mag2exp.ltem.defocus_image#

mag2exp.ltem.defocus_image(phase, /, cs=0, df_length=0.0002, voltage=None, wavelength=None)#

Calculating the defocused image.

The wavefunction of the electrons is created from the magnetic phase shift \(\phi_m\)

\[\psi_0 = e^{i\phi_m},\]

and propagated through the electron microscope to the image plane by use of the Contrast Transfer Function \(T\). The wavefunction at a defocus length \(\Delta f\) in Fourier space is given by

\[\widetilde{\psi}_{\Delta f} = \widetilde{\psi}_0 e^{2 i \pi \lambda k^2 (-\frac{1}{2} \Delta f + \frac{1}{4} C_s \lambda^2 k^2)},\]

where \(\lambda\) is the relativistic wavelength of the electrons, \(C_s\) is the spherical aberration coefficient of the microscope, and \(k\) is the wavevector in Fourier space.

The intensity of an image at a specific defocus is given by

\[I_{\Delta f} = \left\vert \psi_{\Delta f} \right\vert^2.\]

In focus \(I=\left\vert\psi_0\right\vert^2=1\)

Either electron wavelength or acceleration voltage U must be specified.

Parameters:

  • phase (discretisedfield.Field) – Phase of the electrons from LTEM.

  • cs (numbers.Real, optional) – Spherical aberration coefficient.

  • df_length (numbers.Real, optional) – Defocus length in m.

  • voltage (numbers.Real, optional) – Accelerating voltage of electrons in V.

  • wavelength (numbers.Real, optional) – Relativistic wavelength of the electrons in m.

Returns:

Intensity at specified defocus.

Return type:

discretisedfield.Field

Examples

  1. Zero defocus.

>>> import discretisedfield as df
>>> import mag2exp
>>> import numpy as np
>>> mesh = df.Mesh(p1=(-5e-9, -4e-9, -1e-9), p2=(5e-9, 4e-9, 1e-9),
...                    cell=(2e-9, 1e-9, 0.5e-9))
...
>>> def value_fun(point):
...     x, y, z = point
...     if x > 0:
...         return (0,-1, 0)
...     else:
...         return (0, 1, 0)
...
>>> field = df.Field(mesh, nvdim=3, value=value_fun)
>>> phase, ft_phase = mag2exp.ltem.phase(field)
>>> df_img = mag2exp.ltem.defocus_image(phase, cs=0, df_length=0,
...                                     voltage=300e3)
>>> np.allclose(df_img.array, [1])
True

  1. Visualising the phase using matplotlib.

>>> import discretisedfield as df
>>> import mag2exp
>>> mesh = df.Mesh(p1=(-5e-9, -4e-9, -1e-9), p2=(5e-9, 4e-9, 1e-9),
...                    cell=(2e-9, 1e-9, 0.5e-9))
...
>>> def value_fun(point):
...     x, y, z = point
...     if x > 0:
...         return (0, -1, 0)
...     else:
...         return (0, 1, 0)
...
>>> field = df.Field(mesh, nvdim=3, value=value_fun)
>>> phase, ft_phase = mag2exp.ltem.phase(field)
>>> df_img = mag2exp.ltem.defocus_image(phase, cs=8000,
...                                     df_length=0.2e-3,
...                                     voltage=300e3)
>>> df_img.mpl.scalar()
../../_images/mag2exp-ltem-defocus_image-1.png

See also

phase() relativistic_wavelength()