# How to find the center of a paraboloidal wave

1. Mar 9, 2012

### semc

1. The problem statement, all variables and given/known data
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

2. Relevant equations
Equation of paraboloidal wave and transmittance through a thin lens.

3. 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?

2. Mar 12, 2012

### 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.