Reconstruction of off-axis holograms is done numerically and it consists of two mathematical transformations. First, a Fourier transform of the hologram is performed. The resulting complex image consists of the autocorrelation (center band) and two mutually conjugated sidebands. Only one side band is selected by applying a low-pass filter (round mask) centered on the chosen side-band. The central band and the twin side-band are both set to zero. Next, the selected side-band is re-positioned to the center of the complex image and the backward Fourier-transform is applied. The resulting image in the object domain is complex-valued, and thus, the amplitude and phase distributions of the object function are reconstructed.
Electron holography in in-line scheme
Original holographic scheme by Dennis Gabor is inline scheme, which means that reference and object wave share the same optical axis. This scheme is also called point projection holography. An object is placed into divergent electron beam, part of the wave is scattered by the object (object wave) and it interferes with the unscattered wave (reference wave) in detector plane. The spatial coherence in in-line scheme is defined by the size of the electron source. Holography with low-energy electrons (50-1000eV) can be realized in in-line scheme.Inline electron holography scheme.
It is important to shield the interferometric system from electromagnetic fields, as they can induce unwanted phase-shifts due to the Aharonov–Bohm effect. Static fields will result in a fixed shift of the interference pattern. It is clear every component and sample must be properly grounded and shielded from outside noise.
Electron holography is commonly used to study electric and magnetic fields in thin films, as magnetic and electric fields can shift the phase of the interfering wave passing through the sample.
The principle of electron holography can also be applied to interference lithography.
- D. Gabor, A new microscopic principle, Nature 4098, 777 (1948).
- M. E. Haine, T. Mulvey, The formation of the diffraction image with electrons in the Gabor diffraction microscope, J. Opt. Soc. Am. 42, 763 (1952).
- G. Möllenstedt and H. Düker, Beobachtungen und Messungen an Biprisma-Interferenzen mit Elektronenwellen, Zeitschrift für Physik, 145, 377 (1956).
- J. M. Cowley, Twenty forms of electron holography, Ultramicroscopy 41, 335–348 (1992).
- M. Lehmann, H. Lichte, Tutorial on off-axis electron holography, Microsc. Microanal. 8(6), 447–466 (2002).
- H.-W. Fink, W. Stocker and H. Schmid, Holography with low-energy electrons, Phys. Rev. Lett. 65(10), 1204–1206 (1990).
- H. Lichte, Electron holography approaching atomic resolution, Ultramicroscopy 20, 293 (1986).
- A. Tonomura, Applications of electron holography, Rev. Mod. Phys. 59, 639 (1987).
- R. E. Dunin-Borkowski et al., Micros. Res. and Tech. 64, 390 (2004).