Computer-Generated Holography (CGH)

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

Setup

Consider any three points in space -- P₀, P₁, and P₂ -- that do not lie on a straight line. From elementary Euclidian geometry we know that these three points define a plane. They also can be used to specify the vertices of a flat triangular aperture that lies in this plane.

Figure III.1

Let's construct a Cartesian coordinate system in which the X-Y plane is coincident with the plane defined by these three points and orient it as illustrated in Figure III.1, that is, such that,

(Note that, defined in this way, X₁ and Y₂ are necessarily positive numbers, but Y₁ may be either positive or negative.)

Finally, shift the Cartesian coordinate system sideways in the X-Y plane so that P₀ is no longer at the origin but is, instead, at the position,

as illustrated in Fig. III.3. (The rationale for shifting the origin of the coordinate system to this position will be given below.) As a result, the other two points will be located at positions,

We will henceforth refer to the Cartesian coordinate system defined in this way as the aperture coordinate system. In the aperture coordinate system, the equation for the line that connects vertices P₂ and P₁, and thereby forms the "upper" edge of the triangular aperture is,

and the equation for the line that connects vertices P₀ and P₁, and thereby forms the "lower" edge of the triangular aperture is,

Analytical Result

By analogy with earlier chapters, if the triangular aperture is assumed to be uniformly bright (i.e., a_j,k = a₀ dX dY, and a₀ is the brightness per unit area), and the phase f = 0 at all locations on the aperture, then at any position in space (X_a,Y_a,Z_a) as measure in the aperture coordinate system from the aperture origin (as illustrated above in Fig. III.3),

Both of these integrals can be completed in the same fashion as described in earlier chapters for the 1D slit or for the 2D rectangular aperture, but now some care must be taken in establishing the limits of integration because along two of the edges of the triangular aperture, the limiting values of "Y" are now a function of the X-coordinate. However, these functions are readily specified for the "upper" and "lower" edges of the triangle by equations (4) and (5), respectively. Hence, the integral over Y must be performed between the limits Y_lower and Y_upper, where,

and the integral over X must be performed between the limits X_lower and X_upper, where,

Completing the integrals with these limits gives,

where,

Discussion

Now, when comparing equation (10) to equation (I.B-7) -- which is the result for a rectangular aperture -- we see two key differences. First, because a_upper is not equal to a_lower, we are unable to combine the terms containing b into a sinc function. (See, however, what happens when Y_a = 0, as discussed below.) Second, equation (10) does not contain a phase offset factor in either the X or Y coordinate directions analogous to the terms that contain Q₁ and J₁ in (I.B-7). Terms of this nature do not appear here precisely because of our strategic positioning of the origin of the aperture coordinate system as illustrated in Figure III.1 and defined by equation (2).

The analytical expression (10) formally poses problems when either X_a = 0 or Y_a = 0 ( b = 0 ) because of division by zero. However, equation (10) still provides the correct result if we realize that, when X_a = 0,

When Y_a = 0, one can show that equation (10) reduces to the simpler expression,

Finally we note that, via the general coordinate transformations derived in § II.C, we can now use equation (10), above, to specify the complex amplitude of light at any coordinate location on a holographic screen. When employing these coordinate transformations, care must be taken to define the offset coordinates (X_O,Y_O,Z_O) shown in Figure II.3 in terms of the separation between the origin (usually the center) of the hologram and the particular origin of the aperture coordinate system, as specified above. Then, for example, if X_O = Y_O = 0, equation (II.C-4) provides the functions X_a(x_h,y_h), Y_a(x_h,y_h), and Z_a(x_h,y_h), that are required to express the distance L and the angles a_upper, a_lower, and b entirely in terms of coordinate positions across the hologram.