Solution to the twin image problem in holography

While the invention of holography by Dennis Gabor truly constitutes an ingenious concept, it has ever since been troubled by the so called twin image problem limiting the information that can be obtained from a holographic record. Due to symmetry reasons there are always two images appearing in the reconstruction process. Thus, the reconstructed object is obscured by its unwanted out of focus twin image. Especially for emission electron as well as for x- and gamma-ray holography, where the source-object distances are small, the reconstructed images of atoms are very close to their twin images from which they can hardly be distinguished. In some particular instances only, experimental efforts could remove the twin images. More recently, numerical methods to diminish the effect of the twin image have been proposed but are limited to purely absorbing objects failing to account for phase shifts caused by the object. Here we show a universal method to reconstruct a hologram completely free of twin images disturbance while no assumptions about the object need to be imposed. Both, amplitude and true phase distributions are retrieved without distortion.


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The basic setup for holography is depicted in Figure 1 together with the positions of the object and its twin image when an in-line hologram is reconstructed. In the object plane, the twin appears as an out of focus image while in the twin image plane the object appears out of focus. The two images are mirror-symmetric with respect to the point source. In holography with visible light, the object and its twin image can be separated by using parallel beams and subtracting a second hologram from the reconstructed image 1,5 , by employing a beam splitter 6 or introducing additional lenses into the recording and reconstructing scheme 7 . However, lenses are not available for x-ray or gamma-ray holography. In electron emission holography, the close proximity of source and sample also makes it impossible to employ lenses or a beam-splitter between them.
In other schemes, like holography with low energy electrons lenses are to be avoided due to their inherent aberrations. Moreover, in-line holography exhibits high phase sensitivity and is therefore, for coherent low energy electrons 13 and even for high energy electrons 14 , for which DNA molecules represent extremely weak phase objects, the method of choice.
The most widely employed approach to address the twin image problem is to record a set of holograms at different wavelengths 15,16 . However, this method only suppresses but not eliminates the twin image and is experimentally difficult to implement in particular when it comes to record fragile biological molecules subject to radiation damage. So far numerical methods to diminish the effect of the twin image have been restricted to holograms of purely absorbing objects 8,9,10,11,12 , a coarse approximation of physical reality. In this letter we show how the twin image can be eliminated by numerical reconstruction of a hologram without imposing any restrictions on or assumptions about the object to be imaged. gives rise to the object wave, Aexp(ikr 0 )t(r 0 ). The total field at the screen is the sum of the reference and object wave A(R 0 (r s )+O 0 (r s )), where R 0 =exp(ikr s ), and AO 0 (r s ) is the object wave distribution on the screen, which is calculated by solving the Kirchhoff-Helmholtz integral 17 following routine is applied to such normalized holograms making it independent of details of the data acquisition.
The final goal of our method is to reconstruct the distribution of the complex sum (R 0 (r s )+O 0 (r s )). This is achieved by an iterative procedure 18,19,20 which basically boils down to the field propagation back and forth between the screen-and the object-plane, until all artefacts due to the twin image are gone. It includes the following steps: where the amplitude is always given by the square root of the normalized hologram ) , and the phase Ω(r s ) is initially set to kr s -the phase of the known reference wave R 0 =exp(ikr s )and it evolves towards its true value during iteration.
(ii) Back propagation to the object plane is simulated using the Helmholtz-Kirchhoff  Figure 3. The result after 500 iterations shows that residues due to the twin image in the reconstructed absorption and phase distributions are gone.
With this, a novel method to finally solve the twin image problem is established and can now be applied without limitations to wavelength or wave front shapes (planar, or spherical), for imaging objects of arbitrary size, exhibiting absorbing and/or phase shifting properties. From a single holographic record, twin-image free true absorption and phase distributions are iteratively retrieved.