Calculation of a spread light on a surface

Question

Suppose we have a light of power $P$ distributed on a plane $(x,y)$. The distribution of the power is of the form: $$P=f(x,y)$$ If we have a lens conjugating every point of the plane $(x,y)$ in another plane $(X,Y)$ (focal plane) in a way that for every point in the plane $(x,y)$ we have a circle of radius $r$ in $(X,Y)$, what is the distribution of the power in $(X,Y)$? Thanks.

Answer 1

The wave amplitude in the focal plane is the Fourier transform of the wave amplitude in the pupil plane of the lens $L$. Assuming we are dealing with a diffraction-limited lens, the relationship between object amplitude $A_o$ and image amplitude $A_i$ is thus a convolution:

$$A_i(x,y) = \iint_{\mathbb{R}^2} dx' dy' \, L(x',y') A_o(x-x',y-y')$$

Note that $A_0(x,y) = \sqrt{f(x,y)} e^{i \phi(x,y)}$, where $\phi$ is some undetermined phase function - it could be a constant or zero, but it might not be. (This may be specified or determined through experiment.) The power distribution in the conjugate plane would then be $|A_i(x,y)|^2$.