Portland State University PDXScholar
Electrical and Computer Engineering Faculty Publications and Presentations Electrical and Computer Engineering
10-1-1994 Generalized reverse theorems for multipass applications in matrix optics
Lee W. Casperson Portland State University
Anthony A. Tovar
Follow this and additional works at: https://pdxscholar.library.pdx.edu/ece_fac
Part of the Electrical and Computer Engineering Commons Let us know how access to this document benefits ou.y
Citation Details Anthony A. Tovar and Lee W. Casperson, "Generalized reverse theorems for multipass applications in matrix optics," J. Opt. Soc. Am. A 11, 2633-2642 (1994).
This Article is brought to you for free and open access. It has been accepted for inclusion in Electrical and Computer Engineering Faculty Publications and Presentations by an authorized administrator of PDXScholar. Please contact us if we can make this document more accessible: [email protected]. A. A. Tovar and L. W. Casperson Vol. 11, No. lO/October 1994/J. Opt. Soc. Am. A abl I 2633 ri- lu- I ng ng I Generalized reverse theorems for ~ I multipass applications in matrix optics er I
Anthony A. Tovar and Lee W. Casperson
I Department of Electrical Engineering, Portland State University, Portland, Oregon 97207-0751 ,~t I h Received January 3, 1994; revised manuscript received April 5, 1994; accepted April 14, 1994 :e~ I The reverse theorem' that governs backward propagation through optical systems represented by transfer matrices is examined for various matrix theories. We extend several reverse theorems to allow for optical systems represented by matrices that mayor may not be unimodular and that may be 2 X 2 or take on an ~ I augmented 3 X 3 form. As an example, we use the 3 X 3 form of the reverse theorem to study a laser with intracavity misaligned optics. It is shown that, by tilting one of the laser's mirrors, we can align the laser ~ I output arbitrarily, and the mirror tilt angle is calculated. I"~ I 1. INTRODUCTION conjugate mirrors. Similarly, multipass schemes may be used to decrease the transmission bandwidth of a Simple 2 X 2 transfer-matrix methods are commonly filter. Another category of laser applications involves used for studying a wide variety of problems in optics 1 remote sensing and control,19 which may require reverse I and in other areas of engineering and physics. Such a propagation through the optical system. Examples in matrix method exists, for example, to trace the position clude remote sensing of the atmosphere, nondestructive I and the slope of paraxial light rays through optical sys evaluation, adaptive optics, fiber-optic sensors, and mi tems that include lenses, mirrors, lenslike media, and croscopy. When the optical system is represented by a other optical components. Similarly, a 2 X 2 matrix given matrix, then the corresponding matrix that rep I method2,3 is used in Gaussian beam theory, where the resents backward propagation through the system is beam's width and phase front curvature are propagated of interest. This reverse matrix is also important if I through more general optical systems that may include there are established system symmetry requirements or complex lenslike media4,5 and Gaussian apertures.6 The if there is a need for experimental determination of a I Jones calculus matrix method for polarization calcula system matrix. tions may be used for propagating the two Cartesian Based on the examination of several types of optical I electric-field components of TEM plane waves through elements and systems, one is sometimes able to di optical systems that contain birefringent optical elements vine the form of the reverse matrix. However, such I and polarizers.7- 9 The voltage and current transfer a methodology ought not to be necessary, and system characteristics of an electric circuit may be obtained atic procedures are demonstrated to yield the reverse I by the use of two-port network matrices. There are matrix for most conventional matrix theories. The re also 2 X 2 matrix theories that govern light propaga verse matrix is often reported only for the special case of I tion through thin films,10,1l distributed-feedback wave unimodular matrix theories. However, many matrix the guides and lasers,12 and Gaussian light pulses through ories are unimodular only for some special case. For I chirping elements.13,14 Matrix methods may also be example, Jones calculus is unimodular only when the used in the study of quantum mechanics,15 magnetic optical system is lossless and when absolute phase is I circuits, mechanical systems with springs, and com ignored. A notable extension of the Jones calculus ac- puter graphics. 16 The use of a 2 X 2 matrix method ~,.counts, to first order, for polarization-dependent Fres has several advantages over the use of other analytical I nel reflection and refraction for nonnormal incidence. 17 methods and provides an orderly systems approach. I This extended Jones matrix method20 retains the simple Matrix methods encourage a standardization of notation 2 X 2 form but is inherently nonunimodular. Of course, and the use of diagrams. Highly sought-after analogies if the birefringent optical system contains polarizers, become transparent. then the system is represented as a zero-determinant I I Many optical systems contain some type of reflecting matrix, and there is no unique reverse matrix. The element that causes the light signal to propagate through characteristic matrix method for light propagation in i I all or part of an optical system backward. For ex stratified media11 is unimodular only when the media . I ample, standing-wave and bidirectional ring laser os and the bounda:ries are 10ssless.1O Similar restrictions cillators contain optical signals that propagate through are involved in transfer matrices used for distributed their intracavity optics in both directions. Reflective feedback structures12 and fiber ring resonators.21 In the elements may also be used in optical system design in case of Gaussian beams and paraxial rays the unimodu I which some desired effect is to be enhanced. This is larity condition occurs only when the medium at the out the case in multipass amplifier schemes for increased put has the same refractive properties as the medium I amplification 18 or for distortion correction with phase- at the input.22 The generalization of the nonunimodu-
I © I 0740-3232/941102633-10$06.00 1994 Optical Society of America 2634 J. Opt. Soc. Am. A/Vol. 11, No. 10/0ctober 1994 A. A. Tovar and L. W. Casperson lar reverse matrix concept to other matrix theories is tion, whether the signal is injected in the forward or the addressed in this paper. reverse direction, X 2 and Y2 represent the output and X For every 2 X 2 matrix method there is an aug and YI represent the input parameters. The A and D mented matrix that corresponds to a 3 X 3 matrix matrix elements are dimensionless. The units of Bare method. The form of the 3 X 3 matrix of interest here the units of X divided by the units of Y. The units of C is much simpler than the general 3 X 3 matrix. In are the multiplicative inverse of the units of B. both the paraxial ray matrix theory and the Gaussian The ABCD matrix in Eq. (1) may represent forward beam theory the 3 X 3 matrix method permits the de propagation through a single system element, or it may signer to trace paraxial light rays and Gaussian beams refer to the overall system matrix. To obtain this system through misaligned optical systems. I This 3 X 3 for- ~. matrix given the individual element matrices, one need malism may be applied, for example, to the design of " only multiply the system elements in reverse order. This 23 25 pulse compressors. - Similarly, 3 X 3 matrix meth- I can be easily seen from an example. The special case of a ods are necessary for studying electrical circuits that, system consisting of two cascaded elements is considered. contain intranetwork independent voltage and current~' If the first element is given by Eq. (1), then the second is sources. Another example exists in computer graphics, in which operations are performed on subfigures in a pic (2) ture by means of 2 X 2 matrix multiplication. However, to perform translation one needs an augmented 3 X 3 matrix description.16 The total system matrix may be obtained by the substi The purpose of this paper is to generalize the concept tution of Eq. (1) into Eq. (2): of a reverse matrix so that it applies to a variety of op tical systems that are represented by matrices that may be nonunimodular, 3 X 3, or both. A unified overview of 2 X 2 transfer-matrix theory is given in Section 2. In Section 3 the reverse matrix is derived for several opti Then, as we stated above, a system matrix is defined as cal matrix theories. The reverse matrix is also general the product of individual element matrices in reverse or ized for unimodular 3 X 3 matrix theories. In Section 4 der. It follows from induction that, if the system consists the importance of the results is highlighted by application of n elements, then the total system matrix is of the theory to a practical example. In particular, it is demonstrated that a Fabry-Perot laser's output may be (4) arbitrarily aligned, even though it contains tilted intra cavity optics. The alignment procedure simply involves In many matrix theories the determinant of each of the tilting one of the laser mirrors, and the mirror tilt angle system elements is unity, i.e., the matrix for the element is calculated. is unimodular. Since the determinant of a product is the product of the determinants, it follows that, for such a ma 2. GENERAL PROPERTIES OF2 X 2 trix theory, any system matrix will be unimodular. For TRANSFER-MATRIX METHODS nonunimodular matrix theories the determinant usually carries important information. There are several properties common to 2 X 2 transfer It is sometimes of interest that certain properties of a matrix methods, and certain classes of matrices arise in given system be conserved. As an example, it may be several of the theories. In this section we give a unified desired to examine systems for which X 2 *X 2 + Y 2 *Y2 = overview of these general properties to exploit analo Xl *Xl + YI *YI . The class of matrices for which this is gies between matrix theories. These analogies suggest true is called unitary. It can be shown that TT* = T-I several novel elements, including a cross-wiring circuit for a unitary matrix, where the T subscript represents element, intranetwork independent voltage and current transposition (i.e., interchange of the Band C elements) sources in electric circuits, and multiple-input-single and the asterisk represents complex conjugation. Uni output birefringent and distributed-feedback optical tary matrices have the properties that the complex mag systems. In addition to matrix properties, there are nitude of their determinants is unity and that the product several universal matrix operations that may be used . of unitary matrices is a unitary matrix. in system synthesis. These operations include raising a matrix to an arbitrary power, backward propagation A. Specific Matrices through a matrix-represented component, and matrix In the study of matrix optics one finds that there are factorization. For each 2 X 2 matrix theory there is also individual matrices that commonly arise in several ma an associated bilinear transformation. Each of these trix theories. To reinforce analogies between systems, properties is discussed in this section. we find it useful to emphasize these matrices. The propagation formulas for any given 2 X 2 transfer The first matrix to be considered is the identity matrix, matrix theory can be written in the form (5) (1) which changes Ileither the X component nor the Y com where X and Yare the two dependent parameters that ponent of the signal. The identity matrix can be used change through the existence of the system. In our nota- as a continuity condition. However, it is also sometimes III A, A. Tovar and L, W. Casperson VQl. 11, No. lO/October 1994/J. Opt. Soc. Am. A 2635 le highly desirable to design an overall system that does not where the potentially complex X and 8 are defined by the ·1 change the input signal. In this case the system matrix relationships A = D == cos 8 and X == (-B/C)lJ2. In many D has this identity matrix form. This design methodology 2 X 2 transfer-matrix theories there are also individual .,'e is of interest in a Gaussian beam optical system when, for elements that are represented by a matrix of this form. oJ example, a flat untilted mirror is optimum but practical In particular, in the Gaussian beam theory, expression (9) problems force one to position the mirror away from the is used to represent a complex lenslike medium.4,5Simi d end of the laser.22 In the Jones calculus matrix method larly, matrix (9) is used to represent a transmission line y for optical polarization calculations one can imagine an in the electrical circuit theory. Thus it follows that a n optical system that, on account of its nature, distorts the symmetric Gaussian beam optical system can be syn d input polarization. In this case it may be desired to syn thesized with a single complex lenslike medium and a s thesize a system so that the overall system matrix is the symmetric electrical system can be synthesized with a a identity matrix. single transmission line. This matrix [expression (9)] is l. The unimodular matrix common in matrix theories derived from a second-order s differential equation with constant coefficients. If the co (6) efficients are nonconstant, then alternative solutions to the differential equations are of interest.26,27 :) [~ n A potentially important special case of expression (9) changes the X component without changing the Y occurs when X = -1 and 8 is real: signal component. If X is real and positive here, then expression (6) is the matrix representation of a uniform COS 8 medium in Gaussian beam matrix theory.2 .In electrical -sin 8 J. (10) [ sin 8 cos 8 circuit matrix theory, X represents a series impedance. ) A dual of expression (6) is the unimodular matrix This unitary matrix represents a pure rotation about the origin in the XY plane. Such rotation matrices are (7) prevalent in Jones calculus and many other calculations. Ga Other operations in the XY plane include the nonuni which changes the Y signal component without changing modular matrices for mirror reflection across the X axis, the X signal component. It is used to represent a thin lens and/or Gaussian aperture in the Gaussian beam the (11) ory and a shunt impedance in the electrical theory. [~ ~ll Because of the commonness and the importance of matrices (5)-(7), they are viable candidates as ma and mirror reflection across the Y axis, trix primitives from which an arbitrary system may be 22 synthesized. In the paraxial ray and Gaussian beam (12) theories many simple systems are made up of flat mir [~1 n- rors [expression (5)], uniform media [expression (6)], and lenses [expression (7)]. Similarly, in the electric circuit The matrix for mirror reflection across both X and Y theory many two-port systems are composed only of series signal component axes is also a special case of expres [expression (6)] and shunt [expression (7)] impedances. sion (8). It is written as Scaling can be obtained by the not necessarily unimodu lar matrix -1 0 ] (13) [ o -1 '
XX 0 (8) [ o Xy J' and it occurs in several unimodular matrix theories. In both the paraxial ray and Gaussian beam theories, where Xx and xy are the scale factors for the X and Y expression (13) represents a retroreflecting mirror. If signal components, respectively. In the Gaussian beam the analogy with paraxial ray theory is exploited, then theory, expression (8) with Xx = 1 represents a dielec rf~ it is suggested that this matrix can be used to represent tric boundary.6 In the unimodular limit, Xx = Xy-1, , cross wiring in the electrical circuit theory. Similarly, in and a matrix of this form is used to represent an ideal ( the paraxial ray theory, matrix (11) is used to represent transformer in the electrical theory and an anisotropic a phase conjugate mirror. 7 medium in the Jones calculus matrix method. Given the possibility of rotations, mirror reflections, Symmetry is often considered a desirable property in a and scaling in the XY plane, the next natural operation system. For our purposes a symmetric system is one that ,. that arises is translation. However, simple translation causes a signal injected backward to undergo the same in the XY plane cannot be performed with 2 X 2 matrix transformation as one injected in the forward direction. theories. An elegant way to account for translation is to As we will see below, for several types of unimodular augment the 2 X 2 matrix as a 3 X 3 matrix of the form systems the requirement of symmetry implies that A = D. A unimodular matrix in which the diagonal elements are equal (A = D) can be put in the form X2) [A B E](X1) (14) (~2· = ~ ~ ~ ~1' COS 8 X sin 8 ] (9) [ - X -1 sin 8 cos 8 ' In this case the matrix for a simple translation is 2636 J. Opt. Soc. Am. A/Vol. 11, No. 10/0ctober 1994 A. A. Tovar and L. W. Casperson ( Sylvester's theorem takes on the simpler form 1 0 XX] o 1 Xy , (15) [ I o 0 1 COS 8 X sin 8 JS [COS(S8) X sin(sO) ] [ - X-I sin 8 cos 8 = - X-I sin(s8) cos(sO) . I where Xx and Xy are the amounts of translation in the (18) X and Y axes, respectively. In the paraxial ray and I : Gaussian beam theories this translation matrix is Just as the matrix operation that corresponds to interpreted physically as optical element or system .,. Sylvester's theorem has the physical interpretation of misalignment.! Thus, with the 3 X 3 theory, a lens, for .: the cascade of s identical optical systems, other matrix I • example, is allowed to be displaced from the optic axis. / operations can also be interpreted. It can be seen from A 3 X 3 electric theory would permit ideal independent' Eq. (1) that a given system matrix yields the output I voltage and current sources distributed throughout th~ signal given an input signal. If we multiply both sides system. A 3 X 3 Jones calculus may include signal coIJl.!' of Eq. (1) by the inverse of the system matrix, it follows I bining and could account for multiple system inputs. that the matrix inverse can be interpreted as the ma An important property of matrix form (14) is that the trix that yields the input given the output. A reverse I E and F elements do not affect the A - D elements in matrix may be defined as a matrix that yields the input matrix multiplication. Furthermore, the determinant of going in the reverse direction given the output going in I the matrix is simply AD - BC, the same as that of the the reverse direction. Similarly, the inverse of a reverse corresponding 2 X 2 matrix. matrix yields the output going in the reverse direction I given the input going in the reverse direction. These B. System Synthesis matrix interpretations are summarized in Table 1. Design criteria for a given system can be realized as con In this way the reverse matrix is the appropriate ma I straints on the system matrix. These constraints may trix for propagating through a system backward. When often be written in terms of matrix operations. In this a system matrix is equal to its own reverse, the sys I subsection several matrix operations are identified that tem is (by our definition) symmetric. For our purposes permit the system matrix, based on these design criteria, this symmetry may be part of the given design criteria. I to be found. Once the system matrix is known, then it is However, different matrix theories may possess differ of interest to consider procedures to determine the optical ent reverse matrices. For paraxial light rays and Gauss components needed to fulfill these criteria, which is ac ian beams in first-order optical systems and for electrical complished by factorization of the system matrix into ma signals in two-port networks the reverse matrix is29 trix primitives. Each of these primitives represents an optical component available to the optical designer. In (19) this subsection only, an emphasis is placed on unimodu lar matrix theories. In addition to matrix multiplication of individual ma As we will show below, this is also the reverse matrix trices, there are several other meaningful operations that for other unimodular matrix theories in which the Y pa may be performed on individual and system matrices. A rameter is the derivative of the X parameter. If a given first step in the synthesis process may include the inter matrix is equal to its reverse matrix, then the system is pretation of these operations. The first operation con symmetric, and from Eq. (19) it follows that the condition sidered is the sth power of a unimodular matrix, which is for symmetry is A = D. given by the unimodular 2 X 2 special case of Sylvester's As opposed to these matrix theories, the reverse matrix theorem28 : for the unitary form of Jones calculus is!9
A B JS = _l_[A sin(s8) - sin[(s - 1)8] B sin(s8) ] [ C D sin 8 ~ C sin(s8) D sin(s8) - sin[(s - 1)8] , (16) where A+D TR ~ [~ ~ 1 (20) cos 8 == ---. (17) 2 Thus a lossless birefringent optical system is symmetric if Sylvester's theorem is valid not only for positive integer its Jones matrix elements have the property that B = C. powers of matrices but for negative integer powers and As part of the design criteria for a given periodic roots as well.28 However, this corresponds to a potential system, the input signal may be required to repeat after design criterion. Suppose that it is desired to synthesize propagating through s identical subsystems. Indeed, the a known system matrix as the cascade of s identical sub sinusoidal nature of Sylvester's theorem [Eq. (16)] sug systems. The design procedure amounts to taking the gests such a repetitive signal condition. In particular, if sth root (l/s power) of the system matrix and writing it in terms of some set of defined matrix primitives. (21) Because of the somewhat complicated form of Sylvester's theorem, it is useful to consider the special case ofEq. (16) when A = D == cos 8 and X == (_B/C)1I2. then the input signal is reproduced after propagating Here the system matrix is given by expression (9), and through s systems or through a single system s times. ~rson A. A. Tovar and L. W. Casperson Vol. 11, No. 10/0ctober 1994/J. Opt. Soc. Am. A 2637
Table 1. Physical Interpretation of Here y is allowed to be complex. It is interesting to note Some Simple Matrix Operationsa from Eq. (26) that, as opposed to scalars, matrices may System Throughput have an infinite number of roots. Not all the identity lJ matrix factorizations above are given in terms of matrix T Output given input primitives. However, in the last case factorization (24) (18) T-1 Input given output and/or (25) may be inserted into Eq. (26) to accomplish I TR Inputl going backward given outputI going backward this. These results lead to interesting optical systems 3 to TR- 1 Outputl going backward given inputl going backward such as the cat's-eye reflector,1 which, by design, has 1 of the same system matrix as that for a retroreflector ltrix aThe R subscript represents the reverse operation. Input and output [expression (13)]. rom designations are independent of propagation direction. ~put C. Bilinear Transformation ides It may be seen from Eq. (16) that this occurs when 88 = For every 2 X 2 matrix theory there exists an associated ows 2k1T', where k is an integer. In terms of matrix elements, bilinear transformation. If a ratio parameter is defined rna the repetitive signal condition from Eq. (17) is as ~rse .put A+D -2- = cos{2k7T/8) , (22) Z =- X (27) ~ in y' ~rse where we make the restriction that ,;ion then from Eq. (1) the corresponding transformation for ese the ratio parameter is i· o ~ k ~ 8/2 (23)
AZ1 +B ;na- to avoid duplicating solutions. A graphic interpretation (28) , Z2 = CZ + D ,len of the result is given in Ref. 30. 1 :ys When the system matrix, based on design criteria, is This transformation is sometimes called the ABCD law. ses known, one must factor the system matrix in terms of In electrical theory Z is physically interpreted as an ~a. matrix primitives such as expressions (5)-(7). Each of impedance, and in Gaussian beam theory it is related :er these matrix primitives must represent a manufacturable to the width and phase-front radius of curvature of a .ss optical component. If the system matrix is unimodular, Gaussian beam. In Jones calculus it is interpreted as cal 22 then there exist two three-matrix factorizations the ellipse of polarization.32 The ratio parameter is in terpreted as a reflection coefficient in the distributed feed back and fiber ring resonator theories.31 [~ ~ J~ [~ (A -/)/C ][~ ~ J[~ (D -/)/C l 19) Every system has a characteristic ratio parameter Zoo (24) such that, if Zoo is input to the system, the same Zoo is output. Thus, if we constrain the output ratio parameter rix to be equal to the input ratio parameter, it follows from )a A B J .[ 1 0J[1 B J[ 1 0J [ D = (D - B (A - B Eq. (28) and the unimodularity condition AD - BC = 1 'en C 1)/ 1 0 1 1)/ 1 that : is (25) :on in terms of only matrix primitives of the form of A BZ. A D ± { 1 - ( A D (29) I + ~ ~ ~ I. expressions (6) and (7). As we mentioned above, if the rr :'IX system matrix has the property A = D, then a single = exp{±i8) , (30) matrix of the form of expression (9) may be used if it can be realized as a single component. Of course, the factor where ization in Eq. (24) is valid only when C is nonzero, and similarly the factorization in Eq. (25) is valid only when cos 8 =: A + D (31) B is nonzero. If the system matrix elements Band Care 2 both zero or if the system matrix is nonunimodular, then It may be noted that Eq. (31) is identical to Eq. (17). 0) additional factorizations are necessary.22 In the Jones ,~ s If a signal propagates through a system many times or calculus method other matrix primitives are of interest. I through many systems, then Z may approach the value :if As we suggested above, there exist design criteria that, -y Zoo. If Z approaches Zoo, the system is said to be stable '/. demand that the output signal be identical to the input' with respect to Z, and this occurs when the complex Jc signal when the system is periodic. Thus it is useful to magnitude32 ~r consider other factorizations of the identity matrix:" I le IA+ BZool > 1, (32) !~ ,if where IA + BZoo I = 1 represents metastability and IA + [~ ~ J~ ([ -~/l ~ J[~ ~ J)' BZool < 1 represents instability. l) ~ (±[~ nr ~ (±[~ ~1 Jr 3. DERIVATION OF GENERALIZED REVERSE THEOREMS g ~ [y~, ~ ~ _~-l ~ (26) The purpose of this section is to demonstrate a systematic r· r -[ r procedure to obtain the reverse matrix for a given matrix 2638 J. Opt. Soc. Am. A/Vol. 11, No. 10/0ctober 1994 A. A. Tovar and L. W. Casperson theory. The process does not require the usual inspec The matrix TR T represents forward propagation through tion of individual system elements.7,19 The secondary a system followed by reverse propagation through that purpose of this section is to use these systematic proce same system. In standing-wave laser theory this matrix dures to obtain new reverse matrices for several matrix often corresponds to a round trip. Now that some general theories. In particular, the reverse m~trix for 3 X 3 elec properties of reverse matrices have been described, the tric circuit matrices is found here. These results may be specific reverse matrices for several matrix theories will used in studies of reverse propagation through electric cir be discussed. cuits with intranetwork independent current and voltage sources. In the Gaussian beam theory the 2 X 2 re A. Paraxial Ray Matrices verse matrices governing the beam's spot size and phase ,. The purpose of this subsection is to derive the form of front curvature are known. 33 The reverse matrix for i the reverse matrix that applies to paraxial light rays. At nonunimodular 3 X 3 paraxial ray matrices is also found'i some position 'T along the optic axis the X signal vector Previously, only the nonunimodular 2 X 2 form and the component represents the position of a light ray, and the unimodular 3 X 3 form34 had been discussed. The r~ Y component' represents the slope of the light ray. For verse matrix for the Jones calculus matrix method has this derivation it is important to note that Y = dX / d 'T. been reported for unimodular 2 X 2 matrices.7,19 Here The reverse matrix governs the propagation of the the nonunimodular 3 X 3 reverse matrix is derived. This signal going backward and starting from the output. generalization may account for multiple-input optical Therefore the definition of TR from Table 1 is systems with loss or gain. The nonunimodular 3 X 3 reverse matrix is also found for the matrix theories gov TR == Input I going backward given Output I going backward. erning distributed-feedback lasers and waveguides and (36) fiber ring resonators. The reverse theorems do not apply to optical systems As above, the propagation of X and dX / d 'T is governed represented by a zero-determinant matrix. In these sys by the transformation matrix tems there may be several different inputs that yield the same output. Though there are different reverse matri ces for different matrix methods, there are certain uni (37) versal properties that these reverse matrices all share. [ :X] =T[:X] , For example, the reverse of a product of matrices is the a'T 2 a'T 1 product of the reverse of each of these matrices in reverse where T is the system matrix. Multiplying both sides order. In equation form this may be written as of Eq. (37) by T-1 yields the formula for the input ray (33) position and slope given the output ray position and slope:
The justification of Eq. (33) is suggested by Fig. 1. In this figure the matrix T, governing propagation from the a~] = T-l[a~] (38) left-hand side to the right-hand side, is defined as a prod [ a'T 1 a'T 2 uct of submatrices: T4T3T2T1. Similarly, new matrices going from the right-hand side to the left-hand side may The direction of the signal is + 'T. If the signal is propa be defined. However, it is evident from the figure that gating in the reverse direction, then 'T is replaced with these matrices correspond to the reverse matrices TR, T4R , -.'T. Thus Eq. (38) may be rewritten as T3R, T2R , and T 1R . It follows that, to obtain TR, one must multiply the reverse submatrices in reverse order. This conclusion is independent of whether the matrix is 2 X 2 or of higher order and is inqependent of the matrix theory being studied. Similarly, there is no assumption about the determinant except that it is nonzero to ensure the existence of the inverse. For Gaussian beam theory this T property of the unimodular 2 X 2 reverse matrix was dis cussed previously. 33 As a special case of this theorem, suppose that each of the submatrices TI-Ts is identical; then it follows that
(34)
This is the intuitive result that propagation through s EO identical systems backward is the same as propagation backward through a single system s times. Though it was assumed that the exponent s in Eq. (34) is a positive integer, it is valid for any exponent s. Another special case of Eq. (33) is Fig. 1. Schematic demonstration that the reverse of a product of matrices is the product of reverse matrices in reverse order in (35) dependent of matrix theory, i.e., (T4T3T2Tl)R = TIRT2RT3R T4R. A. A. Tovar and L. W. Casperson I Vol. 11, No. 10/0ctober 1994/J. Opt. Soc. Am. A 2639 Multiplying both sides by the matrix in expression (11) and noting that None of the properties unique to paraxial ray matrices I was used in the derivation. Besides the postulating of the existence of an inverse, the only assumption was that I (40) the Y component of signal vector was the 7 derivative of [~ ~1][~ ~1 J~ [b ~ J the X component of the signal vector. Thus this result I reduce Eq. (39) to is not unique to ray matrices. These reverse matrices also apply to the Gaussian beam matrix formalism, the I electric circuit matrix theory, and other Wronskian-type matrix theories, where Y = dX / d 7. X [1 0 1 [1 0 X [ ~ J == 0 -1 JT- 0 -1 J[ ~ J (41) I a(-7) 1 'a(-7) 2 B. Jones Calculus The reverse matrix has been found for the ray matrix I However from the definition of the reverse matrix formalism, the Gaussian beam matrix formalism, and the [Eq. (36)], it follows that the reverse matrix is two-port electric circuit theory. The reverse matrix takes I on the same form for each. However, for the Jones polar ization calculus the reverse matrix is different and must TR = 01 -10 JT _1[10 -10 J. (42) I [ be calculated separately. Rather than derive the 2 X'2 and then the 3 X 3 reverse Since (T-1)2 = (T2)-I, it can be readily seen from Eq. (40) I matrix, we derive the more general 3 X 3 case. The that (TR)2 = (T2)R' By induction, a general property of Jones matrix is allowed to have the form reverse matrices [Eq. (34)] follows. Similarly, Eq. (33) I can be seen. Thus this specific reverse matrix has the A B same property that was shown to be a general property T= C D EJF . [ (45) I of reverse matrices. 001 The specific form of T-1 is well known, and Eq. (42) I can be reduced to The 2 X 2 reverse matrix becomes a simplified special case, where E = F = O. The transformation of the aug 1 mented Jones vectors is I 1 [1 o D - B 0 TR = AD -BC 0 -1 J[ -C A J[ 0-1 J I Ax eXP(i
Table 2. Reverse Matrices for Several Matrix Theories c Transfer Matrix Theory 2 X 2 Reverse Matrix 3 X 3 Reverse Matrix Paraxial ray matrices B Gaussian beam matrices 1 [D BF -DE] Electric circuit matrices AD -BC.C ~] A -(CE - AF) Miscellaneous Wronskian matrices AD ~BC [~ 0 AD-BC
Jones calculus -B 1 [ D BF-DEJ AD - BC -C; -:r AD~BC[~C A CE-AF 0 AD -BC
) Distributed-feedback matrices -B 1 [ CE -AF] 'IA Fiber ring resonator matrices AD -BC LC ;] D BF-DE AD~BC[-~ 0 AD -BC
~i optical activity, however, because, though they have the Multiplying both sides of Eq. (54) by the appropriate ma I same forward matrix, their reverse matrices may differ. trix yields Equation (50) is listed in Table 2. This result was previ ously reported only for unimodular 2 X 2 matrices.19 If the matrix is unimodular, then the corresponding optical system is symmetric if Band C are pure imaginary and if A = D*. For these Jones matrices the reverse matrix has been defined so that the transverse axes are unchanged. Thus it follows from Eq. (36) that Thus the reverse coordinate system is left handed. When this is undesirable, one may, for example, change the sign of Ay in Eq. (48), and the resulting reverse matrix would be given by the complex conjugate ofEq. (44). As a final note, it is interesting that the transpose may be written for nonzero-determinant matrices as The result of this calculation is included in Table 2. In the 2 X 2 special case, Eq. (56) reduces to 0 1 B 0 1 ] TT = (AD - BC) -1 0 J-I[AC D J-I[ -1 0 . (51) [ c TR = AD 1- BC [A-B -CJD . (57) t c. Distributed-Feedback Matrices t As a further example of the methodology for finding '\ the reverse matrix, the reverse matrix is found for the If AD - BC = 1, then the system represented by the i distributed-feedback matrix theory. Here the electric matrix T is symmetric if B = -C. t: field is separated into rightward and leftward waves that t: I form the signal vector. For generality, it is postulated 4. EXAMPLE:, MISALIGNED LASER e; that matrices take the more general form of Eq. (45). I Thus the signal vector is augmented with unity as Standing-wave laser oscillators consist of optical elements that are inevitably out of perfect alignment. These mis alignments, whether they are accidental or intentional, t (52) are crucial to the operation of the laser. Misalignment sensitivity may be examined with the 3 X 3 reverse ma trix of paraxial ray optics. 34 The purpose of this section is to demonstrate that a laser with misaligned intracav Proceeding as in Subsections 3.A and 3.B, we multiply both sides by T-I: ity optics may be effectively aligned by a tilt of one of the laser's mirrors. This problem is well suited to paraxial ray optics and the 3 X 3 reverse matrix. For simplicity only, it is assumed that the laser oscilla (53) tor is operated in its fundamental Gaussian mode. Thus the design condition is that the Gaussian beam emerge from the laser perpendicular to the output coupler. For The signal vector can again be written as a matrix multi this example it is assumed that the flat untilted output plied by the corresponding signal vector traveling in the coupler is the right-hand mirror and that the left-hand opposite direction: mirror is also flat but tilted at some angle (). It is further more assumed that the round-trip Gaussian beam matrix for the laser consists of purely real elements. This is the case when there are no significant apertures and the gain per wavelength of the amplifying medium is small. In fl A. A. Tovar and L. W. Casperson Vol. 11, No. 10/0ctober 1994/J. Opt. Soc. Am. A 2641
only these lossless optical systems the center of a Gauss 5. CONCLUSION ian beam travels along paraxial light ray trajectories.5 Backward propagation through an optical system occurs Thus one may use ray matrix techniques to trace the in a large class of multipass applications, where the op displacement and the slope of the center of the Gauss tical signal traverses the system at least twice. Such ian beam. The paraxial ray matrix for a mirror tilted at an angle applications include remote sensing, nondestructive evaluation, and synthesis of optical delay lines and laser is e oscillators. In the ubiquitous matrix theories considered 1 0 0] here, a 2 X 2 transfer matrix is used to represent for T mirror = 0 1 tan(20) . (58) ward propagation of light through the optical system, [001 and a corresponding reverse matrix is used to represent backward propagation. A general procedure has been The intracavity optics, which may be misaligned, are demonstrated to obtain reverse matrices. The reverse represented by an ABCDEF matrix. If the reference matrix has been found for several generalized theories plane is chosen at the output coupler, then the round-trip that make use of matrices that may be nonunimodular, matrix is possess an important 3 X 3 form, or both. A possible application of the results h~s been demonstrated with an Tround trip = TsystemT mirror (Tsystem)R (59) example. In particular, it was shown that, by tilting a mirror, one may align a laser even though it possesses A BE] [1 0 0] " = C D F 0 1 tan(28) misaligned intracavity optics. [ o 0 1 0 0 1 D B BF ':"'DE ] ACKNOWLEDGMENT X C A -(CE - AF) (60) This study was supported in part by the National Science [ o 0 1 Foundation under grant ECS-9014481. AD + BC 2AB B[2(AF - CE) + tan(20)]] = 2CD AD + BC' D[2(AF - CE) + tan(20)] , REFERENCES [ o 0 1 1. A. Gerrard and J. M. Burch, Introduction to Matrix Methods (61) in Optics (Wiley, New York, 1975). 2. H. Kogelnik, "Imaging of optical modes-resonators with where the unimodularity condition, valid for any laser internal lenses," Bell Syst. Tech. J. 44,455-494 (1965). 3. H. Kogelnik, "On the propagation of Gaussian beams of light oscillator,22 has been used. The oscillation condition through lenslike media including those with a loss or gain states that the displacement and the slope of the center variation," Appl. Opt. 4, 1562-1569 (1965). of the Gaussian beam repeat after some number of round 4. L. W. Casperson and A. Yariv, "The Gaussian mode in optical trips. Furthermore, the design condition requires that resonators with a radial gain profile," Appl. Phys. Lett. 12, the final position and slope be zero, which is satisfied 355-357 (1968). 5. L. W. Casperson, "Gaussian light beams in inhomogeneous when the E and F elements of the round-trip matrix are media," Appl. Opt. 12,2434-2441 (1973). identically zero. From Eq. (61) the E and F elements of 6. L. W. Casperson and S. Lunnam, "Gaussian modes in high the round-trip matrix are zero if B = D = o. However, loss laser resonators," Appl. Opt. 14,1193-1199 (1975). this would violate the unimodularity requirement of the 7. R. C. Jones, "A new calculus for the treatment of optical systems. I. Description and discussion of the calculus," system. The design condition would also be satisfied if J. Opt. Soc. Am. 31,488-493 (1941). 8. R. C. Jones, "A new calculus for the treatment of optical tan(20) = 2(CE - AF) . (62) systems. II. Proof of three general equivalence theorems," J. Opt. Soc. Am. 31,493-499 (1941). In this case the system is effectively aligned even though 9. R. C. Jones, "A new calculus for the treatment of opti the intracavity optics are individually misaligned. It cal systems. III. The Sohncke theory of optical activity," may be noted that a nonlaser reflective optical system J. Opt. Soc. Am. 31, 500-503 (1941). 10. M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, can also be aligned in this fashion. 4' New York, 1980), pp. 66-70. The fact that a complicated optical system with vari I, 11. R. A. Craig, "Method for analysis of the characteristic matrix ous misalignments can be aligned simply by the rotation in optical systems," J. Opt. Soc. Am. A 4, 1092 -1096 (1987). of the feedback mirror is reasonable when the axial ray 12. J. Hong, W. Huang, and T. Makino, "On the transfer ma trix method for distributed-feedback waveguide devices," is considered. If the optical system is aligned, then an J. Lightwave Technol. 10, 1860-1868 (1992). axial light ray (r, r' = 0) will remain an axial light ray at / 13. S. P. Dijaili, A. Dienes, and J. S. Smith, "ABCD matrices for the output. However, in a misaligned optical system an dispersive pulse propagation," IEEE J. Quantum Electron. axial light ray is displaced at the output, and the slope 26, 1158-1164 (1990). may be changed as well. If this output light ray strikes 14. J. L. A. Chilla and O. E. Martinez, "Spatial-temporal analy sis of the self-mode-locked Ti:sapphire laser," J. Opt. Soc. a mirror at normal incidence, then the light ray will re Am. B 10,638-643 (1993). trace its path back through the optical system. Thus the 15. J. L. Rosner, "The Smith chart and quantum mechanics," axial light ray will return to its initial form, though Am. J. Phys. 61, 310-316 (1993). the direction of the ray has changed. This means that 16. R. A. Plastock and G. Kalley, Schaum's Outline Series, The ory and Problems of Computer Graphics (McGraw-Hill, New the overall system is aligned. To ensure that the axial York, 1986). light ray strikes the mirror at normal incidence, we must 17. See, for example, B.-G. Kim and E. Garmire, "Comparison tilt the mirror, and the tilt angle is given by Eq. (62). between the matrix method and the coupled-wave method in 2642 J. Opt. Soc. Am. A/Vol. 11, No. 10/0ctober 1994 A. A. Tovar and L. W. Casperson sr- the analysis of Bragg reflector structures," J. Opt. Soc. Am. 25. O. E. Martinez, "Matrix formalism for dispersive laser cavi I A 9, 132-136 (1992). ties," IEEE J. Quantum Electron. 25, 296-300 (1989). 18. J. Krasinski, D. F. Heller, and Y. B. Band, "Multipass ampli 26. L. W. Casperson, "Beam propagation in tapered quadratic_ fiers using optical circulators," IEEE J. Quantum Electron. index waveguides: analytical solutions," J. Lightwave I 26, 950-958 (1990). TechnoL LT-3, 264-272 (1985). 19. N. Vansteenkiste, P. Vignolo, and A. Aspect, "Optical re 27. L. W. Casperson, "Beam propagation in periodic quadratic_ I versibility theorems for polarization: application to remote index waveguides," AppL Opt. 24,4395-4403 (1985). control of polarization," J. Opt. Soc. Am. A 10,2240-2245 28. P. I. Richards, Manual of Mathematical Physics (Pergamon (1993). New York, 1959). ' I 20. P. Yeh, "Extended Jones matrix method," J. Opt. Soc. Am. 29. A. E. Siegman, "Orthogonality properties of optical resonator 72, 507-513 (1982). eigenmodes," Opt. Commun. 31, 369-373 (1979). I 21. J. Capmany and M. A. Muriel, "A new transfer matrix for 30. L. W. Casperson and P. M. Scheinert, "Multipass reso malism for the analysis of fiber ring resonators: compound nators for annular gain lasers," Opt. Quantum Electron. 13 coupled structures for FDMA demultiplexing," J. Lightwave 193-199 (1981). ' I TechnoL 8, 1904-1919 (1990). 31. R. M. A. Azzam and N. M. Bashara, Ellipsometry and Po 22. L. W. Casperson, "Synthesis of Gaussian beam optical sys- larized Light (North-Holland, New York, 1987), pp. 84-91. tems," AppL Opt. 20, 2243-2249 (1981). jj 32. L. W. Casperson, "Mode stability of lasers and periodic opti I 23. O. E. Martinez, "Matrix formalism for pulse compressors," cal systems," IEEE J. Quantum Electron. QE-I0, 629-634 IEEE J. Quantum Electron. 24, 2530-2536 (1988). (1974). I 24. O. E. Martinez, P. Thiagarajan, M. C. Marconi, and 33. C. Fog, "Synthesis of optical systems," AppL Opt. 21 J. J. Rocca, "Correction to magnified expansion and com 1530-1531 (1982). ' pression of subpicosecond pulses from a frequency-doubled 34. V. Magni, "Multielement stable resonators containing a vari I Nd:YAG laser," IEEE J. Quantum Electron. 26, 1676-1679 able lens," J. Opt. Soc. Am. A 4, 1962-1969 (1987). (1990).