Paraboloidal Wave Conversion via Biconvex Lens

[roeb]

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?

 

[Jhero]

it is important to use mathematical equations and principles to support your findings and conclusions. In this case, we can use the thin lens equation and the concept of wave propagation to show how a biconvex lens can convert a paraboloidal wave centered at point P1 into a paraboloidal wave centered at point P2.

First, let's define the variables in the problem:

- P1: point of origin of the paraboloidal wave
- P2: point of origin of the converted paraboloidal wave
- z1: distance from P1 to the lens
- z2: distance from lens to P2
- f: focal length of the biconvex lens
- A0: amplitude of the paraboloidal wave
- k: wave number
- n: refractive index of the lens material
- thickness: thickness of the lens

Now, let's start by writing the equation for the paraboloidal wave at P1:

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

Next, we can use the thin lens equation to relate z1, z2 and f:

1/z1 + 1/z2 = 1/f

Rearranging this equation, we get:

z2 = f(z1-f)/z1

Now, we can substitute this value of z2 into the equation for U1(r):

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

= A0/(z-z1) exp(-jk(z-f(z1-f)/z1)) exp(-jk(x^2 + y^2)/(2(z-z1))

= A0/(z-z1) exp(-j(kz-kf(z1-f)/z1)) exp(-jk(x^2 + y^2)/(2(z-z1))

= A0/(z-z1) exp(-jkf(z1-f)/z1) exp(-jk(x^2 + y^2)/(2(z-z1))) exp(-jkz)

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

Now