From dae3d64b74dfe43c552e229c7b221bb853be2f3b Mon Sep 17 00:00:00 2001
From: Edoardo Vergerio <edoardo240500@gmail.com>
Date: Thu, 4 Jun 2026 14:16:47 +0200
Subject: [PATCH] added metaprop_ad.py for testing automatic differentiation
 and conic filtering

---
 metaprop_ad.py | 432 +++++++++++++++++++++++++++++++++++++++++++++++++
 1 file changed, 432 insertions(+)
 create mode 100644 metaprop_ad.py

diff --git a/metaprop_ad.py b/metaprop_ad.py
new file mode 100644
index 0000000..fc5845b
--- /dev/null
+++ b/metaprop_ad.py
@@ -0,0 +1,432 @@
+#from jax import numpy as npa, grad
+#from jax.numpy.fft import fft2, ifft2, fftshift, ifftshift
+from typing import List, Tuple, Union
+from autograd import numpy as npa, tensor_jacobian_product, grad as gd
+import meep.adjoint as mpa
+import numpy as np
+import matplotlib.pyplot as plt
+import xarray as xr
+from PIL import Image
+import nlopt
+ArrayLikeType = Union[List, Tuple, np.ndarray]
+
+wavelength = 1.55
+k0 = 2 * np.pi / wavelength
+
+size_x = 300 * wavelength
+size_y = 300 * wavelength
+res_x = 3 #number of pixels per unit-length
+res_y = res_x 
+
+Nx = int(res_x * size_x) + 1
+Ny = int(res_y * size_y) + 1
+S = Nx * Ny
+
+norm_phase_min = 0
+norm_phase_max = 1
+w_norm = np.ones((S)) * 0.5 #normalized phase parameters in [0,1]
+phase_min = 0
+phase_max = 2 * np.pi
+
+xs = np.linspace(-size_x/2, size_x/2, Nx)
+ys = np.linspace(-size_y/2, size_y/2, Ny)
+sampling_period = xs[1] - xs[0]
+
+z_prop = 300 #propagation distance
+
+input_field = np.ones((S))
+
+opt_max_eval = 300
+
+def phase_given_w(w):
+    '''Returns the phase shift given the design parameters w.
+    Args:
+        w: design parameters (flattened)
+    Returns:
+        The phase shift corresponding to the geometrical parameters w (flattened)'''
+    #for simplicity, we linearly map w (0,1) to a phase shift (0, 2pi)
+    return 2 * npa.pi * (w)
+
+def get_pattern() -> np.ndarray:
+    '''Imports the intensity pattern from file'''
+    pattern_name = 'unipd.png'
+    pattern = Image.open('misc/' + pattern_name).convert('L')
+    pattern = np.array(pattern).astype(float)
+
+    pattern = pattern - np.min(pattern)
+    pattern = pattern / np.max(pattern)
+
+    return pattern
+
+def intensity_desired_fn_pattern(
+    xs: list,
+    ys: list,
+    rescale: float = 0.5
+) -> np.ndarray:
+    '''Returns the pattern intensity value at the specified (xs,ys) locations, with a scale factor.
+    Args: 
+        xs: list of x coordinates
+        ys: list of y coordinates
+        rescale: scale factor for the pattern
+    Returns:
+        The flattened pattern intensity value at the specified (xs,ys) locations, with a scale factor
+    '''
+    pattern_values = get_pattern()
+    pattern_values = np.rot90(np.rot90(np.rot90(pattern_values)))
+
+    nx,ny = pattern_values.shape
+    xs_pattern = np.linspace(rescale*min(xs), rescale*max(xs), nx)
+    ys_pattern = np.linspace(rescale*min(ys), rescale*max(ys), ny)
+    pattern_dataArray = xr.DataArray(pattern_values, coords=dict(x=xs_pattern, y=ys_pattern))
+    pattern_interp = pattern_dataArray.interp(x=xs, y=ys)
+
+    pattern_final = npa.nan_to_num(pattern_interp.values, nan=npa.min(pattern_interp))
+    pattern_to_return = npa.nan_to_num(pattern_interp.values, nan=npa.min(pattern_interp)).flatten()
+    
+    #make a plot before returning
+    plt.figure()
+    ax = plt.gca()
+    im = ax.imshow(
+        pattern_final,
+        extent=(min(xs), max(xs), min(ys), max(ys)),
+        origin='lower',
+        cmap='inferno',
+    )
+    ax.set_xlabel('x (um)')
+    ax.set_ylabel('y (um)')
+    ax.set_title('Target Intensity Pattern')
+    plt.colorbar(im, ax=ax, label='Intensity (a.u.)')
+    plt.savefig("results/test1_target_intensity_pattern.pdf")
+    plt.close()
+    return pattern_to_return
+
+target_intensity = intensity_desired_fn_pattern(xs, ys)
+mean_target_intensity = np.mean(target_intensity)
+
+
+filter_radius_um = 1
+def _centered(arr: np.ndarray, newshape: ArrayLikeType) -> np.ndarray:
+    """Formats the output of an FFT to center the zero-frequency component.
+
+    A helper function borrowed from SciPy:
+    https://github.com/scipy/scipy/blob/v1.4.1/scipy/signal/signaltools.py#L263-L270
+
+    Args:
+        arr: output array from an FFT operation.
+        newshape: 1d array with two elements (integers) specifying the dimensions
+            of the array to be returned.
+
+    Returns:
+        The input array with the zero-frequency component as the central element.
+    """
+    newshape = np.asarray(newshape)
+    currshape = np.array(arr.shape)
+    startind = (currshape - newshape) // 2
+    endind = startind + newshape
+    myslice = [slice(startind[k], endind[k]) for k in range(len(endind))]
+
+    return arr[tuple(myslice)]
+
+def _edge_pad(arr: np.ndarray, pad: np.ndarray) -> np.ndarray:
+    """Border-pads the edges of an array.
+
+    Used to preprocess the design weights prior to convolution with the filter.
+    Border padding an image will set the value of each padded pixel equal to
+    the value of the nearest pixel in the image. Used to implement feature-
+    preserving convolution filters that prevent unwanted edge effects.
+
+    Args:
+        arr: 2d array whose borders contain the values to use for padding
+        pad: 2x2 array of integers indicating the size
+            of the borders to pad the array with.
+
+    Returns:
+        A 2d array with border padding.
+    """
+    # fill sides
+    left = npa.tile(arr[0, :], (pad[0][0], 1))
+    right = npa.tile(arr[-1, :], (pad[0][1], 1))
+    top = npa.tile(arr[:, 0], (pad[1][0], 1)).transpose()
+    bottom = npa.tile(arr[:, -1], (pad[1][1], 1)).transpose()
+
+    # fill corners
+    top_left = npa.tile(arr[0, 0], (pad[0][0], pad[1][0]))
+    top_right = npa.tile(arr[-1, 0], (pad[0][1], pad[1][0]))
+    bottom_left = npa.tile(arr[0, -1], (pad[0][0], pad[1][1]))
+    bottom_right = npa.tile(arr[-1, -1], (pad[0][1], pad[1][1]))
+
+    if pad[0][0] > 0 and pad[0][1] > 0 and pad[1][0] > 0 and pad[1][1] > 0:
+        return npa.concatenate(
+            (
+                npa.concatenate((top_left, top, top_right)),
+                npa.concatenate((left, arr, right)),
+                npa.concatenate((bottom_left, bottom, bottom_right)),
+            ),
+            axis=1,
+        )
+    elif pad[0][0] == 0 and pad[0][1] == 0 and pad[1][0] > 0 and pad[1][1] > 0:
+        return npa.concatenate((top, arr, bottom), axis=1)
+    elif pad[0][0] > 0 and pad[0][1] > 0 and pad[1][0] == 0 and pad[1][1] == 0:
+        return npa.concatenate((left, arr, right), axis=0)
+    elif pad[0][0] == 0 and pad[0][1] == 0 and pad[1][0] == 0 and pad[1][1] == 0:
+        return arr
+    else:
+        raise ValueError("At least one of the padding numbers is invalid.")
+
+def _quarter_to_full_kernel(arr: np.ndarray, pad_to: np.ndarray) -> np.ndarray:
+    """Constructs the full kernel from its nonnegative quadrant.
+
+    Args:
+        arr: 2d input array representing the nonnegative quadrant of a
+            filter kernel with C4v symmetry.
+        pad_to: 1d array with two elements (integers) specifying the size
+            of the zero padding.
+
+    Returns:
+        The complete kernel.
+    """
+    pad_size = pad_to - 2 * np.array(arr.shape) + 1
+
+    top = np.zeros((pad_size[0], arr.shape[1]))
+    bottom = np.zeros((pad_size[0], arr.shape[1] - 1))
+    middle = np.zeros((pad_to[0], pad_size[1]))
+
+    top_left = arr[:, :]
+    top_right = npa.flipud(arr[1:, :])
+    bottom_left = npa.fliplr(arr[:, 1:])
+    bottom_right = npa.flipud(
+        npa.fliplr(arr[1:, 1:])
+    )  # equivalent to flip, but flip is incompatible with autograd
+
+    return npa.concatenate(
+        (
+            npa.concatenate((top_left, top, top_right)),
+            middle,
+            npa.concatenate((bottom_left, bottom, bottom_right)),
+        ),
+        axis=1,
+    )
+
+def conic_filter(phase_mask,radius,size_x,size_y,resolution):
+    '''Applies a conic filter to the input phase mask. The filter is defined as a 
+    conic region of a specified radius, where the values are averaged. 
+
+    Args:
+        phase_mask: the input phase mask to be filtered (2D)
+        radius: the radius of the conic filter in micrometers
+        size_x: the size of the phase mask in x direction in micrometers
+        size_y: the size of the phase mask in y direction in micrometers
+        resolution: the resolution of the phase mask in pixels per micrometer
+    Returns:
+        The filtered phase mask (2D)
+    '''
+    Nx = int(round(size_x * resolution)) + 1
+    Ny = int(round(size_y * resolution)) + 1
+    
+    xv = np.arange(0, size_x / 2, 1 / resolution) if resolution > 0 else [0]
+    yv = np.arange(0, size_y / 2, 1 / resolution) if resolution > 0 else [0]
+
+    X, Y = npa.meshgrid(xv, yv)
+    
+    h = npa.where(X**2 + Y**2 < radius**2, (1 - np.sqrt(abs(X**2 + Y**2)) / radius),0)
+    #h = npa.where(X**2 + Y**2 < radius**2, 1, 0)
+    h = _quarter_to_full_kernel(h, 3 * np.array([Nx, Ny]))
+    h = h / npa.sum(h)
+    H = npa.fft.fft2(h)
+
+    phase_mask_2d = phase_mask.reshape(Nx,Ny)
+    phase_mask_padded = _edge_pad(phase_mask_2d, ((Nx, Nx), (Ny, Ny)))
+    phase_mask_fft = npa.fft.fft2(phase_mask_padded)
+    
+    Y = phase_mask_fft * H
+    
+    y = _centered(
+        npa.real(npa.fft.ifft2(Y)),
+        (Nx, Ny)
+    )
+
+    return y.flatten()
+
+def phase_filter(phase_mask):
+    '''Applies a conic filter on the phase mask'''
+    phase_mask_filtered = conic_filter(
+        phase_mask.reshape(Nx,Ny), #row major mapping 
+        filter_radius_um,
+        size_x,
+        size_y,
+        res_x,
+    )
+    return phase_mask_filtered.flatten()
+
+def forward_propagate(phase_mask):
+    '''Propagates the input field, after applying the phase mask
+
+    Args:
+        phase_mask: the phase mask to be applied to the input field (flattened)
+    Returns:        
+        A tuple with output_field (flattened) and corresponding propagation matrix P (2D)
+    '''
+    field_after_mask = input_field * npa.exp(1j * phase_mask)
+    field_after_mask_2d = field_after_mask.reshape((Nx, Ny))
+    field_fft = npa.fft.fftshift(npa.fft.fft2(field_after_mask_2d))
+
+    ks = 2 * npa.pi / sampling_period
+    kappas = npa.arange(-field_fft.shape[0]//2, field_fft.shape[0]//2) * ks / field_fft.shape[0]
+    KX, KY = npa.meshgrid(kappas, kappas)
+    K_parallel = npa.sqrt(KX**2 + KY**2)
+    nu_parallel = K_parallel / (2 * npa.pi)
+    phase_factor = k0 * z_prop * npa.sqrt(1 - (wavelength * nu_parallel) ** 2 + 0*1j)
+
+    P = npa.exp(1j * phase_factor)
+    fft_field_propagated = npa.multiply(field_fft,P) #element wise product
+
+    output_field_2d = npa.fft.ifft2(npa.fft.ifftshift(fft_field_propagated))
+    output_field = output_field_2d.flatten()
+
+    return output_field, P
+
+def adjoint_propagate(x):
+    '''Backpropagates the output field using the adjoint of the propagation matrix P.
+    Args:
+        output_field: the output field to be backpropagated (flattened)
+        P: the propagation matrix used in the forward propagation
+        phase_mask: the phase mask used in the forward propagation (flattened)
+    Returns:
+        The backpropagated field (flattened)
+    '''
+    phase_mask = phase_given_w(x)
+    field_after_mask = input_field * npa.exp(1j * phase_mask)
+    field_after_mask_2d = field_after_mask.reshape((Nx, Ny))
+    field_fft = npa.fft.fftshift(npa.fft.fft2(field_after_mask_2d))
+
+    ks = 2 * npa.pi / sampling_period
+    kappas = npa.arange(-field_fft.shape[0]//2, field_fft.shape[0]//2) * ks / field_fft.shape[0]
+    KX, KY = npa.meshgrid(kappas, kappas)
+    K_parallel = npa.sqrt(KX**2 + KY**2)
+    nu_parallel = K_parallel / (2 * npa.pi)
+    phase_factor = k0 * z_prop * npa.sqrt(1 - (wavelength * nu_parallel) ** 2 + 0*1j)
+
+    P = npa.exp(1j * phase_factor)
+    fft_field_propagated = npa.multiply(field_fft,P) #element wise product
+
+    output_field_2d = npa.fft.ifft2(npa.fft.ifftshift(fft_field_propagated))
+    output_field = output_field_2d.flatten()
+
+    output_intensity = output_field * np.conj(output_field)
+    adjoint_field = (output_intensity - target_intensity) * output_field
+    adjoint_field_2d = adjoint_field.reshape((Nx, Ny))
+    adjoint_field_fft = npa.fft.fftshift(npa.fft.fft2(adjoint_field_2d))
+    P_dagger = P.T.conj()
+    adjoint_field_propagated_fft = adjoint_field_fft * P_dagger #element wise
+    adjoint_field_propagated_2d = npa.fft.ifft2(npa.fft.ifftshift(adjoint_field_propagated_fft))
+    
+    #apply the conjugate of the phase mask to backpropagate through it
+    adjoint_field_propagated = adjoint_field_propagated_2d.flatten() * np.exp(-1j * phase_mask) 
+    return adjoint_field_propagated
+
+phase_mask_filtered = np.zeros((S))
+def _compute_cost(phase_mask):
+
+    output_field, P = forward_propagate(phase_mask)
+    
+    output_intensity = npa.real(output_field * npa.conj(output_field))
+    C_s = (output_intensity - target_intensity) ** 2
+
+    C = npa.sum(C_s)
+
+    return C
+
+opt_history = []
+def cost_fun(x, grad):
+    ''' Cost function to be minimized.
+    Args:
+        x: input normalized parameter field (flattened)
+        grad: gradient of the cost function with respect to x, modified in place (flattened)
+    Returns:
+        The cost function value for the input parameter field x.
+    '''
+    w_norm[:] = x
+
+    phase_mask = phase_given_w(w_norm)
+
+    phase_mask_filtered = phase_filter(phase_mask)
+
+    cost_function_denominator = np.sum(np.ones((S)))
+
+    C = _compute_cost(phase_mask_filtered) / cost_function_denominator
+    opt_history.append(C)
+    #make a plot of the optimization history
+    plt.figure()
+    plt.plot(opt_history)
+    plt.xlabel('Iteration')
+    plt.ylabel('Cost Function Value')
+    plt.title('Optimization History')
+    plt.savefig("results/test1_optimization_history.pdf")
+    plt.close()
+    #first round of backpropagation
+    grad_test = gd(_compute_cost)(phase_mask_filtered)
+
+    #second round of backpropagation
+    full_grad = tensor_jacobian_product(phase_filter)(phase_mask,grad_test)
+    #print("grad from autograd",grad_test)
+    
+    if grad.size > 0:
+        #adj_field_prop = adjoint_propagate(x)
+        #grad[:] = 4 * np.real(-1j * input_field.reshape((Nx,Ny)).T.conj().flatten() * adj_field_prop)
+        grad[:] = full_grad * 2 * np.pi / cost_function_denominator
+    #print("grad from theo",grad)
+    #print("grad diff",grad - grad_test)
+    return C
+
+if __name__ == "__main__":
+    #initialize nlopt solver
+    solver = nlopt.opt(nlopt.LD_CCSAQ, S)
+    solver.set_lower_bounds(norm_phase_min)
+    solver.set_upper_bounds(norm_phase_max)
+    solver.set_min_objective(cost_fun)
+    solver.set_maxeval(opt_max_eval)
+    solver.set_param("dual_ftol_rel", 1e-7)
+    solver.set_param("verbosity",1)
+
+    print("Starting optimization...")
+    w_norm[:] = solver.optimize(w_norm)
+    print("Optimization completed.")
+
+    #verify the results
+    phase_mask = phase_given_w(w_norm)
+    filtered_mask = phase_filter(phase_mask)
+    #save the phase_mask as raw 2D text
+    np.savetxt("results/optimized_phase_mask.txt", filtered_mask.reshape((Nx, Ny)))
+    
+    output_field, _ = forward_propagate(filtered_mask)
+    output_intensity = np.real(output_field * np.conj(output_field))
+    #make a plot of the phase mask, with phase wrap between -pi and pi
+    plt.figure()
+    plt.imshow(phase_mask.reshape((Nx, Ny)), extent=(-size_x/2, size_x/2, -size_y/2, size_y/2), origin='lower') 
+    plt.colorbar()
+    plt.title('Phase Mask')
+    plt.xlabel('x (um)')
+    plt.ylabel('y (um)')
+    plt.savefig("results/test1_optimized_phase_mask.pdf")
+    plt.show()
+    plt.close()
+    #make a subplot to compare the output intensity with the target intensity
+    plt.figure(figsize=(12, 5))
+    plt.subplot(1, 2, 1)
+    plt.imshow(output_intensity.reshape((Nx, Ny)), extent=(-size_x/2, size_x/2, -size_y/2, size_y/2), origin='lower', cmap='inferno')
+    plt.colorbar()
+    plt.clim(0,1)
+    plt.title('Output Intensity')
+    plt.xlabel('x (um)')
+    plt.ylabel('y (um)')
+    plt.subplot(1, 2, 2)
+    plt.imshow(target_intensity.reshape((Nx, Ny)), extent=(-size_x/2, size_x/2, -size_y/2, size_y/2), origin='lower', cmap='inferno')
+    plt.colorbar()
+    plt.clim(0,1)
+    plt.title('Target Intensity')
+    plt.xlabel('x (um)')
+    plt.ylabel('y (um)')
+    plt.tight_layout()
+    plt.savefig("results/test1_comparison_output_target.pdf")
+    #plt.show()
+    plt.close()
\ No newline at end of file