Dependence of CGH reconstruction due to the transfer phase function of angular spectrum method

Computer-generated hologram (CGH) is the method of making holograms with computer. A CGH can be made without preparing and instructing the exclusive optical tool. Therefore, we can make holograms easily and economically compared to the optical method. There are several calculating methods for a CGH such as Fourier transform, Fresnel diffraction and etc. In this study, we discuss on the angular spectrum method which is not limited by the diffraction distance. The Kirchhoff integral has the advantage of making reconstructions similar to optical experiment. However, the angular spectrum method has the disadvantage of accruing an error without proper conditions. These conditions are related to the transfer phase function. Moreover, the transfer phase function is related to the sampling interval and the number of sampling points. Therefore, it is necessary to investigate the relationship about these conditions and reconstruction errors. This study reports on the reconstruction using the angular spectrum method under different conditions about the transfer phase function. Simulation results will be shown on the relationship under these conditions.


Angular spectrum method 1.1 Transfer phase function
Angular spectrum method uses discrete Fourier transform.In this method, hologram is made as follows.First, original image is discrete Fourier transform.Next, multiplying the transfer phase function and this image.Lastly, this image is inverse discrete Fourier transform.
The formula about transfer phase function is written as (1).
In this formula,  is distance between the original image and the hologram. is wavenumber. is wavelength.  and   are spatial frequency.The spatial frequency is inverse proportion to the sampling interval.Therefore, transfer phase function depends on  and the sampling interval.Formula ( 2) is complex amplitude distribution of hologram made of angular spectrum method. (2)

Discretization
As mentioned above, angular spectrum method uses discrete Fourier transform.Therefore, it is need to consider discretization.Formula (3) is Fourier transform about original image.Discretizing formula (3) to calculate formula (4).
In formula (4), N is a number of sampling points.As with original image, discretizing hologram.As a result, formula (5) is got.
Formula (8) is discretize the complex amplitude distribution of the hologram made of angular spectrum method.

Comparing angular spectrum method and Kirchhoff integral
In general, the Kirchhoff integral is used to CGH.Because, this method make reconstructions similar to optical experiment.Therefore, in this section, compering reconstructions made of the Kirchhoff integral and angular spectrum method.Also, viewing the transfer phase function influences for angular spectrum method.

The error depending on distance
For formula (1), the transfer phase function depends on distance between the original image and the hologram.Therefore, examining how reconstructions change according to .In formula (10), E is a variable which shows difference in amplitudes of the original image and the reconstruction.Therefore, the larger the absolute value of formula (10), the smaller the error.4, similar results can be obtained with angular spectrum method.From the above, if formula (7) holds, angular spectrum method will yields the same result as the Kirchhoff integral.

The error depending on sampling interval
Formula (11) is sampling theorem in making hologram.
Fig. 1 is the original image.A number of sampling points of this image is 128*128[pixel].Sampling interval of this image is 20.0[μm].For formula (7), deciding that a number of sampling points of hologram is 128*128[pixel] and sampling interval of hologram is 20.0[μm].Making reconstructions of the Kirchhoff integral and angular spectrum method in the above conditions.Setting  to 0.1, 0.2, 0.4, 0.8[m].Fig. 2 (a) ~ (d) are reconstructions made of the Kirchhoff integral.Fig. 3 (a) ~ (d) are reconstructions made of angular spectrum method.Moreover, Fig. 4 is a graph of the error of reconstructions in decibel notation.Formula (10) is used to find the error.