 ## Contents

Here we outline a sequence of steps that likely should be taken in order to build confidence in the Principal Illustration of computer-generated holography that has been laid out in our accompanying discussion.

## Construct a Solid Cube

The desire, here, is to construct a single 2D holographic aperture that will simultaneously generate all six square sides of a cube. We ultimately will furthermore want to break each square side into a pair of triangles, that is, we want to construct a single 2D holographic aperture that will simultaneously generate twelve appropriately aligned triangles. If we learn how to do this effectively, then we will in principle have a tool that can construct 2D holographic apertures that can generate arbitrarily complex "video game" scenes that are composed of a very large number of triangles. Our desire is to be able to construct any already existing video scene by reading in the coordinates of the triplet of points that make up every imaged triangle.

### Use Summations

STEP 1:   First assume that a square aperture of width "wa" is chopped into 51 × 51 equally spaced points of light, each with a zero phase and uniform brightness. We will refer to this as the aperture's complex amplitude, A1(i,j), where both indices go from 1 to 51. Next, construct a (2D) image plane of width "wi" that is parallel to the aperture and located a distance D >> wa from the aperture, determining by simple brute-force summation the amplitude and phase of the combined light that arrives at each of 51 × 51 locations on the image scene. We will refer to this as the image-plane's complex amplitude, B1(k,l), where again both indices go from 1 to 51. (Given the wavelength of the monochromatic light that is leaving the aperture, and the chosen distance D, pick a ratio of widths, wi/wa, that makes sense.) The result should be that the magnitude, sqrt(B1* B1), closely resembles a 2D sinc function across the image plane.

STEP 2:   Turn this construction inside out. Shrink the image plane down to the size of the original aperture and illuminate it such that its complex amplitude, A2(i,j) = B1(i,j); actually, it may be okay to set all of the phase angles the phase of the light is zero everywhere, but the amplitude of the light is given by the 2D sinc function. This will be the new aperture. Now construct a new (2D) image plane that is parallel to the new aperture and located the same distance D from the aperture, determining by simple brute-force summation the amplitude and phase of the combined light that arrives at each of the 51 × 51 locations on this new image screen. We will refer to this as the image-plane's complex amplitude, B2(k,l). The result should be something quite close to a uniformly illuminated square. This shows us how to light an aperture — in this case, light it with a 2D sinc function — in order for the resulting holographic image to be one uniformly illuminated face of a cube. Let's label this as "side α" of the cube.

STEP 3:   Next, determine what the complex amplitude of the aperture needs to be in order to be able to construct an holographic image of the opposite side of the cube, that is, the side that is parallel to "side α" but a distance wa farther away from the image plane. Let's do this by repeating STEP 1 while replacing the distance "D" with "D+wa". In this case, after performing the simple brute-force summation, let's designate the image-plane's resulting complex amplitude as B3(k,l).