def block_rotation_matrix_D_svwf(l_max, m_max, alpha, beta, gamma, wdsympy=False):
"""Rotation matrix for the rotation of SVWFs between the labratory
coordinate system (L) and a rotated coordinate system (R)
Args:
l_max (int): Maximal multipole degree
m_max (int): Maximal multipole order
alpha (float): First Euler angle, rotation around z-axis, in rad
beta (float): Second Euler angle, rotation around y'-axis in rad
gamma (float): Third Euler angle, rotation around z''-axis in rad
wdsympy (bool): If True, Wigner-d-functions come from the sympy toolbox
Returns:
rotation matrix of dimension [blocksize, blocksize]
"""
b_size = flds.blocksize(l_max, m_max)
rotation_matrix = np.zeros([b_size, b_size], dtype=complex)
for l in range(l_max + 1):
mstop = min(l, m_max)
for m1 in range(-mstop, mstop + 1):
for m2 in range(-mstop, mstop + 1):
rotation_matrix_coefficient = mathfunc.wigner_D(l, m1, m2, alpha, beta, gamma, wdsympy)
for tau in range(2):
n1 = flds.multi_to_single_index(tau, l, m1, l_max, m_max)
n2 = flds.multi_to_single_index(tau, l, m2, l_max, m_max)
rotation_matrix[n1, n2] = rotation_matrix_coefficient
return rotation_matrix
def relative_difference_Tmatrices(Tmat_s, lmax_s, mmax_s, Tmat_l,
lmax_l, mmax_l):
n_list_s = np.zeros(len(Tmat_s), dtype=int)
n_list_l = np.zeros(len(Tmat_s), dtype=int)
idx = 0
for tau in range(2):
for l in range(1, lmax_s + 1):
for m in range(np.max([-l, -mmax_s]),
np.min([l, mmax_s]) + 1):
n_list_s[idx] = flds.multi_to_single_index(
tau, l, m, lmax_s, mmax_s)
n_list_l[idx] = flds.multi_to_single_index(
tau, l, m, lmax_l, mmax_l)
idx += 1
row, column = np.meshgrid(n_list_s, n_list_s)
row2, column2 = np.meshgrid(n_list_l, n_list_l)
TMat_temp = np.zeros([len(Tmat_l), len(Tmat_l)], dtype=complex)
TMat_temp[row2, column2] = Tmat_s[row, column]
return np.linalg.norm(Tmat_l - TMat_temp) / np.linalg.norm(Tmat_l)
def pwe_to_swe_conversion(pwe, l_max, m_max, reference_point):
"""Convert plane wave expansion object to a spherical wave expansion object.
Args:
pwe (PlaneWaveExpansion): Plane wave expansion to be converted
l_max (int): Maximal multipole degree of spherical wave expansion
m_max (int): Maximal multipole order of spherical wave expansion
reference_point (list): Coordinates of reference point in the format [x, y, z]
Returns:
SphericalWaveExpansion object.
"""
if reference_point[2] < pwe.lower_z or reference_point[2] > pwe.upper_z:
raise ValueError('reference point not inside domain of pwe validity')
swe = fex.SphericalWaveExpansion(k=pwe.k, l_max=l_max, m_max=m_max, kind='regular',
reference_point=reference_point, lower_z=pwe.lower_z,
upper_z=pwe.upper_z)
kpgrid = pwe.k_parallel_grid()
agrid = pwe.azimuthal_angle_grid()
kx = kpgrid * np.cos(agrid)
ky = kpgrid * np.sin(agrid)
kz = pwe.k_z_grid()
kzvec = pwe.k_z()
kvec = np.array([kx, ky, kz])
rswe_mn_rpwe = np.array(reference_point) - np.array(pwe.reference_point)
# phase factor for the translation of the reference point from rvec_iS to rvec_S
ejkriSS = np.exp(1j * (kvec[0] * rswe_mn_rpwe[0] + kvec[1] * rswe_mn_rpwe[1] + kvec[2] * rswe_mn_rpwe[2]))
# phase factor times pwe coefficients
gejkriSS = pwe.coefficients * ejkriSS[None, :, :] # indices: pol, jk, ja
ct = kzvec / pwe.k
st = pwe.k_parallel / pwe.k
plm_list, pilm_list, taulm_list = mathfunc.legendre_normalized(ct, st, l_max)
for m in range(-m_max, m_max + 1):
emjma_geijkriSS = np.exp(-1j * m * pwe.azimuthal_angles)[None, None, :] * gejkriSS
for l in range(max(1, abs(m)), l_max + 1):
for tau in range(2):
ak_integrand = np.zeros(kpgrid.shape, dtype=complex)
for pol in range(2):
Bdag = transformation_coefficients_vwf(tau, l, m, pol=pol, pilm_list=pilm_list,
taulm_list=taulm_list, kz=kzvec, dagger=True)
ak_integrand += Bdag[:, None] * emjma_geijkriSS[pol, :, :]
if len(pwe.k_parallel) > 1:
an = np.trapz(np.trapz(ak_integrand, pwe.azimuthal_angles) * pwe.k_parallel, pwe.k_parallel) * 4
else:
an = ak_integrand * 4
swe.coefficients[flds.multi_to_single_index(tau, l, m, swe.l_max, swe.m_max)] = np.squeeze(an)
return swe
def testMulti2SingleSTLM(self):
idcs = []
lmax = 5
mmax = 5
count = 0
for tau in range(2):
for l in range(1, lmax + 1):
for m in range(-l, l + 1):
idcs.append(
flds.multi_to_single_index(tau=tau,
l=l,
m=m,
l_max=lmax,
m_max=mmax))
count += 1
self.assertEqual(idcs, list(range(len(idcs))))
ind_num = flds.blocksize(lmax, mmax)
self.assertEqual(count, ind_num)
idcs = []
lmax = 6
mmax = 3
count = 0
for tau in range(2):
for l in range(1, lmax + 1):
mlim = min(l, mmax)
for m in range(-mlim, mlim + 1):
idcs.append(
flds.multi_to_single_index(tau=tau,
l=l,
m=m,
l_max=lmax,
m_max=mmax))
count += 1
self.assertEqual(idcs, list(range(len(idcs))))
ind_num = flds.blocksize(lmax, mmax)
self.assertEqual(count, ind_num)
def testTmatrix(self):
t = tmt.t_matrix_sphere(omega * n_medium, omega * n_particle, radius, lmax, mmax)
n = flds.multi_to_single_index(tau, l, m, lmax, mmax)
mie = tmt.mie_coefficient(tau, l, omega * n_medium, omega * n_particle, radius)
assert t[n, n] == mie
sphere1 = smuthi.particles.Sphere(radius=100, refractive_index=3, position=[100, 200, 300], l_max=lmax, m_max=mmax)
sphere2 = smuthi.particles.Sphere(radius=100, refractive_index=3, position=[200, -200, 200], l_max=lmax, m_max=mmax)
sphere3 = smuthi.particles.Sphere(radius=200, refractive_index=2+0.2j, position=[200,-200,200], l_max=lmax,
m_max=mmax)
t2 = tmt.t_matrix(vacuum_wavelength=550, n_medium=n_medium, particle=sphere1)
t3 = tmt.t_matrix_sphere(2*np.pi/550 * n_medium, 2*np.pi/550 * 3, 100, lmax, mmax)
np.testing.assert_allclose(t2, t3)
def index(self, i, tau, l, m):
r"""
Args:
i (int): particle number
tau (int): spherical polarization index
l (int): multipole degree
m (int): multipole order
Returns:
Position in a system vector that corresponds to the :math:`(\tau, l, m)` coefficient of the i-th particle.
"""
blocksizes = [
flds.blocksize(particle.l_max, particle.m_max)
for particle in self.particle_list[:i]
]
return sum(blocksizes) + flds.multi_to_single_index(
tau, l, m, self.particle_list[i].l_max,
self.particle_list[i].m_max)
def internal_field_piecewise_expansion(vacuum_wavelength, particle_list, layer_system):
"""Compute a piecewise field expansion of the internal field of spheres.
Args:
vacuum_wavelength (float): vacuum wavelength
particle_list (list): list of smuthi.particles.Particle objects
layer_system (smuthi.layers.LayerSystem): stratified medium
Returns:
internal field as smuthi.field_expansion.PiecewiseFieldExpansion object
"""
intfld = fldex.PiecewiseFieldExpansion()
for particle in particle_list:
if type(particle).__name__ == 'Sphere':
i_part = layer_system.layer_number(particle.position[2])
k_medium = flds.angular_frequency(vacuum_wavelength) * layer_system.refractive_indices[i_part]
k_particle = flds.angular_frequency(vacuum_wavelength) * particle.refractive_index
internal_field = fldex.SphericalWaveExpansion(
k_particle,
l_max=particle.l_max,
m_max=particle.m_max,
kind='regular',
reference_point=particle.position)
internal_field.validity_conditions.append(particle.is_inside)
for tau in range(2):
for l in range(1, particle.l_max+1):
for m in range(-l, l+1):
n = flds.multi_to_single_index(tau, l, m, particle.l_max, particle.m_max)
b_to_c = (tmt.internal_mie_coefficient(tau, l, k_medium, k_particle, particle.radius)
/ tmt.mie_coefficient(tau, l, k_medium, k_particle, particle.radius))
internal_field.coefficients[n] = particle.scattered_field.coefficients[n] * b_to_c
intfld.expansion_list.append(internal_field)
particle.internal_field = internal_field
return intfld
def t_matrix_sphere(k_medium, k_particle, radius, l_max, m_max):
"""T-matrix of a spherical scattering object.
Args:
k_medium (float or complex): Wavenumber in surrounding medium (inverse length unit)
k_particle (float or complex): Wavenumber inside sphere (inverse length unit)
radius (float): Radius of sphere (length unit)
l_max (int): Maximal multipole degree
m_max (int): Maximal multipole order
Returns:
T-matrix as ndarray
"""
t = np.zeros((flds.blocksize(l_max, m_max), flds.blocksize(l_max, m_max)),
dtype=complex)
for tau in range(2):
for m in range(-m_max, m_max + 1):
for l in range(max(1, abs(m)), l_max + 1):
n = flds.multi_to_single_index(tau, l, m, l_max, m_max)
t[n, n] = mie_coefficient(tau, l, k_medium, k_particle, radius)
return t
def __init__(self,
vacuum_wavelength,
particle_list,
layer_system,
k_parallel='default',
resolution=None,
interpolator_kind='linear'):
z_list = [particle.position[2] for particle in particle_list]
assert z_list.count(z_list[0]) == len(z_list)
CouplingMatrixRadialLookup.__init__(self, vacuum_wavelength,
particle_list, layer_system,
k_parallel, resolution)
x_array = np.array(
[particle.position[0] for particle in particle_list])
y_array = np.array(
[particle.position[1] for particle in particle_list])
self.particle_rho_array = np.sqrt(
(x_array[:, None] - x_array[None, :])**2 +
(y_array[:, None] - y_array[None, :])**2)
self.particle_phi_array = np.arctan2(
y_array[:, None] - y_array[None, :],
x_array[:, None] - x_array[None, :])
# contains for each n all positions in the large system arrays that correspond to n:
self.system_vector_index_list = [[] for i in range(self.blocksize)]
# same size as system_vector_index_list, contains the according particle numbers:
self.particle_number_list = [[] for i in range(self.blocksize)]
self.m_list = [None for i in range(self.blocksize)]
for i, particle in enumerate(particle_list):
for m in range(-particle.m_max, particle.m_max + 1):
for l in range(max(1, abs(m)), particle.l_max + 1):
for tau in range(2):
n_lookup = flds.multi_to_single_index(tau=tau,
l=l,
m=m,
l_max=self.l_max,
m_max=self.m_max)
self.system_vector_index_list[n_lookup].append(
self.index(i, tau, l, m))
self.particle_number_list[n_lookup].append(i)
self.m_list[n_lookup] = m
for n in range(self.blocksize):
self.system_vector_index_list[n] = np.array(
self.system_vector_index_list[n])
self.particle_number_list[n] = np.array(
self.particle_number_list[n])
lookup = [[None for i in range(self.blocksize)]
for i2 in range(self.blocksize)]
for n1 in range(self.blocksize):
for n2 in range(self.blocksize):
lookup[n1][n2] = scipy.interpolate.interp1d(
x=self.radial_distance_array,
y=self.lookup_table[:, n1, n2],
kind=interpolator_kind,
axis=-1,
assume_sorted=True)
def matvec(in_vec):
out_vec = np.zeros(shape=in_vec.shape, dtype=complex)
for n1 in range(self.blocksize):
i1 = self.particle_number_list[n1]
idx1 = self.system_vector_index_list[n1]
m1 = self.m_list[n1]
for n2 in range(self.blocksize):
i2 = self.particle_number_list[n2]
idx2 = self.system_vector_index_list[n2]
m2 = self.m_list[n2]
rho = self.particle_rho_array[i1[:, None], i2[None, :]]
phi = self.particle_phi_array[i1[:, None], i2[None, :]]
M = lookup[n1][n2](rho)
M = M * np.exp(1j * (m2 - m1) * phi)
out_vec[idx1] += M.dot(in_vec[idx2])
return out_vec
self.linear_operator = scipy.sparse.linalg.LinearOperator(
shape=self.shape, matvec=matvec, dtype=complex)
def __init__(self,
vacuum_wavelength,
particle_list,
layer_system,
k_parallel='default',
resolution=None,
cuda_blocksize=None,
interpolator_kind='linear'):
if cuda_blocksize is None:
cuda_blocksize = cu.default_blocksize
CouplingMatrixRadialLookup.__init__(self, vacuum_wavelength,
particle_list, layer_system,
k_parallel, resolution)
sys.stdout.write('Prepare CUDA kernel and device lookup data ... ')
sys.stdout.flush()
start_time = time.time()
if interpolator_kind == 'linear':
coupling_source = cusrc.linear_radial_lookup_source % (
self.blocksize, self.shape[0],
self.radial_distance_array.min(), resolution)
elif interpolator_kind == 'cubic':
coupling_source = cusrc.cubic_radial_lookup_source % (
self.blocksize, self.shape[0],
self.radial_distance_array.min(), resolution)
coupling_function = cu.SourceModule(coupling_source).get_function(
"coupling_kernel")
n_lookup_array = np.zeros(self.shape[0], dtype=np.uint32)
m_particle_array = np.zeros(self.shape[0], dtype=np.float32)
x_array = np.zeros(self.shape[0], dtype=np.float32)
y_array = np.zeros(self.shape[0], dtype=np.float32)
i_particle = 0
for i, particle in enumerate(particle_list):
for m in range(-particle.m_max, particle.m_max + 1):
for l in range(max(1, abs(m)), particle.l_max + 1):
for tau in range(2):
# idx = self.index(i, tau, l, m)
i_taulm = flds.multi_to_single_index(
tau, l, m, particle.l_max, particle.m_max)
idx = i_particle + i_taulm
n_lookup_array[idx] = flds.multi_to_single_index(
tau, l, m, self.l_max, self.m_max)
m_particle_array[idx] = m
# scale the x and y position to the lookup resolution:
x_array[idx] = particle.position[0]
y_array[idx] = particle.position[1]
i_particle += flds.blocksize(particle.l_max, particle.m_max)
# lookup as numpy array in required shape
re_lookup = self.lookup_table.real.astype(np.float32)
im_lookup = self.lookup_table.imag.astype(np.float32)
# transfer data to gpu
n_lookup_array_d = cu.gpuarray.to_gpu(n_lookup_array)
m_particle_array_d = cu.gpuarray.to_gpu(m_particle_array)
x_array_d = cu.gpuarray.to_gpu(x_array)
y_array_d = cu.gpuarray.to_gpu(y_array)
re_lookup_d = cu.gpuarray.to_gpu(re_lookup)
im_lookup_d = cu.gpuarray.to_gpu(im_lookup)
sys.stdout.write('done | elapsed: ' +
str(int(time.time() - start_time)) + 's\n')
sys.stdout.flush()
cuda_gridsize = (self.shape[0] + cuda_blocksize - 1) // cuda_blocksize
def matvec(in_vec):
re_in_vec_d = cu.gpuarray.to_gpu(np.float32(in_vec.real))
im_in_vec_d = cu.gpuarray.to_gpu(np.float32(in_vec.imag))
re_result_d = cu.gpuarray.zeros(in_vec.shape, dtype=np.float32)
im_result_d = cu.gpuarray.zeros(in_vec.shape, dtype=np.float32)
coupling_function(n_lookup_array_d.gpudata,
m_particle_array_d.gpudata,
x_array_d.gpudata,
y_array_d.gpudata,
re_lookup_d.gpudata,
im_lookup_d.gpudata,
re_in_vec_d.gpudata,
im_in_vec_d.gpudata,
re_result_d.gpudata,
im_result_d.gpudata,
block=(cuda_blocksize, 1, 1),
grid=(cuda_gridsize, 1))
return re_result_d.get() + 1j * im_result_d.get()
self.linear_operator = scipy.sparse.linalg.LinearOperator(
shape=self.shape, matvec=matvec, dtype=complex)
def __init__(self,
vacuum_wavelength,
particle_list,
layer_system,
k_parallel='default',
resolution=None,
interpolator_kind='cubic'):
if interpolator_kind == 'cubic':
interpolation_order = 3
else:
interpolation_order = 1
CouplingMatrixVolumeLookup.__init__(self, vacuum_wavelength,
particle_list, layer_system,
k_parallel, resolution)
x_array = np.array(
[particle.position[0] for particle in particle_list])
y_array = np.array(
[particle.position[1] for particle in particle_list])
z_array = np.array(
[particle.position[2] for particle in particle_list])
self.particle_rho_array = np.sqrt(
(x_array[:, None] - x_array[None, :])**2 +
(y_array[:, None] - y_array[None, :])**2)
self.particle_phi_array = np.arctan2(
y_array[:, None] - y_array[None, :],
x_array[:, None] - x_array[None, :])
self.particle_sz_array = z_array[:, None] + z_array[None, :]
self.particle_dz_array = z_array[:, None] - z_array[None, :]
# contains for each n all positions in the large system arrays that correspond to n:
self.system_vector_index_list = [[] for i in range(self.blocksize)]
# same size as system_vector_index_list, contains the according particle numbers:
self.particle_number_list = [[] for i in range(self.blocksize)]
self.m_list = [None for i in range(self.blocksize)]
for i, particle in enumerate(particle_list):
for m in range(-particle.m_max, particle.m_max + 1):
for l in range(max(1, abs(m)), particle.l_max + 1):
for tau in range(2):
n_lookup = flds.multi_to_single_index(tau=tau,
l=l,
m=m,
l_max=self.l_max,
m_max=self.m_max)
self.system_vector_index_list[n_lookup].append(
self.index(i, tau, l, m))
self.particle_number_list[n_lookup].append(i)
self.m_list[n_lookup] = m
for n in range(self.blocksize):
self.system_vector_index_list[n] = np.array(
self.system_vector_index_list[n])
self.particle_number_list[n] = np.array(
self.particle_number_list[n])
self.lookup_plus_real = [[None for i in range(self.blocksize)]
for i2 in range(self.blocksize)]
self.lookup_plus_imag = [[None for i in range(self.blocksize)]
for i2 in range(self.blocksize)]
self.lookup_minus_real = [[None for i in range(self.blocksize)]
for i2 in range(self.blocksize)]
self.lookup_minus_imag = [[None for i in range(self.blocksize)]
for i2 in range(self.blocksize)]
for n1 in range(self.blocksize):
for n2 in range(self.blocksize):
self.lookup_plus_real[n1][
n2] = scipy.interpolate.RectBivariateSpline(
x=self.rho_array,
y=self.sum_z_array,
z=self.lookup_table_plus[:, :, n1, n2].real,
kx=interpolation_order,
ky=interpolation_order)
self.lookup_plus_imag[n1][
n2] = scipy.interpolate.RectBivariateSpline(
x=self.rho_array,
y=self.sum_z_array,
z=self.lookup_table_plus[:, :, n1, n2].imag,
kx=interpolation_order,
ky=interpolation_order)
self.lookup_minus_real[n1][
n2] = scipy.interpolate.RectBivariateSpline(
x=self.rho_array,
y=self.diff_z_array,
z=self.lookup_table_minus[:, :, n1, n2].real,
kx=interpolation_order,
ky=interpolation_order)
self.lookup_minus_imag[n1][
n2] = scipy.interpolate.RectBivariateSpline(
x=self.rho_array,
y=self.diff_z_array,
z=self.lookup_table_minus[:, :, n1, n2].imag,
kx=interpolation_order,
ky=interpolation_order)
def matvec(in_vec):
out_vec = np.zeros(shape=in_vec.shape, dtype=complex)
for n1 in range(self.blocksize):
i1 = self.particle_number_list[n1]
idx1 = self.system_vector_index_list[n1]
m1 = self.m_list[n1]
for n2 in range(self.blocksize):
i2 = self.particle_number_list[n2]
idx2 = self.system_vector_index_list[n2]
m2 = self.m_list[n2]
rho = self.particle_rho_array[i1[:, None], i2[None, :]]
phi = self.particle_phi_array[i1[:, None], i2[None, :]]
sz = self.particle_sz_array[i1[:, None], i2[None, :]]
dz = self.particle_dz_array[i1[:, None], i2[None, :]]
Mpl = self.lookup_plus_real[n1][n2].ev(
rho,
sz) + 1j * self.lookup_plus_imag[n1][n2].ev(rho, sz)
Mmn = self.lookup_minus_real[n1][n2].ev(
rho,
dz) + 1j * self.lookup_minus_imag[n1][n2].ev(rho, dz)
M = (Mpl + Mmn) * np.exp(1j * (m2 - m1) * phi)
out_vec[idx1] += M.dot(in_vec[idx2])
return out_vec
self.linear_operator = scipy.sparse.linalg.LinearOperator(
shape=self.shape, matvec=matvec, dtype=complex)
def taxsym_read_tmatrix(filename, l_max, m_max):
"""Export TAXSYM.f90 output to SMUTHI T-matrix.
.. todo:: feedback to adapt particle m_max to nfmds m_max
Args:
filename (str): Name of the file containing the T-matrix output of TAXSYM.f90
l_max (int): Maximal multipole degree
m_max (int): Maximal multipole order
Returns:
T-matrix as numpy.ndarray
"""
with open(nfmds.nfmds_folder + '/TMATFILES/Info' + filename,
'r') as info_file:
info_file_lines = info_file.readlines()
assert 'The scatterer is an axisymmetric particle' in ' '.join(
info_file_lines)
for line in info_file_lines:
if line.split()[0:4] == ['-', 'maximum', 'expansion', 'order,']:
n_rank = int(line.split()[-1][0:-1])
if line.split()[0:5] == ['-', 'number', 'of', 'azimuthal', 'modes,']:
m_rank = int(line.split()[-1][0:-1])
with open(nfmds.nfmds_folder + '/TMATFILES/' + filename, 'r') as tmat_file:
tmat_lines = tmat_file.readlines()
t_nfmds = [[]]
column_index = 0
for line in tmat_lines[3:]:
split_line = line.split()
for i_entry in range(int(len(split_line) / 2)):
if column_index == 2 * n_rank:
t_nfmds.append([])
column_index = 0
t_nfmds[-1].append(
complex(split_line[2 * i_entry]) +
1j * complex(split_line[2 * i_entry + 1]))
column_index += 1
t_matrix = np.zeros(
(flds.blocksize(l_max, m_max), flds.blocksize(l_max, m_max)),
dtype=complex)
for m in range(-m_max, m_max + 1):
n_max_nfmds = n_rank - max(1, abs(m)) + 1
for tau1 in range(2):
for l1 in range(max(1, abs(m)), l_max + 1):
n1 = flds.multi_to_single_index(tau=tau1,
l=l1,
m=m,
l_max=l_max,
m_max=m_max)
l1_nfmds = l1 - max(1, abs(m))
n1_nfmds = 2 * n_rank * abs(m) + tau1 * n_max_nfmds + l1_nfmds
for tau2 in range(2):
for l2 in range(max(1, abs(m)), l_max + 1):
n2 = flds.multi_to_single_index(tau=tau2,
l=l2,
m=m,
l_max=l_max,
m_max=m_max)
l2_nfmds = l2 - max(1, abs(m))
n2_nfmds = tau2 * n_max_nfmds + l2_nfmds
if abs(m) <= m_rank:
if m >= 0:
t_matrix[n1, n2] = t_nfmds[n1_nfmds][n2_nfmds]
else:
t_matrix[n1,
n2] = t_nfmds[n1_nfmds][n2_nfmds] * (
-1)**(tau1 + tau2)
return t_matrix
def direct_coupling_block_pvwf_mediated(vacuum_wavelength, receiving_particle,
emitting_particle, layer_system,
k_parallel):
"""Direct particle coupling matrix :math:`W` for two particles (via plane vector wave functions).
For details, see:
Dominik Theobald et al., Phys. Rev. A 96, 033822, DOI: 10.1103/PhysRevA.96.033822 or arXiv:1708.04808
Args:
vacuum_wavelength (float): Vacuum wavelength :math:`\lambda` (length unit)
receiving_particle (smuthi.particles.Particle): Particle that receives the scattered field
emitting_particle (smuthi.particles.Particle): Particle that emits the scattered field
layer_system (smuthi.layers.LayerSystem): Stratified medium in which the coupling takes place
k_parallel (numpy.array): In-plane wavenumber for plane wave expansion
Returns:
Direct coupling matrix block (numpy array).
"""
if type(receiving_particle).__name__ != 'Spheroid' or type(
emitting_particle).__name__ != 'Spheroid':
raise NotImplementedError(
'plane wave coupling currently implemented only for spheroids')
lmax1 = receiving_particle.l_max
mmax1 = receiving_particle.m_max
assert lmax1 == mmax1, 'PVWF coupling requires lmax == mmax for each particle.'
lmax2 = emitting_particle.l_max
mmax2 = emitting_particle.m_max
assert lmax2 == mmax2, 'PVWF coupling requires lmax == mmax for each particle.'
lmax = max([lmax1, lmax2])
m_max = max([mmax1, mmax2])
blocksize1 = flds.blocksize(lmax1, mmax1)
blocksize2 = flds.blocksize(lmax2, mmax2)
n_medium = layer_system.refractive_indices[layer_system.layer_number(
receiving_particle.position[2])]
# finding the orientation of a plane separating the spheroids
_, _, alpha, beta = spheroids_closest_points(
emitting_particle.semi_axis_a, emitting_particle.semi_axis_c,
emitting_particle.position, emitting_particle.euler_angles,
receiving_particle.semi_axis_a, receiving_particle.semi_axis_c,
receiving_particle.position, receiving_particle.euler_angles)
# positions
r1 = np.array(receiving_particle.position)
r2 = np.array(emitting_particle.position)
r21_lab = r1 - r2 # laboratory coordinate system
# distance vector in rotated coordinate system
r21_rot = np.dot(
np.dot([[np.cos(beta), 0, np.sin(beta)], [0, 1, 0],
[-np.sin(beta), 0, np.cos(beta)]],
[[np.cos(alpha), -np.sin(alpha), 0],
[np.sin(alpha), np.cos(alpha), 0], [0, 0, 1]]), r21_lab)
rho21 = (r21_rot[0]**2 + r21_rot[1]**2)**0.5
phi21 = np.arctan2(r21_rot[1], r21_rot[0])
z21 = r21_rot[2]
# wavenumbers
omega = flds.angular_frequency(vacuum_wavelength)
k = omega * n_medium
kz = flds.k_z(k_parallel=k_parallel,
vacuum_wavelength=vacuum_wavelength,
refractive_index=n_medium)
if z21 < 0:
kz_var = -kz
else:
kz_var = kz
# Bessel lookup
bessel_list = []
for dm in range(mmax1 + mmax2 + 1):
bessel_list.append(scipy.special.jn(dm, k_parallel * rho21))
# legendre function lookups
ct = kz_var / k
st = k_parallel / k
_, pilm_list, taulm_list = sf.legendre_normalized(ct, st, lmax)
# initialize result
w = np.zeros((blocksize1, blocksize2), dtype=complex)
# prefactor
const_arr = k_parallel / (kz * k) * np.exp(1j * (kz_var * z21))
for m1 in range(-mmax1, mmax1 + 1):
for m2 in range(-mmax2, mmax2 + 1):
jmm_eimphi_bessel = 4 * 1j**abs(m2 - m1) * np.exp(
1j * phi21 * (m2 - m1)) * bessel_list[abs(m2 - m1)]
prefactor = const_arr * jmm_eimphi_bessel
for l1 in range(max(1, abs(m1)), lmax1 + 1):
for l2 in range(max(1, abs(m2)), lmax2 + 1):
for tau1 in range(2):
n1 = flds.multi_to_single_index(
tau1, l1, m1, lmax1, mmax1)
for tau2 in range(2):
n2 = flds.multi_to_single_index(
tau2, l2, m2, lmax2, mmax2)
for pol in range(2):
B_dag = trf.transformation_coefficients_vwf(
tau1,
l1,
m1,
pol,
pilm_list=pilm_list,
taulm_list=taulm_list,
dagger=True)
B = trf.transformation_coefficients_vwf(
tau2,
l2,
m2,
pol,
pilm_list=pilm_list,
taulm_list=taulm_list,
dagger=False)
integrand = prefactor * B * B_dag
w[n1, n2] += np.trapz(integrand, k_parallel)
rot_mat_1 = trf.block_rotation_matrix_D_svwf(lmax1, mmax1, 0, beta, alpha)
rot_mat_2 = trf.block_rotation_matrix_D_svwf(lmax2, mmax2, -alpha, -beta,
0)
return np.dot(np.dot(np.transpose(rot_mat_1), w), np.transpose(rot_mat_2))
def direct_coupling_block(vacuum_wavelength, receiving_particle,
emitting_particle, layer_system):
"""Direct particle coupling matrix :math:`W` for two particles. This routine is explicit, but slow.
Args:
vacuum_wavelength (float): Vacuum wavelength :math:`\lambda` (length unit)
receiving_particle (smuthi.particles.Particle): Particle that receives the scattered field
emitting_particle (smuthi.particles.Particle): Particle that emits the scattered field
layer_system (smuthi.layers.LayerSystem): Stratified medium in which the coupling takes place
Returns:
Direct coupling matrix block as numpy array.
"""
omega = flds.angular_frequency(vacuum_wavelength)
# index specs
lmax1 = receiving_particle.l_max
mmax1 = receiving_particle.m_max
lmax2 = emitting_particle.l_max
mmax2 = emitting_particle.m_max
blocksize1 = flds.blocksize(lmax1, mmax1)
blocksize2 = flds.blocksize(lmax2, mmax2)
# initialize result
w = np.zeros((blocksize1, blocksize2), dtype=complex)
# check if particles are in same layer
rS1 = receiving_particle.position
rS2 = emitting_particle.position
iS1 = layer_system.layer_number(rS1[2])
iS2 = layer_system.layer_number(rS2[2])
if iS1 == iS2 and not emitting_particle == receiving_particle:
k = omega * layer_system.refractive_indices[iS1]
dx = rS1[0] - rS2[0]
dy = rS1[1] - rS2[1]
dz = rS1[2] - rS2[2]
d = np.sqrt(dx**2 + dy**2 + dz**2)
cos_theta = dz / d
sin_theta = np.sqrt(dx**2 + dy**2) / d
phi = np.arctan2(dy, dx)
# spherical functions
bessel_h = [
sf.spherical_hankel(n, k * d) for n in range(lmax1 + lmax2 + 1)
]
legendre, _, _ = sf.legendre_normalized(cos_theta, sin_theta,
lmax1 + lmax2)
# the particle coupling operator is the transpose of the SVWF translation operator
# therefore, (l1,m1) and (l2,m2) are interchanged:
for m1 in range(-mmax1, mmax1 + 1):
for m2 in range(-mmax2, mmax2 + 1):
eimph = np.exp(1j * (m2 - m1) * phi)
for l1 in range(max(1, abs(m1)), lmax1 + 1):
for l2 in range(max(1, abs(m2)), lmax2 + 1):
A, B = complex(0), complex(0)
for ld in range(max(abs(l1 - l2), abs(m1 - m2)), l1 +
l2 + 1): # if ld<abs(m1-m2) then P=0
a5, b5 = trf.ab5_coefficients(l2, m2, l1, m1, ld)
A += a5 * bessel_h[ld] * legendre[ld][abs(m1 - m2)]
B += b5 * bessel_h[ld] * legendre[ld][abs(m1 - m2)]
A, B = eimph * A, eimph * B
for tau1 in range(2):
n1 = flds.multi_to_single_index(
tau1, l1, m1, lmax1, mmax1)
for tau2 in range(2):
n2 = flds.multi_to_single_index(
tau2, l2, m2, lmax2, mmax2)
if tau1 == tau2:
w[n1, n2] = A
else:
w[n1, n2] = B
return w