Computer-Generated Holography (CGH)

PHYS4412 (Computational Science II)
Department of Physics & Astronomy
Louisiana State University

As Nishikawa et al. (1997) have indicated [see their equation (5)], the most general expression for the complex amplitude of light that arrives at a given (x,y) position on the image screen from a three-dimensional object whose surface (aperture) is defined by "n" separate, infinitesimal points of light, is,

When the aperture (object surface) is a rectangle that is parallel to the image screen, it is wisest to place the "n" points across the surface in such a way that they form an ordered, 2D rectilinear array having "j" columns and "k" ( = n / j ) rows, in which case these two equations immediately take the form of equations (I.B-1) and (I.B-2), respectively.

Potentially Dangerous Simplifications

In deriving the results reported in Chapters I.A, I.B, and I.C, we generally have employed the following three broad, simplifying assumptions.

1. The linear dimensions of the aperture (object) are very small compared to the distance between the aperture (object) and the image screen;

2. The aperture (object surface) is uniformly bright; and

3. At all points across the aperture (object surface), the phase of the light is zero as it leaves the aperture (object surface).

By adopting the first of these assumptions, we have been able to drop the nonlinear terms in the definition of D_n and then rewrite equation (1) (or its equivalent) in a form that can be solved using standard 1D- or 2D-FFT techniques. This makes the determination of A both straightforward and relatively fast. But it must be understood that a result obtained in this manner is only an approximate one because the nonlinear terms in D_n have been ignored.

When all three of these assumptions are adopted, we have been able to rewrite equation (1) in a form that can be integrated analytically. This, of course, can speed up the determination of A tremendously, but again one has to appreciate that the obtained result is only an approximation. Throughout the next few chapters we will continue to utilize these three simplifying assumptions in an effort to demonstrate how the amplitude A(x,y) can be straightforwardly calculated even when a flat, rectangular aperture (e.g., the surface of a cube) is tilted with respect to the image screen and/or when the aperture (surface of an object) has a triangular shape. These two concepts are critical ones as we develop a technique that will permit the rapid numerical construction of holographic images for arbitrarily shaped and arbitrarily positioned, three-dimensional objects. In later chapters we shall return to a discussion of these enumerated simplifying assumptions in order to ascertain what degree of accuracy and what object features must be sacrificed in order to achieve palatable computational speeds.