Skeleton (category theory)

In optics, a Gires–Tournois etalon is a transparent plate with two reflecting surfaces, one of which has very high reflectivity. Due to multiple-beam interference, light incident on the lower-reflectivity surface of a Gires–Tournois etalon is (almost) completely reflected, but has a phase shift that depends strongly on the wavelength of the light.

The complex amplitude reflectivity of a Gires–Tournois etalon is given by

$r=-\frac{r_1-e^{-i\delta }}{1-r_1{e}^{-i\delta }}$

where r₁ is the complex amplitude reflectivity of the first surface,

$\delta =\frac{4\pi }{\lambda }nt\cos {\theta }_t$

n is the index of refraction of the plate

t is the thickness of the plate

θ_t is the angle of refraction the light makes within the plate, and

λ is the wavelength of the light in vacuum.

Nonlinear phase shift

File:Phase Shift.gif

Nonlinear phase shift Φ as a function of δ for different R values: (a) R = 0, (b) R = 0.1, (c) R = 0.5, and (d) R = 0.9.

Note that $\left|r\right|=1$, independent of $\delta$. This indicates that all the incident energy is reflected and intensity is uniform. However, the multiple reflection causes a nonlinear phase shift $\mathrm{\Phi }$. To show this effect, we assume $r_1$ is real and $r_1=\sqrt{R}$, where $R$ is the intensity reflectivity of the first surface.

Further, define the nonlinear phase shift $\mathrm{\Phi }$ through

$r=e^{i\mathrm{\Phi }}$

and yield

$\tan \left(\frac{\mathrm{\Phi }}{2}\right)=-\frac{1+\sqrt{R}}{1-\sqrt{R}}\tan \left(\frac{\delta }{2}\right)$

For R = 0, no reflection from the first surface and the resultant nonlinear phase shift is equal to the round-trip phase change ($\mathrm{\Phi }=\delta$) – linear response. However, as can be seen, when R is increased, the nonlinear phase shift $\mathrm{\Phi }$ gives the nonlinear response to $\delta$ and shows step-like behavior. Gires–Tournois etalon has applications for laser pulse compression and nonlinear Michelson interferometer.

Gires–Tournois etalons are closely related to Fabry–Pérot etalons.

References

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• Gires–Tournois Interferometer in RP Photonics Encyclopedia of Laser Physics and Technology