Fresnel parallelepiped

The Fresnel parallelepiped (also: Fresnel rhombohedron ) is an optical prism that was introduced by Augustin-Jean Fresnel in 1817 to convert 45 ° linearly polarized light into circularly polarized light.

The function of the parallelepiped is therefore similar to that of a retardation plate , but its defined phase shift is not based on birefringence , but on double total reflection at a certain angle. It has the advantage that, in contrast to the retardation plate, the phase shift hardly depends on the wavelength . ${\displaystyle \Delta n}$

Structure and functionality

Beam path in a Fresnel parallelepiped

The function of the Fresnel parallelepiped is based on a defined phase shift of the two components of the polarized light during total reflection on the inner surface of the prism. For this purpose, 45 ° linearly polarized light is directed perpendicularly to one end of the prism and refracted into the prism without changing direction . It then falls on an inclined longitudinal surface of the prism. If the angle of incidence is greater than the critical angle of total reflection , it is totally reflected there. The resulting phase shift causes the originally linearly polarized light to become elliptically polarized light. To generate circularly polarized light, a second total reflection is therefore necessary before the light exits through the second end face of the prism. ${\displaystyle \alpha }$ ${\displaystyle \alpha _{\text{krit}}}$

For a defined phase shift of (leads from 45 ° -linear to circular polarization), it is necessary that the light hits the totally reflective interfaces at a certain angle . This angle depends on the critical angle of total reflection, which in turn includes the refractive index of the material used: ${\displaystyle \delta =90^{\circ }}$ ${\displaystyle \alpha }$ ${\displaystyle \alpha _{\text{krit}}}$

${\displaystyle \tan {\frac {\delta }{2n}}={\frac {\cos \alpha {\sqrt {\sin ^{2}\alpha -\sin ^{2}\alpha _{\text{krit}}}}}{\sin ^{2}\alpha }}}$

where is the number of total reflections in the parallelepiped. ${\displaystyle n}$

Normally, with a Fresnel parallelepiped, two total reflections occur in the prism ( ). ${\displaystyle n=2\Rightarrow \tan {\frac {\delta }{2n}}=\tan 22{,}5^{\circ }\approx 0{,}4142}$

For a prism made of crown glass with a refractive index of 1.51 and a critical angle of total reflection of ${\displaystyle \alpha _{\text{krit}}=\arcsin \!\left({\frac {1}{1{,}51}}\right)\approx 41{,}47^{\circ }}$

the angle of incidence on the totally reflecting surfaces must therefore be: ${\displaystyle \alpha \approx 54{,}62^{\circ }}$

Individual evidence

1. ^ A. Fresnel: Mémoire sur les modifications que la réflexion imprime à la lumière polarisée . In: Mémoires de l'Académie des sciences de l'Institute de France . tape 11 , 1832, p. 373–434 (The manuscript was sent in on November 10, 1817, and was read on January 7, 1823.).
2. ↑ ^{a b} Heinz Haferkorn: Optics: Physico-Technical Basics And Applications . Wiley-VCH, 2003, ISBN 978-3-527-40372-1 , pp. 436 .
3. ↑ Eugene Hecht: Optics . 5th edition. Oldenbourg Verlag, Munich / Vienna 2009, ISBN 978-3-486-58861-3 , p. 576-577 .