vectorlike representation of multilayers

2386 J. Opt. Soc. Am. A/Vol. 21, No. 12 /December 2004 Barriuso et al.

Vectorlike representation of multilayers

Alberto G. Barriuso, Juan J. Monzon, and Luis L. Sanchez-Soto

Departamento de Optica, Facultad de Fısica, Universidad Complutense, 28040 Madrid, Spain

Jose F. Carinena

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

Received March 29, 2004; revised manuscript received June 2, 2004; accepted June 7, 2004

We use the concept of turns to provide a geometrical representation of the action of any lossless multilayer,which can be considered to be analogous in the unit disk to sliding vectors in Euclidean geometry. This con-struction clearly shows the peculiar effects arising in the composition of multilayers. A simple optical experi-ment revealing the appearance of the Wigner angle is analyzed in this framework. © 2004 Optical Society ofAmerica

OCIS codes: 230.4170, 120.5700, 120.7000, 000.3860.

1. INTRODUCTIONThe search for mathematical constructs to describe physi-cal phenomena has always been a top priority. For ex-ample, the need to describe a direction in space, combinedwith the use of geometry to approach physical problems,brought forth the concept of a vector. The idea that com-plex numbers have a geometrical interpretation as vec-tors lying in a plane led Hamilton to introduce quater-nions for the analysis of three-dimensional space.1 Theprice paid was that the composition of quaternions is notcommutative. Soon after that, it became clear that rota-tions can be represented advantageously by unit quater-nions.

A notion closely related to the Hamilton treatment isthat of turns.2 The turn associated with a rotation ofaxis n through angle q is a directed arc of length q/2 onthe great circle orthogonal to n on the unit sphere. Bymeans of these objects, the composition of rotations is de-scribed through a parallelogramlike law: If these turnsare translated on the great circles until the head of thearc of the first rotation coincides with the tail of the arc ofthe second one, then the turn between the free tail andthe head is associated with the resultant rotation.Hamilton turns are thus analogous in spherical geometryto sliding vectors in Euclidean geometry.

In recent years many concepts of a geometrical naturehave been introduced to gain further insight into the be-havior of layered media. The algebraic basis for thesedevelopments is that the transfer matrix associated witha lossless multilayer is an element of the group SU(1, 1),which is locally isomorphic to the Lorentz group SO(2, 1)in (2 1 1) dimensions. This leads to a natural and com-plete identification between reflection and transmissioncoefficients and the parameters of the corresponding Lor-entz transformation.3,4

Juarez and Santander5 developed a generalization ofHamilton turns to the Lorentz group, while Simon et al.6,7

worked out an equivalent algebraic approach for SU(1, 1),in which they introduced a noncommutative geometricaladdition for these hyperbolic turns that reproduces the

1084-7529/2004/122386-06$15.00 ©

composition law of the group, i.e., for both the reflectionand transmission coefficients. The latter coefficientsseem to be almost totally ignored in the literature. Thegoal of this paper is to show precisely how this formalismaffords a very intuitive image of multilayer optics.

Of course, since the amplitude and phase of a lightbeam can be conveniently represented by a (Euclidean)vector, the properties of layered media have been repre-sented by graphical constructions. These visualizationtools are commonly in use in thin-film design and includethe Smith chart, the admittance diagram,8 the (reflec-tance) circle diagrams,9 and the vector method,10 amongothers. However, the formalism of turns is intrinsic anddoes not rely on any specific representation for the lightwaves. In other words, once the turn for a multilayer isknown, its action on any light state is fully determined.We emphasize that although these techniques can be usedfor quantitative calculations, they cannot compete nowa-days with modern software design, and their great valueis in the visualization of the characteristics of amultilayer.

This paper is organized as follows. In Section 2 wepresent some details of how the action of any multilayercan be seen as a geometrical motion in the unit disk.Each one of these motions can be decomposed in terms oftwo reflections, which justifies the idea of turn introducedin Section 3, where their composition law is also intro-duced through a parallelogram law, in close analogy towhat happens with sliding vectors in Euclidean geometry.The noncommutative character of this law leads to inter-esting phenomena, such as the appearance of extraphases in the composition of multilayers, which is exam-ined in Section 4, where we give a simple though non-trivial example that illustrates clearly how this geometri-cal scheme works in practice.

2. MULTILAYER ACTION IN THE UNITDISKThe theory of reflection and transmission of light bystratified planar structures is of wide interest in optics.

2004 Optical Society of America

Barriuso et al. Vol. 21, No. 12 /December 2004 /J. Opt. Soc. Am. A 2387

A considerable amount of theoretical work has been doneon this topic, a detailed discussion of which can be foundin a number of sources.8,10–14 Nowadays the standardmethod uses a matrix representation of the field in eachmedium, as pioneered by Abeles.15 In this paper we fol-low the elegant approach developed by Hayfield andWhite.16

We deal with a stratified structure that consists of astack of plane-parallel layers sandwiched between twosemi-infinite ambient (a) and substrate (s) media that weshall assume to be identical in order to simplify the cal-culations as much as possible. Hereafter all the mediaare assumed to be lossless, linear, homogeneous, and iso-tropic. We choose the Z axis perpendicular to the bound-aries and directed as in Fig. 1.

A monochromatic, linearly polarized plane wave is in-cident from the ambient making an angle u0 with the nor-mal to the first interface and with amplitude Ea

(1) . Theelectric field is either in the plane of incidence ( p polar-ization) or perpendicular to the plane of incidence (s po-larization). Since the multilayer has two input channels,we consider as well another plane wave of the same fre-quency and polarization and with amplitude Es

(2) incidentfrom the substrate at the same angle u0 . This guaran-tees that all the multiple reflected and transmitted wavessuperpose with the same wave vectors. The overall out-put fields in the ambient and the substrate will be de-noted Ea

(2) and Es(1) , respectively. In the common experi-

mental situation, there is no light incident from thesubstrate @Es

(2) 5 0#, although there are relevant situa-tions in which both input fields are present (e.g., at thebeam splitter in a Michelson interferometer).

The field amplitudes at each side of the multilayer arelinearly related by the equation

S Ea~1!

Ea~2!D 5 MasS Es

~1!

Es~2!D , (2.1)

where the multilayer transfer matrix Mas can be shown tobe14

Mas 5 F 1/Tas Ras* /Tas*

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

b* a* G . (2.2)

Here the complex numbers Ras and Tas , which can be ex-pressed as

Ras 5 uRasuexp~ir!, Tas 5 uTasuexp~it!, (2.3)

are, respectively, the overall reflection and transmissioncoefficients for a wave incident from the ambient. Be-cause uRasu2 1 uTasu2 5 1, we have det Mas 5 uau2 2 ubu2

5 1, and then Mas belongs to the group SU(1, 1).In Ref. 17 we have proposed viewing the multilayer ac-

tion in a relativisticlike framework, giving a formalequivalence between the fields in Eq. (2.1) and the space–time coordinates in a (2 1 1)-dimensional space. Thesecoordinates verify that (x0)2 2 (x1)2 2 (x2)2 5 1, whichdefines a unit two-sheeted hyperboloid characteristic ofthe group SO(2, 1). If one uses stereographic projection,taking the south pole as projection center, the upper sheetof the unit hyperboloid is projected into the unit disk andthe lower sheet into the external region, while infinitygoes to the boundary of the unit disk.

The geodesics in the hyperboloid are intersections withthe hyperboloid of planes passing through the origin.Consequently, hyperbolic lines are obtained from these bystereographic projection, and they correspond to arcs ofcircles that orthogonally cut the boundary of the unitdisk.

In many instances (e.g., in polarization optics13) we areinterested in the transformation properties of field quo-tients rather than the fields themselves. Therefore itseems natural to consider the complex numbers

zs 5 Es~2!/Es

~1! , za 5 Ea~2!/Ea

~1! . (2.4)

The action of the multilayer given in Eq. (2.2) can then beseen as a function za 5 f(zs) that can be appropriatelycalled the multilayer transfer function.18 From a geo-metrical viewpoint, this function defines a transformationof the complex plane, mapping the point zs into the pointza , according to19

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

a 1 bzs, (2.5)

which is a bilinear or Mobius transformation. One canverify that the unit disk, the external region and the unitcircle remain invariant under the multilayer action.Note also that when no light impinges from the substratezs 5 0, then za 5 Ras . It is worth mentioning that thisapproach is quite general, since it provides the transfor-mation of any point in the unit disk for every value of theinput fields Ea

(1) and Es(2) .

To classify the multilayer action, it proves convenientto work out the fixed points of the mapping, that is, thefield configurations such that za 5 zs [ zf in Eq. (2.5)20:

zf 5 F@Mas , zf#, (2.6)

whose solutions are

zf 51

2b(22i Im~a! 6 $@Tr~Mas!#

2 2 4%1/2). (2.7)

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.


When @Tr(Mas)#2 , 4 the multilayer action is elliptic andhas only one fixed point inside the unit disk. Since inEuclidean geometry a rotation is characterized as havingonly one invariant point, this multilayer action can appro-priately be called a hyperbolic rotation.

When @Tr(Mas)#2 . 4 the action is hyperbolic and hastwo fixed points, both on the boundary of the unit disk.The geodesic line joining these two fixed points remainsinvariant, and thus, by analogy with the Euclidean case,this action will be called a hyperbolic translation.

Finally, when @Tr(Mas)#2 5 4 the multilayer action isparabolic and has only one (double) fixed point on theboundary of the unit disk.

Here we will be concerned only with the case@Tr(Mas)#2 . 4, since it is known that any element ofSU(1, 1) can be written (in many ways) as the product oftwo hyperbolic translations.7 The axis of the hyperbolictranslation is the geodesic line joining the two fixedpoints. A point on the axis will be translated to anotherpoint a (hyperbolic) distance21

z 5 2 ln[12 (Tr~Mas! 1 $@Tr~Mas!#

2 2 4%1/2)] (2.8)

along the axis.

3. HYPERBOLIC TURNS AND THEIRCOMPOSITIONIn Euclidean geometry, a translation of magnitude z alonga line g can be seen as the product of two reflections inany two straight lines orthogonal to g separated a dis-tance z/2. This idea can be translated in much the sameway to the unit disk, once the concepts of line and dis-tance are understood in the hyperbolic sense. In conse-quence, any pair of points z1 and z2 on the axis of thetranslation g at a distance z/2 can be chosen as intersec-tions of G1 and G2 (lines orthogonal to g) with g. It isthen natural to associate with the translation an orientedsegment of length z/2 on g, which is otherwise free to slideon g (see Fig. 2). This is analogous to Hamilton’s turnsand will be called a hyperbolic turn Tg,z/2 .

Note that with this construction, an off-axis point suchas zs will be mapped by these two reflections (through anintermediate point z int) to another point za along a curveequidistant from the axis. These other curves, unlike the

Fig. 2. Representation of the sliding turn Tg,z/2 in terms of tworeflections in two lines G1 and G2 orthogonal to the axis of thetranslation g, which has two fixed points zf1 and zf2 . The trans-formation of a typical off-axis point zs is also shown.

axis of translation, are not hyperbolic lines. The essen-tial point is that once the turn is known, the transforma-tion of every point in the unit disk is automatically estab-lished.

Alternatively, we can formulate the concept of turn asfollows. Let Mas be a hyperbolic translation with Tr(Mas)positive (equivalently, Re(a) . 1). Then Mas is positivedefinite, and one can ensure that its square root existsand reads as

~AMas! 51

$2@Re~a! 1 1#%1/2Fa 1 1 b

b* a* 1 1G .(3.1)

This matrix has the same fixed points as Mas , but thetranslated distance is just half that induced by Mas ; i.e.,

z~Mas! 5 2z~AMas!. (3.2)

This suggests that the matrix AMas can be appropriatelyassociated to the turn Tg,z/2 that represents the transla-tion induced by Mas . Therefore we symbolically write

Tg,z/2 ° AMas. (3.3)

One may be tempted to extend the Euclidean composi-tion of concurrent vectors to the problem of hyperbolicturns. Indeed, this can be done quite straightforwardly.5

Let us consider the case of the composition of two of thesemultilayers represented by the matrices M1 and M2 (forsimplicity, we shall henceforth omit the subscript as) ofparameters z1 and z2 along intersecting axes g1 and g2 ,respectively. Take the associated turns Tg1 ,z1/2 andTg2 ,z2/2 and slide them along g1 and g2 until they are‘‘head to tail.’’ Afterward, the turn determined by thefree tail and head is the turn associated to the resultant,which thus represents a translation of parameter z alongthe line g. This construction is shown in Fig. 3, wherethe noncommutative character is also evident.

In Euclidean geometry, the resultant of this parallelo-gram law can be quantitatively determined by a direct ap-plication of the cosine theorem. For any hyperbolic tri-angle with sides of lengths z1 and z2 that make an angleu, we take the expression from any standard book on hy-perbolic geometry21:

cosh z 5 cosh z1 cosh z2 1 sinh z1 sinh z2 cos u.(3.4)

Fig. 3. Composition of two hyperbolic turns Tg1 ,z1/2 and Tg2 ,z2/2

by using a parallelogramlike law when the axes g1 and g2 of thetranslations intersect.


Moreover, for future use we state that the (hyperbolic)area V of the geodesic triangle has the value

tan~V/2! 5tanh~z1/2!tanh~z2/2!sin u

1 2 tanh~z1/2!tanh~z2/2!cos u. (3.5)

4. APPLICATION: REVEALING THEWIGNER ANGLE IN THE UNIT DISKTo show how this formalism can account for the existenceof peculiar effects in the composition of multilayers, weaddress here the question of the Wigner angle in the unitdisk and propose a simple optical experiment to deter-mine this angle.

The Wigner angle emerges in the study of the composi-tion of two noncollinear pure boosts in special relativity:the combination of two such successive boosts cannot re-sult in a pure boost but renders an additional pure rota-tion, usually known as the Wigner rotation22 (the name ofThomas rotation23,24 is sometimes used). In other words,boosts are not a group.

To fix the physical background, consider three frames ofreference K, K8, and K9. Frames K –K8 and K8–K9 haveparallel respective axes. Frame K9 moves with uniformvelocity v2 with respect to K8, which in turn moves withvelocity v1 relative to K. The Lorentz transformationthat connects K with K9 is given by the productL1(v1)L2(v2), which can be decomposed as

L1~v1!L2~v2! 5 L~12!~v!R~C!, (4.1)

where one must be careful to operate in just the same or-der as written. In words, this means that an observer inK sees the axes of K9 rotated relative to the observer’sown axes by a Wigner rotation described by R(C). Theexplicit expression for the axis and angle of this rotationcan be found, e.g., in Ref. 22 and will be worked out nowfrom the perspective of the multilayer action in the unitdisk.

First we observe that any matrix M P SU(1, 1) can beexpressed in a unique way in the form

M 5 HU, (4.2)

where H is positive definite Hermitian and U is unitary.One can check by simple inspection that the explicit formof this (polar) decomposition reads as25

M 5 HU 5 F 1/uTu R* /uTu

R/uTu 1/uTu G3 Fexp~2it! 0

0 exp~it!G . (4.3)

The component H is equivalent to a pure boost, and U isequivalent to a spatial rotation.

It is clear from Eq. (4.3) that @Tr(H)#2 . 4, so it repre-sents a hyperbolic translation. Moreover, one can checkthat its associated fixed points are diametrically locatedon the unit circle, and so the axis of this translation isprecisely the diameter joining them. By writing now

R 5 tanh~z/2!exp~ir!, T 5 sech~z/2!exp~it!,(4.4)

one can easily verify that the matrix H in Eq. (4.3) trans-forms the origin into the complex point R; that is,

F@H, 0# 5 R, F@H21, R# 5 0. (4.5)

In complete analogy with Eq. (4.1) we compose now twomultilayers represented by Hermitian matrices H1 and H2(that is, with zero transmission phase lag t1 5 t2 5 0),and we get, after simple calculations,26

H1H2 5 H~12!U 5 F 1/uT ~12!u R ~12!* /uT ~12!u

R ~12! /uT ~12!u 1/uT ~12!uG

3 Fexp~2iC/2! 0

0 exp~iC/2!G , (4.6)

where

R ~12! 5R1 1 R2

1 1 R1* R2

, T ~12! 5uT1T2u

1 1 R1* R2

,

C

25 arg@T ~12!# 5 arg~1 1 R1R2* !, (4.7)

and the subscripts 1 and 2 refer to the corresponding mul-tilayers. The appearance of an extra unitary matrix inEq. (4.6) is the signature of a Wigner rotation in themultilayer composition and, accordingly, the Wignerangle C is just twice the phase of the transmission coeffi-cient of the compound multilayer.

To view this Wigner angle in the unit disk, let zs be thepoint in the substrate that is transformed by themultilayer H2 into the origin, and let za be the result oftransforming the origin by H1 . According to Eq. (4.5),one has

F@H2 , 2R2# 5 0, F@H1 , 0# 5 R1 . (4.8)

Consider now the (geodesic) triangle defined by thepoints zs , O, and za in Fig. 4. The general formula (3.5)gives for this triangle

V 5 C, (4.9)

which confirms the geometric nature of this Wigner angle,since it can be understood in terms of the area (or equiva-lently, the anholonomy) of a closed circuit.27,28

According to the ideas developed in Section 3 we can re-duce the multilayers H1 and H2 to the associated turnsrepresented by arrows in Fig. 4. The ‘‘head-to-tail’’ ruleapplied to T1 and T2 (for simplicity, we omit in the sub-scripts of these turns the corresponding parameters) im-mediately gives the resulting turn T(12) . However, notethat if we follow the formal prescription shown in Eq.(3.3) and ascribe T1 ° AH1, and T2 ° AH2, we con-clude that the composition law imposes

T~12! ° AH1H2. (4.10)

All these results are independent of the position of theturn. In fact, in Fig. 5 we have put the turn T(12) in dif-ferent positions along the axis g. In every position wehave drawn two radii passing through the head and thetail of T(12) and taken on them twice the hyperbolic dis-tance from the origin. The pairs of points obtained inthis way (such as zs8 and za8) are transformed precisely by


H1H2 . In other words H1H2 can be decomposed in manyways as the composition of two Hermitian matrices, andevery geodesic triangle zs8 O za8 has the same hyperbolicarea C.

It seems pertinent to conclude by showing an experi-mental implementation of the data shown in Fig. 4. Tothis end, we first recall13 that for a single plate of refrac-tive index nj and thickness dj embedded in air (n0 5 1)and illuminated by a monochromatic light of wavelengthin vacuo l, a standard calculation gives the transfer ma-trix M0j0 with the following reflection and transmissioncoefficients:

R0j0 5r0j@1 2 exp~2i2d j!#

1 2 r0j2 exp~2i2d j!

,

T0j0 5t0jt j0 exp~2id j!

1 2 r0j2 exp~2i2d j!

, (4.11)

where r0j and t0j are the Fresnel reflection and transmis-sion coefficients at the interface 0j (which applies to bothp and s polarizations by the simple attachment of a sub-script p or s) and d j is the plate phase thickness

d j 52p

lnjdj cos u j , (4.12)

Fig. 4. Composition of two multilayers represented by Hermit-ian matrices H1 and H2 . H2 maps the point zs 5 2R2 into theorigin, while H1 maps the origin into za 5 R1 . We show also theassociated turns T1 and T2 as well as the resulting turn T(12) ob-tained through the parallelogram law. The compositemultilayer H1H2 transforms the point zs into za . The data ofthe corresponding multilayers are given in the text.

Fig. 5. Same as Fig. 4, but now the resulting turn T(12) has beenslid to three different positions along the axis. The correspond-ing points are also transformed by H1H2 . All the geodesic tri-angles plotted have the same hyperbolic area C.

u j being the angle of refraction in the layer. The transfermatrix for the coherent addition of m of these layers is

M 5 )j51

m

M0j0 . (4.13)

As shown in Fig. 6 we take as the first multilayer H1the lossless system formed by two thin films, one of zincsulphide (with refractive index n1 5 2.3 and thicknessd1 5 80 nm) and the other of cryolite (with refractive in-dex n2 5 1.35 and thickness d2 5 104 nm) deposited on aglass substrate (with refractive index n3 5 1.5 and thick-ness d3 5 1.3 mm) and embedded in air. The light has awavelength in vacuo of l0 5 546 nm and falls from theambient at normal incidence. Such a simple systemcould be manufactured with standard evaporation tech-niques.

We have performed a computer simulation of the per-formance of this multilayer H1 with a standard package,obtaining T1 5 0.9055 and R1 5 0.3736 2 0.2014i,which in turn gives t1 5 0 and r1 5 20.4944 rad.

Our second multilayer H2 is a symmetrical systemformed by two films of zinc sulphide of thickness d45 40 nm separated by a spacer of air with a phase thick-ness d5 5 3.707 rad. For this subsystem we have T25 0.9399 and R2 5 0.3413i, and therefore t2 5 0 andr2 5 p/2 rad.

When these two multilayers are put together by themarked points in Fig. 6, the resulting multilayer has atransmission phase lag of t 5 20.1361 rad, which is justhalf the area of the geodesic triangle zsOza in Fig. 4, aspredicted by the theory.

In summary, we expect that the geometrical approachpresented here will be an interesting tool for representingin a graphical way the action of multilayers. Moreover,the composition law of these turns allows for a clear un-derstanding of the nontrivial effects appearing in thecomposition of multilayers.

We stress that the benefit of this approach lies not inany inherent advantage in terms of efficiency in solvingproblems in layered structures. Rather, we expect thatturns could provide a general and unifying tool to analyzemultilayer performance in an elegant way that, in addi-tion, is closely related to other fields of physics.

ACKNOWLEDGMENTSWe thank Jose Marıa Montesinos and Mariano Santanderfor enlightening discussions.

Corresponding author Luis L. Sanchez-Soto’s e-mailaddress is [email protected].

Fig. 6. Scheme of two Hermitian multilayers H1 and H2 . Thecompound multilayer H1H2 obtained by putting together thesetwo components induces a Wigner rotation of angle 2t.


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