diff --git a/metaprop_fibermode_doublet_epigraph_ego.py b/metaprop_fibermode_doublet_epigraph_ego.py
new file mode 100644
index 0000000..59bef9b
--- /dev/null
+++ b/metaprop_fibermode_doublet_epigraph_ego.py
@@ -0,0 +1,774 @@
+import time
+import multiprocessing
+import os
+
+import pyfftw
+#pyFFTW setup
+n_cpu = multiprocessing.cpu_count()
+pyfftw.config.NUM_THREADS = n_cpu
+pyfftw.interfaces.cache.enable()
+pyfftw.config.PLANNER_EFFORT = 'FFTW_MEASURE'
+
+import nlopt
+
+import numpy as np
+import matplotlib.pyplot as plt
+import matplotlib.patches as patches
+#set the global font size for all plots
+plt.rcParams.update({'font.size': 14})
+
+from scipy.special import jv, kv
+
+from smt.applications.ego import EGO, KRG
+from smt.sampling_methods import LHS
+from smt.design_space import (
+    DesignSpace,
+) 
+
+##############################
+#physics and simulation domain
+wavelength = 1.55e-6
+k0 = 2 * np.pi / wavelength
+
+opt_max_eval = 150
+
+unit_cell_pitch = 400e-9 #equivalent to spatial sampling
+
+Nx = 2048 #pixels per dimension
+Ny = Nx
+S = Nx * Ny
+
+size_x = unit_cell_pitch * Nx #actual physical size
+size_y = unit_cell_pitch * Ny
+
+xs = np.linspace(-size_x/2, size_x/2 - size_x / Nx, Nx)
+ys = np.linspace(-size_y/2, size_y/2 - size_y / Ny, Ny)
+
+X, Y = np.meshgrid(xs, ys)
+rho = np.sqrt(X**2 + Y**2)
+
+sampling_period = xs[1] - xs[0]
+
+d = [0, 0, 0] #d = [d1,d2] d1:distance MS1-MS2, d2: distance MS2-target
+d0 = d[0] #propagation distance: source - MS1
+d1 = d[1] #propagation distance: MS1 - MS2
+d2 = d[2] #propagation distance: MS2 - target
+phase_min = 0
+phase_max = 2 * np.pi
+##############################
+
+##############################
+#optimization algorithm
+epigraph_tolerance = np.array([1e-4]*2)
+#epigraph_tolerance = np.array([1e-4])
+##############################
+
+##############################
+#Source
+beam_waist = 9e-6
+source_spacing = 300e-6
+##############################
+
+##############################
+#plan pyfftw objects for forward and inverse FFTs
+fft_input_array = pyfftw.empty_aligned((Nx, Ny), dtype='complex128', n=16)
+fft_output_array = pyfftw.empty_aligned((Nx, Ny), dtype='complex128', n=16)
+ifft_input_array = pyfftw.empty_aligned((Nx, Ny), dtype='complex128', n=16)
+ifft_output_array = pyfftw.empty_aligned((Nx, Ny), dtype='complex128', n=16)
+
+fft_operator = pyfftw.FFTW(fft_input_array, fft_output_array, axes=(0,1), direction='FFTW_FORWARD', threads=n_cpu, flags=('FFTW_MEASURE','FFTW_DESTROY_INPUT',))
+ifft_operator = pyfftw.FFTW(ifft_input_array, ifft_output_array, axes=(0,1), direction='FFTW_BACKWARD', threads=n_cpu, flags=('FFTW_MEASURE','FFTW_DESTROY_INPUT',))
+##############################
+
+##############################
+#setting up spatial frequencies
+ks = 2 * np.pi / sampling_period
+kappas = np.arange(-Nx//2, Nx//2) * ks / Nx
+KX, KY = np.meshgrid(kappas, kappas)
+K_parallel = np.sqrt(KX**2 + KY**2)
+nu_parallel = K_parallel / (2 * np.pi)
+
+phase_factor_0 = k0 * d[0] * np.sqrt(1 - (wavelength * nu_parallel) ** 2 + 0*1j)
+P_0 = np.exp(1j * phase_factor_0)
+P_0_nat = np.fft.ifftshift(P_0)
+phase_factor_1 = k0 * d[1] * np.sqrt(1 - (wavelength * nu_parallel) ** 2 + 0*1j)
+P_1 = np.exp(1j * phase_factor_1)
+P_1_nat = np.fft.ifftshift(P_1)
+phase_factor_2 = k0 * d[2] * np.sqrt(1 - (wavelength * nu_parallel) ** 2 + 0*1j)
+P_2 = np.exp(1j * phase_factor_2)
+P_2_nat = np.fft.ifftshift(P_2)
+
+P_1_dagger = P_1.T.conj()
+P_2_dagger = P_2.T.conj()
+P_1_dagger_nat = np.fft.ifftshift(P_1).T.conj()
+P_2_dagger_nat = np.fft.ifftshift(P_2).T.conj()
+##############################
+
+##############################
+#Plot zoom parameters [um]
+zoom_x = 60
+zoom_y = 60
+zoom_core_diameter = 2 * 6
+full_view_x = size_x * 1e6 / 2
+full_view_y = size_y * 1e6 / 2
+##############################
+
+def update_phase_factors():
+    print("Updating phase factors using d=",d)
+    phase_factor_0[:,:] = k0 * d[0] * np.sqrt(1 - (wavelength * nu_parallel) ** 2 + 0*1j)
+    P_0[:,:] = np.exp(1j * phase_factor_0)
+    P_0_nat[:,:] = np.fft.ifftshift(P_0)
+    phase_factor_1[:,:] = k0 * d[1] * np.sqrt(1 - (wavelength * nu_parallel) ** 2 + 0*1j)
+    P_1[:,:] = np.exp(1j * phase_factor_1)
+    P_1_nat[:,:] = np.fft.ifftshift(P_1)
+    phase_factor_2[:,:] = k0 * d[2] * np.sqrt(1 - (wavelength * nu_parallel) ** 2 + 0*1j)
+    P_2[:,:] = np.exp(1j * phase_factor_2)
+    P_2_nat[:,:] = np.fft.ifftshift(P_2)
+
+def update_input_fields():
+    input_field_1[:] = propagate_source(source_field_1)
+    input_field_1_2d[:] = input_field_1.reshape((Nx, Ny)) #reshape to 2D for plotting
+
+    #initialize input field 2 (propagated source 2 field)
+    input_field_2[:] = propagate_source(source_field_2)
+    input_field_2_2d[:] = input_field_2.reshape((Nx, Ny)) #reshape to 2D for plotting
+
+    #input field 1 integrated intensity
+    input_field_1_intensity = np.sum(np.abs(input_field_1)**2)
+    print("Input field integrated intensity: %.4e" % input_field_1_intensity)
+
+    #input field 2 integrated intensity
+    input_field_2_intensity = np.sum(np.abs(input_field_2)**2)
+    print("Input field integrated intensity: %.4e" % input_field_2_intensity)
+
+    #make a plot and save the input field
+    #make_2Dplot_of(input_field_1_2d, choose_quantity="amplitude", save_name="input_field_1_amplitude", plot_zoom_x=full_view_x, plot_zoom_y=full_view_y)
+    #make_2Dplot_of(input_field_1_2d, choose_quantity="phase", save_name="input_field_1_phase", plot_zoom_x=full_view_x, plot_zoom_y=full_view_y)
+    #make_2Dplot_of(input_field_2_2d, choose_quantity="amplitude", save_name="input_field_2_amplitude", plot_zoom_x=full_view_x, plot_zoom_y=full_view_y)
+    #make_2Dplot_of(input_field_2_2d, choose_quantity="phase", save_name="input_field_2_phase", plot_zoom_x=full_view_x, plot_zoom_y=full_view_y)
+
+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 * np.pi * (w)
+
+def get_pattern() -> np.ndarray:
+    '''Returns the flattened target fiber mode pattern'''
+    pattern_name = 'LP11_field_distribution.txt'
+    pattern = np.loadtxt('misc/' + pattern_name, dtype=complex)
+
+    return pattern.flatten()
+
+def get_fiber_mode_pattern(mode_list:list)->list:
+    '''Returns the fiber mode pattern for the selected n index.
+    Args:
+        mode_list: the list of the selected fiber mode indices
+    Returns:
+        The fiber mode patterns corresponding to the selected indices returned 
+        as a list of 2D arrays with the same order as the input mode_list
+    '''
+    #check if the modes in mode_list have already been compute by checking
+    #the saved files in the 'results' folder. If they have, load them from the files. If not, compute them and save them to the files.
+    fiber_modes_input_folder = "fiber_modes"
+    output_patterns = []
+    try:
+        for enum_index, n in enumerate(mode_list):
+            #try to load mode index n
+            try:
+                pattern_file = fiber_modes_input_folder + f"/LP{n}1_field_distribution.npy"
+                pattern = np.load(pattern_file)
+                output_patterns.append(pattern)
+                print("Loaded fiber patterns for mode index: ", n)
+            #if mode index n was not found, compute it
+            except FileNotFoundError:
+                if not os.path.exists(fiber_modes_input_folder):
+                        os.makedirs(fiber_modes_input_folder)
+
+                print("Computing fiber patterns for mode index: ", n)
+                lda = 1.55e-6
+
+                n_co = 1.4630 #core refractive index
+                n_cl = 1.4585 #cladding refractive index
+
+                fiber_core_diameter = 12e-6 # [m]
+                a = fiber_core_diameter / 2 #fiber core radius
+
+                #find the v parameter range that corresponds to the 1550 nm wavelength
+                #v_1550 = 2 * np.pi * a / lda * np.sqrt(n_co**2 - n_cl**2)
+                v_1550 = 4.5
+
+                b = np.linspace(1e-3, 1-1e-3, 400000)
+                V_list = np.linspace(2, 5.1, 20)
+                V_obs = v_1550
+                #b, V = np.meshgrid(b_list, V_list)
+
+                b_solutions_at_V_obs = []
+                b_solutions = []    
+                for V in V_list:
+                    LHS = np.sqrt(1-b) * jv(n+1, V * np.sqrt(1-b)) / jv(n, V * np.sqrt(1-b))
+                    RHS = np.sqrt(b) * kv(n+1, V * np.sqrt(b)) / kv(n, V * np.sqrt(b))
+
+                    intersection_index = np.argmax(np.where(np.isclose(LHS, RHS, atol=0.000001),1,0).flatten())
+
+                    b_solutions.append(b[intersection_index])
+
+                LHS = np.sqrt(1-b) * jv(n+1, V_obs * np.sqrt(1-b)) / jv(n, V_obs * np.sqrt(1-b))
+                RHS = np.sqrt(b) * kv(n+1, V_obs * np.sqrt(b)) / kv(n, V_obs * np.sqrt(b))
+                b_solutions_at_V_obs.append(b[np.argmax(np.where(np.isclose(LHS, RHS, atol=0.00001),1,0).flatten())])
+
+                b_mode_sel = b_solutions_at_V_obs[0]
+
+                k_co = 2 * np.pi / lda * n_co
+                k_cl = 2 * np.pi / lda * n_cl
+
+                beta = np.sqrt(b_mode_sel * (k_co**2 - k_cl**2) + k_cl**2)
+
+                chi_co = v_1550 * np.sqrt(1 - b_mode_sel) / a
+                chi_cl = v_1550 * np.sqrt(b_mode_sel) / a
+
+                E_field = np.zeros_like(X, dtype=complex)
+
+                for i in range(Nx):
+                    for j in range(Ny):
+                        r = np.sqrt(X[i,j]**2 + Y[i,j]**2)
+                        if r < 10*a:
+                            phi = np.arctan2(Y[i,j], X[i,j])
+                            if r < a:#inside core
+                                E_field[i,j] = jv(n,chi_co * r) / jv(n, chi_co * a) * np.cos(n * phi)
+                            else:#outside core
+                                E_field[i,j] = kv(n, chi_cl * r) / kv(n, chi_cl * a) * np.cos(n * phi)
+                #save the computed pattern to a file for future use
+                pattern_file = fiber_modes_input_folder + f"/LP{n}1_field_distribution.npy"
+                np.save(pattern_file, E_field)
+                output_patterns.append(E_field)
+        return output_patterns
+    except:
+        raise(Exception("Error in computing fiber patterns"))
+        
+def get_source_field(beam_waist,center_x=0,center_y=0,):
+    '''Returns a normalized Gaussian source field for a given beam waist. Normalization consists in scaling
+    the field such that its integrated intensity is 1.
+    Args:
+        beam_waist: the waist of the Gaussian beam (in meters)
+        center_x: the x-coordinate of the center of the Gaussian beam (in meters)
+        center_y: the y-coordinate of the center of the Gaussian beam (in meters)
+    Returns:
+        The normalized Gaussian source field
+    '''
+
+    if center_x != 0 or center_y != 0:
+        X_shifted, Y_shifted = shift_spatial_grid(center_x, center_y)
+        rho = np.sqrt(X_shifted**2 + Y_shifted**2)
+    else:
+        rho = np.sqrt(X**2 + Y**2)
+
+    gaussian_field = np.exp(-(rho)**2 / (beam_waist)**2) #Gaussian input field, flattened
+    gaussian_field_normalized = (gaussian_field / np.sqrt(np.sum(np.abs(gaussian_field)**2))) #normalize the input field to have power = 1
+    return gaussian_field_normalized
+
+def propagate_source(source_to_propagate):
+    '''Returns the propagated source field (flattened) just before metasurface 1 (MS1). This involves propagating the 
+    source for a distance d0: from its plane of definition up to MS1.'''
+    fft_input_array[:,:] = source_to_propagate.reshape((Nx, Ny))
+    fft_operator()
+    ifft_input_array[:,:] = fft_operator.output_array * P_0_nat
+    ifft_operator()
+    propagated_source_field = ifft_operator.output_array.flatten()
+    return propagated_source_field
+
+def make_1Dplot_of(x_axis,y_axis,legend_labels,plot_zoom_x=zoom_x,save_name="plot_1D",) -> None:
+    '''Makes a 1D plot of the given data and saves it to a PDF file.
+    Args:
+        x_axis: list of x-axis data (list of 1D arrays)
+        y_axis: list of y-axis data (list of 1D arrays)
+        plot_zoom_x: zoom level for the x-axis (in micrometers)
+        save_name: name of the file to save the plot (without extension)
+    Returns:
+        None
+    '''
+    if len(x_axis) != len(y_axis):
+        raise ValueError("Lists x_axis and y_axis must have the same length.")
+    plt.figure()
+    for i in range(len(x_axis)):
+        if len(legend_labels) != len(x_axis):
+            raise ValueError("Length of legend_labels must match the number of plots.")
+        else:
+            plt.plot(x_axis[i], y_axis[i], label=legend_labels[i])
+
+    plt.legend()
+    plt.tight_layout()
+    plt.savefig("results/" + save_name + ".pdf")
+    plt.xlim(-plot_zoom_x, plot_zoom_x)
+    plt.close()
+
+def make_2Dplot_of(given_field,choose_quantity="amplitude",save_name="plot",plot_zoom_x=zoom_x,plot_zoom_y=zoom_y) -> None:
+    '''Makes a plot of the given 2D field in a zoomed region. 
+    Saves the plot with the chosen filename to a PDF file. 
+    Args:
+        given_field: the 2D field to be plotted (flattened)
+        choose_quantity: either "amplitude" or "phase" to select quantity to plot
+        save_name: name of the file to save the plot (without extension)
+        plot_zoom_x: zoom level for the x-axis (in micrometers)
+        plot_zoom_y: zoom level for the y-axis (in micrometers)
+    Returns:
+        None'''
+    
+    given_field_2d = given_field.reshape((Nx, Ny)) #assure the field is 2D before plotting
+    plt.figure()
+    if choose_quantity == "amplitude":
+        plt.imshow(np.abs(given_field_2d), extent=(-size_x*1e6/2, size_x*1e6/2, -size_y*1e6/2, size_y*1e6/2))
+    elif choose_quantity == "phase":
+        plt.imshow((np.angle(given_field_2d)+2*np.pi)%(2*np.pi), extent=(-size_x*1e6/2, size_x*1e6/2, -size_y*1e6/2, size_y*1e6/2), cmap='hsv', vmin=0, vmax=2*np.pi)
+    elif choose_quantity == "phase_mask":
+        plt.imshow((given_field_2d+2*np.pi)%(2*np.pi), extent=(-size_x*1e6/2, size_x*1e6/2, -size_y*1e6/2, size_y*1e6/2), cmap='hsv', vmin=0, vmax=2*np.pi)
+    else:
+        raise ValueError("choose_quantity must be either 'amplitude' or 'phase'")
+
+    plt.xlim(-plot_zoom_x, plot_zoom_x)
+    plt.ylim(-plot_zoom_y, plot_zoom_y)
+    plt.xlabel("x [um]")
+    plt.ylabel("y [um]")
+    plt.colorbar()
+    plt.tight_layout()
+    plt.savefig("results/" + save_name + ".pdf")
+    plt.close()
+
+def make_polarPlot_of(given_field,choose_quantity="amplitude",save_name="polar_plot",plot_zoom_r=zoom_x) -> None:
+    '''Makes a polar plot of the given 2D field in a zoomed region. 
+    Saves the plot with the chosen filename to a PDF file. 
+    Args:
+        given_field: the 2D field to be plotted (flattened)
+        choose_quantity: either "amplitude" or "phase" to select quantity to plot
+        save_name: name of the file to save the plot (without extension)
+        plot_zoom_r: zoom level for the radial axis (in micrometers)
+    Returns:
+        None
+    '''
+    _, ax = plt.subplots(figsize=(6, 6))
+    if choose_quantity == "amplitude":
+        given_field_2d = np.abs(given_field.reshape((Nx, Ny))) #assure the field is 2D before plotting
+        im = ax.imshow(given_field_2d,origin='lower')
+    elif choose_quantity == "phase":
+        given_field_2d = (np.angle(given_field.reshape((Nx, Ny))) + 2*np.pi) % (2*np.pi)#assure the field is 2D before plotting
+        im = ax.imshow(given_field_2d, cmap='hsv', vmin=0, vmax=2*np.pi)
+    else:
+        raise ValueError("choose_quantity must be either 'amplitude' or 'phase'")
+    
+    center_x, center_y = Nx//2, Ny//2
+    radius = plot_zoom_r * 1e-6 / (size_x/2) * (Nx/2) #convert zoom in um to pixels
+    circle = patches.Circle((center_x, center_y), radius, transform=ax.transData)
+    im.set_clip_path(circle)
+    ax.set_xlim(center_x - radius, center_x + radius)
+    ax.set_ylim(center_y - radius, center_y + radius)
+    ax.axis('off')
+    cbar = plt.colorbar(im, ax=ax)
+    plt.savefig("results/" + save_name + ".pdf")
+
+def shift_spatial_grid(shift_amount_x,shift_amount_y):
+    '''Shifts the spatial grid by a given amount. This is useful for centering the source field on the metasurface.
+    Args:
+        shift_amount_x: amount to shift the grid in x direction (in meters)
+        shift_amount_y: amount to shift the grid in y direction (in meters)
+    Returns:
+        the shifted spatial grid (X, Y)'''
+    Xp = X + shift_amount_x
+    Yp = Y + shift_amount_y
+    return Xp, Yp
+    
+###########################
+#initialize source field
+source_field_1 = get_source_field(beam_waist,center_x=-source_spacing/2,center_y=0)
+source_field_1_2d = source_field_1.reshape((Nx, Ny)) #reshape to 2D for plotting
+
+source_field_2 = get_source_field(beam_waist,center_x=source_spacing/2,center_y=0)
+source_field_2_2d = source_field_2.reshape((Nx, Ny)) #reshape to 2D for plotting
+
+#source field integrated intensity
+source_1_power = np.sum(np.abs(source_field_1)**2)
+source_2_power = np.sum(np.abs(source_field_2)**2)
+print("Source 1 field integrated intensity: %.4e" % source_1_power)
+print("Source 2 field integrated intensity: %.4e" % source_2_power)
+#make a plot and save the source field
+#make_2Dplot_of(source_field_1_2d, choose_quantity="amplitude", save_name="source_1_field_amplitude", plot_zoom_x=3*zoom_x, plot_zoom_y=3*zoom_y)
+#make_2Dplot_of(source_field_1_2d, choose_quantity="phase", save_name="source_1_field_phase", plot_zoom_x=3*zoom_x, plot_zoom_y=3*zoom_y)
+#make_2Dplot_of(source_field_2_2d, choose_quantity="amplitude", save_name="source_2_field_amplitude", plot_zoom_x=3*zoom_x, plot_zoom_y=3*zoom_y)
+#make_2Dplot_of(source_field_2_2d, choose_quantity="phase", save_name="source_2_field_phase", plot_zoom_x=3*zoom_x, plot_zoom_y=3*zoom_y)
+###########################
+
+###########################
+#initialize input field 1 (propagated source 1 field)
+input_field_1 = propagate_source(source_field_1)
+input_field_1_2d = input_field_1.reshape((Nx, Ny)) #reshape to 2D for plotting
+
+#initialize input field 2 (propagated source 2 field)
+input_field_2 = propagate_source(source_field_2)
+input_field_2_2d = input_field_2.reshape((Nx, Ny)) #reshape to 2D for plotting
+
+#input field 1 integrated intensity
+input_field_1_intensity = np.sum(np.abs(input_field_1)**2)
+print("Input field integrated intensity: %.4e" % input_field_1_intensity)
+
+#input field 2 integrated intensity
+input_field_2_intensity = np.sum(np.abs(input_field_2)**2)
+print("Input field integrated intensity: %.4e" % input_field_2_intensity)
+
+#list of input fields to be propagated within the optimization loop
+input_field_list = [input_field_1, input_field_2]
+#input_field_list = [input_field_1]
+###########################
+
+###########################
+#initialize target field
+target_mode_indices = [1,2] 
+fiber_mode_pattern_1, fiber_mode_pattern_2 = get_fiber_mode_pattern(target_mode_indices)
+
+target_Efield_1_2d = fiber_mode_pattern_1 / np.sqrt(np.sum(np.abs(fiber_mode_pattern_1)**2))
+target_Efield_1 =  target_Efield_1_2d.flatten() #normalized target field (intensity = 1)
+
+target_Efield_2_2d = fiber_mode_pattern_2 / np.sqrt(np.sum(np.abs(fiber_mode_pattern_2)**2))
+target_Efield_2 =  target_Efield_2_2d.flatten() #normalized target field (intensity = 1)
+
+# Plot: Target fields
+#make_2Dplot_of(target_Efield_1_2d, choose_quantity="amplitude", save_name="target_field_1_amplitude", plot_zoom_x=zoom_x, plot_zoom_y=zoom_y)
+#make_2Dplot_of(target_Efield_1_2d, choose_quantity="phase", save_name="target_field_1_phase", plot_zoom_x=zoom_x, plot_zoom_y=zoom_y)
+#make_2Dplot_of(target_Efield_2_2d, choose_quantity="amplitude", save_name="target_field_2_amplitude", plot_zoom_x=zoom_x, plot_zoom_y=zoom_y)
+#make_2Dplot_of(target_Efield_2_2d, choose_quantity="phase", save_name="target_field_2_phase", plot_zoom_x=zoom_x, plot_zoom_y=zoom_y)
+
+target_field_list = [target_Efield_1, target_Efield_2]
+#target_field_list = [target_Efield_1]
+###########################
+
+def forward_propagate(input_field:np.ndarray,weights:np.ndarray) -> np.ndarray:
+    '''Propagates the input field through the system. Stores results in-place within the intermediate_fields list.
+    Returns:
+        the output field computed at the observation plane
+    '''
+    print("Starting propagating...")
+    intermediate_fields = [None, None]
+    start_time = time.time()
+    phase_mask = phase_given_w(weights)
+    fft_input_array[:,:] = np.fft.ifftshift((input_field * np.exp(1j * phase_mask[:S])).reshape((Nx, Ny)))
+    fft_operator()
+    #print("BEFORE")
+    fft_power = np.sum(np.abs(fft_operator.output_array)**2)/S
+    non_propagating_mask = np.where((wavelength * nu_parallel)**2 > 1, 0, 1)
+    non_propagating_mask = np.fft.ifftshift(non_propagating_mask)
+    propagating_power = np.sum(np.abs(fft_operator.output_array * non_propagating_mask)**2)/S
+    #print("FFT power: %.4e, Propagating power: %.4e, Non-propagating power: %.4e" % (fft_power, propagating_power, fft_power - propagating_power))
+
+    #FIRST PROPAGATION
+    ifft_input_array[:,:] = fft_operator.output_array * P_1_nat #element wise product
+    #print("AFTER")
+
+    fft_power = np.sum(np.abs(ifft_operator.input_array)**2)/S
+    non_propagating_mask = np.where((wavelength * nu_parallel)**2 > 1, 0, 1)
+    non_propagating_mask = np.fft.ifftshift(non_propagating_mask)
+    propagating_power = np.sum(np.abs(ifft_operator.input_array * non_propagating_mask)**2)/S
+    #print("FFT power: %.4e, Propagating power: %.4e, Non-propagating power: %.4e" % (fft_power, propagating_power, fft_power - propagating_power))
+    
+    ifft_operator()
+    intermediate_fields[0] = np.fft.fftshift(ifft_operator.output_array.copy())
+    #print("Field intensity after first phase mask and propagation: %.4e" % np.sum(np.abs(intermediate_fields[0])**2))
+
+    fft_input_array[:,:] = np.fft.ifftshift((np.fft.fftshift(ifft_operator.output_array) * np.exp(1j * phase_mask[S:].reshape((Nx, Ny)))))
+    fft_operator()
+
+    #SECOND PROPAGATION
+    ifft_input_array[:,:] = fft_operator.output_array * P_2_nat #element wise product
+    ifft_operator()
+    intermediate_fields[1] = np.fft.fftshift(ifft_operator.output_array.copy())
+    
+    end_time = time.time()
+    print("Propagation finished in {:.6f} seconds.".format(end_time - start_time))
+
+    #make_2Dplot_of(intermediate_fields[0], choose_quantity="amplitude", save_name="intermediate_field_1_amplitude", plot_zoom_x=full_view_x, plot_zoom_y=full_view_y)
+    #make_2Dplot_of(intermediate_fields[1], choose_quantity="amplitude", save_name="intermediate_field_2_amplitude", plot_zoom_x=full_view_x, plot_zoom_y=full_view_y)
+
+    return intermediate_fields
+
+def adjoint_propagate(input_field_2d,target_Efield_2d,intermediate_fields):
+    '''Backpropagates the output field using the adjoint of the propagation matrix P.
+    Returns:
+        the gradient of the cost function with respect to the phase mask (flattened)
+    '''
+    print("Computing adjoint...")
+    start_time = time.time()
+
+    Phi_1_dagger = np.exp(-1j * phase_mask[:S]).reshape((Nx, Ny))
+    Phi_2_dagger = np.exp(-1j * phase_mask[S:]).reshape((Nx, Ny))
+
+    #compute the adjoint source
+    fft_input_array[:,:] = np.fft.ifftshift(np.sum(intermediate_fields[1] * np.conj(target_Efield_2d)) * target_Efield_2d)
+    fft_operator()
+    ifft_input_array[:,:] = fft_operator.output_array * P_2_dagger_nat
+    ifft_operator()
+    fft_input_array[:,:] = np.fft.ifftshift(np.fft.fftshift(ifft_operator.output_array) * Phi_2_dagger)
+
+    grad_C_phi_2 = 2 * np.real(-1j * intermediate_fields[0].conj() * np.fft.fftshift(fft_input_array)).flatten() 
+
+    fft_operator()
+    ifft_input_array[:,:] = fft_operator.output_array * P_1_dagger_nat #element wise
+    ifft_operator()
+
+    grad_C_phi_1 = 2 * np.real(-1j * input_field_2d.conj() * np.fft.fftshift(ifft_operator.output_array) * Phi_1_dagger).flatten()
+    
+    end_time = time.time()
+    print("Adjoint finished in {:.6f} seconds.".format(end_time - start_time))
+    return np.concatenate((grad_C_phi_1, grad_C_phi_2))
+
+
+def value_and_grad(weights):
+    values = []
+    gradients = []
+
+    phase_mask[:] = phase_given_w(weights) #update phase mask in place
+    for i, input_field in enumerate(input_field_list):
+        
+        intermediate_fields = forward_propagate(input_field, weights)
+        
+        circular_mask = np.where(rho.flatten() < 10*6e-6, 1, 0) 
+        C_s = np.abs(np.sum(intermediate_fields[-1].flatten() * circular_mask * np.conj(target_field_list[i]))) ** 2
+        C = np.real(np.sum(C_s))
+
+        values.append(-C)
+        gradients.append(
+            adjoint_propagate(
+                input_field_2d = input_field.reshape((Nx, Ny)), 
+                target_Efield_2d = target_field_list[i].reshape((Nx, Ny)),
+                intermediate_fields = intermediate_fields)
+        )
+    
+    return np.array(values), np.array(gradients)
+
+obj_values1 = []
+obj_values2 = []
+def epigraph_constraint(result,epigraph_and_weights,gradient)->None:
+    '''Constraint function for the epigraph formulation
+    Args:
+        result: 1D array, modified in place, with the results of this constraint
+            evaluation
+        epigraph_and_weights: 1D array of epigraph (first element) and 
+            design weights (remaining elements)
+        gradient: gradients, modified in place, of the constraint function with respect to the design weights
+    '''
+    epigraph = epigraph_and_weights[0]
+    weights = epigraph_and_weights[1:]
+
+    obj_val, grad = value_and_grad(weights)
+    obj_values1.append(obj_val[0])
+    obj_values2.append(obj_val[1])
+
+    if gradient.size > 0:
+        gradient[:,0] = -1
+        gradient[:,1:] = - grad * (2*np.pi)
+
+    #result of the evaluation of this constraint function
+    result[:] = np.real(obj_val) - epigraph
+
+opt_history = []
+def obj_fun(epigraph_and_weights, grad):
+    '''Objective function for the epigraph formulation
+    Args:
+        epigraph_and_weights: 1D array of epigraph (first element) and 
+            design weights (remaining elements)
+        grad: gradient of the objective function with respect to the design weights, modified in place
+    Returns:
+        The scalar epigraph variable (the objective function)
+    '''
+
+    epigraph = epigraph_and_weights[0]
+
+    if grad.size > 0:
+        grad[0] = 1
+        grad[1:] = 0
+    
+    opt_history.append(epigraph)
+    #make a 1d plot of the optimization history together with the objective values
+    #make_1Dplot_of(x_axis=[np.arange(len(obj_values1)), np.arange(len(obj_values2)), np.arange(len(opt_history))], y_axis=[obj_values1, obj_values2, opt_history], legend_labels=["ob1","ob2","t"], save_name="optimization_history", plot_zoom_x=len(opt_history))
+
+    return epigraph
+
+def initialize_masks():
+    X1,Y1 = shift_spatial_grid(-source_spacing/2,0)
+    rho1 = np.sqrt(X1**2 + Y1**2)
+    circular_mask_1 = np.where(rho1.flatten() < 10*6e-6, 1, 0)
+
+    X2,Y2 = shift_spatial_grid(source_spacing/2,0)
+    rho2 = np.sqrt(X2**2 + Y2**2)
+    circular_mask_2 = np.where(rho2.flatten() < 10*6e-6, 1, 0)
+
+    target_pattern_1_norm_phase = (np.angle(target_Efield_1) + 2 * np.pi) % (2 * np.pi) / (2*np.pi)
+    target_pattern_1_norm_phase = np.roll(target_pattern_1_norm_phase, shift=int(source_spacing/sampling_period/2), axis=0)
+
+    target_pattern_2_norm_phase = (np.angle(target_Efield_2) + 2 * np.pi) % (2 * np.pi) / (2*np.pi)
+    target_pattern_2_norm_phase = np.roll(target_pattern_2_norm_phase, shift=-int(source_spacing/sampling_period/2), axis=0)
+    #on metasurface 1, set initial guess on normalized phase elements
+    w_norm[:S] += circular_mask_1 * target_pattern_1_norm_phase + circular_mask_2 * target_pattern_2_norm_phase
+    #on metasurface 2, generate random initial guess for normalized phase elements
+    #w_norm[S:] += np.random.rand(S) * circular_mask_1 + np.random.rand(S) * circular_mask_2
+
+w_norm = np.zeros((2*S)) #global definition concatenated normalized parameters for both masks
+phase_mask = phase_given_w(w_norm) #global definition 
+epigraph_and_weights = np.zeros((2*S + 1))
+
+def metaprop_solve():
+    ###########################
+    #initialize the normalized phase masks and global phase mask
+    initialize_masks()
+    phase_mask[:] = phase_given_w(w_norm)
+
+    #make an initial plot of the phase masks
+    #make_2Dplot_of(phase_mask[:S].reshape((Nx, Ny)), choose_quantity="phase_mask", save_name="initial_phase_mask_1", plot_zoom_x=full_view_x, plot_zoom_y=full_view_y)
+    #make_2Dplot_of(phase_mask[S:].reshape((Nx, Ny)), choose_quantity="phase_mask", save_name="initial_phase_mask_2", plot_zoom_x=full_view_x, plot_zoom_y=full_view_y)
+
+    #define bounding box for the optimization. The outer mask is fixed to zero phase, while the inner mask can vary between 0 and 2pi phase.
+    #the bounding box is the outer shell of a square with thickness 100 um
+    norm_phase_min = 0.0
+    norm_phase_max = 1.0
+    mask_buffer = 100e-6
+    mask_upper_bounds = norm_phase_max * np.where((np.abs(X.flatten()) > size_x/2 - mask_buffer) | (np.abs(Y.flatten()) > size_y/2 - mask_buffer), 0, 1)
+    mask_upper_bounds = np.concatenate((mask_upper_bounds, mask_upper_bounds),axis=0)
+    mask_lower_bounds = np.zeros_like(mask_upper_bounds)
+
+    epigraph_and_weights[:] = np.insert(w_norm,0,0) #initialize epigraph variable to 0
+    print("#####Shape of epigraph_and_weights: ", epigraph_and_weights.shape)
+    ###########################
+
+    #initialize nlopt solver
+    solver = nlopt.opt(nlopt.LD_CCSAQ, 2*S + 1) #weights (2S) + epigraph (1)
+    epigraph_and_weights_lower_bounds = np.insert(mask_lower_bounds.flatten(),0,-np.inf)
+    epigraph_and_weights_upper_bounds = np.insert(mask_upper_bounds.flatten(),0,+np.inf)
+    solver.set_lower_bounds(epigraph_and_weights_lower_bounds)
+    solver.set_upper_bounds(epigraph_and_weights_upper_bounds)
+    solver.set_min_objective(obj_fun)
+    solver.add_inequality_mconstraint(
+        lambda result_, epigraph_and_weights_, grad_: epigraph_constraint(
+            result_,
+            epigraph_and_weights_,
+            grad_,
+        ),
+        epigraph_tolerance,
+    )
+    solver.set_maxeval(opt_max_eval)
+    solver.set_param("dual_ftol_rel", 1e-5)
+    solver.set_param("verbosity", 1)
+    solver.set_param("inner_maxeval", 5)
+
+    print("Starting optimization...")
+    start = True
+    if start:
+        epigraph_and_weights[:] = solver.optimize(epigraph_and_weights)
+    print("Optimization completed.")
+
+    #verify the results
+    opt_weights = epigraph_and_weights[1:]
+    verify_out_field_1 = forward_propagate(input_field_list[0], opt_weights)[-1]
+    verify_out_field_2 = forward_propagate(input_field_list[1], opt_weights)[-1]
+    all_source_on_field = input_field_list[0] + input_field_list[1]
+    verify_out_field_all = forward_propagate(all_source_on_field, opt_weights)[-1]
+    
+    #make_polarPlot_of(verify_out_field_1, choose_quantity="amplitude", save_name="modeconv1_optimized_output_field_source1_amplitude", plot_zoom_r=zoom_core_diameter)
+    #make_polarPlot_of(verify_out_field_1, choose_quantity="phase", save_name="modeconv1_optimized_output_field_source1_phase", plot_zoom_r=zoom_core_diameter)
+    #make_polarPlot_of(verify_out_field_2, choose_quantity="amplitude", save_name="modeconv1_optimized_output_field_source2_amplitude", plot_zoom_r=zoom_core_diameter)
+    #make_polarPlot_of(verify_out_field_2, choose_quantity="phase", save_name="modeconv1_optimized_output_field_source2_phase", plot_zoom_r=zoom_core_diameter)
+    #make_polarPlot_of(verify_out_field_all, choose_quantity="amplitude", save_name="modeconv1_optimized_output_field_allSource_amplitude", plot_zoom_r=zoom_core_diameter)
+    #make_polarPlot_of(verify_out_field_all, choose_quantity="phase", save_name="modeconv1_optimized_output_field_allSource_phase", plot_zoom_r=zoom_core_diameter)
+
+    #save the output fields for later use
+    saveFields = False
+    if saveFields:
+        np.savetxt("results/modeconv1_optimized_output_field_source1.txt", verify_out_field_1.reshape((Nx, Ny)))
+        np.savetxt("results/modeconv1_optimized_output_field_source2.txt", verify_out_field_2.reshape((Nx, Ny)))
+        np.savetxt("results/modeconv1_optimized_output_field_allSource.txt", verify_out_field_all.reshape((Nx, Ny)))
+  
+    # Reshape fields to 2D for visualization
+    phase_mask_1 = phase_given_w(epigraph_and_weights[1:S+1]).reshape((Nx, Ny))
+    phase_mask_2 = phase_given_w(epigraph_and_weights[S+1:]).reshape((Nx, Ny))    
+    # Plot: Phase masks
+    #make_2Dplot_of(phase_mask_1, choose_quantity="phase_mask", save_name="phase_mask_1", plot_zoom_x=full_view_x, plot_zoom_y=full_view_y)
+    #make_2Dplot_of(phase_mask_2, choose_quantity="phase_mask", save_name="phase_mask_2", plot_zoom_x=full_view_x, plot_zoom_y=full_view_y)
+    
+    #save the phase masks for later use
+    saveMasks = False
+    if saveMasks:
+        np.savetxt("results/optimized_phase_mask_1.txt", phase_mask_1)
+        np.savetxt("results/optimized_phase_mask_2.txt", phase_mask_2)
+
+    print("All plots saved to results/ folder.")
+
+def reset_globals():
+    obj_values1[:] = []
+    obj_values2[:] = []
+    opt_history[:] = []
+    w_norm[:] = np.zeros((2*S)) #reset the normalized phase masks for the next iteration
+
+def metaprop(x):
+    #initialize results file if it does not exist
+
+    if not os.path.exists('results/metaprop_results.txt'):
+        with open('results/metaprop_results.txt', 'w') as f:
+            f.write("d1 [m] d2 [m] d3 [m] ob1 ob2 t\n")
+
+    m, n = x.shape
+    C = np.zeros((m,1))
+    for i in range(m):
+        for j in range(n):
+            d[j] = 1e-6*np.round(x[i,j]) #update the global propagation distances with the new design parameters
+
+        print("Current propagation distances: ", d)
+        update_phase_factors() #update the phase factors for the new propagation distances
+        update_input_fields()
+        metaprop_solve() #solve the inner optimization problem for the given design parameters
+
+        ob1 = obj_values1[-1]
+        ob2 = obj_values2[-1]
+        t = opt_history[-1]
+        #save all the partial results for a specific set of design parameters x to a text file
+        #in append mode, so that we can keep track of the optimization history
+        with open('results/metaprop_results.txt', 'a') as f:
+            f.write(f"{d[0]:.6e} {d[1]:.6e} {d[2]:.6e} {ob1:.6e} {ob2:.6e} {t:.6e}\n")
+        
+        alpha = 0.5
+        C[i] = t + alpha * np.abs(ob1 - ob2) #cost function to be minimized by EGO)
+
+        reset_globals()   
+    return C
+
+def run_ego():
+    ##############################
+    #EGO setup
+    ##############################
+    n_ego_iter = 30
+    xlimits = np.array([[200,700], [200,700], [200,700]])
+    seed = 777
+    design_space = DesignSpace(xlimits, seed=seed)
+    criterion = 'EI'
+    ndoe = 20
+
+    #build DOE
+    sampling = LHS(xlimits=xlimits, seed=seed)
+    xdoe = sampling(ndoe)
+
+    sm = KRG(design_space=design_space, n_start=25, print_global=False)
+    ego = EGO(
+        n_iter=n_ego_iter,
+        criterion=criterion,
+        xdoe=xdoe,
+        surrogate=sm,
+        n_start=25,
+    )
+
+    x_opt, y_opt, ind_best, x_data, y_data = ego.optimize(fun=metaprop)
+    #save the optimization results to a text file
+    
+    np.savetxt('results/ego_opt_results.txt', np.hstack((x_opt,y_opt)))
+
+if __name__ == "__main__":
+    #if the results folder does not exist, create it
+    if not os.path.exists('results'):
+        os.makedirs('results')
+    run_ego()
\ No newline at end of file