Goos-Hänchen effect

The Goos-Hänchen effect is an optical phenomenon in which linearly polarized light undergoes a small shift, parallel to the direction of propagation, when totally internally reflected. This effect is the linear polarization analog of the Imbert-Fedorov effect.

This effect occurs because the reflections of a finite sized beam will interfere along a line transverse to the average propagation direction. As shown in the figure, the superposition of two plane waves with slightly different angles of incidence but with the same frequency or wavelength is given by

$mathbf\{underline\{E(x,z,t)=mathbf\{underline\{E^\{TE/TM\}\; left(\; e^\{jmathbf\{k\}\_1\; cdot\; mathbf\{r\; +\; e^\{jmathbf\{k\}\_2\; cdot\; mathbf\{r\; ight)\; cdot\; e^\{-j\; omega\; t\}$

where

$mathbf\{k\}\_\{1\}\; =\; k\; left(\; cos\{left(\; heta\_0\; +\; Delta\; heta\; ight)\}\; mathbf\{hat\{x\; +\; sin\{left(\; heta\_0\; +\; Delta\; heta\; ight)\}\; mathbf\{hat\{z\; ight)$

and

$mathbf\{k\}\_\{2\}\; =\; k\; left(\; cos\{left(\; heta\_0\; -\; Delta\; heta\; ight)\}\; mathbf\{hat\{x\; +\; sin\{left(\; heta\_0\; -\; Delta\; heta\; ight)\}\; mathbf\{hat\{z\; ight)$

with $k\; =\; omega\; n\_1$.

It can be shown that the two waves generate an interference pattern transverse to the average propagation direction, $mathbf\{k\}\_0\; =\; k\; left(\; cos\{\; heta\_0\}\; mathbf\{hat\{x\; +\; sin\{\; heta\_0\}\; mathbf\{hat\{z\; ight)$ and on the interface along the $(y,z)$ plane.

Both waves are reflected from the surface and undergo phase shifts.

References

* Frederique de Fornel, Evanescent Waves: From Newtonian Optics to Atomic Optics, Springer (2001), pp.12-18