I Validity of Fresnel Approximation

[yucheng]

Wikipedia says that Fresnel diffraction is valid if the Fresnel number is approximately 1. What Fresnel number then is the Fresnel approximation for paraxial-paraboloidal waves valid? It's not mentioned...

Oh I just realized that

$$\frac{N_F \theta_m^2}{4} \ll 1$$

So it depends on the maximum angle... Oops

Anyway, we have the validity condition for Fresnel Approximation $a^4 \ll 4 z^3 \lambda$

So.. for what a is it valid? How small should $a$ be given the relation $\ll$, that is rather arbitrary?

Suppose $z = 1 m$, $\lambda = 633 \\ nm$ (Exercise 2.2-1, Fundamentals of Photonics, Saleh & Teich)

Thanks in advance!

 

[tech99]

My simple understanding as a microwave engineer is that for the case of waves emerging from an aperture, if we are close to the aperture we have Fresnel conditions and if we are far we have Fraunhofer conditions. The two regions are sometimes referred to as the near and far radiation zones. The boundary is ill defined, and occurs at the Rayleigh Distance, which may be defined as (Diameter^2) / 2 lambda. In the Fresnel region we tend to see a parallel beam, but it usually has a waist and can also have hot spots. Maybe it can also have a black dot in the middle. In the Fraunhofer region the beam diverges at an angle defined by the aperture size in wavelengths, and we also see sidelobes, which are mainly defined by the illumination taper of the aperture. The radiation pattern in the Fraunhofer region does not change with distance.
The near radiation zone is not to be confused with the reactive near field, which occurs at fractions of a wavelength from a source, so we need care in using the terminology.

 

[hutchphd]

I like Fresnel is "intermediate far" and Fraunhofer is "far". But of course the near and far radiation zone is a better description.