Early Developments

The first holograms were invented by A. Lohmann .It made use of computer line printers and prodcued using coherent LASER illumination .Th following figure shows one of the first computer-generated hologram and its reconstruction.

 

As one can observe easily, it is very hard to comprehend the actual image from the holograms.
As the developments progressed , more and more sophisticated recording devices were used to create gray scale holograms.

The figure represents an example of grey-scale computer-generated hologram and an image reconstructed from a grey scale hologram.

Implementation approach by the inventors :
The grey scale holograms were recorded using electro-mechanical recording device (a modified faximile machine) with resolution 200 μm and then optically reduced with reduction factor 20 to final resolution 10 μm for optical reconstruction with a laser beam.

Now we'll discuss about the mathematical aspects .Mathematical model of the object is usedto specify the amplitude and phase information of the object wave front which is generated through diffraction,scattering of the light particles and then captured on a recording device .
The types of the models can be distinguished: -

1. analytical models,
2. geometrical or vector graphics

      models that represent objects as compositions of elementary diffracting elements such as point scatters, segments of 2-D or 3-D curves etc. such as polygonal mesh or wire-frame models in 3-Dimensional computer graphics - “raster graphics”, or “bitmap” models that represent objects as 2-D or 3-D arrays of points (object samples /pixels).

Analytical models are used mainly for generating diffractive optical elements.
By using methods of analytical or numerical integration with appropriate oversampling to avoid transformation sampling anomalies and then sampled to generate their sampled version for recording on electronic media.
There are several mathematical modelling techniques but almost of them involve following stages:

 

 

Let's look on an example : Fourier Modelling

 

Consider following ,

If the geometrical dimensions of the observation object and of the observation surface fragment are small compared to the distance Z from the object to the observation plane, the integral relationship in the integral Fresnel transform:  

Hate this much mathematics?

In essence , consider above represents the observed representation of the object as a function of wavefronts and distance from the object  where λ is the illumination wave length. as (x,y) represent the coordinates of the object

On the same line Fourier model uses fourier transform to give that relationship as follows:

learn more about Fourier transform 
In the next chapter we'll discuss other models and further details into encoding and recording process.

Ref- L. Yaroslavsky. Introduction to Digital Holography                                                                                                          
                                                                                        - Kartik Mehetre