The idea behind holography was first suggested by Denis Gabor in the late 1940's. However as there were no lasers which could provide the highly monochromatic and spatially coherent light which was required, the idea didnt really take off till sometime in the late 1960's.

The primary problem here is to reconstruct 3 dimensional images of objects. A photograph is not truly 3 dimensional. If it had been, by moving your head you should have been able to see different views(which you obviously cant). Holography is a method used to create '3 dimensional' photographs of objects.

Say that you have a source of light on one side and between you and the light source is a screen with an aperture. For example you might be within a room and the screen might be the wall with a window. The Fresnel Kirchoff theory of diffraction now states that if you are given the complex amplitude of the light over the aperture you can determine the amplitude at any point behind the screen. So if I were to tell you the intensity and the phase of the light across the window, you could use that to calculate the intensity and the phase at all points within the room. This now brings out the problem with conventional photography:A photographic plate records only the intensity, not the phase of light, and this is why a lot of information is lost.

Gabor came up with an ingenious way to solve this problem. Current techniques use a variant of his method and this WU deals with these variants.
Briefly speaking the light from the object is allowed to interfere with a spatially coherent laser beam, and the interference pattern is recorded on a photographic plate. The intensity of the interference pattern depends on the difference in phase between the object wave and the reference wave. As the phase of the reference wave is known(this is why spatial coherence is required), we can always recalculate the phase of the object wave! Thus, in a nutshell, faced with a problem of recording both the intensity and the phase and given a device which could record only the phase( a photographic plate) Gabor invented a a configuration in which all the information was stored in the intensity and none in the phase!

Actually no calculation is required to reconstruct the object wave. Simply shining the same reference beam on the developed hologram will lead to the formation of a virtual image and a real image of the original object. The Mathematical details are given below.

Let us establish local Cartesian coordinates on the photographic plate. Lets say the object and the reference waves have the form
Object wave:aei*p(x,y)
Reference wave:aei*p1(x,y)
The resultant intensity due to this is
I(x,y) = A2 + a2+ A*a*cos(p1(x,y)-p(x,y))
a is generally small compared to A so we have
I(x,y) = A2+2*A*a*cos(p1(x,y)-p(x,y))
When the plate is developed, the transmission coeffiecient of the positive is
t(x,y) = 1 - g*I(x,y)
Thus if the original wave(see above) is passed through the photographic plate the resultant wave may be written as
A(1-g*A2)ei*p1(x,y) + gA2a(ep(x,y) + e2p1(x,y)-p(x,y))
The first term above is just a continuation of the reference wave. The first term inside the bracket is the virtual image of the object(which is what we want). If p1 is a constant, then the second term in the brackets represents a real image, otherwise it is a distorted image. Both the real image and the continuation of the reference wave are at distinct angles and so do not affect the visibility of the virtual image.