Fresnel rhomb

A Fresnel rhomb is an optical prism that introduces a 90° phase difference between two perpendicular components of polarization, by means of two total internal reflections.

Conversely, if the angle of incidence and reflection is fixed, the phase difference introduced by the rhomb depends only on its refractive index, which typically varies only slightly over the visible spectrum.

[2] During that time he deployed it in crucial experiments involving polarization, birefringence, and optical rotation,[3][4][5] all of which contributed to the eventual acceptance of his transverse-wave theory of light.

[Note 1] Light passing through a Fresnel rhomb undergoes two total internal reflections at the same carefully chosen angle of incidence.

Two Fresnel rhombs can be used in tandem (usually cemented to avoid reflections at their interface) to achieve the function of a half-wave plate.

In the latter case, the reflection coefficients for the s and p components are real, and are conveniently expressed by Fresnel's sine law[11] and Fresnel's tangent law[12] where θi is the angle of incidence and θt is the angle of refraction (with subscript t for transmitted), and the sign of the latter result is a function of the convention described above.

Combining the complementarity with Snell's law yields θi = arctan(1/n) as Brewster's angle for dense-to-rare incidence.

[Note 2] That completes the information needed to plot δs and δp for all angles of incidence in Fig.

In 1813, Biot established that one case studied by Arago, namely quartz cut perpendicular to its optic axis, was actually a gradual rotation of the plane of polarization with distance.

Without (yet) explicitly invoking transverse waves, this theory treated the light as consisting of two perpendicularly polarized components.

[16] In 1817, Fresnel noticed that plane-polarized light seemed to be partly depolarized by total internal reflection, if initially polarized at an acute angle to the plane of incidence.

The memoir of November 1817 [1] bears the undated marginal note: "I have since replaced these two coupled prisms by a parallelepiped in glass."

[19] In that memoir,[4] Fresnel reported that if polarized light was fully "depolarized" by a rhomb, its properties were not further modified by a subsequent passage through an optically rotating medium, whether that medium was a crystal or a liquid or even his own emulator; for example, the light retained its ability to be repolarized by a second rhomb.

As an engineer of bridges and roads, and as a proponent of the wave theory of light, Fresnel was still an outsider to the physics establishment when he presented his parallelepiped in March 1818.

In July he submitted the great memoir on diffraction that immortalized his name in elementary physics textbooks.

In 1819 came the announcement of the prize for the memoir on diffraction, the publication of the Fresnel–Arago laws, and the presentation of Fresnel's proposal to install "stepped lenses" in lighthouses.

[21] The experimental confirmation was reported in a "postscript" to the work in which Fresnel expounded his mature theory of chromatic polarization, introducing transverse waves.

[24]) Meanwhile, by April 1822, Fresnel accounted for the directions and polarizations of the refracted rays in birefringent crystals of the biaxial class – a feat that won the admiration of Pierre-Simon Laplace.

In a memoir on stress-induced birefringence (now called photoelasticity) read in September 1822,[25] Fresnel reported an experiment involving a row of glass prisms with their refracting angles in alternating directions, and with two half-prisms at the ends, making the whole assembly rectangular.

When the prisms facing the same way were compressed in a vise, objects viewed through the length of the assembly appeared double.

2 above, which shows that the phase difference δ is more sensitive to the refractive index for smaller angles of incidence.)

[32] In summary, the invention of the rhomb was not a single event in Fresnel's career, but a process spanning a large part of it.

[33] But Fresnel's treatment of total internal reflection seems to have been the first occasion on which a physical meaning was attached to the argument of a complex number.

Fig. 1 : Cross-section of a Fresnel rhomb (blue) with graphs showing the p component of vibration ( parallel to the plane of incidence) on the vertical axis, vs. the s component ( square to the plane of incidence and parallel to the surface ) on the horizontal axis. If the incoming light is linearly polarized, the two components are in phase (top graph). After one reflection at the appropriate angle, the p component is advanced by 1/8 of a cycle relative to the s component (middle graph). After two such reflections, the phase difference is 1/4 of a cycle (bottom graph), so that the polarization is elliptical with axes in the s and p directions. If the s and p components were initially of equal magnitude, the initial polarization (top graph) would be at 45° to the plane of incidence, and the final polarization (bottom graph) would be circular .
Fig. 2 : Phase advance at "internal" reflections for refractive indices of 1.55, 1.5, and 1.45 ("internal" relative to "external"). Beyond the critical angle, the p (red) and s (blue) polarizations undergo unequal phase shifts on total internal reflection; the macroscopically observable difference between these shifts is plotted in black.
Augustin-Jean Fresnel (1788–1827)