On Birefringence Phenomena, Associated with Total Reflection B

On birefringence phenomena, associated with total reflection B. Julia, A. Neveu

To cite this version:

B. Julia, A. Neveu. On birefringence phenomena, associated with total reflection. Journal de Physique, 1973, 34 (5-6), pp.335-340. 10.1051/jphys:01973003405-6033500. jpa-00207388

HAL Id: jpa-00207388 https://hal.archives-ouvertes.fr/jpa-00207388 Submitted on 1 Jan 1973

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Classification Physics Abstracts

08.10 - 08.20 - 18.10

ON BIREFRINGENCE PHENOMENA, ASSOCIATED WITH TOTAL REFLECTION

B. JULIA and A. NEVEU Laboratoire de Physique Théorique et Hautes Energies, Orsay (*) (Reçu le 10 novembre 1972)

Résumé. 2014 Nous étudions les propriétés de biréfringence de la réflexion totale d’un faisceau lumineux monochromatique, sur le plan de séparation de deux milieux homogènes isotropes, par des méthodes de déphasage. Ceci explique à la fois l’effet Goos-Hänchen longitudinal dans lequel une source de lumière monochromatique non polarisée donne deux images de polarisations rectilignes orthogonales, et l’effet transverse étudié récemment par Imbert, pour lequel les deux images sont polarisées circulairement. Nous donnons une méthode simple pour déterminer les polarisations et les positions des images d’une source cylindrique.

Abstract. 2014 We study by phase-shift methods the birefringence properties of total reflection of a light beam at the separation plane between two isotropic homogeneous media. This includes the Goos-Hänchen longitudinal effect in wich a source of unpolarized light is split into two images, each of them linearly polarized, as well as the transverse shift recently investigated experimentally by Imbert, where the effect is between left and right circular polarizations. We give a general simple method for determining the polarizations and positions of the images.

Introduction. - The existence of birefringence pro- unphysical point in the computation of the various perties of total reflection of light on the separation effects with the Poynting vector is the use of plane plane of two isotropic homogeneous media has been waves of a finite width, neglecting what may occur known for some time [1 ], and experimental evidence near the edges. For these reasons, an approach using was first demonstrated by Goos and Hanchez [2]. phase-shifts seems safer to us, in particular because These authors considered a large number of successive it is more closely related both to the exact solution of total reflections of a pencil of light between two Maxwell’s equations, and to the mechanism of for- parallel planes, and found that a source of unpola- mation of images in geometrical optics. rized light is split into two images, one image being In this paper, we show that one can compute all polarized parallel to the incidence plane, the other effects which have been experimentally observed by perpendicular. Using a different apparatus, Imbert [3] using the exact solutions of Maxwell’s equations for has recently found evidence for a splitting between the reflection of polarized plane waves on the sepa- right and left circular polarizations (see also [10]) : in ration plane of two homogeneous isotropic media, his experiment, the successive total reflections of the together with some simple geometrical optics approxi- pencil of unpolarized light take place on the sides of a mations, which turn out to be always valid in expe- regular dielectric prism. rimental situations. We deal with the usual longitu- These effects, which are of the order of one wave- dinal Goos-Hânchen effect in section 1, and with length at each reflection, have been computed by Imbert’s transverse shift in sections 2 and 3. It turns out various methods [4], [6], [7] among which [4] the use that the geometrical arrangement of the experimental of Poynting vectors is very popular. In our opinion apparatus is of crucial importance in the determination however, if the Poynting vector may be useful to of the polarizations of the two images of an un pola- obtain an order of magnitude of the effect, it cannot rized source. provide a completely consistent treatment of the problem : in particular because it does not tell clearly 1. Longitudinal shift (Goos-Hânchen). - The lon- which polarization states are relevant ; another gitudinal shift is the easiest to compute, owing to the simpler geometrical structure of the system : we consider (Fig. 1) a rectilinear source of light S (narrow (*) Laboratoire associé au Centre National de la Recherche slit for instance), emitting a pencil of of small Scientifique. Postal address : Laboratoire de Physique Théorique light This is reflected on the et Hautes Energies, Bâtiment 211, Université de Paris-Sud, aperture Brn. pencil totally 91405 Orsay, France. plane surface between the vacuum and an isotropic

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01973003405-6033500 336 medium with index n. This separation plane is parallel to the source ; Oy is its intersection with the plane in which figure 1 is drawn. Let i be the mean angle of incidence of the narrow pencil emitted by S. We can decompose this pencil into a linear superposition of plane waves, which have slightly different incidence angles on Oy. Each plane wave can then be reflected on the surface Oy, accord- ing to Maxwell’s theory. We then consider the set of the reflected plane waves. In the approximation of geometrical optics (to be justified later), these reflected waves recombine by superposition to a pencil which, in lowest order in Brn, comes from the virtual cylindrical source 1 : 1 is just the center of curvature of the envelope of the reflected plane waves. 1 will in general be different from S’, the symmetric of S with respect to Oy, and its position will depend on the polarization, thus giving rise to the effect. FIG. 1. - Goos-Hânchen Reflection It is straight forward to compute the location of I : coordinates in the S’ its (S’ X, Y) system (see Fig. 1) a finite portion of the caustic surface, the separation are : of polarizations is complete only if the distance between the two images is larger than their spreading out :

If this condition is not satisfied, one can still see the where À is the wavelength of the light in vacuo, and 5(i) Goos-Hânchen effect by using light polarized, perpen- the phase shift of a plane wave at total reflection, as dicular or parallel, to the incidence plane. computed from Maxwell’s theory. Going to the expres- Let us now briefly discuss the validity of the approxi- sion of 5 in terms of i and n for polarizations perpen- mation of geometrical optics used in our derivation dicular and parallel to the incidence plane [5], one of formulae (1)-(4). Indeed, it is in principle unsafe finds the explicit expressions : to use geometrical optics to obtain a result like (1)-(4) which is of the order of a wavelength, that is to say of the same order of magnitude as diffraction. However, in the experimental apparatus, the beam is reflected N times between two reflection planes like Oy, in order to amplify the effect. In this process, it is clear that indeed phase-shifts add up, whereas the magnitude of diffraction, which is set by the aperture of the beam, remains constant. Thanks to this multiplying effect, the two images are split by N(L11.L - 11 111), which can be made much larger than diffraction. Hence, whereas it would be meaningless for a single reflection, our of B"1’) approximation geometrical optics is justified for the full experiment. As i varies, I1 and III define two caustic surfaces Transverse shift and which are the images of S through our anisotropic 2. approximate analysis of optical system. Imbert’s experiment. - In [3], Imbert describes a nice Several remarks are in order : formulae (2) and (4) experiment which shows that total reflection can also do not seem to exist in the literature on the subject. give rise to a splitting between the two circular pola- Our results for the experimentally observed quan- risations. He also computes this effect by using the tities L1 LL and All1 agree with those of Hora [6] and Poynting vector of the evanescent wave. What we Boulware [7] who also derives them via a phase-shift want to show in this section is that the effect can be analysis. However, they differ by a factor cos2 i/cos2 ij, computed by using phases, just as the longitudinal with sin il = 1/n from the formula of Renard [8] and effect of section 1. The origin of this new effect on Imbert who, however, derive them using a heuristic circular polarizations is more subtle than the effect of [3] ° argument with Poynting vectors. section 1, and rests essentially on the rather complex Since the image of a beam with a finite aperture Brn is geometrical structure of Imbert’s experiment. 337

In this section, we shall only give an approximate arrangement is such that the light follows a helical treatment of the effect, leaving the complete analysis path in the prism. The view of that path from under is for the next section. Indeed, we shall now show that drawn on figure 3. If the phase shifts at each reflection the geometry of Imbert’s experiment implies that the are zero, a linearly-polarized plane-wave remains splitting of the two circular polarizations is already linearly polarized. However, the direction of the pola- present, even if the phase shifts at each reflection are rization vector is in general différent for each ray of the zero, independent from the incidence angle for both reflected beam. linear polarizations, perpendicular and parallel to the incidence plane. Since in the experiment of [3] all incidence-angles are very close to the limiting angle, the phase shifts of both transverse and parallel polarizations are indeed quite small. However, they vary rapidly when the incidence angle varies, so that the treatment of this section is only approximate in that sense : we shall find that it accounts for about two thirds of the experimental result ; a more refined treatment, taking into account the variation of phase shifts, will be presented in the next section. We now describe (Fig. 2) the geometry of the experiment, following Imbert’s notations whenever pos- sible. The source S is a line in the horizontal plane H. It emits a cylindrical narrow pencil of light with the angle 0’ with respect to the horizontal plane H. The light emitted by S is totally reflected on the faces of a FiG. 3. - Imbert’s Apparatus (from under) vertical prism of glass of index n = 1.8. The cross- This contrasts with the result of an ideal metallic section of this prism is an equilateral triangle. The reflection, where one of the phase shifts is zero, and ...... the other is 7:, in which case there is no effect. It is then easy to get convinced that the polarizations which should be used are not the linear polarizations, but the circular polarizations which diagonalize the S-matrix, in Imbert’s experiment. In fact, after one reflection they are conserved for all the rays of the beam. For all rays issued from the source, let us take the vector no orthogonal to the source and to the beam as the origin of phases. For the first reflection, phases are to be counted with respect to the normal ni to the incidence plane. This vector ni depends on 0’, the angle of the ray and of the horizontal plane. Let qJ 1 ((}’) be the angle between no and ni, and n2 the symmetric of no with respect to the reflection plane. If cot(- cvt) is the phase of a right (left) circular wave with respect to no before the reflection, it is rot - 2 qJ 1 ( - rot - 2 qJ 1) with respect to n2 after the reflection. Hence, total reflection induces for each circular polarization a phase shift which depends on the ray of the beam. For the second reflection, the images Si of S, and n2 of no play the same role as S and no for the first reflection. By studying the geometry of the system, one also finds that the angle ç between two successive incidence planes is ç = 2 qJ 1. Hence, after N reflections, the phase shift of a right (left) circular ray is NqJ((}’) (- Nç(0’)). By the same argument as for the longitudinal shift of section 1, one finds that the source is then split into two images, right and left polarized, which are separated from each other by 2 d, with

FIG. 2. - Imbert’s Apparatus (perspective) 338

The experimental set-up of [3] gives : produced by the right circular component of the two linearly polarized images of section 1. From this example, we can induce the general cri- terion for the choice of polarizations to which the so that one finds (1) geometrical optics treatment of section 1 and 2 can be applied ; we restrict ourselves to cylindrical beams, which is enough if the source is a slit or a rectilinear or, numerically, object as in Imbert’s experiment (2). In this case, considering the incident beam as a superposition of plane waves, the action of the optical system on each This is to be compared with Imbert’s experimental plane wave can be described by a two-by-two matrix B : value [3] ] if the polarization vector of the incident plane wave is the polarization vector of the outgoing plane We thus see that our approximate treatment which aBP/ , neglects the variation of l5 with the incidence angle wave is For a incident the gives an effect which, though too small, is of the cylindrical beam, of B a . correct order magnitude. matrix B depends only on one parameter, 0’, which is the angle of the rays of the beam with some reference 3. Complète description of the transverse shift. - plane. It is clear from the discussion of the preceding Let us now come to a more rigorous treatment of the transverse shift, in which we shall take into account section that the polarization states to which one the variation of phase shifts with the incidence angle. aBP/ can are those such that the In the first two preceding sections, we have used the apply geometrical optics approximation of geometrical optics to reconstruct the outgoing polarization states are independent images from the envelope of the outgoing plane waves. B(e’)W a from for an overall when 0’ the However, we are dealing with a system in which pola- 0’, except phase, spans beam : for other choice of the incident rization plays a crucial role, and standard geometrical any polariza- the would over the optics do not tell us which polarizations of the out- tion, polarization vary rapidly so that going waves should be used to determine the position outgoing beam, simple geometrical optics and nature of the images. would be inapplicable. In other words, one should look for the such at the first order Let us see, in an example, the kind of problems one polarizations that, in e runs into when one does not choose the polarization (3), to which one applies geometrical optics appropriately : in section 1, we could imagine to send an incoherent light on the apparatus, and at the exit, to pick up a circular with a filter. given polarization (say, right) (2) We hope to deal with the more complex case of spherical 1 to The envelope method of section applied blindly waves in a later publication. this set of outgoing right circularly-polarized plane (3) Here we derive rigorously a formulation in terms of geo- waves would then give a single image, half way metrical optics from Maxwell’s equations (the crude approxi- between the two images of linearly polarized light. mation À = 0 would give J = 0). This would contradict the result of section 1, which is 1) We must divide the general problem into two problems with one parameter (the eikonal) each. that one should observe the circular precisely right 2) We can write ([9] p. 119) E and H in the general form component of these two images. This paradox is easily solved : the two images found in section 1 are observable if are well with to they separated respect omitting the phase factor e-irot ,(e and h are complex vectors). diffraction effects : the beam must have a large enough 3) In a homogeneous medium of index n, Maxwell’s equa- and the constant over this tions become aperture, intensity relatively - aperture. One then finds that over such an aperture the filtered right circular light would vary a lot in intensity, with passing through zero many times. Hence, such an amplitude cannot be interpreted as coming from a single image, but precisely as the inference pattern and the analogous equations with h. We use the eikonal 8 for geometrical optics : it means K = 0. We can also impose L = 0 (1) Ly is the shift normal to the slit S ; it corresponds to a i. e. polarization is constant along each « ray » [9]. Conclusion : We must then have Ae = 0 to verify Maxwell’s shift L’ y normal to the incidence plane, or = cos Ly qJ /2 equations precisely That is the case if (more Â2 lellael « 1 ). e(r) = Cte, that is if the only variation of E is in the eikonal that result agrees with Schilling’s [8]. It can be shown to hold for any prism whose cross-section is a regular polygon 339 or and the simplifying fact that

one easily finds that the eigenstates of B-1 B’ are indeed the left and right polarizations when ô L--- 0, ) is thus an eigenvector of B-1 B’, with eigenva- and that the distance between the two circularly ( BP/ polarized images is lue ia. B being a unitary matrix, a is real. This eigen- value is then related to the displacement L1 of the image of the source by the formula

One can translate this in terms of the displacement per reflection :

A general polarization state of the incident beam should then be decomposed as a superposition of the From formula (8), and using the fact that b N 0, one two polarizations @ which solve eq. (6) and (Plal , ("2)l’2 finds one then observes two images, for such a general polarization, with polarizations B which (fila 1 and BP2 a2 are - It is now to separated by L111 d 2. straightforward In this equation, dç/d0’ is the geometrical term already apply those ideas to Imbert’s experiment : at each given by section 2. One also has in Imbert’s experi- reflection inside the prism, the relevant matrix A ment : includes not only the phase shifts given by the solutions of Maxwell’s equations, but also takes care of the rotation 9 of successive incidence planes due to and, i being close to the limiting angle ii, one can use the peculiar geometry of the experiment. One then the approximate formulae : finds :

Hence (4) where ôjj (b.1) is the phase shift of a plane wave polarized parallel (perpendicular) to the incidence plane, b = t (bll - b .1). qJ and ô are functions of B’ through the geometry of figure 2. The total transition matrix of or numerically, the apparatus is This value may be compared with Imbert’s result : where N is the number of reflections inside the prism. Introducing the angle u by

The ratio of the two values is 1.03 for n = 1.8 and i in with results. one finds = il good agreement experimental

Conclusion. - We have shown that simple argu- ments using classical optics and a careful analysis of the geometrical structure of Imbert’s experiment can yield a direct computation and interpretation of his where results in terms of the phase shifts at total reflection computed directly from Maxwell’s equations : these methods also apply to the longitudinal Goos-Hânchen effect, and explain how a single phenomenon (total Using Imbert’s experimental values : internal reflection) from a single-unpolarized source

(4) That result was independently derived by D. Boulware [7], 340 can give two images which are either linearly pola- Beauregard for many informations about their expe- rized (Goos-Hânchen) or circularly polarized (Imbert). riment and their calculations.

Acknowledgements. - We are grateful to NOTE : While this work was in progress, we received Pr. C. Bouchiat for a discussion which is at the origin a preprint by D. Boulware [7] in which a phase-shift of this work, and to Dr. C. Imbert and 0. Costa de analysis of Imbert’s experiment is also proposed.

References

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