Why Is the Paraboloidal Wavefront Approximation Valid?

[yucheng]

Homework Statement

Try plotting....

Relevant Equations

$$U(\vec{r}) \approx \frac{A_0}{z} exp(-jkz) \exp(-jk\frac{x^2 + y^2}{2z})$$

We can either plot the real part of the complex amplitude, or the wavefront.

However, how is wavefront meaningful for varying amplitude? In order to plot the paraboloid, we must vary $z$, which varies the amplitude $\frac{A_0}{z}$. Unless the amplitude is varies little, i.e. $1/z$ approximately constant within $\Delta z = \lambda$?

In the book Fundamentals of Photonics, Saleh & Teich, the author mentions that the phase of the second exponential function serves to bend the planar wavefronts into paraboloidal surfaces i.e. $frac{x^2 + y^2}{2z} = \text{const}$, however, shouldn't it be $z + frac{x^2 + y^2}{2z} = \text{const}$ when plotting surfaces of constant phase i.e. wavefronts?

The result should look like this.

Thanks in advance!

 

[TSny]

From the link, they are considering spherical waves where $$U(\vec r) = \frac{A_0}{r}e^{-jkr}.$$ The phase is $kr$. The true wavefronts are all spherical. For large distances from $r = 0$, small patches of the wavefronts can be approximated by paraboloid-shaped wavefronts. (For very large distances, the wavefronts can be approximated by plane wavefronts.)

Consider moving out along the $z$ axis to some point $P$ with coordinates $(x, y, z) = (0, 0, z_0)$. The phase of the wave at that point will be $kz_0$. We look for points in the vicinity of $P$ for which the wave has the same phase. We assume $z_0$ is large enough so that points in the vicinity of $P$ will have coordinates $(x, y, z)$ satisfying $x \ll z$ and $y \ll z$. For these points, $$r = \sqrt{x^2 + y^2 + z^2} \approx z + \frac{x^2+y^2}{2z} .$$ The condition that the phase $kr$ at these points be the same as at $P$ is $$k\left[ z + \frac{x^2+y^2}{2z}\right ] = k z_0$$ From this show that points near $P$ on the wavefront passing through $P$ have coordinates $(x, y, z)$ which satisfy $$z-z_0 \approx -\frac{x^2+y^2}{2z_0} .$$ Describe the shape of the locus of points $(x, y, z)$ satisfying $$z-z_0 = -\frac{x^2+y^2}{2z_0} .$$

 

[yucheng]

$$z-z_0 \approx -\frac{x^2+y^2}{2z_0}$$

I guess this is the key. But why would it be a valid approximation though? Let me try a quantitative argument...