Rigorous diffraction theory states that if you know a scalar function that satisfies the helmholtz equation, $u(\vec{x}$, at a plane (for example the xy-Plane), given as $u(\vec{x}) = u_0(x,y)$, then you can calculate it at every other point by convoluting it with the function $\partial_z \frac{e^{ik(|\vec{x}|)}}{|\vec{x}|}$ (that would be Rayleigh-Sommerfeld-Diffraction Theory).

This would be äquivalent to a Fourier-Decomposition of $u(\vec{x})$ into plane waves with wave-number $k$, and propagating these through space. In this framework, you would get the solution by calculating the fouriertranform $\tilde{u}(\vec{k})$: $$\tilde{u}(\vec{k}) = e^{iz\sqrt{k^2-k_x^2 - k_y^2}} \tilde{u}_0(k_x, k_y)$$

In this case, the function $\partial_z \frac{e^{ik(|\vec{x}|)}}{|\vec{x}|}$ is exactly the fouriertransform of $e^{iz\sqrt{k^2-k_x^2 - k_y^2}}$.

The paraxial approximation assumes that there are only components of the solution with $k_x << k_z$ and $k_y << k_z$. If this is the case, we can calculate the fouriertransform as: $$\tilde{u}(\vec{k}) = e^{i(zk - \frac{k_x^2+k_y^2}{2k})} \tilde{u}_0(k_x, k_y)$$

Equivalently, obtain the solution $u(\vec{x})$ by convoluting with the function $$h(x, y, z) = \frac{-i k}{z} e^{i(kz + \frac{k(x^2 + y^2)}{2 z})} \\ u(x,y,z) = \int dx\prime \int dy\prime u_0(x\prime,y\prime) h(x - x\prime, y - y\prime, z) $$ which is exactly the way to calculate diffraction patterns in Fresnel diffraction.

Question 1: Does that mean, Fresnel diffraction is equivalent to the assumption that only paraxial waves contribute to a function?

Question 2: Does that mean, the outcome of fresnel diffraction $u(x,y,z)$ suffices the paraxial wave equation?

Question 3 (most important): Does that mean, fresnel diffraction of the function $u_0(x,y) = A_0 e^{-\frac{(x^2 + y^2)}{w_0^2}}$ will yield the exact formula of a gaussian beam? I tried to calculate it, but I failed: When I convolute $u_0(x,y) = A_0 e^{-\frac{(x^2 + y^2)}{w_0^2}}$ with $h(x, y, z) = \frac{-i k}{z} e^{i(kz + \frac{k(x^2 + y^2)}{2 z})}$, I don't arrive at the formula for a gaussian beam. I still think that this should be possible. Does somebody know where this is expliccitly calculated?