View source for The Calculus of Holography

==A Simplified Analysis of Holography==

Assume a photographic plate in the xy plane:

Given a reference beam:

[[Image:HoloEQ1.gif]]

where [[Image:HoloEQ2.gif]] is the amplitude as a complex number retaining the phase information.

The reflected object beam is:

[[Image:HoloEQ3.gif]]

At the plate the fringe amplitude is given by:

[[Image:HoloEQ4.gif]]

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

(Note: [[Image: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.)

Thus,

[[Image:HoloEQ5.gif]]

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

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

[[Image:HoloEQ11.gif]]

i.e.

[[Image:HoloEQ12.gif]]

or

[[Image:HoloEQ13.gif]]

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