geometrical setting for the classification of multilayers

Monzon et al. Vol. 19, No. 5 /May 2002/J. Opt. Soc. Am. A 985

Geometrical setting for the classificationof multilayers

Juan J. Monzon, Teresa Yonte, and Luis L. Sanchez-Soto

Departamento de Optica, Facultad de Ciencias Fısicas, Universidad Complutense, 28040 Madrid, Spain

Jose F. Carinena

Departamento de Fısica Teorica, Facultad de Ciencias, Universidad de Zaragoza, 50009 Zaragoza, Spain

Received July 30, 2001; revised manuscript received October 3, 2001; accepted October 19, 2001

We elaborate on the consequences of the factorization of the transfer matrix of any lossless multilayer in termsof three basic matrices of simple interpretation. By considering the bilinear transformation that this transfermatrix induces in the complex plane, we introduce the concept of multilayer transfer function and study itsproperties in the unit disk. In this geometrical setting, our factorization translates into three actions that canbe viewed as the basic components for understanding the multilayer behavior. Additionally, we introduce asimple trace criterion that allows us to classify multilayers into three types with properties closely related toone (and only one) of these three basic matrices. We apply this approach to analyze some practical examplesthat are typical of these types of matrices. © 2002 Optical Society of America

OCIS codes: 000.3860, 120.5700, 120.7000, 230.4170.

1. INTRODUCTIONLayered media play an important role in many applica-tions in modern optics, especially in relation to optical fil-ters and the like. Therefore it is not surprising that thetopics covered in most of the textbooks on the subject usea mixture of design, manufacture, and applications, deal-ing only with the basic physics needed to carry out prac-tical computations.1

However, for a variety of reasons layered media havephysical relevance on their own.2,3 As any linear systemwith two input and two output channels, any multilayercan be described in terms of a 2 3 2 transfer matrix. Infact, it has been recently established that for a losslessmultilayer this transfer matrix is an element of the groupSU(1, 1).4,5 From this perspective, it is precisely the ab-stract composition law of SU(1, 1) that is ultimately re-sponsible for the interesting composition law of the reflec-tion and transmission coefficients.6,7

This purely algebraic result is certainly remarkable.But as soon as one realizes that SU(1, 1) is also the basicgroup of hyperbolic geometry,8 it is tempting to look for anexpanded geometrical interpretation of the multilayer ac-tion. Moreover, given the role played by geometricalideas in all branches of physics, particularly in specialrelativity, it is easy to see that this approach might pro-vide deeper insights into the behavior of a multilayer in awider unifying framework that can yield fruitful analo-gies with other physical phenomena.

Accordingly, we have proposed9 to view the action ofany lossless multilayer as a bilinear transformation onthe unit disk, obtained by stereographic projection of theunit hyperboloid of SU(1, 1). This kind of bilinear repre-sentation has been discussed in detail for the Poincaresphere in polarization optics10,11 and for Gaussian beam

0740-3232/2002/050985-07$15.00 ©

propagation12; it is also useful in laser mode locking andoptical pulse transmission.13

In spite of these achievements, the behavior of an arbi-trary lossless stack could still become cumbersome to in-terpret in physical terms. In fact, in practice it is usualto work directly with the numerical values of a matrix ob-tained from the experiment, which cannot be directly re-lated to the inner multilayer structure. To remedy thissituation, we have resorted recently14 to the Iwasawa de-composition, which provides a remarkable factorization ofthe matrix representing any multilayer (no matter howcomplicated it is) as the product of three matrices ofsimple interpretation.

At the geometrical level, such a decomposition trans-lates directly into the classification of three basic actionsin the unit disk that are studied in this paper and thatcan be considered as the building blocks for understand-ing multilayers. Moreover, we have also shown15 thatthe trace of the transfer matrix allows for classification ofmultilayers into three different types with properties veryclose to those appearing in the Iwasawa decomposition.

In this paper we go one step further and exploit thisnew classification to study several practical examplesthat are representative of each type. This shows thepower of the method and at the same time allows for adeeper understanding of layered media. As a direct ap-plication, we treat the relevant case of symmetric multi-layers, finding a precise criterion for zero-reflectance con-ditions. Nevertheless, we stress that the benefit of thisformulation does not lie in any inherent advantage interms of efficiency in solving problems in layered struc-tures. Rather, we expect that the formalism presentedhere can provide a general and unifying tool to analyzemultilayer performance in an elegant and concise way

2002 Optical Society of America

986 J. Opt. Soc. Am. A/Vol. 19, No. 5 /May 2002 Monzon et al.

that, additionally, has application to other fields of phys-ics and that seems to be more than merely interesting.

2. TRANSFER MATRIX FOR A LOSSLESSMULTILAYERWe first briefly summarize the essential ingredients ofmultilayer optics that we shall need for our purposes.10

The configuration is a stratified structure, illustrated inFig. 1, that consists of a stack of 1 ,..., j ,..., m plane-parallel lossless layers sandwiched between two semi-infinite ambient (a) and substrate (s) media, which weshall assume to be identical, since this is the common ex-perimental case. Hereafter all the media are assumed tobe lossless, linear, homogeneous, and isotropic.

We consider an incident, monochromatic, linearly po-larized, plane wave from the ambient that makes anangle u0 with the normal to the first interface and hasamplitude Ea

(1) . The electric field is either in the planeof incidence ( p polarization) or perpendicular to the planeof incidence (s polarization). We consider as well anotherplane wave of the same frequency and polarization, withamplitude Es

(2) , incident from the substrate at the sameangle u0 .16

As a result of multiple reflections at all the interfaces,we have a backward-traveling plane wave in the ambient,denoted Ea

(2) , and a forward-traveling plane wave in thesubstrate, denoted Es

(1) . If we take the field amplitudesas a vector of the form

E 5 S E ~1!

E ~2!D , (1)

which applies to both ambient and substrate media, thenthe amplitudes at each side of the multilayer are relatedby the 2 3 2 complex matrix Mas we shall call themultilayer transfer matrix17 in the form

Ea 5 MasEs . (2)

Fig. 1. Wave vectors of the input [Ea(1) and Es

(2)] and output[Ea

(2) and Es(1)] fields in a multilayer sandwiched between two

identical semi-infinite ambient and substrate media.

The matrix Mas can be shown to be of the form6

Mas 5 F 1/Tas Ras* /Tas*

Ras /Tas 1/Tas* G [ F a b

b* a* G , (3)

where the complex numbers

Ras 5 uRasuexp~ir!, Tas 5 uTasuexp~it!, (4)

are, respectively, the overall reflection and transmissioncoefficients for a wave incident from the ambient. Notethat we have det Mas 5 11, which is equivalent touRasu2 1 uTasu2 5 1. Then the set of lossless multilayermatrices reduces to the group SU(1, 1) whose elementsdepend on three independent real parameters.

The identity matrix corresponds to Tas 5 1 and Ras5 0, so it represents an antireflection system withouttransmission phase shift. The matrix that describes theoverall system obtained by putting two multilayers to-gether is the product of the matrices representing eachone of them, taken in the appropriate order. So two mul-tilayers, which are inverse, when composed give an anti-reflection system.

If we denote by Rsa and Tsa the overall reflection andtransmission coefficients for a wave incident from thesubstrate (physically this corresponds to the samemultilayer taken in the reverse order) one can verify that4

TasTsa 2 RasRsa 5 exp~i2t!, Rsa 5 2Ras* exp~i2t!,(5)

which is a generalization of the well-known Stokesrelations3 for the overall stack.

On the other hand, it is worth noting that while the pic-ture of a multilayer taken in the reverse order is clear, atfirst sight it is not so easy to imagine the inverse of thatmultilayer. However, using Eqs. (5), one can obtain

Mas21 5 Msa* , (6)

which remedies this drawback.

3. BASIC FACTORIZATION FORMULTILAYERS: THE IWASAWADECOMPOSITIONMany matrix factorizations have been considered in theliterature,18–20 the goal of all of them being to decomposea matrix as a unique product of other matrices of simplerinterpretation. In particular, given the essential roleplayed by the Iwasawa decomposition both in fundamen-tal studies and in applications to several fields (especiallyin optics), one is tempted also to investigate its role inmultilayer optics.

Without embarking on mathematical subtleties, we es-tablish the Iwasawa decomposition as follows21: Any el-ement of a (noncompact, semisimple) Lie group can bewritten as an ordered product of three elements, takenone each from a maximal compact subgroup K, a maximalAbelian subgroup A, and a maximal nilpotent subgroup N.Furthermore, such a decomposition is global and unique.

For a lossless multilayer matrix Mas P SU(1, 1), thedecomposition reads as14

Mas 5 K~f !A~j!N~n!, (7)


where

K~f ! 5 Fexp~if/2! 0

0 exp~2if/2!G ,

A~j! 5 F cosh~j/2! i sinh~j/2!

2i sinh~j/2! cosh~j/2!G ,

N~n! 5 F1 2 in/2 n/2

n/2 1 1 in/2G . (8)

The parameters f, j, and n are given in terms of the ele-ments of the multilayer matrix by

f/2 5 arg~a 1 ib!,

j/2 5 ln~1/ua 1 ibu!,

n/2 5 Re~ab* !/ua 1 ibu2, (9)

and their ranges are j, n P R, and 22p < f < 2p.Therefore, given a priori limits on a and b (i.e., on Tas andRas), one could easily establish the corresponding limitson f, j, and n, and vice versa.

Now we can interpret the physical action of the matri-ces appearing in Eq. (7). The symbol K(f) represents thefree propagation of the fields E in the ambient mediumthrough an optical phase thickness of f/2, which reducesto a mere shift of the origin of phases. Alternatively, thiscan be seen as an antireflection system. The second ma-trix A(j) represents a symmetric multilayer with realtransmission coefficient TA 5 sech j/2 and reflectionphase shift rA 5 6p/2. There are many ways to get thisperformance, perhaps the simplest one is a Fabry–Perotsystem composed of two identical plates separated by atransparent spacer. By adjusting the refractive indicesand the thicknesses of the media, one can always get thedesired values (see Sec. 7). Finally, the third matrix N(n)represents a system having TN 5 cos tN exp(itN) andRN 5 sin tN exp(itN), with tan tN 5 n/2. The simplestway to accomplish this task is by an asymmetric two-layer system.

4. A REMARKABLE CASE: SYMMETRICMULTILAYERSSo far, we have discussed various properties of arbitrarylossless multilayers. However, the particular case ofsymmetric structures deserves a lot of attention. In thissection we show how the Iwasawa decomposition providesa useful tool to deal with them.

We recall that for a symmetric stack the reflection andtransmission coefficients are the same whether light is in-cident on one side or the other of the multilayer. In con-sequence, one has Ras 5 Rsa , and the generalized Stokesrelations [Eq. (5)] give then the well-known result22–24

r 2 t 5 6p/2. (10)

This implies that the element b in the transfer matrix[Eq. (3)] is a pure imaginary number. Therefore the ma-trix Mas depends only on two real parameters, which inturn means that f, j, and n [see Eqs. (9)] are not indepen-dent. In fact, a straightforward calculation shows thatthey must fulfill the constraint

n 5 @exp~j! 2 1#tan~f/2!; (11)

thus Ras can be written as

Ras 5 exp~2if !tanh~j/2!@tan~f/2! 2 i#

1 2 i tanh~j/2! tan~f/2!. (12)

Particular care has been paid to the characterization ofzero-reflectance conditions for these symmetricsystems.25,26 Equation (12) allows us to express the lociof zero Ras by the simple condition

j 5 0, (13)

which by Eq. (11) implies that n 5 0 and imposes the con-dition that in this case Mas reduces trivially to a matrixK(f).

The stability of these nonreflecting configurations hasbeen studied by Lekner.26 Indeed, by using a continuityargument, he has shown that ‘‘almost all partial reflectorswith symmetric profiles which are close in parameterspace to a profile which has reflectivity zeros, will alsohave reflectivity zeros’’ (Ref. 26, p. 319). We intend toshow how our formalism allows for a more precise crite-rion.

To this end let us assume a symmetric multilayer sat-isfying initially condition (13). Now, suppose that someparameter (refractive index, thickness, angle of incidence,etc.) that we shall denote generically by l is varied. Ob-viously, admissible variations must preserve the symme-try of the system.

The variation of l induces changes in Ras and so in fand j. The new multilayer will also have zero Ras if theparameters satisfy dRas /dl 5 0; that is,

dRas

dl5

]Ras

]fU

j50

df

dl1

]Ras

]jU

j50

dj

dl5 0. (14)

Using Eq. (12), we get that ]Ras /]f uj50 is identicallyzero, while ]Ras /]juj50 never vanishes. We concludethen that the condition we are looking for is

dj

dl5 0. (15)

This result fully characterizes the partial reflectors in-voked by Lekner and can be of practical importance forthe design of robust antireflection systems.

5. MULTILAYER TRANSFER FUNCTION INTHE UNIT DISKIn many instances (e.g., in polarization optics10) we areinterested in the transformation properties of quotients ofvariables rather than in the variables themselves. Inconsequence, it seems natural to consider the complexnumbers

z 5 E ~2!/E ~1!, (16)

for both ambient and substrate. From a geometricalviewpoint Eq. (2) defines a transformation of the complexplane C, mapping the point zs into the point za , accordingto

za 5 F@Mas , zs# 5b* 1 a* zs

a 1 bzs. (17)


Thus the action of the multilayer can be seen as a func-tion za 5 *(zs) that can be appropriately called themultilayer transfer function.9 The action of the inversematrix Mas

21 is zs 5 F@Mas21, za#.

These bilinear transformations define an action of thegroup SU(1, 1) on the complex plane C. The complexplane appears then decomposed into three regions thatremain invariant under the action of the group: the unitdisk, its boundary and the external region.27

The Iwasawa decomposition has an immediate transla-tion in this geometrical framework, and one is led to treatseparately the action of each one of the matrices appear-ing in this decomposition. To this end, it is worth notingthat the group SU(1, 1) that we are considering appearsalways as a group of transformations of the complexplane. The concept of orbit is especially appropriate forobtaining an intuitive view of the corresponding action.We recall that, given a point z, its orbit is the set of pointsz8 obtained from z by the action of all the elements of thegroup. In Fig. 2 we have plotted some typical orbits foreach one of the subgroups of matrices K(f), A(j), and N(n).For matrices K(f) the orbits are circumferences centeredat the origin and passing by z. For A(j) they are arcs ofcircumference running from point 1i to point 2i throughz. Finally, for the matrices N(n) the orbits are circumfer-ences passing through point 1i and joining points zand 2z* .

6. TRACE CRITERION FORCLASSIFICATION OF MULTILAYERSTo go beyond this geometrical picture of multilayers, letus introduce the following classification: A matrix is oftype K when @Tr(Mas)#2 , 4, is of type A when@Tr(Mas)#2 . 4, and finally is of type N when @Tr(Mas)#2

5 4. To gain insight into this classification, let us alsointroduce the fixed points28 of a transfer matrix as thepoints in the complex plane that are invariant under theaction of Mas , i.e.,

z 5 F@Mas, z#, (18)

whose solutions are

z 52 i Im~a! 6 $@Re~a!#2 2 1%1/2

b. (19)

Since we have

Tr~Mas! 5 2 Re~a! 52 cos t

uTasu, (20)

we can easily check that the matrices of type K have twofixed points, one inside and the other outside the unit

Fig. 2. Plot of several orbits in the unit disk of the elements ofthe Iwasawa decomposition K(f), A(j), and N(n) for the group ofmultilayer transfer matrices.

disk, both related by a inversion; the matrices of type Ahave two fixed points both on the boundary of the unitdisk; and, finally, the matrices of type N have only one(double) fixed point on the boundary of the unit disk.

Now the origin of the notation for these types of matri-ces should be clear: If one considers the Iwasawa decom-position [Eq. (7)], one can see that the matrices K(f) are oftype K with the origin as the fixed point in the unit disk,matrices A(j) are of type A with fixed points 1i and 2i,and matrices N(n) are of type N with the double fixedpoint 1i. Of course, this is in agreement with the orbitsshown in Fig. 2.

To proceed further, let us note that by taking the con-jugate of Mas with any matrix C P SU(1, 1) we obtain an-other multilayer matrix, i.e.,

Mas 5 CMasC21, (21)

such that Tr(Mas) 5 Tr(Mas). The fixed points of Mas arethen the image by C of the fixed points of Mas . If wewrite the matrix C as

C 5 F c1 c2

c2* c1*G , (22)

the matrix elements of Mas (denoted by carets) and thoseof Mas are related by

a 5 auc1u2 2 a* uc2u2 2 2i Im~bc1c2* !,

b 5 bc12 2 b* c2

2 2 2ic1c2 Im~a!. (23)

For our classification viewpoint it is essential to re-mark that if a multilayer has a transfer matrix of type K,A, or N, one can always find a family of matrices C suchthat Mas in Eq. (21) is just a matrix K(f ), A(j), or N(n),respectively. The explicit construction of this family ofmatrices is easy: It suffices to impose that C transformthe fixed points of Mas into the corresponding fixed pointsof K(f ), A(j), or N(n). By way of example let us con-sider the case when Mas is of type K and its fixed pointinside the unit disk is zf . Then one should have

F@CMasC21, 0# 5 F@CMas , zf# 5 F@C, zf# 5 0. (24)

Solving this equation one gets directly

c1 51

~1 2 uzfu2!1/2 exp~id!, c2 5 2c1zf* , (25)

where d is a real, free parameter. The same procedureapplies to the other two cases.

Since the matrix Mas belongs to one of the subgroupsK(f ), A(j), or N(n) of the Iwasawa decomposition, and allthese subgroups are, in our special case, Abelian and uni-parametric, we have that

Mas~m1!Mas~m2! 5 Mas~m1 1 m2!, (26)

where m represents the adequate parameter f, j, or n.Therefore, when dealing with a periodic layered system,whose matrix is obtained as the Nth power of the basicperiod, we have

MasN 5 C21Mas

N ~m!C 5 C21Mas~Nm!C, (27)


where Mas is now the matrix of the basic period. This isa remarkable result,29 and our procedure highlights thatit does not depend on the explicit form of the basic period.

We wish to point out that the trace criterion has beenpreviously introduced30,31 to treat light propagation in pe-riodic structures. In fact, in that approach the values ofthe trace separate the band-stop from the band-pass re-gions of a period stratification.

Finally, to broaden the physical picture of this classifi-cation, let us transform Eq. (2) by the unitary matrix

U 51

A2F1 i

i 1G . (28)

Then we can rewrite it alternatively as

Ea 5 MasEs , (29)

where the new field vectors E and the new multilayer ma-trix Mas are obtained by conjugation by U.

One can easily check that det Mas 5 11 and all its el-ements are real numbers. Therefore Mas belongs to thegroup SL(2, R) that underlies the structure of the cel-ebrated ABCD law in first-order optics.32–35

By transforming the Iwasawa decomposition [Eq. (7)]by U, we get the corresponding one for SL(2, R), which hasbeen previously worked out36:

Mas 5 K~f !A~j!N~n!, (30)

where

K~f ! 5 F cos~f/2! sin~f/2!

2sin~f/2! cos~f/2!G ,

A~j! 5 Fexp~j/2! 0

0 exp~2j/2!G ,

N~n! 5 F1 0

n 1G . (31)

The physical action of these matrices is clear. Let us con-sider all of them ABCD matrices in geometrical opticsthat apply to position x and momentum p (direction) co-ordinates of a ray in a transverse plane. These are thenatural phase-space variables of ray optics. Then K(f)would represent a rotation in these variables, A(j) a mag-nifier that scales x up to the factor m 5 exp(j/2) and pdown by the same factor, and N(n) the action of a lens ofpower n.34

In the multilayer picture, E(1) can be seen as the corre-sponding x, while E(2) can be seen as the corresponding p.Then the key result of this discussion is that when themultilayer transfer matrix has @Tr(Mas)#2

Œ4, one canfind in a direct way a family of matrices that gives a newvector basis such that the action of the multilayer, whenviewed in such a basis, is exclusively rotationlike, or mag-nifierlike, or lenslike.15

7. SIMPLE EXAMPLES AND CONCLUDINGREMARKSWe conclude by showing how our approach works in somepractical examples. Perhaps the best way of starting isto consider the simplest layered structure that one can

imagine: a single film sandwiched between the same am-bient and substrate media. In spite of its simplicity, itcontains the essential physical ingredients of multilayeroptics.

We consider a single transparent film (medium 1) of re-fractive index n1 and thickness d1 embedded in air (me-dium 0). For this system we have10

Tas [ T010 5~1 2 r01

2 !exp~2ib1!

1 2 r012 exp~2i2b1!

, (32)

where r01 is the Fresnel reflection coefficient at the inter-face 01 and b1 5 (2pn1d1 cos u1)/l is the plate phasethickness (here l is the wavelength in vacuo of the inci-dent light and u1 is the refraction angle). Accordingly,we get

@Tr~M010!#2 5 4 cos2 b1 < 4, (33)

and the equality holds only in resonance conditions, i.e.,when uT010u 5 ucos b1u 5 1. In consequence, the matrix ofa single film is always of type K.

Let us consider now two films (1 and 2) described by thematrices M010 and M020 , respectively. The compound sys-tem obtained by putting them together is described by theproduct of these matrices, M010M020 .4,6,10 In conclusion,since any layered stack can be viewed as the compositionof single films, this shows that any multilayer matrix isgenerated by the product of matrices of type K. Take intoaccount that the product of two matrices of type K (or A orN) can have trace Œ4.

On the other hand, as we stated at the end of Section 3,to get a pure matrix A(j) one should consider a Fabry–Perot-like system formed by two identical plates—each ofthem with phase thickness b1—separated by a spacer ofair with phase thickness b2 . If we take as initial condi-tion that in the substrate zs 5 0.4 exp(2ip/3), then astandard calculation gives Tas and Ras , and from themwe obtain the value za 5 20.44 1 0.49i with the param-eters indicated in Fig. 3. Obviously, from these (experi-mental) data alone we cannot infer at all the possiblepath for this discrete transformation.

However, the Iwasawa decomposition remedies this se-rious drawback: From the geometrical meaning dis-cussed above, and once we know the values of f, j, and n[which are easily computed from Eqs. (9)], we establish,by the ordered application of the matrices K(f), A(j), andN(n), that the trajectory from zs to za is well definedthrough the corresponding orbits, as shown in Fig. 3.

Moreover—and this is the important principle we wishto extract from this simple example—if in some experi-ment the values of zs to za are measured, one can find ina unique way, no matter how complicated the multilayeris, the three arcs of orbits that connect the initial and fi-nal points in the unit disk.

In Fig. 4 we have plotted the values of the parametersf, j, and n for this system when b2 is varied between 0and p. We have also plotted the values of @Tr(Mas)#2. Itis evident that the system can be of every type dependingon the value of b2 . The two marked points determinespecial behaviors in agreement with Eq. (11) for symmet-ric systems: For the left one, f 5 n 5 0 and the systemis represented by a pure matrix A(j); for the right one,


j 5 n 5 0 and it is represented by a matrix K(f): Thatis, the system is an antireflection stack with Tas5 exp(2if/2). Note that this system can never be rep-resented by a matrix N(n), because it is symmetric.

To show the characteristic properties of an asymmetricsystem, we consider, as indicated in Section 3, the sim-

Fig. 3. Geometrical representation in the unit disk of the actionof a symmetric system made up of two identical plates (n15 1.7, d1 5 1 mm, u0 5 p/4, l 5 0.6888 mm and s-polarizedlight) separated by a spacer of phase thickness b2 5 3 rad. Thepoint zs 5 0.4 exp(2ip/3) is transformed by the system into thepoint za 5 20.44 1 0.49i. We indicate the three orbits given bythe Iwasawa decomposition and, as a thick line, the trajectory as-sociated with the multilayer action.

Fig. 4. Plot of the values of @Tr(Mas)#2 and of the parameters f,j, and n in the Iwasawa decomposition for the same system as inFig. 3, as a function of b2 .

Fig. 5. Same plot as in Fig. 4 but for an asymmetric systemmade up of a glass plate (n1 5 1.5 and b1 5 2.75 rad) coatedwith a zinc sulfide film (n2 5 2.3).

plest one constituted by a two-layer stack made of a glassplate (with phase thickness b1) coated with a film of zincsulfide with phase thickness b2 . In Fig. 5 we have plot-ted the values of f, j, and n, as well as @Tr(Mas)#2, with b1fixed and b2 varied between 0 and p. From our analysisabove, it is clear that only in the marked point do we havef 5 j 5 0 and @Tr(Mas)#2 5 4, so the system is repre-sented by a pure matrix N(n). In contrast to the previousexample, this system can never be represented by a ma-trix A(j), because it is asymmetric.

In summary, we propose that the geometric frameworkpresented here provides an appropriate tool for analyzingand classifying multilayer performance in an elegant andconcise way that, in addition, could be readily applied toother fields of physics.

Luis L. Sanchez-Soto’s e-mail address is [email protected].

REFERENCES AND NOTES1. H. A. Macleod, Thin-film Optical Filters (Adam Hilger, Bris-

tol, UK, 1986).2. P. Yeh, Optical Waves in Layered Media (Wiley, New York,

1988).3. J. Lekner, Theory of Reflection (Kluwer Academic, Dor-

drecht, The Netherlands, 1987).4. J. J. Monzon and L. L. Sanchez-Soto, ‘‘Lossless multilayers

and Lorentz transformations: more than an analogy,’’ Opt.Commun. 162, 1–6 (1999).

5. J. J. Monzon and L. L. Sanchez-Soto, ‘‘Fully relativisticlikeformulation of multilayer optics,’’ J. Opt. Soc. Am. A 16,2013–2018 (1999).

6. J. J. Monzon and L. L. Sanchez-Soto, ‘‘Origin of the Thomasrotation that arises in lossless multilayers,’’ J. Opt. Soc.Am. A 16, 2786–2792 (1999).

7. J. J. Monzon and L. L. Sanchez-Soto, ‘‘A simple optical dem-onstration of geometric phases from multilayer stacks: theWigner angle as an anholonomy,’’ J. Mod. Opt. 48, 21–34(2001).

8. H. S. M. Coxeter, Non-Euclidean Geometry (University ofToronto Press, Toronto, 1968).

9. T. Yonte, J. J. Monzon, L. L. Sanchez-Soto, J. F. Carinena,and C. Lopez-Lacasta, ‘‘Understanding multilayers from ageometrical viewpoint,’’ J. Opt. Soc. Am. A 19, 603–609(2002).

10. R. M. A. Azzam and N. M. Bashara, Ellipsometry and Po-larized Light (North-Holland, Amsterdam, 1987), Sec. 4.6.

11. D. Han, Y. S. Kim, and M. E. Noz, ‘‘Polarization optics andbilinear representations of the Lorentz group,’’ Phys. Lett. A219, 26–32 (1996).

12. H. Kogelnik, ‘‘Imaging of optical modes–resonators with in-ternal lenses,’’ Bell Syst. Tech. J. 44, 455–494 (1965).

13. M. Nakazawa, J. H. Kubota, A. Sahara, and K. Tamura,‘‘Time-domain ABCD matrix formalism for laser mode-locking and optical pulse transmission,’’ IEEE J. Sel. Top.Quantum Electron. 34, 1075–1081 (1998).

14. J. J. Monzon, T. Yonte, and L. L. Sanchez-Soto, ‘‘Basic fac-torization for multilayers,’’ Opt. Lett. 26, 370–372 (2001).

15. L. L. Sanchez-Soto, J. J. Monzon, T. Yonte, and J. F. Car-inena, ‘‘Simple trace criterion for classification of multilay-ers,’’ Opt. Lett. 26, 1400–1402 (2001).

16. When ambient (0) and substrate (m 1 1) media are differ-ent, the angles u0 and um11 are connected by Snell’s law,n0 sin u0 5 nm11 sin um11 , where nj denotes the refractiveindex of the jth medium.

17. I. Ohlıdal and D. Franta, ‘‘Ellipsometry of thin film sys-tems,’’ in Progress in Optics XLI, E. Wolf, ed. (North-Holland, Amsterdam, 2000), p. 181.

18. H. H. Arsenault and B. Macukow, ‘‘Factorization of the


transfer matrix for symmetrical optical systems,’’ J. Opt.Soc. Am. 73, 1350–1359 (1983).

19. S. Abe and J. T. Sheridan, ‘‘Optical operations on wave func-tions as the Abelian subgroups of the special affine Fouriertransformation,’’ Opt. Lett. 19, 1801–1803 (1994).

20. J. Shamir and N. Cohen, ‘‘Root and power transformationsin optics,’’ J. Opt. Soc. Am. A 12, 2415–2423 (1995).

21. S. Helgason, Differential Geometry, Lie Groups and Sym-metric Spaces (Academic, New York, 1978).

22. V. Degiorgio, ‘‘Phase shift between the transmitted and re-flected optical fields of a semireflecting lossless mirror isp/2,’’ Am. J. Phys. 48, 81–82 (1980).

23. A. Zeilinger, ‘‘General properties of lossless beam splittersin interferometry,’’ Am. J. Phys. 49, 882–883 (1981).

24. Z. Y. Ou and L. Mandel, ‘‘Derivation of reciprocity relationsfor a beam splitter from energy balance,’’ Am. J. Phys. 57,66–67 (1989).

25. J. Lekner, ‘‘Nonreflecting stratifications,’’ Can. J. Phys. 68,738–742 (1989).

26. J. Lekner, ‘‘The phase relation between reflected and trans-mitted waves, and some consequences,’’ Am. J. Phys. 58,317–320 (1990).

27. A. Perelomov, Generalized Coherent States and Their Appli-cations (Springer, Berlin, 1986).

28. V. Bargmann, ‘‘Irreducible unitary representations of theLorentz group,’’ Ann. Math. 48, 568–640 (1947).

29. M. Born and E. Wolf, Principles of Optics, 7th ed. (Cam-bridge U. Press, Cambridge, UK, 1999) Sect. 1.6.5.

30. J. Lekner, ‘‘Light in periodically stratified media,’’ J. Opt.Soc. Am. A 11, 2892–2899 (1994).

31. J. Lekner, ‘‘Omnidirectional reflection by multilayer dielec-tric mirrors,’’ J. Opt. A Pure Appl. Opt. 2, 349–352 (2000).

32. H. Bacry and M. Cadilhac, ‘‘The metaplectic group and Fou-rier optics,’’ Phys. Rev. A 23, 2533–2536 (1981).

33. M. Nazarathy and J. Shamir, ‘‘First order systems–a ca-nonical operator representation: lossless systems,’’ J. Opt.Soc. Am. 72, 356–364 (1982).

34. E. C. G. Sudarshan, N. Mukunda, and R. Simon, ‘‘Realiza-tion of first order optical systems using thin lenses,’’ Opt.Acta 32, 855–872 (1985).

35. R. Simon, N. Mukunda, and E. C. G. Sudarshan, ‘‘Partiallycoherent beams and a generalized abcd-law,’’ Opt. Com-mun. 65, 322–328 (1988).

36. R. Simon, E. C. G. Sudarshan, and N. Mukunda, ‘‘General-ized rays in first order optics: transformation properties ofGaussian Schell-model fields,’’ Phys. Rev. A 29, 3273–3279(1984).

geometrical setting for the classification of multilayers

Documents