Parabolic Waves

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Homework Statement



Show that a paraboloidal wave centered at the point P1 (x,y, z1) is converted by a biconvex lens of focal length f into a paraboloidal wave centered about P2 where 1/z1 + 1/z2 = 1/f.

Homework Equations





The Attempt at a Solution


if U1(r) = A0/(z-z1) exp(-jk(z-z1)) exp(-jk(x^2 + y^2)/(2(z-z1))

Now I also know that the transmittance thru a thin lens is given by
t(x,y) = h_0 exp( jk (x^2+y^2)/2f) where h_0 = exp(-jnk*thickness)

So I would think that U2(r) = t(x,y) * U1(r)
Unfortunately, if I multiply the two together I don't really get anything that I would find useful.

My question is - how do I get z1 and z2 so that I can show that I am satisfying the imaging equation?
 

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