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. Thus the rhomb functions as if it were a wideband quarter-wave plate: whereas a birefringent quarter-wave plate introduces a 90° phase difference for a single wavelength of light, the phase difference introduced by the rhomb depends only on its refractive index, which typically varies only slightly over the visible range of wavelengths (see Dispersion). The material of which the rhomb is made (typically glass) is specifically not birefringent.
The rhomb usually takes the form of a right parallelepiped — that is, a right parallelogram-based prism. If the incident ray is perpendicular to one of the smaller rectangular faces, the angle of incidence and reflection at the next face 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°.
The Fresnel rhomb is named after its inventor, the French physicist Augustin-Jean Fresnel invented the rhomb in stages between 1817  and 1823, during which time he deployed it in crucial experiments which led to his successful theory of light as a transverse wave.
An incident electromagnetic wave (such as light) consists of transverse vibrations in the electric and magnetic fields; these are proportional to and at right angles to each other so we will just consider the electric field. When striking an interface, the electric field oscillations can be resolved into two perpendicular components, known as the s and p components; their electric fields are, repectively, normal and parallel to the plane of incidence. These are also called, respectively, the TE (transverse electric) and TM (transverse magnetic) components.
After entering a Fresnel rhomb, the light undergoes two internal reflections. 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 phase shift of 1/4 of a cycle (90°; π/2) is effected. The result is a change in the state of polarization of a polarized input beam. This can be used, for instance, to change in between linear and circular polarization.
For a general input polarization, the net effect of the rhomb is identical to that of a quarter-wave plate, which is less expensive and more commonly used for that application. The rhomb, however, achieves a more precise phase retardation over a large wavelength range.
The operation of the Fresnel rhomb rests on the phase shift light encounters when undergoing total internal reflection which is unequal between the two polarization components. The phase shifts themselves are not normally observable, but the relative phase shift is very observable in terms of its effect on a light beam's state of polarization. Reflection of light at a plane interface is governed by the Fresnel (amplitude) coefficients whose calculation reveals these phase shifts (along with a reflection coefficient of unity magnitude) for internal reflections beyond the critical angle. These phase shifts and especially their difference (in black) are plotted here versus internal incidence angle for three different refractive indices.
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 latter arrangement, unlike a single Fresnel rhomb, has the additional feature that the exiting beam can be collinear with the original incident beam.
Chromatic polarization — the appearance of colors when polarized light is passed through a slice of doubly-refractive crystal followed by an analyzer — was discovered in 1811 by François Arago and 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. He went on to discover that certain liquids, including turpentine (térébenthine), shared this property (see Optical rotation).
In 1816, Augustin-Jean 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.
Stage 1: Coupled prisms (1817)
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. By including total internal reflection in a chromatic-polarization experiment, he found that the apparently depolarized light was a mixture of components polarized parallel and perpendicular to the plane of incidence, and that the total reflection introduced a phase difference between them. Choosing an appropriate angle of incidence (not yet exactly specified) gave a phase difference of 1/8 of a cycle. Two such reflections from the "parallel faces" of "two coupled prisms" gave a phase difference of 1/4 of a cycle. In that case, if the light was initially polarized at 45° to the plane of incidence and reflection, it appeared to be completely depolarized after the two reflections. These findings were reported in a memoir submitted and read to the French Academy of Sciences in November 1817.
In a "supplement" dated January 1818, Fresnel reported that optical rotation could be emulated by passing the polarized light through a pair of "coupled prisms", followed by an ordinary birefringent lamina sliced parallel to its axis, with the axis at 45° to the plane of reflection of the prisms, followed by a second pair of prisms at 90° to the first. This was the first experimental evidence of a mathematical relation between optical rotation and birefringence.
Stage 2: Parallelepiped (1818)
The memoir of November 1817  bears the undated marginal note: "I have since replaced these two coupled prisms by a parallelepiped in glass." A dated reference to the parallelepiped form — the form that we would now recognize as a Fresnel rhomb — is found in a memoir which Fresnel read to the Academy on 30 March 1818, and which was subsequently lost until 1846. In that memoir, 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. But he was increasingly difficult to ignore. In April 1818 he claimed priority for the Fresnel integrals. 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.
Equations (23) and (24), above, are known respectively as Fresnel's sine law and Fresnel's tangent law. Fresnel derived formulae equivalent to these in 1821, by modeling light waves as transverse elastic waves with vibrations perpendicular to what had previously been called the plane of polarization. He promptly confirmed by experiment that the equations correctly predicted the direction of polarization of the reflected beam when the incident beam was polarized at 45° to the plane of incidence, for light incident from air onto glass or water. The experimental confirmation was reported in a "postscript" to the work in which Fresnel expounded his mature theory of chromatic polarization, based on transverse waves. Details of the derivation were given later, in a memoir read to the Academy in January 1823. The derivation combined conservation of energy with continuity of the tangential vibration at the interface, but failed to allow for any condition on the normal component of vibration. (The first derivation from electromagnetic principles was given by Hendrik Lorentz in 1875.)
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.
At the end of a memoir on stress-induced birefringence, read in September 1822, Fresnel proposed an experiment involving a Fresnel rhomb, for the purpose of verifying that optical rotation is a form of birefringence. This experiment, like the one on stress-induced birefringence, required a row of prisms with their refracting angles in alternating directions, with two half-prisms at the ends, making the whole assembly rectangular. Fresnel predicted that if the prisms were cut from monocrystalline quartz with their optic axes aligned along the row, and with alternating directions of optical rotation, an object seen by looking along the common optic axis would give two images, which would seem unpolarized if viewed through an analyzer alone; but if viewed through a Fresnel rhomb, they would be polarized at ±45° to the plane of reflection.
Confirmation of this prediction was reported in a memoir submitted in December 1822, in which Fresnel coined the terms linear polarization, circular polarization, and elliptical polarization. In the experiment, the Fresnel rhomb revealed that the two images were circularly polarized in opposite directions, and the separation of the images showed that the different (circular) polarizations propagated at different speeds. To obtain a visible separation, Fresnel needed only one 14°-152°-14° prism and two half-prisms.
In the latter memoir, Fresnel explained optical rotation by noting that linearly-polarized light could be resolved into two circularly-polarized components rotating in opposite directions. If these components propagated at slightly different speeds, the phase difference between them — and therefore the orientation of their linearly-polarized resultant — would vary continuously with distance.
Stage 3: Calculation of angles (1823)
Thus, by December 1822, Fresnel had exploited the observable characteristics of the rhomb in experiments, and had at last introduced the terminology with which we now describe the purpose of the rhomb, but was yet to account for the angle at which the rhomb must be cut in order to serve that purpose. In the memoir of January 1823, in which he gave the detailed derivations of his formulae for rs and rp he found that for angles of incidence greater than the critical angle, the resulting coefficients were complex with unit magnitude. Noting that the magnitude represented the amplitude ratio as usual, he guessed that the ,argument represented the phase shift, and verified the hypothesis by experiment.
For glass with a refractive index of 1.51, Fresnel calculated that a 45° phase difference between the two coefficients required an angle of incidence of 48°37' or 54°37'. He cut a rhomb to the latter angle and found that it performed as expected.
Seeking more severe experimental tests of his theory, Fresnel calculated and verified the angle of incidence that would give a 90° phase difference after three reflections at the same angle, and four reflections at the same angle. In each case there were two solutions, and in each case he reported that the larger angle of incidence gave an accurate circular polarization (for an initial linear polarization at 45° to the plane of reflection). For the case of three reflections he also tested the smaller angle, but found that it gave some coloration due to the proximity of the critical angle and its slight dependence on wavelength. (Compare Fig. 4 above, which shows that the phase difference δ is more sensitive to the refractive index for smaller angles of incidence.) Most convincingly, Fresnel predicted and verified that four total internal reflections at 68°27' would give an accurate circular polarization if two of the reflections had water as the external medium while the other two had air, but not if the reflecting surfaces were all wet or all dry.
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. Arguably, the calculation of the phase shift on total internal reflection marked not only the completion of his theory of the rhomb, but also the essential completion of his reconstruction of physical optics on the transverse-wave hypothesis (see Augustin-Jean Fresnel).
The calculation of the phase shift was also a landmark in the application of complex numbers. Euler had pioneered the use of complex exponents in solutions of ordinary differential equations, on the understanding that the real part of the solution was the relevant part. 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. According to Salomon Bochner,
We think that this was the first time that complex numbers or any other mathematical objects which are "nothing-but-symbols" were put into the center of an interpretative context of "reality," and it is an extraordinary fact that this interpretation, although the first of its kind, stood up so well to verification by experiment and to the later "maxwellization" of the entire theory. In very loose terms one can say that this was the first time in which "nature" was abstracted from "pure" mathematics, that is from a mathematics which had not been previously abstracted from nature itself.
Notes and references
- A. Fresnel, "Mémoire sur les modifications que la réflexion imprime à la lumière polarisée" ("Memoir on the modifications that reflection impresses on polarized light"), signed & submitted 10 November 1817, read 24 November 1817; printed in Fresnel, 1866, pp. 441–85, including pp. 452 (rediscovery of depolarization by total internal reflection), 455 (two reflections, "coupled prisms", "parallelepiped in glass"), 467–8 (phase difference per reflection); see also p. 487, note 1, for the date of reading (confirmed by Kipnis, 1991, p. 217n).
- A. Fresnel, "Mémoire sur la loi des modifications que la réflexion imprime à la lumière polarisée" ("Memoir on the law of the modifications that reflection impresses on polarized light"), read 7 January 1823; reprinted in Fresnel, 1866, pp. 767–99 (full text, published 1831), pp. 753–62 (extract, published 1823). See especially pp. 773 (sine law), 757 (tangent law), 760–61 and 792–6 (angles of total internal reflection for given phase differences).
- Jenkins & White, 1976, p. 532.
- Darrigol, 2012, pp. 193–6, 290.
- Darrigol, 2012, p. 206.
- This effect had been previously discovered by Brewster, but not yet adequately reported. See: "On a new species of moveable polarization", [Quarterly] Journal of Science and the Arts, vol. 2, no. 3, 1817, p. 213; T. Young, "Chromatics", Supplement to the Fourth, Fifth, and Sixth Editions of the Encyclopædia Britannica, vol. 3 (first half, issued February 1818), pp. 141–63, at p. 157; Lloyd, 1834, p. 368.
- Darrigol, 2012, p. 207.
- A. Fresnel, "Supplément au Mémoire sur les modifications que la réflexion imprime à la lumière polarisée" ("Supplement to the Memoir on the modifications that reflection impresses on polarized light"), signed 15 January 1818, submitted for witnessing 19 January 1818; printed in Fresnel, 1866, pp. 487–508.
- Buchwald, 1989, pp. 223, 336; on the latter page, a "prism" means a Fresnel rhomb or equivalent.
- Kipnis, 1991, pp. 207n, 217n; Buchwald, 1989, p. 461, ref. 1818d.
- A. Fresnel, "Mémoire sur les couleurs développées dans les fluides homogènes par la lumière polarisée" ("Memoir on colors developed in homogeneous fluids by polarized light"), read 30 March 1818 (according to Kipnis, 1991, p. 217), published 1846; reprinted in Fresnel, 1866, pp. 655–83, especially pp. 659–62.
- Whittaker, 1910, p. 134; Darrigol, 2012, p. 213.
- Buchwald, 1989, pp. 390–91; Fresnel, 1866, pp. 646–8.
- A. Fresnel, "Note sur le calcul des teintes que la polarisation développe dans les lames cristallisées" ("Note on the calculation of hues that polarization develops in crystalline laminae"), Annales de Chimie et de Physique, vol. 17, pp. 102–12 (May 1821), 167–96 (June 1821), 312–16 ("Postscript", July 1821); reprinted in Fresnel, 1866, pp. 609–48.
- Buchwald, 1989, pp. 391–3; Whittaker, 1910, pp. 133–5.
- Buchwald, 1989, p. 392.
- A. Fresnel, "Note sur la double réfraction du verre comprimé" ("Note on the double refraction of compressed glass"), read 16 September 1822, published 1822; reprinted in Fresnel, 1866, pp. 713–18.
- A. Fresnel, "Mémoire sur la double réfraction que les rayons lumineux éprouvent en traversant les aiguilles de cristal de roche suivant les directions parallèles à l'axe" ("Memoir on the double refraction that light rays undergo in traversing the needles of rock crystal [quartz] in directions parallel to the axis"), signed & submitted 9 December 1822; reprinted in Fresnel, 1866, pp. 731–51 (full text, published 1825), pp. 719–29 (extract, published 1823). On the publication dates, see also Buchwald, 1989, p. 462, ref. 1822b.
- Buchwald, 1989, pp. 230–31; Fresnel, 1866, p. 744.
- Fresnel, 1866, pp. 737–9. Cf. Whewell, 1857, pp. 356–8; Jenkins & White, 1976, pp. 589–90.
- Buchwald, 1989, p. 442; Fresnel, 1866, p. 749.
- Lloyd, 1834, pp. 369–70; Buchwald, 1989, pp. 393–4, 453; Fresnel, 1866, pp. 781–96.
- Fresnel, 1866, pp. 760–61, 792–3.
- Fresnel, 1866, pp. 761, 793–6; Whewell, 1857, p. 359.
- Bochner, 1963, pp. 198–9.
- Bochner, 1963, p. 200; punctuation unchanged.
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- S. Bochner, "The significance of some basic mathematical conceptions for physics", Isis, vol. 54, no. 2 (June 1963), pp. 179–205; jstor.org/stable/228537.
- J.Z. Buchwald, 1989, The Rise of the Wave Theory of Light: Optical Theory and Experiment in the Early Nineteenth Century, University of Chicago Press.
- R.E. Collin, 1966, Foundations for Microwave Engineering, Tokyo: McGraw-Hill.
- O. Darrigol, 2012, A History of Optics: From Greek Antiquity to the Nineteenth Century, Oxford.
- R. Fitzpatrick, 2013, Oscillations and Waves: An Introduction, Boca Raton, FL: CRC Press.
- R. Fitzpatrick, 2013a, "Total Internal Reflection", University of Texas at Austin, accessed 14 March 2018.
- A. Fresnel, 1866 (ed. H. de Senarmont, E. Verdet, and L. Fresnel), Oeuvres complètes d'Augustin Fresnel, Paris: Imprimerie Impériale (3 vols., 1866–70), vol. 1 (1866).
- F.A. Jenkins and H.E. White, 1976, Fundamentals of Optics, 4th Ed., New York: McGraw-Hill.
- N. Kipnis, 1991, History of the Principle of Interference of Light, Basel: Birkhäuser.
- H. Lloyd, 1834, "Report on the progress and present state of physical optics", Report of the Fourth Meeting of the British Association for the Advancement of Science (held at Edinburgh in 1834), London: J. Murray, 1835, pp. 295–413.
- W. Whewell, 1857, History of the Inductive Sciences: From the Earliest to the Present Time, 3rd Ed., London: J.W. Parker & Son, vol. 2.
- E.T. Whittaker, 1910, A History of the Theories of Aether and Electricity: From the Age of Descartes to the Close of the Nineteenth Century, London: Longmans, Green, & Co.