Fresnel approximations

The Fresnel approximations consist in assuming that the first 2 terms of the square root Taylor expansion are sufficient to correctly represent the phase, provided that z is large enough:

The weighting factor can then be written as:

which means that for a distance z large enough, the maximum phase change due to the first neglected term has to be << 1 rad: 

This condition will be realized if:

The superposition integral (III-1) then becomes:

We can develop the quadratic terms in the exponent:

Therefore, except for the phase and amplitude multiplying factors which are independent of x1 and y1, we can calculate the diffracting wave amplitude by calculating the FT:

This FT has to be evaluated at frequencies: