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S4conventions.py
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S4conventions.py
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# -*- coding: utf-8 -*-
"""
(c) 2015 Steven Byrnes
This script was used to help write and debug the software, it serves no
external purpose. Specifically, S4 software returns complex amplitudes but
doesn't explain well the meaning of the amplitudes. This file is figuring it
out, by checking against the electric and magnetic fields.
I posted some of what I learned at
https://github.com/victorliu/S4/pull/25/files
"""
import math
import numpy as np
from numpy import array
import matplotlib.pyplot as plt
import cmath
import grating, gratingdata
from random import random
# my units package
# https://pypi.python.org/pypi/numericalunits
# just run the command "pip install numericalunits"
import numericalunits as nu
from numericalunits import m, nm, um
pi = math.pi
inf = float('inf')
degree = pi / 180
def array_almost_equal(a,b):
a,b = np.array(a), np.array(b)
return (abs(a-b)).max() <= 1e-9 * (abs(a) + abs(b)).max()
def sp_polarization(kx, ky, kz, n):
"""If an amplitude in "s" or "p" polarization is 1, find the E and H field
that S4 would output. (...which has H in arbitrary units, E in the same units
but with a factor of Z0 (impedance of free space)).
n is the index."""
assert n==1 # Haven't worked out the n>1 case
if kx == ky == 0:
# warning: program treats -0.0 differently from 0.0...
Ep = [1,0,0]
Es = [0,1,0]
Hp = [0,1,0]
Hs = [-1,0,0]
return [np.array(v) for v in (Es,Ep,Hs,Hp)]
k = (kx**2 + ky**2 + kz**2)**0.5
Es = [-ky/(kx**2+ky**2)**0.5, kx/(kx**2+ky**2)**0.5, 0]
Ep = [kx*kz/(k * (kx**2+ky**2)**0.5),
ky*kz/(k * (kx**2+ky**2)**0.5),
-(kx**2+ky**2)**0.5 / k]
Es=np.array(Es) ; Ep = np.array(Ep)
# alternative / derivation
Es_unnorm = np.cross([0,0,1], [kx,ky,kz])
Ep_unnorm = np.cross(Es_unnorm, [kx,ky,kz])
Es_alt = Es_unnorm / sum(Es_unnorm**2)**0.5
Ep_alt = Ep_unnorm / sum(Ep_unnorm**2)**0.5
assert array_almost_equal(Es, Es_alt)
assert array_almost_equal(Ep, Ep_alt)
Hp = Es
Hs = -Ep
return Es,Ep,Hs,Hp
def xy_polarization(kx,ky,kz, n):
"""For output amplitudes, S4 uses a different polarization scheme, which
we'll call 'x' and 'y' polarization.
There are a couple funny things about it: (1) The unit-amplitude fields are
actually not normalized, (2) The x and y are not orthogonal (except if
kx=0 or ky=0). But it's actually kinda nice in other ways: Mainly, the fact
that it has no discontinuous changes, unlike s and p near normal. So I'm
actually using it myself by choice for incoming waves
(see grating.characterize()).
Anyway, if an x-polarized or y-polarized amplitude is 1, find the E and H
field that S4 would output.
n is the index of refraction"""
if kx == ky == 0:
signkz = math.copysign(1,kz)
E_xpol = [signkz/n,0,0]
E_ypol = [0,-signkz/n,0]
H_xpol = [0,1,0]
H_ypol = [1,0,0]
return [np.array(v) for v in (E_xpol,E_ypol,H_xpol,H_ypol)]
k = (kx**2 + ky**2 + kz**2)**0.5
H_xpol = [0,1,-ky/kz]
E_xpol = [(ky**2+kz**2)/(k*kz*n), -kx*ky/(k*kz*n), -kx/(k*n)]
H_ypol = [1,0,-kx/kz]
E_ypol = [kx*ky/(k*kz*n), (-kx**2-kz**2)/(k*kz*n), ky/(k*n)]
# alternative / derivation
E_xpol_alt = np.cross(H_xpol, [kx/k, ky/k, kz/k]) / n
E_ypol_alt = np.cross(H_ypol, [kx/k, ky/k, kz/k]) / n
assert array_almost_equal(E_xpol, E_xpol_alt)
assert array_almost_equal(E_ypol, E_ypol_alt)
return [np.array(v) for v in (E_xpol,E_ypol,H_xpol,H_ypol)]
def x_from_sp(kx,ky,kz,n):
"""how to create incoming x polarization from combining s and p. Just
checking my math here"""
assert n==1
Es,Ep,Hs,Hp = sp_polarization(kx,ky,kz,n=n)
E_xpol,E_ypol,H_xpol,H_ypol = xy_polarization(kx,ky,kz,n=n)
k = (kx**2 + ky**2 + kz**2)**0.5
p_coef = kx/(kx**2+ky**2)**0.5
s_coef = -ky*k/(kz*(kx**2+ky**2)**0.5)
assert array_almost_equal(p_coef*Hp + s_coef*Hs, H_xpol)
assert array_almost_equal(p_coef*Ep + s_coef*Es, E_xpol)
def y_from_sp(kx,ky,kz,n):
"""how to create incoming y polarization by combining s and p. Just
checking my math here"""
assert n==1
Es,Ep,Hs,Hp = sp_polarization(kx,ky,kz,n=n)
E_xpol,E_ypol,H_xpol,H_ypol = xy_polarization(kx,ky,kz,n=n)
k = (kx**2 + ky**2 + kz**2)**0.5
p_coef = -ky/(kx**2+ky**2)**0.5
s_coef = -kx*k/(kz*(kx**2+ky**2)**0.5)
assert array_almost_equal(p_coef*Hp + s_coef*Hs, H_ypol)
assert array_almost_equal(p_coef*Ep + s_coef*Es, E_ypol)
def arbitrary_from_xy(Hx,Hy,kx,ky,kz,n):
"""how to recreate an arbitrary field by combing x and y polarizations.
Just checking my math here. Oh, actually this is obvious."""
E_xpol,E_ypol,H_xpol,H_ypol = xy_polarization(kx,ky,kz,n=n)
x_coef = Hy
y_coef = Hx
assert array_almost_equal((x_coef*H_xpol + y_coef*H_ypol)[0:2], [Hx,Hy])
x_from_sp(random(),random(),random(),n=1)
y_from_sp(random(),random(),random(),n=1)
arbitrary_from_xy(random(),random(),random(), random(), random(),n=random())
"""The rest of the file is for making S4 output (1) Complex amplitudes of
propagating diffraction orders, and (2) Actual E and H fields at particular
points. Then find the formula for calculating (2) given (1)."""
def read_fields(mygrating, target_wavelength=580*nm):
"""When grating.lua is set up to output the complex amplitudes and then
real-space fields, this function is for parsing that output.
Again, this function will not work if you don't uncomment certain lines
in grating.lua."""
# Assuming lua is set up to spit out near-field...
ux = math.sin(mygrating.get_angle_in_air(580*nm)) - 0.1
uy = 0.123
process = mygrating.run_lua_initiate(ux_min=ux, ux_max=ux, uy_min=uy,
uy_max=uy, u_steps=1, wavelength=600*nm)
output, error = process.communicate()
output = output.decode()
grating_amplitude_data = []
lines = output.split('\r\n')
for i,line in enumerate(lines):
if line[0:6] == 'Fields':
break
split = line.split()
#print(split)
if len(split) > 0:
grating_amplitude_data.append({'wavelength': float(split[0]),
's_or_p': split[1],
'ux':float(split[2]),
'uy':float(split[3]),
'ox':int(split[4]),
'oy':int(split[5]),
'ampfy':float(split[6]) + 1j * float(split[7]),
'ampfx':float(split[8]) + 1j * float(split[9]),
'ampry':float(split[10]) + 1j * float(split[11]),
'amprx':float(split[12]) + 1j * float(split[13])})
field_data = []
for line in lines[i+1:]:
if line != '':
field_data.append([float(x) for x in line.split('\t')])
field_data = array(field_data)
x_list = np.unique(field_data[:,0]) * um
y_list = np.unique(field_data[:,1]) * um
z = field_data[0,2] * um
assert (field_data[:,2] * um == z).all()
# E[ix, iy, 2] is Ez at the point x_list[ix],y_list[iy]
E = np.empty(shape=(len(x_list), len(y_list),3), dtype=complex)
H = np.empty(shape=(len(x_list), len(y_list),3), dtype=complex)
for row in field_data:
ix = np.argmin(abs(x_list/um - row[0]))
iy = np.argmin(abs(y_list/um - row[1]))
E[ix,iy,0] = row[3] + 1j * row[9]
E[ix,iy,1] = row[4] + 1j * row[10]
E[ix,iy,2] = row[5] + 1j * row[11]
H[ix,iy,0] = row[6] + 1j * row[12]
H[ix,iy,1] = row[7] + 1j * row[13]
H[ix,iy,2] = row[8] + 1j * row[14]
return E,H,x_list,y_list,z,grating_amplitude_data
def E_from_amplitudes(x, y, z, grating_amplitude_list, mygrating):
"""z is relative to air-cylinder interface, i.e. the start of the first S4
layer.
Note, you need to set pol by hand in the first line to agree with
grating.lua"""
pol = 's' # TODO - read from lua. For now, set it by hand.
assert len({x['wavelength'] for x in grating_amplitude_list}) == 1
wavelength_in_nm = grating_amplitude_list[0]['wavelength']
output_amplitude_list = [e for e in grating_amplitude_list if e['s_or_p'] == pol]
#num_orders = len(output_amplitude_list)
z_above_cyl = z - mygrating.cyl_height
# not interested in evanescent crap
assert z_above_cyl > 3*um or z < -3*um
kvac = 2*pi / (wavelength_in_nm * nm)
n_glass = mygrating.n_glass if mygrating.n_glass > 0 else grating.n_glass(wavelength_in_nm)
kglass = kvac * n_glass
E = np.array([0,0,0], dtype=complex)
H = np.array([0,0,0], dtype=complex)
ktotal = kglass if z > 0 else kvac
kz_sign = +1 if z > 0 else -1
for d in output_amplitude_list:
ux,uy,ox,oy,ampfy,ampfx,ampry,amprx = d['ux'],d['uy'],d['ox'],d['oy'],d['ampfy'],d['ampfx'],d['ampry'],d['amprx']
kx_incoming = ux * kvac
ky_incoming = uy * kvac
kx = kx_incoming + mygrating.grating_kx * ox
ky = ky_incoming + 2*pi/mygrating.lateral_period * oy
kz = kz_sign * (ktotal**2 - kx**2 - ky**2)**0.5
if kz.imag != 0:
#the TIR-type orders are propagaing when z>0 but evanescent in air
assert z<0
continue
if z > 0:
E_fx,E_fy,H_fx,H_fy = xy_polarization(kx,ky,kz,n_glass)
E += ((ampfy * E_fy
+ ampfx * E_fx)
* cmath.exp(1j * kx * x)
* cmath.exp(1j * ky * y)
* cmath.exp(1j * kz * z_above_cyl))
H += ((ampfy * H_fy
+ ampfx * H_fx)
* cmath.exp(1j * kx * x)
* cmath.exp(1j * ky * y)
* cmath.exp(1j * kz * z_above_cyl))
else:
E_rx,E_ry,H_rx,H_ry = xy_polarization(kx,ky,kz,1)
E += ((ampry * E_ry
+ amprx * E_rx)
* cmath.exp(1j * kx * x)
* cmath.exp(1j * ky * y)
* cmath.exp(1j * kz * z))
H += ((ampry * H_ry
+ amprx * H_rx)
* cmath.exp(1j * kx * x)
* cmath.exp(1j * ky * y)
* cmath.exp(1j * kz * z))
if z<0:
kx = kx_incoming
ky = ky_incoming
kz = (ktotal**2 - kx**2 - ky**2)**0.5
assert kz.imag == 0
Es,Ep,Hs,Hp = sp_polarization(kx,ky,kz,1)
if pol == 's':
amplitude_s = 1
amplitude_p = 0
else:
amplitude_s = 0
amplitude_p = 1
E += ((amplitude_s * Es
+ amplitude_p * Ep)
* cmath.exp(1j * kx * x)
* cmath.exp(1j * ky * y)
* cmath.exp(1j * kz * z))
H += ((amplitude_s * Hs
+ amplitude_p * Hp)
* cmath.exp(1j * kx * x)
* cmath.exp(1j * ky * y)
* cmath.exp(1j * kz * z))
return E,H
########## Test #######
mygrating = gratingdata.mygrating31a
#mygrating = gratingdata.mygrating45n
mygrating.lateral_period *= 2.7
#mygrating.n_glass = 1.2
E,H,x_list,y_list,z,grating_amplitude_data = read_fields(mygrating, target_wavelength=600*nm)
print('z/um', z/um)
ix = 16
iy = 12
x = x_list[ix]
y = y_list[iy]
E_from_amps, H_from_amps = E_from_amplitudes(x,y,z,grating_amplitude_data,mygrating)
print('Hopefully all of the following are equal to 1.0...')
print('E ratio x', E_from_amps[0] / E[ix,iy,0])
print('E ratio y', E_from_amps[1] / E[ix,iy,1])
print('E ratio z', E_from_amps[2] / E[ix,iy,2])
print('H ratio x', H_from_amps[0] / H[ix,iy,0])
print('H ratio y', H_from_amps[1] / H[ix,iy,1])
print('H ratio z', H_from_amps[2] / H[ix,iy,2])