Definition of far field for gaussian laser beam

Although the definition given for "far-field" doesn't seem very strict, it is the strictest definition that can be given.

The problem here is similar to the following problem: any (one-dimensional) function, if you zoom in closely enough, looks like a straught line, and can be approximated by the first two terms (constant and linear) in the Taylor expansion. But "how close" is "close enough" for that approximation to be valid?

The reality is that the Gaussian beam diameter never expands exactly linearly, just like an arbitrary function may never be exactly linear if you zoom in close enough. But as you get further from the focus, the deviations from linearity get increasingly small, so that eventually you can probably ignore them for your calculational purposes.

The statement "the beam expands linearly in the far field" is a statement that, when $z gg xz_R$, the deviation from linear expansion is small enough that it is generally safe to ignore. If you had a more precise application you wouldn't ignore those corrections.

The longitudinal scale parameter for a Gaussian beam is the Rayleigh range: $$ z_R = frac{pi w_0^2}{lambda} , $$ where $w_0$ is the radium of the beam at its waist and $lambda$ is the wavelength.

Therefore, at propagation distances $zgg z_R$ from the waist, the Gaussian beam behaves approximately like a spherical wave, within a beam divergence angle of $$ theta=frac{lambda}{pi w_0} , $$ which means that the beam radius expands approximately linearly. The far-field is therefore considered to be the region where $zgg z_R$.