Shufflebrain

When objects distort the phase and amplitude of light, the resulting warps add up not to an image of the object, but to the object's transform. The eye's optical system--cornea and lens--transform the transform into the object's image. The convolution theorem shows that the Fourier transform of the Fourier transform of an object yields the image. I'm truly sorry about the double talk. But the convolution theorem explains how the eye, a projector or a microscope can turn the wave's warps into an image: the objects transform the carrier waves, and the optical system of the eye transforms the transforms from transform to perceptual space. Now I must apologize for triple and quadruple talk. But think back a moment to the halos. They were transforms of transforms right at your very own finger tips. Because your eyes performed a third transformation, you saw the transforms instead of the edges of your fingers.

Just as the Mercator projection and the globe are different ways of representing the same thing, so Fourier transforms and Fourier series give us different perspectives on periodic phenomena. We can use what we learned in chapter 7 to understand the ideas we're formulating now. Think of the visible features of a face, a dewdrop or a stand of pines as a potential compound wave in three dimensions. The interaction between carrier waves and objects is comparable to Fourier analysis--to the dissection of the compound wave into a series of its components, except in transform space instead of perceptual space. In other words, the first transformation is much like producing a Fourier series. The second transformation--the one that occurs at the eye and shifts the components from transform to perceptual space--is comparable to Fourier synthesis, to synthesis of the series of components into a compound wave.

The hologram captures the transform of an object, not the object's image. The interference of object and reference waves shifts their components into transform space. To conceptualize the reaction, let's form our imagery around waves in perceptual space, but let's use reasoning alone for events that occur in transform space. Visualize the components of the compound object wave as being strung out in a row, as we might draw them on a piece of graph paper in perceptual space. Along come the reference wave and collide and interfere with each component. Each collision produces a daughter wave whose phase and amplitude are the algebraic sums of the phase and amplitude of the particular component plus those for the reference wave; or the reference wave and each components superimpose on each other. But the reactions take place in transform space, remember, and not in the perceptual space we use to assist our imagery. Therefore, we would observe not the image carried in the object wave, but an interference pattern, the transform--the hologram!

By imagining the components of a transform to be a Fourier series, we provide ourselves with something to "picture." This analogy could create the false impression that the object wave loses its continuity, that each component congeals into an isolated little unit in transform space. A continuum is a system in which the parts aren't separated, and the hologram is a continuum. We can appreciate its continuous nature by observing the diffuse-light hologram, any arbitrarily selected piece of which will reconstruct a whole image of the scene. Although the reference beam must act upon each of the object wave's components, as in our conceptualization, the interference pattern represents the whole.

Of course no interference patterns or holograms can develop unless the reference and object wave have a "well-define" phase relationship (coherency). Recall that the optical holographer, using Young's and Fresnel's old tricks, produces "well-defined" phase relationships by deriving object and reference waves from the same coherent source. What do we really mean, though, by "well-defined?" To say "coherency" or "in step" is merely to shift words around[4] without really answering the question. Instead, let me invoke a strange but powerful theorem of topology that will take us into the general meaning of "well-defined." The theorem is known as Brouwer's fixed-point theorem.

Fixed-point theorem is at the foundation of several mathematical ideas, and it is implicit in a great many more. The interested reader will enjoy Shinbrot's excellent article on the subject (see bibliography). Here, I will flat-footedly state the theorem without probing its simple but tricky proof[5].