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fraunhofer.py
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"""
fraunhofer.py: calculates 2D Fraunhofer diffraction (via Fourier Transform)
"""
__author__ = "Manuel Sanchez del Rio"
__contact__ = "[email protected]"
__copyright = "ESRF, 2016"
import numpy as np
#
# wavefront definitions
#
def wavefront_initialize(pixelsize_h=1e-6,pixelsize_v=1e-6,npixels_h=1024,npixels_v=1024,amplitude_value=0.0):
#
#create array at object (aperture) plane
#
amplitude = np.zeros((npixels_h,npixels_v)) # amplitude map
amplitude += amplitude_value
p_i_h = np.arange(npixels_h) * pixelsize_h
p_x = (p_i_h - 0.5 * (p_i_h[-1] - p_i_h[0]) )
p_i_v = np.arange(npixels_v) * pixelsize_v
p_y = (p_i_v - 0.5 * (p_i_v[-1] - p_i_v[0]) )
return p_x,p_y,amplitude
def wavefront_aperture(p_x,p_y,amplitude,diameter=40e-6,type=0):
# aperture_type: 0=circular, 1=Square, 2=Gaussian
p_xx = p_x[:, np.newaxis]
p_yy = p_y[np.newaxis, :]
filter = np.zeros_like(amplitude)
if type == 0: # Circular aperture
radius = (diameter/2)
print("radius=%f um"%(1e6*radius))
filter_illuminated_indices = np.where(p_xx**2 + p_yy**2 < radius**2)
if filter_illuminated_indices[0].size ==0:
print("Warning: wavefront_aperture(): Nothing goes trough the aperture")
else:
filter[filter_illuminated_indices] = 1.0
elif type == 1: # square
radius = (diameter/2)
print("radius=%f um"%(1e6*radius))
filter_illuminated_indices = np.where( (np.abs(p_xx) < radius) & (np.abs(p_yy) < radius))
if filter_illuminated_indices[0].size ==0:
print("Warning: wavefront_aperture(): Nothing goes trough the aperture")
else:
filter[filter_illuminated_indices] = 1.0
elif type == 2: # Gaussian
sigma = diameter/2.35
print("source sigma=%f um"%(1e6*sigma))
rho2 = p_xx**2 + p_yy**2
#TODO: add Gaussian amplitude
filter = np.sqrt(np.exp(-rho2/2/sigma**2)) # Gaussian in intensity, so srrt for amplitude
filter = np.exp(-rho2/2/sigma**2) # Gaussian amplitude
else:
raise ValueError("Aperture type (shape) not valid")
return p_x,p_y,amplitude*filter
#
# tools
#
def propagator2d(x,y,z,method="fraunhofer",wavelength=1e-10,propagation_distance=1.0,return_angles=0):
#
# interface to different propagators
#
from timeit import default_timer as timer
t_start = timer()
if method == "fraunhofer":
x1,y1,z1 = propagator2d_fraunhoffer(x,y,z,wavelength=wavelength)
if return_angles:
pass
else:
x1 *= propagation_distance
y1 *= propagation_distance
elif method == "fourier_convolution":
x1,y1,z1 = propagator2d_fourier_convolution(x,y,z,propagation_distance=propagation_distance,wavelength=wavelength)
if return_angles:
x1 /= propagation_distance
y1 /= propagation_distance
elif method == "integral":
x1,y1,z1 = propagator2d_integral(x,y,z,propagation_distance=propagation_distance,wavelength=wavelength)
if return_angles:
x1 /= propagation_distance
y1 /= propagation_distance
elif method == "srw":
x1,y1,z1 = propagator2d_srw(x,y,z,propagation_distance=propagation_distance,wavelength=wavelength)
if return_angles:
x1 /= propagation_distance
y1 /= propagation_distance
else:
raise Exception("method %s not implemented"%method)
t_end = timer()
print("Elapsed time in propagation calculations: %5.3f ms"%((t_end-t_start)*1e3))
print("Shapes in propagation calculations: before: ",z.shape," after: ",z1.shape)
print("Limits in propagation calculations H: before: ",x[0],x[-1]," after: ",x1[0],x1[-1]," points: ",x.shape)
print("Limits in propagation calculations V: before: ",y[0],y[-1]," after: ",y1[0],y1[-1]," points: ",y.shape)
return x1,y1,z1
def propagator2d_srw(p_x,p_y,amplitude,propagation_distance=1.0,wavelength=1e-10):
#
# convolving with the Fresnel kernel via SRW package
#
import srwlib
from NumpyToSRW import numpyArrayToSRWArray, SRWWavefrontFromElectricField, SRWEFieldAsNumpy
# srw_amplituder = numpyArrayToSRWArray(amplitude)
# print(type(srw_amplituder))
srw_wfr = SRWWavefrontFromElectricField(p_x[0], p_x[-1], amplitude,
p_y[0], p_y[-1], np.zeros_like(amplitude),
12396.0/(wavelength*1e10), 1.0, 1.0, 1e-3, 1.0, 1e-3)
print(type(srw_wfr))
#
# propagation
#
optDrift = srwlib.SRWLOptD(propagation_distance) #Drift space
# 0 1 2 3 4 5 6 7 8 9 10 11
# propagParDrift = [1, 1, 1., 0, 0, 1., 1., 1., 1., 0, 0, 0]
propagParDrift = [0, 0, 1., 0, 0, 1., 1., 1., 1., 0, 0, 0]
#Wavefront Propagation Parameters:
#[0]: Auto-Resize (1) or not (0) Before propagation
#[1]: Auto-Resize (1) or not (0) After propagation
#[2]: Relative Precision for propagation with Auto-Resizing (1. is nominal)
#[3]: Allow (1) or not (0) for semi-analytical treatment of the quadratic (leading) phase terms at the propagation
#[4]: Do any Resizing on Fourier side, using FFT, (1) or not (0)
#[5]: Horizontal Range modification factor at Resizing (1. means no modification)
#[6]: Horizontal Resolution modification factor at Resizing
#[7]: Vertical Range modification factor at Resizing
#[8]: Vertical Resolution modification factor at Resizing
#[9]: Type of wavefront Shift before Resizing (not yet implemented)
#[10]: New Horizontal wavefront Center position after Shift (not yet implemented)
#[11]: New Vertical wavefront Center position after Shift (not yet implemented)
optBL = srwlib.SRWLOptC([optDrift], [propagParDrift]) #"Beamline" - Container of Optical Elements (together with the corresponding wavefront propagation instructions)
print(' Simulating Electric Field Wavefront Propagation bu SRW ... ', end='\n')
srwlib.srwl.PropagElecField(srw_wfr, optBL)
amplitude2 = SRWEFieldAsNumpy(srw_wfr)
amplitude2 = amplitude2[0,:,:,0]
print("Amplitude shape before:",amplitude.shape,"; after: ",amplitude2.shape)
# fft = np.fft.fft2(amplitude)
#
# # frequency for axis 1
# pixelsize = p_x[1] - p_x[0]
# npixels = p_x.size
# freq_nyquist = 0.5/pixelsize
# freq_n = np.linspace(-1.0,1.0,npixels)
# freq_x = freq_n * freq_nyquist
# # freq = freq * wavelength
#
# # frequency for axis 2
# pixelsize = p_y[1] - p_y[0]
# npixels = p_y.size
# freq_nyquist = 0.5/pixelsize
# freq_n = np.linspace(-1.0,1.0,npixels)
# freq_y = freq_n * freq_nyquist
# # freq_y = freq_y * wavelength
#
# freq_xy = np.array(np.meshgrid(freq_y,freq_x))
#
# fft *= np.exp((-1.0j) * np.pi * wavelength * propagation_distance *
# np.fft.fftshift(freq_xy[0]*freq_xy[0] + freq_xy[1]*freq_xy[1]) )
# ifft = np.fft.ifft2(fft)
p_x2 = np.linspace(srw_wfr.mesh.xStart, srw_wfr.mesh.xFin, srw_wfr.mesh.nx)
p_y2 = np.linspace(srw_wfr.mesh.yStart, srw_wfr.mesh.yFin, srw_wfr.mesh.ny)
return p_x2,p_y2,amplitude2
def propagator2d_integral(p_x,p_y,amplitude,propagation_distance=1.0,wavelength=1e-10,shuffle_interval=1e-5):
#
# Fresnel-Kirchhoff integral (neglecting inclination factor)
#
det_x = p_x.copy()
det_y = p_y.copy()
#
# manual
#
# p_xy = np.zeros((2,p_x.size,p_y.size))
# det_xy = np.zeros((2,det_x.size,det_y.size))
# for i in range(p_x.size):
# for j in range(p_y.size):
# p_xy[0,i,j] = p_x[i]
# p_xy[1,i,j] = p_y[j]
# det_xy[0,i,j] = det_x[i]
# det_xy[1,i,j] = det_y[j]
#
# np
#
p_xy = np.array(np.meshgrid(p_y,p_x))
det_xy = np.array(np.meshgrid(det_y,det_x))
amplitude_propagated = np.zeros_like(amplitude,dtype='complex')
wavenumber = 2 * np.pi / wavelength
for i in range(det_x.size):
for j in range(det_y.size):
if shuffle_interval == 0:
rd_x = 0.0
rd_y = 0.0
else:
rd_x = (np.random.rand(p_x.size,p_y.size)-0.5)*shuffle_interval
rd_y = (np.random.rand(p_x.size,p_y.size)-0.5)*shuffle_interval
r = np.sqrt( np.power(p_xy[0,:,:] + rd_x - det_xy[0,i,j],2) +
np.power(p_xy[1,:,:] + rd_y - det_xy[1,i,j],2) +
np.power(propagation_distance,2) )
amplitude_propagated[i,j] = (amplitude / r * np.exp(1.j * wavenumber * r)).sum()
return det_x,det_y,amplitude_propagated # .reshape((det_x.size,det_y.size))
def propagator2d_fourier_convolution(p_x,p_y,image,propagation_distance=1.0,wavelength=1e-10):
#
# convolving with the Fresnel kernel via FFT multiplication
#
fft = np.fft.fft2(image)
# frequency for axis 1
pixelsize = p_x[1] - p_x[0]
npixels = p_x.size
freq_nyquist = 0.5/pixelsize
freq_n = np.linspace(-1.0,1.0,npixels)
freq_x = freq_n * freq_nyquist
# freq = freq * wavelength
# frequency for axis 2
pixelsize = p_y[1] - p_y[0]
npixels = p_y.size
freq_nyquist = 0.5/pixelsize
freq_n = np.linspace(-1.0,1.0,npixels)
freq_y = freq_n * freq_nyquist
# freq_y = freq_y * wavelength
freq_xy = np.array(np.meshgrid(freq_y,freq_x))
fft *= np.exp((-1.0j) * np.pi * wavelength * propagation_distance *
np.fft.fftshift(freq_xy[0]*freq_xy[0] + freq_xy[1]*freq_xy[1]) )
# fft = np.fft.fftshift(fft)
# fft *= np.exp((-1.0j) * np.pi * wavelength * propagation_distance *
# (freq_xy[0]*freq_xy[0] + freq_xy[1]*freq_xy[1]) )
# fft = np.fft.ifftshift(fft)
ifft = np.fft.ifft2(fft)
return p_x.copy(),p_y.copy(),ifft
def propagator2d_fraunhoffer(p_x,p_y,image,wavelength=1e-10):
"""
Fraunhoffer propagator
:param x: x array of spatial coordinates in meters
:param y: y array of spatial coordinates in meters
:param complax_amplitude array: shape: [n_points_x,n_points_y]
:param wavelength: photon wavelength in meters
:return: three arrays with the propagated pattern : angle_x [rad], angle_y, complex_amplitude.
"""
#
#compute Fourier transform
#
F1 = np.fft.fft2(image) # Take the fourier transform of the image.
# Now shift the quadrants around so that low spatial frequencies are in
# the center of the 2D fourier transformed image.
F2 = np.fft.fftshift( F1 )
# frequency for axis 1
pixelsize = p_x[1] - p_x[0]
npixels = p_x.size
freq_nyquist = 0.5/pixelsize
freq_n = np.linspace(-1.0,1.0,npixels)
freq_x = freq_n * freq_nyquist
freq_x *= wavelength
# frequency for axis 2
pixelsize = p_y[1] - p_y[0]
npixels = p_y.size
freq_nyquist = 0.5/pixelsize
freq_n = np.linspace(-1.0,1.0,npixels)
freq_y = freq_n * freq_nyquist
freq_y *= wavelength
return freq_x,freq_y,F2
def line_image(image,horizontal_or_vertical='H'):
if horizontal_or_vertical == "H":
npixels = image.shape[0]
tmp = image[:,image.shape[1]/2]
else:
npixels = image.shape[1]
tmp = image[image.shape[0]/2,:]
return tmp
def line_fwhm(line):
#
#CALCULATE fwhm in number of abscissas bins (supposed on a regular grid)
#
tt = np.where(line>=max(line)*0.5)
if line[tt].size > 1:
# binSize = x[1]-x[0]
FWHM = (tt[0][-1]-tt[0][0])
return FWHM
else:
return -1
#
# plotting tools
#
def plot_show():
import matplotlib.pylab as plt
plt.show()
def plot_image(mymode,theta,psi,title="TITLE",xtitle=r"X [$\mu m$]",ytitle=r"Y [$\mu m$]",cmap=None,show=1):
import matplotlib.pylab as plt
fig = plt.figure()
# cmap = plt.cm.Greys
plt.imshow(mymode.T,origin='lower',extent=[theta[0],theta[-1],psi[0],psi[-1]],cmap=cmap)
plt.colorbar()
ax = fig.gca()
ax.set_xlabel(xtitle)
ax.set_ylabel(ytitle)
plt.title(title)
if show: plt.show()
def plot(*positional_parameters,title="",xtitle="",ytitle="",show=1,legend=None,color=None):
import matplotlib.pylab as plt
n_arguments = len(positional_parameters)
if n_arguments == 0:
return
fig = plt.figure()
if n_arguments == 1:
y = positional_parameters[0]
x = np.arange(y.size)
plt.plot(x,y,label=legend)
elif n_arguments == 2:
x = positional_parameters[0]
y = positional_parameters[1]
plt.plot(x,y,label=legend,color=color)
elif n_arguments == 4:
x1 = positional_parameters[0]
y1 = positional_parameters[1]
x2 = positional_parameters[2]
y2 = positional_parameters[3]
if legend != None:
legend1 = legend[0]
legend2 = legend[1]
else:
legend1 = None
legend2 = None
if color != None:
color1 = color[0]
color2 = color[1]
else:
color1 = None
color2 = None
plt.plot(x1,y1,label=legend1,color=color1)
plt.plot(x2,y2,label=legend2,color=color2)
else:
"Incorrect number of arguments, plotting only two first arguments"
x = positional_parameters[0]
y = positional_parameters[1]
plt.plot(x,y,label=legend)
if legend != None:
ax = plt.subplot(111)
ax.legend(bbox_to_anchor=(1.1, 1.05))
plt.title(title)
plt.xlabel(xtitle)
plt.ylabel(ytitle)
if show:
plt.show()
def main():
#
# inputs (in SI)
#
wavelength = 1.24e-10
aperture_diameter = 40e-6 # if Gaussian, aperture_diameter = 2.35*sigma
aperture_type = 2 # 0=circular, 1=Square, 2=Gaussian (sigma = diameter/2.35)
pixelsize_x = 1e-6
pixelsize_y = pixelsize_x
npixels_x = 1024 # 200 #
npixels_y = npixels_x # 50 #
propagation_distance = 30.0
# method = "fourier_convolution"
method = "fraunhofer"
# method = "integral"
# method = "srw"
#
# calculations
#
# get a wavefront
p_x,p_y,amplitude = wavefront_initialize(pixelsize_x,pixelsize_y,npixels_x,npixels_y,amplitude_value=1.0)
# set aperture
p_x,p_y,amplitude = wavefront_aperture(p_x,p_y,amplitude,diameter=aperture_diameter,type=aperture_type)
#plot aperture
plot_image(np.abs(amplitude)**2,p_x*1e6,p_y*1e6, show=0,
title="aperture intensity, Diameter=%5.1f um"%(1e6*aperture_diameter),xtitle="X [um]",ytitle="Y [um]")
#
# propagation
#
angle_x, angle_y, amplitude_propagated = propagator2d(p_x,p_y,amplitude,method=method,wavelength=wavelength,
propagation_distance=propagation_distance,return_angles=1)
# angle_x, angle_y, amplitude_propagated = propagator2d_fraunhoffer(p_x,p_y,amplitude,wavelength=wavelength)
if method == "fraunhofer":
print("Fraunhoffer diffraction valid for distances > > a^2/lambda = %f m"%((aperture_diameter/2)**2/wavelength))
plot_image(np.abs(amplitude_propagated)**2,angle_x*1e6,angle_y*1e6, show=0,
title="Diffracted intensity (%s)"%method,xtitle="X [urad]",ytitle="Y [urad]")
#
# extract profiles and calculate theoretical ones
#
# retrieve H and V profiles
horizontal_intensity_profile = line_image(np.abs(amplitude_propagated)**2,horizontal_or_vertical='H')
horizontal_intensity_profile /= horizontal_intensity_profile.max()
vertical_intensity_profile = line_image(np.abs(amplitude_propagated)**2,horizontal_or_vertical='V')
vertical_intensity_profile /= vertical_intensity_profile.max()
# theoretical profile
if aperture_type == 0: #circular, also display analytical values
from scipy.special import jv
x = (2*np.pi/wavelength) * (aperture_diameter/2) * angle_x
y = (2*np.pi/wavelength) * (aperture_diameter/2) * angle_y
U_vs_theta_x = 2*jv(1,x)/x
U_vs_theta_y = 2*jv(1,y)/y
I_vs_theta_x = U_vs_theta_x**2
I_vs_theta_y = U_vs_theta_y**2
elif aperture_type == 1: # square
x = (2*np.pi/wavelength) * (aperture_diameter/2) * angle_x
y = (2*np.pi/wavelength) * (aperture_diameter/2) * angle_y
U_vs_theta_x = 2*np.sin(x)/x
U_vs_theta_y = 2*np.sin(y)/y
I_vs_theta_x = U_vs_theta_x**2
I_vs_theta_y = U_vs_theta_y**2
I_vs_theta_x /= I_vs_theta_x.max()
I_vs_theta_y /= I_vs_theta_y.max()
elif aperture_type == 2: #Gaussian
sigma = aperture_diameter/2.35
sigma_ft = 1.0 / sigma * wavelength / (2.0 * np.pi)
# Factor 2.0 is because we wwant intensity (amplitude**2)
I_vs_theta_x = np.exp( -2.0*(angle_x**2/sigma_ft**2/2) )
I_vs_theta_y = np.exp( -2.0*(angle_y**2/sigma_ft**2/2) )
fwhm_intensity_profile_horizontal = line_fwhm(horizontal_intensity_profile) * (angle_x[1]-angle_x[0])
fwhm_intensity_profile_vertical = line_fwhm(vertical_intensity_profile) * (angle_y[1]-angle_y[0])
fwhm_theoretical_profile_horizontal = line_fwhm(I_vs_theta_x) * (angle_x[1]-angle_x[0])
fwhm_theoretical_profile_vertical = line_fwhm(I_vs_theta_y) * (angle_y[1]-angle_y[0])
#
# calculate widths
#
print("HORIZONTAL FWHM (%s) : %f urad, FWHM theoretical: %f urad, 1.22*wavelength/Diameter: %f urad"%(
method,
1e6*fwhm_intensity_profile_horizontal,1e6*fwhm_theoretical_profile_horizontal,1e6*1.22*wavelength/aperture_diameter))
print("VERTICAL FWHM (%s) : %f urad, FWHM theoretical: %f urad, 1.22*wavelength/Diameter: %f urad"%(
method,
1e6*fwhm_intensity_profile_vertical,1e6*fwhm_theoretical_profile_vertical,1e6*1.22*wavelength/aperture_diameter))
print("HORIZONTAL (4pi/lambda) sigma sigma' : (%s): %f, theoretical: %f "%(method,
4*np.pi / wavelength * fwhm_intensity_profile_horizontal/2.35 * aperture_diameter/2.35,
4*np.pi / wavelength * fwhm_theoretical_profile_horizontal/2.35 * aperture_diameter/2.35 ))
# plot profiles
plot( angle_x*1e6, horizontal_intensity_profile, angle_x*1e6, I_vs_theta_x, show=0,
legend=["profile","theory"],color=["red","black"],
title="Horizontal profile of diffracted intensity (%s)"%method,xtitle='theta [urad]',ytitle='Diffracted intensity [a.u.]')
plot( angle_y*1e6, vertical_intensity_profile, angle_y*1e6, I_vs_theta_y, show=1,
legend=["profile","theory"],color=["red","black"],
title="Vertical profile of diffracted intensity (%s)"%method,xtitle='theta [urad]',ytitle='Diffracted intensity [a.u.]')
if __name__ == "__main__":
main()