From ceec9d8af0889ce1b82882aac280c0eb656b7cf7 Mon Sep 17 00:00:00 2001 From: Ardavan Oskooi Date: Mon, 26 Feb 2024 23:53:23 -0800 Subject: [PATCH 01/13] tutorial example for computing the adjoint gradient of a level set --- doc/docs/Python_Tutorials/Adjoint_Solver.md | 11 + .../levelset_gradient_backpropagation.png | Bin 0 -> 156632 bytes doc/docs/images/levelset_jacobian_matrix.png | Bin 0 -> 92448 bytes .../images/levelset_weights_and_jacobian.png | Bin 0 -> 69982 bytes .../binary_grating_levelset.py | 311 ++++++++++++++++++ 5 files changed, 322 insertions(+) create mode 100644 doc/docs/images/levelset_gradient_backpropagation.png create mode 100644 doc/docs/images/levelset_jacobian_matrix.png create mode 100644 doc/docs/images/levelset_weights_and_jacobian.png create mode 100644 python/examples/adjoint_optimization/binary_grating_levelset.py diff --git a/doc/docs/Python_Tutorials/Adjoint_Solver.md b/doc/docs/Python_Tutorials/Adjoint_Solver.md index 63441e673..b65d204a7 100644 --- a/doc/docs/Python_Tutorials/Adjoint_Solver.md +++ b/doc/docs/Python_Tutorials/Adjoint_Solver.md @@ -804,6 +804,17 @@ if __name__ == "__main__": ) ``` +Binary Grating as a Level Set +----------------------------- + +This is a demonstration of computing the gradient of a level set using the adjoint solver. + +![](../images/levelset_gradient_backpropagation.png#center) + +![](../images/levelset_jacobian_matrix.png#center) + +![](../images/levelset_weights_and_jacobian.png#center) + Compact Notebook Tutorials of Basic Features -------------------------------------------- diff --git a/doc/docs/images/levelset_gradient_backpropagation.png b/doc/docs/images/levelset_gradient_backpropagation.png new file mode 100644 index 0000000000000000000000000000000000000000..3d6d01c1b1b1a8962e931347266277d44d1a5a23 GIT binary patch literal 156632 zcmeFZWmwbi|29sF2#Sb;G}7H6jf#MzbTgE0=}rTb?q(__9iwX);Q(uU6; zsXroz89X9C{JLkMcK12Ex`5g3dxp=+(gk~u_19S+YdsEY#-6W4#V>M8L1adFBUF*u 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/dev/null +++ b/python/examples/adjoint_optimization/binary_grating_levelset.py @@ -0,0 +1,311 @@ +"""Validates the adjoint gradient of a level set. + +This is a 2D test which uses subpixel smoothing of the MaterialGrid for +computing the gradient of the diffraction efficiency of the first transmitted +order of a 1D grating. The adjoint gradient is validated using the brute-force +finite difference via the directional derivative. The grating structure is +represented as a level set. +""" + +from enum import Enum +from typing import Tuple + +from autograd.extend import primitive, defvjp +from autograd import numpy as anp +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 +from scipy.interpolate import interp1d +import skfmm + + +RESOLUTION_UM = 200 +WAVELENGTH_UM = 0.53 +GRATING_PERIOD_UM = 1.2 +PADDING_UM = 1.5 + +Polarization = Enum("Polarization", "S P") + + +def design_region_to_meshgrid( + design_region_size: mp.Vector3, nx: int, ny: int +) -> Tuple[anp.ndarray, anp.ndarray]: + """Returns the 2D coordinates of the meshgrid for the design region.""" + + xcoord = anp.linspace(-0.5 * design_region_size.x, +0.5 * design_region_size.x, nx) + ycoord = anp.linspace(-0.5 * design_region_size.y, +0.5 * design_region_size.y, ny) + xv, yv = anp.meshgrid(xcoord, ycoord, indexing="ij") + + return xv, yv + + +def signed_distance(weights: anp.ndarray) -> anp.ndarray: + """Maps the 2D weights using a signed-distance function.""" + + # Create signed distance function. + sd = skfmm.distance(weights - 0.5) + + # Linear interpolation of zero-levelset onto 0.5-levelset. + xp = [anp.min(sd.flatten()), 0, anp.max(sd.flatten())] + yp = [0, 0.5001, 1] + + return anp.interp(sd.flatten(), xp, yp) + + +@primitive +def levelset_and_smoothing( + grating_height_um: float, + grating_duty_cycle: float, + design_region_size: mp.Vector3, + nx: int, + ny: int, +) -> anp.ndarray: + """Generates the grating as a levelset with signed-distance smoothing.""" + + xv, yv = design_region_to_meshgrid(design_region_size, nx, ny) + + weights = anp.where( + anp.abs(yv) <= 0.5 * grating_duty_cycle * GRATING_PERIOD_UM, + anp.where(xv <= xv[0][0] + grating_height_um, 1, 0), + 0, + ) + + smoothed_weights = signed_distance(weights) + + return smoothed_weights + + +def levelset_and_smoothing_vjp( + ans: anp.ndarray, + grating_height_um: float, + grating_duty_cycle: float, + design_region_size: mp.Vector3, + nx: int, + ny: int, +) -> anp.ndarray: + """Computes the vector-Jacobian product.""" + + xv, yv = design_region_to_meshgrid(design_region_size, nx, ny) + + # pixel dimensions + delta_x = design_region_size.x / (nx - 1) + + weights = anp.where( + anp.abs(yv) <= 0.5 * grating_duty_cycle * GRATING_PERIOD_UM, + anp.where(xv <= xv[0][0] + grating_height_um + delta_x, 1, 0), + 0, + ) + + jacobian = (signed_distance(weights) - ans) / delta_x + print(f"jacobian:, {anp.linalg.norm(jacobian)}") + + extent = [xv[0][0], xv[0][-1], yv[0][0], yv[0][-1]] + fig, ax = plt.subplots() + im = ax.imshow( + anp.transpose(jacobian.reshape(nx, ny)), + extent=extent, + cmap="hot_r", + interpolation="none", + aspect="equal", + ) + ax.set_title("gradient") + 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("grating_gradient.png", dpi=150, bbox_inches="tight") + + return lambda g: anp.tensordot(g, jacobian, axes=1) + + +def grating_1d( + pol: Polarization, + grating_height_um: float, + design_region_size: mp.Vector3, + nx: int, + ny: int, +) -> mpa.OptimizationProblem: + """Sets up the adjoint optimization of a 1D grating.""" + + frequency = 1 / WAVELENGTH_UM + substrate_um = 2.0 + pml_um = 1.0 + pml_layers = [mp.PML(direction=mp.X, thickness=pml_um)] + + n_glass = 1.5 + glass = mp.Medium(index=n_glass) + + size_x_um = pml_um + substrate_um + grating_height_um + PADDING_UM + pml_um + size_y_um = GRATING_PERIOD_UM + cell_size = mp.Vector3(size_x_um, size_y_um, 0) + + k_point = mp.Vector3() + + if pol.name == "S": + eig_parity = mp.ODD_Z + src_cmpt = mp.Ez + else: + eig_parity = mp.EVEN_Z + src_cmpt = mp.Hz + + src_pt = mp.Vector3(-0.5 * size_x_um + pml_um, 0, 0) + sources = [ + mp.Source( + mp.GaussianSource(frequency, fwidth=0.1 * frequency), + component=src_cmpt, + center=src_pt, + size=mp.Vector3(0, size_y_um, 0), + ) + ] + + matgrid = mp.MaterialGrid( + mp.Vector3(nx, ny), + mp.air, + glass, + weights=anp.ones((nx, ny)), + do_averaging=True, + beta=anp.inf, + ) + + matgrid_region = mpa.DesignRegion( + matgrid, + volume=mp.Volume( + center=mp.Vector3( + (-0.5 * size_x_um + pml_um + substrate_um + 0.5 * design_region_size.x), + 0, + 0, + ), + size=design_region_size, + ), + ) + + geometry = [ + mp.Block( + material=glass, + size=mp.Vector3(pml_um + substrate_um, mp.inf, mp.inf), + center=mp.Vector3(-0.5 * size_x_um + 0.5 * (pml_um + substrate_um), 0, 0), + ), + mp.Block( + material=matgrid, + size=matgrid_region.size, + center=matgrid_region.center, + ), + ] + + sim = mp.Simulation( + resolution=RESOLUTION_UM, + cell_size=cell_size, + boundary_layers=pml_layers, + k_point=k_point, + sources=sources, + geometry=geometry, + ) + + tran_pt = mp.Vector3(0.5 * size_x_um - pml_um, 0, 0) + + order_y = 1 + kdiff = mp.Vector3( + (frequency**2 - (order_y / GRATING_PERIOD_UM) ** 2) ** 0.5, + order_y / GRATING_PERIOD_UM, + 0, + ) + print(f"kdiff = ({kdiff.x:.5f}, {kdiff.y:.5f}, {kdiff.z:.5f})") + + obj_args = [ + mpa.EigenmodeCoefficient( + sim, + mp.Volume( + center=tran_pt, + size=mp.Vector3(0, size_y_um, 0), + ), + mode=1, + kpoint_func=lambda *not_used: kdiff, + eig_parity=eig_parity, + eig_vol=mp.Volume(center=tran_pt, size=mp.Vector3(0, 1 / RESOLUTION_UM, 0)), + ), + ] + + def obj_func(tran_mon): + return anp.abs(tran_mon) ** 2 + + opt = mpa.OptimizationProblem( + simulation=sim, + objective_functions=obj_func, + objective_arguments=obj_args, + design_regions=[matgrid_region], + frequencies=[frequency], + ) + + return opt + + +if __name__ == "__main__": + grating_height_um = 0.52 + grating_duty_cycle = 0.65 + height_perturbation_um = 2.0 / RESOLUTION_UM + + design_region_size = mp.Vector3( + grating_height_um + PADDING_UM, GRATING_PERIOD_UM, 0 + ) + design_region_resolution_um = int(2 * RESOLUTION_UM) + nx = int(design_region_size.x * design_region_resolution_um) + 1 + ny = int(design_region_size.y * design_region_resolution_um) + 1 + + opt = grating_1d( + Polarization.P, + grating_height_um, + design_region_size, + nx, + ny, + ) + + smoothed_design_weights = levelset_and_smoothing( + grating_height_um, + grating_duty_cycle, + design_region_size, + nx, + ny, + ) + + obj_value_unperturbed, grad_unperturbed = opt( + [smoothed_design_weights], + need_gradient=True, + ) + + fig, ax = plt.subplots() + opt.plot2D(init_opt=False, ax=ax) + if mp.am_master(): + fig.savefig("grating_1D_plot2D.png", dpi=150, bbox_inches="tight") + + defvjp(levelset_and_smoothing, levelset_and_smoothing_vjp) + grad_backprop = tensor_jacobian_product(levelset_and_smoothing, 0)( + grating_height_um, + grating_duty_cycle, + design_region_size, + nx, + ny, + grad_unperturbed, + ) + + perturbed_design_weights = levelset_and_smoothing( + grating_height_um + height_perturbation_um, + grating_duty_cycle, + design_region_size, + nx, + ny, + ) + perturbed_design_weights = perturbed_design_weights.flatten() + + obj_value_perturbed, _ = opt([perturbed_design_weights], need_gradient=False) + + adj_directional_deriv = height_perturbation_um * grad_backprop + fnd_directional_deriv = obj_value_perturbed[0] - obj_value_unperturbed[0] + rel_err = abs( + (fnd_directional_deriv - adj_directional_deriv) / fnd_directional_deriv + ) + print( + f"directional-deriv:, {fnd_directional_deriv:.8f} (finite difference), " + f"{adj_directional_deriv:.8f} (adjoint), {rel_err:.6f} (error)" + ) From d1ea7b5deefb96b1b31abae4e38d37a3261766d0 Mon Sep 17 00:00:00 2001 From: Ardavan Oskooi Date: Wed, 28 Feb 2024 23:13:46 -0800 Subject: [PATCH 02/13] update figures and add results for S and P polarization --- doc/docs/Python_Tutorials/Adjoint_Solver.md | 14 ++++++++++++++ .../levelset_gradient_backpropagation.png | Bin 156632 -> 154595 bytes .../images/levelset_weights_and_jacobian.png | Bin 69982 -> 89540 bytes 3 files changed, 14 insertions(+) diff --git a/doc/docs/Python_Tutorials/Adjoint_Solver.md b/doc/docs/Python_Tutorials/Adjoint_Solver.md index b65d204a7..7a22bf7ea 100644 --- a/doc/docs/Python_Tutorials/Adjoint_Solver.md +++ b/doc/docs/Python_Tutorials/Adjoint_Solver.md @@ -809,6 +809,20 @@ Binary Grating as a Level Set This is a demonstration of computing the gradient of a level set using the adjoint solver. +Results show good agreement between the adjoint gradient and the finite difference for the $\mathcal{S}$ polarization but *not* the $\mathcal{P}$ polarization. The accuracy does not improve with increasing resolution. + +The simulation script is in [python/examples/adjoint_optimization/binary_grating_levelset.py](https://github.com/NanoComp/meep/tree/master/python/examples/adjoint_optimization/binary_grating_levelset.py). + +**$\mathcal{S}$ polarization** +``` +directional-deriv:, -0.00394839 (finite difference), -0.00355608 (adjoint), 0.099361 (error) +``` + +**$\mathcal{P}$ polarization** +``` +directional-deriv:, -0.00090782 (finite difference), -0.00382003 (adjoint), 3.207894 (error) +``` + ![](../images/levelset_gradient_backpropagation.png#center) ![](../images/levelset_jacobian_matrix.png#center) diff --git a/doc/docs/images/levelset_gradient_backpropagation.png b/doc/docs/images/levelset_gradient_backpropagation.png index 3d6d01c1b1b1a8962e931347266277d44d1a5a23..134f5299801f55728212af65f5dc7a5e58fc517c 100644 GIT binary patch delta 116655 zcmYhjby!u~_dN`vB8noQbV)ZzN*+|C5jk`t-6-8#L8Mj8a9JCx2t!vPNb 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with downsampling from a high-resolution grid --- .../binary_grating_levelset.py | 345 ++++++++++-------- 1 file changed, 190 insertions(+), 155 deletions(-) diff --git a/python/examples/adjoint_optimization/binary_grating_levelset.py b/python/examples/adjoint_optimization/binary_grating_levelset.py index 0c858b6a9..850c4add1 100644 --- a/python/examples/adjoint_optimization/binary_grating_levelset.py +++ b/python/examples/adjoint_optimization/binary_grating_levelset.py @@ -1,146 +1,198 @@ -"""Validates the adjoint gradient of a level set. +"""Computes the adjoint gradient of a level set. -This is a 2D test which uses subpixel smoothing of the MaterialGrid for -computing the gradient of the diffraction efficiency of the first transmitted -order of a 1D grating. The adjoint gradient is validated using the brute-force -finite difference via the directional derivative. The grating structure is +This is a 2D example for computing the gradient of the diffraction efficiency +of the first transmitted order of a 1D grating with respect to the grating +height. The adjoint gradient is validated using the brute-force finite +difference via the directional derivative. The grating structure is represented as a level set. """ from enum import Enum -from typing import Tuple +from typing import Callable, Tuple from autograd.extend import primitive, defvjp -from autograd import numpy as anp +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 -from scipy.interpolate import interp1d -import skfmm +import numpy as np +from scipy.interpolate import RegularGridInterpolator -RESOLUTION_UM = 200 +RESOLUTION_UM = 100 WAVELENGTH_UM = 0.53 -GRATING_PERIOD_UM = 1.2 -PADDING_UM = 1.5 +GRATING_PERIOD_UM = 1.28 +GRATING_HEIGHT_UM = 0.52 +GRATING_DUTY_CYCLE = 0.65 +GRATING_HEIGHT_PERTURBATION_UM = 2.0 / RESOLUTION_UM +DIFFRACTION_ORDER = 1 +SUBSTRATE_UM = 1.5 +PML_UM = 1.0 +AIR_UM = 1.0 +N_GLASS = 1.5 +DESIGN_REGION_RESOLUTION_UM = 10 * RESOLUTION_UM Polarization = Enum("Polarization", "S P") +design_region_size = mp.Vector3( + GRATING_HEIGHT_UM + GRATING_HEIGHT_PERTURBATION_UM, GRATING_PERIOD_UM, 0 +) +nx_design_grid = int(design_region_size.x * DESIGN_REGION_RESOLUTION_UM) + 1 +ny_design_grid = int(design_region_size.y * DESIGN_REGION_RESOLUTION_UM) + 1 +nx_sim_grid = int(design_region_size.x * RESOLUTION_UM) + 1 +ny_sim_grid = int(design_region_size.y * RESOLUTION_UM) + 1 -def design_region_to_meshgrid( - design_region_size: mp.Vector3, nx: int, ny: int -) -> Tuple[anp.ndarray, anp.ndarray]: - """Returns the 2D coordinates of the meshgrid for the design region.""" +DEBUG_OUTPUT = False - xcoord = anp.linspace(-0.5 * design_region_size.x, +0.5 * design_region_size.x, nx) - ycoord = anp.linspace(-0.5 * design_region_size.y, +0.5 * design_region_size.y, ny) - xv, yv = anp.meshgrid(xcoord, ycoord, indexing="ij") - - return xv, yv +def design_region_to_meshgrid(nx: int, ny: int) -> Tuple[np.ndarray, np.ndarray]: + """Returns the 2D coordinates of the meshgrid for the design region. -def signed_distance(weights: anp.ndarray) -> anp.ndarray: - """Maps the 2D weights using a signed-distance function.""" + Args: + nx, ny: the number of grid points in the x and y directions, respectively. - # Create signed distance function. - sd = skfmm.distance(weights - 0.5) + Returns: + The coordinates of the x and y grid points (2D arrays) as a 2-tuple. + """ - # Linear interpolation of zero-levelset onto 0.5-levelset. - xp = [anp.min(sd.flatten()), 0, anp.max(sd.flatten())] - yp = [0, 0.5001, 1] + xcoord = np.linspace(-0.5 * design_region_size.x, +0.5 * design_region_size.x, nx) + ycoord = np.linspace(-0.5 * design_region_size.y, +0.5 * design_region_size.y, ny) + xv, yv = np.meshgrid(xcoord, ycoord, indexing="ij") - return anp.interp(sd.flatten(), xp, yp) + return xv, yv @primitive -def levelset_and_smoothing( - grating_height_um: float, - grating_duty_cycle: float, - design_region_size: mp.Vector3, - nx: int, - ny: int, -) -> anp.ndarray: - """Generates the grating as a levelset with signed-distance smoothing.""" - - xv, yv = design_region_to_meshgrid(design_region_size, nx, ny) - - weights = anp.where( - anp.abs(yv) <= 0.5 * grating_duty_cycle * GRATING_PERIOD_UM, - anp.where(xv <= xv[0][0] + grating_height_um, 1, 0), +def levelset_and_smoothing(grating_height_um: float) -> np.ndarray: + """Returns the density weights for a binary grating as a levelset. + + Args: + grating_height_um: the height of the grating. + + Returns: + The density weights as a flattened (1D) array. + """ + + xv, yv = design_region_to_meshgrid(nx_design_grid, ny_design_grid) + weights = np.where( + np.abs(yv) <= 0.5 * GRATING_DUTY_CYCLE * GRATING_PERIOD_UM, + np.where(xv <= xv[0][0] + grating_height_um, 1, 0), 0, ) + xcoord = np.linspace( + -0.5 * design_region_size.x, +0.5 * design_region_size.x, nx_design_grid + ) + ycoord = np.linspace( + -0.5 * design_region_size.y, +0.5 * design_region_size.y, ny_design_grid + ) + interp = RegularGridInterpolator((xcoord, ycoord), weights) + + # Smooth the design weights by downsampling from the design grid + # to the simulation grid using bilinear interpolation. + sim_grid_xv, sim_grid_yv = design_region_to_meshgrid(nx_sim_grid, ny_sim_grid) + smoothed_weights = interp((sim_grid_xv, sim_grid_yv)) + + if DEBUG_OUTPUT: + fig, ax = plt.subplots() + im = ax.imshow( + np.transpose(smoothed_weights), + cmap="binary", + interpolation="none", + aspect="equal", + ) + ax.set_title(r"$\rho_{smooth-levelset}(h)$") + 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("smoothed_levelset.png", dpi=150, bbox_inches="tight") - smoothed_weights = signed_distance(weights) - - return smoothed_weights + return smoothed_weights.flatten() def levelset_and_smoothing_vjp( - ans: anp.ndarray, + ans: np.ndarray, grating_height_um: float, - grating_duty_cycle: float, - design_region_size: mp.Vector3, - nx: int, - ny: int, -) -> anp.ndarray: - """Computes the vector-Jacobian product.""" +) -> Callable[[np.ndarray], np.ndarray]: + """Returns a function for computing the vector-Jacobian product. + + The Jacobian is computed manually using a finite difference. - xv, yv = design_region_to_meshgrid(design_region_size, nx, ny) + Args: + ans: the design weights for the smoothed grating (no perturbation). + grating_height_um: the height of the grating. - # pixel dimensions - delta_x = design_region_size.x / (nx - 1) + Returns: + An anonymous function for computing the vector-Jacobian product. + """ - weights = anp.where( - anp.abs(yv) <= 0.5 * grating_duty_cycle * GRATING_PERIOD_UM, - anp.where(xv <= xv[0][0] + grating_height_um + delta_x, 1, 0), + xv, yv = design_region_to_meshgrid(nx_design_grid, ny_design_grid) + weights = np.where( + np.abs(yv) <= 0.5 * GRATING_DUTY_CYCLE * GRATING_PERIOD_UM, + np.where( + xv <= xv[0][0] + grating_height_um + GRATING_HEIGHT_PERTURBATION_UM, 1, 0 + ), 0, ) - - jacobian = (signed_distance(weights) - ans) / delta_x - print(f"jacobian:, {anp.linalg.norm(jacobian)}") - - extent = [xv[0][0], xv[0][-1], yv[0][0], yv[0][-1]] - fig, ax = plt.subplots() - im = ax.imshow( - anp.transpose(jacobian.reshape(nx, ny)), - extent=extent, - cmap="hot_r", - interpolation="none", - aspect="equal", + xcoord = np.linspace( + -0.5 * design_region_size.x, +0.5 * design_region_size.x, nx_design_grid + ) + ycoord = np.linspace( + -0.5 * design_region_size.y, +0.5 * design_region_size.y, ny_design_grid ) - ax.set_title("gradient") - 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("grating_gradient.png", dpi=150, bbox_inches="tight") + interp = RegularGridInterpolator((xcoord, ycoord), weights) + + # Smooth the design weights by downsampling from the design grid + # to the simulation grid using bilinear interpolation. + xv_sim_grid, yv_sim_grid = design_region_to_meshgrid(nx_sim_grid, ny_sim_grid) + smoothed_weights = interp((xv_sim_grid, yv_sim_grid)) + smoothed_weights = smoothed_weights.flatten() + + jacobian = (smoothed_weights - ans) / GRATING_HEIGHT_PERTURBATION_UM + + if DEBUG_OUTPUT: + fig, ax = plt.subplots() + im = ax.imshow( + np.transpose(jacobian.reshape(nx_sim_grid, ny_sim_grid)), + cmap="inferno", + interpolation="none", + aspect="equal", + ) + ax.set_title(r"$\partial \rho_{smooth-levelset} / \partial h$") + 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("gradient_smooth_levelset.png", dpi=150, bbox_inches="tight") - return lambda g: anp.tensordot(g, jacobian, axes=1) + return lambda g: np.tensordot(g, jacobian, axes=1) -def grating_1d( - pol: Polarization, - grating_height_um: float, - design_region_size: mp.Vector3, - nx: int, - ny: int, -) -> mpa.OptimizationProblem: - """Sets up the adjoint optimization of a 1D grating.""" +def grating_1d(pol: Polarization) -> mpa.OptimizationProblem: + """Sets up the adjoint optimization of a 1D grating. - frequency = 1 / WAVELENGTH_UM - substrate_um = 2.0 - pml_um = 1.0 - pml_layers = [mp.PML(direction=mp.X, thickness=pml_um)] + Args: + pol: the polarization state (S or P). - n_glass = 1.5 - glass = mp.Medium(index=n_glass) + Returns: + A meep.adjoint.OptimizationProblem object for the simulation. + """ - size_x_um = pml_um + substrate_um + grating_height_um + PADDING_UM + pml_um + frequency = 1 / WAVELENGTH_UM + pml_layers = [mp.PML(direction=mp.X, thickness=PML_UM)] + glass = mp.Medium(index=N_GLASS) + size_x_um = ( + PML_UM + + SUBSTRATE_UM + + GRATING_HEIGHT_UM + + GRATING_HEIGHT_PERTURBATION_UM + + AIR_UM + + PML_UM + ) size_y_um = GRATING_PERIOD_UM cell_size = mp.Vector3(size_x_um, size_y_um, 0) - k_point = mp.Vector3() if pol.name == "S": @@ -150,7 +202,7 @@ def grating_1d( eig_parity = mp.EVEN_Z src_cmpt = mp.Hz - src_pt = mp.Vector3(-0.5 * size_x_um + pml_um, 0, 0) + src_pt = mp.Vector3(-0.5 * size_x_um + PML_UM, 0, 0) sources = [ mp.Source( mp.GaussianSource(frequency, fwidth=0.1 * frequency), @@ -161,19 +213,18 @@ def grating_1d( ] matgrid = mp.MaterialGrid( - mp.Vector3(nx, ny), + mp.Vector3(nx_sim_grid, ny_sim_grid), mp.air, glass, - weights=anp.ones((nx, ny)), - do_averaging=True, - beta=anp.inf, + weights=np.ones((nx_sim_grid, ny_sim_grid)), + do_averaging=False, ) matgrid_region = mpa.DesignRegion( matgrid, volume=mp.Volume( center=mp.Vector3( - (-0.5 * size_x_um + pml_um + substrate_um + 0.5 * design_region_size.x), + (-0.5 * size_x_um + PML_UM + SUBSTRATE_UM + 0.5 * design_region_size.x), 0, 0, ), @@ -184,8 +235,8 @@ def grating_1d( geometry = [ mp.Block( material=glass, - size=mp.Vector3(pml_um + substrate_um, mp.inf, mp.inf), - center=mp.Vector3(-0.5 * size_x_um + 0.5 * (pml_um + substrate_um), 0, 0), + size=mp.Vector3(PML_UM + SUBSTRATE_UM, mp.inf, mp.inf), + center=mp.Vector3(-0.5 * size_x_um + 0.5 * (PML_UM + SUBSTRATE_UM), 0, 0), ), mp.Block( material=matgrid, @@ -203,15 +254,13 @@ def grating_1d( geometry=geometry, ) - tran_pt = mp.Vector3(0.5 * size_x_um - pml_um, 0, 0) + tran_pt = mp.Vector3(0.5 * size_x_um - PML_UM, 0, 0) - order_y = 1 kdiff = mp.Vector3( - (frequency**2 - (order_y / GRATING_PERIOD_UM) ** 2) ** 0.5, - order_y / GRATING_PERIOD_UM, + (frequency**2 - (DIFFRACTION_ORDER / GRATING_PERIOD_UM) ** 2) ** 0.5, + DIFFRACTION_ORDER / GRATING_PERIOD_UM, 0, ) - print(f"kdiff = ({kdiff.x:.5f}, {kdiff.y:.5f}, {kdiff.z:.5f})") obj_args = [ mpa.EigenmodeCoefficient( @@ -227,8 +276,8 @@ def grating_1d( ), ] - def obj_func(tran_mon): - return anp.abs(tran_mon) ** 2 + def obj_func(mode_coeff): + return npa.abs(mode_coeff) ** 2 opt = mpa.OptimizationProblem( simulation=sim, @@ -242,70 +291,56 @@ def obj_func(tran_mon): if __name__ == "__main__": - grating_height_um = 0.52 - grating_duty_cycle = 0.65 - height_perturbation_um = 2.0 / RESOLUTION_UM - - design_region_size = mp.Vector3( - grating_height_um + PADDING_UM, GRATING_PERIOD_UM, 0 - ) - design_region_resolution_um = int(2 * RESOLUTION_UM) - nx = int(design_region_size.x * design_region_resolution_um) + 1 - ny = int(design_region_size.y * design_region_resolution_um) + 1 - - opt = grating_1d( - Polarization.P, - grating_height_um, - design_region_size, - nx, - ny, - ) + # Only the S polarization is supported (for now). + opt = grating_1d(Polarization.S) - smoothed_design_weights = levelset_and_smoothing( - grating_height_um, - grating_duty_cycle, - design_region_size, - nx, - ny, - ) + smoothed_design_weights = levelset_and_smoothing(GRATING_HEIGHT_UM) - obj_value_unperturbed, grad_unperturbed = opt( - [smoothed_design_weights], - need_gradient=True, + obj_val_unperturbed, grad_unperturbed = opt( + [smoothed_design_weights], need_gradient=True ) - fig, ax = plt.subplots() - opt.plot2D(init_opt=False, ax=ax) - if mp.am_master(): - fig.savefig("grating_1D_plot2D.png", dpi=150, bbox_inches="tight") + if DEBUG_OUTPUT: + fig, ax = plt.subplots() + im = ax.imshow( + np.transpose(grad_unperturbed.reshape(nx_sim_grid, ny_sim_grid)), + cmap="inferno", + interpolation="none", + aspect="equal", + ) + ax.set_title(r"$\partial F / \partial \rho_{smooth-levelset}$") + 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( + "gradient_wrt_smoothed_design_weights.png", dpi=150, bbox_inches="tight" + ) + + fig, ax = plt.subplots() + opt.plot2D(init_opt=False, ax=ax) + if mp.am_master(): + fig.savefig("sim_layout.png", dpi=150, bbox_inches="tight") defvjp(levelset_and_smoothing, levelset_and_smoothing_vjp) grad_backprop = tensor_jacobian_product(levelset_and_smoothing, 0)( - grating_height_um, - grating_duty_cycle, - design_region_size, - nx, - ny, + GRATING_HEIGHT_UM, grad_unperturbed, ) perturbed_design_weights = levelset_and_smoothing( - grating_height_um + height_perturbation_um, - grating_duty_cycle, - design_region_size, - nx, - ny, + GRATING_HEIGHT_UM + GRATING_HEIGHT_PERTURBATION_UM ) perturbed_design_weights = perturbed_design_weights.flatten() - obj_value_perturbed, _ = opt([perturbed_design_weights], need_gradient=False) + obj_val_perturbed, _ = opt([perturbed_design_weights], need_gradient=False) - adj_directional_deriv = height_perturbation_um * grad_backprop - fnd_directional_deriv = obj_value_perturbed[0] - obj_value_unperturbed[0] + adj_directional_deriv = GRATING_HEIGHT_PERTURBATION_UM * grad_backprop + fnd_directional_deriv = obj_val_perturbed[0] - obj_val_unperturbed[0] rel_err = abs( (fnd_directional_deriv - adj_directional_deriv) / fnd_directional_deriv ) print( - f"directional-deriv:, {fnd_directional_deriv:.8f} (finite difference), " + f"dir-deriv:, {fnd_directional_deriv:.8f} (finite difference), " f"{adj_directional_deriv:.8f} (adjoint), {rel_err:.6f} (error)" ) From b25480d8805c559f84a140bd519c2b2df8760ea5 Mon Sep 17 00:00:00 2001 From: Ardavan Oskooi Date: Wed, 19 Jun 2024 16:55:35 -0700 Subject: [PATCH 04/13] update figures for backpropagation calculation and Jacobian matrix --- doc/docs/Python_Tutorials/Adjoint_Solver.md | 14 -------------- .../levelset_gradient_backpropagation.png | Bin 154595 -> 105703 bytes doc/docs/images/levelset_jacobian_matrix.png | Bin 92448 -> 70298 bytes .../images/levelset_weights_and_jacobian.png | Bin 89540 -> 0 bytes 4 files changed, 14 deletions(-) delete mode 100644 doc/docs/images/levelset_weights_and_jacobian.png diff --git a/doc/docs/Python_Tutorials/Adjoint_Solver.md b/doc/docs/Python_Tutorials/Adjoint_Solver.md index 7a22bf7ea..af315b660 100644 --- a/doc/docs/Python_Tutorials/Adjoint_Solver.md +++ b/doc/docs/Python_Tutorials/Adjoint_Solver.md @@ -809,26 +809,12 @@ Binary Grating as a Level Set This is a demonstration of computing the gradient of a level set using the adjoint solver. -Results show good agreement between the adjoint gradient and the finite difference for the $\mathcal{S}$ polarization but *not* the $\mathcal{P}$ polarization. The accuracy does not improve with increasing resolution. - The simulation script is in [python/examples/adjoint_optimization/binary_grating_levelset.py](https://github.com/NanoComp/meep/tree/master/python/examples/adjoint_optimization/binary_grating_levelset.py). -**$\mathcal{S}$ polarization** -``` -directional-deriv:, -0.00394839 (finite difference), -0.00355608 (adjoint), 0.099361 (error) -``` - -**$\mathcal{P}$ polarization** -``` -directional-deriv:, -0.00090782 (finite difference), -0.00382003 (adjoint), 3.207894 (error) -``` - ![](../images/levelset_gradient_backpropagation.png#center) ![](../images/levelset_jacobian_matrix.png#center) -![](../images/levelset_weights_and_jacobian.png#center) - Compact Notebook Tutorials of Basic Features -------------------------------------------- diff --git a/doc/docs/images/levelset_gradient_backpropagation.png b/doc/docs/images/levelset_gradient_backpropagation.png index 134f5299801f55728212af65f5dc7a5e58fc517c..a731ee3f5c3ccc61c394f26dbf196f391582a502 100644 GIT binary patch literal 105703 zcmdSA_d8r))IO|42uVmnlmwB4DAA%tFCmED88s2niRhgmqDSvWA8m**x*?6|-C(qY z(S{jq1~bZge7?`~2fWw&{pEFC%yG`lo^#e-d+oLE`(81++Nw9GS*S@!NN%XBDeIGv zklT`wkabaA1C@a*WRm{c>*u#m%pUK^~ol{NoFqIn*N=XRK)H)8PmPQer7FrCO}=@%Eschk zyZ8YWw|-r*iQzA_iptZ^7Bm|=Dx=Pd^z@2`%E^fjijL@FuiU<^6d;%@+afW#&*Sy~9QMqI z(Oh1?JiT_+;=lJ^{*hSz@xRBdq>TS(3?|#b8!bZrd+gR7^*>`Xd6EnI+J^}n|9)c| zo_8N*hub2R4K!YWUl3^i3>aA5Wuf-MT6q&|wcO@OwDa|?7pocR?r*o93K^T{NmD-N zhw(RWgNhFho-HRxgb}fnv9WoX0{_jdr=%ORZ4GY_kzdG@%jTJp79m%h z)Sz5<`D&7r1!TNMSXZ~OqvrJs#2U)@E)!LwOC@}MWW?M5b&dEnT9YpP)})846)-Wk z0XOh$YOoJiWuG#kb4F%>Msma{f`YjJ^nydl+I>i0mf)L~LTNj2cXv+@wEKM0k8vix z>xDioXaU0O_||FXx-nYY_}Li1;UrV(x(_!+m#sBYx#-n*FzxA?)}7R+2jK1u{<+PL z(`WCOf)#Sb@u*DUwHEc%FbKFQxP5IXzkmU7KQWriJ1J)m`O211Z6;!~B)9bgQtoK5 zybi9kIqJHotTWLk>o|VsZQ_;Tj0m$1A-#R4vP^*c^S+D|{b(D^)e*H?5xskaFUNQ{ zJ!1*C?cI0|DI;ZO77Y=2p0(zPo@Zu~nSnRL-r7$DS=tZlk|f4o?~hBO^=@^UihcN? z$(4rbI;xq0>`#BnMYH&ku3N3oAv^%rk1Z)xmL6*J=F!h9tx#+KQTT=c>hGU?!a(Uz z%V}*L?1);^AK0xl@Cp)nwt?Gy|KQ&5JK2{DkvtrW)RfR6i&U%5{-jk|nmyfiD1luetJm4xnm0bV)OS#Sd{7k}X})=_8__Lh&MBKkKxSaj|AqX_&R zI)wvWaThgD$!}h;@HzuUf;U-B+8JBeBNsf-!_;UU9MecA7%)TmyEE9mmCQr zRT@R)?ekg)~`q3Mh`fX~dnA#&_`CmxrK64ybdY#GJ-uW)aLB7(U381-d1TZz zVecxsC)$5DOUt*A##kLi#gm{AA{0*`9%WjGBE@%b&2Cen@$gf9Y3R>rShFDCoqt5^ zrp4i5&2=GXlSdOp5x5<2u(5PqS=kmc@qQ)T zvfXDtpt4d)GnY#_Uup|2;mZ39VT}yE?>0~OX6zwHjszmqveAk2m&gcJa)AKEc4zJ@ z>Iql*2RZu@W9y}*rKY2xlW%mM>*KP@Iw4yK$cVJC-i6$U4~-K+dn*15@)xB?HL&yD zzlRIonHtQ+*iJ@FZ+N1pzxvkUr6wTc3+-r~iHSRYSlJ!#{(f$o&yN;E!Jzx+$1ehF zkmqL#V7cygtW=Y{`CEJPJ6yWe6FFjrf$mz4Jl=cTh@^))PO$lk*~S>q6^IVyP4;Q_ zcurZ?<9&@P_1aoY%M0n{-W)N)1cBg;vs-9)Zf0VYzUsBD1|EY46S$N{21^fMMtsXV z&A9bc$=(cp1&H?Ko4TpC037e`=!@>I3s-Z3on3da%A~Z-*u%EEE$_|S`FEOJ3|Q0B zNDkKY_$DaXM`e|hCJmRnF3OxwP?HRT_LhlKj&oqv0;%d`Kyo(4cv^bQ=y9z<#KbnBg zad8cBc1{(Q?uHS4Lp@ln_h4%XA*MEN((nw~al*AbBte+&^iSobw(>8-Yktnpp}x+d zd}QFPm^+!4HaeeL@^aQ>?`_{(vb90d4E_o0mVqIPo&-;fs6hPG!~Y)RCflUd;!}b~K_3 zS`g4ru(#AzAWgt;*9(S<>FM2ikL{f9Wk$nX?^${#HhzIxT&n@u5q&l{yMILCv6=Ot z_{)epFxxOUz{uH>qvk5^ zL1eZG3Yfo4<^lV^l;-Zw7U`}rnpYtl5742otc}@N}IX?4<3-e~vA7WSDgD zmY-}GKHj4@EGwpAtRUtXH;;U}QR)5RH?@;+)-#pB+Qy?L-;V0qM{g3H#=%0)S*7Q|O$DwK<1R@)+}Cbi&3IZAZN)?V zI!I>s2hHS=&d|vLz2g3N${l1~4z>$X%a0k4V#9zMxi1LSbz}yRar~=kXo*s;{$Ftk z6YC+&3tBxE%l3@W=h_-2Ih*Nxd`Vy`>f+jrlo?YNbkSP6N#moklVj5n2%E?|UB2`b z`SU_I)#T*M<_$eNw&mS6kB#98-3cb}>>}O!iMxlB7=7lu<@u`SdGb2`WBW;=qp;B?=;>Tv zs_0IPab4uaIiugUaIA}?pu0QoQ|&w|Q72^v+ks5yq$m8`g_6&(t-a@G&y4T2N`IuX zZH{mAhLvbTO<#x3WoJ)6ay~n4*lz-j(UZo?w77%!o48WxQn?)6<=+G4?H-El>_>L)eZmt(nIez3MQ~>z!j1%oxQnhe{B;B#?t`6fW{-i6 z(Q}rSDTz9j!WY`HRoDah3%w<6{w-^ZJRie;VcU&71JCU&JRTfcdyzH!10<>v7*Y^y z>~9U+O1oEm0ir6locrt)ZQYMNCsgBoGRBzrSj060k-P7M^(-_rHs$UyLg2$IN!8e1 zB$6P<`%NO}-=idluJxi9QQSOwi{kRupV?#^n?uk2QLa!GgDa7{IEH^o@@dVXhk84% z{3XKAdb}#Rrd;Bu-g9zJ_dWs)Wwg*X-Ojz4&}}rfJxlMc8}nTORs{0DIMC^(OfYYI zAKpe^xLk*^*BTSV|&$`x_Mj+CM zQ3YmPKa>k=eA1&;Oi(^D%czAw*n^jcpQV1WiXQEd}^G!S1@t-hHauHGGZ}+;N?f zv+v2`6#&xy>jeeV?R1jQD%xt|*}`BmNbeyK=_8rEtJBpyd<|?JzwRQN90j|Mtk8}Q z2_t;G#bl*rxf0W~Ez?Ci0%)i2Q^R~w(Cy^H7hCk%wN2*?xNI<_y#MexJ|wGgMn zMwwZW0*(h^pnEkjF=IEsykEO66x6aqH2Lr^62BkK_U&BP$yH{!(g5Wgq640Gt7m>b z+Pb1rm3o(d!?c3oBcg`XDzx*|83cV}+pAA%)z_E0g)A#Ir8Rs6g>do3^)wmY*YK-; 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This is useful for applications involving [shape optimization](https://en.wikipedia.org/wiki/Shape_optimization) (or geometry parameterization). + +As a demonstration, we will compute the gradient of the diffraction efficiency of the first transmitted order of a 1D binary grating with respect to the grating height. The accuracy of the adjoint gradient is validated using a brute-force finite difference via the directional derivative. + +The calculation of the diffraction efficiency involves [mode decomposition](Python_User_Interface/#mode-decomposition) of planewaves which is described in [Tutorial/Transmissive Diffraction Spectrum for Planewave at Normal Incidence](Mode_Decomposition.md#transmissive-diffraction-spectrum-for-planewave-at-normal-incidence). An important aspect of specifying the `meep.adjoint.EigenmodeCoefficient` object for diffraction orders (planewaves) in 2D is to directly specify the wavevector using the `kpoint_func` parameter. Additionally, it is necessary to specify the `eig_vol` parameter to have a *length of one pixel in the periodic direction* in order for MPB to correctly interpret this wavevector in the Cartesian basis rather than the reciprocal-lattice basis defined by the grating period. + +``` +RESOLUTION_UM = 50 +dir-deriv:, -0.02535355 (finite difference), -0.01469133 (adjoint), 0.420542 (error) + +RESOLUTION_UM = 100 +dir-deriv:, -0.00604817 (finite difference), -0.00473221 (adjoint), 0.217580 (error) + +RESOLUTION_UM = 200 +dir-deriv:, -0.00284452 (finite difference), -0.00252470 (adjoint), 0.112432 (error) + +RESOLUTION_UM = 400 +dir-deriv:, -0.00140221 (finite difference), -0.00132065 (adjoint), 0.058165 (error) + +RESOLUTION_UM = 800 +dir-deriv:, -0.00069606 (finite difference), -0.00067547 (adjoint), 0.029583 (error) +``` The simulation script is in [python/examples/adjoint_optimization/binary_grating_levelset.py](https://github.com/NanoComp/meep/tree/master/python/examples/adjoint_optimization/binary_grating_levelset.py). @@ -815,6 +836,10 @@ The simulation script is in [python/examples/adjoint_optimization/binary_grating ![](../images/levelset_jacobian_matrix.png#center) +![](../images/levelset_adjoint_gradient.png#center) + +![](../images/levelset_adjoint_gradient_error_vs_resolution.png#center) + Compact Notebook Tutorials of Basic Features -------------------------------------------- diff --git a/doc/docs/images/levelset_adjoint_gradient.png b/doc/docs/images/levelset_adjoint_gradient.png new file mode 100644 index 0000000000000000000000000000000000000000..d1066bf3fe38f85e3a1a721d60e3956a0012cd8b GIT binary patch literal 135218 zcmY&=1z40_*EJwWinO3Z$Iu}PLrY0_x0Hy~&>)S3gdimig0x6WOAOtDBHhy64gZ8v;aSV3$;1iJ|MNwQ@u&>{gyFyc5GTOb+IlO3`k!Zr69h?8 z5UTv|Gal0{hS6XJ{{51;DC;0803N=E-D@wX_l3`El|Kb34}>J#!aeN*aPH!ED$)%hG|6~DhZUK{?5IgrXP 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b/python/examples/adjoint_optimization/binary_grating_levelset.py index 850c4add1..3af695b2e 100644 --- a/python/examples/adjoint_optimization/binary_grating_levelset.py +++ b/python/examples/adjoint_optimization/binary_grating_levelset.py @@ -26,7 +26,7 @@ GRATING_PERIOD_UM = 1.28 GRATING_HEIGHT_UM = 0.52 GRATING_DUTY_CYCLE = 0.65 -GRATING_HEIGHT_PERTURBATION_UM = 2.0 / RESOLUTION_UM +GRATING_HEIGHT_PERTURBATION_UM = 1.0 / RESOLUTION_UM DIFFRACTION_ORDER = 1 SUBSTRATE_UM = 1.5 PML_UM = 1.0 From 01235c320ffd6f428852c0f1de2ecd8838b7f189 Mon Sep 17 00:00:00 2001 From: Ardavan Oskooi Date: Thu, 20 Jun 2024 06:50:47 -0700 Subject: [PATCH 06/13] more details on defining planewave using EigenmodeCoefficients object --- doc/docs/Python_Tutorials/Adjoint_Solver.md | 2 +- doc/docs/images/levelset_jacobian_matrix.png | Bin 80200 -> 80040 bytes 2 files changed, 1 insertion(+), 1 deletion(-) diff --git a/doc/docs/Python_Tutorials/Adjoint_Solver.md b/doc/docs/Python_Tutorials/Adjoint_Solver.md index 8106c1987..f96bfec6b 100644 --- a/doc/docs/Python_Tutorials/Adjoint_Solver.md +++ b/doc/docs/Python_Tutorials/Adjoint_Solver.md @@ -811,7 +811,7 @@ It is also possible to compute the gradient of a [level set](https://en.wikipedi As a demonstration, we will compute the gradient of the diffraction efficiency of the first transmitted order of a 1D binary grating with respect to the grating height. The accuracy of the adjoint gradient is validated using a brute-force finite difference via the directional derivative. -The calculation of the diffraction efficiency involves [mode decomposition](Python_User_Interface/#mode-decomposition) of planewaves which is described in [Tutorial/Transmissive Diffraction Spectrum for Planewave at Normal Incidence](Mode_Decomposition.md#transmissive-diffraction-spectrum-for-planewave-at-normal-incidence). An important aspect of specifying the `meep.adjoint.EigenmodeCoefficient` object for diffraction orders (planewaves) in 2D is to directly specify the wavevector using the `kpoint_func` parameter. Additionally, it is necessary to specify the `eig_vol` parameter to have a *length of one pixel in the periodic direction* in order for MPB to correctly interpret this wavevector in the Cartesian basis rather than the reciprocal-lattice basis defined by the grating period. +The calculation of the diffraction efficiency involves [mode decomposition](Python_User_Interface/#mode-decomposition) of planewaves which is described in [Tutorial/Transmissive Diffraction Spectrum for Planewave at Normal Incidence](Mode_Decomposition.md#transmissive-diffraction-spectrum-for-planewave-at-normal-incidence). Specifying a diffraction order (planewave) in 2D using the `meep.adjoint.EigenmodeCoefficient` object involves three components: (1) wavevector using the `kpoint_func` parameter, (2) polarization using the `eig_parity` parameter, and (3) `eig_vol` parameter to have a *length of one pixel in the periodic direction*. The latter is necessary for MPB to correctly interpret the wavevector in the Cartesian basis rather than the reciprocal-lattice basis defined by the grating period. ``` RESOLUTION_UM = 50 diff --git a/doc/docs/images/levelset_jacobian_matrix.png b/doc/docs/images/levelset_jacobian_matrix.png index ee54a9ffd0430a1586d0377688d0d9825e1a4a77..c52e15180f369ff5d302ae76e61aa3ed2dac3779 100644 GIT binary patch literal 80040 zcmce;Wmwf)*Eebx2#RznT~g919n#(1-5m-7(k;>g0@BhA0@5wgB`w`uXRQ0Vp7*^! zoUi9xmwUUn;9BcH=a^&sY7!(XE&B8c?vq=$Zaozj6OzAm>z?GTTX#AhJ%B%n%^pjF z-yYkFsX5%bg&K+Y=eCD(0~P!sj-#-uqk@gGql=!s(JdDj7dkU*3kL%|TO&Ffdy}MX zZroe9UfvQH;#YJ{-kfpq)EsmBdo_3Hi*q>Z@#+!g4>oc1ktDJcG_GS#qm;gLsnQLv z6Dih-KJVxPm*EtdyZ!RN)BB+R z`i2$^p`Swk_sjR@pAhR3|JMuJ-ha;U{Qvyzd;HbDkG}lZmtSh~-;?~WFFz}KsS^_O zQQ(y&>y&!3#DBl=XH!c{&HbBEQr-P-n&RDD887qK(q#YjTDHUDq-ej_4x?|H$Zx-R zUmKtvu6=r@o+ACMJ~FF^cF`gK5n9Uu1#Lnk80hNV#WV#)O zQ;;fcIIl{edZza1MsyeYkXWoLvq-pfh(w_Pan$ru{?*7|`)V8@tS1a}{w@cpuhiR%V<;_Za_cdLp1=e1wm(3*pFcLi+hG_}cG-A~y$ 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zoj4#*i5UiE+dUYQ%<9JVX@>5~#9UGogTFke)U_*3=Jbz<_4yCfTRwI@v<j_ zEGYtChKJ9*ab>R{w~~q3bE;H&B;Xc^|F5!WzQ@7ccSZekYrlh$xv6znf%!HLcYV5SNl~14RoL7vxJ1N_%-)% zu8J-2zg7#@gBnHd_wyYm0;#IE7lm=bFv+o7kWb1%4yI0FQR?&qRA(pGRd}x9Y3V-P zAwN3fXxHiBy!nQ9_%gLiF1nANaRoC!+jYxgE?F&Q<OfG?4xc`rU~a;{k3QFnpjF;mL%A`y$>3cCMnPV zHrrkm*}0R>%ONS!c6B9bI9UFY0F1bNIPA=~^C=gN|M#75*w8;PL~ZzhB59(05V!YV zQwDUUwtuiot6}< zW#w%7CR6_zzQ_-TZ2Ld|`v1Gc4NU*F>fjeUFEgCW->gYPd`|zq)-7vf%eUWJx$GAw l7a6tcpAY{3*^mAsXyVLjclMWeANn>aS{et`bNAX_{9nl)q7(oC From 027458b10619d5b3737c4b0dcd6c610cd0d3f138 Mon Sep 17 00:00:00 2001 From: Ardavan Oskooi Date: Fri, 21 Jun 2024 06:48:59 -0700 Subject: [PATCH 07/13] add text describing simulation setup and results --- doc/docs/Python_Tutorials/Adjoint_Solver.md | 366 +++++++++++++++++++- 1 file changed, 358 insertions(+), 8 deletions(-) diff --git a/doc/docs/Python_Tutorials/Adjoint_Solver.md b/doc/docs/Python_Tutorials/Adjoint_Solver.md index f96bfec6b..7272d61a2 100644 --- a/doc/docs/Python_Tutorials/Adjoint_Solver.md +++ b/doc/docs/Python_Tutorials/Adjoint_Solver.md @@ -807,11 +807,25 @@ if __name__ == "__main__": Adjoint Gradient of a Level Set ------------------------------- -It is also possible to compute the gradient of a [level set](https://en.wikipedia.org/wiki/Level_set) (i.e., a discontinuous function) using the density-based adjoint solver. This is useful for applications involving [shape optimization](https://en.wikipedia.org/wiki/Shape_optimization) (or geometry parameterization). +It is also possible to compute the gradient of a [level set](https://en.wikipedia.org/wiki/Level_set) (i.e., a discontinuous function) using the density-based adjoint solver. This is useful for applications involving [shape optimization](https://en.wikipedia.org/wiki/Shape_optimization) or geometry parameterization. -As a demonstration, we will compute the gradient of the diffraction efficiency of the first transmitted order of a 1D binary grating with respect to the grating height. The accuracy of the adjoint gradient is validated using a brute-force finite difference via the directional derivative. +As a demonstration, we will compute the gradient of the diffraction efficiency of the first transmitted order with $\mathcal{S}$ polarization (electric field out of the plane, i.e. $E_z$) of a 1D binary grating with respect to the grating height. The accuracy of the adjoint gradient is validated using a brute-force finite difference via the directional derivative. -The calculation of the diffraction efficiency involves [mode decomposition](Python_User_Interface/#mode-decomposition) of planewaves which is described in [Tutorial/Transmissive Diffraction Spectrum for Planewave at Normal Incidence](Mode_Decomposition.md#transmissive-diffraction-spectrum-for-planewave-at-normal-incidence). Specifying a diffraction order (planewave) in 2D using the `meep.adjoint.EigenmodeCoefficient` object involves three components: (1) wavevector using the `kpoint_func` parameter, (2) polarization using the `eig_parity` parameter, and (3) `eig_vol` parameter to have a *length of one pixel in the periodic direction*. The latter is necessary for MPB to correctly interpret the wavevector in the Cartesian basis rather than the reciprocal-lattice basis defined by the grating period. +The calculation of the diffraction efficiency involves [mode decomposition](Python_User_Interface/#mode-decomposition) of planewaves which is described in [Tutorial/Transmissive Diffraction Spectrum for Planewave at Normal Incidence](Mode_Decomposition.md#transmissive-diffraction-spectrum-for-planewave-at-normal-incidence). An important aspect of setting up this simulation is specifying a diffraction order (planewave) in 2D using the `meep.adjoint.EigenmodeCoefficient` object. This involves three components: (1) the wavevector via the `kpoint_func` parameter, (2) the polarization via the `eig_parity` parameter, and (3) the `eig_vol` parameter must be set to have a *length of one pixel in the periodic direction*. The latter is necessary for MPB to correctly interpret the wavevector in the Cartesian basis rather than the reciprocal-lattice basis defined by the grating period. + +The calculation of the gradient of the diffraction efficiency ($F$) with respect to the grating height ($h$) involves using the chain rule to obtain a product of two derivatives: (1) $\partial F / \partial \rho$ and (2) $\partial \rho / \partial h$, where $\rho$ are the density weights (2D grid of $N_x \times N_y = N$ points) used to represent the binary grating as a levelset. (1) is the adjoint gradient computed by Meep. (2) requires the levelset to be *smoothed* to ensure it is differentiable. A non-smoothed levelset is generally not differentiable. (2) is implemented in Autograd using a custom vector-Jacobian product (vJP) which is used as part of reverse-mode differentiation (backpropagation) to compute $\partial F / \partial h$. An overview of this calculation is shown below. + +![](../images/levelset_gradient_backpropagation.png#center) + +The derivative $\partial \rho / \partial h$ can be approximated by a finite difference using a one-pixel perturbation applied to the grating height. The calculation is summarized in the schematic below. There are two function evaluations of $\rho(h)$ for which the cost is negligible. Smoothing of the levelset can be performed using a number of different methods including a [signed-distance function](https://en.wikipedia.org/wiki/Signed_distance_function) or convolution filter. In this example, the smoothing is based on downsampling the levelset from a high-resolution grid (10X the resolution of the simulation grid) to the lower-resolution simulation grid using bilinear interpolation. Note that only the boundary pixels are nonzero in the Jacobian matrix in this case. + +![](../images/levelset_jacobian_matrix.png#center) + +A schematic of the simulation layout showing the design region containing the grating as a levelset is shown below. The adjoint gradient $\partial F / \partial \rho$ computed by Meep at a resolution of 400 pixels/$\mu$m is shown next to this schematic. + +![](../images/levelset_adjoint_gradient.png#center) + +The simulation script is in [python/examples/adjoint_optimization/binary_grating_levelset.py](https://github.com/NanoComp/meep/tree/master/python/examples/adjoint_optimization/binary_grating_levelset.py). Running this script at five different resolutions produces the output below for the directional derivative computed using the finite difference and the adjoint gradient as well as the relative error from these two results. ``` RESOLUTION_UM = 50 @@ -830,15 +844,351 @@ RESOLUTION_UM = 800 dir-deriv:, -0.00069606 (finite difference), -0.00067547 (adjoint), 0.029583 (error) ``` -The simulation script is in [python/examples/adjoint_optimization/binary_grating_levelset.py](https://github.com/NanoComp/meep/tree/master/python/examples/adjoint_optimization/binary_grating_levelset.py). +A logarithmic plot of the relative error vs. grid resolution based on these results demonstrates linear convergence. This is expected for fields right on the boundary of a discontinuous interface. -![](../images/levelset_gradient_backpropagation.png#center) +Note: because of a likely bug in the adjoint gradients for the $\mathcal{P}$ polarization (electric field in the $xy$ plane), there is a large error in the adjoint gradient which [does not decrease with resolution](https://github.com/NanoComp/meep/pull/2792#issuecomment-2171942477). This is why this demonstration only involved the $\mathcal{S}$ polarization. -![](../images/levelset_jacobian_matrix.png#center) +![](../images/levelset_adjoint_gradient_error_vs_resolution.png#center) -![](../images/levelset_adjoint_gradient.png#center) +```py +from enum import Enum +from typing import Callable, Tuple -![](../images/levelset_adjoint_gradient_error_vs_resolution.png#center) +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 +from scipy.interpolate import RegularGridInterpolator + + +RESOLUTION_UM = 100 +WAVELENGTH_UM = 0.53 +GRATING_PERIOD_UM = 1.28 +GRATING_HEIGHT_UM = 0.52 +GRATING_DUTY_CYCLE = 0.65 +GRATING_HEIGHT_PERTURBATION_UM = 1.0 / RESOLUTION_UM +DIFFRACTION_ORDER = 1 +SUBSTRATE_UM = 1.5 +PML_UM = 1.0 +AIR_UM = 1.0 +N_GLASS = 1.5 +DESIGN_REGION_RESOLUTION_UM = 10 * RESOLUTION_UM + +Polarization = Enum("Polarization", "S P") + +design_region_size = mp.Vector3( + GRATING_HEIGHT_UM + GRATING_HEIGHT_PERTURBATION_UM, GRATING_PERIOD_UM, 0 +) +nx_design_grid = int(design_region_size.x * DESIGN_REGION_RESOLUTION_UM) + 1 +ny_design_grid = int(design_region_size.y * DESIGN_REGION_RESOLUTION_UM) + 1 +nx_sim_grid = int(design_region_size.x * RESOLUTION_UM) + 1 +ny_sim_grid = int(design_region_size.y * RESOLUTION_UM) + 1 + +DEBUG_OUTPUT = False + + +def design_region_to_meshgrid(nx: int, ny: int) -> Tuple[np.ndarray, np.ndarray]: + """Returns the 2D coordinates of the meshgrid for the design region. + + Args: + nx, ny: the number of grid points in the x and y directions, respectively. + + Returns: + The coordinates of the x and y grid points (2D arrays) as a 2-tuple. + """ + + xcoord = np.linspace(-0.5 * design_region_size.x, +0.5 * design_region_size.x, nx) + ycoord = np.linspace(-0.5 * design_region_size.y, +0.5 * design_region_size.y, ny) + xv, yv = np.meshgrid(xcoord, ycoord, indexing="ij") + + return xv, yv + + +@primitive +def levelset_and_smoothing(grating_height_um: float) -> np.ndarray: + """Returns the density weights for a binary grating as a levelset. + + Args: + grating_height_um: the height of the grating. + + Returns: + The density weights as a flattened (1D) array. + """ + + xv, yv = design_region_to_meshgrid(nx_design_grid, ny_design_grid) + weights = np.where( + np.abs(yv) <= 0.5 * GRATING_DUTY_CYCLE * GRATING_PERIOD_UM, + np.where(xv <= xv[0][0] + grating_height_um, 1, 0), + 0, + ) + xcoord = np.linspace( + -0.5 * design_region_size.x, +0.5 * design_region_size.x, nx_design_grid + ) + ycoord = np.linspace( + -0.5 * design_region_size.y, +0.5 * design_region_size.y, ny_design_grid + ) + interp = RegularGridInterpolator((xcoord, ycoord), weights) + + # Smooth the design weights by downsampling from the design grid + # to the simulation grid using bilinear interpolation. + sim_grid_xv, sim_grid_yv = design_region_to_meshgrid(nx_sim_grid, ny_sim_grid) + smoothed_weights = interp((sim_grid_xv, sim_grid_yv)) + + if DEBUG_OUTPUT: + fig, ax = plt.subplots() + im = ax.imshow( + np.transpose(smoothed_weights), + cmap="binary", + interpolation="none", + aspect="equal", + ) + ax.set_title(r"$\rho_{smooth-levelset}(h)$") + 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("smoothed_levelset.png", dpi=150, bbox_inches="tight") + + return smoothed_weights.flatten() + + +def levelset_and_smoothing_vjp( + ans: np.ndarray, + grating_height_um: float, +) -> Callable[[np.ndarray], np.ndarray]: + """Returns a function for computing the vector-Jacobian product. + + The Jacobian is computed manually using a finite difference. + + Args: + ans: the design weights for the smoothed grating (no perturbation). + grating_height_um: the height of the grating. + + Returns: + An anonymous function for computing the vector-Jacobian product. + """ + + xv, yv = design_region_to_meshgrid(nx_design_grid, ny_design_grid) + weights = np.where( + np.abs(yv) <= 0.5 * GRATING_DUTY_CYCLE * GRATING_PERIOD_UM, + np.where( + xv <= xv[0][0] + grating_height_um + GRATING_HEIGHT_PERTURBATION_UM, 1, 0 + ), + 0, + ) + xcoord = np.linspace( + -0.5 * design_region_size.x, +0.5 * design_region_size.x, nx_design_grid + ) + ycoord = np.linspace( + -0.5 * design_region_size.y, +0.5 * design_region_size.y, ny_design_grid + ) + interp = RegularGridInterpolator((xcoord, ycoord), weights) + + # Smooth the design weights by downsampling from the design grid + # to the simulation grid using bilinear interpolation. + xv_sim_grid, yv_sim_grid = design_region_to_meshgrid(nx_sim_grid, ny_sim_grid) + smoothed_weights = interp((xv_sim_grid, yv_sim_grid)) + smoothed_weights = smoothed_weights.flatten() + + jacobian = (smoothed_weights - ans) / GRATING_HEIGHT_PERTURBATION_UM + + if DEBUG_OUTPUT: + fig, ax = plt.subplots() + im = ax.imshow( + np.transpose(jacobian.reshape(nx_sim_grid, ny_sim_grid)), + cmap="inferno", + interpolation="none", + aspect="equal", + ) + ax.set_title(r"$\partial \rho_{smooth-levelset} / \partial h$") + 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("gradient_smooth_levelset.png", dpi=150, bbox_inches="tight") + + return lambda g: np.tensordot(g, jacobian, axes=1) + + +def grating_1d(pol: Polarization) -> mpa.OptimizationProblem: + """Sets up the adjoint optimization of a 1D grating. + + Args: + pol: the polarization state (S or P). + + Returns: + A meep.adjoint.OptimizationProblem object for the simulation. + """ + + frequency = 1 / WAVELENGTH_UM + pml_layers = [mp.PML(direction=mp.X, thickness=PML_UM)] + glass = mp.Medium(index=N_GLASS) + size_x_um = ( + PML_UM + + SUBSTRATE_UM + + GRATING_HEIGHT_UM + + GRATING_HEIGHT_PERTURBATION_UM + + AIR_UM + + PML_UM + ) + size_y_um = GRATING_PERIOD_UM + cell_size = mp.Vector3(size_x_um, size_y_um, 0) + k_point = mp.Vector3() + + if pol.name == "S": + eig_parity = mp.ODD_Z + src_cmpt = mp.Ez + else: + eig_parity = mp.EVEN_Z + src_cmpt = mp.Hz + + src_pt = mp.Vector3(-0.5 * size_x_um + PML_UM, 0, 0) + sources = [ + mp.Source( + mp.GaussianSource(frequency, fwidth=0.1 * frequency), + component=src_cmpt, + center=src_pt, + size=mp.Vector3(0, size_y_um, 0), + ) + ] + + matgrid = mp.MaterialGrid( + mp.Vector3(nx_sim_grid, ny_sim_grid), + mp.air, + glass, + weights=np.ones((nx_sim_grid, ny_sim_grid)), + do_averaging=False, + ) + + matgrid_region = mpa.DesignRegion( + matgrid, + volume=mp.Volume( + center=mp.Vector3( + (-0.5 * size_x_um + PML_UM + SUBSTRATE_UM + 0.5 * design_region_size.x), + 0, + 0, + ), + size=design_region_size, + ), + ) + + geometry = [ + mp.Block( + material=glass, + size=mp.Vector3(PML_UM + SUBSTRATE_UM, mp.inf, mp.inf), + center=mp.Vector3(-0.5 * size_x_um + 0.5 * (PML_UM + SUBSTRATE_UM), 0, 0), + ), + mp.Block( + material=matgrid, + size=matgrid_region.size, + center=matgrid_region.center, + ), + ] + + sim = mp.Simulation( + resolution=RESOLUTION_UM, + cell_size=cell_size, + boundary_layers=pml_layers, + k_point=k_point, + sources=sources, + geometry=geometry, + ) + + tran_pt = mp.Vector3(0.5 * size_x_um - PML_UM, 0, 0) + + kdiff = mp.Vector3( + (frequency**2 - (DIFFRACTION_ORDER / GRATING_PERIOD_UM) ** 2) ** 0.5, + DIFFRACTION_ORDER / GRATING_PERIOD_UM, + 0, + ) + + obj_args = [ + mpa.EigenmodeCoefficient( + sim, + mp.Volume( + center=tran_pt, + size=mp.Vector3(0, size_y_um, 0), + ), + mode=1, + kpoint_func=lambda *not_used: kdiff, + eig_parity=eig_parity, + eig_vol=mp.Volume(center=tran_pt, size=mp.Vector3(0, 1 / RESOLUTION_UM, 0)), + ), + ] + + def obj_func(mode_coeff): + return npa.abs(mode_coeff) ** 2 + + opt = mpa.OptimizationProblem( + simulation=sim, + objective_functions=obj_func, + objective_arguments=obj_args, + design_regions=[matgrid_region], + frequencies=[frequency], + ) + + return opt + + +if __name__ == "__main__": + # Only the S polarization is supported (for now). + opt = grating_1d(Polarization.S) + + smoothed_design_weights = levelset_and_smoothing(GRATING_HEIGHT_UM) + + obj_val_unperturbed, grad_unperturbed = opt( + [smoothed_design_weights], need_gradient=True + ) + + if DEBUG_OUTPUT: + fig, ax = plt.subplots() + im = ax.imshow( + np.transpose(grad_unperturbed.reshape(nx_sim_grid, ny_sim_grid)), + cmap="inferno", + interpolation="none", + aspect="equal", + ) + ax.set_title(r"$\partial F / \partial \rho_{smooth-levelset}$") + 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( + "gradient_wrt_smoothed_design_weights.png", dpi=150, bbox_inches="tight" + ) + + fig, ax = plt.subplots() + opt.plot2D(init_opt=False, ax=ax) + if mp.am_master(): + fig.savefig("sim_layout.png", dpi=150, bbox_inches="tight") + + defvjp(levelset_and_smoothing, levelset_and_smoothing_vjp) + grad_backprop = tensor_jacobian_product(levelset_and_smoothing, 0)( + GRATING_HEIGHT_UM, + grad_unperturbed, + ) + + perturbed_design_weights = levelset_and_smoothing( + GRATING_HEIGHT_UM + GRATING_HEIGHT_PERTURBATION_UM + ) + perturbed_design_weights = perturbed_design_weights.flatten() + + obj_val_perturbed, _ = opt([perturbed_design_weights], need_gradient=False) + + adj_directional_deriv = GRATING_HEIGHT_PERTURBATION_UM * grad_backprop + fnd_directional_deriv = obj_val_perturbed[0] - obj_val_unperturbed[0] + 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)" + ) +``` Compact Notebook Tutorials of Basic Features -------------------------------------------- From 32691cbc43f42711fd9b87ffe83a0e06ecce76ef Mon Sep 17 00:00:00 2001 From: "Steven G. Johnson" Date: Fri, 28 Jun 2024 15:08:33 -0400 Subject: [PATCH 08/13] 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 7272d61a2..b87bd715b 100644 --- a/doc/docs/Python_Tutorials/Adjoint_Solver.md +++ b/doc/docs/Python_Tutorials/Adjoint_Solver.md @@ -807,7 +807,7 @@ if __name__ == "__main__": Adjoint Gradient of a Level Set ------------------------------- -It is also possible to compute the gradient of a [level set](https://en.wikipedia.org/wiki/Level_set) (i.e., a discontinuous function) using the density-based adjoint solver. This is useful for applications involving [shape optimization](https://en.wikipedia.org/wiki/Shape_optimization) or geometry parameterization. +It is also possible to compute the derivative of the Meep outputs with respect to a geometric parameter via a [level-set](https://en.wikipedia.org/wiki/Level_set) formulation (an implicit-function representation of a material discontinuity) using the density-based adjoint solver. This is useful for applications involving [shape optimization](https://en.wikipedia.org/wiki/Shape_optimization) of explicitly parameterized geometries. As a demonstration, we will compute the gradient of the diffraction efficiency of the first transmitted order with $\mathcal{S}$ polarization (electric field out of the plane, i.e. $E_z$) of a 1D binary grating with respect to the grating height. The accuracy of the adjoint gradient is validated using a brute-force finite difference via the directional derivative. From 1d332e11ba4c1b07464c1232c65a3385b76800b5 Mon Sep 17 00:00:00 2001 From: "Steven G. Johnson" Date: Fri, 28 Jun 2024 15:09:35 -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 b87bd715b..ee200853f 100644 --- a/doc/docs/Python_Tutorials/Adjoint_Solver.md +++ b/doc/docs/Python_Tutorials/Adjoint_Solver.md @@ -809,7 +809,7 @@ Adjoint Gradient of a Level Set It is also possible to compute the derivative of the Meep outputs with respect to a geometric parameter via a [level-set](https://en.wikipedia.org/wiki/Level_set) formulation (an implicit-function representation of a material discontinuity) using the density-based adjoint solver. This is useful for applications involving [shape optimization](https://en.wikipedia.org/wiki/Shape_optimization) of explicitly parameterized geometries. -As a demonstration, we will compute the gradient of the diffraction efficiency of the first transmitted order with $\mathcal{S}$ polarization (electric field out of the plane, i.e. $E_z$) of a 1D binary grating with respect to the grating height. The accuracy of the adjoint gradient is validated using a brute-force finite difference via the directional derivative. +As a demonstration, we will compute the derivative of the diffraction efficiency of the first transmitted order with $\mathcal{S}$ polarization (electric field out of the plane, i.e. $E_z$) of a 1D binary grating with respect to the grating height. The accuracy of the adjoint derivative is validated using a brute-force finite-difference approximation. The calculation of the diffraction efficiency involves [mode decomposition](Python_User_Interface/#mode-decomposition) of planewaves which is described in [Tutorial/Transmissive Diffraction Spectrum for Planewave at Normal Incidence](Mode_Decomposition.md#transmissive-diffraction-spectrum-for-planewave-at-normal-incidence). An important aspect of setting up this simulation is specifying a diffraction order (planewave) in 2D using the `meep.adjoint.EigenmodeCoefficient` object. This involves three components: (1) the wavevector via the `kpoint_func` parameter, (2) the polarization via the `eig_parity` parameter, and (3) the `eig_vol` parameter must be set to have a *length of one pixel in the periodic direction*. The latter is necessary for MPB to correctly interpret the wavevector in the Cartesian basis rather than the reciprocal-lattice basis defined by the grating period. From 20d042b4686f3b2b08f1ee4bf70a38137e8aa1f8 Mon Sep 17 00:00:00 2001 From: "Steven G. Johnson" Date: Fri, 28 Jun 2024 15:12:39 -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 ee200853f..d73881769 100644 --- a/doc/docs/Python_Tutorials/Adjoint_Solver.md +++ b/doc/docs/Python_Tutorials/Adjoint_Solver.md @@ -813,7 +813,7 @@ As a demonstration, we will compute the derivative of the diffraction efficiency The calculation of the diffraction efficiency involves [mode decomposition](Python_User_Interface/#mode-decomposition) of planewaves which is described in [Tutorial/Transmissive Diffraction Spectrum for Planewave at Normal Incidence](Mode_Decomposition.md#transmissive-diffraction-spectrum-for-planewave-at-normal-incidence). An important aspect of setting up this simulation is specifying a diffraction order (planewave) in 2D using the `meep.adjoint.EigenmodeCoefficient` object. This involves three components: (1) the wavevector via the `kpoint_func` parameter, (2) the polarization via the `eig_parity` parameter, and (3) the `eig_vol` parameter must be set to have a *length of one pixel in the periodic direction*. The latter is necessary for MPB to correctly interpret the wavevector in the Cartesian basis rather than the reciprocal-lattice basis defined by the grating period. -The calculation of the gradient of the diffraction efficiency ($F$) with respect to the grating height ($h$) involves using the chain rule to obtain a product of two derivatives: (1) $\partial F / \partial \rho$ and (2) $\partial \rho / \partial h$, where $\rho$ are the density weights (2D grid of $N_x \times N_y = N$ points) used to represent the binary grating as a levelset. (1) is the adjoint gradient computed by Meep. (2) requires the levelset to be *smoothed* to ensure it is differentiable. A non-smoothed levelset is generally not differentiable. (2) is implemented in Autograd using a custom vector-Jacobian product (vJP) which is used as part of reverse-mode differentiation (backpropagation) to compute $\partial F / \partial h$. An overview of this calculation is shown below. +The calculation of the derivative of the diffraction efficiency ($F$) with respect to the grating height ($h$) involves using the chain rule to obtain a product of two derivatives (Jacobians): (1) $\partial F / \partial \rho$ and (2) $\partial \rho / \partial h$, where $\rho$ are the density weights (2D grid of $N_x \times N_y = N$ points) used to represent the binary grating as a level set. (1) is the adjoint gradient computed by Meep. (2) requires the level set to be *smoothed* to ensure it that the derivative $\partial \rho / \partial h$ exists. (2) is implemented in Autograd using a custom vector–Jacobian product (vJP) which is used as part of reverse-mode differentiation (backpropagation) to compute $\partial F / \partial h$. An overview of this calculation is shown below. ![](../images/levelset_gradient_backpropagation.png#center) From 7d57be8a0eefd86fec9d5e11298029061b7fb02e Mon Sep 17 00:00:00 2001 From: "Steven G. Johnson" Date: Fri, 28 Jun 2024 15:21:35 -0400 Subject: [PATCH 11/13] 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 d73881769..bf46217fd 100644 --- a/doc/docs/Python_Tutorials/Adjoint_Solver.md +++ b/doc/docs/Python_Tutorials/Adjoint_Solver.md @@ -846,7 +846,7 @@ dir-deriv:, -0.00069606 (finite difference), -0.00067547 (adjoint), 0.029583 (er A logarithmic plot of the relative error vs. grid resolution based on these results demonstrates linear convergence. This is expected for fields right on the boundary of a discontinuous interface. -Note: because of a likely bug in the adjoint gradients for the $\mathcal{P}$ polarization (electric field in the $xy$ plane), there is a large error in the adjoint gradient which [does not decrease with resolution](https://github.com/NanoComp/meep/pull/2792#issuecomment-2171942477). This is why this demonstration only involved the $\mathcal{S}$ polarization. +Currently, we recommend using this procedure only for the $\mathcal{S}$ ($E_z$) polarization, since for the for the $\mathcal{P}$ polarization (electric field in the $xy$ plane), there appear to be large discretization errors in the adjoint gradient which [we are currently investigating](https://github.com/NanoComp/meep/pull/2792#issuecomment-2171942477). This is why this demonstration only involved the $\mathcal{S}$ polarization. ![](../images/levelset_adjoint_gradient_error_vs_resolution.png#center) From 89d66df3db0defa9d885fa174616fcf877e9b053 Mon Sep 17 00:00:00 2001 From: "Steven G. Johnson" Date: Fri, 28 Jun 2024 15:21:51 -0400 Subject: [PATCH 12/13] 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 bf46217fd..cd998a1a7 100644 --- a/doc/docs/Python_Tutorials/Adjoint_Solver.md +++ b/doc/docs/Python_Tutorials/Adjoint_Solver.md @@ -817,7 +817,7 @@ The calculation of the derivative of the diffraction efficiency ($F$) with respe ![](../images/levelset_gradient_backpropagation.png#center) -The derivative $\partial \rho / \partial h$ can be approximated by a finite difference using a one-pixel perturbation applied to the grating height. The calculation is summarized in the schematic below. There are two function evaluations of $\rho(h)$ for which the cost is negligible. Smoothing of the levelset can be performed using a number of different methods including a [signed-distance function](https://en.wikipedia.org/wiki/Signed_distance_function) or convolution filter. In this example, the smoothing is based on downsampling the levelset from a high-resolution grid (10X the resolution of the simulation grid) to the lower-resolution simulation grid using bilinear interpolation. Note that only the boundary pixels are nonzero in the Jacobian matrix in this case. +The derivative $\partial \rho / \partial h$ can be approximated by a finite difference using a one-pixel perturbation applied to the grating height; this is computationally convenient because it greatly simplifies the construction of $\rho(h)$ as explained below. (A finite difference involves two function evaluations of $\rho(h)$, but the cost for this is negligible since it involves no Meep simulations.) The construction of $\rho(h)$ involves two steps: first, constructing a simple binary image $b(h)$ at a high resolution; and second, smoothing $b(h)$ into a continuous level-set function $\rho(h)$. This smoothing of the image can be performed using a number of different methods including a [signed-distance function](https://en.wikipedia.org/wiki/Signed_distance_function) or convolution filter. In this example, the smoothing is based simply on downsampling the image from a high-resolution grid (10X the resolution of the simulation grid) to the lower-resolution simulation grid using bilinear interpolation, which leads to "gray" pixels at the boundaries between materials in a way that changes continuously with $h$. Only these boundary pixels have nonzero derivatives in the Jacobian $\partial\rho/\partial h$ in this case. This calculation is summarized in the schematic below. ![](../images/levelset_jacobian_matrix.png#center) From 5675f24c52c94fb289f4b7836355d78d7908ed9d Mon Sep 17 00:00:00 2001 From: "Steven G. Johnson" Date: Fri, 28 Jun 2024 15:23:54 -0400 Subject: [PATCH 13/13] Update doc/docs/Python_Tutorials/Adjoint_Solver.md --- doc/docs/Python_Tutorials/Adjoint_Solver.md | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/doc/docs/Python_Tutorials/Adjoint_Solver.md b/doc/docs/Python_Tutorials/Adjoint_Solver.md index cd998a1a7..7e246115d 100644 --- a/doc/docs/Python_Tutorials/Adjoint_Solver.md +++ b/doc/docs/Python_Tutorials/Adjoint_Solver.md @@ -804,8 +804,8 @@ if __name__ == "__main__": ) ``` -Adjoint Gradient of a Level Set -------------------------------- +Derivatives with Respect to Shape Parameters +-------------------------------------------- It is also possible to compute the derivative of the Meep outputs with respect to a geometric parameter via a [level-set](https://en.wikipedia.org/wiki/Level_set) formulation (an implicit-function representation of a material discontinuity) using the density-based adjoint solver. This is useful for applications involving [shape optimization](https://en.wikipedia.org/wiki/Shape_optimization) of explicitly parameterized geometries.

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