# The Calculus of Holography

## A Simplified Analysis of Holography

Assume a photographic plate in the xy plane:

Given a reference beam:

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

Thus,

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

When we reconstruct the hologram, , with the reference beam , so that the transmitted light has the complex magnitude ,

i.e.

or

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