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I am studying the derivation of the Fresnel-Kirchhoff integral. It puzzles me that one of the conditions imposed is a phase shift of $\pi/2$ between the incident and diffracted wave. Where does this phase shift arise from?

random person
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  • I would think that this is the Guoy phase shift, but I'm not in a position to say for sure. The Guoy shift is usually discussed in the context of converging beams, but I would think that the physical principles would apply here, as I think the effect is due to transverse field confinement – garyp Oct 16 '22 at 13:53
  • I've found a somewhat intuitive explanation in chapter 18 of "Introduction to Optics" (Frank Pedrotti), which uses Fresnel zones to explain it, but I was wondering if there is a more fundamental way of seeing it – random person Oct 16 '22 at 14:41

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I used to think that it is caused by the boundary current induced by the diffracting wave. This handwaving answer might make some sense if we only talked about metal screens but what about if the screen is an insulator or absorber...

The closest I found to be an explanation is in Goodman: Introduction to Fourier Optics, 2nd ed., Chapter 3.7 and 3.8, see in Goodman where he writes about the secondary sources in the aperture. Here $P_0$ is the source of the monochromatic spherical wave and $P_1$ is in the aperture.

The secondary source at $P_1$ has the following properties:

  1. It has a complex amplitude that is proportional to the amplitude of the excitation U(P1) at the corresponding point.
  2. It has an amplitude that is inversely proportional to $A$, or equivalently directly proportional to the optical frequency $\nu$.
  3. It has a phase that leads the phase of the incident wave by 90°, as indicated by the factor 1/j.

"A reasonable explanation of the second and third properties would be as follows. Wave motion from the aperture to the observation point takes place by virtue of changes of the field in the aperture. In the next section we will see more explicitly that the field at $P_0$ contributed by a secondary source at $P_1$ depends on the time-rate-of-change of the field at $P_1$ Since our basic monochromatic field disturbance is a clockwise rotating phasor of the form $exp(- j2\pi \nu t)$, the derivative of this function will be proportional to both $\nu$ and to $-j = 1/j$."

And then in Chapter 3.8 Goodman writes the polychromatic integral in time domain as (I changed the notation of the velocity of light from $v$ to $c$ so it is not confused with $\nu$):

$$u(P_0,t) = \int \int_{\Sigma} \frac{cos(\vec n, \vec r_{01})}{2\pi c r_{01}}\frac{d}{dt}u\left(P_1,t-\frac{r_{01}}{c}\right)ds \tag{3.57}\label{3.57}$$

Goodman naturally interprets this as that the diffracted wave depends on the time derivative of the aperture distribution. The problem with this argument is that he derives $\eqref{3.57}$ from having already gotten the "$j$" in front of the Fresnel-Kirchhoff integral. So we are back where we started... But even if we started the derivation in time domain why would the the diffracted wave depend on the time derivative of the aperture distribution?

@garyp says the $\pi/2$ shift caused buy the aperture may be related to the Gouy phase shift through the transverse constriction. The experiments of Gouy, Sagnac and Fabry showed a shift of $\pi$ in the focal region. Ditchburn discusses this experiment in Chapter 7 of "Light" and explains it using the Fresnel construction.

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