# Gires–Tournois etalon

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

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

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

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

${\displaystyle \delta ={\frac {4\pi }{\lambda }}nt\cos \theta _{t}}$
n is the index of refraction of the plate
t is the thickness of the plate
θt is the angle of refraction the light makes within the plate, and
λ is the wavelength of the light in vacuum.

## Nonlinear effective phase shift

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.

Suppose that ${\displaystyle r_{1}}$ is real. Then ${\displaystyle |r|=1}$, independent of ${\displaystyle \delta }$. This indicates that all the incident energy is reflected and intensity is uniform. However, the multiple reflection causes a nonlinear phase shift ${\displaystyle \Phi }$.

To show this effect, we assume ${\displaystyle r_{1}}$ is real and ${\displaystyle r_{1}={\sqrt {R}}}$, where ${\displaystyle R}$ is the intensity reflectivity of the first surface. Define the effective phase shift ${\displaystyle \Phi }$ through

${\displaystyle r=e^{i\Phi }.}$

One obtains

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

For R = 0, no reflection from the first surface and the resultant nonlinear phase shift is equal to the round-trip phase change (${\displaystyle \Phi =\delta }$) – linear response. However, as can be seen, when R is increased, the nonlinear phase shift ${\displaystyle \Phi }$ gives the nonlinear response to ${\displaystyle \delta }$ 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

• F. Gires, and P. Tournois (1964). "Interferometre utilisable pour la compression d'impulsions lumineuses modulees en frequence". C. R. Acad. Sci. Paris. 258: 6112–6115. (An interferometer useful for pulse compression of a frequency modulated light pulse.)
• Gires–Tournois Interferometer in RP Photonics Encyclopedia of Laser Physics and Technology