Kx, Ky = km.K_matrix_cubic_2D(beta_x, beta_y, ax, ay, P, Q)
#from here, we can actually dtermine omega from our specification of kx and ky
# omega = c0*np.sqrt(beta_x**2+beta_y**2)/L0;
#
# ## we can do a determination of a dispersive medium from here
# #drude for example
#eps_drude = 1-omega_p**2/(omega**2-cmath.sqrt(-1)*gamma*omega);
#print(eps_drude); eps_tracker.append(eps_drude);
epsilon = np.ones((Nx, Ny))
halfy = int(Ny / 2)
epsilon[halfy - 100:halfy + 100, :] = 12
## =============== Convolution Matrices ==============
E_r = cm.convmat2D(epsilon, P, Q)
## ===========================================================
eigenvalues, eigenvectors, A_matrix = eg.PWEM2D_TE(Kx, Ky, E_r)
#what are the eigenvalue units
omega_eig_store.append(np.sqrt(abs(np.real(eigenvalues))))
plt.plot(beta_x * np.ones((PQ, )),
np.sort(np.sqrt(abs(np.real(eigenvalues)))), '.b')
plt.plot(beta_x * np.ones((PQ, )),
np.sort(np.sqrt(abs(np.imag(eigenvalues)))), '.r')
## plot the light line
plt.plot(kx_scan, abs(kx_scan))
plt.show()
#print(str(nx1)+', '+str(nx2))
ER[nx1:nx2+1, int(ny_ind), 0] = er1;
# plt.imshow(ER[:,:,0])
# plt.colorbar()
# plt.show()
# Af = np.fft.fftshift(np.fft.fft2(ER[:,:,0]));
# plt.figure();
# plt.subplot(121)
# plt.imshow(np.log(abs(Af))); #note that the fft HAS A HUGE RANGE OF VALUES, which can become a source of inaccuracy
# plt.colorbar()
# plt.subplot(122)
# plt.imshow(np.abs(ER[:,:,0]))
# plt.show();
## conv matrices of the 1st
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]);
# ============== build high resolution circle ================== Nx = 512; Ny = 512; A = np.ones((Nx,Ny)); ci = int(Nx/2); cj= int(Ny/2); cr = (radius/a)*Nx; I,J=np.meshgrid(np.arange(A.shape[0]),np.arange(A.shape[1])); dist = np.sqrt((I-ci)**2 + (J-cj)**2); A[np.where(dist<cr)] = e_r; #visualize structure plt.imshow(A); plt.show() ## =============== Convolution Matrices ============== E_r = cm.convmat2D(A, P,Q) print(E_r.shape) print(type(E_r)) plt.figure(); plt.imshow(abs(E_r), cmap = 'jet'); plt.colorbar() plt.show() ## =============== K Matrices ========================= beta_x = beta_y = 0; plt.figure(); ## check K-matrices for normal icnidence Kx, Ky = km.K_matrix_cubic_2D(0,0, a, a, P, Q); np.set_printoptions(precision = 3)
Nx = 512
Ny = 512
A = e_r * np.ones((Nx, Ny))
A = A.astype('complex')
x1 = int(Nx / 2) - 50
x2 = int(Nx / 2) + 50
y1 = int(Ny / 2) - 50
y2 = int(Ny / 2) + 50
A[:, y1:y2] = eps_drude
## A METALLIC HOLE...the fact that we have to recalculate the convolution
# plt.imshow(np.abs(A));
# plt.show();
## =============== Convolution Matrices ==============
E_r = cm.convmat2D(A, N, M)
NM = (2 * N + 1) * (2 * M + 1)
## ================== GEOMETRY OF THE LAYERS AND CONVOLUTIONS ==================##
thickness_slab = 0.2
# in units of L0;
ER = [E_r]
UR = [np.identity(NM)]
layer_thicknesses = [thickness_slab]
# this retains SI unit convention
#source parameters
theta = 0 * degrees
#%elevation angle
phi = 0 * degrees
#print(str(nx1)+', '+str(nx2))
ER[nx1:nx2+1, int(ny_ind), 0] = er1;
# plt.imshow(ER[:,:,0])
# plt.colorbar()
# plt.show()
Af = np.fft.fftshift(np.fft.fft2(ER[:,:,0]));
plt.figure();
plt.subplot(121)
plt.imshow(np.log(abs(Af))); #note that the fft HAS A HUGE RANGE OF VALUES, which can become a source of inaccuracy
plt.colorbar()
plt.subplot(122)
plt.imshow(np.abs(ER[:,:,0]))
plt.show();
## conv matrices of the 1st
E_conv = np.matrix(cm.convmat2D(ER[:, :, 0], PQ[0], PQ[1]));
np.set_printoptions(precision = 4)
print(E_conv)
mu_conv = np.matrix(np.identity(NH));
## Build the second layer (uniform)
URC2 = np.matrix(np.identity(NH))
ERC2= erd*np.matrix(np.identity(NH));
ER = [E_conv, ERC2];
UR = [mu_conv, URC2];
wavelengths = np.linspace(0.5, 5, 501)*centimeters
ref = list(); trans = list();
for i in range(len(wavelengths)): #in SI units
## Specify number of fourier orders to use: #sometimes can get orders which make the system singular N = 2 M = 2 NM = (2 * N + 1) * (2 * M + 1) # ============== build high resolution circle ================== E_r = e_layer * np.matrix(np.identity(NM)) U_r = np.matrix(np.identity(NM)) #check that the convmat returns an identity matrix eps_grid = np.ones((512, 512)) er_check = e_layer * cm.convmat2D(eps_grid, N, M) er_fft = np.fft.fftshift(np.fft.fft(np.ones((3, 3)))) / (9) ## ================== GEOMETRY OF THE LAYERS AND CONVOLUTIONS ==================## thickness_slab = 0.76 * L0 #100 nm; ER = [E_r] UR = [U_r] layer_thicknesses = [thickness_slab] #this retains SI unit convention ## =============== Simulation Parameters ========================= ## set wavelength scanning range wavelengths = L0 * np.linspace(0.25, 1, 1000) #500 nm to 1000 nm #LHI fails for small wavelengths? kmagnitude_scan = 2 * np.pi / wavelengths
A = e_r * np.ones((Nx, Ny))
A = A.astype('complex')
A2 = e_r * np.ones((Nx, Ny))
A2 = A2.astype('complex')
x1 = int(Nx / 2) - 50
x2 = int(Nx / 2) + 50
y1 = int(Ny / 2) - 50
y2 = int(Ny / 2) + 50
A[:, y1:y2] = eps_drude
## A METALLIC HOLE...the fact that we have to recalculate the convolution
A2[x1:x2, :] = eps_drude
# plt.imshow(np.abs(A));
# plt.show();
## =============== Convolution Matrices ==============
E_r = cm.convmat2D(A, N, M)
E_r2 = cm.convmat2D(A2, N, M)
NM = (2 * N + 1) * (2 * M + 1)
## ================== GEOMETRY OF THE LAYERS AND CONVOLUTIONS ==================##
thickness_slab = 0.2
# in units of L0;
ER = [E_r, e_r * np.identity(NM), E_r2]
UR = [np.identity(NM), np.identity(NM),
np.identity(NM)]
layer_thicknesses = [thickness_slab, thickness_slab, thickness_slab]
# this retains SI unit convention
#source parameters
theta = 0 * degrees
#%elevation angle
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)
