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functions.py
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functions.py
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import numpy as np
import scipy as sp
import scipy.linalg
import scipy.special
import scipy.optimize
import scipy.sparse
import sksparse.cholmod
from pathos.multiprocessing import ProcessingPool as Pool
def sinc(x):
if np.isclose(x,0):
return 1.0
else:
return np.sin(x)/x
def getAnglePlane_v2(v1, v2, n):
'''
Calculate rotation angle used in A matrix calculation
See Appendix D for more inforamtion
'''
# Special cases
cosine = np.dot(v1,v2)
if cosine >= 1 or np.isclose(cosine, 1):
return 0
elif cosine <= -1 or np.isclose(cosine, -1):
return np.pi
theta = np.arccos(cosine)
# check if n is in the same direction as v1xv2
k = np.cross(v1, v2)
k = k/np.linalg.norm(k)
alignment = np.dot(k,n)
if(np.isclose(alignment, 1)):
alpha = theta
else:
alpha = -theta
return alpha
def getT(k1, k2, params, N_lim = 5):
'''
Calculate far-field single scattering matrix (A) using Mie theory
k1 = incident wavevector
k2 = outgoing wavevector
params must contain:
'x': size parameter
'm': relative refractive index
'type': scattering type (rayleigh, mie or chiral)
if 'type' is 'chiral' then params must also contain 'mL' and 'mR',
the relative refractive indices for left and right handed circularly polarized light
N_lim is the number of terms used in calculating the scattering coefficients. Increase for greater accuracy.
'''
k1 = k1/np.linalg.norm(k1)
k2 = k2/np.linalg.norm(k2)
for i, component in enumerate(k1):
if np.isclose(component, 0):
k1[i] = 0
for i, component in enumerate(k2):
if np.isclose(component, 0):
k2[i] = 0
x = params['x']
m = params['m']
M_type = params['type']
if M_type == 'chiral':
mL = params['mL']
mR = params['mR']
theta_i = np.arccos(k1[2])
phi_i = np.arctan2(k1[1], k1[0])
theta_s = np.arccos(k2[2])
phi_s = np.arctan2(k2[1], k2[0])
cos_theta_scattering = np.dot(k1, k2)
if np.allclose(k1, np.array([0,0,-1])):
e_theta_i = np.array([1,0,0])
e_phi_i = np.array([0,-1,0])
else:
e_theta_i = np.array([np.cos(theta_i)*np.cos(phi_i), np.cos(theta_i)*np.sin(phi_i), -np.sin(theta_i)])
e_phi_i = np.array([-np.sin(phi_i), np.cos(phi_i), 0])
if np.allclose(k2, np.array([0,0,-1])):
e_theta_s = np.array([1,0,0])
#e_phi_s = np.array([0,-1,0])
else:
e_theta_s = np.array([np.cos(theta_s)*np.cos(phi_s), np.cos(theta_s)*np.sin(phi_s), -np.sin(theta_s)])
#e_phi_s = np.array([-np.sin(phi_s), np.cos(phi_s), 0])
if np.allclose(k1, k2):
e_par_i = e_theta_i
e_par_s = e_theta_s
e_per = e_phi_i
alpha1 = 0
alpha2 = 0
elif np.allclose(k1, -k2):
e_par_i = e_theta_i
e_par_s = -e_theta_s
e_per = e_phi_i
alpha1 = 0
alpha2 = np.pi
else:
e_per = np.cross(k1, k2)
e_per = e_per/np.linalg.norm(e_per)
e_par_i = np.cross(e_per,k1)
e_par_i = e_par_i/np.linalg.norm(e_par_i)
e_par_s = np.cross(e_per,k2)
e_par_s = e_par_s/np.linalg.norm(e_par_s)
alpha1 = getAnglePlane_v2(e_theta_i, e_par_i, k1)
alpha2 = getAnglePlane_v2(e_par_s, e_theta_s, k2)
T1 = np.array([[np.cos(alpha1), np.sin(alpha1)],[-np.sin(alpha1), np.cos(alpha1)]])
T2 = np.array([[1,0],[0,-1]])
T4 = np.array([[1,0],[0,-1]])
T5 = np.array([[np.cos(alpha2), np.sin(alpha2)],[-np.sin(alpha2), np.cos(alpha2)]])
if M_type == 'rayleigh':
a1 = -1j*x**3*(m**2 - 1)/(m**2 + 2)
S1 = a1*1
S2 = a1*cos_theta_scattering
T3 = np.array([[S2, 0],[0, S1]])
elif M_type == 'mie':
mu = np.dot(k1,k2)
psi_x = sp.special.riccati_jn(N_lim, x)
psi_mx = sp.special.riccati_jn(N_lim, m*x)
phi_x = sp.special.riccati_yn(N_lim, x)
xi_x = [psi + 1j*phi for psi, phi in zip(psi_x, phi_x)]
S1 = 0
S2 = 0
for n in range(1, N_lim):
a = (m*psi_mx[0][n]*psi_x[1][n] - psi_x[0][n]*psi_mx[1][n]) / (m*psi_mx[0][n]*xi_x[1][n] - xi_x[0][n]*psi_mx[1][n])
b = (psi_mx[0][n]*psi_x[1][n] - m*psi_x[0][n]*psi_mx[1][n]) / (psi_mx[0][n]*xi_x[1][n] - m*xi_x[0][n]*psi_mx[1][n])
S1 = S1 + (2*n+1)/(n*(n+1))*(a*pi(n,mu) + b*tau(n,mu))
S2 = S2 + (2*n+1)/(n*(n+1))*(a*tau(n,mu) + b*pi(n,mu))
T3 = np.array([[S2, 0],[0, S1]])
elif M_type == 'chiral':
mu = np.dot(k1,k2)
psi_x = sp.special.riccati_jn(N_lim, x)
psi_mLx = sp.special.riccati_jn(N_lim, mL*x)
psi_mRx = sp.special.riccati_jn(N_lim, mR*x)
phi_x = sp.special.riccati_yn(N_lim, x)
xi_x = [psi + 1j*phi for psi, phi in zip(psi_x, phi_x)]
S1 = 0
S2 = 0
S3 = 0
for n in range(1, N_lim):
new_pi = pi(n,mu)
new_tau = tau(n,mu)
WL = m*psi_mLx[0][n]*xi_x[1][n] - xi_x[0][n]*psi_mLx[1][n]
WR = m*psi_mRx[0][n]*xi_x[1][n] - xi_x[0][n]*psi_mRx[1][n]
VL = psi_mLx[0][n]*xi_x[1][n] - m*xi_x[0][n]*psi_mLx[1][n]
VR = psi_mRx[0][n]*xi_x[1][n] - m*xi_x[0][n]*psi_mRx[1][n]
AL = m*psi_mLx[0][n]*psi_x[1][n] - psi_x[0][n]*psi_mLx[1][n]
AR = m*psi_mRx[0][n]*psi_x[1][n] - psi_x[0][n]*psi_mRx[1][n]
BL = psi_mLx[0][n]*psi_x[1][n] - m*psi_x[0][n]*psi_mLx[1][n]
BR = psi_mRx[0][n]*psi_x[1][n] - m*psi_x[0][n]*psi_mRx[1][n]
a = (VR*AL + VL*AR)/(WL*VR + VL*WR)
b = (WL*BR + WR*BL)/(WL*VR + VL*WR)
c = 1j*(WR*AL - WL*AR)/(WL*VR + VL*WR)
S1 = S1 + (2*n+1)/(n*(n+1))*(a*new_pi + b*new_tau)
S2 = S2 + (2*n+1)/(n*(n+1))*(a*new_tau + b*new_pi)
S3 = S3 + (2*n+1)/(n*(n+1))*c*(new_pi + new_tau)
T3 = np.array([[S2, S3],[-S3, S1]])
T = -T5@T4@T3@T2@T1
return T
def Rsym(M):
'''
Returns the 'reciprocal operator' applied to a matrix as defined in
https://journals.aps.org/prresearch/abstract/10.1103/PhysRevResearch.3.013129
'''
x, y = np.shape(M)
b = np.ones((y,x), dtype = np.complex128)
b[1::2,::2] = -1.0
b[::2,1::2] = -1.0
out = np.multiply(M.T,b)
return out
def mie_cross_section(x, m, k, N_lim = 20):
psi_x = sp.special.riccati_jn(N_lim, x)
psi_mx = sp.special.riccati_jn(N_lim, m*x)
phi_x = sp.special.riccati_yn(N_lim, x)
xi_x = [psi + 1j*phi for psi, phi in zip(psi_x, phi_x)]
C_sca = 0
C_ext = 0
for n in range(1, N_lim):
a = (m*psi_mx[0][n]*psi_x[1][n] - psi_x[0][n]*psi_mx[1][n]) / (m*psi_mx[0][n]*xi_x[1][n] - xi_x[0][n]*psi_mx[1][n])
b = (psi_mx[0][n]*psi_x[1][n] - m*psi_x[0][n]*psi_mx[1][n]) / (psi_mx[0][n]*xi_x[1][n] - m*xi_x[0][n]*psi_mx[1][n])
C_sca = C_sca + (2*n + 1)*(np.absolute(a)**2 + np.absolute(b)**2)
C_ext = C_ext + (2*n + 1)*np.real(a+b)
C_sca = 2*np.pi * C_sca / k**2
C_ext = 2*np.pi * C_ext / k**2
return C_sca, C_ext
def M2S(M):
'''
Converts a transfer matrix into a scattering matrix
'''
N, N = np.shape(M)
N = int(N/4)
alpha = M[0:2*N, 0:2*N]
gamma = M[2*N:4*N, 0:2*N]
delta = M[2*N:4*N, 2*N:4*N]
beta = M[0:2*N, 2*N:4*N]
deltainv = sp.linalg.inv(delta)
r = -deltainv@gamma
t = alpha - beta@deltainv@gamma
t2 = deltainv
r2 = beta@deltainv
S = np.block([[r, t2],[t, r2]])
return S
# Returns the scattering matrix for a system composed of S1 then S2
def S_cascade(S1,S2,sigmap):
'''
Calculates the scattering matrix of a system composed of two contiguous scattering media
with scattering matrices S1 and S2
See appendix B for more information
'''
N, N = np.shape(S1)
size = int(N/2)
r_1 = S1[0:size, 0:size]
t_1 = S1[size:N, 0:size]
t2_1 = S1[0:size, size:N]
r2_1 = S1[size:N, size:N]
r_2 = S2[0:size, 0:size]
t_2 = S2[size:N, 0:size]
t2_2 = S2[0:size, size:N]
r2_2 = S2[size:N, size:N]
Q = sp.linalg.inv(np.identity(size) - r2_1@r_2)
r = r_1 + t2_1@r_2@Q@t_1
r2 = r2_2 + t_2@Q@r2_1@t2_2
t = t_2@Q@t_1
t2 = sigmap@Rsym(t)@sigmap
S = np.block([[r,t2],[t,r2]])
return S
def S_sampler_svd(statistics, sigma_p):
'''
Randomly generates a scattering matrix.
Statistics is the output of the get_statistics function
'''
mean_t = statistics['mean t']
mean_r = statistics['mean r']
mean_r2 = statistics['mean r2']
chol_t = statistics['chol t']
chol_r = statistics['chol r']
chol_r2 = statistics['chol r2']
n_modes = len(mean_t)
t = np.zeros((2*n_modes, 2*n_modes), dtype=np.complex128)
r = np.zeros((2*n_modes, 2*n_modes), dtype=np.complex128)
r2 = np.zeros((2*n_modes, 2*n_modes), dtype=np.complex128)
size = np.shape(chol_t)[0]
random_numbers = np.random.randn(size)
modified_t = chol_t@random_numbers
reals_t = modified_t[0:int(size/2)]
imags_t = modified_t[int(size/2):int(size)]
size = np.shape(chol_r)[0]
random_numbers = np.random.randn(size)
modified_r = chol_r@random_numbers
reals_r = modified_r[0:int(size/2)]
imags_r = modified_r[int(size/2):int(size)]
size = np.shape(chol_r2)[0]
random_numbers = np.random.randn(size)
modified_r2 = chol_r2@random_numbers
reals_r2 = modified_r2[0:int(size/2)]
imags_r2 = modified_r2[int(size/2):int(size)]
#helper function
def h(i, n_modes):
row = np.floor(i/n_modes)
col = i%n_modes
return int(row), int(col)
for i in range(n_modes**2):
row, col = h(i,n_modes)
if row == col:
t[2*row, 2*col] = mean_t[row][0,0]
t[2*row, 2*col+1] = mean_t[row][0,1]
t[2*row+1, 2*col] = mean_t[row][1,0]
t[2*row+1, 2*col+1] = mean_t[row][1,1]
r[2*row, 2*col] = reals_r[4*i] + 1j*imags_r[4*i] + mean_r[row][0,0]
r[2*row, 2*col+1] = reals_r[4*i+1] + 1j*imags_r[4*i+1] + mean_r[row][0,1]
r[2*row+1, 2*col] = reals_r[4*i+2] + 1j*imags_r[4*i+2] + mean_r[row][1,0]
r[2*row+1, 2*col+1] = reals_r[4*i+3] + 1j*imags_r[4*i+3] + mean_r[row][1,1]
r2[2*row, 2*col] = reals_r2[4*i] + 1j*imags_r2[4*i] + mean_r2[row][0,0]
r2[2*row, 2*col+1] = reals_r2[4*i+1] + 1j*imags_r2[4*i+1] + mean_r2[row][0,1]
r2[2*row+1, 2*col] = reals_r2[4*i+2] + 1j*imags_r2[4*i+2] + mean_r2[row][1,0]
r2[2*row+1, 2*col+1] = reals_r2[4*i+3] + 1j*imags_r2[4*i+3] + mean_r2[row][1,1]
else:
t[2*row, 2*col] = reals_t[4*i] + 1j*imags_t[4*i]
t[2*row, 2*col+1] = reals_t[4*i+1] + 1j*imags_t[4*i+1]
t[2*row+1, 2*col] = reals_t[4*i+2] + 1j*imags_t[4*i+2]
t[2*row+1, 2*col+1] = reals_t[4*i+3] + 1j*imags_t[4*i+3]
r[2*row, 2*col] = reals_r[4*i] + 1j*imags_r[4*i]
r[2*row, 2*col+1] = reals_r[4*i+1] + 1j*imags_r[4*i+1]
r[2*row+1, 2*col] = reals_r[4*i+2] + 1j*imags_r[4*i+2]
r[2*row+1, 2*col+1] = reals_r[4*i+3] + 1j*imags_r[4*i+3]
r2[2*row, 2*col] = reals_r2[4*i] + 1j*imags_r2[4*i]
r2[2*row, 2*col+1] = reals_r2[4*i+1] + 1j*imags_r2[4*i+1]
r2[2*row+1, 2*col] = reals_r2[4*i+2] + 1j*imags_r2[4*i+2]
r2[2*row+1, 2*col+1] = reals_r2[4*i+3] + 1j*imags_r2[4*i+3]
#Symmetrize (r and r2 should already be symmetrized to that these steps have no effect...
# but just to clean out any numerical inaccuracies)
r = ( r +sigma_p@Rsym(r)@sigma_p )/2
r2 = ( r2 + sigma_p@Rsym(r2)@sigma_p)/2
t2 = sigma_p@Rsym(t)@sigma_p
I = np.identity(2*n_modes)
t_svd = t + I
t2_svd = t2 + I
r_svd = r
r2_svd = r2
S_svd = np.block([[r_svd,t2_svd],[t_svd,r2_svd]])
u, s, vh = sp.linalg.svd(S_svd)
S = u@vh
return S
def S2M(S):
'''
Converts a scattering matrix into a transfer matrix
'''
N, N = np.shape(S)
N = int(N/4)
r = S[0:2*N, 0:2*N]
t = S[2*N:4*N, 0:2*N]
r2 = S[2*N:4*N, 2*N:4*N]
t2 = S[0:2*N, 2*N:4*N]
t2inv = sp.linalg.inv(t2)
alpha = t - r2@t2inv@r
beta = r2@t2inv
gamma = -t2inv@r
delta = t2inv
M = np.block([[alpha, beta],[gamma, delta]])
return M
def mode_sample_cartesian(dx,dy):
'''
Produces a cartesian grid of wavevector modes for given lattice spacings dx and dy
'''
mode_list = []
Nx = int(np.ceil(1/dx))
Ny = int(np.ceil(1/dy))
x = np.linspace(-Nx*dx, Nx*dx, 2*Nx + 1)
y = np.linspace(Ny*dy, -Ny*dy, 2*Ny + 1)
xv, yv = np.meshgrid(x, y)
ny, nx = np.shape(xv)
# Loop through mode centres
for j in range(ny):
for i in range(nx):
x = xv[j,i]
y = yv[j,i]
if(x**2 + y**2 < 1):
mode_list.append(np.array([x,y]))
n_modes = len(mode_list)
weight = np.pi/n_modes
return mode_list, weight
def get_mode_keys(mode_list):
'''
The mode keys help keep track of the incident and outgoing wavevectors of different matrix blocks
For example, t_key[j,i] contains and outgoing and incident wavevectors of the (j,i) 2x2 block of
the transmission matrix
'''
N_modes = len(mode_list)
t_key = np.zeros((N_modes, N_modes), dtype='object')
t2_key = np.zeros((N_modes, N_modes), dtype='object')
r_key = np.zeros((N_modes, N_modes), dtype='object')
r2_key = np.zeros((N_modes, N_modes), dtype='object')
for i in range(N_modes):
for j in range(N_modes):
in_mode_left = mode_list[j].copy()
in_mode_right = mode_list[j].copy()
k_z_in = np.sqrt(1 - in_mode_left[0]**2 - in_mode_left[1]**2)
in_mode_left = np.append(in_mode_left, [k_z_in])
in_mode_right = np.append(in_mode_right, [-k_z_in])
out_mode_left = mode_list[i].copy()
out_mode_right = mode_list[i].copy()
k_z_out = np.sqrt(1 - out_mode_left[0]**2 - out_mode_left[1]**2)
out_mode_left = np.append(out_mode_left, [-k_z_out])
out_mode_right = np.append(out_mode_right, [k_z_out])
t_key[i][j] = [out_mode_right, in_mode_left]
t2_key[i][j] = [out_mode_left, in_mode_right]
r_key[i][j] = [out_mode_left, in_mode_left]
r2_key[i][j] = [out_mode_right, in_mode_right]
return {'r': r_key, 't': t_key, 'r2': r2_key, 't2': t2_key}
## Parallelised construction of the covariance matrix
def get_statistics(params):
'''
Generates lists of the mean matrices and cholesky decompositions of the correlation matrices
associated with elements of the scattering matrix.
The output of this function is to be used in S_sampler_svd
Note: the slowest part of this function uses multiprocessing for improved performance
'''
N_cores = params['n cores']
k = params['k']
N = len(params['keys']['t'])
At_array = np.ndarray(shape=(N,N), dtype='object')
Ar_array = np.ndarray(shape=(N,N), dtype='object')
Ar2_array = np.ndarray(shape=(N,N), dtype='object')
n = params['n']
tolerance = params['tol']
weight = params['weight']
tkey = params['keys']['t']
rkey = params['keys']['r']
r2key = params['keys']['r2']
mode_list = params['modes']
print('Calculating A matrices...')
# Calculate A matrices from Mie theory
for i in range(N):
for j in range(N):
# Transmission
kin = tkey[j,i][1]
kout = tkey[j,i][0]
A = getT(kin, kout, params)
At_array[j,i] = A
# Reflection
kin = rkey[j,i][1]
kout = rkey[j,i][0]
A = getT(kin, kout, params)
Ar_array[j,i] = A
# Reflection 2
kin = r2key[j,i][1]
kout = r2key[j,i][0]
A = getT(kin, kout, params)
Ar2_array[j,i] = A
# Find experimental cross section
css_x = []
for i in range(N):
newcs_x = 0
for j in range(N):
# sigma calculation
size = mode_list[j][1]
cos_theta = np.sqrt(1 - np.linalg.norm(mode_list[j][0])**2)
newcs_x = newcs_x + np.linalg.norm(At_array[j,i][:,0])**2 * size / (k**2*cos_theta)
newcs_x = newcs_x + np.linalg.norm(Ar_array[j,i][:,0])**2 * size / (k**2*cos_theta)
css_x.append(newcs_x)
# Work out thickness that ensures the strength of the mean field in any direction
# is less than the tolerance (default value of tolerance is 0.1)
Ls = []
for i in range(N):
cos_theta_i = np.sqrt(1 - np.linalg.norm(mode_list[i][0])**2)
Lnew = tolerance*k**2*cos_theta_i / (np.max(sp.linalg.svdvals(At_array[i,i])) * 2*np.pi*n)
Ls.append(Lnew)
L = np.min(Ls)
#########
# Means #
#########
mean_t = []
mean_r = []
mean_r2 = []
for i in range(N):
kin_z = tkey[i,i][1][2]
new_mean_t = 2*np.pi*n*L/(k*k*kin_z) * At_array[i,i]
new_mean_r = 2*np.pi*n*L/(k*k*kin_z) * Ar_array[i,i] *sinc(kin_z*k*L)
new_mean_r2 = 2*np.pi*n*L/(k*k*kin_z) * Ar2_array[i,i] *sinc(kin_z*k*L)
mean_t.append(new_mean_t)
mean_r.append(new_mean_r)
mean_r2.append(new_mean_r2)
####################
# Helper functions #
####################
# h1 takes the (j,i) coordinates of an element in t or r and converts it to the number from
# 0 to (2N)^2 that corresponds to it in the covariacne matrix
def h1(j,i,Nmodes):
return 4*i + 4*Nmodes*j
# h2 takes a pair of elements in t (in t coordinates) and returns the coordinates of the correlation between those elements
# in the sparse matrix
def h2(j,i,q,p,Nmodes):
x = h1(j,i,Nmodes)
y = h1(q,p,Nmodes)
return (x,y)
print('Setting up level sets...')
################################################
# Algorithm for building covariance matrices
# 1) Build differences matrix
difference_matrix = np.zeros((N,N,2), dtype=float)
for j in range(N):
for i in np.arange(j+1,N):
difference_matrix[j,i,:] = np.array(mode_list[i][0]) - np.array(mode_list[j][0])
difference_matrix[i,j,:] = -difference_matrix[j,i,:]
# 2) Find level sets of differences matrix
diagonal_level_set = [(i,i) for i in range(N)]
level_sets = []
conjugate_level_sets = []
difference_list = []
for j in range(N):
for i in np.arange(j+1,N):
# new_set flag dictates whether or not a new level set needs to be constructed
new_set = True
new_difference = difference_matrix[j,i,:]
indices = (j,i)
reverse_indices = (i,j)
# Check if new_difference belongs to any of the level sets
for index, difference in enumerate(difference_list):
if np.allclose(difference, new_difference):
# new_difference belongs to the level set with index
# no need to make a new level set
level_sets[index].append(indices)
conjugate_level_sets[index].append(reverse_indices)
new_set = False
break
if np.allclose(difference, -new_difference):
# the reverse of new_difference belongs to the level set with index
# no need to make a new level set
level_sets[index].append(reverse_indices)
conjugate_level_sets[index].append(indices)
new_set = False
break
if(new_set):
difference_list.append(new_difference)
new_level_set = [indices]
new_conjugate_level_set = [reverse_indices]
level_sets.append(new_level_set)
conjugate_level_sets.append(new_conjugate_level_set)
# 3) Loop over level sets and build up data array to be inserted into sparse matrix
data_array_t = np.zeros((0,3),dtype='object')
data_array_t_diagonals = np.zeros((0,3),dtype='object')
data_array_r = np.zeros((0,3),dtype='object')
data_array_r_diagonals = np.zeros((0,3),dtype='object')
data_array_r2 = np.zeros((0,3),dtype='object')
data_array_r2_diagonals = np.zeros((0,3),dtype='object')
pseudo_data_array_t = np.zeros((0,3),dtype='object')
pseudo_data_array_t_diagonals = np.zeros((0,3),dtype='object')
pseudo_data_array_r = np.zeros((0,3),dtype='object')
pseudo_data_array_r_diagonals = np.zeros((0,3),dtype='object')
pseudo_data_array_r2 = np.zeros((0,3),dtype='object')
pseudo_data_array_r2_diagonals = np.zeros((0,3),dtype='object')
print('Diagonal sets...')
# Deal with diagonal_level_set
for first_index in range(N):
first_indices = diagonal_level_set[first_index]
i = first_indices[1]
j = first_indices[0]
for second_index in np.arange(first_index, N):
second_indices = diagonal_level_set[second_index]
p = second_indices[1]
q = second_indices[0]
row, col = h2(j,i,q,p,N)
in_first_t = np.array(tkey[j,i][1])
out_first_t = np.array(tkey[j,i][0])
in_second_t = np.array(tkey[q,p][1])
out_second_t = np.array(tkey[q,p][0])
kzi = in_first_t[2]
kzj = out_first_t[2]
kzp = in_second_t[2]
kzq = out_second_t[2]
################
# Transmission
A_vec_ji = np.array([At_array[j,i][0,0], At_array[j,i][0,1], At_array[j,i][1,0], At_array[j,i][1,1]])
A_vec_qp = np.array([At_array[q,p][0,0], At_array[q,p][0,1], At_array[q,p][1,0], At_array[q,p][1,1]])
# Regular
C = np.outer(A_vec_ji, np.conj(A_vec_qp))
H = C * n*L/k**2 * weight * 1/np.sqrt(kzi*kzj*kzp*kzq) * sinc(k*L/2*((kzi-kzj)-(kzp-kzq)))
new_data_entries_t = np.zeros((16,3),dtype='object')
new_data_entries_t[:,0] = np.array([row+0]*4 + [row+1]*4 + [row+2]*4 + [row+3]*4)
new_data_entries_t[:,1] = np.array([col+0,col+1,col+2,col+3]*4)
new_data_entries_t[:,2] = H.reshape(1,16)
if(i==p and j==q):
data_array_t_diagonals = np.vstack([data_array_t_diagonals, new_data_entries_t])
else:
data_array_t = np.vstack([data_array_t, new_data_entries_t])
# Pseudo
C = np.outer(A_vec_ji, A_vec_qp)
H = C * n*L/k**2 * weight * 1/np.sqrt(kzi*kzj*kzp*kzq) * sinc(k*L/2*((kzi-kzj)+(kzp-kzq)))
new_data_entries_t = np.zeros((16,3),dtype='object')
new_data_entries_t[:,0] = np.array([row+0]*4 + [row+1]*4 + [row+2]*4 + [row+3]*4)
new_data_entries_t[:,1] = np.array([col+0,col+1,col+2,col+3]*4)
new_data_entries_t[:,2] = H.reshape(1,16)
if(i==p and j==q):
pseudo_data_array_t_diagonals = np.vstack([pseudo_data_array_t_diagonals, new_data_entries_t])
else:
pseudo_data_array_t = np.vstack([pseudo_data_array_t, new_data_entries_t])
###############
# Reflection
A_vec_ji = np.array([Ar_array[j,i][0,0], Ar_array[j,i][0,1], Ar_array[j,i][1,0], Ar_array[j,i][1,1]])
A_vec_qp = np.array([Ar_array[q,p][0,0], Ar_array[q,p][0,1], Ar_array[q,p][1,0], Ar_array[q,p][1,1]])
# Regular
C = np.outer(A_vec_ji, np.conj(A_vec_qp))
H = C * n*L/k**2 * weight * 1/np.sqrt(kzi*kzj*kzp*kzq) * sinc(k*L/2*((kzi+kzj)-(kzp+kzq)))
new_data_entries_r = np.zeros((16,3),dtype='object')
new_data_entries_r[:,0] = np.array([row+0]*4 + [row+1]*4 + [row+2]*4 + [row+3]*4)
new_data_entries_r[:,1] = np.array([col+0,col+1,col+2,col+3]*4)
new_data_entries_r[:,2] = H.reshape(1,16)
if(i==p and j==q):
data_array_r_diagonals = np.vstack([data_array_r_diagonals, new_data_entries_r])
else:
data_array_r = np.vstack([data_array_r, new_data_entries_r])
# Pseudo
C = np.outer(A_vec_ji, A_vec_qp)
H = C * n*L/k**2 * weight * 1/np.sqrt(kzi*kzj*kzp*kzq) * sinc(k*L/2*((kzi+kzj)+(kzp+kzq)))
new_data_entries_r = np.zeros((16,3),dtype='object')
new_data_entries_r[:,0] = np.array([row+0]*4 + [row+1]*4 + [row+2]*4 + [row+3]*4)
new_data_entries_r[:,1] = np.array([col+0,col+1,col+2,col+3]*4)
new_data_entries_r[:,2] = H.reshape(1,16)
if(i==p and j==q):
pseudo_data_array_r_diagonals = np.vstack([pseudo_data_array_r_diagonals, new_data_entries_r])
else:
pseudo_data_array_r = np.vstack([pseudo_data_array_r, new_data_entries_r])
################
# Reflection 2
A_vec_ji = np.array([Ar2_array[j,i][0,0], Ar2_array[j,i][0,1], Ar2_array[j,i][1,0], Ar2_array[j,i][1,1]])
A_vec_qp = np.array([Ar2_array[q,p][0,0], Ar2_array[q,p][0,1], Ar2_array[q,p][1,0], Ar2_array[q,p][1,1]])
# Regular
C = np.outer(A_vec_ji, np.conj(A_vec_qp))
H = C * n*L/k**2 * weight * 1/np.sqrt(kzi*kzj*kzp*kzq) * sinc(k*L/2*((-kzi-kzj)-(-kzp-kzq)))
new_data_entries_r2 = np.zeros((16,3),dtype='object')
new_data_entries_r2[:,0] = np.array([row+0]*4 + [row+1]*4 + [row+2]*4 + [row+3]*4)
new_data_entries_r2[:,1] = np.array([col+0,col+1,col+2,col+3]*4)
new_data_entries_r2[:,2] = H.reshape(1,16)
if(i==p and j==q):
data_array_r2_diagonals = np.vstack([data_array_r2_diagonals, new_data_entries_r2])
else:
data_array_r2 = np.vstack([data_array_r2, new_data_entries_r2])
# Pseudo
C = np.outer(A_vec_ji, A_vec_qp)
H = C * n*L/k**2 * weight * 1/np.sqrt(kzi*kzj*kzp*kzq) * sinc(k*L/2*((-kzi-kzj)+(-kzp-kzq)))
new_data_entries_r2 = np.zeros((16,3),dtype='object')
new_data_entries_r2[:,0] = np.array([row+0]*4 + [row+1]*4 + [row+2]*4 + [row+3]*4)
new_data_entries_r2[:,1] = np.array([col+0,col+1,col+2,col+3]*4)
new_data_entries_r2[:,2] = H.reshape(1,16)
if(i==p and j==q):
pseudo_data_array_r2_diagonals = np.vstack([pseudo_data_array_r2_diagonals, new_data_entries_r2])
else:
pseudo_data_array_r2 = np.vstack([pseudo_data_array_r2, new_data_entries_r2])
print('Non diagonal sets...')
#args looks like
# [sets, conjugate_sets]
def get_correlation_stacks(args):
level_sets = args[0]
conjugate_level_sets = args[1]
data_array_t = np.zeros((0,3),dtype='object')
data_array_t_diagonals = np.zeros((0,3),dtype='object')
data_array_r = np.zeros((0,3),dtype='object')
data_array_r_diagonals = np.zeros((0,3),dtype='object')
data_array_r2 = np.zeros((0,3),dtype='object')
data_array_r2_diagonals = np.zeros((0,3),dtype='object')
pseudo_data_array_t = np.zeros((0,3),dtype='object')
pseudo_data_array_t_diagonals = np.zeros((0,3),dtype='object')
pseudo_data_array_r = np.zeros((0,3),dtype='object')
pseudo_data_array_r_diagonals = np.zeros((0,3),dtype='object')
pseudo_data_array_r2 = np.zeros((0,3),dtype='object')
pseudo_data_array_r2_diagonals = np.zeros((0,3),dtype='object')
weight = mode_list[0][1]
# Non-diagonal level sets
for level_set, conjugate_level_set in zip(level_sets, conjugate_level_sets):
# 3a) Loop over elements in the first set
for first_index, first_indices in enumerate(level_set):
i = first_indices[1]
j = first_indices[0]
# 3aa) triangular loop over remaining elements in the first set
# Regular correlations between level_set elements
for second_index in np.arange(first_index, len(level_set)):
second_indices = level_set[second_index]
p = second_indices[1]
q = second_indices[0]
row, col = h2(j,i,q,p,N)
in_first_t = np.array(tkey[j,i][1])
out_first_t = np.array(tkey[j,i][0])
in_second_t = np.array(tkey[q,p][1])
out_second_t = np.array(tkey[q,p][0])
kzi = in_first_t[2]
kzj = out_first_t[2]
kzp = in_second_t[2]
kzq = out_second_t[2]
################
# Transmission
A_vec_ji = np.array([At_array[j,i][0,0], At_array[j,i][0,1], At_array[j,i][1,0], At_array[j,i][1,1]])
A_vec_qp = np.array([At_array[q,p][0,0], At_array[q,p][0,1], At_array[q,p][1,0], At_array[q,p][1,1]])
C = np.outer(A_vec_ji, np.conj(A_vec_qp))
H = C * n*L/k**2 * weight * 1/np.sqrt(kzi*kzj*kzp*kzq) * sinc(k*L/2*((kzi-kzj)-(kzp-kzq)))
new_data_entries_t = np.zeros((16,3),dtype='object')
new_data_entries_t[:,0] = np.array([row+0]*4 + [row+1]*4 + [row+2]*4 + [row+3]*4)
new_data_entries_t[:,1] = np.array([col+0,col+1,col+2,col+3]*4)
new_data_entries_t[:,2] = H.reshape(1,16)
if(i==p and j==q):
data_array_t_diagonals = np.vstack([data_array_t_diagonals, new_data_entries_t])
else:
data_array_t = np.vstack([data_array_t, new_data_entries_t])
###############
# Reflection
A_vec_ji = np.array([Ar_array[j,i][0,0], Ar_array[j,i][0,1], Ar_array[j,i][1,0], Ar_array[j,i][1,1]])
A_vec_qp = np.array([Ar_array[q,p][0,0], Ar_array[q,p][0,1], Ar_array[q,p][1,0], Ar_array[q,p][1,1]])
C = np.outer(A_vec_ji, np.conj(A_vec_qp))
H = C * n*L/k**2 * weight * 1/np.sqrt(kzi*kzj*kzp*kzq) * sinc(k*L/2*((kzi+kzj)-(kzp+kzq)))
new_data_entries_r = np.zeros((16,3),dtype='object')
new_data_entries_r[:,0] = np.array([row+0]*4 + [row+1]*4 + [row+2]*4 + [row+3]*4)
new_data_entries_r[:,1] = np.array([col+0,col+1,col+2,col+3]*4)
new_data_entries_r[:,2] = H.reshape(1,16)
if(i==p and j==q):
data_array_r_diagonals = np.vstack([data_array_r_diagonals, new_data_entries_r])
else:
data_array_r = np.vstack([data_array_r, new_data_entries_r])
################
# Reflection 2
A_vec_ji = np.array([Ar2_array[j,i][0,0], Ar2_array[j,i][0,1], Ar2_array[j,i][1,0], Ar2_array[j,i][1,1]])
A_vec_qp = np.array([Ar2_array[q,p][0,0], Ar2_array[q,p][0,1], Ar2_array[q,p][1,0], Ar2_array[q,p][1,1]])
C = np.outer(A_vec_ji, np.conj(A_vec_qp))
H = C * n*L/k**2 * weight * 1/np.sqrt(kzi*kzj*kzp*kzq) * sinc(k*L/2*((-kzi-kzj)-(-kzp-kzq)))
new_data_entries_r2 = np.zeros((16,3),dtype='object')
new_data_entries_r2[:,0] = np.array([row+0]*4 + [row+1]*4 + [row+2]*4 + [row+3]*4)
new_data_entries_r2[:,1] = np.array([col+0,col+1,col+2,col+3]*4)
new_data_entries_r2[:,2] = H.reshape(1,16)
if(i==p and j==q):
data_array_r2_diagonals = np.vstack([data_array_r2_diagonals, new_data_entries_r2])
else:
data_array_r2 = np.vstack([data_array_r2, new_data_entries_r2])
# 3ab) full loop over all elements in the second set
# All pseudo correlations
for second_index, second_indices in enumerate(conjugate_level_set):
p = second_indices[1]
q = second_indices[0]
row, col = h2(j,i,q,p,N)
in_first_t = np.array(tkey[j,i][1])
out_first_t = np.array(tkey[j,i][0])
in_second_t = np.array(tkey[q,p][1])
out_second_t = np.array(tkey[q,p][0])
kzi = in_first_t[2]
kzj = out_first_t[2]
kzp = in_second_t[2]
kzq = out_second_t[2]
################
# Transmission
A_vec_ji = np.array([At_array[j,i][0,0], At_array[j,i][0,1], At_array[j,i][1,0], At_array[j,i][1,1]])
A_vec_qp = np.array([At_array[q,p][0,0], At_array[q,p][0,1], At_array[q,p][1,0], At_array[q,p][1,1]])
C = np.outer(A_vec_ji, A_vec_qp)
H = C * n*L/k**2 * weight * 1/np.sqrt(kzi*kzj*kzp*kzq) * sinc(k*L/2*((kzi-kzj)+(kzp-kzq)))
new_data_entries_t = np.zeros((16,3),dtype='object')
new_data_entries_t[:,0] = np.array([row+0]*4 + [row+1]*4 + [row+2]*4 + [row+3]*4)
new_data_entries_t[:,1] = np.array([col+0,col+1,col+2,col+3]*4)
new_data_entries_t[:,2] = H.reshape(1,16)
if(i==p and j==q):
pseudo_data_array_t_diagonals = np.vstack([pseudo_data_array_t_diagonals, new_data_entries_t])
else:
pseudo_data_array_t = np.vstack([pseudo_data_array_t, new_data_entries_t])
###############
# Reflection
A_vec_ji = np.array([Ar_array[j,i][0,0], Ar_array[j,i][0,1], Ar_array[j,i][1,0], Ar_array[j,i][1,1]])
A_vec_qp = np.array([Ar_array[q,p][0,0], Ar_array[q,p][0,1], Ar_array[q,p][1,0], Ar_array[q,p][1,1]])
C = np.outer(A_vec_ji, A_vec_qp)
H = C * n*L/k**2 * weight * 1/np.sqrt(kzi*kzj*kzp*kzq) * sinc(k*L/2*((kzi+kzj)+(kzp+kzq)))
new_data_entries_r = np.zeros((16,3),dtype='object')
new_data_entries_r[:,0] = np.array([row+0]*4 + [row+1]*4 + [row+2]*4 + [row+3]*4)
new_data_entries_r[:,1] = np.array([col+0,col+1,col+2,col+3]*4)
new_data_entries_r[:,2] = H.reshape(1,16)
if(i==p and j==q):
pseudo_data_array_r_diagonals = np.vstack([pseudo_data_array_r_diagonals, new_data_entries_r])
else:
pseudo_data_array_r = np.vstack([pseudo_data_array_r, new_data_entries_r])
################
# Reflection 2
A_vec_ji = np.array([Ar2_array[j,i][0,0], Ar2_array[j,i][0,1], Ar2_array[j,i][1,0], Ar2_array[j,i][1,1]])
A_vec_qp = np.array([Ar2_array[q,p][0,0], Ar2_array[q,p][0,1], Ar2_array[q,p][1,0], Ar2_array[q,p][1,1]])
C = np.outer(A_vec_ji, A_vec_qp)
H = C * n*L/k**2 * weight * 1/np.sqrt(kzi*kzj*kzp*kzq) * sinc(k*L/2*((-kzi-kzj)+(-kzp-kzq)))
new_data_entries_r2 = np.zeros((16,3),dtype='object')
new_data_entries_r2[:,0] = np.array([row+0]*4 + [row+1]*4 + [row+2]*4 + [row+3]*4)
new_data_entries_r2[:,1] = np.array([col+0,col+1,col+2,col+3]*4)
new_data_entries_r2[:,2] = H.reshape(1,16)
if(i==p and j==q):
pseudo_data_array_r2_diagonals = np.vstack([pseudo_data_array_r2_diagonals, new_data_entries_r2])
else:
pseudo_data_array_r2 = np.vstack([pseudo_data_array_r2, new_data_entries_r2])
# 3b) Loop over elements in the second set
for first_index, first_indices in enumerate(conjugate_level_set):
i = first_indices[1]
j = first_indices[0]
# 3ba) triangular loop over reamaining elements in the second set
# Regular correlations between conjugate_level_set elements
for second_index in np.arange(first_index, len(conjugate_level_set)):
second_indices = conjugate_level_set[second_index]
p = second_indices[1]
q = second_indices[0]
row, col = h2(j,i,q,p,N)
in_first_t = np.array(tkey[j,i][1])
out_first_t = np.array(tkey[j,i][0])
in_second_t = np.array(tkey[q,p][1])
out_second_t = np.array(tkey[q,p][0])
kzi = in_first_t[2]
kzj = out_first_t[2]
kzp = in_second_t[2]
kzq = out_second_t[2]
################
# Transmission
A_vec_ji = np.array([At_array[j,i][0,0], At_array[j,i][0,1], At_array[j,i][1,0], At_array[j,i][1,1]])
A_vec_qp = np.array([At_array[q,p][0,0], At_array[q,p][0,1], At_array[q,p][1,0], At_array[q,p][1,1]])
C = np.outer(A_vec_ji, np.conj(A_vec_qp))
H = C * n*L/k**2 * weight * 1/np.sqrt(kzi*kzj*kzp*kzq) * sinc(k*L/2*((kzi-kzj)-(kzp-kzq)))
new_data_entries_t = np.zeros((16,3),dtype='object')
new_data_entries_t[:,0] = np.array([row+0]*4 + [row+1]*4 + [row+2]*4 + [row+3]*4)
new_data_entries_t[:,1] = np.array([col+0,col+1,col+2,col+3]*4)
new_data_entries_t[:,2] = H.reshape(1,16)
if(i==p and j==q):
data_array_t_diagonals = np.vstack([data_array_t_diagonals, new_data_entries_t])
else:
data_array_t = np.vstack([data_array_t, new_data_entries_t])
###############
# Reflection
A_vec_ji = np.array([Ar_array[j,i][0,0], Ar_array[j,i][0,1], Ar_array[j,i][1,0], Ar_array[j,i][1,1]])
A_vec_qp = np.array([Ar_array[q,p][0,0], Ar_array[q,p][0,1], Ar_array[q,p][1,0], Ar_array[q,p][1,1]])
C = np.outer(A_vec_ji, np.conj(A_vec_qp))
H = C * n*L/k**2 * weight * 1/np.sqrt(kzi*kzj*kzp*kzq) * sinc(k*L/2*((kzi+kzj)-(kzp+kzq)))
new_data_entries_r = np.zeros((16,3),dtype='object')
new_data_entries_r[:,0] = np.array([row+0]*4 + [row+1]*4 + [row+2]*4 + [row+3]*4)
new_data_entries_r[:,1] = np.array([col+0,col+1,col+2,col+3]*4)
new_data_entries_r[:,2] = H.reshape(1,16)
if(i==p and j==q):
data_array_r_diagonals = np.vstack([data_array_r_diagonals, new_data_entries_r])
else:
data_array_r = np.vstack([data_array_r, new_data_entries_r])
################
# Reflection 2
A_vec_ji = np.array([Ar2_array[j,i][0,0], Ar2_array[j,i][0,1], Ar2_array[j,i][1,0], Ar2_array[j,i][1,1]])