Validity of wave front curvature calculation for a gaussian beam depending on the focusing conditions

Question

The curvature of a gaussian beam is usually given by $$R(z)=z\left(1+\left(\frac{z_r}{z}\right)^2\right)$$ with $z_r$ the Rayleigh length. Is there a limitation for that curvature (similar to usual beam propagation methods such as the NLSE) depending on the focusing angle/NA of the lens which is focusing the gaussian beam, or is that equation valid for $\text{NA}\in[0,\infty[$?

Answer 1

Gaussian beams are solutions of the paraxial wave equation, which is only valid under the paraxial approximation. In other words, it is assumed that beam divergence angle is small. The beam divergence angle is given by $$ \theta = \frac{\lambda}{\pi w_0} , $$ where $w_0$ is the beam radius at the waist and $\lambda$ is the wave length.

When the numerical aperture (NA) becomes too big, this approximation is not valid anymore.