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. If the incident beam is linearly polarized at 45° to the plane of incidence and reflection, the emerging beam is circularly polarized, and vice versa. If the incident beam is linearly polarized at some other inclination, the emerging beam is elliptically polarized with one principal axis in the plane of reflection, and vice versa.
The rhomb usually takes the form of a right parallelepiped, or in other words, a solid with six parallelogram faces (a square is to a cube as a parallelogram is to a parallelepiped). If the incident ray is perpendicular to one of the smaller rectangular faces, the angle of incidence and reflection at both of the longer faces is equal to the acute angle of the parallelogram. This angle is chosen so that each reflection introduces a phase difference of 45° between the components polarized parallel and perpendicular to the plane of reflection. For a given, sufficiently high refractive index, there are two angles meeting this criterion; for example, an index of 1.5 requires an angle of 50.2° or 53.3°.
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. Thus the rhomb functions as if it were a wideband quarter-wave plate in contrast to a conventional birefringent (doubly-refractive) quarter-wave plate, whose phase difference is more sensitive to the frequency (color) of the light. The material of which the rhomb is made usually glass is specifically not birefringent.
The Fresnel rhomb is named after its inventor, the French physicist Augustin-Jean Fresnel, who developed the device in stages between 1817 and 1823. During that time he deployed it in crucial experiments involving polarization, birefringence, and optical rotation, all of which contributed to the eventual acceptance of his transverse-wave theory of light.
Operation
Incident electromagnetic waves (such as light) consist of transverse vibrations in the electric and magnetic fields; these are proportional to and at right angles to each other and may therefore be represented by (say) the electric field alone. When striking an interface, the electric field oscillations can be resolved into two perpendicular components, known as the s and p components, which are parallel to the surface and the plane of incidence, respectively; in other words, the s and p components are respectively square and parallel to the plane of incidence.The s originally comes from the German , meaning "perpendicular" (to the plane of incidence). The alternative mnemonics in the text are perhaps more suitable for English speakers.
Light passing through a Fresnel rhomb undergoes two total internal reflections at the same carefully chosen angle of incidence. After one such reflection, the p component is advanced by 1/8 of a cycle (45°; π/4 radians) relative to the s component. With two such reflections, a relative phase shift of 1/4 of a cycle (90°; π/2) is obtained.Jenkins & White, 1976, p.532. The word relative is critical: as the wavelength is very small compared with the dimensions of typical apparatus, the individual phase advances suffered by the s and p components are not readily observable, but the difference between them is easily observable through its effect on the state of polarization of the emerging light.
If the incoming light is linearly polarized (plane-polarized), the s and p components are initially in phase; hence, after two reflections, "the p component is 90° ahead in phase", so that the polarization of the emerging light is elliptical with principal axes in the s and p directions (Fig.1). Similarly, if the incoming light is elliptically polarized with axes in the s and p directions, the emerging light is linearly polarized.
In the special case in which the incoming s and p components not only are in phase but also have equal magnitudes, the initial linear polarization is at 45° to the plane of incidence and reflection, and the final elliptical polarization is circular. If the circularly polarized light is inspected through an analyzer (second polarizer), it seems to have been completely "depolarized", because its observed brightness is independent of the orientation of the analyzer. But if this light is processed by a second rhomb, it is repolarized at 45° to the plane of reflection in that rhomb a property not shared by ordinary (unpolarized) light.
Related devices
For a general input polarization, the net effect of the rhomb is identical to that of a birefringent (doubly-refractive) quarter-wave plate, except that a simple birefringent plate gives the desired 90° separation at a single frequency, and not (even approximately) at widely different frequencies, whereas the phase separation given by the rhomb depends on its refractive index, which varies only slightly over a wide frequency range (see Dispersion). 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. The tandem arrangement, unlike a single Fresnel rhomb, has the additional feature that the emerging beam can be collinear with the original incident beam.
Theory
In order to specify the phase shift on reflection, we must choose a sign convention for the reflection coefficient, which is the ratio of the reflected amplitude to the incident amplitude. In the case of the s components, for which the incident and reflected vibrations are both normal (perpendicular) to the plane of incidence, the obvious choice is to say that a positive reflection coefficient, corresponding to zero phase shift, is one for which the incident and reflected fields have the same direction (no reversal; no "inversion"). In the case of the p components, this article adopts the convention that a positive reflection coefficient is one for which the incident and reflected fields are inclined towards the same medium. We may then cover both cases by saying that a positive reflection coefficient is one for which the direction of the field vector normal to the plane of incidence (the electric vector for the s polarization, or the magnetic vector for the p polarization) is unchanged by the reflection. (But the reader should be warned that some authors use a different convention for the p components, with the result that the stated phase shift differs by 180° from the value given here.)
With the chosen sign convention, the phase advances on total internal reflection, for the s and p components, are respectively given byCf. Jenkins & White, 1976, p.529.
and
where θ is the angle of incidence, and is the refractive index of the internal (optically denser) medium relative to the external (optically rarer) medium. (Some authors, however, use the reciprocal refractive index,Examples include Born & Wolf (1970, p.49, eqs.60) and Stratton (1941, p.499, eqs.43). Furthermore, Born & Wolf define and as arguments rather than phase shifts, causing a change of sign. so that their expressions for the phase shifts look different from the above.)
The phase advance of the p component relative to the s component is then given byStratton, 1941, p.500, eq.(44). The corresponding expression in Born & Wolf (1970, p.50) is the other way around because the terms represent arguments rather than phase shifts.
This is plotted in black in Fig.2, for angles of incidence exceeding the critical angle, for three values of the refractive index. It can be seen that a refractive index of 1.45 is not enough to give a 45° phase difference, whereas a refractive index of 1.5 is enough (by a slim margin) to give a 45° phase difference at two angles of incidence: about 50.2° and 53.3°.
For θ greater than the critical angle, the phase shifts on total reflection are deduced from complex values of the reflection coefficients. For completeness, Fig.2 also shows the phase shifts on partial reflection, for θ less than the critical angle. In the latter case, the reflection coefficients for the s and p components are real, and are conveniently expressed by Fresnel's sine lawFresnel, 1866, pp.773,789n; Hecht, 2002, p.115, eq.(4.42).
and Fresnel's tangent lawFresnel, 1866, pp.757,789n; Hecht, 2002, p.115, eq.(4.43).
where θ 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.Also used in the histories by Whittaker (1910, p.134) and Darrigol (2012, p.213), and in the texts by Born & Wolf (1970, p.40, eqs.21a) and Jenkins & White (1976, p.524, eqs.25a). (We can now see a disadvantage of that convention, namely that the two coefficients have opposite signs as we approach normal incidence; the corresponding advantage is that they have the same signs at grazing incidence.)
By Fresnel's sine law, is positive for all angles of incidence with a transmitted ray (since for dense-to-rare incidence), giving a phase shift of zero. But, by his tangent law, is negative for small angles (that is, near normal incidence), and changes sign at Brewster's angle, where θ and θt are complementary. Thus the phase shift is 180° for small θ but switches to 0° at Brewster's angle. Combining the complementarity with Snell's law yields as Brewster's angle for dense-to-rare incidence.The more familiar formula is for rare-to-dense incidence. In both cases, is the refractive index of the denser medium relative to the rarer medium.
That completes the information needed to plot and for all angles of incidence in Fig.2, in which is in red and in blue. On the angle-of-incidence scale (horizontal axis), Brewster's angle is where (red) falls from 180° to 0°, and the critical angle is where both and (red and blue) start to rise again. To the left of the critical angle is the region of partial reflection; here both reflection coefficients are real (phase 0° or 180°) with magnitudes less than 1. To the right of the critical angle is the region of total reflection; there both reflection coefficients are complex with magnitudes equal to 1.
In Fig.2, the phase difference is computed by a final subtraction; but there are other ways of expressing it. Fresnel himself, in 1823,Buchwald, 1989, pp.394, 453; Fresnel, 1866, pp.759, 786–787, 790. gave a formula for . Born and Wolf (1970, p.50) derive an expression for and find its maximum analytically.
(For derivations of eqs.() to () above, see Total internal reflection, especially and .)
History
Background
Augustin-Jean Fresnel came to the study of total internal reflection through his research on polarization. In 1811, François Arago discovered that polarized light was apparently "depolarized" in an orientation-dependent and color-dependent manner when passed through a slice of birefringent crystal: the emerging light showed colors when viewed through an analyzer (second polarizer). Chromatic polarization, as this phenomenon came to be called, was more thoroughly investigated in 1812 by Jean-Baptiste Biot. 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.Darrigol, 2012, pp.193–196, 290. He went on to discover that certain liquids, including turpentine (térébenthine), shared this property (see Optical rotation).
In 1816, Fresnel offered his first attempt at a wave-based theory of chromatic polarization. Without (yet) explicitly invoking transverse waves, this theory treated the light as consisting of two perpendicularly polarized components.Darrigol, 2012, p. 206.
Stage 1: Coupled prisms (1817)