“Improved Computer-Generated Holograms for Optical Filtering”

by Chung-Kai Hsueh

October 1979

Several improvements in computer-generated holograms (CGH) are presented. Among these are a phase coding technique, a new type of binary hologram and a way of making continuous holograms. Emphasis is given to holograms that are used in a filtering system with a large input format.

CGH's are widely used in optical filtering systems. However, due to the sampled nature of the CGH, several diffraction orders occur in the output plane. This may result in overlapping of different orders in the output plane if the size of the input object is large. This problem can be avoided by sampling the filter at a rate determined by the sum of the input size and the impulse response size. This is essentially an oversampling in the transform domain. To eliminate the need for oversampling we describe a way of making continuous holograms. Two binary patterns corresponding to the amplitude and phase parts of the Fourier transform are made according to the sampling rate determined by the extent of the impulse response. These two patterns are low-pass filtered and recorded on color film by using illuminations of different colors. A complex-valued hologram is thus made. Pure amplitude or pure phase holograms can be made in a similar manner. This hologram reconstructs only a single order. When used as a filter, the limitation on the input size is therefore eliminated. However due to the non-ideal interpolation, background noise and some high order artifacts are present.

The oversampling problem can also be alleviated by using two other methods described in this thesis. They are the cell size expansion method and hologram interpolation with the discrete Fourier transform (DFT). The cell size expansion method reduces the intensity at the high orders but works only for small impulse responses. Hologram interpolation with the DFT increases the separation of different orders and therefore allow a larger input format. A correct procedure for doing interpolation by the DFT is presented.

Binary holograms have the advantages of easy fabrication and high signal-to-noise ratio. A new binary hologram called the double- phase hologram (DPH) is described. In one of the configurations, the reconstruction appears further away from the central bright spot. It is therefore more suitable than other binary holograms for a filtering system with a large input format although the efficiency is lower. Phase coding methods and their effects are also studied. When the hologram is used for reconstructing an image or is used in an incoherent system, only the intensity is of concern. The phase of the impulse response can be manipulated in order to reduce the dynamic range of the Fourier transform. However, due to the use of the DFT in making the hologram, the phase coding increases the aliasing errors and the impulse response has speckle-like noise. An iterative phase coding method is described which uses constrained bandwidth and discards the high frequency components. This method is capable of reducing the speckle-like noise while the dynamic range is also considerably reduced.

To demonstrate the application of CGH's, a character recognition system is presented. The phase-only matched filter is shown to be better than the matched filter in both efficiency and discrimination power. The phase-only matched filters are implemented by both the sampled kinoform method and continuous kinoform method. Experimental results are shown.