Visualizing the Paraboloidal Wave

In summary, the wavefronts are approximated by paraboloids when the amplitude is small, but the true wavefronts are spherical when the amplitude is large.
  • #1
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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!
 
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  • #2
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 $$z-z_0 = -\frac{x^2+y^2}{2z_0} .$$
 
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  • #3
TSny said:
$$z-z_0 \approx -\frac{x^2+y^2}{2z_0}$$
I guess this is the key. But why would it be a valid approximation though? Let me try a quantitative argument...
 

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