Kinoform synthesis with DSRT method
INTRODUCTION
Synthesis of diffractive optical elements[1-3] is one of the vital tasks of computer optics. Diffractive optical elements (DOE) are widely used in different areas of science and technology [1-10]. Among different kinds of DOE kinoforms[1,11] are most effective for many applications. Kinoform is synthesized phase DOE which allows to reconstruct image by its illumination with plane wave similarly to hologram. Unlike hologram however, kinoform forms only one diffraction order containing reconstructed image. Thus kinoform has maximum theoretical diffraction efficiency. Kinoforms are generally used as focusers, wavefront correctors, optical switches and transducers in image processing systems[1,3]. For dynamical display of synthesized DOE (including kinoforms) spatial light modulators (SLMs[2,13]) are widely used. There is no analytical solution to kinoform synthesis problem, therefore various iterative methods providing relatively small error of resulting intensity distribution are used[1,11-12]. However with restrictions on quantity of addressable phase levels this error becomes significant. The goal of this work is to apply direct search with random trajectory method (DSRT) [14,15] to task of kinoform synthesis for obtaining kinoforms with low deviation of light intensity distribution from desired one.
Rest of the paper organized as follows. In Section 2 application of DSRT method to kinoform synthesis task is described. In Section 3 results of numerical experiments on kinoforms synthesis with DSRT method and its comparison with conventional Gerchberg-Saxton method are presented. Main results are given in Conclusion.
Kinoform synthesis with DSRT method
DSRT method is slow but simple, effective and extremely versatile. It is simple direct search method similar to direct binary search method for binary holograms generation [16,17]. Main difference is that proposed direct search with random trajectory (DSRT) method is designed to process arrays with multiple gradations. For the first time we used it to generate matrices with specified power spectrum distribution [14]. Then we utilized it to generate keys with special properties for image optical encryption in spatially incoherent light [15]. DSRT method allows for easy change of objective criterion to obtain matrix with desired properties. And now we have applied it for kinoform synthesis task.
*holo@pico.mephi.ru
DSRT method can work with virtually any initial phase distribution, but it was determined that better and faster results are obtained if in place of initial phase distribution kinoform generated with conventional method is used. In other words, it is better to use DSRT method in conjunction with another method of kinoform synthesis [18]. For that part we used well-known Gerchberg-Saxton method [19].
Procedure of kinoform generation with DSRT method is described below:
1) Initial kinoform is synthesized with conventional method.
2) Random trajectory of consecutive processing of all kinoform elements is generated. Then iterative process begins.
3) Iteration consists of following steps. Each kinoform element sequentially in accordance with generated trajectory has its value changed to one that provides better value of objective criterion, i.e. lower deviation of reconstructed from desired one. If current value of kinoform element provides best objective criterion value then it left unchanged.
4) After all kinoform elements are processed, iteration repeats until stagnation is reached.
Since in this paper we synthesize kinoforms for incoherent light applications and therefore only light intensity distribution matters, as objective criterion we used normalized standard deviation (NSTD) [20] between reconstructed image and desired one:
, (1)
where E[ξ,η] and F[ξ,η] are matrices with intensities values of the initial and the reconstructed images respectively, [ξ,η] are discrete coordinates, and N×N is quantity of image pixels.
Let’s assess calculation cost of DSRT method. So, for kinoform with N×N elements one iteration consists of N2 subiterations. After each subiteration single kinoform element might be changed if its change leads to objective criterion improvement. As it can be seen procedure is time consuming. In order to fulfill single iteration for kinoform with N×N elements and m phase levels by simply running through all possible values of every element
(2)
Fourier transforms is required. It is a lot of calculations. However we have found that there is no need to go throw all possible values of kinoform elements because dependency of objective criterion value on kinoform element value virtually always have single extremum. Consequently, we can divide range of possible kinoform element values in two, then find which one contains element value which provides best objection criterion value, then divide it in two again and so on until exact element value is found. This lowers number of Fourier transforms required for single iteration to approximately
. (3)
It can be seen from Equation (3) that calculation speed gain increases with quantity of phase levels. In case of 4 phase levels speed gain is only 1.1 times, but if we have 16 phase levels, then speed gain is 3 times already. 256 available phase levels lead to 27 times decrease in calculation time.
Now, if fast Fourier transform is used then approximately:
(4)
basic multiply-and-accumulate operations is required (in case when fast Fourier transform is used) to perform one iteration.
Based on Equation (4), dependencies of quantities of required basic multiply-and-accumulate operations for one iteration of DSRT method on kinoform size N for different quantities of phase levels are presented in Fig. 1.
Figure. 1. Dependencies of quantities of required basic multiply-and-accumulate operations for one iteration of DSRT method on kinoform size for different quantities of phase levels
It can be seen that calculation cost increases drastically with kinoform size increase. For instance, 10 times increase in kinoform size leads to approximately 10000 times increase in calculation cost.