def run_TMM_simulation(wavelengths, polarization_amplitudes, k_inc, theta, phi, ER, UR, layer_thicknesses,\
transmission_medium, incident_medium):
"""
:param wavelengths:
:param k_inc: holds kx, ky of the incident wave-vector (which is enough to specify kz since k0 is specified by wavelength
:param theta:
:param phi:
:param ER: relative dielectric constants of each layer
:param UR: relative permeability of each layer
:param layer_thicknesses:
:param transmission_medium:
:param incident_medium:
:return:
"""
assert len(layer_thicknesses) == len(ER) == len(UR); "number of layer parameters not the same"
ref = [];
trans = [];
I = np.matrix(np.eye(2, 2)); # unit 2x2 matrix
[e_r, m_r] = incident_medium;
[e_t, m_t] = transmission_medium;
n_i = np.sqrt(e_r*m_r);
[kx, ky] = k_inc;
normal_vector = np.array([0, 0, -1]) # positive z points down;
ate_vector = np.matrix([0, 1, 0]); # vector for the out of plane E-field
## ================= specify gap media ========================##
e_h = 1;
m_h = 1;
Pg, Qg, kzg = pq.P_Q_kz(kx, ky, e_h, m_h)
Wg = I; # Wg should be the eigenmodes of the E field, which paparently is the identity, yes for a homogeneous medium
sqrt_lambda = cmath.sqrt(-1) * Wg;
# remember Vg is really Qg*(Omg)^-1; Vg is the eigenmodes of the H fields
Vg = Qg * Wg * (sqrt_lambda) ** -1;
## ========================================== ##
[pte, ptm] = polarization_amplitudes;
for i in range(len(wavelengths)): # in SI units
## initialize global scattering matrix: should be a 4x4 identity so when we start the redheffer star, we get I*SR
Sg11 = np.matrix(np.zeros((2, 2)));
Sg12 = np.matrix(np.eye(2, 2));
Sg21 = np.matrix(np.eye(2, 2));
Sg22 = np.matrix(np.zeros((2, 2))); # matrices
Sg = np.block(
[[Sg11, Sg12], [Sg21, Sg22]]); # initialization is equivelant as that for S_reflection side matrix
### ================= Working on the Reflection Side =========== ##
Pr, Qr, kzr = pq.P_Q_kz(kx, ky, e_r, m_r)
## ============== values to keep track of =======================##
S_matrices = list();
kz_storage = [kzr];
X_storage = list();
## ==============================================================##
# define vacuum wavevector k0
lam0 = wavelengths[i]; # k0 and lam0 are related by 2*pi/lam0 = k0
k0 = 2*np.pi/lam0;
## modes of the layer
Om_r = np.matrix(cmath.sqrt(-1) * kzr * I);
X_storage.append(Om_r);
W_ref = I;
V_ref = Qr * Om_r.I; # can't play games with V like with W because matrices for V are complex
## calculating A and B matrices for scattering matrix
Ar, Br = sm.A_B_matrices(Wg, W_ref, Vg, V_ref);
S_ref, Sr_dict = sm.S_R(Ar, Br); # scatter matrix for the reflection region
S_matrices.append(S_ref);
Sg, D_r, F_r = rs.RedhefferStar(Sg, S_ref);
## go through the layers
for i in range(len(ER)):
# ith layer material parameters
e = ER[i];
m = UR[i];
# longitudinal k_vector
P, Q, kzl = pq.P_Q_kz(kx, ky, e, m)
kz_storage.append(kzl)
## E-field modes that can propagate in the medium
W_i = I;
## corresponding H-field modes.
Om = cmath.sqrt(-1) * kzl * I;
X_storage.append(Om)
V_i = Q * np.linalg.inv(Om);
# now defIne A and B
A, B = sm.A_B_matrices(Wg, W_i, Vg, V_i);
# calculate scattering matrix
S_layer, Sl_dict = sm.S_layer(A, B, layer_thicknesses[i], k0, Om)
S_matrices.append(S_layer);
## update global scattering matrix using redheffer star
Sg, D_i, F_i = rs.RedhefferStar(Sg, S_layer);
##========= Working on the Transmission Side==============##
Pt, Qt, kz_trans = pq.P_Q_kz(kx, ky, e_t, m_t);
kz_storage.append(kz_trans);
Om = cmath.sqrt(-1) * kz_trans * I;
Vt = Qt * np.linalg.inv(Om);
# get At, Bt
At, Bt = sm.A_B_matrices(Wg, I, Vg, Vt)
ST, ST_dict = sm.S_T(At, Bt)
S_matrices.append(ST);
# update global scattering matrix
Sg, D_t, F_t = rs.RedhefferStar(Sg, ST);
K_inc_vector = n_i * k0 * np.matrix([np.sin(theta) * np.cos(phi), \
np.sin(theta) * np.sin(phi), np.cos(theta)]);
# cinc is the c1+
E_inc, cinc, Polarization = ic.initial_conditions(K_inc_vector, theta, normal_vector, pte, ptm)
## COMPUTE FIELDS
Er = Sg[0:2, 0:2] * cinc; # S11; #(cinc = initial mode amplitudes), cout = Sg*cinc; #2d because Ex, Ey...
Et = Sg[2:, 0:2] * cinc; # S21
Er = np.squeeze(np.asarray(Er));
Et = np.squeeze(np.asarray(Et));
Erx = Er[0];
Ery = Er[1];
Etx = Et[0];
Ety = Et[1];
# apply the grad(E) = 0 equation to get z components
Erz = -(kx * Erx + ky * Ery) / kzr;
Etz = -(kx * Etx + ky * Ety) / kz_trans; ## using divergence of E equation here
# add in the Erz component to vectors
Er = np.matrix([Erx, Ery, Erz]); # a vector
Et = np.matrix([Etx, Ety, Etz]);
R = np.linalg.norm(Er) ** 2;
T = np.linalg.norm(Et) ** 2;
ref.append(R);
trans.append(T);
return ref, trans
def run_TMM_anisotropic(wavelengths, polarization_amplitudes, theta, phi, ER, UR, layer_thicknesses,\
transmission_medium, incident_medium):
ref = [];
trans = []
wvlen_scan = np.linspace(0.5, 4, 1000);
thickness = 0.5; # 1 layer
for wvlen in wvlen_scan:
k0 = 2*np.pi/wvlen;
kx = np.sin(theta)*np.cos(phi);
ky = np.sin(theta)*np.sin(phi);
# we will build the system by rows?
a11 = -1j*(ky*mu_tensor[1,2]/mu_tensor[2,2] + kx*(epsilon_tensor[2,0]/epsilon_tensor[2,2]))
a12 = 1j*kx*(mu_tensor[1,2]/mu_tensor[2,2] - epsilon_tensor[2,1]/epsilon_tensor[2,2]);
a13 = kx*ky/epsilon_tensor[2,2] + mu_tensor[1,0] - mu_tensor[1,2]*mu_tensor[2,0]/mu_tensor[2,2];
a14 = -kx**2/epsilon_tensor[2,2] + mu_tensor[1,1]- mu_tensor[1,2]*mu_tensor[2,1]/mu_tensor[2,2];
a21 = 1j* ky *(mu_tensor[0,2]/mu_tensor[2,2] - epsilon_tensor[2,0]/epsilon_tensor[2,2]);
a22 = -1j * kx*(mu_tensor[0,2]/mu_tensor[2,2]) +ky *(epsilon_tensor[2,1]/epsilon_tensor[2,2]);
a23 = ky**2/epsilon_tensor[2,2] - mu_tensor[0,0] + mu_tensor[0,2]*mu_tensor[2,0]/mu_tensor[2,2];
a24 = -kx*ky/epsilon_tensor[2,2] - mu_tensor[0,1] + mu_tensor[0,2]*mu_tensor[2,1]/mu_tensor[2,2];
a31 = (kx*ky/mu_tensor[2,2] + epsilon_tensor[1,0] - epsilon_tensor[1,2]*epsilon_tensor[2,0]/epsilon_tensor[2,2])
a32 = (-kx**2/mu_tensor[2,2] +epsilon_tensor[1,1] - epsilon_tensor[1,2]*epsilon_tensor[2,1]/epsilon_tensor[2,2]);
a33 = -1j*(ky*(epsilon_tensor[1,2]/epsilon_tensor[2,2])+kx*(mu_tensor[2,0]/mu_tensor[2,2]));
a34 = 1j*kx*(epsilon_tensor[1,2]/epsilon_tensor[2,2]-mu_tensor[2,1]/mu_tensor[2,2] )
a41 = ky**2/mu_tensor[2,2] - epsilon_tensor[0,0] + epsilon_tensor[0,2]*epsilon_tensor[2,0]/epsilon_tensor[2,2];
a42 = -kx*ky/mu_tensor[2,2] - epsilon_tensor[0,1] + epsilon_tensor[0,2]*epsilon_tensor[2,1]/epsilon_tensor[2,2];
a43 = 1j*ky*(epsilon_tensor[0,2]/epsilon_tensor[2,2]-mu_tensor[2,0]/mu_tensor[2,2] );
a44 = -1j*(kx*(epsilon_tensor[0,2]/epsilon_tensor[2,2])+ky*(mu_tensor[2,1]/mu_tensor[2,2]));
A = np.matrix([[a11, a12, a13, a14],
[a21, a22, a23, a24],
[a31, a32, a33, a34],
[a41, a42, a43, a44]]);
#print(np.linalg.cond(A))
eigenvals, eigenmodes = np.linalg.eig(A);
rounded_eigenvals = np.round(eigenvals, 3)
sorted_eigs, sorted_inds = nonHermitianEigenSorter(np.round(eigenvals,10));
## ========================================================
W_i = eigenmodes[0:2, sorted_inds];
V_i = eigenmodes[2:, sorted_inds];
Om = np.matrix(np.diag(sorted_eigs) );
#print(np.round(W_i,3), np.round(V_i,3), np.round(Om,3))
#then what... match boundary conditions... try using the gaylord formulation.
# or use the scattering matrix formalism, where we still, technically deal with all field components...
Sg11 = np.matrix(np.zeros((2, 2))); Sg12 = np.matrix(np.eye(2, 2));
Sg21 = np.matrix(np.eye(2, 2)); Sg22 = np.matrix(np.zeros((2, 2))); # matrices
Sg = np.block([[Sg11, Sg12], [Sg21, Sg22]]); # initialization is equivelant as that for S_reflection side matrix
### ================= Working on the Reflection Side =========== ##
Pr, Qr, kzr = pq.P_Q_kz(kx, ky, e_r, m_r)
## ============== values to keep track of =======================##
S_matrices = list();
kz_storage = [kzr];
X_storage = list();
## ==============================================================##
# define vacuum wavevector k0
lam0 = wvlen; # k0 and lam0 are related by 2*pi/lam0 = k0
k0 = 2*np.pi/lam0;
## modes of the layer
Om_r = np.matrix(cmath.sqrt(-1) * kzr * I);
X_storage.append(Om_r);
W_ref = I;
V_ref = Qr * Om_r.I; # can't play games with V like with W because matrices for V are complex
#print(Om_r)
## calculating A and B matrices for scattering matrix
Ar, Br = sm.A_B_matrices(Wg, W_ref, Vg, V_ref);
S_ref, Sr_dict = sm.S_R(Ar, Br); # scatter matrix for the reflection region
S_matrices.append(S_ref);
Sg, D_r, F_r = rs.RedhefferStar(Sg, S_ref);
# longitudinal k_vector
## ============ WORKING INSIDE ANISOTROPIC LAYER ================#
# now defIne A and B
Al, Bl = sm.A_B_matrices(Wg, W_i, Vg, V_i);
# calculate scattering matrix
S_layer, Sl_dict = sm.S_layer(Al, Bl, thickness, k0, Om)
S_matrices.append(S_layer);
## update global scattering matrix using redheffer star
Sg, D_i, F_i = rs.RedhefferStar(Sg, S_layer);
##========= Working on the Transmission Side==============##
Pt, Qt, kz_trans = pq.P_Q_kz(kx, ky, e_t, m_t);
kz_storage.append(kz_trans);
Omt = cmath.sqrt(-1) * kz_trans * I;
Vt = Qt * np.linalg.inv(Omt);
# get At, Bt
At, Bt = sm.A_B_matrices(Wg, I, Vg, Vt)
ST, ST_dict = sm.S_T(At, Bt)
S_matrices.append(ST);
# update global scattering matrix
Sg, D_t, F_t = rs.RedhefferStar(Sg, ST);
K_inc_vector = n_i * k0 * np.matrix([np.sin(theta) * np.cos(phi), \
np.sin(theta) * np.sin(phi), np.cos(theta)]);
# cinc is the c1+
E_inc, cinc, Polarization = ic.initial_conditions(K_inc_vector, theta, normal_vector, pte, ptm)
## COMPUTE FIELDS
Er = Sg[0:2, 0:2] * cinc; # S11; #(cinc = initial mode amplitudes), cout = Sg*cinc; #2d because Ex, Ey...
Et = Sg[2:, 0:2] * cinc; # S21
Er = np.squeeze(np.asarray(Er));
Et = np.squeeze(np.asarray(Et));
Erx = Er[0]; Ery = Er[1]; Etx = Et[0]; Ety = Et[1];
# apply the grad(E) = 0 equation to get z components, this equation comes out of the longitudinal equation
# or the divergence equation and is valid since the transmission region is VACUUM (as is the reflection region)
Erz = -(kx * Erx + ky * Ery) / kzr; #uses the divergence law
Etz = -(kx * Etx + ky * Ety) / kz_trans; ## using divergence of E equation here
# add in the Erz component to vectors
Er = np.matrix([Erx, Ery, Erz]); # a vector
Et = np.matrix([Etx, Ety, Etz]);
R = np.linalg.norm(Er) ** 2;
T = np.linalg.norm(Et) ** 2;
ref.append(R);
trans.append(T);
return ref, trans;