + ) + ] + + sim = mp.Simulation( + resolution=RESOLUTION_UM, + cell_size=cell_size, + dimensions=1, + boundary_layers=pml_layers, + sources=sources, + geometry=geometry, + ) + + tran_pt = mp.Vector3(0, 0, 0.5 * size_z_um - PML_UM) + + obj_args = [ + mpa.FourierFields(sim, mp.Volume(center=tran_pt), mp.Ex), + mpa.FourierFields(sim, mp.Volume(center=tran_pt), mp.Hy), + ] + + def obj_func(dft_ex, dft_hy): + return npa.real(npa.conj(dft_ex) * dft_hy) / norm_flux + + opt = mpa.OptimizationProblem( + simulation=sim, + objective_functions=obj_func, + objective_arguments=obj_args, + design_regions=[matgrid_region], + frequencies=[frequency], + ) + + return opt + + +if __name__ == "__main__": + layer_thickness_um = np.array( + [0.3, 0.1, 0.2, 0.1, 0.6, 0.4, 0.2, 0.1, 0.4, 0.3, 0.2] + ) + + smoothed_design_weights = levelset_and_smoothing(layer_thickness_um) + + norm_flux = input_flux() + + opt = multilayer_stack(norm_flux) + + obj_val_unperturbed, grad_unperturbed = opt( + [smoothed_design_weights], need_gradient=True + ) + print(f"obj_val_unperturbed[0] = {obj_val_unperturbed[0]}") + + defvjp(levelset_and_smoothing, levelset_and_smoothing_vjp) + grad_backprop = tensor_jacobian_product(levelset_and_smoothing, 0)( + layer_thickness_um, grad_unperturbed + ) + + if DEBUG_OUTPUT: + fig, ax = plt.subplots() + ax.plot(grad_unperturbed) + ax.set_title(r"$\partial F / \partial \rho_{smooth-levelset}$") + if mp.am_master(): + fig.savefig( + "gradient_wrt_smoothed_design_weights.png", dpi=150, bbox_inches="tight" + ) + + perturbation_um = 1e-3 * np.ones(NUM_LAYERS) + perturbed_design_weights = levelset_and_smoothing( + layer_thickness_um + perturbation_um + ) + perturbed_design_weights = perturbed_design_weights.flatten() + + obj_val_perturbed, _ = opt([perturbed_design_weights], need_gradient=False) + print(f"obj_val_perturbed[0] = {obj_val_perturbed[0]}") + + adj_directional_deriv = (perturbation_um[None, :] @ grad_backprop)[0] + fnd_directional_deriv = obj_val_perturbed[0] - obj_val_unperturbed[0] + print(f"adj_directional_deriv:, {adj_directional_deriv}") + print(f"fnd_directional_deriv:, {fnd_directional_deriv}") + rel_err = abs( + (fnd_directional_deriv - adj_directional_deriv) / fnd_directional_deriv + ) + print( + f"dir-deriv:, {fnd_directional_deriv:.8f} (finite difference), " + f"{adj_directional_deriv:.8f} (adjoint), {rel_err:.6f} (error)" + ) From b4be365c1d2b50bb37785a5f636fdffaec8f0b9e Mon Sep 17 00:00:00 2001 From: Ardavan Oskooi Date: Sun, 7 Jul 2024 17:38:59 -0700 Subject: [PATCH 02/13] add 0.5 correction factor to adjoint gradient as well as docstrings --- doc/docs/Python_Tutorials/Adjoint_Solver.md | 6 + .../adjoint_optimization/multilayer_opt.py | 211 +++++++++--------- 2 files changed, 108 insertions(+), 109 deletions(-) diff --git a/doc/docs/Python_Tutorials/Adjoint_Solver.md b/doc/docs/Python_Tutorials/Adjoint_Solver.md index 1de1b7103..9ae94d062 100644 --- a/doc/docs/Python_Tutorials/Adjoint_Solver.md +++ b/doc/docs/Python_Tutorials/Adjoint_Solver.md @@ -1190,6 +1190,12 @@ if __name__ == "__main__": ) ``` +Shape Optimization of a Multilayer Stack +---------------------------------------- + +We extend the demonstration of the shape derivative from the previous tutorial to perform a shape optimization of a multilayer stack. + + Compact Notebook Tutorials of Basic Features -------------------------------------------- diff --git a/python/examples/adjoint_optimization/multilayer_opt.py b/python/examples/adjoint_optimization/multilayer_opt.py index 4d4bf420b..14b9de7ae 100644 --- a/python/examples/adjoint_optimization/multilayer_opt.py +++ b/python/examples/adjoint_optimization/multilayer_opt.py @@ -1,9 +1,13 @@ """Shape optimization of a multilayer stack. The 1D stack consists of alternating materials of index N_LAYER_1 and N_LAYER_2 -in the arrangement: N_LAYER_1, N_LAYER_2, N_LAYER_1, N_LAYER_2, ..., N_LAYER_1. +in the arrangement: N_LAYER_2, N_LAYER_1, N_LAYER_2, N_LAYER_1, ..., N_LAYER_2. -The design parameters are the N layer thicknesses: [t_1, t_2, ..., t_N]. +The design parameters are the N layer thicknesses: [t_1, t_2, ..., t_N]. N must +be odd. + +Note: N_LAYER_1 must be vacuum because the left and right sides of the stack +are vacuum. This is a limitation of the current setup. """ from typing import Callable @@ -11,44 +15,39 @@ from autograd.extend import primitive, defvjp from autograd import numpy as npa from autograd import tensor_jacobian_product -from mpl_toolkits.axes_grid1 import make_axes_locatable -import matplotlib.pyplot as plt import meep as mp import meep.adjoint as mpa +import nlopt import numpy as np -RESOLUTION_UM = 200 +RESOLUTION_UM = 800 WAVELENGTH_UM = 1.0 AIR_UM = 1.0 PML_UM = 1.0 -NUM_LAYERS = 11 +NUM_LAYERS = 9 MIN_LAYER_UM = 0.1 -MAX_LAYER_UM = 0.9 -N_LAYER_1 = 3.5 -N_LAYER_2 = 1.0 # TODO (oskooi): support arbitrary use case (not just air). +MAX_LAYER_UM = 0.5 +N_LAYER_1 = 1.0 # TODO (oskooi): support arbitrary materials. +N_LAYER_2 = 1.3 LAYER_PERTURBATION_UM = 1.0 / RESOLUTION_UM DESIGN_REGION_RESOLUTION_UM = 10 * RESOLUTION_UM - -DEBUG_OUTPUT = True - -design_region_size = mp.Vector3(0, 0, NUM_LAYERS * MAX_LAYER_UM) -nz_design_grid = int(design_region_size.z * DESIGN_REGION_RESOLUTION_UM) + 1 -nz_sim_grid = int(design_region_size.z * RESOLUTION_UM) + 1 -num_unit_cells = int(0.5 * (NUM_LAYERS - 1)) +DESIGN_REGION_SIZE = mp.Vector3(0, 0, NUM_LAYERS * MAX_LAYER_UM) +NZ_DESIGN_GRID = int(DESIGN_REGION_SIZE.z * DESIGN_REGION_RESOLUTION_UM) + 1 +NZ_SIM_GRID = int(DESIGN_REGION_SIZE.z * RESOLUTION_UM) + 1 def design_region_to_grid(nz: int) -> np.ndarray: - """Returns the 1D coordinates of the grid for the design region. + """Returns the coordinates of the grid for the design region. Args: nz: number of grid points. Returns: - The coordinates of the grid points. + The 1D coordinates of the grid points. """ - z_grid = np.linspace(-0.5 * design_region_size.z, +0.5 * design_region_size.z, nz) + z_grid = np.linspace(-0.5 * DESIGN_REGION_SIZE.z, +0.5 * DESIGN_REGION_SIZE.z, nz) return z_grid @@ -64,12 +63,12 @@ def levelset_and_smoothing(layer_thickness_um: np.ndarray) -> np.ndarray: The density weights as a flattened (1D) array. """ - air_padding_um = 0.5 * (design_region_size.z - np.sum(layer_thickness_um)) + air_padding_um = 0.5 * (DESIGN_REGION_SIZE.z - np.sum(layer_thickness_um)) - weights = np.zeros(nz_design_grid) - z_grid = design_region_to_grid(nz_design_grid) + weights = np.zeros(NZ_DESIGN_GRID) + z_design_grid = design_region_to_grid(NZ_DESIGN_GRID) - # Air padding at left edge. + # Air padding at left edge z_start = 0 z_end = int(air_padding_um * DESIGN_REGION_RESOLUTION_UM) weights[z_start:z_end] = 0 @@ -80,22 +79,14 @@ def levelset_and_smoothing(layer_thickness_um: np.ndarray) -> np.ndarray: weights[z_start:z_end] = 1 if (j % 2 == 0) else 0 z_start = z_end - # Air padding at right edge. - weights[z_start:] = 0 + # Air padding at right edge + z_end = z_start + int(air_padding_um * DESIGN_REGION_RESOLUTION_UM) + weights[z_start:z_end] = 0 # Smooth the design weights by downsampling from the design grid # to the simulation grid using bilinear interpolation. - z_sim_grid = design_region_to_grid(nz_sim_grid) - smoothed_weights = np.interp(z_sim_grid, z_grid, weights) - - if DEBUG_OUTPUT: - fig, ax = plt.subplots() - ax.plot(z_sim_grid, smoothed_weights, "b-") - ax.plot(z_grid, weights, "r-") - ax.set_xlabel(r"z ($\mu$m)") - ax.set_ylabel("design weights (smoothed)") - if mp.am_master(): - fig.savefig("multilayer_stack_levelset.png", dpi=150, bbox_inches="tight") + z_sim_grid = design_region_to_grid(NZ_SIM_GRID) + smoothed_weights = np.interp(z_sim_grid, z_design_grid, weights) return smoothed_weights.flatten() @@ -105,17 +96,18 @@ def levelset_and_smoothing_vjp( ) -> Callable[[np.ndarray], np.ndarray]: """Returns a function for computing the vector-Jacobian product.""" - air_padding_um = 0.5 * (design_region_size.z - np.sum(layer_thickness_um)) + total_layer_thickness_um = np.sum(layer_thickness_um) + air_padding_um = 0.5 * (DESIGN_REGION_SIZE.z - total_layer_thickness_um) - jacobian = np.zeros((nz_sim_grid, NUM_LAYERS)) + jacobian = np.zeros((NZ_SIM_GRID, NUM_LAYERS)) - z_grid = design_region_to_grid(nz_design_grid) - z_sim_grid = design_region_to_grid(nz_sim_grid) + z_design_grid = design_region_to_grid(NZ_DESIGN_GRID) + z_sim_grid = design_region_to_grid(NZ_SIM_GRID) for i in range(NUM_LAYERS): - weights = np.zeros(nz_design_grid) + weights = np.zeros(NZ_DESIGN_GRID) - # Air padding at left edge. + # Air padding at left edge z_start = 0 z_end = int(air_padding_um * DESIGN_REGION_RESOLUTION_UM) weights[z_start:z_end] = 0 @@ -129,40 +121,26 @@ def levelset_and_smoothing_vjp( weights[z_start:z_end] = 1 if (j % 2 == 0) else 0 z_start = z_end - # Air padding at right edge. - weights[z_start:] = 0 + # Air padding at right edge + z_end = z_start + int(air_padding_um * DESIGN_REGION_RESOLUTION_UM) + weights[z_start:z_end] = 0 # Smooth the design weights by downsampling from the design grid # to the simulation grid using bilinear interpolation. - smoothed_weights = np.interp(z_sim_grid, z_grid, weights) + smoothed_weights = np.interp(z_sim_grid, z_design_grid, weights) jacobian[:, i] = (smoothed_weights - ans) / LAYER_PERTURBATION_UM - if DEBUG_OUTPUT: - fig, ax = plt.subplots() - im = ax.imshow( - np.transpose(jacobian), - cmap="inferno", - interpolation="none", - aspect="equal", - ) - ax.set_title(r"$\partial \rho_{smooth-levelset} / \partial t$") - divider = make_axes_locatable(ax) - cax = divider.append_axes("right", size="5%", pad=0.05) - fig.colorbar(im, cax=cax) - if mp.am_master(): - fig.savefig("multilayer_stack_gradient.png", dpi=150, bbox_inches="tight") - return lambda g: np.tensordot(g, jacobian, axes=1) def input_flux() -> float: - """Returns the flux generated by the source.""" + """Returns the flux generated by the source in vacuum.""" frequency = 1 / WAVELENGTH_UM pml_layers = [mp.PML(direction=mp.Z, thickness=PML_UM)] - size_z_um = PML_UM + AIR_UM + design_region_size.z + AIR_UM + PML_UM + size_z_um = PML_UM + AIR_UM + DESIGN_REGION_SIZE.z + AIR_UM + PML_UM cell_size = mp.Vector3(0, 0, size_z_um) src_cmpt = mp.Ex @@ -185,9 +163,7 @@ def input_flux() -> float: tran_pt = mp.Vector3(0, 0, 0.5 * size_z_um - PML_UM) - flux_mon = sim.add_flux( - frequency, 0, 1, mp.FluxRegion(center=tran_pt, size=mp.Vector3()) - ) + flux_mon = sim.add_flux(frequency, 0, 1, mp.FluxRegion(center=tran_pt)) sim.run( until_after_sources=mp.stop_when_fields_decayed(25.0, src_cmpt, tran_pt, 1e-6) @@ -211,7 +187,7 @@ def multilayer_stack(norm_flux: float) -> mpa.OptimizationProblem: frequency = 1 / WAVELENGTH_UM pml_layers = [mp.PML(direction=mp.Z, thickness=PML_UM)] - size_z_um = PML_UM + AIR_UM + design_region_size.z + AIR_UM + PML_UM + size_z_um = PML_UM + AIR_UM + DESIGN_REGION_SIZE.z + AIR_UM + PML_UM cell_size = mp.Vector3(0, 0, size_z_um) src_cmpt = mp.Ex @@ -228,16 +204,16 @@ def multilayer_stack(norm_flux: float) -> mpa.OptimizationProblem: mat_2 = mp.Medium(index=N_LAYER_2) matgrid = mp.MaterialGrid( - mp.Vector3(0, 0, nz_sim_grid), + mp.Vector3(0, 0, NZ_SIM_GRID), mat_1, mat_2, - weights=np.ones(nz_sim_grid), + weights=np.ones(NZ_SIM_GRID), do_averaging=False, ) matgrid_region = mpa.DesignRegion( matgrid, - volume=mp.Volume(center=mp.Vector3(), size=design_region_size), + volume=mp.Volume(center=mp.Vector3(), size=DESIGN_REGION_SIZE), ) geometry = [ @@ -263,6 +239,15 @@ def multilayer_stack(norm_flux: float) -> mpa.OptimizationProblem: ] def obj_func(dft_ex, dft_hy): + """Objective function for the optimization. + + Args: + dft_ex, dft_hy: the discrete Fourier-transformed Ex and Hy fields. + + Returns: + The transmittance through the stack. + """ + return npa.real(npa.conj(dft_ex) * dft_hy) / norm_flux opt = mpa.OptimizationProblem( @@ -276,53 +261,61 @@ def obj_func(dft_ex, dft_hy): return opt -if __name__ == "__main__": - layer_thickness_um = np.array( - [0.3, 0.1, 0.2, 0.1, 0.6, 0.4, 0.2, 0.1, 0.4, 0.3, 0.2] - ) +def nlopt_obj_func(layer_thickness_um: np.ndarray, grad: np.ndarray) -> float: + """Objective function for NLopt. - smoothed_design_weights = levelset_and_smoothing(layer_thickness_um) + Args: + layer_thickness_um: thickness of the layers in the stack. + grad: the gradient with respect to the layer thicknesses, modified in + place. - norm_flux = input_flux() + Returns: + The value of the objective function evaluated at the given layer thicknesses. + """ - opt = multilayer_stack(norm_flux) + design_weights = levelset_and_smoothing(layer_thickness_um) - obj_val_unperturbed, grad_unperturbed = opt( - [smoothed_design_weights], need_gradient=True - ) - print(f"obj_val_unperturbed[0] = {obj_val_unperturbed[0]}") + obj_val, gradient = opt([design_weights]) defvjp(levelset_and_smoothing, levelset_and_smoothing_vjp) - grad_backprop = tensor_jacobian_product(levelset_and_smoothing, 0)( - layer_thickness_um, grad_unperturbed + gradient_backpropagate = tensor_jacobian_product(levelset_and_smoothing, 0)( + layer_thickness_um, gradient ) - if DEBUG_OUTPUT: - fig, ax = plt.subplots() - ax.plot(grad_unperturbed) - ax.set_title(r"$\partial F / \partial \rho_{smooth-levelset}$") - if mp.am_master(): - fig.savefig( - "gradient_wrt_smoothed_design_weights.png", dpi=150, bbox_inches="tight" - ) - - perturbation_um = 1e-3 * np.ones(NUM_LAYERS) - perturbed_design_weights = levelset_and_smoothing( - layer_thickness_um + perturbation_um - ) - perturbed_design_weights = perturbed_design_weights.flatten() + # TODO (oskooi): determine why factor of 0.5 is necessary. + gradient_backpropagate = 0.5 * gradient_backpropagate - obj_val_perturbed, _ = opt([perturbed_design_weights], need_gradient=False) - print(f"obj_val_perturbed[0] = {obj_val_perturbed[0]}") + if grad.size > 0: + grad[:] = np.squeeze(gradient_backpropagate) - adj_directional_deriv = (perturbation_um[None, :] @ grad_backprop)[0] - fnd_directional_deriv = obj_val_perturbed[0] - obj_val_unperturbed[0] - print(f"adj_directional_deriv:, {adj_directional_deriv}") - print(f"fnd_directional_deriv:, {fnd_directional_deriv}") - rel_err = abs( - (fnd_directional_deriv - adj_directional_deriv) / fnd_directional_deriv - ) - print( - f"dir-deriv:, {fnd_directional_deriv:.8f} (finite difference), " - f"{adj_directional_deriv:.8f} (adjoint), {rel_err:.6f} (error)" - ) + print(f"iteration:, {obj_val[0]}, {layer_thickness_um}") + + return obj_val[0] + + +if __name__ == "__main__": + algorithm = nlopt.LD_LBFGS + + # Lower and upper bounds for layer thicknesses. + lower_bound = MIN_LAYER_UM * np.ones(NUM_LAYERS) + upper_bound = MAX_LAYER_UM * np.ones(NUM_LAYERS) + + solver = nlopt.opt(algorithm, NUM_LAYERS) + solver.set_lower_bounds(lower_bound) + solver.set_upper_bounds(upper_bound) + solver.set_min_objective(nlopt_obj_func) + solver.set_ftol_abs(0.001) + solver.set_xtol_abs(1 / RESOLUTION_UM) + solver.set_maxeval(20) + + norm_flux = input_flux() + opt = multilayer_stack(norm_flux) + + layer_thickness_um = 0.2 * np.ones(NUM_LAYERS) + optimal_layer_thickness_um = solver.optimize(layer_thickness_um) + optimal_obj_val = solver.last_optimum_value() + result = solver.last_optimize_result() + + print(f"optimal_obj_val:, {optimal_obj_val}") + print(f"optimal_layer_thickness_um:, {optimal_layer_thickness_um}") + print(f"result:, {result}") From e82e8e9b4f6089dc8cf200159ea68110a8a68cab Mon Sep 17 00:00:00 2001 From: Ardavan Oskooi Date: Sun, 28 Jul 2024 21:52:34 -0700 Subject: [PATCH 03/13] switch optimization problem from single wavelength to multi wavelength (minimax) --- .../adjoint_optimization/multilayer_opt.py | 265 +++++++++++------- 1 file changed, 159 insertions(+), 106 deletions(-) diff --git a/python/examples/adjoint_optimization/multilayer_opt.py b/python/examples/adjoint_optimization/multilayer_opt.py index 14b9de7ae..f44414445 100644 --- a/python/examples/adjoint_optimization/multilayer_opt.py +++ b/python/examples/adjoint_optimization/multilayer_opt.py @@ -1,4 +1,4 @@ -"""Shape optimization of a multilayer stack. +"""Shape optimization of a multilayer stack over a broad bandwidth. The 1D stack consists of alternating materials of index N_LAYER_1 and N_LAYER_2 in the arrangement: N_LAYER_2, N_LAYER_1, N_LAYER_2, N_LAYER_1, ..., N_LAYER_2. @@ -6,11 +6,14 @@ The design parameters are the N layer thicknesses: [t_1, t_2, ..., t_N]. N must be odd. -Note: N_LAYER_1 must be vacuum because the left and right sides of the stack -are vacuum. This is a limitation of the current setup. +The design objective involves minimizing the largest integral of the DFT fields +in the multilayer stack over two wavelengths (worst-case optimization). + +Tutorial Reference: +https://meep.readthedocs.io/en/latest/Python_Tutorials/Adjoint_Solver/#shape-optimization-of-a-multilayer-stack """ -from typing import Callable +from typing import Callable, List, Tuple from autograd.extend import primitive, defvjp from autograd import numpy as npa @@ -22,23 +25,31 @@ RESOLUTION_UM = 800 -WAVELENGTH_UM = 1.0 AIR_UM = 1.0 PML_UM = 1.0 NUM_LAYERS = 9 MIN_LAYER_UM = 0.1 MAX_LAYER_UM = 0.5 -N_LAYER_1 = 1.0 # TODO (oskooi): support arbitrary materials. +N_LAYER_1 = 1.0 N_LAYER_2 = 1.3 LAYER_PERTURBATION_UM = 1.0 / RESOLUTION_UM +DESIGN_WAVELENGTHS_UM = (0.95, 1.05) DESIGN_REGION_RESOLUTION_UM = 10 * RESOLUTION_UM -DESIGN_REGION_SIZE = mp.Vector3(0, 0, NUM_LAYERS * MAX_LAYER_UM) -NZ_DESIGN_GRID = int(DESIGN_REGION_SIZE.z * DESIGN_REGION_RESOLUTION_UM) + 1 -NZ_SIM_GRID = int(DESIGN_REGION_SIZE.z * RESOLUTION_UM) + 1 +DESIGN_REGION_UM = mp.Vector3(0, 0, NUM_LAYERS * MAX_LAYER_UM) +NZ_DESIGN_GRID = int(DESIGN_REGION_UM.z * DESIGN_REGION_RESOLUTION_UM) + 1 +NZ_SIM_GRID = int(DESIGN_REGION_UM.z * RESOLUTION_UM) + 1 +MAX_OPT_ITERATIONS = 50 + +num_wavelengths = len(DESIGN_WAVELENGTHS_UM) +frequencies = [1 / wavelength_um for wavelength_um in DESIGN_WAVELENGTHS_UM] + + +def str_from_list(list_: List[float]) -> str: + return "[" + ", ".join(f"{val:.4f}" for val in list_) + "]" def design_region_to_grid(nz: int) -> np.ndarray: - """Returns the coordinates of the grid for the design region. + """Returns the coordinates of the 1D grid for the design region. Args: nz: number of grid points. @@ -46,8 +57,7 @@ def design_region_to_grid(nz: int) -> np.ndarray: Returns: The 1D coordinates of the grid points. """ - - z_grid = np.linspace(-0.5 * DESIGN_REGION_SIZE.z, +0.5 * DESIGN_REGION_SIZE.z, nz) + z_grid = np.linspace(-0.5 * DESIGN_REGION_UM.z, 0.5 * DESIGN_REGION_UM.z, nz) return z_grid @@ -62,11 +72,9 @@ def levelset_and_smoothing(layer_thickness_um: np.ndarray) -> np.ndarray: Returns: The density weights as a flattened (1D) array. """ - - air_padding_um = 0.5 * (DESIGN_REGION_SIZE.z - np.sum(layer_thickness_um)) + air_padding_um = 0.5 * (DESIGN_REGION_UM.z - np.sum(layer_thickness_um)) weights = np.zeros(NZ_DESIGN_GRID) - z_design_grid = design_region_to_grid(NZ_DESIGN_GRID) # Air padding at left edge z_start = 0 @@ -86,6 +94,7 @@ def levelset_and_smoothing(layer_thickness_um: np.ndarray) -> np.ndarray: # Smooth the design weights by downsampling from the design grid # to the simulation grid using bilinear interpolation. z_sim_grid = design_region_to_grid(NZ_SIM_GRID) + z_design_grid = design_region_to_grid(NZ_DESIGN_GRID) smoothed_weights = np.interp(z_sim_grid, z_design_grid, weights) return smoothed_weights.flatten() @@ -97,7 +106,7 @@ def levelset_and_smoothing_vjp( """Returns a function for computing the vector-Jacobian product.""" total_layer_thickness_um = np.sum(layer_thickness_um) - air_padding_um = 0.5 * (DESIGN_REGION_SIZE.z - total_layer_thickness_um) + air_padding_um = 0.5 * (DESIGN_REGION_UM.z - total_layer_thickness_um) jacobian = np.zeros((NZ_SIM_GRID, NUM_LAYERS)) @@ -134,67 +143,27 @@ def levelset_and_smoothing_vjp( return lambda g: np.tensordot(g, jacobian, axes=1) -def input_flux() -> float: - """Returns the flux generated by the source in vacuum.""" - - frequency = 1 / WAVELENGTH_UM - pml_layers = [mp.PML(direction=mp.Z, thickness=PML_UM)] - - size_z_um = PML_UM + AIR_UM + DESIGN_REGION_SIZE.z + AIR_UM + PML_UM - cell_size = mp.Vector3(0, 0, size_z_um) - - src_cmpt = mp.Ex - src_pt = mp.Vector3(0, 0, -0.5 * size_z_um + PML_UM) - sources = [ - mp.Source( - mp.GaussianSource(frequency, fwidth=0.1 * frequency), - component=src_cmpt, - center=src_pt, - ) - ] - - sim = mp.Simulation( - resolution=RESOLUTION_UM, - cell_size=cell_size, - dimensions=1, - boundary_layers=pml_layers, - sources=sources, - ) - - tran_pt = mp.Vector3(0, 0, 0.5 * size_z_um - PML_UM) - - flux_mon = sim.add_flux(frequency, 0, 1, mp.FluxRegion(center=tran_pt)) - - sim.run( - until_after_sources=mp.stop_when_fields_decayed(25.0, src_cmpt, tran_pt, 1e-6) - ) - - flux = mp.get_fluxes(flux_mon)[0] - - return flux - - -def multilayer_stack(norm_flux: float) -> mpa.OptimizationProblem: +def multilayer_stack() -> mpa.OptimizationProblem: """Sets up the adjoint optimization of a multilayer stack. - Args: - norm_flux: input flux to use for normalizing the objective function. - Returns: - A meep.adjoint.Optimization object for the simulation. + A `meep.adjoint.Optimization` callable object. """ + pml_layers = [mp.PML(thickness=PML_UM)] - frequency = 1 / WAVELENGTH_UM - pml_layers = [mp.PML(direction=mp.Z, thickness=PML_UM)] - - size_z_um = PML_UM + AIR_UM + DESIGN_REGION_SIZE.z + AIR_UM + PML_UM + size_z_um = PML_UM + AIR_UM + DESIGN_REGION_UM.z + AIR_UM + PML_UM cell_size = mp.Vector3(0, 0, size_z_um) + frequency_center = np.mean(frequencies) + + # Set source bandwidth to be larger than the range of design wavelengths. + frequency_width = 1.2 * (np.max(frequencies) - np.min(frequencies)) + src_cmpt = mp.Ex src_pt = mp.Vector3(0, 0, -0.5 * size_z_um + PML_UM) sources = [ mp.Source( - mp.GaussianSource(frequency, fwidth=0.1 * frequency), + mp.GaussianSource(frequency_center, fwidth=frequency_width), component=src_cmpt, center=src_pt, ) @@ -213,7 +182,7 @@ def multilayer_stack(norm_flux: float) -> mpa.OptimizationProblem: matgrid_region = mpa.DesignRegion( matgrid, - volume=mp.Volume(center=mp.Vector3(), size=DESIGN_REGION_SIZE), + volume=mp.Volume(center=mp.Vector3(), size=DESIGN_REGION_UM), ) geometry = [ @@ -231,91 +200,175 @@ def multilayer_stack(norm_flux: float) -> mpa.OptimizationProblem: geometry=geometry, ) - tran_pt = mp.Vector3(0, 0, 0.5 * size_z_um - PML_UM) - obj_args = [ - mpa.FourierFields(sim, mp.Volume(center=tran_pt), mp.Ex), - mpa.FourierFields(sim, mp.Volume(center=tran_pt), mp.Hy), + mpa.FourierFields( + sim, volume=matgrid_region.volume, component=mp.Ex, yee_grid=False + ) ] - def obj_func(dft_ex, dft_hy): + def obj_func(dft_ex: np.ndarray) -> Tuple[float]: """Objective function for the optimization. Args: - dft_ex, dft_hy: the discrete Fourier-transformed Ex and Hy fields. + dft_ex: the discrete Fourier-transformed Ex fields as a 2D array of + dimension (num_wavelengths, NX_DESIGN_GRID * NY_DESIGN_GRID). Returns: - The transmittance through the stack. + A tuple of the integrals of Ex in the stack for each wavelength. """ - - return npa.real(npa.conj(dft_ex) * dft_hy) / norm_flux + return npa.sum(npa.absolute(dft_ex) ** 2, axis=1) opt = mpa.OptimizationProblem( simulation=sim, objective_functions=obj_func, objective_arguments=obj_args, design_regions=[matgrid_region], - frequencies=[frequency], + frequencies=frequencies, ) return opt -def nlopt_obj_func(layer_thickness_um: np.ndarray, grad: np.ndarray) -> float: - """Objective function for NLopt. +def obj_func(epigraph_and_layer_thickness_um: np.ndarray, grad: np.ndarray) -> float: + """Objective function for the epigraph formulation. Args: - layer_thickness_um: thickness of the layers in the stack. - grad: the gradient with respect to the layer thicknesses, modified in - place. + epigraph_and_layer_thickness_um: 1D array containing epigraph variable + (first element) and design weights (remaining elements). + grad: the gradient as a flattened (1D) array, modified in place. Returns: - The value of the objective function evaluated at the given layer thicknesses. + The scalar epigraph variable. """ + epigraph = epigraph_and_layer_thickness_um[0] + + if grad.size > 0: + grad[0] = 1 + grad[1:] = 0 + return epigraph + + +def epigraph_constraint( + result: float, epigraph_and_layer_thickness_um: np.ndarray, gradient: np.ndarray +) -> None: + """Constraint function for the epigraph formulation. + + Args: + result: the result of the function evaluation, modified in place. + epigraph_and_layer_thickness_um: 1D array containing the epigraph variable + (first element) and the layer thicknesses (remaining elements). + gradient: the Jacobian matrix with dimensions (num_wavelengths, + 1 + NUM_LAYERS), modified in place. + """ + epigraph = epigraph_and_layer_thickness_um[0] + layer_thickness_um = epigraph_and_layer_thickness_um[1:] design_weights = levelset_and_smoothing(layer_thickness_um) - obj_val, gradient = opt([design_weights]) + obj_val, grad = opt([design_weights]) + # Backpropagate the gradients through the levelset construction function. + grad_backpropagate = np.zeros((NUM_LAYERS, num_wavelengths)) defvjp(levelset_and_smoothing, levelset_and_smoothing_vjp) - gradient_backpropagate = tensor_jacobian_product(levelset_and_smoothing, 0)( - layer_thickness_um, gradient - ) + for k in range(num_wavelengths): + grad_backpropagate[:, k] = tensor_jacobian_product(levelset_and_smoothing, 0)( + layer_thickness_um, grad[:, k] + ) - # TODO (oskooi): determine why factor of 0.5 is necessary. - gradient_backpropagate = 0.5 * gradient_backpropagate + # TODO (oskooi): determine why factor of 0.5 is necessary for correctness. + grad_backpropagate = 0.5 * grad_backpropagate - if grad.size > 0: - grad[:] = np.squeeze(gradient_backpropagate) + if gradient.size > 0: + gradient[:, 0] = -1 # gradient w.r.t. epigraph variable + gradient[:, 1:] = grad_backpropagate.T - print(f"iteration:, {obj_val[0]}, {layer_thickness_um}") + result[:] = obj_val - epigraph - return obj_val[0] + print( + f"iteration:, {current_iteration[0]:2d}, {str_from_list(result)}, " + f"{str_from_list(layer_thickness_um)}" + ) + + obj_func_history.append(obj_val) + epigraph_variable_history.append(epigraph) + current_iteration[0] += 1 if __name__ == "__main__": - algorithm = nlopt.LD_LBFGS + opt = multilayer_stack() # Lower and upper bounds for layer thicknesses. lower_bound = MIN_LAYER_UM * np.ones(NUM_LAYERS) upper_bound = MAX_LAYER_UM * np.ones(NUM_LAYERS) - solver = nlopt.opt(algorithm, NUM_LAYERS) + rng = np.random.RandomState() + layer_thickness_um = MIN_LAYER_UM + rng.rand(NUM_LAYERS) * ( + MAX_LAYER_UM - MIN_LAYER_UM + ) + + # Execute a single forward run before the start of the optimization and + # set the initial epigraph variable to slightly larger than the + # largest value of the objective function over the wavelengths. + design_weights = levelset_and_smoothing(layer_thickness_um) + epigraph, _ = opt([design_weights], need_gradient=False) + epigraph_initial = 1.05 * np.amax(epigraph) + + epigraph_and_layer_thickness_um = np.concatenate( + ([epigraph_initial], layer_thickness_um) + ) + + # Insert bounds for the epigraph variable. + lower_bound = np.insert(lower_bound, 0, 0) + upper_bound = np.insert(upper_bound, 0, +np.inf) + + tolerance_epigraph = [1e-3] * num_wavelengths + + solver = nlopt.opt(nlopt.LD_MMA, NUM_LAYERS + 1) solver.set_lower_bounds(lower_bound) solver.set_upper_bounds(upper_bound) - solver.set_min_objective(nlopt_obj_func) - solver.set_ftol_abs(0.001) - solver.set_xtol_abs(1 / RESOLUTION_UM) - solver.set_maxeval(20) + solver.set_min_objective(obj_func) + solver.set_maxeval(MAX_OPT_ITERATIONS) + solver.add_inequality_mconstraint(epigraph_constraint, tolerance_epigraph) + solver.set_ftol_rel(1e-3) - norm_flux = input_flux() - opt = multilayer_stack(norm_flux) + obj_func_history = [] + epigraph_variable_history = [] + current_iteration = [0] - layer_thickness_um = 0.2 * np.ones(NUM_LAYERS) - optimal_layer_thickness_um = solver.optimize(layer_thickness_um) + epigraph_and_layer_thickness_um = solver.optimize(epigraph_and_layer_thickness_um) optimal_obj_val = solver.last_optimum_value() - result = solver.last_optimize_result() + return_code = solver.last_optimize_result() print(f"optimal_obj_val:, {optimal_obj_val}") - print(f"optimal_layer_thickness_um:, {optimal_layer_thickness_um}") - print(f"result:, {result}") + print( + "optimal_layer_thickness_um:, " + f"{str_from_list(epigraph_and_layer_thickness_um[1:])}" + ) + print(f"return_code:, {return_code}") + + # Save important optimization parameters and output for post processing. + with open("optimal_design.npz", "wb") as datafile: + np.savez( + datafile, + RESOLUTION_UM=RESOLUTION_UM, + AIR_UM=AIR_UM, + PML_UM=PML_UM, + NUM_LAYERS=NUM_LAYERS, + MIN_LAYER_UM=MIN_LAYER_UM, + MAX_LAYER_UM=MAX_LAYER_UM, + N_LAYER_1=N_LAYER_1, + N_LAYER_2=N_LAYER_2, + DESIGN_WAVELENGTHS_UM=DESIGN_WAVELENGTHS_UM, + DESIGN_REGION_UM=DESIGN_REGION_UM, + DESIGN_REGION_RESOLUTION_UM=DESIGN_REGION_RESOLUTION_UM, + NZ_DESIGN_GRID=NZ_DESIGN_GRID, + NZ_SIM_GRID=NZ_SIM_GRID, + MAX_OPT_ITERATIONS=MAX_OPT_ITERATIONS, + obj_func_history=obj_func_history, + epigraph_variable_history=epigraph_variable_history, + epigraph_variable=epigraph_and_layer_thickness_um[0], + layer_thickness_um=epigraph_and_layer_thickness_um[1:], + tolerance_epigraph=tolerance_epigraph, + optimal_obj_val=optimal_obj_val, + return_code=return_code, + ) From 6258b46f5d68c135c8362d12f70377d424160f2d Mon Sep 17 00:00:00 2001 From: Ardavan Oskooi Date: Thu, 1 Aug 2024 06:54:20 -0700 Subject: [PATCH 04/13] add text and figures --- doc/docs/Python_Tutorials/Adjoint_Solver.md | 380 +++++++++++++++++- .../adjoint_optimization/multilayer_opt.py | 4 +- 2 files changed, 381 insertions(+), 3 deletions(-) diff --git a/doc/docs/Python_Tutorials/Adjoint_Solver.md b/doc/docs/Python_Tutorials/Adjoint_Solver.md index 9ae94d062..1c1e104a8 100644 --- a/doc/docs/Python_Tutorials/Adjoint_Solver.md +++ b/doc/docs/Python_Tutorials/Adjoint_Solver.md @@ -1193,7 +1193,385 @@ if __name__ == "__main__": Shape Optimization of a Multilayer Stack ---------------------------------------- -We extend the demonstration of the shape derivative from the previous tutorial to perform a shape optimization of a multilayer stack. +We extend the demonstration of the shape derivative from the previous tutorial to perform a shape optimization of a multilayer stack over a broad bandwidth. The 1D design problem involves finding the layer thicknesses for a fixed number of layers (9) which minimize the largest transmittance through the stack at two wavelengths: $\lambda_1$ = 0.95 μm and $\lambda_2$ = 1.05 μm. However, rather than use the transmittance as the objective function, we use the field decay via the integral of the fields in the stack ($\int |E_x|^2$). This provides more information to the optimizer which generally produces better results. For reference, the analytic solution at a *single* wavelength of $\lambda$ = 1.0 μm is a quarter-wavelength stack with thicknesses $\lambda / (4 n_1)$ = 0.25 and $\lambda / (4 n_2)$ = 0.1923. + +(For lots of layers and a single wavelength, you can easily get a non-quarter-wave structure with very low transmission, and once you get to that point the gradients will be almost zero so probably it will converge very slowly towards a quarter-wave solution because the optimizer is generally very slow to try to reduce transmission from 1e-3 to 1e-6, for example.) + +The stack consists of alternating materials of refractive index $n_1$ = 1.0 and $n_2$ = 1.3. The layers are arranged as $n_2$, $n_1$, $n_2$, $n_1$, ..., $n_2$. The thickness of all layers is constrained to be in the range of 0.1 μm to 0.5 μm. The thickness constraint is trivial to implement in this case because it applies directly to a geometric feature rather than to the density weights. + +The worst-case optimization is implemented using the [epigraph formulation](https://nlopt.readthedocs.io/en/latest/NLopt_Introduction/#equivalent-formulations-of-optimization-problems). A plot of the objective function vs. iteration number for the two wavelengths is shown below. Also included in this plot is the epigraph variable. The initial layer thicknesses are chosen randomly. The optimizer seems to converge to a local optima within ~25 iterations. In this case, the optimizer ran for thirty iterations before reaching the threshold tolerance for the objective value. Since these are 1D simulations, the runtime for each design iteration is generally fast (about ten seconds using two cores of Intel Xeon i7-7700K CPU @ 4.20GHz). + +The plot shows that the optimizer which uses the Method of Moving Asymptotes (MMA) algorithm takes some steps that make things *worse* during the early iterations. This is a common phenomenon in many algorithms — initially, the optimizer takes steps that are too large and then has to backtrack. After a few iterations, the algorithm has more information about a "good" step size. + +![](../images/multilayer_opt_obj_func_history.png#center) + +A plot of the $|E_x|^2$ fields in the stack (design region) for the optimal design is shown below. The plot shows that while the DFT fields for each of the two wavelengths are decaying through the stack, the transmittance values are quite different: T($\lambda$ = 0.95 μm) = 0.3551 vs. T($\lambda$ = 1.05 μm) = 0.8098. This result is consistent with the plot of the objective function vs. iteration number which shows that, because of the minimax setup, the stack is being optimized for $\lambda$ = 0.95 μm and *not* $\lambda$ = 1.05 μm. We observed this trend for various local optima. This seems to suggest that the optimizer targeting a single wavelength of a broad bandwdith design problem may be a general feature for this class of optimization problem. Also, the magnitude of $|E_x|^2$ at the right edge of the stack is clearly larger for $\lambda$ = 1.05 μm than $\lambda$ = 0.95 μm. This is again consistent with the transmittance being larger for $\lambda$ = 1.05 than for $\lambda$ = 0.95 μm. + +![](../images/broadband_field_decay_in_multilayer_stack.png#center) + +The script is [python/examples/adjoint_optimization/multilayer_opt.py](https://github.com/NanoComp/meep/tree/master/python/examples/adjoint_optimization/multilayer_opt.py). + +```py +from typing import Callable, List, Tuple + +from autograd.extend import primitive, defvjp +from autograd import numpy as npa +from autograd import tensor_jacobian_product +import meep as mp +import meep.adjoint as mpa +import nlopt +import numpy as np + + +RESOLUTION_UM = 800 +AIR_UM = 1.0 +PML_UM = 1.0 +NUM_LAYERS = 9 +MIN_LAYER_UM = 0.1 +MAX_LAYER_UM = 0.5 +N_LAYER_1 = 1.0 +N_LAYER_2 = 1.3 +LAYER_PERTURBATION_UM = 1.0 / RESOLUTION_UM +DESIGN_WAVELENGTHS_UM = (0.95, 1.05) +DESIGN_REGION_RESOLUTION_UM = 10 * RESOLUTION_UM +DESIGN_REGION_UM = mp.Vector3(0, 0, NUM_LAYERS * MAX_LAYER_UM) +NZ_DESIGN_GRID = int(DESIGN_REGION_UM.z * DESIGN_REGION_RESOLUTION_UM) + 1 +NZ_SIM_GRID = int(DESIGN_REGION_UM.z * RESOLUTION_UM) + 1 +MAX_OPT_ITERATIONS = 50 + +num_wavelengths = len(DESIGN_WAVELENGTHS_UM) +frequencies = [1 / wavelength_um for wavelength_um in DESIGN_WAVELENGTHS_UM] + + +def str_from_list(list_: List[float]) -> str: + return "[" + ", ".join(f"{val:.4f}" for val in list_) + "]" + + +def design_region_to_grid(nz: int) -> np.ndarray: + """Returns the coordinates of the 1D grid for the design region. + + Args: + nz: number of grid points. + + Returns: + The 1D coordinates of the grid points. + """ + z_grid = np.linspace(-0.5 * DESIGN_REGION_UM.z, 0.5 * DESIGN_REGION_UM.z, nz) + + return z_grid + + +@primitive +def levelset_and_smoothing(layer_thickness_um: np.ndarray) -> np.ndarray: + """Returns the density weights for a multilayer stack as a levelset. + + Args: + layer_thickness_um: thickness of each layer in the stack. + + Returns: + The density weights as a flattened (1D) array. + """ + air_padding_um = 0.5 * (DESIGN_REGION_UM.z - np.sum(layer_thickness_um)) + + weights = np.zeros(NZ_DESIGN_GRID) + + # Air padding at left edge + z_start = 0 + z_end = int(air_padding_um * DESIGN_REGION_RESOLUTION_UM) + weights[z_start:z_end] = 0 + + z_start = z_end + for j in range(NUM_LAYERS): + z_end = z_start + int(layer_thickness_um[j] * DESIGN_REGION_RESOLUTION_UM) + weights[z_start:z_end] = 1 if (j % 2 == 0) else 0 + z_start = z_end + + # Air padding at right edge + z_end = z_start + int(air_padding_um * DESIGN_REGION_RESOLUTION_UM) + weights[z_start:z_end] = 0 + + # Smooth the design weights by downsampling from the design grid + # to the simulation grid using bilinear interpolation. + z_sim_grid = design_region_to_grid(NZ_SIM_GRID) + z_design_grid = design_region_to_grid(NZ_DESIGN_GRID) + smoothed_weights = np.interp(z_sim_grid, z_design_grid, weights) + + return smoothed_weights.flatten() + + +def levelset_and_smoothing_vjp( + ans: np.ndarray, layer_thickness_um: np.ndarray +) -> Callable[[np.ndarray], np.ndarray]: + """Returns a function for computing the vector-Jacobian product.""" + + total_layer_thickness_um = np.sum(layer_thickness_um) + air_padding_um = 0.5 * (DESIGN_REGION_UM.z - total_layer_thickness_um) + + jacobian = np.zeros((NZ_SIM_GRID, NUM_LAYERS)) + + z_design_grid = design_region_to_grid(NZ_DESIGN_GRID) + z_sim_grid = design_region_to_grid(NZ_SIM_GRID) + + for i in range(NUM_LAYERS): + weights = np.zeros(NZ_DESIGN_GRID) + + # Air padding at left edge + z_start = 0 + z_end = int(air_padding_um * DESIGN_REGION_RESOLUTION_UM) + weights[z_start:z_end] = 0 + + z_start = z_end + for j in range(NUM_LAYERS): + layer_um = layer_thickness_um[j] + if j == i: + layer_um += LAYER_PERTURBATION_UM + z_end = z_start + int(layer_um * DESIGN_REGION_RESOLUTION_UM) + weights[z_start:z_end] = 1 if (j % 2 == 0) else 0 + z_start = z_end + + # Air padding at right edge + z_end = z_start + int(air_padding_um * DESIGN_REGION_RESOLUTION_UM) + weights[z_start:z_end] = 0 + + # Smooth the design weights by downsampling from the design grid + # to the simulation grid using bilinear interpolation. + smoothed_weights = np.interp(z_sim_grid, z_design_grid, weights) + + jacobian[:, i] = (smoothed_weights - ans) / LAYER_PERTURBATION_UM + + return lambda g: np.tensordot(g, jacobian, axes=1) + + +def multilayer_stack() -> mpa.OptimizationProblem: + """Sets up the adjoint optimization of a multilayer stack. + + Returns: + A `meep.adjoint.Optimization` callable object. + """ + pml_layers = [mp.PML(thickness=PML_UM)] + + size_z_um = PML_UM + AIR_UM + DESIGN_REGION_UM.z + AIR_UM + PML_UM + cell_size = mp.Vector3(0, 0, size_z_um) + + frequency_center = np.mean(frequencies) + + # Set source bandwidth to be larger than the range of design wavelengths. + frequency_width = 1.2 * (np.max(frequencies) - np.min(frequencies)) + + src_cmpt = mp.Ex + src_pt = mp.Vector3(0, 0, -0.5 * size_z_um + PML_UM) + sources = [ + mp.Source( + mp.GaussianSource(frequency_center, fwidth=frequency_width), + component=src_cmpt, + center=src_pt, + ) + ] + + mat_1 = mp.Medium(index=N_LAYER_1) + mat_2 = mp.Medium(index=N_LAYER_2) + + matgrid = mp.MaterialGrid( + mp.Vector3(0, 0, NZ_SIM_GRID), + mat_1, + mat_2, + weights=np.ones(NZ_SIM_GRID), + do_averaging=False, + ) + + matgrid_region = mpa.DesignRegion( + matgrid, + volume=mp.Volume(center=mp.Vector3(), size=DESIGN_REGION_UM), + ) + + geometry = [ + mp.Block( + material=matgrid, size=matgrid_region.size, center=matgrid_region.center + ) + ] + + sim = mp.Simulation( + resolution=RESOLUTION_UM, + cell_size=cell_size, + dimensions=1, + boundary_layers=pml_layers, + sources=sources, + geometry=geometry, + ) + + obj_args = [ + mpa.FourierFields( + sim, volume=matgrid_region.volume, component=mp.Ex, yee_grid=True + ) + ] + + def obj_func(dft_ex: np.ndarray) -> Tuple[float]: + """Objective function for the optimization. + + Args: + dft_ex: the discrete Fourier-transformed Ex fields as a 2D array of + dimension (num_wavelengths, NX_DESIGN_GRID * NY_DESIGN_GRID). + + Returns: + A tuple of the integrals of Ex in the stack for each wavelength. + """ + return npa.sum(npa.absolute(dft_ex) ** 2, axis=1) + + opt = mpa.OptimizationProblem( + simulation=sim, + objective_functions=obj_func, + objective_arguments=obj_args, + design_regions=[matgrid_region], + frequencies=frequencies, + ) + + return opt + + +def obj_func(epigraph_and_layer_thickness_um: np.ndarray, grad: np.ndarray) -> float: + """Objective function for the epigraph formulation. + + Args: + epigraph_and_layer_thickness_um: 1D array containing epigraph variable + (first element) and design weights (remaining elements). + grad: the gradient as a flattened (1D) array, modified in place. + + Returns: + The scalar epigraph variable. + """ + epigraph = epigraph_and_layer_thickness_um[0] + + if grad.size > 0: + grad[0] = 1 + grad[1:] = 0 + + return epigraph + + +def epigraph_constraint( + result: float, epigraph_and_layer_thickness_um: np.ndarray, gradient: np.ndarray +) -> None: + """Constraint function for the epigraph formulation. + + Args: + result: the result of the function evaluation, modified in place. + epigraph_and_layer_thickness_um: 1D array containing the epigraph variable + (first element) and the layer thicknesses (remaining elements). + gradient: the Jacobian matrix with dimensions (num_wavelengths, + 1 + NUM_LAYERS), modified in place. + """ + epigraph = epigraph_and_layer_thickness_um[0] + layer_thickness_um = epigraph_and_layer_thickness_um[1:] + design_weights = levelset_and_smoothing(layer_thickness_um) + + obj_val, grad = opt([design_weights]) + + # Backpropagate the gradients through the levelset construction function. + grad_backpropagate = np.zeros((NUM_LAYERS, num_wavelengths)) + defvjp(levelset_and_smoothing, levelset_and_smoothing_vjp) + for k in range(num_wavelengths): + grad_backpropagate[:, k] = tensor_jacobian_product(levelset_and_smoothing, 0)( + layer_thickness_um, grad[:, k] + ) + + # TODO (oskooi): determine why factor of 0.5 is necessary for correctness. + grad_backpropagate = 0.5 * grad_backpropagate + + if gradient.size > 0: + gradient[:, 0] = -1 # gradient w.r.t. epigraph variable + gradient[:, 1:] = grad_backpropagate.T + + result[:] = obj_val - epigraph + + print( + f"iteration:, {current_iteration[0]:2d}, {str_from_list(result)}, " + f"{str_from_list(layer_thickness_um)}" + ) + + obj_func_history.append(obj_val) + epigraph_variable_history.append(epigraph) + current_iteration[0] += 1 + + +if __name__ == "__main__": + opt = multilayer_stack() + + # Lower and upper bounds for layer thicknesses. + lower_bound = MIN_LAYER_UM * np.ones(NUM_LAYERS) + upper_bound = MAX_LAYER_UM * np.ones(NUM_LAYERS) + + rng = np.random.RandomState() + layer_thickness_um = MIN_LAYER_UM + rng.rand(NUM_LAYERS) * ( + MAX_LAYER_UM - MIN_LAYER_UM + ) + + # Execute a single forward run before the start of the optimization and + # set the initial epigraph variable to slightly larger than the + # largest value of the objective function over the wavelengths. + design_weights = levelset_and_smoothing(layer_thickness_um) + epigraph, _ = opt([design_weights], need_gradient=False) + epigraph_initial = 1.05 * np.amax(epigraph) + + epigraph_and_layer_thickness_um = np.concatenate( + ([epigraph_initial], layer_thickness_um) + ) + + # Insert bounds for the epigraph variable. + lower_bound = np.insert(lower_bound, 0, 0) + upper_bound = np.insert(upper_bound, 0, np.inf) + + tolerance_epigraph = [1e-3] * num_wavelengths + + solver = nlopt.opt(nlopt.LD_MMA, NUM_LAYERS + 1) + solver.set_lower_bounds(lower_bound) + solver.set_upper_bounds(upper_bound) + solver.set_min_objective(obj_func) + solver.set_maxeval(MAX_OPT_ITERATIONS) + solver.add_inequality_mconstraint(epigraph_constraint, tolerance_epigraph) + solver.set_ftol_rel(1e-3) + + obj_func_history = [] + epigraph_variable_history = [] + current_iteration = [0] + + epigraph_and_layer_thickness_um = solver.optimize(epigraph_and_layer_thickness_um) + optimal_obj_val = solver.last_optimum_value() + return_code = solver.last_optimize_result() + + print(f"optimal_obj_val:, {optimal_obj_val}") + print( + "optimal_layer_thickness_um:, " + f"{str_from_list(epigraph_and_layer_thickness_um[1:])}" + ) + print(f"return_code:, {return_code}") + + # Save important optimization parameters and output for post processing. + with open("optimal_design.npz", "wb") as datafile: + np.savez( + datafile, + RESOLUTION_UM=RESOLUTION_UM, + AIR_UM=AIR_UM, + PML_UM=PML_UM, + NUM_LAYERS=NUM_LAYERS, + MIN_LAYER_UM=MIN_LAYER_UM, + MAX_LAYER_UM=MAX_LAYER_UM, + N_LAYER_1=N_LAYER_1, + N_LAYER_2=N_LAYER_2, + DESIGN_WAVELENGTHS_UM=DESIGN_WAVELENGTHS_UM, + DESIGN_REGION_UM=DESIGN_REGION_UM, + DESIGN_REGION_RESOLUTION_UM=DESIGN_REGION_RESOLUTION_UM, + NZ_DESIGN_GRID=NZ_DESIGN_GRID, + NZ_SIM_GRID=NZ_SIM_GRID, + MAX_OPT_ITERATIONS=MAX_OPT_ITERATIONS, + obj_func_history=obj_func_history, + epigraph_variable_history=epigraph_variable_history, + epigraph_variable=epigraph_and_layer_thickness_um[0], + layer_thickness_um=epigraph_and_layer_thickness_um[1:], + tolerance_epigraph=tolerance_epigraph, + optimal_obj_val=optimal_obj_val, + return_code=return_code, + ) +``` Compact Notebook Tutorials of Basic Features diff --git a/python/examples/adjoint_optimization/multilayer_opt.py b/python/examples/adjoint_optimization/multilayer_opt.py index f44414445..9c81fe480 100644 --- a/python/examples/adjoint_optimization/multilayer_opt.py +++ b/python/examples/adjoint_optimization/multilayer_opt.py @@ -202,7 +202,7 @@ def multilayer_stack() -> mpa.OptimizationProblem: obj_args = [ mpa.FourierFields( - sim, volume=matgrid_region.volume, component=mp.Ex, yee_grid=False + sim, volume=matgrid_region.volume, component=mp.Ex, yee_grid=True ) ] @@ -319,7 +319,7 @@ def epigraph_constraint( # Insert bounds for the epigraph variable. lower_bound = np.insert(lower_bound, 0, 0) - upper_bound = np.insert(upper_bound, 0, +np.inf) + upper_bound = np.insert(upper_bound, 0, np.inf) tolerance_epigraph = [1e-3] * num_wavelengths From 7e23b98b93399bfff0a69e691e06723e5dc5d828 Mon Sep 17 00:00:00 2001 From: Ardavan Oskooi Date: Thu, 1 Aug 2024 06:54:58 -0700 Subject: [PATCH 05/13] add figures --- ...roadband_field_decay_in_multilayer_stack.png | Bin 0 -> 70683 bytes .../images/multilayer_opt_obj_func_history.png | Bin 0 -> 85811 bytes 2 files changed, 0 insertions(+), 0 deletions(-) create mode 100644 doc/docs/images/broadband_field_decay_in_multilayer_stack.png create mode 100644 doc/docs/images/multilayer_opt_obj_func_history.png diff --git a/doc/docs/images/broadband_field_decay_in_multilayer_stack.png b/doc/docs/images/broadband_field_decay_in_multilayer_stack.png new file mode 100644 index 0000000000000000000000000000000000000000..bab0776f62efd52875421e04dce0e92f15e5437c GIT binary patch literal 70683 zcmcG$1z43`*Dksc5hX-KN=iv3rKG_i1*8;_4i%N|4n>eoDV1(TLb_49Lj-9R5TrY# 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a/doc/docs/Python_Tutorials/Adjoint_Solver.md +++ b/doc/docs/Python_Tutorials/Adjoint_Solver.md @@ -1193,19 +1193,19 @@ if __name__ == "__main__": Shape Optimization of a Multilayer Stack ---------------------------------------- -We extend the demonstration of the shape derivative from the previous tutorial to perform a shape optimization of a multilayer stack over a broad bandwidth. The 1D design problem involves finding the layer thicknesses for a fixed number of layers (9) which minimize the largest transmittance through the stack at two wavelengths: $\lambda_1$ = 0.95 μm and $\lambda_2$ = 1.05 μm. However, rather than use the transmittance as the objective function, we use the field decay via the integral of the fields in the stack ($\int |E_x|^2$). This provides more information to the optimizer which generally produces better results. For reference, the analytic solution at a *single* wavelength of $\lambda$ = 1.0 μm is a quarter-wavelength stack with thicknesses $\lambda / (4 n_1)$ = 0.25 and $\lambda / (4 n_2)$ = 0.1923. +We extend the demonstration of the shape derivative from the previous tutorial to perform shape optimization of a multilayer stack over a broad bandwidth. The 1D design problem involves finding the layer thicknesses for a fixed number of layers (9) which minimize the largest transmittance through the stack at two wavelengths: $\lambda_1$ = 0.95 μm and $\lambda_2$ = 1.05 μm. However, rather than use the transmittance (or reflectance, if maximizing) as the objective function, we use the field decay via the integral of the fields in the stack ($\int |E_x|^2$). This provides more information to the optimizer which generally produces better results. For reference, the analytic solution at a *single* wavelength of $\lambda$ = 1.0 μm is a [quarter-wavelength stack](https://en.wikipedia.org/wiki/Distributed_Bragg_reflector) with layer thicknesses $\lambda / (4 n_1)$ = 0.25 and $\lambda / (4 n_2)$ = 0.1923. -(For lots of layers and a single wavelength, you can easily get a non-quarter-wave structure with very low transmission, and once you get to that point the gradients will be almost zero so probably it will converge very slowly towards a quarter-wave solution because the optimizer is generally very slow to try to reduce transmission from 1e-3 to 1e-6, for example.) +(For lots of layers and a single wavelength, the optimizer can easily find a non-quarter-wavelength structure with very low transmission, and once it gets to that point the gradients will be almost zero so probably it will converge very slowly towards a quarter-wavelength solution because the optimizer is generally very slow to try to reduce transmittance from 1e-3 to 1e-6, for example.) -The stack consists of alternating materials of refractive index $n_1$ = 1.0 and $n_2$ = 1.3. The layers are arranged as $n_2$, $n_1$, $n_2$, $n_1$, ..., $n_2$. The thickness of all layers is constrained to be in the range of 0.1 μm to 0.5 μm. The thickness constraint is trivial to implement in this case because it applies directly to a geometric feature rather than to the density weights. +The stack consists of two materials of alternating refractive index $n_1$ = 1.0 and $n_2$ = 1.3. The layers are arranged as $n_2$, $n_1$, $n_2$, $n_1$, ..., $n_2$. The thickness of all layers is constrained to be in the range of 0.1 to 0.5 μm. The thickness constraint is trivial to implement in this case because it applies directly to a geometric feature rather than to the density weights. -The worst-case optimization is implemented using the [epigraph formulation](https://nlopt.readthedocs.io/en/latest/NLopt_Introduction/#equivalent-formulations-of-optimization-problems). A plot of the objective function vs. iteration number for the two wavelengths is shown below. Also included in this plot is the epigraph variable. The initial layer thicknesses are chosen randomly. The optimizer seems to converge to a local optima within ~25 iterations. In this case, the optimizer ran for thirty iterations before reaching the threshold tolerance for the objective value. Since these are 1D simulations, the runtime for each design iteration is generally fast (about ten seconds using two cores of Intel Xeon i7-7700K CPU @ 4.20GHz). +The worst-case optimization is implemented using the [epigraph formulation](https://nlopt.readthedocs.io/en/latest/NLopt_Introduction/#equivalent-formulations-of-optimization-problems) and the Method of Moving Asymptotes (MMA) algorithm. A plot of the objective function vs. iteration number for the two wavelengths is shown below. Also included in this plot is the epigraph variable. The initial layer thicknesses are chosen randomly. The optimizer seems to converge to a local optima within ~25 iterations. In this case, the optimizer ran for the maximum number of iterations (50). The nine layer thicknesses (μm) of the optimal design are: 0.1664, 0.2452, 0.1399, 0.2719, 0.1786, 0.3060, 0.4952, 0.1688, 0.2306. Since these are 1D simulations, the runtime for each design iteration is generally fast (about ten seconds using two cores of Intel Xeon i7-7700K CPU @ 4.20GHz). -The plot shows that the optimizer which uses the Method of Moving Asymptotes (MMA) algorithm takes some steps that make things *worse* during the early iterations. This is a common phenomenon in many algorithms — initially, the optimizer takes steps that are too large and then has to backtrack. After a few iterations, the algorithm has more information about a "good" step size. +The plot shows that the optimizer takes some steps that make things *worse* during the early iterations. This is a common phenomenon in many algorithms — initially, the optimizer takes steps that are too large and then has to backtrack. After a few iterations, the algorithm has more information about a "good" step size. ![](../images/multilayer_opt_obj_func_history.png#center) -A plot of the $|E_x|^2$ fields in the stack (design region) for the optimal design is shown below. The plot shows that while the DFT fields for each of the two wavelengths are decaying through the stack, the transmittance values are quite different: T($\lambda$ = 0.95 μm) = 0.3551 vs. T($\lambda$ = 1.05 μm) = 0.8098. This result is consistent with the plot of the objective function vs. iteration number which shows that, because of the minimax setup, the stack is being optimized for $\lambda$ = 0.95 μm and *not* $\lambda$ = 1.05 μm. We observed this trend for various local optima. This seems to suggest that the optimizer targeting a single wavelength of a broad bandwdith design problem may be a general feature for this class of optimization problem. Also, the magnitude of $|E_x|^2$ at the right edge of the stack is clearly larger for $\lambda$ = 1.05 μm than $\lambda$ = 0.95 μm. This is again consistent with the transmittance being larger for $\lambda$ = 1.05 than for $\lambda$ = 0.95 μm. +A plot of the $|E_x|^2$ fields in the stack (design region) for the optimal design is shown below. The plot shows that while the DFT fields for each of the two wavelengths are decaying through the stack, the transmittance values are quite different: T($\lambda$ = 0.95 μm) = 0.3140 vs. T($\lambda$ = 1.05 μm) = 0.7353. This result is consistent with the plot of the objective function vs. iteration number which shows that, because of the minimax setup, the stack is being optimized for $\lambda$ = 0.95 μm and *not* $\lambda$ = 1.05 μm. We observed this trend for various local optima. This seems to suggest that optimizing one of several wavelengths may be a general feature for this particular class of problem. Also, the magnitude of $|E_x|^2$ at the right edge of the stack is larger for $\lambda$ = 1.05 μm than $\lambda$ = 0.95 μm. This is consistent with the transmittance being larger for $\lambda$ = 1.05 than for $\lambda$ = 0.95 μm. ![](../images/broadband_field_decay_in_multilayer_stack.png#center) @@ -1520,14 +1520,14 @@ if __name__ == "__main__": lower_bound = np.insert(lower_bound, 0, 0) upper_bound = np.insert(upper_bound, 0, np.inf) - tolerance_epigraph = [1e-3] * num_wavelengths + epigraph_tolerance = [1e-3] * num_wavelengths solver = nlopt.opt(nlopt.LD_MMA, NUM_LAYERS + 1) solver.set_lower_bounds(lower_bound) solver.set_upper_bounds(upper_bound) solver.set_min_objective(obj_func) solver.set_maxeval(MAX_OPT_ITERATIONS) - solver.add_inequality_mconstraint(epigraph_constraint, tolerance_epigraph) + solver.add_inequality_mconstraint(epigraph_constraint, epigraph_tolerance) solver.set_ftol_rel(1e-3) obj_func_history = [] From 37065a28c548295792b4bcc7a9886ad39f5076e3 Mon Sep 17 00:00:00 2001 From: Ardavan Oskooi Date: Thu, 1 Aug 2024 11:33:15 -0700 Subject: [PATCH 07/13] update figures with new design --- 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zU#m%n)15IGF@cuvtn6HdhTou2fHnU?sYXvqF6p81M9VlMSlEh zdgjcT?10wfWQ}dd;&gcB+^ZcbStAcWFUKMaJ`*&0!tyw_4AA8qzLkiuaLtOJA8O$| z(X#KcL6=}^0T2?f%bT<kC|-Bs!R8 zt;5E6oUVeL``STTlfs*FWV^7}ErS0DAd{tdN+!(Z)Cw1gV zxr?iqm>7$I2vFDz*_7<+hR?2#QpE!k6nacR*k`VIg3v-veR;^8J7bqKa%^hT@aS-w zG^08zEfK1*fzO{0Dl1bl??{_f-gg4^j69OC_MKTP|i{v%cZM7x%1YX2@pj{)hVi5n8)!dYP z*M@z?xmQXJqh5WWZy+l&Wejq4Kw3}WJQ|KkYJW3}H4nbzbh-b0vkcoZ5$n*(yWMW7 zYbwyRKL<^0w{JgZD5S@R$=XeL1`*)>fGtL$czVPdX+huYSiC`uXpiCJWdS{iQ8=HH z=R^TJEh9r4Q~5SDCB~@d6~?oF;h0YZg9j>$IK;qP(9rdWN=Oi=x);ILlP%k!S#4{^L2d*v5V8p;Z90NT80E zb&e)L9MPDFik?FKE&}dJ-6gKL*0v7~&sLT1dT;@ZM(gk_yJG1Sf Date: Sun, 4 Aug 2024 15:50:26 -0700 Subject: [PATCH 08/13] update script, text, and two figures with new results and add schematic of design problem --- doc/docs/Python_Tutorials/Adjoint_Solver.md | 246 +++++++++++------- ...adband_field_decay_in_multilayer_stack.png | Bin 70565 -> 62194 bytes .../multilayer_opt_obj_func_history.png | Bin 77083 -> 49503 bytes .../images/multilayer_stack_shape_opt.png | Bin 0 -> 64122 bytes .../adjoint_optimization/multilayer_opt.py | 181 +++++++------ 5 files changed, 264 insertions(+), 163 deletions(-) create mode 100644 doc/docs/images/multilayer_stack_shape_opt.png diff --git a/doc/docs/Python_Tutorials/Adjoint_Solver.md b/doc/docs/Python_Tutorials/Adjoint_Solver.md index ca5356f6b..8cb41c69a 100644 --- a/doc/docs/Python_Tutorials/Adjoint_Solver.md +++ b/doc/docs/Python_Tutorials/Adjoint_Solver.md @@ -1193,25 +1193,30 @@ if __name__ == "__main__": Shape Optimization of a Multilayer Stack ---------------------------------------- -We extend the demonstration of the shape derivative from the previous tutorial to perform shape optimization of a multilayer stack over a broad bandwidth. The 1D design problem involves finding the layer thicknesses for a fixed number of layers (9) which minimize the largest transmittance through the stack at two wavelengths: $\lambda_1$ = 0.95 μm and $\lambda_2$ = 1.05 μm. However, rather than use the transmittance (or reflectance, if maximizing) as the objective function, we use the field decay via the integral of the fields in the stack ($\int |E_x|^2$). This provides more information to the optimizer which generally produces better results. For reference, the analytic solution at a *single* wavelength of $\lambda$ = 1.0 μm is a [quarter-wavelength stack](https://en.wikipedia.org/wiki/Distributed_Bragg_reflector) with layer thicknesses $\lambda / (4 n_1)$ = 0.25 and $\lambda / (4 n_2)$ = 0.1923. +We extend the demonstration of the shape derivative from the previous tutorial to perform shape optimization of a multilayer stack over a broad bandwidth. The 1D design problem is shown in the schematic below and involves finding the layer thicknesses for a fixed number of layers (9) which minimize the largest transmittance (or alternatively, maximize the smallest reflectance) at two wavelengths: $\lambda_1$ = 0.95 μm and $\lambda_2$ = 1.05 μm. However, rather than use the transmittance as the objective function, we use the *field decay* via (the logarithm of) the integral of the fields in the stack ($\int |E_x|^2$). This provides more information to the optimizer which should produce better results. -(For lots of layers and a single wavelength, the optimizer can easily find a non-quarter-wavelength structure with very low transmission, and once it gets to that point the gradients will be almost zero so probably it will converge very slowly towards a quarter-wavelength solution because the optimizer is generally very slow to try to reduce transmittance from 1e-3 to 1e-6, for example.) +The stack consists of two materials of alternating refractive index $n_A$ = 1.3 and $n_B$ = 1.0. The layers are arranged as $n_A$, $n_B$, $n_A$, $n_B$, ..., $n_A$. The semi-infinite regions to the left and right of the stack are vacuum. -The stack consists of two materials of alternating refractive index $n_1$ = 1.0 and $n_2$ = 1.3. The layers are arranged as $n_2$, $n_1$, $n_2$, $n_1$, ..., $n_2$. The thickness of all layers is constrained to be in the range of 0.1 to 0.5 μm. The thickness constraint is trivial to implement in this case because it applies directly to a geometric feature rather than to the density weights. +![](../images/multilayer_stack_shape_opt.png#center) -The worst-case optimization is implemented using the [epigraph formulation](https://nlopt.readthedocs.io/en/latest/NLopt_Introduction/#equivalent-formulations-of-optimization-problems) and the Method of Moving Asymptotes (MMA) algorithm. A plot of the objective function vs. iteration number for the two wavelengths is shown below. Also included in this plot is the epigraph variable. The initial layer thicknesses are chosen randomly. The optimizer seems to converge to a local optima within ~25 iterations. In this case, the optimizer ran for the maximum number of iterations (50). The nine layer thicknesses (μm) of the optimal design are: 0.1664, 0.2452, 0.1399, 0.2719, 0.1786, 0.3060, 0.4952, 0.1688, 0.2306. Since these are 1D simulations, the runtime for each design iteration is generally fast (about ten seconds using two cores of Intel Xeon i7-7700K CPU @ 4.20GHz). +A reference design to compare our results against is a [quarter-wavelength stack](https://en.wikipedia.org/wiki/Distributed_Bragg_reflector). The mean frequency of $\lambda_1$ and $\lambda_2$ is $\lambda \approx$ 1.0 μm for which the quarter-wavelength layer thicknesses are approximately $\lambda / (4 n_A)$ = 0.19 μm and $\lambda / (4 n_B)$ = 0.25 μm. These values are used to specify upper and lower bounds for the layer thicknesses. This is important given the non-convex nature of this particular design problem. -The plot shows that the optimizer takes some steps that make things *worse* during the early iterations. This is a common phenomenon in many algorithms — initially, the optimizer takes steps that are too large and then has to backtrack. After a few iterations, the algorithm has more information about a "good" step size. +(For several layers and a single wavelength, the optimizer can easily find a non-quarter-wavelength structure with very low transmission, and once it gets to that point the gradients will be almost zero so probably it will converge very slowly towards a quarter-wavelength solution because the optimizer is generally very slow to try to reduce transmittance from 1e-3 to 1e-6, for example.) + +The worst-case optimization is implemented using the [epigraph formulation](https://nlopt.readthedocs.io/en/latest/NLopt_Introduction/#equivalent-formulations-of-optimization-problems) and the Conservative Convex Separable Approximation (CCSA) algorithm. The optimization is run ten times with random initial designs from which the local optima with the smallest objective value is chosen. For the run involving the best design, a plot of the objective function vs. iteration number for the two wavelengths is shown below. Also included in this plot is the epigraph variable. These results demonstrate that the stack is being optimized for $\lambda$ = 0.95 μm and *not* $\lambda$ = 1.05 μm. We observed this trend for various local optima. The optimizer converges to the local optima using 13 iterations. The nine layer thicknesses (μm) of this design are: 0.1822, 0.2369, 0.1822, 0.2369, 0.1822, 0.2369, 0.1910, 0.2504, 0.1931. Since these are 1D simulations, the runtime for each design iteration is generally fast (about ten seconds using one core of an Intel Xeon i7-7700K CPU @ 4.20GHz). + +In some cases (not shown), the optimizer may take some steps that make things *worse* during the early iterations. This is a common phenomenon in many algorithms — initially, the optimizer takes steps that are too large and then has to backtrack. After a few iterations, the algorithm has more information about a "good" step size. ![](../images/multilayer_opt_obj_func_history.png#center) -A plot of the $|E_x|^2$ fields in the stack (design region) for the optimal design is shown below. The plot shows that while the DFT fields for each of the two wavelengths are decaying through the stack, the transmittance values are quite different: T($\lambda$ = 0.95 μm) = 0.3140 vs. T($\lambda$ = 1.05 μm) = 0.7353. This result is consistent with the plot of the objective function vs. iteration number which shows that, because of the minimax setup, the stack is being optimized for $\lambda$ = 0.95 μm and *not* $\lambda$ = 1.05 μm. We observed this trend for various local optima. This seems to suggest that optimizing one of several wavelengths may be a general feature for this particular class of problem. Also, the magnitude of $|E_x|^2$ at the right edge of the stack is larger for $\lambda$ = 1.05 μm than $\lambda$ = 0.95 μm. This is consistent with the transmittance being larger for $\lambda$ = 1.05 than for $\lambda$ = 0.95 μm. +Finally, a plot of $|E_x|^2$ in the stack (design region) normalized by the fields from the quarter-wavelength stack is shown below. The plot shows that while the DFT fields for each of the two wavelengths are decaying through the stack, the transmittance values are different: T($\lambda_1$ = 0.95 μm) = 0.2574 vs. T($\lambda_2$ = 1.05 μm) = 0.3734. The transmittance for the quarter-wavelength stack at $\lambda \approx$ 1.0 μm is 0.2522. Note that the magnitude of $|E_x|^2$ at the right edge of the stack is larger for $\lambda_2$ = 1.05 μm than $\lambda_1$ = 0.95 μm. This is consistent with the transmittance being larger for $\lambda_2$ = 1.05 than for $\lambda_1$ = 0.95 μm. ![](../images/broadband_field_decay_in_multilayer_stack.png#center) The script is [python/examples/adjoint_optimization/multilayer_opt.py](https://github.com/NanoComp/meep/tree/master/python/examples/adjoint_optimization/multilayer_opt.py). ```py +import copy from typing import Callable, List, Tuple from autograd.extend import primitive, defvjp @@ -1227,17 +1232,16 @@ RESOLUTION_UM = 800 AIR_UM = 1.0 PML_UM = 1.0 NUM_LAYERS = 9 -MIN_LAYER_UM = 0.1 -MAX_LAYER_UM = 0.5 -N_LAYER_1 = 1.0 -N_LAYER_2 = 1.3 +N_LAYER = (1.0, 1.3) LAYER_PERTURBATION_UM = 1.0 / RESOLUTION_UM DESIGN_WAVELENGTHS_UM = (0.95, 1.05) +MAX_LAYER_UM = max(DESIGN_WAVELENGTHS_UM) / (4 * min(N_LAYER)) DESIGN_REGION_RESOLUTION_UM = 10 * RESOLUTION_UM DESIGN_REGION_UM = mp.Vector3(0, 0, NUM_LAYERS * MAX_LAYER_UM) NZ_DESIGN_GRID = int(DESIGN_REGION_UM.z * DESIGN_REGION_RESOLUTION_UM) + 1 NZ_SIM_GRID = int(DESIGN_REGION_UM.z * RESOLUTION_UM) + 1 -MAX_OPT_ITERATIONS = 50 +MAX_OPT_ITERATIONS = 30 +NUM_OPT_REPEAT = 10 num_wavelengths = len(DESIGN_WAVELENGTHS_UM) frequencies = [1 / wavelength_um for wavelength_um in DESIGN_WAVELENGTHS_UM] @@ -1256,7 +1260,11 @@ def design_region_to_grid(nz: int) -> np.ndarray: Returns: The 1D coordinates of the grid points. """ - z_grid = np.linspace(-0.5 * DESIGN_REGION_UM.z, 0.5 * DESIGN_REGION_UM.z, nz) + z_grid = np.linspace( + -0.5 * DESIGN_REGION_UM.z, + 0.5 * DESIGN_REGION_UM.z, + nz + ) return z_grid @@ -1300,7 +1308,8 @@ def levelset_and_smoothing(layer_thickness_um: np.ndarray) -> np.ndarray: def levelset_and_smoothing_vjp( - ans: np.ndarray, layer_thickness_um: np.ndarray + ans: np.ndarray, + layer_thickness_um: np.ndarray ) -> Callable[[np.ndarray], np.ndarray]: """Returns a function for computing the vector-Jacobian product.""" @@ -1368,25 +1377,30 @@ def multilayer_stack() -> mpa.OptimizationProblem: ) ] - mat_1 = mp.Medium(index=N_LAYER_1) - mat_2 = mp.Medium(index=N_LAYER_2) + mat_1 = mp.Medium(index=N_LAYER[0]) + mat_2 = mp.Medium(index=N_LAYER[1]) matgrid = mp.MaterialGrid( mp.Vector3(0, 0, NZ_SIM_GRID), mat_1, mat_2, weights=np.ones(NZ_SIM_GRID), - do_averaging=False, + do_averaging=False ) matgrid_region = mpa.DesignRegion( matgrid, - volume=mp.Volume(center=mp.Vector3(), size=DESIGN_REGION_UM), + volume=mp.Volume( + center=mp.Vector3(), + size=DESIGN_REGION_UM + ), ) geometry = [ mp.Block( - material=matgrid, size=matgrid_region.size, center=matgrid_region.center + material=matgrid, + size=matgrid_region.size, + center=matgrid_region.center ) ] @@ -1396,12 +1410,15 @@ def multilayer_stack() -> mpa.OptimizationProblem: dimensions=1, boundary_layers=pml_layers, sources=sources, - geometry=geometry, + geometry=geometry ) obj_args = [ mpa.FourierFields( - sim, volume=matgrid_region.volume, component=mp.Ex, yee_grid=True + sim, + volume=matgrid_region.volume, + component=mp.Ex, + yee_grid=True ) ] @@ -1413,22 +1430,26 @@ def multilayer_stack() -> mpa.OptimizationProblem: dimension (num_wavelengths, NX_DESIGN_GRID * NY_DESIGN_GRID). Returns: - A tuple of the integrals of Ex in the stack for each wavelength. + A tuple of the log of the integrals of Ex in the stack for each + wavelength. """ - return npa.sum(npa.absolute(dft_ex) ** 2, axis=1) + return npa.log(npa.sum(npa.absolute(dft_ex)**2, axis=1)) opt = mpa.OptimizationProblem( simulation=sim, objective_functions=obj_func, objective_arguments=obj_args, design_regions=[matgrid_region], - frequencies=frequencies, + frequencies=frequencies ) return opt -def obj_func(epigraph_and_layer_thickness_um: np.ndarray, grad: np.ndarray) -> float: +def obj_func( + epigraph_and_layer_thickness_um: np.ndarray, + grad: np.ndarray +) -> float: """Objective function for the epigraph formulation. Args: @@ -1449,12 +1470,14 @@ def obj_func(epigraph_and_layer_thickness_um: np.ndarray, grad: np.ndarray) -> f def epigraph_constraint( - result: float, epigraph_and_layer_thickness_um: np.ndarray, gradient: np.ndarray + result: float, + epigraph_and_layer_thickness_um: np.ndarray, + gradient: np.ndarray ) -> None: """Constraint function for the epigraph formulation. Args: - result: the result of the function evaluation, modified in place. + result: evaluation of the constraint function, modified in place. epigraph_and_layer_thickness_um: 1D array containing the epigraph variable (first element) and the layer thicknesses (remaining elements). gradient: the Jacobian matrix with dimensions (num_wavelengths, @@ -1471,7 +1494,8 @@ def epigraph_constraint( defvjp(levelset_and_smoothing, levelset_and_smoothing_vjp) for k in range(num_wavelengths): grad_backpropagate[:, k] = tensor_jacobian_product(levelset_and_smoothing, 0)( - layer_thickness_um, grad[:, k] + layer_thickness_um, + grad[:, k] ) # TODO (oskooi): determine why factor of 0.5 is necessary for correctness. @@ -1496,81 +1520,123 @@ def epigraph_constraint( if __name__ == "__main__": opt = multilayer_stack() - # Lower and upper bounds for layer thicknesses. - lower_bound = MIN_LAYER_UM * np.ones(NUM_LAYERS) - upper_bound = MAX_LAYER_UM * np.ones(NUM_LAYERS) + # Specify lower and upper bounds for layer thicknesses based on layer + # thicknesses for a quarter-wavelength stack at the mean frequency. + mean_wavelength_um = 1 / np.mean(1 / np.array(DESIGN_WAVELENGTHS_UM)) + mean_layer_thickness_um = np.zeros(2) + mean_layer_thickness_um[0] = mean_wavelength_um / (4 * N_LAYER[0]) + mean_layer_thickness_um[1] = mean_wavelength_um / (4 * N_LAYER[1]) + layer_thickness_um_lower_bound = np.zeros(NUM_LAYERS) + layer_thickness_um_upper_bound = np.zeros(NUM_LAYERS) + fraction_thickness = 0.08 + layer_thickness_um_lower_bound[0::2] = ((1 - fraction_thickness) * + mean_layer_thickness_um[1]) + layer_thickness_um_upper_bound[0::2] = ((1 + fraction_thickness) * + mean_layer_thickness_um[1]) + layer_thickness_um_lower_bound[1::2] = ((1 - fraction_thickness) * + mean_layer_thickness_um[0]) + layer_thickness_um_upper_bound[1::2] = ((1 + fraction_thickness) * + mean_layer_thickness_um[0]) - rng = np.random.RandomState() - layer_thickness_um = MIN_LAYER_UM + rng.rand(NUM_LAYERS) * ( - MAX_LAYER_UM - MIN_LAYER_UM + # Insert bounds for the epigraph variable. + epigraph_and_layer_thickness_um_lower_bound = np.insert( + layer_thickness_um_lower_bound, 0, 0 + ) + epigraph_and_layer_thickness_um_upper_bound = np.insert( + layer_thickness_um_upper_bound, 0, np.inf ) + epigraph_tolerance = [0.1] * num_wavelengths - # Execute a single forward run before the start of the optimization and - # set the initial epigraph variable to slightly larger than the - # largest value of the objective function over the wavelengths. - design_weights = levelset_and_smoothing(layer_thickness_um) - epigraph, _ = opt([design_weights], need_gradient=False) - epigraph_initial = 1.05 * np.amax(epigraph) + min_obj_val = np.inf + min_epigraph_and_layer_thickness_um = np.zeros(NUM_LAYERS + 1) + obj_func_history = [] + epigraph_variable_history = [] - epigraph_and_layer_thickness_um = np.concatenate( - ([epigraph_initial], layer_thickness_um) - ) + for j in range(NUM_OPT_REPEAT): + rng = np.random.RandomState() + layer_thickness_um = ( + layer_thickness_um_lower_bound + rng.rand(NUM_LAYERS) * + (layer_thickness_um_upper_bound - layer_thickness_um_lower_bound) + ) - # Insert bounds for the epigraph variable. - lower_bound = np.insert(lower_bound, 0, 0) - upper_bound = np.insert(upper_bound, 0, np.inf) + # Execute a single forward run before the start of the optimization and + # set the initial epigraph variable to slightly larger than the + # largest value of the objective function over the wavelengths. + design_weights = levelset_and_smoothing(layer_thickness_um) + epigraph, _ = opt( + [design_weights], + need_gradient=False + ) + fraction_max_epigraph = 0.2 + epigraph_initial = (1 + fraction_max_epigraph) * np.amax(epigraph) + epigraph_and_layer_thickness_um_upper_bound[0] = epigraph_initial + + epigraph_and_layer_thickness_um = np.concatenate( + ( + [epigraph_initial], + layer_thickness_um + ) + ) - epigraph_tolerance = [1e-3] * num_wavelengths + solver = nlopt.opt(nlopt.LD_CCSAQ, NUM_LAYERS + 1) + solver.set_lower_bounds(epigraph_and_layer_thickness_um_lower_bound) + solver.set_upper_bounds(epigraph_and_layer_thickness_um_upper_bound) + solver.set_min_objective(obj_func) + solver.set_maxeval(MAX_OPT_ITERATIONS) + solver.add_inequality_mconstraint( + epigraph_constraint, + epigraph_tolerance + ) + solver.set_ftol_rel(0.02) - solver = nlopt.opt(nlopt.LD_MMA, NUM_LAYERS + 1) - solver.set_lower_bounds(lower_bound) - solver.set_upper_bounds(upper_bound) - solver.set_min_objective(obj_func) - solver.set_maxeval(MAX_OPT_ITERATIONS) - solver.add_inequality_mconstraint(epigraph_constraint, epigraph_tolerance) - solver.set_ftol_rel(1e-3) + obj_func_history[:] = [] + epigraph_variable_history[:] = [] + current_iteration = [0] - obj_func_history = [] - epigraph_variable_history = [] - current_iteration = [0] + epigraph_and_layer_thickness_um = solver.optimize( + epigraph_and_layer_thickness_um + ) + optimal_obj_val = solver.last_optimum_value() + return_code = solver.last_optimize_result() - epigraph_and_layer_thickness_um = solver.optimize(epigraph_and_layer_thickness_um) - optimal_obj_val = solver.last_optimum_value() - return_code = solver.last_optimize_result() + print(f"optimal_obj_val:, {j:2d}, {optimal_obj_val}, {return_code}") + print( + f"optimal_layer_thickness_um:, {j:2d}, " + f"{str_from_list(epigraph_and_layer_thickness_um[1:])}" + ) + + if optimal_obj_val < min_obj_val: + min_obj_val = optimal_obj_val + min_epigraph_and_layer_thickness_um = epigraph_and_layer_thickness_um + min_obj_func_history = copy.deepcopy(obj_func_history) + min_epigraph_variable_history = copy.deepcopy(epigraph_variable_history) + print(f"new_minima:, {j:2d}, {min_obj_val}") - print(f"optimal_obj_val:, {optimal_obj_val}") - print( - "optimal_layer_thickness_um:, " - f"{str_from_list(epigraph_and_layer_thickness_um[1:])}" - ) - print(f"return_code:, {return_code}") # Save important optimization parameters and output for post processing. - with open("optimal_design.npz", "wb") as datafile: - np.savez( - datafile, - RESOLUTION_UM=RESOLUTION_UM, - AIR_UM=AIR_UM, - PML_UM=PML_UM, - NUM_LAYERS=NUM_LAYERS, - MIN_LAYER_UM=MIN_LAYER_UM, - MAX_LAYER_UM=MAX_LAYER_UM, - N_LAYER_1=N_LAYER_1, - N_LAYER_2=N_LAYER_2, - DESIGN_WAVELENGTHS_UM=DESIGN_WAVELENGTHS_UM, - DESIGN_REGION_UM=DESIGN_REGION_UM, - DESIGN_REGION_RESOLUTION_UM=DESIGN_REGION_RESOLUTION_UM, - NZ_DESIGN_GRID=NZ_DESIGN_GRID, - NZ_SIM_GRID=NZ_SIM_GRID, - MAX_OPT_ITERATIONS=MAX_OPT_ITERATIONS, - obj_func_history=obj_func_history, - epigraph_variable_history=epigraph_variable_history, - epigraph_variable=epigraph_and_layer_thickness_um[0], - layer_thickness_um=epigraph_and_layer_thickness_um[1:], - tolerance_epigraph=tolerance_epigraph, - optimal_obj_val=optimal_obj_val, - return_code=return_code, - ) + np.savez( + "optimal_design.npz", + RESOLUTION_UM=RESOLUTION_UM, + AIR_UM=AIR_UM, + PML_UM=PML_UM, + NUM_LAYERS=NUM_LAYERS, + N_LAYER=N_LAYER, + DESIGN_WAVELENGTHS_UM=DESIGN_WAVELENGTHS_UM, + MAX_LAYER_UM=MAX_LAYER_UM, + DESIGN_REGION_UM=DESIGN_REGION_UM, + DESIGN_REGION_RESOLUTION_UM=DESIGN_REGION_RESOLUTION_UM, + NZ_DESIGN_GRID=NZ_DESIGN_GRID, + NZ_SIM_GRID=NZ_SIM_GRID, + MAX_OPT_ITERATIONS=MAX_OPT_ITERATIONS, + epigraph_and_layer_thickness_um_lower_bound=epigraph_and_layer_thickness_um_lower_bound, + epigraph_and_layer_thickness_um_upper_bound=epigraph_and_layer_thickness_um_upper_bound, + obj_func_history=min_obj_func_history, + epigraph_variable_history=min_epigraph_variable_history, + epigraph_variable=min_epigraph_and_layer_thickness_um[0], + layer_thickness_um=min_epigraph_and_layer_thickness_um[1:], + epigraph_tolerance=epigraph_tolerance, + optimal_obj_val=min_obj_val + ) ``` diff --git a/doc/docs/images/broadband_field_decay_in_multilayer_stack.png b/doc/docs/images/broadband_field_decay_in_multilayer_stack.png index badb25173da81fb19d0d403472bed15ea566e036..3a9f373a95e37dc4d717047761770d808754e432 100644 GIT binary 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z&ftxWTBFF=ATD`ni1v*wv4nGx++Jp*1(ZquM|ozgKmTz15I-9)5H?uh^6~oYSJguY z5dVpe0LMHhJ~it1FJgD#T(}Vq;WL$$A;3Ub_|}y6jl0w7vIpDWPG^><`z1T~4weT) zs0~HR)(9fRJtXRaY@V%c-mo%cBKzrZ^j_UB(V~zmx6KLIHJ-kcKVph5=5@dK=B++e z^ToD-nRM&mP}E~4GmP@X4o5(MqP9GnW1>2Xv(xHNMP7GFeu-y+h?Dv~`wgqVNwt3-9vo(QezU5T9Nr$aIhK;|=Ae}Iji8bE5`fFD-EhMBIe zE-^p_QvMr1^W5@+Fu-6y+Jfp?bg7Szw9%74-2!y`N6BNm0Q5#v_sGNHQ~+H-a8O-C z!-7PrnTrOI7(hUdcza7%xJdvQ)!uH;Pky6PEm!KmJ|BQre&(s3-qf@Mnkv2JjPUKJ zMI=#FhY4`8|AmV|+4 kqXf3rIC%Cc mpa.OptimizationProblem: ) ] - mat_1 = mp.Medium(index=N_LAYER_1) - mat_2 = mp.Medium(index=N_LAYER_2) + mat_1 = mp.Medium(index=N_LAYER[0]) + mat_2 = mp.Medium(index=N_LAYER[1]) matgrid = mp.MaterialGrid( mp.Vector3(0, 0, NZ_SIM_GRID), @@ -214,9 +214,10 @@ def obj_func(dft_ex: np.ndarray) -> Tuple[float]: dimension (num_wavelengths, NX_DESIGN_GRID * NY_DESIGN_GRID). Returns: - A tuple of the integrals of Ex in the stack for each wavelength. + A tuple of the log of the integrals of Ex in the stack for each + wavelength. """ - return npa.sum(npa.absolute(dft_ex) ** 2, axis=1) + return npa.log(npa.sum(npa.absolute(dft_ex) ** 2, axis=1)) opt = mpa.OptimizationProblem( simulation=sim, @@ -255,7 +256,7 @@ def epigraph_constraint( """Constraint function for the epigraph formulation. Args: - result: the result of the function evaluation, modified in place. + result: evaluation of the constraint function, modified in place. epigraph_and_layer_thickness_um: 1D array containing the epigraph variable (first element) and the layer thicknesses (remaining elements). gradient: the Jacobian matrix with dimensions (num_wavelengths, @@ -297,78 +298,112 @@ def epigraph_constraint( if __name__ == "__main__": opt = multilayer_stack() - # Lower and upper bounds for layer thicknesses. - lower_bound = MIN_LAYER_UM * np.ones(NUM_LAYERS) - upper_bound = MAX_LAYER_UM * np.ones(NUM_LAYERS) + # Specify lower and upper bounds for layer thicknesses based on layer + # thicknesses for a quarter-wavelength stack at the mean frequency. + mean_wavelength_um = 1 / np.mean(1 / np.array(DESIGN_WAVELENGTHS_UM)) + mean_layer_thickness_um = np.zeros(2) + mean_layer_thickness_um[0] = mean_wavelength_um / (4 * N_LAYER[0]) + mean_layer_thickness_um[1] = mean_wavelength_um / (4 * N_LAYER[1]) + layer_thickness_um_lower_bound = np.zeros(NUM_LAYERS) + layer_thickness_um_upper_bound = np.zeros(NUM_LAYERS) + fraction_thickness = 0.05 + layer_thickness_um_lower_bound[0::2] = ( + 1 - fraction_thickness + ) * mean_layer_thickness_um[1] + layer_thickness_um_upper_bound[0::2] = ( + 1 + fraction_thickness + ) * mean_layer_thickness_um[1] + layer_thickness_um_lower_bound[1::2] = ( + 1 - fraction_thickness + ) * mean_layer_thickness_um[0] + layer_thickness_um_upper_bound[1::2] = ( + 1 + fraction_thickness + ) * mean_layer_thickness_um[0] - rng = np.random.RandomState() - layer_thickness_um = MIN_LAYER_UM + rng.rand(NUM_LAYERS) * ( - MAX_LAYER_UM - MIN_LAYER_UM + # Insert bounds for the epigraph variable. + epigraph_and_layer_thickness_um_lower_bound = np.insert( + layer_thickness_um_lower_bound, 0, 0 + ) + epigraph_and_layer_thickness_um_upper_bound = np.insert( + layer_thickness_um_upper_bound, 0, np.inf ) + epigraph_tolerance = [0.01] * num_wavelengths - # Execute a single forward run before the start of the optimization and - # set the initial epigraph variable to slightly larger than the - # largest value of the objective function over the wavelengths. - design_weights = levelset_and_smoothing(layer_thickness_um) - epigraph, _ = opt([design_weights], need_gradient=False) - epigraph_initial = 1.05 * np.amax(epigraph) + min_obj_val = np.inf + min_epigraph_and_layer_thickness_um = np.zeros(NUM_LAYERS + 1) + obj_func_history = [] + epigraph_variable_history = [] - epigraph_and_layer_thickness_um = np.concatenate( - ([epigraph_initial], layer_thickness_um) - ) + for j in range(NUM_OPT_REPEAT): + rng = np.random.RandomState() + layer_thickness_um = layer_thickness_um_lower_bound + rng.rand(NUM_LAYERS) * ( + layer_thickness_um_upper_bound - layer_thickness_um_lower_bound + ) - # Insert bounds for the epigraph variable. - lower_bound = np.insert(lower_bound, 0, 0) - upper_bound = np.insert(upper_bound, 0, np.inf) + # Execute a single forward run before the start of the optimization and + # set the initial epigraph variable to slightly larger than the + # largest value of the objective function over the wavelengths. + design_weights = levelset_and_smoothing(layer_thickness_um) + epigraph, _ = opt([design_weights], need_gradient=False) + fraction_max_epigraph = 0.2 + epigraph_initial = (1 + fraction_max_epigraph) * np.amax(epigraph) + epigraph_and_layer_thickness_um_upper_bound[0] = epigraph_initial + + epigraph_and_layer_thickness_um = np.concatenate( + ([epigraph_initial], layer_thickness_um) + ) - tolerance_epigraph = [1e-3] * num_wavelengths + solver = nlopt.opt(nlopt.LD_CCSAQ, NUM_LAYERS + 1) + solver.set_lower_bounds(epigraph_and_layer_thickness_um_lower_bound) + solver.set_upper_bounds(epigraph_and_layer_thickness_um_upper_bound) + solver.set_min_objective(obj_func) + solver.set_maxeval(MAX_OPT_ITERATIONS) + solver.add_inequality_mconstraint(epigraph_constraint, epigraph_tolerance) + solver.set_ftol_rel(0.05) - solver = nlopt.opt(nlopt.LD_MMA, NUM_LAYERS + 1) - solver.set_lower_bounds(lower_bound) - solver.set_upper_bounds(upper_bound) - solver.set_min_objective(obj_func) - solver.set_maxeval(MAX_OPT_ITERATIONS) - solver.add_inequality_mconstraint(epigraph_constraint, tolerance_epigraph) - solver.set_ftol_rel(1e-3) + obj_func_history[:] = [] + epigraph_variable_history[:] = [] + current_iteration = [0] - obj_func_history = [] - epigraph_variable_history = [] - current_iteration = [0] + epigraph_and_layer_thickness_um = solver.optimize( + epigraph_and_layer_thickness_um + ) + optimal_obj_val = solver.last_optimum_value() + return_code = solver.last_optimize_result() - epigraph_and_layer_thickness_um = solver.optimize(epigraph_and_layer_thickness_um) - optimal_obj_val = solver.last_optimum_value() - return_code = solver.last_optimize_result() + print(f"optimal_obj_val:, {j:2d}, {optimal_obj_val}, {return_code}") + print( + f"optimal_layer_thickness_um:, {j:2d}, " + f"{str_from_list(epigraph_and_layer_thickness_um[1:])}" + ) - print(f"optimal_obj_val:, {optimal_obj_val}") - print( - "optimal_layer_thickness_um:, " - f"{str_from_list(epigraph_and_layer_thickness_um[1:])}" - ) - print(f"return_code:, {return_code}") + if optimal_obj_val < min_obj_val: + min_obj_val = optimal_obj_val + min_epigraph_and_layer_thickness_um = epigraph_and_layer_thickness_um + min_obj_func_history = copy.deepcopy(obj_func_history) + min_epigraph_variable_history = copy.deepcopy(epigraph_variable_history) # Save important optimization parameters and output for post processing. - with open("optimal_design.npz", "wb") as datafile: - np.savez( - datafile, - RESOLUTION_UM=RESOLUTION_UM, - AIR_UM=AIR_UM, - PML_UM=PML_UM, - NUM_LAYERS=NUM_LAYERS, - MIN_LAYER_UM=MIN_LAYER_UM, - MAX_LAYER_UM=MAX_LAYER_UM, - N_LAYER_1=N_LAYER_1, - N_LAYER_2=N_LAYER_2, - DESIGN_WAVELENGTHS_UM=DESIGN_WAVELENGTHS_UM, - DESIGN_REGION_UM=DESIGN_REGION_UM, - DESIGN_REGION_RESOLUTION_UM=DESIGN_REGION_RESOLUTION_UM, - NZ_DESIGN_GRID=NZ_DESIGN_GRID, - NZ_SIM_GRID=NZ_SIM_GRID, - MAX_OPT_ITERATIONS=MAX_OPT_ITERATIONS, - obj_func_history=obj_func_history, - epigraph_variable_history=epigraph_variable_history, - epigraph_variable=epigraph_and_layer_thickness_um[0], - layer_thickness_um=epigraph_and_layer_thickness_um[1:], - tolerance_epigraph=tolerance_epigraph, - optimal_obj_val=optimal_obj_val, - return_code=return_code, - ) + np.savez( + "optimal_design.npz", + RESOLUTION_UM=RESOLUTION_UM, + AIR_UM=AIR_UM, + PML_UM=PML_UM, + NUM_LAYERS=NUM_LAYERS, + N_LAYER=N_LAYER, + DESIGN_WAVELENGTHS_UM=DESIGN_WAVELENGTHS_UM, + MAX_LAYER_UM=MAX_LAYER_UM, + DESIGN_REGION_UM=DESIGN_REGION_UM, + DESIGN_REGION_RESOLUTION_UM=DESIGN_REGION_RESOLUTION_UM, + NZ_DESIGN_GRID=NZ_DESIGN_GRID, + NZ_SIM_GRID=NZ_SIM_GRID, + MAX_OPT_ITERATIONS=MAX_OPT_ITERATIONS, + epigraph_and_layer_thickness_um_lower_bound=epigraph_and_layer_thickness_um_lower_bound, + epigraph_and_layer_thickness_um_upper_bound=epigraph_and_layer_thickness_um_upper_bound, + obj_func_history=min_obj_func_history, + epigraph_variable_history=min_epigraph_variable_history, + epigraph_variable=min_epigraph_and_layer_thickness_um[0], + layer_thickness_um=min_epigraph_and_layer_thickness_um[1:], + epigraph_tolerance=epigraph_tolerance, + optimal_obj_val=min_obj_val, + ) From 0fc477bd913baba5aa3533271b6e01e004af1d9b Mon Sep 17 00:00:00 2001 From: "Steven G. Johnson" Date: Fri, 9 Aug 2024 15:13:59 -0400 Subject: [PATCH 09/13] Update doc/docs/Python_Tutorials/Adjoint_Solver.md --- doc/docs/Python_Tutorials/Adjoint_Solver.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/doc/docs/Python_Tutorials/Adjoint_Solver.md b/doc/docs/Python_Tutorials/Adjoint_Solver.md index 8cb41c69a..f8aa1d877 100644 --- a/doc/docs/Python_Tutorials/Adjoint_Solver.md +++ b/doc/docs/Python_Tutorials/Adjoint_Solver.md @@ -1193,7 +1193,7 @@ if __name__ == "__main__": Shape Optimization of a Multilayer Stack ---------------------------------------- -We extend the demonstration of the shape derivative from the previous tutorial to perform shape optimization of a multilayer stack over a broad bandwidth. The 1D design problem is shown in the schematic below and involves finding the layer thicknesses for a fixed number of layers (9) which minimize the largest transmittance (or alternatively, maximize the smallest reflectance) at two wavelengths: $\lambda_1$ = 0.95 μm and $\lambda_2$ = 1.05 μm. However, rather than use the transmittance as the objective function, we use the *field decay* via (the logarithm of) the integral of the fields in the stack ($\int |E_x|^2$). This provides more information to the optimizer which should produce better results. +We extend the demonstration of the shape derivative from the previous tutorial to perform shape optimization of a multilayer stack over a broad bandwidth. The 1D design problem is shown in the schematic below and involves finding the layer thicknesses for a fixed number of layers (9) which minimize the integrated field intensity $\int |E_x|^2$, which demonstrates a `FourierFields` objective function. (This is equivalent to minimizing absorption if $\varepsilon$ had a small imaginary part, and is related but not precisely equivalent to minimizing transmission or maximizing reflection.) In particular, we minimize the worst case (largest) of the intensities at two wavelengths: $\lambda_1$ = 0.95 μm and $\lambda_2$ = 1.05 μm, to roughly emulate a broadband problem. The stack consists of two materials of alternating refractive index $n_A$ = 1.3 and $n_B$ = 1.0. The layers are arranged as $n_A$, $n_B$, $n_A$, $n_B$, ..., $n_A$. The semi-infinite regions to the left and right of the stack are vacuum. From 00823ffda6e791a98a33f9de52b1f80f27d1f683 Mon Sep 17 00:00:00 2001 From: "Steven G. Johnson" Date: Fri, 9 Aug 2024 15:15:02 -0400 Subject: [PATCH 10/13] Update doc/docs/Python_Tutorials/Adjoint_Solver.md --- doc/docs/Python_Tutorials/Adjoint_Solver.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/doc/docs/Python_Tutorials/Adjoint_Solver.md b/doc/docs/Python_Tutorials/Adjoint_Solver.md index f8aa1d877..9537b392d 100644 --- a/doc/docs/Python_Tutorials/Adjoint_Solver.md +++ b/doc/docs/Python_Tutorials/Adjoint_Solver.md @@ -1199,7 +1199,7 @@ The stack consists of two materials of alternating refractive index $n_A$ = 1.3 ![](../images/multilayer_stack_shape_opt.png#center) -A reference design to compare our results against is a [quarter-wavelength stack](https://en.wikipedia.org/wiki/Distributed_Bragg_reflector). The mean frequency of $\lambda_1$ and $\lambda_2$ is $\lambda \approx$ 1.0 μm for which the quarter-wavelength layer thicknesses are approximately $\lambda / (4 n_A)$ = 0.19 μm and $\lambda / (4 n_B)$ = 0.25 μm. These values are used to specify upper and lower bounds for the layer thicknesses. This is important given the non-convex nature of this particular design problem. +A reference design to compare our results against is a [quarter-wavelength stack](https://en.wikipedia.org/wiki/Distributed_Bragg_reflector). The mean wavelength of $\lambda_1$ and $\lambda_2$ is $\lambda = 1.0 μm for which the quarter-wavelength layer thicknesses are $\lambda / (4 n_A)$ = 0.19 μm and $\lambda / (4 n_B)$ = 0.25 μm. These values are used to specify upper and lower bounds for the layer thicknesses. This is important given the non-convex nature of this particular design problem. (For several layers and a single wavelength, the optimizer can easily find a non-quarter-wavelength structure with very low transmission, and once it gets to that point the gradients will be almost zero so probably it will converge very slowly towards a quarter-wavelength solution because the optimizer is generally very slow to try to reduce transmittance from 1e-3 to 1e-6, for example.) From 73f3a0a35b749ac837a4c89be0cf7cbffb3545ad Mon Sep 17 00:00:00 2001 From: "Steven G. Johnson" Date: Fri, 9 Aug 2024 15:16:10 -0400 Subject: [PATCH 11/13] Update doc/docs/Python_Tutorials/Adjoint_Solver.md --- doc/docs/Python_Tutorials/Adjoint_Solver.md | 1 - 1 file changed, 1 deletion(-) diff --git a/doc/docs/Python_Tutorials/Adjoint_Solver.md b/doc/docs/Python_Tutorials/Adjoint_Solver.md index 9537b392d..f18e6d323 100644 --- a/doc/docs/Python_Tutorials/Adjoint_Solver.md +++ b/doc/docs/Python_Tutorials/Adjoint_Solver.md @@ -1201,7 +1201,6 @@ The stack consists of two materials of alternating refractive index $n_A$ = 1.3 A reference design to compare our results against is a [quarter-wavelength stack](https://en.wikipedia.org/wiki/Distributed_Bragg_reflector). The mean wavelength of $\lambda_1$ and $\lambda_2$ is $\lambda = 1.0 μm for which the quarter-wavelength layer thicknesses are $\lambda / (4 n_A)$ = 0.19 μm and $\lambda / (4 n_B)$ = 0.25 μm. These values are used to specify upper and lower bounds for the layer thicknesses. This is important given the non-convex nature of this particular design problem. -(For several layers and a single wavelength, the optimizer can easily find a non-quarter-wavelength structure with very low transmission, and once it gets to that point the gradients will be almost zero so probably it will converge very slowly towards a quarter-wavelength solution because the optimizer is generally very slow to try to reduce transmittance from 1e-3 to 1e-6, for example.) The worst-case optimization is implemented using the [epigraph formulation](https://nlopt.readthedocs.io/en/latest/NLopt_Introduction/#equivalent-formulations-of-optimization-problems) and the Conservative Convex Separable Approximation (CCSA) algorithm. The optimization is run ten times with random initial designs from which the local optima with the smallest objective value is chosen. For the run involving the best design, a plot of the objective function vs. iteration number for the two wavelengths is shown below. Also included in this plot is the epigraph variable. These results demonstrate that the stack is being optimized for $\lambda$ = 0.95 μm and *not* $\lambda$ = 1.05 μm. We observed this trend for various local optima. The optimizer converges to the local optima using 13 iterations. The nine layer thicknesses (μm) of this design are: 0.1822, 0.2369, 0.1822, 0.2369, 0.1822, 0.2369, 0.1910, 0.2504, 0.1931. Since these are 1D simulations, the runtime for each design iteration is generally fast (about ten seconds using one core of an Intel Xeon i7-7700K CPU @ 4.20GHz). From 06a3dc3e9cdec36b09cc2e369d7a8fb37fed952d Mon Sep 17 00:00:00 2001 From: "Steven G. Johnson" Date: Fri, 9 Aug 2024 15:16:24 -0400 Subject: [PATCH 12/13] Update doc/docs/Python_Tutorials/Adjoint_Solver.md --- doc/docs/Python_Tutorials/Adjoint_Solver.md | 1 - 1 file changed, 1 deletion(-) diff --git a/doc/docs/Python_Tutorials/Adjoint_Solver.md b/doc/docs/Python_Tutorials/Adjoint_Solver.md index f18e6d323..ccb7bc9cc 100644 --- a/doc/docs/Python_Tutorials/Adjoint_Solver.md +++ b/doc/docs/Python_Tutorials/Adjoint_Solver.md @@ -1201,7 +1201,6 @@ The stack consists of two materials of alternating refractive index $n_A$ = 1.3 A reference design to compare our results against is a [quarter-wavelength stack](https://en.wikipedia.org/wiki/Distributed_Bragg_reflector). The mean wavelength of $\lambda_1$ and $\lambda_2$ is $\lambda = 1.0 μm for which the quarter-wavelength layer thicknesses are $\lambda / (4 n_A)$ = 0.19 μm and $\lambda / (4 n_B)$ = 0.25 μm. These values are used to specify upper and lower bounds for the layer thicknesses. This is important given the non-convex nature of this particular design problem. - The worst-case optimization is implemented using the [epigraph formulation](https://nlopt.readthedocs.io/en/latest/NLopt_Introduction/#equivalent-formulations-of-optimization-problems) and the Conservative Convex Separable Approximation (CCSA) algorithm. The optimization is run ten times with random initial designs from which the local optima with the smallest objective value is chosen. For the run involving the best design, a plot of the objective function vs. iteration number for the two wavelengths is shown below. Also included in this plot is the epigraph variable. These results demonstrate that the stack is being optimized for $\lambda$ = 0.95 μm and *not* $\lambda$ = 1.05 μm. We observed this trend for various local optima. The optimizer converges to the local optima using 13 iterations. The nine layer thicknesses (μm) of this design are: 0.1822, 0.2369, 0.1822, 0.2369, 0.1822, 0.2369, 0.1910, 0.2504, 0.1931. Since these are 1D simulations, the runtime for each design iteration is generally fast (about ten seconds using one core of an Intel Xeon i7-7700K CPU @ 4.20GHz). In some cases (not shown), the optimizer may take some steps that make things *worse* during the early iterations. This is a common phenomenon in many algorithms — initially, the optimizer takes steps that are too large and then has to backtrack. After a few iterations, the algorithm has more information about a "good" step size. From 1df09afeb9dbca18ae63fac0e67fd6165059ba42 Mon Sep 17 00:00:00 2001 From: "Steven G. Johnson" Date: Fri, 9 Aug 2024 15:18:39 -0400 Subject: [PATCH 13/13] Update doc/docs/Python_Tutorials/Adjoint_Solver.md --- doc/docs/Python_Tutorials/Adjoint_Solver.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/doc/docs/Python_Tutorials/Adjoint_Solver.md b/doc/docs/Python_Tutorials/Adjoint_Solver.md index ccb7bc9cc..897e91b48 100644 --- a/doc/docs/Python_Tutorials/Adjoint_Solver.md +++ b/doc/docs/Python_Tutorials/Adjoint_Solver.md @@ -1201,7 +1201,7 @@ The stack consists of two materials of alternating refractive index $n_A$ = 1.3 A reference design to compare our results against is a [quarter-wavelength stack](https://en.wikipedia.org/wiki/Distributed_Bragg_reflector). The mean wavelength of $\lambda_1$ and $\lambda_2$ is $\lambda = 1.0 μm for which the quarter-wavelength layer thicknesses are $\lambda / (4 n_A)$ = 0.19 μm and $\lambda / (4 n_B)$ = 0.25 μm. These values are used to specify upper and lower bounds for the layer thicknesses. This is important given the non-convex nature of this particular design problem. -The worst-case optimization is implemented using the [epigraph formulation](https://nlopt.readthedocs.io/en/latest/NLopt_Introduction/#equivalent-formulations-of-optimization-problems) and the Conservative Convex Separable Approximation (CCSA) algorithm. The optimization is run ten times with random initial designs from which the local optima with the smallest objective value is chosen. For the run involving the best design, a plot of the objective function vs. iteration number for the two wavelengths is shown below. Also included in this plot is the epigraph variable. These results demonstrate that the stack is being optimized for $\lambda$ = 0.95 μm and *not* $\lambda$ = 1.05 μm. We observed this trend for various local optima. The optimizer converges to the local optima using 13 iterations. The nine layer thicknesses (μm) of this design are: 0.1822, 0.2369, 0.1822, 0.2369, 0.1822, 0.2369, 0.1910, 0.2504, 0.1931. Since these are 1D simulations, the runtime for each design iteration is generally fast (about ten seconds using one core of an Intel Xeon i7-7700K CPU @ 4.20GHz). +The worst-case optimization is implemented using the [epigraph formulation](https://nlopt.readthedocs.io/en/latest/NLopt_Introduction/#equivalent-formulations-of-optimization-problems) and the Conservative Convex Separable Approximation (CCSA) algorithm. The optimization is run ten times with random initial designs from which the local optimum with the smallest objective value is chosen. For the run involving the best design, a plot of the objective function vs. iteration number for the two wavelengths is shown below. Also included in this plot is the epigraph variable. These results demonstrate that the performance of the stack is limited by that of the shorter wavelength $\lambda$ = 0.95 μm, so the $\lambda$ = 1.05 μm performance does not directly affect the result. We observed this trend for various local optima. The optimizer converges to the local optimum after 13 iterations. The nine layer thicknesses of this design are (in μm): 0.1822, 0.2369, 0.1822, 0.2369, 0.1822, 0.2369, 0.1910, 0.2504, 0.1931. Since these are 1D simulations, the runtime for each design iteration is generally fast (about ten seconds using one core of an Intel Xeon i7-7700K CPU @ 4.20GHz). In some cases (not shown), the optimizer may take some steps that make things *worse* during the early iterations. This is a common phenomenon in many algorithms — initially, the optimizer takes steps that are too large and then has to backtrack. After a few iterations, the algorithm has more information about a "good" step size.

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