optics.py
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# -*- coding: utf-8 -*-
"""
Created on Fri Jun 22 16:22:17 2018
@author: david
"""
import numpy as np
import scipy as sp
import scipy.ndimage
import matplotlib
import math
import matplotlib.pyplot as plt
#adjust the components of E0 so that they are orthogonal to d
#if d is 2D, the z component is calculated such that ||[dx, dy, dz]||=1
def orthogonalize(E, d):
if d.shape[-1] == 2:
dz = 1 - d[..., 0]**2 - d[..., 1]**2
d = np.append(d, dz)
s = np.cross(E, d)
dE = np.cross(d, s)
return dE
#return dE/np.linalg.norm(dE) * np.linalg.norm(E)
def intensity(E):
Econj = np.conj(E)
I = np.sum(E*Econj, axis=-1)
return np.real(I)
class planewave:
#implement all features of a plane wave
# k, E, frequency (or wavelength in a vacuum)--l
# try to enforce that E and k have to be orthogonal
#initialization function that sets these parameters when a plane wave is created
def __init__ (self, k, E):
self.k = np.array(k)
self.E = E
if ( np.linalg.norm(k) > 1e-15 and np.linalg.norm(E) >1e-15):
s = np.cross(k, E) #compute an orthogonal side vector
s = s / np.linalg.norm(s) #normalize it
Edir = np.cross(s, k) #compute new E vector which is orthogonal
self.k = np.array(k)
self.E = Edir / np.linalg.norm(Edir) * np.linalg.norm(E)
def __str__(self):
return str(self.k) + "\n" + str(self.E) #for verify field vectors use print command
def kmag(self):
return math.sqrt(self.k[0] ** 2 + self.k[1] ** 2 + self.k[2] ** 2)
#function that renders the plane wave given a set of coordinates
def evaluate(self, X, Y, Z):
k_dot_r = self.k[0] * X + self.k[1] * Y + self.k[2] * Z #phase term k*r
ex = np.exp(1j * k_dot_r) #E field equation = E0 * exp (i * (k * r)) here we simply set amplitude as 1
Ef = ex[..., None] * self.E
return Ef
class objective:
#initialize an objective lens
# sin_alpha is the sine of the angle subtended by the objective (numerical aperture in a refractive index of 1.0)
# n is the real refractive index of the imaging media (default air)
# N is the resolution of the 2D objective map (field at the input and output aperture)
def __init__(self, sin_alpha, n, N):
#set member variables
self.sin_alpha = sin_alpha
self.n = n
#calculate the transfer function from the back aperture to the front aperture
self.N = N
self._CI(N)
self._CR(N)
#calculates the basis vectors describing the mapping between the back and front aperture
def _calc_bases(self, N):
#calculate the (sx, sy) coordinates (angular spectrum) corresponding to each position in the entrance/exit pupil
s = np.linspace(-self.sin_alpha, self.sin_alpha, N)
[SX, SY] = np.meshgrid(s, s)
S_SQ = SX**2 + SY**2 #calculate sx^2 + sy^2
BAND_M = S_SQ <= self.sin_alpha**2 #find a mask for pixels within the bandpass of the objective
self.BAND_M = BAND_M
#pull out pixels that pass through the objective bandpass
SX = SX[BAND_M]
SY = SY[BAND_M]
S_SQ = S_SQ[BAND_M]
SZ = np.sqrt(1 - S_SQ) #calculate sz = sqrt(1 - sx^2 - sy^2) in normalized (-1, 1) coordinates
#calculate the apodization functions for each polarization
self.FS = 1.0 / np.sqrt(SZ)
self.FP = self.FS
#calculate the S and P basis vectors
THETA = np.arctan2(SY, SX)
SX_SXSY = np.cos(THETA)
SY_SXSY = np.sin(THETA)
v_size = (len(SX), 3)
VS_VSP = np.zeros(v_size)
VP = np.zeros(v_size)
VPP = np.zeros(v_size)
VS_VSP[:, 0] = -SY_SXSY
VS_VSP[:, 1] = SX_SXSY
VP[:, 0] = -SX_SXSY
VP[:, 1] = -SY_SXSY
SXSY = np.sqrt(S_SQ)
VPPZ = np.zeros(SXSY.shape)
DC_M = SXSY > 0
VPPZ[DC_M] = S_SQ[DC_M] / SXSY[DC_M]
VPP[:, 0] = -SX_SXSY * SZ
VPP[:, 1] = -SY_SXSY * SZ
VPP[:, 2] = VPPZ
return [VS_VSP, VP, VPP]
#calculate the transfer function that maps the back aperture field to the front aperture field (Eq. 19 in Davis et al.)
def _CI(self, N):
[VS_VSP, VP, VPP] = self._calc_bases(N)
VSP_VST = np.matmul(VS_VSP[:, :, None], VS_VSP[:, None, :])
VPP_VPT = np.matmul(VPP[:, :, None], VP[:, None, :])
self.CI = self.FS[:, None, None] * VSP_VST + self.FP[:, None, None] * VPP_VPT
def _CR(self, N):
[VS_VSP, VP, VPP] = self._calc_bases(N)
VSP_VST = np.matmul(VS_VSP[:, :, None], VS_VSP[:, None, :])
VP_VPPT = np.matmul(VP[:, :, None], VPP[:, None, :])
self.CR = 1.0 / self.FS[:, None, None] * VSP_VST + 1.0 / self.FP[:, None, None] * VP_VPPT
#condenses a field defined at the back aperture of the objective
# E is a 3xNxN image of the field at the back aperture
def back2front(self, E):
Ein = E[self.BAND_M, :, None]
Efront = np.zeros((self.N, self.N, 3), dtype=np.complex128)
Efront[self.BAND_M, :] = np.squeeze(np.matmul(self.CI, Ein))
return Efront
#propagates a field from the front aperture of the objective to the back aperture
# E is a 3xNxN image of the field at the front aperture
def front2back(self, E):
Ein = E[self.BAND_M, :, None]
Eback = np.zeros((self.N, self.N, 3), dtype=np.complex128)
Eback[self.BAND_M, :] = np.squeeze(np.matmul(self.CR, Ein))
return Eback
#calculates the image at the focus given the image at the back aperture
# E is the field at the back aperture of the objective
# k0 is the wavenumber in a vacuum (2pi/lambda_0)
def focus(self, E, k0, RX, RY, RZ):
#calculate the angular spectrum at the front aperture
Ef = self.back2front(E)
#generate an array describing the valid components of S
s = np.linspace(-self.sin_alpha, self.sin_alpha, self.N)
[SX, SY] = np.meshgrid(s, s)
S = [SX[self.BAND_M], SY[self.BAND_M]]
SXSY_SQ = S[0]**2 + S[1]**2
S.append(np.sqrt(1 - SXSY_SQ))
S = np.array(S)
#calculate the k vector (accounting for both k0 and the refractive index of the immersion medium)
K = k0 * self.n * np.transpose(S)[:, None, :]
#calculate the dot product of K and R
R = np.moveaxis(np.array([RX, RY, RZ]), 0, -1)[..., None, :, None]
K_DOT_R = np.squeeze(np.matmul(K, R))
EX = np.exp(1j * K_DOT_R)[..., None]
Ewaves = EX * Ef[self.BAND_M]
Efocus = np.sum(Ewaves, -2)
return Efocus
class gaussian_beam:
#generate a beam
# (Ex, Ey) specifies the polarization
# p specifies the focal point
# l is the free-space wavelength
# sigma_p is the standard deviation of the Gaussian beam at the focal point
# amp is the amplitude of the DC component of the Gaussian beam
def __init__(self, x_polar, y_polar, lambda_0=1, amp=1, sigma_p=None, sigma_k=None):
self.E = [x_polar, y_polar]
self.l = lambda_0
#calculate the standard deviation of the Gaussian function in k-space
if(sigma_p is None and sigma_k is None):
self.sigma = 2 * math.pi / lambda_0
elif(sigma_p is not None):
self.sigma = 1.0 / sigma_p
else:
self.sigma = sigma_k
#self.s = self.sigma_p2k(sigma_p)
self.A = amp
#calculates the maximum k-space frequency supported by this beam
def kmax(self):
return 2 * np.pi / self.l
#calculate the standard deviation of the Gaussian function in k space given the std at the focal point
# def sigma_p2k(self, sigma):
# this code is based on: http://www.robots.ox.ac.uk/~az/lectures/ia/lect2.pdf
# f(r) = 1 / (2 pi sigma^2) e^(-r^2 / (2 sigma^2))
# F(kx, ky) = f( |k_par| ) = e^( -2 pi^2 |k_par|^2 sigma^2 )
# Basically, the Fourier transform of a normally distributed Gaussian is
# a non-scaled Gaussian with standard deviation 1/(2 pi sigma)
#return 1.0/(2 * math.pi * sigma)
#return 1.0 / sigma
def kspace(self, Kx, Ky):
#sigma = self.s * (np.pi / self.l)
#G = 1 / (2 * np.pi * sigma **2) * np.exp(- (0.5/(sigma**2)) * (Kx ** 2 + Ky ** 2))
G = self.A * np.exp(- (Kx ** 2 + Ky ** 2) / (2 * self.sigma ** 2))
#calculate k-vectors outside of the band-limit
B = ((Kx ** 2 + Ky **2) > self.kmax() ** 2)
#calculate Kz (required for calculating the Ez component of E)
#Kz = np.lib.scimath.sqrt(self.kmax() ** 2 - Kx**2 - Ky**2)
Ex = self.E[0] * G
Ey = self.E[1] * G
#Ez = -Kx/Kz * Ex
Ez = np.zeros(Ex.shape)
#set all values outside of the band limit to zero
Ex[B] = 0
Ey[B] = 0
Ez[B] = 0
return [Ex, Ey, Ez]
# The PSF at the focal point is the convolution of a Bessel function and a Gaussian
# This function calculates the parameters of the Gaussian function in terms of a*e^{-x^2 / (2*c^2)}
# a (return) is the scale factor for the Gaussian
# c (return) is the standard deviation
def psf_gaussian(self):
# this code is based on: http://mathworld.wolfram.com/FourierTransformGaussian.html
#a = math.sqrt( 2 * math.pi * (self.s ** 2) )
#c = 1.0 / (2 * math.pi * self.s)
a = math.sqrt( (self.sigma ** 2) )
c = 1.0 / (self.sigma)
return [a, c]
# The PSF at the focal point is the convolution of a Bessel function and a Gaussian
# This function calculates the parameters of the bessel function in terms of a J_1(pi * a * x) / x
def psf_bessel(self):
return self.kmax() / math.pi
# calculate the beam point spread function at the focal plane
# D is the image extent for the square: (-D, D, -D, D)
# N is the number of spatial samples along one dimension of the image (NxN)
# order is the mathematical order of the calculation (in terms of the resolution of the FFT)
def psf(self, D, N):
[a, c] = self.psf_gaussian()
alpha = self.psf_bessel()
#calculate the sample positions along the x-axis
x = np.linspace(-D, D, N)
dx = x[1] - x[0]
[X, Y] = np.meshgrid(x, x)
R = np.sqrt(X**2 + Y**2)
bessel = alpha * sp.special.j1(math.pi * alpha * R) / R
return a * sp.ndimage.filters.gaussian_filter(bessel, c / dx)
#return the k-space transform of the beam at p as an NxN array
def kspace_image(self, N):
#calculate the band limit [-2*pi/lambda, 2*pi/lambda]
kx = np.linspace(-self.kmax(), self.kmax(), N)
ky = kx
#generate a meshgrid for the field
[Kx, Ky] = np.meshgrid(kx, ky)
#generate a Gaussian with a standard deviation of sigma * pi/lambda
return self.kspace(Kx, Ky)
# calculate the practical band limit of the beam such that
# all frequencies higher than the returned values are less than epsilon
def practical_band_limit(self, epsilon = 0.001):
return min(self.sigma * math.sqrt(math.log(1.0/epsilon)), self.kmax())
#return a plane wave decomposition of the beam using N plane waves
def decompose(self, N):
PW = [] #create an empty list of plane waves
# sample a cylinder and project those samples onto a unit sphere
theta = np.random.uniform(0, 2*np.pi, N)
# get the practical band limit to minimize sampling
band_limit = self.practical_band_limit() / self.kmax()
phi = np.arccos(np.random.uniform(1 - band_limit, 1, N))
kx = -self.kmax() * np.sin(phi) * np.cos(theta)
ky = -self.kmax() * np.sin(phi) * np.sin(theta)
mc_weight = 2 * math.pi / N
E0 = self.kspace(kx, ky)
kz = np.sqrt(self.kmax() ** 2 - kx**2 - ky**2)
for i in range(0, N):
k = [kx[i], ky[i], kz[i]]
E = [E0[0][i] * mc_weight, E0[1][i] * mc_weight, E0[2][i] * mc_weight]
w = planewave(k, E)
PW.append(w)
return PW
class layersample:
#generate a layered sample based on a list of layer refractive indices
# layer_ri is a list of complex refractive indices for each layer
# z is the list of boundary positions along z
def __init__(self, layer_ri, z):
self.n = np.array(layer_ri).astype(np.complex128)
self.z = np.array(z)
#calculate the index of the field component associated with a layer
# l is the layer index [0, L)
# c is the component (x=0, y=1, z=2)
# d is the direction (0 = transmission, 1 = reflection)
def i(self, l, c, d):
i = l * 6 + d * 3 + c - 3
return i
# generate the linear system corresponding to this layered sample and plane wave
# s is the direction vector scaled by the refractive index
# k0 is the free-space k-vector
# E is the field vector for the incident plane wave
def generate_linsys(self, s, k0, E):
#allocate space for the matrix
L = len(self.n)
M = np.zeros((6*(L-1), 6*(L-1)), dtype=np.complex128)
#allocate space for the RHS vector
b = np.zeros(6*(L-1), dtype=np.complex128)
#initialize a counter for the equation number
ei = 0
#calculate the sz component for each layer
self.sz = np.zeros(L, dtype=np.complex128)
for l in range(L):
self.sz[l] = np.sqrt(self.n[l]**2 - s[0]**2 - s[1]**2)
#set constraints based on Gauss' law
for l in range(0, L):
#sz = np.sqrt(self.n[l]**2 - s[0]**2 - s[1]**2)
#set the upward components for each layer
# note that layer L-1 does not have a downward component
# David I, Equation 7
if l != L-1:
M[ei, self.i(l, 0, 1)] = s[0]
M[ei, self.i(l, 1, 1)] = s[1]
M[ei, self.i(l, 2, 1)] = -self.sz[l]
ei = ei+1
#set the downward components for each layer
# note that layer 0 does not have a downward component
# Davis I, Equation 6
if l != 0:
M[ei, self.i(l, 0, 0)] = s[0]
M[ei, self.i(l, 1, 0)] = s[1]
M[ei, self.i(l, 2, 0)] = self.sz[l]
ei = ei+1
#enforce a continuous field across boundaries
for l in range(1, L):
sz0 = self.sz[l-1]
sz1 = self.sz[l]
A = np.exp(1j * k0 * sz0 * (self.z[l] - self.z[l-1]))
if l < L - 1:
dl = self.z[l] - self.z[l+1]
arg = -1j * k0 * sz1 * dl
B = np.exp(arg)
#if this is the second layer, use the simplified equations that account for the incident field
if l == 1:
M[ei, self.i(0, 0, 1)] = 1
M[ei, self.i(1, 0, 0)] = -1
if L > 2:
#print(-B, M[ei, self.i(1, 0, 1)])
M[ei, self.i(1, 0, 1)] = -B
b[ei] = -A * E[0]
ei = ei + 1
M[ei, self.i(0, 1, 1)] = 1
M[ei, self.i(1, 1, 0)] = -1
if L > 2:
M[ei, self.i(1, 1, 1)] = -B
b[ei] = -A * E[1]
ei = ei + 1
M[ei, self.i(0, 2, 1)] = s[1]
M[ei, self.i(0, 1, 1)] = sz0
M[ei, self.i(1, 2, 0)] = -s[1]
M[ei, self.i(1, 1, 0)] = sz1
if L > 2:
M[ei, self.i(1, 2, 1)] = -B*s[1]
M[ei, self.i(1, 1, 1)] = -B*sz1
b[ei] = A * sz0 * E[1] - A * s[1]*E[2]
ei = ei + 1
M[ei, self.i(0, 0, 1)] = -sz0
M[ei, self.i(0, 2, 1)] = -s[0]
M[ei, self.i(1, 0, 0)] = -sz1
M[ei, self.i(1, 2, 0)] = s[0]
if L > 2:
M[ei, self.i(1, 0, 1)] = B*sz1
M[ei, self.i(1, 2, 1)] = B*s[0]
b[ei] = A * s[0] * E[2] - A * sz0 * E[0]
ei = ei + 1
#if this is the last layer, use the simplified equations that exclude reflections from the last layer
elif l == L-1:
M[ei, self.i(l-1, 0, 0)] = A
M[ei, self.i(l-1, 0, 1)] = 1
M[ei, self.i(l, 0, 0)] = -1
ei = ei + 1
M[ei, self.i(l-1, 1, 0)] = A
M[ei, self.i(l-1, 1, 1)] = 1
M[ei, self.i(l, 1, 0)] = -1
ei = ei + 1
M[ei, self.i(l-1, 2, 0)] = A*s[1]
M[ei, self.i(l-1, 1, 0)] = -A*sz0
M[ei, self.i(l-1, 2, 1)] = s[1]
M[ei, self.i(l-1, 1, 1)] = sz0
M[ei, self.i(l, 2, 0)] = -s[1]
M[ei, self.i(l, 1, 0)] = sz1
ei = ei + 1
M[ei, self.i(l-1, 0, 0)] = A*sz0
M[ei, self.i(l-1, 2, 0)] = -A*s[0]
M[ei, self.i(l-1, 0, 1)] = -sz0
M[ei, self.i(l-1, 2, 1)] = -s[0]
M[ei, self.i(l, 0, 0)] = -sz1
M[ei, self.i(l, 2, 0)] = s[0]
ei = ei + 1
#otherwise use the full set of boundary conditions
else:
M[ei, self.i(l-1, 0, 0)] = A
M[ei, self.i(l-1, 0, 1)] = 1
M[ei, self.i(l, 0, 0)] = -1
M[ei, self.i(l, 0, 1)] = -B
ei = ei + 1
M[ei, self.i(l-1, 1, 0)] = A
M[ei, self.i(l-1, 1, 1)] = 1
M[ei, self.i(l, 1, 0)] = -1
M[ei, self.i(l, 1, 1)] = -B
ei = ei + 1
M[ei, self.i(l-1, 2, 0)] = A*s[1]
M[ei, self.i(l-1, 1, 0)] = -A*sz0
M[ei, self.i(l-1, 2, 1)] = s[1]
M[ei, self.i(l-1, 1, 1)] = sz0
M[ei, self.i(l, 2, 0)] = -s[1]
M[ei, self.i(l, 1, 0)] = sz1
M[ei, self.i(l, 2, 1)] = -B*s[1]
M[ei, self.i(l, 1, 1)] = -B*sz1
ei = ei + 1
M[ei, self.i(l-1, 0, 0)] = A*sz0
M[ei, self.i(l-1, 2, 0)] = -A*s[0]
M[ei, self.i(l-1, 0, 1)] = -sz0
M[ei, self.i(l-1, 2, 1)] = -s[0]
M[ei, self.i(l, 0, 0)] = -sz1
M[ei, self.i(l, 2, 0)] = s[0]
M[ei, self.i(l, 0, 1)] = B*sz1
M[ei, self.i(l, 2, 1)] = B*s[0]
ei = ei + 1
return [M, b]
#create a matrix for a single plane wave specified by k and E
# d = [dx, dy] are the x and y coordinates of the normalized direction of propagation
# k0 is the free space wave number (2 pi / lambda0)
# E is the electric field vector
def solve1(self, d, k0, E):
#s is the plane wave direction scaled by the refractive index
s = np.array(d) * self.n[0]
#store the matrix and RHS vector (for debugging)
[self.M, self.b] = self.generate_linsys(s, k0, E)
#self.M = M
#self.b = b
#evaluate the linear system
P = np.linalg.solve(self.M, self.b)
#save the results (also for debugging)
self.P = P
#store the coefficients for each layer
L = len(self.n) # calculate the number of layers
self.Pt = np.zeros((3, L), np.complex128)
self.Pr = np.zeros((3, L), np.complex128)
for l in range(L):
if l == 0:
self.Pt[:, 0] = [E[0], E[1], E[2]]
else:
px = P[self.i(l, 0, 0)]
py = P[self.i(l, 1, 0)]
pz = P[self.i(l, 2, 0)]
self.Pt[:, l] = [px, py, pz]
if l == L-1:
self.Pr[:, L-1] = [0, 0, 0]
else:
px = P[self.i(l, 0, 1)]
py = P[self.i(l, 1, 1)]
pz = P[self.i(l, 2, 1)]
self.Pr[:, l] = [px, py, pz]
#store values required for evaluation
#store k
self.k = k0
#store sx and sy
self.s = np.array([s[0], s[1]])
self.solved = True
#evaluate a solved homogeneous substrate
def evaluate(self, X, Y, Z):
if not self.solved:
print("ERROR: the layered substrate hasn't been solved")
return
#this code is a bit cumbersome and could probably be optimized
# Basically, it vectorizes everything by creating an image
# such that the value at each pixel is the corresponding layer
# that the pixel resides in. That index is then used to calculate
# the field within the layer
#allocate space for layer indices
LI = np.zeros(Z.shape, dtype=np.int)
#find the layer index for each sample point
L = len(self.z)
LI[Z < self.z[0]] = 0
for l in range(L-1):
idx = np.logical_and(Z > self.z[l], Z <= self.z[l+1])
LI[idx] = l
LI[Z > self.z[-1]] = L - 1
#calculate the appropriate phase shift for the wave transmitted through the layer
Ph_t = np.exp(1j * self.k * self.sz[LI] * (Z - self.z[LI]))
#calculate the appropriate phase shift for the wave reflected off of the layer boundary
LIp = LI + 1
LIp[LIp >= L] = 0
Ph_r = np.exp(-1j * self.k * self.sz[LI] * (Z - self.z[LIp]))
# print(Ph_r)
Ph_r[LI >= L-1] = 0
#calculate the phase shift based on the X and Y positions
Ph_xy = np.exp(1j * self.k * (self.s[0] * X + self.s[1] * Y))
#apply the phase shifts
Et = self.Pt[:, LI] * Ph_t[:, :]
Er = self.Pr[:, LI] * Ph_r[:, :]
#add everything together coherently
E = (Et + Er) * Ph_xy[:, :]
#return the electric field
return np.moveaxis(E, 0, -1)
def example_psf():
#specify the angle of accepted waves (NA if n = 1)
sin_angle = 0.8
#refractive index of the imaging medium
n = 1.0
#specify the size of the spectral domain (resolution of the front/back aperture)
N = 100
#create a new objective, specifying the NA (angle * refractive index) and resolution of the transform
O = objective(sin_angle, n, N)
#specify the size of the spatial domain
D = 10
#specify the resolution of the spatial domain
M = 100
#free-space wavelength
l0 = 1
#calculate the free-space wavenumber
k0 = 2 * np.pi * l0
#specify the field incident at the aperture
pap = np.array([1, 0, 0]) #polarization
sap = np.array([0, 0, 1]) #incident angle of the plane wave (will be normalized)
sap = sap / np.linalg.norm(sap) #calculate the normalized direction vector for the plane wave
k0ap = sap * k0 #calculate the k-vector of the plane wave incident on the aperture
#create a plane wave
pw = planewave(k0ap, pap)
#generate the domain for the simulation (where the focus will be simulated)
x = np.linspace(-D, D, M)
z = np.linspace(-D, D, M)
[RX, RZ] = np.meshgrid(x, z)
RY = np.zeros(RX.shape)
#create an image of the back aperture (just an incident plane wave)
xap = np.linspace(-D, D, N)
[XAP, YAP] = np.meshgrid(xap, xap)
ZAP = np.zeros(XAP.shape)
#evaluate the plane wave at the back aperture to produce a 2D image of the field
EAP = pw.evaluate(XAP, YAP, ZAP)
#calculate the field at the focus (coordinates are specified in RX, RY, and RZ)
Efocus = O.focus(EAP, k0, RX, RY, RZ)
#display an image of the calculated field
plt.figure()
plt.imshow(intensity(Efocus))
plt.colorbar()
return Efocus
#This sample code produces a field similar to Figure 2 in Davis et al., 2010
def example_layer():
#set the material properties
depths = [-50, -15, 15, 40] #specify the position (first boundary) of each layer (first boundary is where the field is defined)
n = [1.0, 1.4+1j*0.05, 1.4, 1.0] #specify the refractive indices of each layer
#create a layered sample
layers = layersample(n, depths)
#set the input light parameters
d = np.array([0.5, 0]) #direction of propagation of the plane wave
#d = d / np.linalg.norm(d) #normalize this direction vector
l0 = 5 #specify the wavelength in free-space
k0 = 2 * np.pi / l0 #calculate the wavenumber in free-space
E0 = [1, 1, 0] #specify the E vector
E0 = orthogonalize(E0, d) #make sure that both vectors are orthogonal
#solve for the substrate field
layers.solve1(d, k0, E0)
#set the simulation domain
N = 512
M = 1024
D = [-40, 80, 0, 60]
x = np.linspace(D[2], D[3], N)
z = np.linspace(D[0], D[1], M)
[X, Z] = np.meshgrid(x, z)
Y = np.zeros(X.shape)
E = layers.evaluate(X, Y, Z)
Er = np.real(E)
I = intensity(E)
plt.figure()
plt.set_cmap("afmhot")
matplotlib.rcParams.update({'font.size': 32})
plt.subplot(1, 4, 1)
plt.imshow(Er[..., 0], extent=(D[3], D[2], D[1], D[0]))
plt.title("Ex")
plt.subplot(1, 4, 2)
plt.imshow(Er[..., 1], extent=(D[3], D[2], D[1], D[0]))
plt.title("Ey")
plt.subplot(1, 4, 3)
plt.imshow(Er[..., 2], extent=(D[3], D[2], D[1], D[0]))
plt.title("Ez")
plt.subplot(1, 4, 4)
plt.imshow(I, extent=(D[3], D[2], D[1], D[0]))
plt.title("I")
def example_psflayer():
#specify the angle of accepted waves (NA if n = 1)
sin_angle = 0.7
#refractive index of the imaging medium
n = 1.0
#specify the size of the spectral domain (resolution of the front/back aperture)
N = 31
#create a new objective, specifying the NA (angle * refractive index) and resolution of the transform
O = objective(sin_angle, n, N)
#specify the size of the spatial domain
D = 5
#specify the resolution of the spatial domain
M = 200
#free-space wavelength
l0 = 1
#calculate the free-space wavenumber
k0 = 2 * np.pi /l0
#specify the field incident at the aperture
pap = np.array([1, 0, 0]) #polarization
sap = np.array([0, 0, 1]) #incident angle of the plane wave (will be normalized)
sap = sap / np.linalg.norm(sap) #calculate the normalized direction vector for the plane wave
k0ap = sap * k0 #calculate the k-vector of the plane wave incident on the aperture
#create a plane wave
pw = planewave(k0ap, pap)
#create an image of the back aperture (just an incident plane wave)
xap = np.linspace(-D, D, N)
[XAP, YAP] = np.meshgrid(xap, xap)
ZAP = np.zeros(XAP.shape)
#evaluate the plane wave at the back aperture to produce a 2D image of the field
Eback = pw.evaluate(XAP, YAP, ZAP)
Efront = O.back2front(Eback)
#set the material properties
depths = [0, -1, 0, 2] #specify the position (first boundary) of each layer (first boundary is where the field is defined)
n = [1.0, 1.4+1j*0.05, 1.4, 1.0] #specify the refractive indices of each layer
#create a layered sample
L = layersample(n, depths)
#generate the domain for the simulation (where the focus will be simulated)
x = np.linspace(-D, D, M)
z = np.linspace(-D, D, M)
[RX, RZ] = np.meshgrid(x, z)
RY = np.zeros(RX.shape)
i = 0
ang = np.linspace(-sin_angle, sin_angle, N)
for yi in range(N):
dy = ang[yi]
for xi in range(N):
dx = ang[xi]
dxdy_sq = dx**2 + dy**2
if dxdy_sq <= sin_angle**2:
d = np.array([dx, dy])
e = Efront[xi, yi, :]
L.solve1(d, k0, e)
if i == 0:
E = L.evaluate(RX, RY, RZ)
i = i + 1
else:
E = E + L.evaluate(RX, RY, RZ)
plt.imshow(intensity(E))
def example_spp():
l = 0.647
k0 = 2 * np.pi / l
n_oil = 1.51
n_mica = 1.56
n_gold = 0.16049 + 1j * 3.5726
n_water = 1.33
#all units are in um
d_oil = 100
d_mica = 80
d_gold = 0.05
#specify the parameters for the layered sample
zp = [0, d_oil, d_oil + d_mica, d_oil + d_mica + d_gold]
n = [n_oil, n_mica, n_gold, n_water]
#generate a layered sample
L = layersample(n, zp)
N = 10000
min_theta = -90
max_theta = 90
DEG = np.linspace(min_theta, max_theta, N)
I = np.zeros(N)
for i in range(N):
theta = DEG[i] * np.pi / 180
#create a plane wave
x = np.sin(theta)
z = np.cos(theta)
d = [x, 0]
#calculate the E vector by finding a value orthogonal to d
E0 = [-z, 0, x]
#solve for the layered sample
L.solve1(d, k0, E0)
Er = L.Pr[:, 0]
I[i] = intensity(Er)
plt.plot(I)