Abstract
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.
2
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.
7
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.
14
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
15,16
, 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. following routine is applied to such normalized holograms making it independent of details of the data acquisition.
15,16
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: (ii) Back propagation to the object plane is simulated using the Helmholtz-Kirchhoff We then acquire the altered phase value from this new sum and use it as the input phase value for the next iteration starting at step (i). We would like to point out that the first iteration already reconstructs the complex object with the same quality as the conventional hologram reconstruction routines. Further iterations eventually lead to the elimination of the twin term. There are no limitation on the object surrounding, object's size or on the object properties being weak absorbing or weak/strong phase shifting. The object and its surrounding can be anything: for instance, an extended biological molecule which is identified only by a weak phase shift of the incident beam, or an arrangement of heavy atoms acting as point-like absorbing centres.
15,16
We now test the routine, first with a simulated hologram, then by using experimental holograms. A hologram of an extended object with a maximum absorption a(r 0 ) of 80% of an incident beam and a maximum phase shift ϕ(r 0 ) of 3 radians was simulated by 6 using the Helmholtz-Kirchhoff formula 17 and reconstructed with our method described Figure 3 . The result after 500 iterations shows that residues due to the twin image in the reconstructed absorption and phase distributions are gone.
15,16
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.

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