Exploring the Input/Output Relationship of Fourier Optics

[roeb]

Homework Statement

I'm trying to figure out the input/output relationship for Fourier Optics:

If we have a plane at z = 0 with the complex amplitude f(x,y) = U(x,y,0) and the complex amplitude at the output g(x,y) = U(x,y,d) at z = d.

My question is if f(x,y) = 1, how do I explicitly determine what g(x,y) is? My book gives the following integral (among others)

g(x,y) = [tex]H_0 \int_{-\inf}^{\inf} \int_{-\inf}^{\inf} F(v_x, v_y) exp( j \pi \lambda d (v_x^2 + v_y^2)) exp(-j2 \pi (v_x x + v_y y)) dv_x dv_y[/tex]

I guess my main problem is that I'm not even sure what g(x,y) is supposed to look like, I know the Fourier transform of f(x,y) -> F(vx,vy) = delta(v_x) delta(v_y), but I'm not really sure what to do next. How do I integrate the two exponentials?

Does anyone have any textbook suggestions for this? Goodman doesn't seem to go in detail on this topic and I can't really find any other examples.

Thanks,
roeb

 

[roeb]

I was able to solve the problem,

Thanks for looking

-roeb