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Fig. 3. Flowchart of the diffraction limited reconstruction process. The measured cross-term M( x, y) sampled across the sparse pixel grid of the camera sensor in a) is Fourier transformed( FT). Because of the Fourier holography scheme, the Fourier transform has the appearance of b) a( blurry) reconstruction of the object wave field U O and therefore centers the energy at the approximate location of the object across coordinates u = nkz, v = gkz. This blurry reconstruction is then zero-padded( ZP) in c) by the factor q = Dp / d, where Dp is the pixel pitch of the sensor and d is the diffraction limit. The zero-padded reconstruction is then Fourier transformed( FT �1) back into the hologram plane d). This results in a precise sinc-interpolated copy of the cross-term M in d). By multiplicative modulation with the analytical structure of the reference wave R( x, y) in e), the complex wave field U H is fully recreated in the sensor domain across the dense grid. The wave field is then precisely reconstructed using for example the angular spectrum method( ASM) f).
A spherical reference wave works well to prevent averaging of frequencies during recording, but the assumption that the reconstruction process can be performed through a simple Fourier transform is still subject to the validity of equation( 5) and therefore no more precise than the Fresnel propagation formula. Consequently, we need to find a more precise way to reconstruct the wave field in the object plane that does not rely on the Fresnel approximation.
2.3 Diffraction-limited reconstruction
In the following, we will solve the reconstruction problem in a two-step process, which is also shown schematically in the flow chart in Figure 3. Similarly to the scheme of lensless
Fourier holography, we will assume the use of a spherical reference wave with known structure during recording. In the first step, we interpolate the measured cross term M( x, y). This is realized by Fourier transforming the cross term shown in Figure 3a into Fourier space shown in Figure 3b and applying zero-padding by a factor of q = Dp / d in Figure 3c, whereDp is the pixel pitch of the sensor and d is the diffraction limit. The zero-padding in Fourier space corresponds to a sinc-interpolation of the cross term M( x, y) which is a perfect reconstruction of the original function if the underlying cross term M( x, y) is band-limited [ 30 ]. This can be exemplified by the fact, that the Fourier transform of M( x, y) corresponds to propagating the wave field into the object plane by applying equation( 5) and obtaining an approximate reconstruction of the object.