Electron Holography

HyperSpy provides the user with a signal class which can be used to process electron holography data:

It inherits from Signal2D class and thus can use all of its functionality. The usage of the class is explained in the following sections.

The HologramImage class

The HologramImage class is designed to contain images acquired via electron holography.

To transform a Signal2D (or subclass) into a HologramImage use:

>>> im.set_signal_type('hologram')

Reconstruction of holograms

The detailed description of electron holography and reconstruction of holograms can be found in literature [Gabor1948], [Tonomura1999], [McCartney2007], and [Joy1993]. Fourier based reconstruction of off-axis holograms (includes finding a side band in FFT, isolating and filtering it, recenter and calculate inverse Fourier transform) can be performed using the reconstruct_phase() method which returns a Complex2D class, containing the reconstructed electron wave. The reconstruct_phase() method takes sideband position and size as parameters:

>>> import hyperspy.api as hs
>>> im =  hs.datasets.example_signals.object_hologram()
>>> wave_image = im.reconstruct_phase(sb_position=(<y>, <x>),
...                                   sb_size=sb_radius)

The parameters can be found automatically by calling following methods:

>>> sb_position = im.estimate_sideband_position(ap_cb_radius=None,
...                                             sb='lower')
>>> sb_size = im.estimate_sideband_size(sb_position)

estimate_sideband_position() method searches for maximum of intensity in upper or lower part of FFT pattern (parameter sb) excluding the middle area defined by ap_cb_radius. estimate_sideband_size() method calculates the radius of the sideband filter as half of the distance to the central band which is commonly used for strong phase objects. Alternatively, the sideband filter radius can be recalculate as 1/3 of the distance (often used for weak phase objects) for example:

>>> sb_size = sb_size * 2 / 3

To reconstruct the hologram with a vacuum reference wave, the reference hologram should be provided to the method either as Hyperspy’s HologramImage or as a nparray:

>>> reference_hologram = hs.datasets.example_signals.reference_hologram()
>>> wave_image = im.reconstruct_phase(reference_hologram,
...                                   sb_position=sb_position,
...                                   sb_size=sb_sb_size)

Using the reconstructed wave, one can access its amplitude and phase (also unwrapped phase) using amplitude and phase properties (also the unwrapped_phase() method):

>>> wave_image.unwrapped_phase().plot()

Unwrapped phase image.

Additionally, it is possible to change the smoothness of the sideband filter edge (which is by default set to 5% of the filter radius) using parameter sb_smoothness.

Both sb_size and sb_smoothness can be provided in desired units rather than pixels (by default) by setting sb_unit value either to mrad or nm for milliradians or inverse nanometers respectively. For example:

>>> wave_image = im.reconstruct_phase(reference_hologram,
...                                   sb_position=sb_position, sb_size=30,
...                                   sb_smoothness=0.05*30,sb_unit='mrad')

Also the reconstruct_phase() method can output wave images with desired size (shape). By default the shape of the original hologram is preserved. Though this leads to oversampling of the output wave images, since the information is limited by the size of the sideband filter. To avoid oversampling the output shape can be set to the diameter of the sideband as follows:

>>> wave_image = im.reconstruct_phase(reference_hologram,
...                                   sb_position=sb_position,
...                                   sb_size=sb_sb_size,
...                                   output_shape=(2*sb_size, 2*sb_size))

Note that the reconstruct_phase() method can be called without parameters, which will cause their automatic assignment by estimate_sideband_position() and estimate_sideband_size() methods. This, however, is not recommended for not experienced users.