# Algorithms

Algorithms for wavefront propagation, digital holography and iterative phase retrieval

For the past two decades I have been developing novel methods and the related algorithms for imaging with coherent light, electrons, X-rays, terahertz and other waves.

Practical algorithms for simulation of the wavefront propagation, simulation and reconstruction of in-line digital holograms, for plane and spherical waves are described in ref. . Iterative phase retrieval algorithms and their practical implementation to experimental diffraction patterns are discussed refs.[2,3].

In 2021 I showed that a three-dimensional sample distribution can be reconstructed from a single-shot intensity distribution, such as diffraction pattern or a hologram, by applying multi-slice iterative phase retrieval algorithms , fig. 1.

In 2007, I found a solution to the twin image problem in holography via applying an iterative phase retrieval algorithm , fig. 2, highlighted in New Scientist, PhysOrg. Twin image problem is a long-standing problem in holography, known since holography was invented by Denis Gabor in 1947 . Fig. 1. Simulated 2D electron diffraction pattern of bilayer twisted graphene (BLG) shown in inverted logarithmic scale (a). (b) Modelled BLG distribution with defects in one layer. (c) and (d) The phase distributions of the reconstructed transmission functions of the individual layers, values range from 0 to 0.24 rad. Scalebar in (b) - (d) is 1 nm . Fig. 2. Solution to the twin image in holography. Left: Reconstructed object with the artifact twin image. Right: Twin-image free reconstructed object.

## References

1. T. Latychevskaia and H.-W. Fink, Practical algorithms for simulation and reconstruction of digital in-line holograms, Appl. Opt. 54, 2424–2434 (2015), matlab codes for plane and spherical waves.
2. T. Latychevskaia, Iterative phase retrieval in coherent diffractive imaging: practical issues, Appl. Opt. 57, 7187–7197 (2018), matlab code.
3. T. Latychevskaia, Iterative phase retrieval for digital holography, J. Opt. Soc. Am. A 36, D31–D40 (2019).
4. T. Latychevskaia, Three-dimensional structure from single two-dimensional diffraction intensity measurement, Phys. Rev. Lett. 127, 063601 (2021), matlab code.
5. T. Latychevskaia and H.-W. Fink, Solution to the twin image problem in holography, Phys. Rev. Lett. 98, 233901 (2007), matlab codes for plane and spherical waves.
6. D. Gabor, A new microscopic principle, Nature 161, 777–778 (1948).