## Build the second layer (uniform) URC2 = (np.identity(NH)) ERC2= erd*(np.identity(NH)); ## BUILD THE K_MATRIX kx_inc = n_i * np.sin(theta) * np.cos(phi); ky_inc = n_i * np.sin(theta) * np.sin(phi); # constant in ALL LAYERS; ky = 0 for normal incidence kz_inc = cmath.sqrt(n_i**2 - kx_inc ** 2 - ky_inc ** 2); Kx, Ky = km.K_matrix_cubic_2D(kx_inc, ky_inc, k0, Lx, Ly, PQ[0], PQ[1]); #gap media Wg, Vg, Kzg = hl.homogeneous_module(Kx, Ky, 2); ## Get Kzr and Kztrans Wr, Vr, Kzr = hl.homogeneous_module(Kx, Ky, er1); Ar, Br = sm.A_B_matrices_half_space(Wr, Wg, Vr, Vg); #make sure this order is right Sr, Sr_dict = sm.S_R(Ar, Br) Sg = Sr_dict; ## =================================================================## ## First LAYER (homogeneous) ## =======================================================================## P, Q, Kzl = pq.P_Q_kz(Kx, Ky, E_conv, mu_conv) omega_sq = P @ Q; ## no gaurantees this is hermitian or symmetric W1, lambda_matrix = em.eigen_W(omega_sq) V1 = em.eigen_V(Q, W1, lambda_matrix)
def run_RCWA_2D(lam0, theta, phi, ER, UR, layer_thicknesses, lattice_constants,
pte, ptm, N, M, e_half):
'''
:param lam0:
:param theta: incident angle
:param phi: incident angle (azimuthal)
:param ER: list of convolution matrices for each layer
:param UR: list of convolution matrices for each layer
:param layer_thicknesses: list of thicknesses of each layer
:param lattice_constants: [Lx, Ly] 2 element array containing lattice constants of the 2D unit cell
:param pte: te mode amplitude
:param ptm: tm mode amplitude
:param N: num orders for x direction
:param M: num orders for y direction
:param e_half: [e_r e_t], dielectric constants of the reflection and transmission spaces
:return:
'''
## convention specifications
normal_vector = np.array([0, 0, -1]) # positive z points down;
ate_vector = np.array([0, 1, 0])
# vector for the out of plane E-field
## ===========================
Lx = lattice_constants[0]
Ly = lattice_constants[1]
NM = (2 * N + 1) * (2 * M + 1)
# define vacuum wavevector k0
k0 = 2 * np.pi / lam0
## ============== values to keep track of =======================##
S_matrices = list()
kz_storage = list()
## ==============================================================##
m_r = 1
e_r = e_half[0]
## incident wave properties, at this point, everything is in units of k_0
n_i = np.sqrt(e_r * m_r)
# actually, in the definitions here, kx = k0*sin(theta)*cos(phi), so kx, ky here are normalized
kx_inc = n_i * np.sin(theta) * np.cos(phi)
ky_inc = n_i * np.sin(theta) * np.sin(phi)
# constant in ALL LAYERS; ky = 0 for normal incidence
kz_inc = cmath.sqrt(e_r * 1 - kx_inc**2 - ky_inc**2)
# remember, these Kx and Ky come out already normalized
Kx, Ky = km.K_matrix_cubic_2D(kx_inc, ky_inc, k0, Lx, Ly, N, M)
# Kx and Ky are diagonal but have a 0 on it
## =============== K Matrices for gap medium =========================
## specify gap media (this is an LHI so no eigenvalue problem should be solved
e_h = 1
Wg, Vg, Kzg = hl.homogeneous_module(Kx, Ky, e_h)
### ================= Working on the Reflection Side =========== ##
Wr, Vr, kzr = hl.homogeneous_module(Kx, Ky, e_r)
kz_storage.append(kzr)
## calculating A and B matrices for scattering matrix
# since gap medium and reflection media are the same, this doesn't affect anything
Ar, Br = sm.A_B_matrices(Wg, Wr, Vg, Vr)
## s_ref is a matrix, Sr_dict is a dictionary
S_ref, Sr_dict = sm.S_R(Ar, Br)
# scatter matrix for the reflection region
S_matrices.append(S_ref)
Sg = Sr_dict
## go through the layers
for i in range(len(ER)):
# ith layer material parameters
e_conv = ER[i]
mu_conv = UR[i]
# longitudinal k_vector
P, Q, kzl = pq.P_Q_kz(Kx, Ky, e_conv, mu_conv)
kz_storage.append(kzl)
Gamma_squared = P @ Q
## E-field modes that can propagate in the medium, these are well-conditioned
W_i, lambda_matrix = em.eigen_W(Gamma_squared)
V_i = em.eigen_V(Q, W_i, lambda_matrix)
# now defIne A and B, slightly worse conditoined than W and V
A, B = sm.A_B_matrices(W_i, Wg, V_i, Vg)
# ORDER HERE MATTERS A LOT because W_i is not diagonal
# calculate scattering matrix
Li = layer_thicknesses[i]
S_layer, Sl_dict = sm.S_layer(A, B, Li, k0, lambda_matrix)
S_matrices.append(S_layer)
## update global scattering matrix using redheffer star
Sg_matrix, Sg = rs.RedhefferStar(Sg, Sl_dict)
##========= Working on the Transmission Side==============##
m_t = 1
e_t = e_half[1]
Wt, Vt, kz_trans = hl.homogeneous_module(Kx, Ky, e_t)
# get At, Bt
# since transmission is the same as gap, order does not matter
At, Bt = sm.A_B_matrices(Wg, Wt, Vg, Vt)
ST, ST_dict = sm.S_T(At, Bt)
S_matrices.append(ST)
# update global scattering matrix
Sg_matrix, Sg = rs.RedhefferStar(Sg, ST_dict)
## finally CONVERT THE GLOBAL SCATTERING MATRIX BACK TO A MATRIX
K_inc_vector = n_i * np.array([np.sin(theta) * np.cos(phi), \
np.sin(theta) * np.sin(phi), np.cos(theta)])
E_inc, cinc, Polarization = ic.initial_conditions(K_inc_vector, theta,
normal_vector, pte, ptm,
N, M)
# print(cinc.shape)
# print(cinc)
cinc = np.linalg.inv(Wr) @ cinc
## COMPUTE FIELDS: similar idea but more complex for RCWA since you have individual modes each contributing
reflected = Wr @ Sg['S11'] @ cinc
transmitted = Wt @ Sg['S21'] @ cinc
rx = reflected[0:NM, :]
# rx is the Ex component.
ry = reflected[NM:, :]
#
tx = transmitted[0:NM, :]
ty = transmitted[NM:, :]
# longitudinal components; should be 0
rz = np.linalg.inv(kzr) @ (Kx @ rx + Ky @ ry)
tz = np.linalg.inv(kz_trans) @ (Kx @ tx + Ky @ ty)
## we need to do some reshaping at some point
## apparently we're not done...now we need to compute 'diffraction efficiency'
r_sq = np.square(np.abs(rx)) + np.square(np.abs(ry)) + np.square(
np.abs(rz))
t_sq = np.square(np.abs(tx)) + np.square(np.abs(ty)) + np.square(
np.abs(tz))
R = np.real(kzr) @ r_sq / np.real(kz_inc)
#division by a scalar
T = np.real(kz_trans) @ t_sq / (np.real(kz_inc))
return np.sum(R), np.sum(T)
## need a simulation which can return the field profiles inside the structure
## incident wave properties, at this point, everything is in units of k_0
n_i = np.sqrt(e_r*m_r);
# actually, in the definitions here, kx = k0*sin(theta)*cos(phi), so kx, ky here are normalized
kx_inc = n_i * np.sin(theta) * np.cos(phi);
ky_inc = n_i * np.sin(theta) * np.sin(phi); # constant in ALL LAYERS; ky = 0 for normal incidence
kz_inc = cmath.sqrt(e_r * 1 - kx_inc ** 2 - ky_inc ** 2);
#remember, these Kx and Ky come out already normalized
Kx, Ky = km.K_matrix_cubic_2D(kx_inc, ky_inc, k0, a, a, N, M); #Kx and Ky are diagonal but have a 0 on it
## density Kx and Ky (for now)
## =============== K Matrices for gap medium =========================
## specify gap media (this is an LHI so no eigenvalue problem should be solved
e_h = 1; m_h = 1;
Wg, Vg, Kzg = hl.homogeneous_module(Kx, Ky, e_h)
### ================= Working on the Reflection Side =========== ##
Wr, Vr, kzr = hl.homogeneous_module(Kx, Ky, e_r); kz_storage.append(kzr)
## calculating A and B matrices for scattering matrix
# since gap medium and reflection media are the same, this doesn't affect anything
Ar, Br = sm.A_B_matrices(Wg, Wr, Vg, Vr);
## s_ref is a matrix, Sr_dict is a dictionary
S_ref, Sr_dict = sm.S_R(Ar, Br); #scatter matrix for the reflection region
S_matrices.append(S_ref);
Sg = Sr_dict;
Q_storage = list(); P_storage= list();
## go through the layers
# actually, in the definitions here, kx = k0*sin(theta)*cos(phi), so kx, ky here are normalized
kx_inc = n_i * np.sin(theta) * np.cos(phi)
ky_inc = n_i * np.sin(theta) * np.sin(phi)
# constant in ALL LAYERS; ky = 0 for normal incidence
kz_inc = cmath.sqrt(e_r * 1 - kx_inc**2 - ky_inc**2)
#remember, these Kx and Ky come out already normalized
Kx, Ky = km.K_matrix_cubic_2D(kx_inc, ky_inc, k0, a, a, N, M)
#Kx and Ky are diagonal but have a 0 on it
## density Kx and Ky (for now)
## =============== K Matrices for gap medium =========================
## specify gap media (this is an LHI so no eigenvalue problem should be solved
e_h = 1
m_h = 1
Wg, Vg, Kzg = hl.homogeneous_module(Kx, Ky, e_h)
### ================= Working on the Reflection Side =========== ##
Wr, Vr, kzr = hl.homogeneous_module(Kx, Ky, e_r)
kz_storage.append(kzr)
## calculating A and B matrices for scattering matrix
# since gap medium and reflection media are the same, this doesn't affect anything
Ar, Br = sm.A_B_matrices(Wg, Wr, Vg, Vr)
## s_ref is a matrix, Sr_dict is a dictionary
S_ref, Sr_dict = sm.S_R(Ar, Br)
#scatter matrix for the reflection region
S_matrices.append(S_ref)
Sg = Sr_dict
def fun():
'''
run the RCWA simulation...do not use this code...
:return:
'''
meters = 1
centimeters = 1e-2 * meters
degrees = np.pi / 180
# Source parameters
lam0 = 2 * centimeters
theta = 0
phi = 0
pte = 1
# te polarized
ptm = 0
normal_vector = np.array([0, 0, -1]) # positive z points down;
ate_vector = np.array([0, 1, 0])
# vector for the out of plane E-field
k0 = 2 * np.pi / lam0
print('k0: ' + str(k0))
# structure parameters
# reflection 1 and transmission 2
ur1 = 1
er1 = 2
n_i = np.sqrt(ur1 * er1)
ur2 = 1
er2 = 9
## second layer
urd = 1
erd = 6
# dimensions of the unit cell
Lx = 1.75 * centimeters
Ly = 1.5 * centimeters
# thickness of layers
d1 = 0.5 * centimeters
d2 = 0.3 * centimeters
w = 0.8 * Ly
# RCWA parameters
Nx = 512
Ny = round(Nx * Ly / Lx)
PQ = [1, 1]
# number of spatial harmonics
NH = (2 * (PQ[0]) + 1) * (2 * (PQ[1]) + 1)
## =========================== BUILD DEVICE ON GRID ==================================##
dx = Lx / Nx
dy = Ly / Ny
xa = np.linspace(0, Lx, Nx)
xa = xa - np.mean(xa)
ya = np.linspace(0, Ly, Ny)
ya = ya - np.mean(ya)
# initialize layers
UR = np.ones((Nx, Ny, 2))
# interestin
ER = erd * np.ones((Nx, Ny, 2))
L = [d1, d2]
# Build the triangle
h = 0.5 * np.sqrt(3) * w
ny = int(np.round(h / dy))
# discrete height
ny1 = np.round((Ny - ny) / 2)
ny2 = ny1 + ny - 1
print(str(ny1) + ', ' + str(ny2))
for ny_ind in np.arange(ny1, ny2 + 1):
# build the triangle slice wise
f = (ny_ind - ny1) / (ny2 - ny1)
# fractional occupation;
nx = int(round(f * (w / Lx) * Nx))
# x width
nx1 = 1 + int(np.floor((Nx - nx) / 2))
nx2 = int(nx1 + nx)
# print(str(nx1)+', '+str(nx2))
ER[nx1:nx2 + 1, int(ny_ind), 0] = er1
E_conv = (cm.convmat2D(ER[:, :, 0], PQ[0], PQ[1]))
np.set_printoptions(precision=4)
print(E_conv)
mu_conv = (np.identity(NH))
## Build the second layer (uniform)
URC2 = (np.identity(NH))
ERC2 = erd * (np.identity(NH))
## BUILD THE K_MATRIX
kx_inc = n_i * np.sin(theta) * np.cos(phi)
ky_inc = n_i * np.sin(theta) * np.sin(phi)
# constant in ALL LAYERS; ky = 0 for normal incidence
kz_inc = cmath.sqrt(n_i**2 - kx_inc**2 - ky_inc**2)
Kx, Ky = km.K_matrix_cubic_2D(kx_inc, ky_inc, k0, Lx, Ly, PQ[0], PQ[1])
# gap media
Wg, Vg, Kzg = hl.homogeneous_module(Kx, Ky, 2)
## Get Kzr and Kztrans
Wr, Vr, Kzr = hl.homogeneous_module(Kx, Ky, er1)
Ar, Br = sm.A_B_matrices_half_space(Wr, Wg, Vr, Vg)
# make sure this order is right
Sr, Sr_dict = sm.S_R(Ar, Br)
Sg = Sr_dict
## =================================================================##
## First LAYER (homogeneous)
## =======================================================================##
P, Q, Kzl = pq.P_Q_kz(Kx, Ky, E_conv, mu_conv)
omega_sq = P @ Q
## no gaurantees this is hermitian or symmetric
W1, lambda_matrix = em.eigen_W(omega_sq)
V1 = em.eigen_V(Q, W1, lambda_matrix)
A1, B1 = sm.A_B_matrices(W1, Wg, V1, Vg)
S1, S1_dict = sm.S_layer(A1, B1, d1, k0, lambda_matrix)
Sg_matrix, Sg = rs.RedhefferStar(Sg, S1_dict)
## =================================================================##
## SECOND LAYER (homogeneous)
## =======================================================================##
##check with PQ formalism, which is unnecessary
P2, Q2, Kz2_check = pq.P_Q_kz(Kx, Ky, ERC2, URC2)
omega_sq_2 = P2 @ Q2
W2, lambda_matrix_2 = em.eigen_W(
omega_sq_2) # somehow lambda_matrix is fine but W is full of errors
V2 = em.eigen_V(Q2, W2, lambda_matrix_2)
A2, B2 = sm.A_B_matrices(W2, Wg, V2, Vg)
S2, S2_dict = sm.S_layer(A2, B2, d2, k0, lambda_matrix_2)
Sg_matrix, Sg = rs.RedhefferStar(Sg, S2_dict)
## TRANSMISSION LAYER
# #create ST
Wt, Vt, Kzt = hl.homogeneous_module(Kx, Ky, er2)
At, Bt = sm.A_B_matrices_half_space(Wt, Wg, Vt, Vg)
# make sure this order is right
St, St_dict = sm.S_T(At, Bt)
### FUCKKKKKKKKKKKKKKKK
Sg_matrix, Sg = rs.RedhefferStar(Sg, St_dict)
print('final Sg')
print(Sg['S11'])
## ================START THE SSCATTERING CALCULATION ==========================##
K_inc_vector = n_i * np.array([np.sin(theta) * np.cos(phi), \
np.sin(theta) * np.sin(phi), np.cos(theta)])
E_inc, cinc, Polarization = ic.initial_conditions(K_inc_vector, theta,
normal_vector, pte, ptm,
PQ[0], PQ[1])
## COMPUTE FIELDS: similar idea but more complex for RCWA since you have individual modes each contributing
reflected = Wr @ Sg['S11'] @ cinc
# reflection coefficients for every mode...
transmitted = Wt @ Sg['S21'] @ cinc
## these include only (rx, ry), (tx, ty), which is okay as these are the only components for normal incidence in LHI
rx = reflected[0:NH, :]
ry = reflected[NH:, :]
tx = transmitted[0:NH, :]
ty = transmitted[NH:, :]
# longitudinal components; should be 0
rz = np.linalg.inv(Kzr) @ (Kx @ rx + Ky @ ry)
tz = np.linalg.inv(Kzt) @ (Kx @ tx + Ky @ ty)
print('rx')
print(rx)
print('ry')
print(ry)
print('rz')
print(rz)
## apparently we're not done...now we need to compute 'diffraction efficiency'
r_sq = np.square(np.abs(rx)) + np.square(np.abs(ry)) + np.square(
np.abs(rz))
t_sq = np.square(np.abs(tx)) + np.square(np.abs(ty)) + np.square(
np.abs(tz))
R = np.real(Kzr) @ r_sq / np.real(kz_inc)
T = np.real(Kzt) @ t_sq / (np.real(kz_inc))
print('final R vector-> matrix')
print(np.reshape(R, (3, 3)))
# should be 3x3
print('final T vector/matrix')
print(np.reshape(T, (3, 3)))
print('final reflection: ' + str(np.sum(R)))
print('final transmission: ' + str(np.sum(T)))
print('sum of R and T: ' + str(np.sum(R) + np.sum(T)))
## if the sum isn't 1, that's a PROBLEM
t1 = time.time()
return np.sum(R), np.sum(T)