Wavefront examples

%matplotlib notebook
import matplotlib.pyplot as plt
#from matplotlib import rcParams
#rcParams['figure.figsize'] = (9, 6)  # Figure size for inline display
import numpy as np
from scipy.misc import face
from pynx.wavefront import *

Near field propagation of a simple 20x200 microns slit

w = Wavefront(d=np.ones((512, 512), dtype=np.complex64), pixel_size=1e-6, wavelength=1.5e-10)
w = RectangularMask(width=200e-6, height=20e-6) * w
w = PropagateNearField(0.5) * w
# w = PropagateFRT(3) * w
w = ImshowAbs(title="Near field propagation (0.5m) of a 20x200 microns aperture") * w

Near field propagation of a simple 40x200 microns slit (live update)

# of 0.2 m propagation
w = Wavefront(d=np.ones((512, 512), dtype=np.complex64), pixel_size=1e-6, wavelength=1.5e-10)
w = RectangularMask(width=200e-6, height=40e-6) * w

# Perform 15 near field propagation of 0.2m steps, displaying the complex wavefront each time
# the **15 expression allows to repeat the series of operators 15 times.
w = (ImshowRGBA(fig_num=40) * PropagateNearField(0.2))**15 * w

Fractional Fourier transform propagation (3m) of a 43x200 microns slit

The Fractional Fourier transform allows to continuously propagate from the near-field regime (where the pixel size remains equal to the original one) to the far field regime (pixel size proportional to the propagation distance).

Note however that the wavefront cannot be propagated further, for lack of validity of the propagated phases.

w = Wavefront(d=np.ones((1024, 1024), dtype=np.complex64), pixel_size=1.3e-6, wavelength=1.e-10)
w = RectangularMask(width=200e-6, height=43e-6) * w

w = PropagateFRT(10) * w
w = ImshowAbs(title="Fractional Fourier transform propagation (3m) of a 43x200 microns slit") * w

Single slit propagation comparison

See Jacques et al (Phys. Rev. B86 (2012), 144117) single slit setup - here with simulated 1 micron pixel

Compare with figure 7 for a=43,88,142,82 microns

plt.figure(figsize=(10, 6))
for a in np.array([22, 43, 88, 142, 182]) * 1e-6:
    w = Wavefront(d=np.ones((1024, 1024,), dtype=np.complex64), wavelength=1e-10, pixel_size=1.3e-6)
    w = RectangularMask(width=1, height=a) * w

    # w = PropagateNearField(3) * w
    w = PropagateFRT(3) * w
    # w = PropagateFarField(3) * w
    icalc = np.fft.fftshift(abs(w.get()[0])).mean(axis=1) ** 2
    x, y = w.get_x_y()
    plt.plot(np.fft.fftshift(y) * 1e6, icalc, label='a=%5.1f µm' % (a * 1e6))
    plt.text(0, icalc[len(icalc) // 2], 'a=%5.1f µm' % (a * 1e6))
    print('a=%5.1fum, dark spot at a^2/(2pi*lambda)=%5.2fm, I[0]=%5.2f' % (
        2 * a * 1e6, (2 * a) ** 2 / (2 * np.pi * w.wavelength), icalc[len(icalc) // 2]))
plt.title("Propagation of a slit at 3 meters, wavelength= 0.1nm")
plt.legend()
plt.xlim(-100, 100)
plt.xlabel('X (µm)')

a= 44.0um, dark spot at a^2/(2pi*lambda)= 3.08m, I[0]= 0.00
a= 86.0um, dark spot at a^2/(2pi*lambda)=11.77m, I[0]= 0.00
a=176.0um, dark spot at a^2/(2pi*lambda)=49.30m, I[0]= 0.00
a=284.0um, dark spot at a^2/(2pi*lambda)=128.37m, I[0]= 0.00
a=364.0um, dark spot at a^2/(2pi*lambda)=210.87m, I[0]= 0.00

Text(0.5, 0, 'X (µm)')

Propagation of a stack of A x 200 microns apertures, varying A

w = Wavefront(d=np.ones((16, 512, 512), dtype=np.complex64), pixel_size=1e-6, wavelength=1.5e-10)
x, y = w.get_x_y()
d = w.get()
for i in range(16):
    a = 5e-6 / 2 * (i + 1)
    d[i] = (abs(y) < a) * (abs(x) < 100e-6)
w.set(d)
w = PropagateFRT(1.2) * w
plt.figure(figsize=(10, 6))
x, y = w.get_x_y()
x *= 1e6
y *= 1e6
for i in range(16):
    plt.subplot(4, 4, i + 1)
    plt.imshow(abs(np.fft.fftshift(w.get()[i])), extent=(x.min(), x.max(), y.min(), y.max()), origin='lower')
    plt.title("A=%dµm" % (10 * (i + 1)))
    if i >= 12:
        plt.xlabel('X (µm)')
    if i % 4 == 0:
        plt.ylabel('Y (µm)')
    plt.xlim(-150, 150)
    plt.ylim(-100, 100)
plt.suptitle("Fractional Fourier propagation (0.5m) of a A x 200 µm aperture")

Text(0.5, 0.98, 'Fractional Fourier propagation (0.5m) of a A x 200 µm aperture')

# Display one wavefront (i=..) in the stack
ImshowRGBA(i=1)*w

<pynx.wavefront.wavefront.Wavefront at 0x164150460>

Near field propagation of a simple 40x200 microns slit, focused (live update)

The ThinLens operator is used to focus he wavefront exiting from the slit

w = Wavefront(d=np.ones((512, 512), dtype=np.complex64), pixel_size=1e-6, wavelength=1.5e-10)
w = RectangularMask(width=200e-6, height=40e-6) * w

w = (ImshowRGBA(fig_num=60) * PropagateNearField(dz=0.1))**40 * ThinLens(2) * w

# Near field propagation of a simple 40x200 microns slit, displaying the propagated wavefront by steps
# of 0.2 m propagation
w = Wavefront(d=np.ones((512, 512), dtype=np.complex64), pixel_size=1e-6, wavelength=1.5e-10)
w = (ImshowAbs(fig_num=70) * PropagateNearField(dz=0.1))**20 * RectangularMask(width=100e-6, height=40e-6) * w

Example near field propagation

Note: there are actually 3 images from RGB components, hence the i=0 choice for the display

w = Wavefront(d="hubble", pixel_size=1e-6, wavelength=1.5e-10)
w = ImshowAbs(i=0) * PropagateNearField(dz=1.2) * w

Simulate a transmission image

d0 = face()[:,:768,0]
wavelength= 1.5e-10
delta = 1e-6
beta = 1e-9
thickness = 1e-7  # base thickness, to be multiplied by image value 0..255.
pixel_size = 1e-6
mu = 4 * np.pi * beta / wavelength
k = 2 * np.pi / wavelength
print("       mu * t = %f\nk * delta * t = %f" % (mu * thickness, k * delta * thickness))
d0 = np.exp(1j * k * (-delta + 1j * beta) * thickness * d0)

w = Wavefront(d=np.fft.fftshift(d0), pixel_size=pixel_size, wavelength=wavelength)
w = ImshowAbs() * PropagateNearField(dz=1.2) * w

       mu * t = 0.000008
k * delta * t = 0.004189