The Calculus of Holography

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A Simplified Analysis of Holography

Assume a photographic plate in the xy plane:

Given a reference beam:


where HoloEQ2.gif is the amplitude as a complex number retaining the phase information.

The reflected object beam is:


At the plate the fringe amplitude is given by:


because the square of the magnitude of a complex number is product with its complex conjugate.

(Note: PhotographyEQ1.gif is the amplitude of a photograph. See how we lose the imaginary part of the equation, this is when we lose the phase.)



The first and second terms are intensities of the reference and object beams. The third and forth terms are the magnitude and phase of HoloEQ6.gif.

When we reconstruct the hologram, HoloEQ7.gif, with the reference beam HoloEQ8.gif, so that the transmitted light has the complex magnitude HoloEQ9.gif,






where HoloEQ14.gif and is the zero order beam (it passes straight through the hologram). HoloEQ15.gif is the intensity of the reference beam and HoloEQ16.gif is the virtual image. The third term, HoloEQ17.gif, is the real image. It is important to notice that its amplitude is the complex conjugate of HoloEQ18.gif. (We have to flip the plate to make the conjugate or real image.)