ABCD matrices as similarity transformations of Wigner matrices and periodic systems in optics

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1 S. Başkal and Y. S. Kim Vol. 6, No. 9/September 009/ J. Opt. Soc. Am. A 049 ABCD matrices as similarity transformations of Wigner matrices and periodic systems in optics S. Başkal 1, * and Y. S. Kim 1 Department of Physics, Middle East Technical University, Ankara, Turkey Department of Physics, University of Maryland, College Park, Maryland 074, USA *Corresponding author: baskal@newton.physics.metu.edu.tr Received April 14, 009; accepted July 15, 009; posted July 30, 009 (Doc. ID ); published August 4, 009 It is shown that every ray transfer matrix, often called the ABCD matrix, can be written as a similarity transformation of one of the Wigner matrices that dictate the internal space time symmetries of relativistic particles, while the transformation matrix is a rotation preceded by a squeeze. The implementation of this mathematical procedure is described, and how it facilitates the calculations for scattering processes in periodic systems is explained. Multilayer optics and resonators such as laser cavities are discussed in detail. For both cases, the one-cycle transfer matrix is written as a similarity transformation of one of the Wigner matrices, rendering the computation of the ABCD matrix for an arbitrary number of cycles tractable. 009 Optical Society of America OCIS codes: , , INTRODUCTION The ABCD matrices have proved to be very useful in depicting quite a variety of optical phenomena such as ray tracing in geometrical optics, propagation of paraxial waves with Gaussian apertures in wave optics, in resonators such as lasers [1], or even in dealing with two-port networks (PN) in the telephone industry []. Perhaps the most prominent advantage of using matrices is to calculate the overall ABCD matrix of the system composed of different cascaded optical or electrical elements just by matrix multiplications. If the system has some conserved quantities or stays in a stable state, for instance when the refractive indices at the input and output planes are the same, or when the laser cavity is stable [1], then such conditions render the determinant of the ABCD matrix as one. In the case of PN, the transfer function remains unaltered if the points of excitation and response are interchanged [3]. Apart from those that are purely real as in ray optics, in general the elements of the ABCD matrices are complex, although there are systems that can particularly be arranged to yield real matrices, or the constituent complex matrices can be made real by a similarity transformation. The two-by-two complex matrices with unit determinant form the group SL,C, which is the covering group of SO 3,1, whose four-by-four matrix representations correspond to Lorentz transformations. It is well established by now that SL,C, or its subgroups such as SU 1,1 and Sp, provides the underlying mathematics of classical and quantum optics. Specifically, they play a pivotal role in polarization optics [4], interferometers [5,6], lens optics [7,8], multilayer optics [9], and laser cavities [10], as well as squeezed states of light [11,1]. On the basis of the vast amount of literature accumulated on the subject, it is easy to observe now that a common mathematical formulation can be established between the physics of concrete setups composed of lenses or lossless multilayers whose system matrices belong to SL,C or its subgroups that are isomorphic to SO,1 and special relativity, despite the fact that there does not seem to be an apparent relation between those distinct subjects at first glance [13,14]. Successful treatment of stratified media in the context of periodic systems can be traced back to the work of Abelès [15], and since then various matrix or group theoretical methods have been exploited by many authors [16 0]. Mathematical induction can also be a conceivable approach to a periodic system, it is possible to assume first that the ABCD matrix is known for n cycles, and then compute the system for n+1 cycles. Recently, one of us has studied multilayer optics based on exploitation of the properties of the Lorentz group, the cycle had to start from the midpoint of one of the layers [9]. Earlier again, we had to deal with a similar inconvenience of starting the beam cycle from the midpoint between the two mirrors while calculating the ABCD matrix for laser cavities using the method of Wigner s little group [10]. In this paper, all these restrictions and inconveniences are eliminated by starting the cycles from arbitrary points. For this purpose, we shall show that the real ABCD matrix can be cast into one of the Wigner matrices by a similarity transformation, and that the similarity transformation is a rotation followed by a squeeze. This mathematical result proves to be very useful in calculating the overall matrix for periodic systems raising the ABCD matrix of one cycle to its nth power is necessary. We shall study laser cavities and multilayer optics in this context. In both cases, the multicycle system will be reduced to one cycle. In Section, we construct a similarity transformation that will bring the ABCD matrix into the form of one of /09/ /$ Optical Society of America

2 050 J. Opt. Soc. Am. A/ Vol. 6, No. 9/ September 009 S. Başkal and Y. S. Kim the four Wigner matrices. It is shown that the transformation matrix is a rotation matrix followed by a squeeze matrix. It is also shown that these four different Wigner matrices can be combined into one analytical expression. The present formalism is applicable to one-dimensional periodic systems. In Section 3, we study laser cavities in detail. In Section 4, we show how useful Wigner matrices are in computing the scattering matrix for multilayer optics.. SIMILARITY TRANSFORMATIONS The ray transfer matrix, usually known as the ABCD matrix, is a two-by-two matrix M = A B 1 C D whose determinant is AD BC=1 for lossless systems. Since its elements are real, it has three independent parameters. This unimodular matrix can be decomposed into three one-parameter matrices as M = R 1 B R, which is known as the Bargmann decomposition of the Sp matrices [1]. A more manageable form can be obtained when this decomposition is rotated with a rotation matrix R as which becomes M R = R MR, M R = R R 1 B R R, or with the addition of the rotation parameters is expressed in the form M R = R B R, = , = 1. 6 Here the rotation matrices R or R are of the form cos / sin / R =, 7 sin / cos / and B is the two-by-two matrix representation of boosts along the x direction, cosh / B = sinh / 8 sinh / cosh /. Wigner s little groups are the subgroups of the Poincaré group, whose transformations leave the four-momentum of a relativistic particle invariant. The two-by-two representation of this group dictates the internal space time symmetries of relativistic particles [13,]. Various matrix identities from little groups have been exploited within the context of optics, specially motivated by the desire to find a common mathematical ground between the two seemingly unrelated branches of physics; namely, special relativity and optics. This particular identity [8], R B R = S W S, imported from little groups is by no means an exception, S = / e / e is the two-by-two matrix representation of boosts along the z direction, and W is collectively called the Wigner matrices constituting the set W = R *,B,N ±, 9 11 with N = 1 N 0 1, + = Now, the rotated ABCD matrix can be expressed as M R = S W S, and in view of Eq. (3), the ray matrix becomes Therefore, we have M = R S W S R. M = Z, W Z 1,, the transformation matrix is a product of a rotation and squeeze in the following order: Z, = R S. 16 If we are to deal with periodic systems, ultimately we shall have the burden of taking the n th power of the ABCD matrix of one complete cycle to obtain the overall system matrix M n. Fortunately, we can circumvent this difficulty by observing that the Wigner matrices have the following desirable property: Finally, we have W n = R n *,B n,n ± n. M n = Z, W n Z 1, In order to give the relations between the Bargmann parameters, which are actually related to the physical parameters of the system under consideration, explicit forms of Eq. (9) are needed. Let us denote the left-hand side of this identity as D L, which after matrix multiplications explicitly reads as cosh cos cosh sin sinh D L, =. cosh sin sinh cosh cos Thus in view of Eq. (5), we have M R = D L, This matrix has two independent parameters, and the diagonal elements are the same. Now, there are three cases to be distinguished:

3 S. Başkal and Y. S. Kim Vol. 6, No. 9/September 009/ J. Opt. Soc. Am. A 051 (i) If the diagonal elements are smaller than one, then cosh sin sinh and the off-diagonal elements have opposite signs. So we should use R * as the Wigner matrix; thus the right-hand side of Eq. (9) becomes cos * e sin * e. 1 sin * cos * The relation between the Bargmann and the little group parameters are cos * = cosh cos, sin + tanh e = sin tanh. (ii) If the diagonal elements are greater than one, then cosh sin sinh and the off-diagonal elements have the same sign. So we should use B as the Wigner matrix; thus the right-hand side of Eq. (9) is of the form cosh e sinh e, 3 sinh cosh with the relation between the parameters as cosh = cosh cos, sin + tanh e = sin + tanh. 4 (iii) Either of the off-diagonal elements passes through zero while going from a positive number to a negative number, meanwhile cosh =1; then either of those elements vanishes, so one of N ± * suits as the Wigner matrix. Thus the right-hand side of the identity becomes 1 e or e 1 5 with e = sinh, or for the latter with e = sinh. The system matrix M can be expressed in terms of the little group matrix of Eq. (19) and R as M = R D L, R, 6 and the relations between the elements of this matrix and the Bargmann parameters are found as A = cos cosh + sin sinh, D = cos cosh sin sinh, B = sin cosh cos sinh, C = sin cosh cos sinh. 7 To end this section we emphasize that, although the procedure presented above is established within the framework of real unimodular ABCD matrices of ray optics, the results are general and can also be applied to different areas of physics whose system matrices can be converted to those of the form of ray optics. 3. LASER CAVITIES A laser cavity consists of two concave mirrors separated by distance s. The mirror matrix takes the form /R 1, R is the radius of the concave mirror. The separation matrix is 1 s If we start the cycle from one of the two mirrors one complete cycle consists of 1 0 /R 1 1 s /R 1 1 s If we start the cycle at a position d from the mirror, then one complete cycle becomes 1 d /R 1 1 s /R 1 1 s d Thus, half a cycle can be written as L 1 = 1 d /R 1 1 s d, and matrix multiplication yields L 1 = 1 d/r 1 d/r s d + d, 33 /R 1 s d /R keeping in mind that one complete cycle consists of two repeated applications of the half-cycle (see Fig. 1), which is L 1. However, the off-diagonal elements of L 1 have different dimensions, while the diagonal elements are dimensionless. In order to deal with this problem, we write this expression as a similarity transformation and so L = L = KL 1 K 1, 34 K = 1/ s 0 0 s, 35 1 d/r 1 d/r s d /s + d/s. 36 s/r 1 s d /R The elements of this matrix are now dimensionless. At this point we choose a unit system all distances are measured in terms of the mirror separation with s=1 and use notations a=d/ s and b=/ R. Then the normalized matrix for a half-cycle can be written as L = 1 ab 1 ab 1 a 37 b 1 b 1 a. We can now rotate L with R,

4 05 J. Opt. Soc. Am. A/ Vol. 6, No. 9/ September 009 S. Başkal and Y. S. Kim e = f + g f g. 4 Thus with the expressions of the little group parameters *, and the rotation angle given in terms of the physical parameters a and b of the cavity, and also with W w =R *, then half a cycle L starting from an arbitrary plane in the cavity can be written as in Eq. (14) or as in Eq. (15). Then, for n cycles the overall matrix becomes L n and explicitly reads as L n = Z, R 4n * Z 1,. 43 When the cycle starts from the midpoint in the cavity, a=1/ and so the half-cycle matrix becomes 1 b/ 1 b/4 L = 44 b 1 b/, and the angle becomes zero. We do not need the rotation matrices that have provided the arbitrariness of the starting plane for the beam in the cavity as in the case we have illustrated above. The relations between the little group parameters and the physical parameters of the cavity are calculated as cos * =1 b, e = 4 b 4b. 45 Fig. 1. Optical rays in a laser cavity. (a) Multiple cycles in a laser cavity are equivalent to the beam going through multiple lenses, for which one cavity cycle corresponds to the propagation of light through a subsystem of two lenses. (b) A laser cavity consisting of two concave mirrors with separation s. In our earlier paper [10], the cycle in the cavity had to start from s/, the midpoint between the two mirrors. Now, the cycle can start any, s d, including the first mirror. b ab tan = 1 b ab 1 a, 38 and the rotated matrix L R =R L a,b R becomes = L R 1 1 h f + g, 39 f + g h This is the result we obtained in our previous paper on laser cavities [10]. The point of this paper is that we can start the cycle from an arbitrary plane by introducing the parameter. When the cycle begins from one of the lenses, a=0, then the matrix for the half-cycle becomes L = b 1 b. In this case we need the rotation matrix R with tan = 1 b. b Now, the relation between the parameters are cos * =1 b, 47 h =1 b, f = 1+b ab 1 a, e = 1+b + b + 1 b 1/ 1+b b + 1 b 1/. 48 As before we can proceed with Eq. (43) to evaluate the overall matrix of the laser cavity with n cycles. g = b ab + 1 b 1+a a 1/. Now comparing Eq. (39) and Eq. (1) we have and cos * =1 b MULTILAYER OPTICS In multilayer optics, we have to consider two beams moving in opposite directions, one of which is the incident beam and the other is the reflected beam [3]. We can represent them as a two-component column matrix E +e ikz E e ikz, 49 the upper and lower components correspond to the incoming and reflected beams, respectively. For a given frequency, the wavenumber depends on the index of refraction. Thus, if the beam travels along the distance d,

5 S. Başkal and Y. S. Kim Vol. 6, No. 9/September 009/J. Opt. Soc. Am. A 053 the column matrix should be multiplied by the two-by-two matrix [3] P j = ei j / 0 0 e i /, 50 j j /=k j d with j denoting each different medium. If the beam propagates along the first medium and meets the boundary at the second medium, it will be partially reflected and partially transmitted. The boundary matrix is [3] B = cosh / sinh / 51 sinh / cosh /, with cosh / =1/t 1, sinh / = r 1 /t 1, 5 t 1 and r 1 are the transmission and reflection coefficients, respectively, and they satisfy r 1 +t 1 =1. The boundary matrix for the second-to-first medium is the inverse of the above matrix. Therefore, one complete cycle, starting from the second medium, consists of M = B P 1 B P. 53 This complex valued matrix M can be cast into a real matrix by a similarity transformation as C = 1 M = CMC 1, ei /4 ei /4 54 e i /4 e i /4. 55 This transforms the boundary matrix B of Eq. (51) to a squeeze matrix S of Eq. (10) and the phase shift matrices P j of Eq. (50) to rotation matrices R j of Eq. (7). Therefore, we have M = S R 1 S R. 56 In view of the Wigner little group identity of Eq. (9) we rewrite this as M 1 = R 1 B R 1 R, 57 the relations between the parameters are found to be cosh = cosh 1 cos 1 / tanh 1/, cos 1 = cos 1 /cosh 1 cos 1 / tanh 1/. Now the matrix M 1 can be simplified to with M = R 1 B R = and is apparently in the form of Eq. (), we can easily proceed as described in Section 1. The result of this section is illustrated in Fig.. Fig.. Multilayer consisting of two different refractive indices. In an earlier paper by Georgieva and Kim [9], the cycle started from the midpoint of the second medium. Now, it can start any including the boundary between the media. 5. CONCLUDING REMARKS We have shown that the ray transfer matrix can be cast into one of the four one-parameter Wigner matrices through a similarity transformation, and the similarity transformation is a rotation matrix followed by a squeeze matrix. The logarithmic property of these one-parameter Wigner matrices is transmitted to its similarity transformation through SWS 1 n = SW n S 1, 6 and facilitates the calculations for repeated applications by multiplying the parameter by an integer. We have carried out the procedure for laser cavities and multilayer systems. It can also find applications in a variety of other periodic systems, such as one-dimensional scattering problems in quantum mechanics [4,5], especially in condensed-matter physics. We can expand our scope to look into applications in space time symmetries of elementary particles in view of the fact that the Wigner matrices used in this paper are from Wigner s 1939 paper on symmetries in the Lorentz-covariant world [,6]. Although in this paper the polarization of light waves was not taken into account, they have transverse electric and magnetic components, and there are many interesting aspects when the propagation media are not invariant under rotations around the propagation axis. Here, we have dealt with the mathematical properties of the rotation and squeeze matrices that are contained in the decomposition of the ray transfer matrix. If these matrices are applied to the transverse directions, the rotation matrix generates rotations of the two-component polarization vector. The squeeze matrix on the other hand causes asymmetric dissipations of the amplitude. Of course, the combination of these two effects leads to interesting results [7]. REFERENCES 1. A. E. Siegman, Lasers (University Science Books, 1986).. S. Galli and T. Banwell, A novel approach to the modeling of the indoor power line channel Part II: Transfer function and its properties, IEEE Trans. Power Deliv. 0, (005). 3. L. Weinberg, Network Analysis and Synthesis (McGraw- Hill, 196). 4. D. Han, Y. S. Kim, and M. E. Noz, Wigner rotations and Iwasawa decompositions in polarization optics, Phys. Rev. E 60, (1999).

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