Visualizing the Paraboloidal Wave

• yucheng
In summary, the wavefronts are approximated by paraboloids when the amplitude is small, but the true wavefronts are spherical when the amplitude is large.

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.

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