Question

What quantity must be small to ensure the validity of the Fresnel approximation (Use the second-order term of the Taylor series expansion)?

Answer 1

The electric field diffraction pattern at a point (x, y, z) is given by Huygen principle:

is the electric field at the aperture,

k is the wavenumber

i is the imaginary unit.

The main problem for solving the integral is the expression of r. we can simplify the algebra by introducing the substitution:

Substituting into the expression for r, we find:

Using binomial

We can express r as

If we consider all the terms of binomial series, then there is no approximation.Let us substitute this expression in the argument of the exponential within the integral; the key to the Fresnel approximation is to assume that the third term is very small and can be ignored and henceforth any higher orders. In order to make this possible, it has to contribute to the variation of the exponential for an almost null term. In other words, it has to be much smaller than the period of the complex exponential; i.e., :

expressing k in terms of the wavelength,

we get the following relationship:

So

If this condition holds true for all values of x, x' , y and y' , then we can ignore the third term in the Taylor expression. Furthermore, if the third term is negligible, then all terms of higher order will be even smaller, so we can ignore them as well.

For applications involving optical wavelengths, the wavelength λ is typically many orders of magnitude smaller than the relevant physical dimensions. In particular:

We can then approximate the expression with only the first two terms:



This equation, then, is the Fresnel approximation.