Fresnel formulas as Lorentz transformations

Transcription

1 J. J. Monzón and L. L. Sánchez-Soto Vol. 7, No. 8/August 2000/J. Opt. Soc. Am. A 475 Fresnel formulas as Lorentz transformations Juan José Monzón and Luis L. Sánchez-Soto Departamento de Óptica, Facultad de Ciencias Físicas, Universidad Complutense, Madrid, Spain Received October 9, 999; revised manuscript received April 3, 2000; accepted April 28, 2000 From a matrix formulation of the boundary conditions we obtain the fundamental invariant for an interface and a remarkably simple factorization of the interface matrix, which enables us to express the Fresnel coefficients in a new and compact form. This factorization allows us to recast the action of an interface between transparent media as a hyperbolic rotation. By exploiting the local isomorphism between SL(2, C) and the (3 )-dimensional restricted Lorentz group SO(3, ), we construct the equivalent Lorentz transformation that describes any interface Optical Society of America [S (00)0208-4] OCIS codes: , , , INTRODUCTION In a series of interesting and pioneering papers, 4 Vigoureux has drawn attention to the possibility of presenting the optics of stratified planar media in a relativisticlike form. In fact, he pointed out a formal analogy between field amplitudes at a multilayer and coordinates in a ( )-dimensional space time, arriving at a similarity between the composition law of reflected amplitudes and the composition law of velocities. However, he noted that the essential difference was that reflection coefficients are complex quantities, whereas velocities are real quantities. In recent papers 5 7 we have settled the basis for a correct approach to the subject, showing 5 that the matrix describing a general multilayer is an element of the group SL(2, C) of unimodular 2 2 complex matrices, which has six independent parameters. This group is locally isomorphic to the (3 )-dimensional Lorentz group SO(3, ), and, in consequence, by a standard procedure we were able to associate with each multilayer matrix a restricted Lorentz transformation that also depends on six parameters; namely, the three parameters of a spatial rotation and the three components of a boost along an arbitrary direction. This leads to a natural and complete identification between reflection and transmission coefficients of the multilayer and the corresponding parameters of the Lorentz transformation. Furthermore, for the special case of lossless multilayers 6,7 the associated matrix is an element of the group SU(, ), which has three independent parameters and is a subgroup of SL(2, C). This group SU(, ) is locally isomorphic to the (2 )-dimensional Lorentz group SO(2, ), and by the same procedure a multilayer matrix is now equivalent to a restricted Lorentz transformation that also depends on three parameters: the angle of a rotation in the plane and the two components of a boost along an arbitrary direction in the same plane. The multilayer matrix, which represents the overall reflection and transmission properties of a stratified structure, can be always expressed as a product of interface and propagation matrices 8,9 describing the effects of the individual interfaces and layers, respectively. The propagation matrices are diagonal, and, in consequence, their equivalent Lorentz transformation can be easily worked out. In contrast, for the interface matrices this is a more involved problem. Thus the keystone for a correct and deep understanding of any multilayer system is the question of reflection and transmission of light in a single planar interface. Of course, this is a problem treated in any optics textbook, and its solution is expressed in terms of the famous Fresnel formulas. After the remarkable contributions by Azzam, 0 it seems almost impossible to say anything new about Fresnel formulas. However, from the perspective of this paper, two natural questions emerge: Can an interface be described by an equivalent Lorentz transformation? And, if this were the case, would that transformation be the simplest one, i.e., an element of the ( )-dimensional Lorentz group SO(, )? This would justify the existence of the chain SO(n,)(n, 2, 3) in progression from a single interface (n ) to lossless multilayers (n 2) and, finally, to the more complicated absorbing stacks (n 3). Our aim in this paper is precisely to answer these relevant questions. To this end in Section 2 we present a hitherto unsuspectedly simple matrix formulation of the boundary conditions, which lead to a fundamental invariance that is at the heart of the analogy between optics and relativity. 4 In Section 3 we exploit this formulation to recast the action of an interface (and therefore the Fresnel formulas) in a new and physically appealing form that, in Section 4, is reinterpreted as a hyperbolic rotation, whose geometric invariant is identified with the fundamental invariant of the interface. In Section 5 we use the general correspondence between SL(2, C) and SO(3, ) to express the action of the interface as the product of a one-dimensional boost by a spatial rotation in a (3 )-dimensional space time. For transparent media, the rotation is absent, and we have only a boost in SO(, ), as expected. Finally, our conclusions are summarized in Section MATRIX FORMULATION OF BOUNDARY CONDITIONS Let two homogeneous isotropic semi-infinite media, described by complex refractive indices N 0 and N, be sepa /2000/ $ Optical Society of America

2 476 J. Opt. Soc. Am. A/ Vol. 7, No. 8/ August 2000 J. J. Monzón and L. L. Sánchez-Soto rated by a plane boundary. The Z axis is chosen perpendicular to the boundary and directed as in Fig.. We assume an incident monochromatic, linearly polarized plane wave from medium 0, which makes an angle 0 with the Z axis and has amplitude E () 0. The electric field is either in the plane of incidence ( p polarization) or perpendicular to the plane of incidence (s polarization). This wave gives rise to a reflected wave E (r) 0 in medium 0 and a transmitted wave E (t) 0 in medium that makes an angle with the Z axis. The angles of incidence 0 and refraction must be related by Snell s law: N 0 sin 0 N sin. (2.) If media 0 and are transparent (so that N 0 and N are real numbers) and no total reflection occurs, the angles 0 and are also real, and the above picture of how a plane wave is reflected and refracted at the interface is simple. However, when either one or both media are absorbing, the angles 0 and become, in general, complex and the discussion continues to hold only formally, but the physical picture of the fields becomes complicated. 2 We consider as well another plane wave of the same frequency and polarization and of amplitude E (), incident from medium at an angle, as indicated in Fig.. In the same way, we shall denote by E (r) and E (t) the reflected and transmitted amplitudes of the corresponding waves. The complex amplitudes of the total output fields at opposite points immediately above and below the interface will be called E () 0 and E (), respectively. The wave vectors of all waves lie in the plane of incidence, and when the incident waves are p or s polarized, all plane waves excited by the incident waves have the same polarization. The amplitude E () in medium is E () (t) E (r) t 0 () r 0 E (), (2.2) while the amplitude () in medium 0 is () (r) E (t) r 0 () t 0 E (), (2.3) where t 0 and r 0 are the Fresnel transmission and reflection coefficients for interface 0 and t 0 and r 0 refer to the corresponding coefficients for interface 0. These Fresnel coefficients are determined by matching the tangential components of E and H fields across the interface. 3 Our main goal in this section is to express these boundary conditions in an alternative form that will allow us to interpret its structure in more appealing terms. To this end, let us note that for p polarization the boundary conditions at interface 0 for the field incident from medium 0 can be recast in the form () cos 0 () N 0 () cos E () N E cos 0 cos N 0 N cos cos 0 N N 0 E (r) 0 (t). (2.4) Similarly, for the field incident from medium, we have E (r) E (t). (2.5) By using Eqs. (2.2) and (2.3), we arrive at cos 0 cos 0 N 0 N 0 E () 0 () cos cos N N E () E (). (2.6) We wish to emphasize that this alternative formula is a fairly simple expression of the boundary conditions but introduces remarkable simplicity and symmetry. In components, Eq. (2.6) reduces to cos 0 () () cos E () E (), N 0 () () N E () E (), (2.7) which are the boundary conditions expressed in an invariant form. The product of these two equations leads us to the fundamental invariant N 0 cos 0 () 2 () 2 N cos E () 2 E () 2, (2.8) which was assumed as a basic axiom by Vigoureux and Grossel in Ref. 4. This invariant is, in general, a complex quantity and does not reduce to the law of conservation of energy on the interface, because E () i 2 appears instead of E () i 2. Its meaning will be elucidated in following sections. A procedure similar gives for s polarization N 0 cos 0 N 0 cos 0 E () 0 () E () N cos N cos which in components reads now as E (), (2.9) N 0 cos 0 () () N cos E () E (), () () E () E (). (2.0) Fig.. Wave vectors of the incident, reflected, and transmitted fields at interface 0. The product of them is again the fundamental invariant [Eq. (2.8)], which is therefore independent of the state of polarization. Note that, for p polarization, each one of the matrices appearing in the invariance expressed by Eq. (2.6) can be factorized in the form (i 0,):

3 J. J. Monzón and L. L. Sánchez-Soto Vol. 7, No. 8/August 2000/J. Opt. Soc. Am. A 477 cos i cos i N i N i cos i 0 0 N i I 0 s R /4T 0 st sr/4 R /4 T i N cos /N 0 cos 0 0 (3.3) p2r/4, (2.) 0 R/4, where the diagonal matrix T i ( p) (henceforth the letter p or s in a vector or a matrix will indicate the corresponding polarization) involves only the physical variables of each for p and s polarizations, respectively. This factorization is based solely on the boundary conditions and leads us to the conclusion that the interface medium, while R(/4) represents the matrix of a rotation matrix must be of the form of angle /4. In consequence, Eq. (2.6) takes the form T 0 pr/4 () () T pr/4 E () I E () 0 a b (3.4). (2.2) b a, where a and b are complex numbers. This general result In an analogous way, for s polarization, Eq. (2.9) can be implies that the following relations must hold in Eq. (3.2): recast as r 0 r 0, T 0 sr/4 () () T sr/4 E () E (), (2.3) r 0 r 0 t 0 t 0, (3.5) where now which apply to both p and s polarizations by the simple attachment of a label to all the coefficients. This constitutes an alternative algebraic demonstration of the well- N i cos i N i cos i N i cos i 0 0 known Stokes relations without the necessity of resorting to the usual time-reversal argument. 5,6 In consequence, we can write T i s 2R/4. (2.4) Moreover, taking into account Eqs. (2.) and (2.4), it is easy to show that T i s N i 0 0 /N i p. (2.5) it In our opinion, this is a remarkable result that, as far as we know, has been never stated, in spite of its simplicity. It ensures that all the properties for s-polarized fields can be obtained from those of p-polarized fields by the action of a squeezing matrix, 4 whose squeezing factor is precisely the refractive index of medium i and is independent of the angle of incidence. 3. INTERFACE MATRIX AND RENORMALIZED FIELD AMPLITUDES The matrix formulation of the boundary conditions developed in Section 2 yields invariant properties across the interface. However, the standard way of discussing this topic is by relating the field amplitudes at each side of the interface. This relation is expressed as 8 () () I 0 E () E (), (3.) where I 0 is called the interface matrix and, from Eqs. (2.2) and (2.3), is given by I 0 t 0 r 0 (3.2) r 0 t 0 t 0 r 0 r 0. On the other hand, from Eqs. (2.2) and (2.3), the interface matrix I 0 can be expressed in the very compact form I 0 p R /4T 0 pt pr/4 R /4 cos /cos N /N 0R/4, I 0 t 0 r 0. (3.6) r 0 From the mathematical point of view, it is a simple task to check that the set of interface matrices is an Abelian group with respect to the matrix multiplication. The inverse matrix satisfies I 0 I 0 and then describes the interface taken in the reverse order. For both basic polarizations we have det I 0 p det I 0 s N cos. (3.7) N 0 cos 0 For important reasons that will become clear in Section 4, it is adequate to renormalize the field amplitudes to ensure that the interface matrix always has unit determinant. To this end, let us define e 0 () N 0 cos 0 (), e () N cos E (). (3.8) Accordingly, the action of the interface is described now by e 0 () e 0 () i 0 e () e (), (3.9) where the renormalized interface matrix can be written as i 0 R /4 / R/4 2 0 / 0 0 / 0 0 / 0 0 / 0, (3.0) and the factor 0 has the values /2 N 0 cos 0 p N cos 0, (3.a)

4 478 J. Opt. Soc. Am. A/ Vol. 7, No. 8/ August 2000 J. J. Monzón and L. L. Sánchez-Soto 0 s N /2 0 cos 0 N cos. (3.b) Another simple way of expressing these relations is 0 p 0 s cos 0 cos, 0 s 0 p N 0. (3.2) N It is now evident from Eq. (3.0) that the renormalized interface matrix satisfies det i 0, as desired, and can always be expressed as a squeezing matrix of parameter 0 postmultiplied and premultiplied by a rotation of angle /4 and its inverse. Furthermore, the set i ij of the renormalized interface matrices inherits the structure of Abelian group. Moreover, by comparing Eq. (3.6) with Eq. (3.9), we can reinterpret i 0 in terms of renormalized Fresnel coefficients as where i 0 tˆ 0 rˆ 0 rˆ 0, (3.3) tˆ / 0, It is clear from Eqs. (3.4) that the single parameter 0 gives all the information about the interface, even for absorbing media. In Fig. 2 we have plotted the behavior of 0 ( p) and 0 (s) as a function of the angle of incidence 0 for an interface air glass (N 0 /N 2/3) and, for the purpose of comparison, the corresponding values of r 0 ( p) and r 0 (s). Given the properties of the interface matrices, we have i 0 i 0 ; i.e., the inverse describes the interface taken in the reverse order. Thus 0 / 0, and it follows that tˆ 0 tˆ 0, rˆ 0 rˆ 0. (3.7) 4. THE ORIGIN OF THE FUNDAMENTAL INVARIANT FOR AN INTERFACE The definition of the renormalized interface matrix in Eq. (3.0) may appear, at first sight, rather artificial. In this section we shall interpret its meaning by recasting it in an appropriate form that will reveal the origin of the rotation matrices R(/4). To simplify the discussion as much as possible, let us assume that we are dealing with an interface between two transparent media when no total reflection occurs. In this relevant case, the Fresnel reflection and transmission coefficients, and therefore 0, are real numbers. If we introduce a new parameter by which satisfy rˆ 0 0 / 0 0 / 0, (3.4) rˆ 0 2 tˆ 0 2. (3.5) The Fresnel coefficients can be obtained from the renormalized ones as t 0 N /2 cos N 0 cos 0 tˆ 0, r 0 rˆ 0. (3.6) Fig. 2. Plot of the factor 0 and r 0 as functions of the angle of incidence 0 (in degrees) for both p and s polarizations for an air glass interface (N 0, N.5). The marked points correspond to the Brewster angle. 0 exp/2, (4.) then the action of the interface can be expressed as e 0 () e 0 () cosh/2 sinh/2 sinh/2 cosh/2 e () e (), (4.2) where the renormalized Fresnel coefficients are now tˆ 0 cosh/2, rˆ 0 tanh/2. (4.3) Given the importance of this new reformulation of the action of an interface, some comments seem pertinent. First, it is clear that the reflection coefficient can be always expressed as a hyperbolic tangent, whose composition law is very simple. In fact, such an important result was first derived by Khashan 7 and is the origin of several techniques to treat the reflection coefficient of layered structures, 8 including bilinear or quotient functions, 9 that are of the form [Eqs. (3.4)]. However, the transmission coefficient for these structures seems to be (almost) safely ignored in the literature, because it behaves as a hyperbolic secant, whose composition law is more involved. In spite of its apparent simplicity and elegance, the transformation [Eq. (4.2)] can lead to misconceptions and misunderstandings. One may be tempted to identify this action as a Lorentz transformation with a velocity equal to rˆ 0 ina()-dimensional space time. However, this point of view is wrong, because fields are, in general, complex quantities, whereas space time coordinates are

5 J. J. Monzón and L. L. Sánchez-Soto Vol. 7, No. 8/August 2000/J. Opt. Soc. Am. A 479 always real numbers. The correct link with special relativity will be established in Section 5. Now the meaning of the rotation R(/4) can be put forward in a clear way. To this end, note that the transformation [Eq. (4.2)] is formally a hyperbolic rotation of angle /2 acting on the complex variables e (), e (). As is usual in hyperbolic geometry, 20 it is convenient to study this transformation in a coordinate frame whose new axes are the bisecting lines of the original one. In other words, in this frame whose axes are rotated /4 with respect to the original one, the new coordinates are ẽ() e() ẽ () R/4 e () (4.4) for both 0 and media, and the action of the interface is represented by a pure squeezing matrix ẽ 0 () ẽ 0 () / ẽ () ẽ (). (4.5) Furthermore, the product of these complex coordinates remains constant. In the case of Lorentz transformations for energy-momentum, this invariant is the famous Einstein relation E 2 p 2 m 2. In our problem, we have or ẽ 0 () ẽ 0 () ẽ () ẽ (), (4.6) e () 0 2 e () 0 2 e () 2 e () 2, (4.7) which is just the fundamental invariant [Eq. (2.8)] written in the renormalized field variables and whose meaning is nothing but the hyperbolic invariant of the complex field transformation. 5. LORENTZ TRANSFORMATION DESCRIBING AN INTERFACE The Abelian group of renormalized interface matrices is a subgroup of the group SL(2, C) of 2 2 complex matrices with unit determinant, and it is well known that this group is homomorphic to the Lorentz group. To show this important point, let us recall some wellestablished facts about (3 )-dimensional Lorentz transformations. In the usual Minkowski space we introduce the 4-vectors of components (x 0, x, x 2, x 3 ), where x 0 ct. Then a Lorentz transformation is a transformation between two coordinate frames, x x, (5.) (the greek indices run from 0 to 3), such that the pseudo- Euclidean form x 0 2 x 2 x 2 2 x 3 2 (5.2) remains invariant. In particular, we are interested in dealing with the restricted Lorentz group, i.e., the group of the Lorentz transformations that have det and do not reverse the direction of time ( 0 0 ), which is usually denoted by SO(3, ). We recall the mentioned correspondence between SL(2, C) and SO(3, ) explicitly in the following form 2 : Let be the set of four Hermitian matrices 0 I (the identity) and k (the Pauli matrices). With each 4-vector x we associate the 2 2 Hermitian matrix X by X x x0 x 3 x ix 2 x ix 2 x 0 x 3, (5.3) in such a way that Then the transfor- Now let W be a matrix of SL(2, C). mation x 2 trx. (5.4) X WXW, (5.5) where the superscript denotes the Hermitian conjugate, induces a Lorentz transformation on the coefficients x. In fact, it is easy to find it explicitly from Eq. (5.5) as 22 W 2 tr W W. (5.6) This equation can be easily solved to obtain W from a given. Clearly, the matrices W and W generate the same, so this homomorphism is two to one. The matrices W form a two-dimensional spinor representation of the restricted Lorentz group SO(3, ); i.e., 2 W( )W( 2 ) W( 2 ). Now we are in a good position to show the equivalent Lorentz transformation that describes an interface. To this end, we recall that a restricted Lorentz transformation can always be decomposed into a product of a threedimensional spatial rotation R and a boost L along an arbitrary direction n 23 : LR. (5.7) The analogous factorization in SL(2, C) is provided by the polar decomposition, which ensures that any matrix W SL(2, C) can be expressed in a unique way in the form 5 W HU, (5.8) where H is positive definite Hermitian and U is unitary. Under the homomorphism previously discussed, H generates a boost and U a rotation, in agreement with Eq. (5.7). In the case of a general interface, and considering the role played by the parameter introduced in Section 4, it proves convenient to rewrite the complex number 0 as 0 0 expi exp/2expi. (5.9) Then the renormalized interface matrix (3.0) can be recast in the form i 0 cosh/2 sinh/2 cos i sin sinh/2 cosh/2 HU. i sin cos (5.0) As explained above, the Hermitian component H generates a boost and the unitary component U generates a spatial rotation. By use of the explicit form Eq. (5.6), it is easy to verify that the Lorentz transformation associated with the interface matrix is

6 480 J. Opt. Soc. Am. A/ Vol. 7, No. 8/ August 2000 J. J. Monzón and L. L. Sánchez-Soto 0 0 x 2 x 3. Taking into account that in our case the invariant interval is lightlike, we have a stronger conserved 0 0 LR quantity, namely, x 0 2 x 2 x 2 2 x where (5.) 0 0 cos 2 sin 0 0 sin 2 cos 2, / 0 2 cosh, 0 2 / tanh, (5.2) 2 / 0 which establishes the desired identification of the parameters of the matrix i 0 with the corresponding ones for the Lorentz transformation. Furthermore, from the basic field variables (), (5.3) e e() e for both 0 and media, we can construct the associated coherency matrix 24 defined as e E e e () 2 e () e () * e () *e () e () 2, (5.4) which transforms as i 0 E i 0, (5.5) i.e., the same law as the space time coordinates X in Eq. (5.5). In consequence, we can obtain the corresponding space time coordinates for the field variables as tr(e )/2, that is, 3 x0 x x 2 x e() 2 e () 2 /2 Ree () *e () Ime () *e () (5.6) e () 2 e () /2, 2 for both 0 and media. The temporal coordinate is the semi-sum of the fluxes, while the fourth one is the semidifference of the fluxes. The interval remains invariant, x 0 2 x 2 x 2 2 x 3 2 x 0 2 x 2 x 2 2 x 3 2 0, (5.7) and it is lightlike. From the explicit form of the equivalent Lorentz transformation in Eq. (5.), it is clear that the boost acts only on the coordinates x 0 and x, while the rotation acts only on the coordinates x 2 and x 3, and therefore their actions do not mix these pairs of coordinates. In other words, the matrices L and R operate in separated blocks, and in consequence we have x 0 2 x 2 x 0 2 x 2, x 2 2 x 3 2 x 2 2 x 3 2, (5.8) which corresponds to the well-known invariants for a boost in the direction x and a rotation in the plane x 0 2 x 2 x 2 2 x 3 2. (5.9) If we put this equation in terms of field variables, we have e 0 () 2 e 0 () 2 /2 2 Ree 0 () *e 0 () 2 e () 2 e () /2 2 Ree () *e () 2 e 0 () 2 e 0 () /2 2 Ime 0 () *e 0 () 2 e () 2 e () 2 /2 2 Ime () *e () 2. (5.20) When both media are transparent and no total reflection occurs, the Fresnel coefficients are real numbers and 0; i.e., no rotation appears. In such a case the action of the interface reduces to a pure boost along the axis x, and, since the coordinate x 3 is unchanged, the energy flux is conserved across the interface. Note that in this case the boost along the axis x takes a form similar to the hyperbolic rotation [Eq. (4.2)], but with two important differences: First, the boost acts on the coordinates e () 2 e () 2 /2, Ree () *e () and not on e (), e () ; and second, the rapidity of the boost is and not /2. 6. CONCLUSIONS We have formulated the boundary conditions for a planar interface in a matrix form, showing that the renormalized interface matrix always appears as a squeezing matrix postmultiplied and premultiplied by a rotation of angle /4 and its inverse. Moreover, we have demonstrated that all the properties for s-polarized fields can be obtained from those of p-polarized fields by the action of a squeezing matrix, whose squeezing factor is precisely the refractive index of the medium and is independent of the angle of incidence. We have shown that the set of interface matrices is an Abelian subgroup of SL(2, C). By using the local isomorphism with the (3 )-dimensional restricted Lorentz group SO(3, ), we have studied the equivalent Lorentz transformation that describes any interface. For transparent media and when no total reflection occurs, this Lorentz transformation is just a pure one-dimensional boost in SO(, ) and the invariance of the interval gives the flux conservation, as expected from physical considerations. REFERENCES. J. M. Vigoureux, Polynomial formulation of reflection and transmission by stratified planar structures, J. Opt. Soc. Am. A 8, (99). 2. J. M. Vigoureux, Use of Einstein s addition law in studies of reflection by stratified planar structures, J. Opt. Soc. Am. A 9, (992). 3. J. M. Vigoureux, The reflection of light by planar stratified media: the grupoid of amplitudes and a phase Thomas precession, J. Phys. A 26, (993). 4. J. M. Vigoureux and Ph. Grossel, A relativistic-like presen-

7 J. J. Monzón and L. L. Sánchez-Soto Vol. 7, No. 8/August 2000/J. Opt. Soc. Am. A 48 tation of optics in stratified planar media, Am. J. Phys. 6, (993). 5. J. J. Monzón and L. L. Sánchez-Soto, Fully relativisticlike formulation of multilayer optics, J. Opt. Soc. Am. A 6, (999). 6. J. J. Monzón and L. L. Sánchez-Soto, Lossless multilayers and Lorentz transformations: more than an analogy, Opt. Commun. 62, 6 (999). 7. J. J. Monzón and L. L. Sánchez-Soto, Origin of the Thomas rotation that arises in lossless multilayers, J. Opt. Soc. Am. A 6, (999). 8. R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 987), Sec J. J. Monzón and L. L. Sánchez-Soto, On the concept of absorption for a Fabry Perot interferometer, Am. J. Phys. 64, (996). 0. R. M. A. Azzam, Transformation of Fresnel s interface reflection and transmission coefficients between normal and oblique incidence, J. Opt. Soc. Am. 69, (979); Direct relation between Fresnel s interface reflection coefficients for the parallel and perpendicular polarizations, J. Opt. Soc. Am. 69, (979) and references therein.. J. J. Monzón and L. L. Sánchez-Soto, Algebraic structure of Fresnel reflection and transmission coefficients at an interface, Optik (Stuttgart) 0, (999). 2. J. M. Stone, Radiation and Optics (McGraw-Hill, New York, 963), Chap M. Born and E. Wolf, Principles of Optics (Pergamon, Oxford, UK, 980), Sec A. Perelomov, Generalized Coherent States and Their Applications (Springer-Verlag, Berlin, 986). 5. P. Hariharan, Optical Interferometry (Academic, Sydney, 985), Appendix A4. 6. J. Lekner, Theory of Reflection (Kluwer Academic, Dordrecht, The Netherlands, 987), Chap M. A. Khashan, A Fresnel formula for dielectric multilayer mirrors, Optik (Stuttgart) 54, (979). 8. S. W. Corzine, R. H. Yan, and L. A. Coldren, A tanhsubstitution technique for the analysis of abrupt and graded interface multilayer dielectric stacks, IEEE J. Quantum Electron. 27, (99). 9. I. V. Lindell and A. H. Sihvola, The quotient function and its applications, Am. J. Phys. 66, (998). 20. A. Mischenko and A. Fomenko, A Course of Differential Geometry and Topology (Mir, Moscow, 988), Sec A. O. Barut and R. Ra czka, Theory of Group Representations and Applications (PWN-Polish Scientific, Warszaw, 977), Chap A. O. Barut, Electrodynamics and Classical Theory of Fields and Particles (Dover, New York, 980), Chap J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, New York, 998). 24. M. Born and E. Wolf, Principles of Optics (Pergamon, Oxford, UK, 980), Sec. 0.8.