Chapter 3 : Theory

3.1 Formation and reconstruction of a Hologram^{6 7 14}

These expressions are given as exponential functions

O = E_oe^(io)

R = E_re^(ir)

where

O - Object wave

R - Reference wave

I - Intensity

E - Amplitude

- Phase angle

We can write the formula for the intensity in the hologram

I = (O + R) (O* + R*) (3.1a)

= (E_oe^io + E_re^ir ) ^. (E_oe^-io + E_re^-ir ) (3.1b)

= E_o²e^io-io + E_oEre^i(o-r) + E_rE_oe^-(o-r) + E_r²e^ir-ir (3.1c)

= E_o² + E_r² + E_oE_re^i(o-r) + E_oE_re^-i(o-r) (3.1d)

The phase information is contained in the two last terms.

While an ordinary picture records only the intensity distribution in the object, the hologram contains also information about the phase. It means all information about the object is saved in the hologram, and we get an image in 3 dimensions.

A transmission distribution can be carried out for when the film plate is exposed
Where

T = Transmission distribution on the filmplate

T₀ Transmission constant for the filmplate

= Film parameter

I = Exposure intensity

t = Exposure time

We have the following function for the transmission distribution

T = T₀ - It (3.2)

We put in the formula for the intensity (3.1d) into formula (3.2), and we get for the transmission distribution

T = T₀ - (E_o² + E_r² + E_oE_re^i(o-r) + E_oE_re^-i(o-r) )t (3.3a)

T = T₀ - t (E_o² + E_r² )- t EoEr(e^i(o-r) + e^-i(o-r) ) (3.3b)

This equation describes the exposure of a holographic filmplate. This means that the equation describes a hologram.

For the reconstruction of the hologram, we have to illuminate the holographic plate with a beam which is similar or nearly similar to the reference beam.

The reconstruction can then be expressed as

E_R = E_r e^iR T (3.4)

The use of R instead of r is justified as the reconstruction beam is not necessarily the same as the reference beam.

E_R = Er e^iR T₀ - Er e^iR t (E_o² + E_r²) - Er e^iR tE_oE_r (e^i(o-r) + e^-i(o-r) ) (3.5a)

This equation can also be written as

E_R = -tE_rE_o²e^{i _R} + E_re^iR (T₀- tE_r²) (3.5b)

- tE_oE_r²e^i(o-r+R)

- tE_oE_r²e^i(-o+r+_R)

The first term of the equation (3.5b) represents a wave which travels in E_re^iR direction.²⁵

E_r is a wave which spreads from the object and is not constant. The wave is a function of (x,y).

The second term represents the virtual image. If the phase of the reference and reconstruction wave is equal, the second term will be identical to the object wave, with the exception of the amplitude. We have also reconstructed the object completely.

The third term represent the real image. Except for small angles ( _r) will it not be possible to see the virtual and the real image at the same time. From the equation (3.5b) we can see that the phase of the object wave ( _o ) is positive for the real image and negative for the virtual image. This means that the virtual image and the real image lie on opposite sides of the holographic filmplate.

The real image is created by waves which travel in the positive direction. This direction is the same as the direction of the reference beam.

The virtual image is created by waves which travel in the negative direction, which is in the opposite direction to the reference beam.

Figure 3-1 Reconstruction of a hologram