221 lines
7.7 KiB
Python
221 lines
7.7 KiB
Python
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"""
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Functions for performing near-to-farfield transformation (and the reverse).
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"""
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from typing import Dict, List
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import numpy
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from numpy.fft import fft2, fftshift, fftfreq, ifft2, ifftshift
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from numpy import pi
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def near_to_farfield(E_near: List[numpy.ndarray],
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H_near: List[numpy.ndarray],
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dx: float,
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dy: float,
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padded_size: List[int] = None
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) -> Dict[str]:
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"""
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Compute the farfield, i.e. the distribution of the fields after propagation
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through several wavelengths of uniform medium.
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The input fields should be complex phasors.
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:param E_near: List of 2 ndarrays containing the 2D phasor field slices for the transverse
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E fields (e.g. [Ex, Ey] for calculating the farfield toward the z-direction).
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:param H_near: List of 2 ndarrays containing the 2D phasor field slices for the transverse
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H fields (e.g. [Hx, hy] for calculating the farfield towrad the z-direction).
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:param dx: Cell size along x-dimension, in units of wavelength.
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:param dy: Cell size along y-dimension, in units of wavelength.
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:param padded_size: Shape of the output. A single integer `n` will be expanded to `(n, n)`.
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Powers of 2 are most efficient for FFT computation.
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Default is the smallest power of 2 larger than the input, for each axis.
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:returns: Dict with keys
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'E_far': Normalized E-field farfield; multiply by
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(i k exp(-i k r) / (4 pi r)) to get the actual field value.
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'H_far': Normalized H-field farfield; multiply by
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(i k exp(-i k r) / (4 pi r)) to get the actual field value.
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'kx', 'ky': Wavevector values corresponding to the x- and y- axes in E_far and H_far,
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normalized to wavelength (dimensionless).
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'dkx', 'dky': step size for kx and ky, normalized to wavelength.
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'theta': arctan2(ky, kx) corresponding to each (kx, ky).
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This is the angle in the x-y plane, counterclockwise from above, starting from +x.
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'phi': arccos(kz / k) corresponding to each (kx, ky).
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This is the angle away from +z.
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"""
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if not len(E_near) == 2:
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raise Exception('E_near must be a length-2 list of ndarrays')
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if not len(H_near) == 2:
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raise Exception('H_near must be a length-2 list of ndarrays')
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s = E_near[0].shape
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if not all(s == f.shape for f in E_near + H_near):
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raise Exception('All fields must be the same shape!')
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if padded_size is None:
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padded_size = (2**numpy.ceil(numpy.log2(s))).astype(int)
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if not hasattr(padded_size, '__len__'):
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padded_size = (padded_size, padded_size)
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En_fft = [fftshift(fft2(fftshift(Eni), s=padded_size)) for Eni in E_near]
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Hn_fft = [fftshift(fft2(fftshift(Hni), s=padded_size)) for Hni in H_near]
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# Propagation vectors kx, ky
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k = 2 * pi
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kxx = 2 * pi * fftshift(fftfreq(padded_size[0], dx))
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kyy = 2 * pi * fftshift(fftfreq(padded_size[1], dy))
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kx, ky = numpy.meshgrid(kxx, kyy, indexing='ij')
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kxy2 = kx * kx + ky * ky
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kxy = numpy.sqrt(kxy2)
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kz = numpy.sqrt(k * k - kxy2)
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sin_th = ky / kxy
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cos_th = kx / kxy
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cos_phi = kz / k
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sin_th[numpy.logical_and(kx == 0, ky == 0)] = 0
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cos_th[numpy.logical_and(kx == 0, ky == 0)] = 1
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# Normalized vector potentials N, L
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N = [-Hn_fft[1] * cos_phi * cos_th + Hn_fft[0] * cos_phi * sin_th,
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Hn_fft[1] * sin_th + Hn_fft[0] * cos_th]
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L = [ En_fft[1] * cos_phi * cos_th - En_fft[0] * cos_phi * sin_th,
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-En_fft[1] * sin_th - En_fft[0] * cos_th]
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E_far = [-L[1] - N[0],
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L[0] - N[1]]
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H_far = [-E_far[1],
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E_far[0]]
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theta = numpy.arctan2(ky, kx)
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phi = numpy.arccos(cos_phi)
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theta[numpy.logical_and(kx == 0, ky == 0)] = 0
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phi[numpy.logical_and(kx == 0, ky == 0)] = 0
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# Zero fields beyond valid (phi, theta)
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invalid_ind = kxy2 >= k * k
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theta[invalid_ind] = 0
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phi[invalid_ind] = 0
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for i in range(2):
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E_far[i][invalid_ind] = 0
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H_far[i][invalid_ind] = 0
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outputs = {
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'E': E_far,
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'H': H_far,
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'dkx': kx[1]-kx[0],
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'dky': ky[1]-ky[0],
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'kx': kx,
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'ky': ky,
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'theta': theta,
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'phi': phi,
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}
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return outputs
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def far_to_nearfield(E_far: List[numpy.ndarray],
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H_far: List[numpy.ndarray],
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dkx: float,
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dky: float,
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padded_size: List[int] = None
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) -> Dict[str]:
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"""
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Compute the farfield, i.e. the distribution of the fields after propagation
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|
|
through several wavelengths of uniform medium.
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|
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The input fields should be complex phasors.
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:param E_far: List of 2 ndarrays containing the 2D phasor field slices for the transverse
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E fields (e.g. [Ex, Ey] for calculating the nearfield toward the z-direction).
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Fields should be normalized so that
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E_far = E_far_actual / (i k exp(-i k r) / (4 pi r))
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:param H_far: List of 2 ndarrays containing the 2D phasor field slices for the transverse
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H fields (e.g. [Hx, hy] for calculating the nearfield toward the z-direction).
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Fields should be normalized so that
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H_far = H_far_actual / (i k exp(-i k r) / (4 pi r))
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:param dkx: kx discretization, in units of wavelength.
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:param dky: ky discretization, in units of wavelength.
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:param padded_size: Shape of the output. A single integer `n` will be expanded to `(n, n)`.
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Powers of 2 are most efficient for FFT computation.
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Default is the smallest power of 2 larger than the input, for each axis.
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:returns: Dict with keys
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'E': E-field nearfield
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'H': H-field nearfield
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'dx', 'dy': spatial discretization, normalized to wavelength (dimensionless)
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"""
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if not len(E_far) == 2:
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raise Exception('E_far must be a length-2 list of ndarrays')
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if not len(H_far) == 2:
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raise Exception('H_far must be a length-2 list of ndarrays')
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s = E_far[0].shape
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if not all(s == f.shape for f in E_far + H_far):
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raise Exception('All fields must be the same shape!')
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if padded_size is None:
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padded_size = (2**numpy.ceil(numpy.log2(s))).astype(int)
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if not hasattr(padded_size, '__len__'):
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padded_size = (padded_size, padded_size)
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k = 2 * pi
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kxs = fftshift(fftfreq(s[0], 1/(s[0] * dkx)))
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kys = fftshift(fftfreq(s[0], 1/(s[1] * dky)))
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kx, ky = numpy.meshgrid(kxs, kys, indexing='ij')
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kxy2 = kx * kx + ky * ky
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kxy = numpy.sqrt(kxy2)
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kz = numpy.sqrt(k * k - kxy2)
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sin_th = ky / kxy
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cos_th = kx / kxy
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cos_phi = kz / k
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sin_th[numpy.logical_and(kx == 0, ky == 0)] = 0
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cos_th[numpy.logical_and(kx == 0, ky == 0)] = 1
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# Zero fields beyond valid (phi, theta)
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invalid_ind = kxy2 >= k * k
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theta[invalid_ind] = 0
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phi[invalid_ind] = 0
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for i in range(2):
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E_far[i][invalid_ind] = 0
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H_far[i][invalid_ind] = 0
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# Normalized vector potentials N, L
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L = [0.5 * E_far[1],
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-0.5 * E_far[0]]
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N = [L[1],
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-L[0]]
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En_fft = [-( L[0] * sin_th + L[1] * cos_phi * cos_th)/cos_phi,
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-(-L[0] * cos_th + L[1] * cos_phi * sin_th)/cos_phi]
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Hn_fft = [( N[0] * sin_th + N[1] * cos_phi * cos_th)/cos_phi,
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(-N[0] * cos_th + N[1] * cos_phi * sin_th)/cos_phi]
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for i in range(2):
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En_fft[i][cos_phi == 0] = 0
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Hn_fft[i][cos_phi == 0] = 0
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E_near = [ifftshift(ifft2(ifftshift(Ei), s=padded_size)) for Ei in En_fft]
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H_near = [ifftshift(ifft2(ifftshift(Hi), s=padded_size)) for Hi in Hn_fft]
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dx = 2 * pi / (s[0] * dkx)
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dy = 2 * pi / (s[0] * dky)
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outputs = {
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'E': E_near,
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'H': H_near,
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'dx': dx,
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'dy': dy,
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}
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return outputs
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