Keywords: holography, computer generated hologram, Helmholtz equation, paraxial propagation, Fourier-based algorithm, finite difference method.

Holography is one of the three-dimensional viewing techniques and uses the physical phenomena of interference and diffraction of light waves to record on a recording medium and to reconstruct a three-dimensional image. It is a two-step process: recording process and reconstruction process.

The first is the registration of the hologram of an object with the aid of a digital camera or photographic medium realized on the laboratory stand. The second is to obtain the holographic image of the original object registered by the recording medium [2,3]. The development of computer technology allows the transfer of both the recording process and the construction process into the computer. The main purpose of this paper is to develop a program which generates artificial holograms using numerical methods and these computer generated holograms are numerically reconstructed. The fundamental problem in computational holography is the computation of the optical light field distribution which arises over the entire three-dimenional space. The result of this computation is used to reconstruct the original object at the display. The computation techniques used affect the quality of the reconstructed images.

The most commonly used numerical method to solve the scalar wave equation is the Fourier-based method developed by Goodman [1]. This approach is attractive since it requires little programming effort. The evaluation of the solution can be done by a FFT at each spatial position for which the field distribution is desired. Alternatively, an algorithm to solve the problem directly in the time domain is proposed here, which, consists of the discretization of the differential operator in the scalar wave equation, such as the finite-difference method (FDM).

In this paper two different beam propagation methods are presented to simulate the propagation of the light field under the paraxial condition, the first is based on the fast Fourier transformation (FFT-BPM) [4] and the second based on the finite-difference method (FD-BPM) [4]. The beam propagation method (FFT-BPM) proved to be an efficient tool for solving this type of problem. However, in application of this method to problems with a very large cross-section one has to cope with the increased storage and reduced efficiency. The second numerical method to solve the wave equation is to use a finite difference approximation [4], based on the Peaceman-Rachford scheme. Following the finite-difference method (FDM) the wave equation is replaced by a system of linear equations. The resulting three-diagonal systems of equations are solved by some direct and iterative procedures [4].

This paper is organised as follows: In Section 2 a brief introduction to computational holography is given. In Section 3 the solution of the diffraction problem as a Fresnel-Kirchhoff integral is introduced and the numerical method to compute this integral developed. In Section 4 the parabolic wave equation and a finite-difference scheme to solve this equation on a three-dimensional grid are introduced. In Section 5 the results of Fourier-based calculations to finite-difference calculations are compared.

1

J.W. Goodman, "Introduction to Fourier Optics", 2nd edition, McGraw-Hill Companies, Inc., New York, 1996. doi:10.1117/1.601121

2

R.W. Kronrod, N.S. Merzlyakov, L.P. Yaroslavskii, "Reconstruction of a hologram with a computer", Sov. J. Tech. Phys., 17, 333-334, 1972.

3

U. Schnars, "Direct phase determination in hologram interferometry with use of digitally recorded holograms", J. Opt. Soc. Am. A., 11, 2011-2015, 1994. doi:10.1364/JOSAA.11.002011

4

C. Fuhse, "X-ray waveguides and waveguide-based lensless imaging", PhD thesis, University of Göttingen, 2006.

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