How to find the center of a paraboloidal wave

[semc]

Homework Statement

Show that the paraboloidal wave centered at the point z₁ is converted by a lens of focal length f into a paraboloidal wave centered about z₂ where 1/z₁+1/z₂ =1/f

Homework Equations

Equation of paraboloidal wave and transmittance through a thin lens.

The Attempt at a Solution

I multiplied the paraboloidal wave and the transmittance through the thin lens and I got
U(r)=Aexp(-jnkd)exp[jk(\frac{x^2+y^2}{2f}-z)]
But how do I continue to show that this paraboloidal wave is centered at f?

 

[sunjin09]

It looks like your lens is located at z=0; the parabolic wave centered at z is basically a spherical wave centered at z1 where z1 is faraway from the lens, i.e., z1^2>>x^2+y^2, so that the wavefront at the lens ~ exp[ik(x^2+y^2)/(2z1)]. (Taylor expansion of the phase ikR where R^2=x^2+y^2+z^2) Now multiply by the lens phase exp[ik(x^2+y^2)/(2f)] and figure out where the center is. Double check the sign conventions since I haven't done these for years.