Holography with Hard X-Rays
Contents of this document:
General holography
In general the name holography is used for imaging techniques which are sensitive for recording the phase of a wave which was formerly interacting with an object. Holograms with visible light are usually recorded to obtain a three dimensional information of the surface of opaque objects. While visible light is reflected or scattered from the object surface the X-rays are mostly transmitted through the objects or absorbed in it. For that reason holography with X-rays gives information about the change of the phase and the intensity of the radiation which is transmitted through the object. The change of phase and amplitude which is recordable is always an integrated value over the beam path through the sample. And one obtains always a two dimensional projection of the three dimensional object. Of course a three dimensional information from the sample from X-ray holograms can be obtained combining holography with tomography (see tomography).
The interest for doing X-ray holography is therefore not mainly in obtaining three dimensional sample information. The reason is that the phase contrast at short wavelength is much higher than the absorption contrast. This is especially for samples containing light elements (reference). Recent applications and developments are dedicated to high resolution microscopic imaging using holographic methods.
Practical holography
The aim of a holographical experiment is to record simultaneously the amplitude and the phase distribution of a wave which is coming from an object. This is in contrast with the conventional photography which records only the amplitude (intensity) of the object wave. But having both informations one is able to completely reconstruct the object wave in all details. In case of optical holograpy a three dimensional information about the object wave is obtained which is not possible in photography. The practical realisation of a holographical experiment is the following. The wave diffracted by the object (object wave) is coherently superimposed with a well defined reference wave. As a result one obtain a system of interference fringes which can be recorded on a film, CCD camera or another suited detector. This interference pattern which looks completely different as a photography is called a hologram. The distance of the interference fringes in a hologram is proportional to the mutual phase difference of the both waves at this particular point. In a second step one can reconstruct the object wave using the known intensity and phase distribution of the reference wave in the hologram plane.
The reconstruction of the object wave from the hologram can be done in different ways:
- Optical reconstruction by illuminating the recorded film hologram with coherent light and using the hologram as diffraction grating. The observed real image is the reconstruction of the object.
- Numerical reconstruction by using a mathematical model to calculate the diffraction pattern which is created by illumination of the hologram. (Or other way round - calculating that object wave which gives together with the reference wave the recorded hologram)
Types of holography
According to the shape of the reference wave we distinguish holography with plane and with spherical reference wave.
Plane reference wave
The hologram of a single object point having a plane reference wave will be discussed. The object point sends out a spherical object wave and the interference pattern consists of concentric circles with decreasing distances to the outside. The same interference pattern is known as Fresnel zones. The optical reconstruction from this hologram is done with with a plane reference wave falling on the hologram acting on his part as a diffraction grating. It gives a focal spot from the +1st order diffracted wave and a diverging wave from the -1st order diffracted wave which is the same as a virtual focal point at the opposite side of the hologram. Higher order diffracted waves are neglected here. The lateral resolution in the reconstruction is given by the spatial resolution of the recording media which determines the smallest recordable zone width in the hologram.
Spherical reference wave
In the case of a spherical reference wave the hologram of a single object point the interference pattern can be described in analogy to the double slit experiment. The reference and the object wave have the same radius of curvature and a wide spaced interference pattern is expected. The hologram which is observed in a plane which is parallel to the plane containing the object and the reference point consists of interference fringes with a fringe distance Df which is given by Df=lambda x/s. Here lambda is the wavelength, x the object-to-hologram distance and s the reference-to-object distance.
The resolution in the reconstructed object wave is not limited by the detector resolution but by the size of the detector or the aperture of the hologram because small structures gives large distances of interference fringes. The resolution is additional limited by the size of the reference source.
It can be mathematical shown, that the reconstruction of the hologram with a spherical reference wave which is placed in the plane of the object can be done by a two dimensional inverse Fourier transformation. Due to this fact this particular method is called Fourier-transform holography.
References
- W. Leitenberger and A. Snigirev, Fourier-transform holography with coherent hard X-rays. Published in : X-ray Microscopy 1999 , eds. A. Warwick and D. T. Attwood (American Institute of Physics Press, Washington, 2000).
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