Phase-space foundations of electron holography

We present a unified formalism for describing various forms of electron holography in quantum mechanical phase space including their extensions to quantum-state reconstructions. The phase-space perspective allows for taking into account partial coherence as well as the quantum mechanical detection process typically hampering the unique reconstruction of a wave function. We elaborate on the limitations imposed by the electron optical elements of the transmission electron microscope as well as the scattering at the target. The results provide the basis for vastly extending the scope of electron holographic techniques towards analyzing partially coherent signals such as inelastically scattered electrons or electron pulses used in ultrafast transmission electron microscopy.

Figure 1

Different representations of the density operator as the ambiguity function, density matrix, and Wigner function. The relation among the three representations as well as the outcome of an intensity measurement is indicated.

Figure 2

Principle optical setup of in-line holography. Accordingly, the defocus interval is ideally bounded by the focal and image plane.

Figure 3

Combined effect of partial spatial and chromatic incoherence in combination with defocus and chromatic aberration (see Ref. [51] for details) on the average quantum state (i.e., with mean energy and lateral momentum). Spatial incoherence in combination with a defocus leads to a momentum-invariant convolution, which therefore may be deconvolved from the recorded image. Chromatic incoherence acts as a momentum-dependent convolution, which therefore cannot be deconvolved from a single measurement.

Figure 4

In-line holography as quantum-state reconstruction. The shear series of the defocused Wigner function is shown in the top row. The corresponding tilt series obtained by suitably scaling the phase space, (24), is depicted in the bottom row.

Figure 5

Simplified optical setup for off-axis electron holography incorporating three holographic shifts dependent on the biprism voltage.

Figure 6

Quantum-state [ambiguity function; Eq. (34)] reconstruction by off-axis holography measuring the cross section ${\tilde{W}}_{-+}(\mathbf{q},\mathbf{p}=0)$ of the sideband (sb) for different holographic shifts $\left(\mathbf{d}\right)$. A sufficiently high carrier frequency $\left({\mathbf{q}}_c\right)$ permits separation between centerbands (cb) and sidebands.

Figure 7

Astigmatic TIE setup in one lateral dimension. The defocus step is denoted $\delta z$. The differential shift of the intensity with respect to the neighboring image point is indicated by the arrow.

Figure 8

Generalized TIE as quantum-state reconstruction. Moments with increasing order correspond to a Taylor expansion of the ambiguity function.

Figure 9

STEM-DPC setup in one lateral dimension. The shift of the momentum's barycenter recorded in the far field of the specimen is indicated by the arrow.

Figure 10

Ptychography as (de)convolution of the Wigner transformed axial scattering operator. Accordingly, the smaller (i.e., purer) the probe's Wigner function phase-space volume, the larger the corresponding low pass truncating the corresponding ambiguity function of the axial scattering operator (Fourier transform of the Wigner transform).