How can we determine the spatial filter for a 2f:2f filtering setup in optics?

[forty]

Consider a 2f:2f filtering setup with f = 1000mm. The system is illuminated with a uniform plane wave of uni amplitude and wavelength [tex]\lambda[/tex] = 1.0[tex]\mu[/tex]m. The input transparency (object) has amplitude transmittance g(x,y) and the spatial filter has amplitude transmittance s(x,y).

Write an expression relating the complex amplitude q(x,y) in the image plane to g(x,y) and s(x,y).

Determine s(x,y) such that q(x,y) = [tex]\nabla[/tex]²g(x,y) for arbitrary g(x,y).

So an incident plane wave will exit the transparency with a field given by E_oe^ig(x,y) (but unit amplitude so E_o = 1) and the same for the filter e^is(x,y)

So wouldn't it just be the multiple of the 2? q(x,y) = e^{i(g(x,y)+s(x,y))}

I'm really not sure about that... I can't seem to figure out the 2nd part using that which leads me to thinking its completely wrong :S

Any help on this greatly appreciated!

 

[NateD]

For the second part, we note that the second derivative of g(x,y) is the Laplacian of g(x,y). Thus, we can solve for s(x,y) by setting the Laplacian of q(x,y) equal to the Laplacian of g(x,y). We obtain the following expression:s(x,y) = - \nabla^2 g(x,y) + 2\pi f \lambda g(x,y)Where \nabla^2 is the Laplacian operator and f and \lambda are as defined in the problem.