def __prepare(self, structure, source, harmonics):
# initialise parameters for the computation
Nharm = harmonics.Nharm
nr1 = np.sqrt(structure.UR2 * structure.ER2)
self.k_inc = source.K0 * nr1 * np.array(([
np.sin(source.THETA) * np.cos(source.PHI),
np.sin(source.THETA) * np.sin(source.PHI),
np.cos(source.THETA)
]))
self.k_inc /= source.K0
self.k_ref = np.array(([self.k_inc[0], self.k_inc[1], -self.k_inc[2]]))
nr2 = np.sqrt(structure.UR1 * structure.ER1)
self.k_trn = np.array(([self.k_ref[0], self.k_ref[1], 0]))
self.k_trn[2] = np.sqrt(source.K0 * source.K0 * nr2 * nr2 -
self.k_trn[0] * self.k_trn[0] -
self.k_trn[1] * self.k_trn[1])
self.kx_mat = np.zeros((Nharm, Nharm), dtype=complex)
self.ky_mat = np.zeros((Nharm, Nharm), dtype=complex)
kz_0_mat = np.zeros((Nharm, Nharm), dtype=complex)
self.kz_ref_mat = np.zeros((Nharm, Nharm), dtype=complex)
self.kz_trn_mat = np.zeros((Nharm, Nharm), dtype=complex)
for i, i_harm in enumerate(harmonics.P_vec):
for j, j_harm in enumerate(harmonics.Q_vec):
pos = i * harmonics.P_range + j
self.kx_mat[pos, pos] = self.k_inc[0] - j_harm * 2 * np.pi / (
structure.Lx * source.K0)
self.ky_mat[pos, pos] = self.k_inc[1] - i_harm * 2 * np.pi / (
structure.Ly * source.K0)
kz_0_mat[pos, pos] = np.conj(
np.lib.scimath.sqrt(1 - (self.kx_mat[pos, pos])**2 -
(self.ky_mat[pos, pos])**2))
self.kz_ref_mat[pos, pos] = -np.conj(np.lib.scimath.sqrt(np.conj(structure.ER2)*np.conj(structure.UR2)-\
self.kx_mat[pos, pos]**2-self.ky_mat[pos, pos]**2))
self.kz_trn_mat[pos, pos] = np.conj(np.lib.scimath.sqrt(np.conj(structure.ER1)*np.conj(structure.UR1)-\
self.kx_mat[pos, pos]**2-self.ky_mat[pos, pos]**2))
self.I_mat = np.diag(np.ones(Nharm))
self.I_big_mat = np.diag(np.ones(2 * Nharm))
self.zeros_mat = np.zeros((Nharm, Nharm))
zeros_big_mat = np.zeros((2 * Nharm, 2 * Nharm))
self.W0_mat = np.diag(np.ones(2 * Nharm))
P0_mat = np.zeros((2 * Nharm, 2 * Nharm), dtype=complex)
P0_mat[0:Nharm, 0:Nharm] = matmul(self.kx_mat, self.ky_mat)
P0_mat[0:Nharm,
Nharm:2 * Nharm] = self.I_mat - matmul(self.kx_mat, self.kx_mat)
P0_mat[Nharm:2 * Nharm,
0:Nharm] = matmul(self.ky_mat, self.ky_mat) - self.I_mat
P0_mat[Nharm:2 * Nharm,
Nharm:2 * Nharm] = -matmul(self.ky_mat, self.kx_mat)
Q0_mat = P0_mat
lambda0_mat = np.concatenate((np.concatenate(
(1j * kz_0_mat, self.zeros_mat),
axis=1), np.concatenate((self.zeros_mat, 1j * kz_0_mat), axis=1)))
self.V0_mat = matmul(Q0_mat, np.linalg.inv(lambda0_mat))
S_global = np.concatenate((np.concatenate(
(zeros_big_mat, self.I_big_mat),
axis=1), np.concatenate((self.I_big_mat, zeros_big_mat), axis=1)))
return S_global
def __compute_substrate(self, structure, S_global):
kx, ky = self.k_inc[0], self.k_inc[1]
# take substrate into account
_, v_trn_mat = self.__calc_layer_params(structure.UR2, structure.ER2, kx, ky)
vg_mat = self.__calc_gap_layer_params(kx, ky)
a_trn_mat = UNIT_MAT_2D + matmul(np.linalg.inv(vg_mat), v_trn_mat)
b_trn_mat = UNIT_MAT_2D - matmul(np.linalg.inv(vg_mat), v_trn_mat)
s_trn_11_mat = matmul(b_trn_mat, np.linalg.inv(a_trn_mat))
s_trn_12_mat = 0.5*(a_trn_mat - matmul(b_trn_mat,
np.linalg.inv(a_trn_mat), b_trn_mat))
s_trn_21_mat = 2*np.linalg.inv(a_trn_mat)
s_trn_22_mat = -matmul(np.linalg.inv(a_trn_mat), b_trn_mat)
s_trn_mat = np.concatenate((np.concatenate((s_trn_11_mat, s_trn_12_mat), axis=1),
np.concatenate((s_trn_21_mat, s_trn_22_mat), axis=1)))
S_global = redheffer_star_prod(S_global, s_trn_mat, UNIT_MAT_2D)
return S_global
def __calc_s_mat(self, layer_thickness, ur, er, kx, ky, K0):
omegai_mat, vi_mat = self.__calc_layer_params(ur, er, kx, ky)
vg_mat = self.__calc_gap_layer_params(kx, ky)
ai_mat = UNIT_MAT_2D + np.matmul(np.linalg.inv(vi_mat), vg_mat)
bi_mat = UNIT_MAT_2D - np.matmul(np.linalg.inv(vi_mat), vg_mat)
xi_mat = np.diag(np.exp(np.diag(omegai_mat)*K0*layer_thickness))
ai_inv_mat = np.linalg.inv(ai_mat)
di_mat = ai_mat - matmul(xi_mat, bi_mat, ai_inv_mat, xi_mat, bi_mat)
di_inv_mat = np.linalg.inv(di_mat)
s_11_mat = matmul(di_inv_mat, matmul(
xi_mat, bi_mat, ai_inv_mat, xi_mat, ai_mat) - bi_mat)
s_12_mat = matmul(di_inv_mat, xi_mat, ai_mat
- matmul(bi_mat, ai_inv_mat, bi_mat))
# S_12 = S_21, S_11 = S_22
s_mat = np.concatenate((np.concatenate((s_11_mat, s_12_mat), axis=1),
np.concatenate((s_12_mat, s_11_mat), axis=1)))
return s_mat
def __get_R_T(self, structure, source, S_global):
z_unit_vec = np.array(([0, 0, 1]))
alpha_te = np.cross(z_unit_vec, self.k_inc)
alpha_te = alpha_te/np.linalg.norm(alpha_te)
alpha_tm = np.cross(alpha_te, self.k_inc)
alpha_tm = alpha_tm/np.linalg.norm(alpha_tm)
E_inc = source.norm_P_TM*alpha_tm + source.norm_P_TE*alpha_te
c_inc = np.array(([E_inc[0], E_inc[1], 0, 0]))
c_ret = matmul(S_global, c_inc)
e_field_ref = np.array(([c_ret[0], c_ret[1], 0]))
e_field_ref[2] = -(self.k_inc[0]*e_field_ref[0]+self.k_inc[1]*e_field_ref[1])/self.k_inc[2]
k_trn = np.array(([self.k_inc[0], self.k_inc[1], 0]))
nr2 = np.sqrt(structure.ER2*structure.UR2)
k_trn[2] = np.sqrt(source.K0*source.K0*nr2*nr2 - k_trn[0]*k_trn[0] - k_trn[1]*k_trn[1])
e_field_trn = np.array(([c_ret[2], c_ret[3], 0]))
e_field_trn[2] = -(k_trn[0]*e_field_trn[0]+k_trn[1]*e_field_trn[1])\
/(k_trn[2])
R = round((np.linalg.norm(e_field_ref))**2, 4)
T = round((np.linalg.norm(e_field_trn))**2*((k_trn[2]/structure.UR2).real)\
/((self.k_inc[2]/structure.UR1).real), 4)
return R, T
def __get_R_T(self, structure, source, harmonics, S_global):
Nharm = harmonics.Nharm
# computing input and output fields
delta_vec = np.zeros(Nharm)
delta_vec[int(np.floor(Nharm / 2))] = 1
z_unit_vec = np.array(([0, 0, 1]))
alpha_TE = np.cross(z_unit_vec, self.k_inc)
alpha_TE = alpha_TE / np.linalg.norm(alpha_TE)
alpha_TM = np.cross(alpha_TE, self.k_inc)
alpha_TM = alpha_TM / np.linalg.norm(alpha_TM)
E_inc = np.zeros(2 * Nharm, dtype=complex)
E_inc[0:Nharm] = (source.P_TM * alpha_TM +
source.P_TE * alpha_TE)[0] * delta_vec
E_inc[Nharm:2 * Nharm] = (source.P_TM * alpha_TM +
source.P_TE * alpha_TE)[1] * delta_vec
c_inc = matmul(np.linalg.inv(self.W_ref_mat), E_inc)
E_ref_xy = matmul(self.W_ref_mat, S_global[0:2 * Nharm, 0:2 * Nharm],
c_inc)
E_trn_xy = matmul(self.W_trn_mat, S_global[2 * Nharm:4 * Nharm,
0:2 * Nharm], c_inc)
E_ref_z = -matmul(np.linalg.inv(self.kz_ref_mat),\
matmul(self.kx_mat, E_ref_xy[0:Nharm]) +\
matmul(self.ky_mat, E_ref_xy[Nharm:2*Nharm]))
E_trn_z = -matmul(np.linalg.inv(self.kz_trn_mat),\
matmul(self.kx_mat, E_trn_xy[0:Nharm]) +\
matmul(self.ky_mat, E_trn_xy[Nharm:2*Nharm]))
r_sq = np.zeros((Nharm), dtype=complex)
t_sq = np.zeros((Nharm), dtype=complex)
for i in range(0, Nharm):
r_sq[i] = E_ref_xy[i]*np.conj(E_ref_xy[i]) +\
E_ref_xy[Nharm+i]*np.conj(E_ref_xy[Nharm+i]) +\
E_ref_z[i]*np.conj(E_ref_z[i])
t_sq[i] = E_trn_xy[i]*np.conj(E_trn_xy[i]) +\
E_trn_xy[Nharm+i]*np.conj(E_trn_xy[Nharm+i]) +\
E_trn_z[i]*np.conj(E_trn_z[i])
R = -np.real(matmul(self.kz_ref_mat, r_sq) / structure.UR2)
R /= np.real(self.k_inc[2] / structure.UR2) #
T = np.real(matmul(self.kz_trn_mat, t_sq) / structure.UR1)
T /= np.real(self.k_inc[2] / structure.UR1)
return R, T
def __compute_substrate(self, structure, harmonics, S_global):
Nharm = harmonics.Nharm
kx_mat = self.kx_mat
ky_mat = self.ky_mat
# transmission region
Q_trn_mat = np.zeros((2 * Nharm, 2 * Nharm), dtype=complex)
Q_trn_mat[0:Nharm, 0:Nharm] = matmul(kx_mat, ky_mat)
Q_trn_mat[0:Nharm, Nharm:2*Nharm] = structure.UR1*structure.ER1*self.I_mat -\
matmul(kx_mat, kx_mat)
Q_trn_mat[Nharm:2*Nharm, 0:Nharm] = matmul(ky_mat, ky_mat) -\
structure.UR1*structure.ER1*self.I_mat
Q_trn_mat[Nharm:2 * Nharm, Nharm:2 * Nharm] = -matmul(ky_mat, kx_mat)
Q_trn_mat /= structure.UR1
self.W_trn_mat = np.diag(np.ones(2 * Nharm))
lambda_trn_mat = np.concatenate(
(np.concatenate((1j * self.kz_trn_mat, self.zeros_mat), axis=1),
np.concatenate((self.zeros_mat, 1j * self.kz_trn_mat), axis=1)))
V_trn_mat = matmul(Q_trn_mat, np.linalg.inv(lambda_trn_mat))
A_trn_mat = matmul(np.linalg.inv(self.W0_mat), self.W_trn_mat) +\
matmul(np.linalg.inv(self.V0_mat), V_trn_mat)
B_trn_mat = matmul(np.linalg.inv(self.W0_mat), self.W_trn_mat) -\
matmul(np.linalg.inv(self.V0_mat), V_trn_mat)
S_trn_11 = matmul(B_trn_mat, np.linalg.inv(A_trn_mat))
S_trn_12 = 0.5 * (
A_trn_mat - matmul(B_trn_mat, np.linalg.inv(A_trn_mat), B_trn_mat))
S_trn_21 = 2 * np.linalg.inv(A_trn_mat)
S_trn_22 = -matmul(np.linalg.inv(A_trn_mat), B_trn_mat)
S_trn = np.concatenate((np.concatenate((S_trn_11, S_trn_12), axis=1),
np.concatenate((S_trn_21, S_trn_22), axis=1)))
S_global = redheffer_star_prod(S_global, S_trn, self.I_big_mat)
return S_global
def __compute_superstrate(self, structure, harmonics, S_global):
Nharm = harmonics.Nharm
kx_mat = self.kx_mat
ky_mat = self.ky_mat
# reflection region
Q_ref_mat = np.zeros((2 * Nharm, 2 * Nharm), dtype=complex)
Q_ref_mat[0:Nharm, 0:Nharm] = matmul(kx_mat, ky_mat)
Q_ref_mat[0:Nharm, Nharm:2*Nharm] = structure.UR2*structure.ER2*self.I_mat -\
matmul(kx_mat, kx_mat)
Q_ref_mat[Nharm:2*Nharm, 0:Nharm] = matmul(ky_mat, ky_mat) -\
structure.UR2*structure.ER2*self.I_mat
Q_ref_mat[Nharm:2 * Nharm, Nharm:2 * Nharm] = -matmul(ky_mat, kx_mat)
Q_ref_mat /= structure.UR2
self.W_ref_mat = np.diag(np.ones(2 * Nharm))
lambda_ref_mat = np.concatenate(
(np.concatenate((-1j * self.kz_ref_mat, self.zeros_mat), axis=1),
np.concatenate((self.zeros_mat, -1j * self.kz_ref_mat), axis=1)))
V_ref_mat = matmul(Q_ref_mat, np.linalg.inv(lambda_ref_mat))
A_ref_mat = matmul(np.linalg.inv(self.W0_mat), self.W_ref_mat) +\
matmul(np.linalg.inv(self.V0_mat), V_ref_mat)
B_ref_mat = matmul(np.linalg.inv(self.W0_mat), self.W_ref_mat) -\
matmul(np.linalg.inv(self.V0_mat), V_ref_mat)
S_ref_11 = -matmul(np.linalg.inv(A_ref_mat), B_ref_mat)
S_ref_12 = 2 * np.linalg.inv(A_ref_mat)
S_ref_21 = 0.5 * (
A_ref_mat - matmul(B_ref_mat, np.linalg.inv(A_ref_mat), B_ref_mat))
S_ref_22 = matmul(B_ref_mat, np.linalg.inv(A_ref_mat))
S_ref = np.concatenate((np.concatenate((S_ref_11, S_ref_12), axis=1),
np.concatenate((S_ref_21, S_ref_22), axis=1)))
S_global = redheffer_star_prod(S_ref, S_global, self.I_big_mat)
return S_global
def __compute_layers(self, structure, source, harmonics, S_global):
Nharm = harmonics.Nharm
kx_mat = self.kx_mat
ky_mat = self.ky_mat
# iterating through layers
for i in range(0, structure.num_layers):
ERC = structure.erc_vec[i]
URC = structure.urc_vec[i]
thickness = structure.L[i]
P_mat = np.zeros((2 * Nharm, 2 * Nharm), dtype=complex)
P_mat[0:Nharm, 0:Nharm] = matmul(kx_mat, np.linalg.inv(ERC),
ky_mat)
P_mat[0:Nharm, Nharm:2 *
Nharm] = URC - matmul(kx_mat, np.linalg.inv(ERC), kx_mat)
P_mat[Nharm:2 * Nharm,
0:Nharm] = matmul(ky_mat, np.linalg.inv(ERC), ky_mat) - URC
P_mat[Nharm:2 * Nharm, Nharm:2 *
Nharm] = -matmul(ky_mat, np.linalg.inv(ERC), kx_mat)
Q_mat = np.zeros((2 * Nharm, 2 * Nharm), dtype=complex)
Q_mat[0:Nharm, 0:Nharm] = matmul(kx_mat, np.linalg.inv(URC),
ky_mat)
Q_mat[0:Nharm, Nharm:2 *
Nharm] = ERC - matmul(kx_mat, np.linalg.inv(URC), kx_mat)
Q_mat[Nharm:2 * Nharm,
0:Nharm] = matmul(ky_mat, np.linalg.inv(URC), ky_mat) - ERC
Q_mat[Nharm:2 * Nharm, Nharm:2 *
Nharm] = -matmul(ky_mat, np.linalg.inv(URC), kx_mat)
Omega_sq_mat = matmul(P_mat, Q_mat)
v_vec, W_mat = np.linalg.eig(Omega_sq_mat)
lambda_mat = np.diag(np.conj(np.sqrt(v_vec)))
V_mat = matmul(Q_mat, W_mat, np.linalg.inv(lambda_mat))
A_mat = matmul(np.linalg.inv(W_mat), self.W0_mat) +\
matmul(np.linalg.inv(V_mat), self.V0_mat)
B_mat = matmul(np.linalg.inv(W_mat), self.W0_mat) -\
matmul(np.linalg.inv(V_mat), self.V0_mat)
X_mat = np.diag(
np.exp(-np.diag(lambda_mat) * source.K0 * thickness))
S_11 = matmul(np.linalg.inv(A_mat -\
matmul(X_mat, B_mat, np.linalg.inv(A_mat), X_mat, B_mat)),
(matmul(X_mat, B_mat, np.linalg.inv(A_mat), X_mat, A_mat) - B_mat))
S_12 = matmul(np.linalg.inv(A_mat -\
matmul(X_mat, B_mat, np.linalg.inv(A_mat), X_mat, B_mat)),
X_mat, (A_mat - matmul(B_mat, np.linalg.inv(A_mat), B_mat)))
S = np.concatenate((np.concatenate(
(S_11, S_12), axis=1), np.concatenate((S_12, S_11), axis=1)))
S_global = redheffer_star_prod(S_global, S, self.I_big_mat)
return S_global