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Wikipedia and other sources claim that the internal reflection underwater outside Snell's window is circularly polarized. What is the mechanism that causes this circular polarization?

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The claim is true only for the following conditions:

  1. Reflexion that is highly grazing, i.e. the incident wave is almost parallel to the interface and the angle of incidence approaches $\pi/2$;
  2. The incident light is linearly polarized and the plane of polarization is exactly halfway between the $s$ and $p$ polarization planes, i.e. so that the electric field makes an angle of $45^\circ$ with the intersection line between the transverse plane of propagation and the interface.

In short, at glancing angles, the TIR mechanism mimics the action of a quarter wave plate.

I show the above is true with the Fresnel equations in my answer to the Physics SE question How does one calculate the polarization state of random light after total internal reflection (read the answer to about halfway, where I discuss the grazing angle case).

The mechanism is that the Goos-Hänchen phase shift differs for the $s$ and $p$ states. It is a little hard to give an intuitive explanation; one has to refer to the Fresnel equations (as I do in my answer referred to above). But remember that TIR doesn't happen at the interface; the field tunnels beyond the interface a small distance as an evanescent field, as I explain in my answer here. So it is intuitively clear that there is an effective plane of reflexion a small distance beyond the physical interface owing to the nonzero tunnelling i.e. turnaround distance, and this distance happens to be a quarter of a wavelength at highly glancing incidence angles.

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