Nondetour phase computer-generated holograms: an improved variation Roy M. Matic and Eric W. Hansen

Dartmouth College, Thayer School of Engineering, Han­over, New Hampshire 03755. Received 8 February 1982. 0003-6935/82/132324-02$01.00/0. © 1982 Optical Society of America.

In a recent paper Gallagher and Bucklew1 proposed non­detour phase (NDPH) projection type computer-generated holograms. This Letter presents a variation of the projection NDPH which has a slightly higher diffraction efficiency in the reconstructed order and requires less time to compute.

We begin by reviewing Gallagher and Bucklew's results. Using a slightly different approach than that of Ref. 1, the Fourier transform of the projection NDPH (the reconstructed image) is

where h(x,y) = the reconstructed image, T — sample cell size, L = the number of subcells per cell,

Hmnk = A[(n + k/L)T,mT] cos{θ[(n + k/L)T,mT] - (2πk)/L}. μ(cos{θ[(n + k/L)T,mT] - (2πk)/L}),

A(.) = the unit step function, A(u,v) = amplitude of the Fourier transform of

g(x,y), θ(u,v) = phase of the Fourier transform of g{x,y),

and g(x,y) = the desired function encoded by the

NDPH. The Hmnk values are the non-negative projections of the am­plitude of the Fourier transform onto unit vectors at angles 2πk/L. Dropping the sinc terms for simplicity and substi­tuting for Hmnk yield

where F-1 denotes the inverse Fourier transform, * denotes convolution, and δ( . , . ) is the Dirac delta function. To eval­uate the above expression the following identities are used:

Substituting these in the expression for h(x,y) yields an ex­pression with five summations, only one of which contains the amplitude and phase characteristics of the desired function. h(x,y) can be expressed as

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Fig. 1. Reconstructed image of a nondetour phase amplitude holo­gram using 12 subcells per sample cell.

When n = 1, the first diffracted order, the other terms equal zero, and h(x,y) reduces to

The desired function g(x,y) appears in the first order multi­plied by a constant = 1/4. For n = - 1 , g(-x,-y) appears in the - 1 order. For n other than ± 1 , the phase of the function is distorted, and therefore g(x,y) is not reconstructed in any other order.

Note the difference in constant factors between the result derived here and that found in Ref. 1. The differences are a result of a shift term not used in this derivation and an error in Ref. 1. The error, a missing factor of 1/2, occurs in the identity found on the bottom left of page 4269 of Ref. 1. When this factor is multiplied into the result in Ref. 1, it be­comes identical to the result derived here.

The variation of the projection NDPH of Ref. 1 presented in this Letter uses different values for Hmnk:

These Hmnk values are non-negative samples of the amplitude of the Fourier transform and not the projections. Appro­priately, this hologram may be termed a nondetour phase amplitude hologram. Using these Hmnk in the above analysis yields

Substituting the same expansion for the unit step function (3) results in three summations, only one of which contains the amplitude and phase characteristics of the desired function. h(x,y) can therefore be expressed as

When n = 1, the first diffracted order, the other terms go to zero, and h(x,y) reduces to

As in the previous case, the desired function is in the +1 order and is reflected in the - 1 order. In this case, the desired function is multiplied by a constant = 1/Π rather than 1/4 yielding a 27% increase in amplitude of the desired function in the first order. This makes sense since the Hmnk values plotted for this variation are generally larger than the Hmnk values in the projection NDPH.

This variation of NDPH holograms also has an advantage in computational speed. The Gallagher and Bucklew Hm n k values require multiplication by a cosine term. The NDPH variation presented here does not; as a result, the calculation of the Hmnk values is faster. Calculations done on a Honey­well level 66 computer indicate that, once the amplitude and phase of the function are known, the calculation of the Hmnk values is 2.8 times faster for the NDPH variation presented here than for the Gallagher and Bucklew NDPH.

Holograms of the letter P were made using this technique. The hologram shown in Fig. 1 was made on a Tektronix 4662 flatbed plotter with a total of 21 × 21 sample cells, with each sample cell divided into 12 subcells. Each sample cell was 6.36 × 6.36 mm. The plotted hologram was photoreduced 34 times onto Kodak Technical Pan film and developed using standard darkroom techniques. The reconstructed image was photo­graphed using Kodak Plus-× Pan Professional film 4147.

This work was supported by the National Science Foun­dation grant ECS-8006904.

Roy M. Matic is currently with Creare Innovations, Inc., Hanover, N.H.

Reference 1. N. C. Gallagher, Jr., and J. A. Bucklew, Appl. Opt. 19, 4266

(1980).

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