[Numpy-discussion] Horizontal lines in diffraction image (NumPy FFT)
Matthieu Brucher
matthieu.brucher at gmail.com
Wed Aug 6 05:32:20 EDT 2008
Exactly. Using FFT to do a convolution should be done after some
signal processing readings ;) (That's why I hate FFT to do signal
processing as well).
Matthieu
2008/8/6 Nadav Horesh <nadavh at visionsense.com>:
>
> I did about the same thing 9 year ago (in python of course). If I can recall correctly, you need to double the arrays size (with padding of 0) in order to avoid this artifact. I think that its origin is that the DFT is equivalent to periodic boundary conditions.
>
> Nadav.
>
> -----הודעה מקורית-----
> מאת: numpy-discussion-bounces at scipy.org בשם Matthias Hillenbrand
> נשלח: ד 06-אוגוסט-08 05:26
> אל: numpy-discussion at scipy.org
> נושא: [Numpy-discussion] Horizontal lines in diffraction image (NumPy FFT)
>
> Hello,
>
> I am trying to calculate the propagation of a Gaussian beam with an
> angular spectrum propagation algorithm.
> My program does the following steps:
> 1. Generation and multiplication of the Gaussian beam, an aperture,
> and a lens -> u
> 2. FFT(u)
> 3. Multiplication of FFT(u) with the angular spectrum kernel H
> 4. IFFT(FU*H)
>
> Steps 2-4 are repeated 1000 times to show the propagation of the beam
> in z-direction
>
> Unfortunately the calculation results in a lot of horizontal lines
> that should not be there. The program produces reasonable results
> besides this problem.
>
> It is especially interesting that an equivalent calculation with
> Matlab or Octave has the same results without these artifacts. Both
> calculations take approximately 90 seconds.
>
> I am not sure whether I am doing something wrong or there is a
> difference in the FFT codes. I assume that the problem has something
> to do with the propagation kernel H but I cannot see a difference
> between both programs.
>
> Thank you very much for your help!
>
> Matthias
>
>
> ---------------------------------------------------------------------------------------------------
> Python code with comments (version without comments below)
> ---------------------------------------------------------------------------------------------------
>
> from pylab import *
> from numpy.lib import scimath #Complex roots
> import os
>
> #########################################
> ########## Defintion of the input variables #########
> #########################################
>
>
> ##General variables
> lam = 532.e-9 #wavelength in m
> k = 2*pi/lam #wave number
>
> ##Computational array
> arr_size = 80.e-3 #size of the computation array in m
> N = 2**17 #points of the 1D array
>
> ##Aperture
> ap = 40.e-3 #aperture size of the 1D array in m
>
> ##Gaussian beam
> w0 =10.e-3 #waist radius of the gaussian beam in m
>
> ##Lenses (definition of the focal length)
> n = 1.56 #refractive index of the glass
> f = .1 #focal length in m
> Phi = 1/f #focal power of the lens
> r2 = 1.e18 #radius of the second lens surface
> r1 = 1 / ( Phi/(n-1) + 1/r2 ) #computation of the radius of the
> first lens surface
>
> ##z distances
> zmin = 0.99*f #initial z position
> zmax = 1.01*f #final z position
> zsteps = 1000 #number of computated positions
> ROI = 1000 #Region of interest in the diffraction image
>
> ##############################################
> ############### Program execution 1D ################
> ###############################################
>
> x = linspace(-arr_size/2,arr_size/2,N)
> A = ones(N)
> A[where(ap/2<=abs(x))] = 0 #Definition of the aperture
> G = exp(- x**2/w0**2) #Generation of the Gaussian beam
>
> delta = -r1*(1 - scimath.sqrt(1 - x**2/r1**2))
> + r2*(1 - scimath.sqrt(1 - x**2/r2**2))
> m = (r1**2 <= x**2 )
> delta[m] = 0 #correction in case of negative roots
> m = (r2**2 <= x**2 )
> delta[m] = 0 #correction in case of negative roots
> lens = exp(1j * 2 * pi / lam * (n-1) * delta) #Calculation
> of the lens phase function
>
> u = A*G*lens #Complex amplitude before beam propagation
>
> ############################################
> ########### Start angular spectrum method ###########
>
> deltaf = 1/arr_size #spacing in frequency
> domain
> fx = r_[-N/2*deltaf:(N/2)*deltaf:deltaf] #whole frequency domain
> FU = fftshift(fft(u)) #1D FFT
> U = zeros((zsteps,ROI)) #Generation of the image array
> z = linspace(zmin,zmax,zsteps) #Calculation of the
> different z positions
>
> for i in range(zsteps):
> H = exp(1j * 2 * pi / lam * z[i] * scimath.sqrt(1-(lam*fx)**2))
> U[i] = (ifft(fftshift(FU*H)))[N/2 - ROI/2 : N/2 + ROI/2]
> if i%10 == 0:
> t = 'z position: %4i' % i
> print t
>
> ############ End angular spectrum method ############
> #############################################
>
> res = abs(U)
>
> imshow(res)
> show()
>
>
>
>
>
>
> ----------------------------------------------------------
> Python code without comments
> ----------------------------------------------------------
>
> from pylab import *
> from numpy.lib import scimath #Complex roots
> import os
>
> lam = 532.e-9
> k = 2*pi/lam
>
> arr_size = 80.e-3
> N = 2**17
>
> ap = 40.e-3
>
> w0 =10.e-3
>
> n = 1.56
> f = .1
> Phi = 1/f
> r2 = 1.e18
> r1 = 1 / ( Phi/(n-1) + 1/r2 )
>
> zmin = 0.99*f
> zmax = 1.01*f
> zsteps = 1000
> ROI = 1000
>
> x = linspace(-arr_size/2,arr_size/2,N)
> A = ones(N)
> A[where(ap/2<=abs(x))] = 0
> G = exp(- x**2/w0**2)
>
> delta = -r1*(1 - scimath.sqrt(1 - x**2/r1**2))
> + r2*(1 - scimath.sqrt(1 - x**2/r2**2))
> m = (r1**2 <= x**2 )
> delta[m] = 0
> m = (r2**2 <= x**2 )
> delta[m] = 0
> lens = exp(1j * 2 * pi / lam * (n-1) * delta)
>
> u = A*G*lens
>
> deltaf = 1/arr_size
> fx = r_[-N/2*deltaf:(N/2)*deltaf:deltaf]
> FU = fftshift(fft(u))
> U = zeros((zsteps,ROI))
> z = linspace(zmin,zmax,zsteps)
>
> for i in range(zsteps):
> H = exp(1j * 2 * pi / lam * z[i] * scimath.sqrt(1-(lam*fx)**2))
> U[i] = (ifft(fftshift(FU*H)))[N/2 - ROI/2 : N/2 + ROI/2]
> if i%10 == 0:
> t = 'z position: %4i' % i
> print t
>
> res = abs(U)
>
> imshow(res)
> show()
>
>
>
>
>
>
> ----------------------------------------------------------
> Matlab, Octave code
> ----------------------------------------------------------
>
> lam = 532.e-9
> k = 2*pi/lam
>
> arr_size = 80.e-3
> N = 2^17
>
> ap = 40.e-3
>
> w0 =10.e-3
>
> n = 1.56
> f = .1
> Phi = 1/f
> r2 = 1.e18
> r1 = 1 / ( Phi/(n-1) + 1/r2 )
>
> zmin = 0.99*f
> zmax = 1.01*f
> zsteps = 1000
> ROI = 1000
>
> x=linspace(-arr_size/2,arr_size/2,N);
> A = ones(1,N);
> A(find(ap/2<=abs(x)))=0;
> G = exp(- x.^2/w0^2);
>
> delta = -r1*(1 - sqrt(1 - x.^2/r1^2)) + r2*(1 - sqrt(1 - x.^2/r2^2));
> delta(find(r1.^2 <= x.^2 ))=0;
> delta(find(r2.^2 <= x.^2 ))=0;
> lens = exp(1j * 2 * pi / lam * (n-1) * delta);
> u = A.*G.*lens;
>
> deltaf = 1/arr_size;
> fx = [-N/2*deltaf:deltaf:(N/2-1)*deltaf];
> FU = fftshift(fft(u));
> U = zeros(zsteps,ROI);
> z = linspace(zmin,zmax,zsteps);
>
> for i=1:zsteps
> H = exp(1j * 2 * pi / lam * z(i) * sqrt(1-lam.^2*fx.^2));
> U(i,:) = ifft(fftshift(FU.*H))(N/2 - ROI/2 : N/2 + ROI/2-1);
> end
>
> imagesc(abs(U))
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--
French PhD student
Website : http://matthieu-brucher.developpez.com/
Blogs : http://matt.eifelle.com and http://blog.developpez.com/?blog=92
LinkedIn : http://www.linkedin.com/in/matthieubrucher
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