(1980) Yeh, P_Optics of anisotropic layered media..A new 4x4 matrix algebra-Surface Science 96(1–3) 41-53

7/22/2019 (1980) Yeh, P_Optics of anisotropic layered media..A new 4x4 matrix algebra-Surface Science 96(13) 41-53

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Surface Science 96 (1980) 41-530 North-Holland Publishing Company

OPTICS OF ANISOTROPIC LAYERED MEDIA: A NEW 4 X 4 MATRIXALGEBRAPochi YEHRockw ell Int ernat ional Science Cent er Thousand Oaks Cali forn ia 91360 USA

Received 20 August 1979

A new 4 X4 matrix algebra, which combines and generalizes Abel&s 2 X 2 matrix methodand Jones 2 X 2 matrix method, is introduced to investigate plane-wave propagation in anarbitrarily anisotropic medium. In this new method, each layer of finite thickness is representedby a propagation matrix which is diagonal and consists of the phase excursions of the fourpartial plane waves. Each side of an interface is represented by a dynamical matrix that dependson the direction of the eigenpolarizations in the anisotropic medium.

1 Introduction

Anisotropic thin films have become increasingly important in a number ofmodern optical systems such as guided-wave propagation in integrated optics, nar-row-band birefringent filters and, many semiconductor devices. Many of theseactive and passive optical devices require the epitaxial growth of anisotropic thinfilms. The design characteristics of these devices are strongly dependent on theunderstanding of electromagnetic propagation in anisotropic layered media. Inaddition, ellipsometry of anisotropic multilayer systems also requires a preciseunderstanding of electromagnetic propagation in these media. Ellipsometry ofanisotropic multilayer systems may be an ultimate optical technique for the charac-terization of the anisotropic thin film properties, which include the orientation ofthe optical axes, the refractive indices, and the film thicknesses.

A general theory of electromagnetic propagation in isotropic layered media andthe 2 X 2 matrix method are described in the pioneering analysis of Abel&s [l]. Asimilar theory on the electromagnetic propagation in periodic layered media waspublished by Yeh et al. [2]. Several interesting new phenomena are predicted andobserved experimentally in these media; these include Bragg waveguiding [3,4],optical surface waves [5,6] and injection lasers [7,8]. A general theory on thepropagation of electromagnetic radiation in birefringent layered media, especiallythe Sole layered media [9], was recently published by Yeh [lo]; new phenomenasuch as the exchange Bragg scattering and the oscillatory evanescent waves arefound in these media [lO,ll]. A 4 X4 differential matrix method has been devel-

41


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42 P. Yeh / Opt ics of anisotropic layered mediaoped by Berreman [12,13] to study the reflection and the transmission of an arbi-trarily polarized light by stratified anisotropic planar structures.

In the case of isotropic layered media, the electromagnetic radiation can bedivided into two independent (uncoupled) modes: s-modes (with electric fieldvector E perpendicular to the plane of incidence) and p-modes (with electric fieldvector E parallel to the plane of incidence). Since they are uncoupled, the matrixmethod involves the manipulation of 2 X 2 matrices only. In the case of birefringentlayered media, the electromagnetic radiation consists of four partial waves, Modecoupling takes place at the interface where an incident plane wave produces waveswith different polarization states due to the anisotropy of the layers. As a result,4 X4 matrices are needed in the tnatrix method.

Before proceeding with the many applications envisaged for birefringent layeredmedia, it is necessary to understand precisely and in detail the nature of electro-magnetic wave propagation in these media. Although a number of special cases havebeen analyzed, a simple and general theory is not available.

The case of plane wave incidence at a plane interface between two biaxial mediahas been solved by many workers [14-l 61; however, these results are not in auseful form for treating electromagnetic propagation in birefringent multilayermedia. The case of plane wave propagation in a three-layer structure consisting ofbirefringment media has also been solved [17-201; however, in each case, otherthan in ref. /20], the structure involves at least one isotropic layer. Even the resultsobtained in ref. [ZO] are not systematic enough for treating the general tnultilayerbirefringent media. Holmes and Feucht [Zl] treated the reflection and transmissionproperties of a stack of birefringent crystals theoretically. Their analysis isrestricted to the case where one principal axis is normal to the plane of incidenceand the incident waves have their electric fields polarized parallel to the same plane.Recently, Stamnes and Sherman [22] obtained exact solutions for the reflected andtransmitted fields which result when an arbitrary electromagnetic wave is incidenton a plane interface separating two uniaxial media, Their results, however, are notin a useful form for treating birefringent layered media.

This paper describes a general theory of electromagnetic plane wave propagationin birefringent layered media. The theoretical approach is general, so that manysituations considered previously will be shown to be special cases of the formalism.New concepts of dynamical matrix as well as propagation matrix are introduced:this makes it possible to write out the transfer matrix in terms of diagram represen-tation.

2. Propagation of plane waves in homogeneous anisotropic mediaFirst a brief review of the propagation of plane waves in homogeneous aniso-

tropic media is in order. The approach and formulation are different from the tradi-tional one [23] because of the demands of their appli~tion in this problem. The


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44 P Yeh /Optics ofanisotropic layered media

Fig. 1. Graphic method to determine the propagation constants from the normal surface

with respect to the origin of the axes. Drawing a line from the tip of the vector c& +0-9 parallel to the z direction yields, in general, four points of intersection. Thesefour wave vectors k, = ok + /3j + y$ all lie in the plane of incidence, which alsoremains the same throughout the layered medium because o( and fl are constant.However, the four group velocities associated with these partial waves are, ingeneral, not lying in the plane of incidence. If all the four wave vectors k, are real,two of them have group velocities with positive z component and the other twohave group velocities with negative z component. The z component of the groupvelocity vanishes when Y,, becomes complex. The polarization of these waves isgiven by

where u = 1, 2, 3, 4 and NOs are the normalization constants such that fi. p = 1.The electric field of the plane electromagnetic waves can thus be written

4

E=~A,ri,exp[i(ax+&+y,z-wt)].=1

(6)Partial waves with complex propagation vectors cannot exist in an infinite


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P. Yeh / Op t ics ofani sotropi c lay ered medi a 45

homogeneous birefringent medium. If the medium is semi-infinite, the exponenti-ally damped partial waves are legitimate solutions near the interface, and the fieldenvelope decays exponentially as a function of z, where z is the distance from theinterface. These exponentially damped partial waves are called evanescent waves.The evanescent waves in birefringent media in general have complex ys, i.e., y =yK t iyt. In a uniaxially birefringent medium, the ordinary evanescent wave has apurely imaginary y. If the three principal dielectric constants are all real, then thesepartial waves with complex ys can be shown to have their Poynting vectors parallelto the interface. In other words, the energy is flowing parallel to the interface andthe propagation is lossless as it should be. A mathematical proof is given in appen-dix A of ref. [lo] for the special case of extraordinary evanescent waves in auniaxially birefringent medium.

3. Matrix methodThe 4 X4 matrix algebra, which analyzes the propagation of monochromatic

plane waves in birefringent layered media, can now be introduced. The approach isgeneral so that the results will be used later on for many special cases of propaga-tion in anisotropic layered media. The materials are assumed to be nonmagnetic sothat 1_1= onstant throughout the whole layered medium. The dielectric permittivitytensor e in xyz coordinates is given by

40) z


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46 i? Y eh / Opti cs ofani sotropie la yer ed m edi a

The cofumn vectors are not independent of each other. They are related throughthe continuity conditions at the interfaces. As a matter of fact, only one vector (orany four components of four different vectors) can be arbitrarily chosen. Themagnetic field distribution is obtained by using Maxwells equations and is given by

4ff=g, A.(n)(i,(12)expCi[axfpy+y,(n)(z-z,)-wtl},B*(n) = (QWYW) Q,(n) 2k,(n) = a.2 + pj + y*(n) 3 INote that the ijO(n are no longer unit vectors.

(9)

(10)(1x1

Imposing the continuity of EX, E.V,H,, and H.,, at the interface z = z,,_t Ieads to

5 A&r - l)ri,(n - 1) *g= 5 A.(n)~~i,(n).~exp[-iy,(n)t,], (12)u=r 0=1

$r A,@ - l)&(fi - 1) *P= kr ~~(~z)~~(~) -.PexpI-ly&)f,l, (13)

@$ A& - I)?& - 1) *.%= 5 A,(n) ci,(n)* .? exp[-kyo(n)r,] , (14)0=1

Q$r A& - l)c?,(n - 1)-P = ~~n,(i,)g~~)-pexpi-iroDI)f,,i,where f, = z, -- z, , (12= 1,2, . . N.

These four equations can be rewritten as a matrix equation

(151

= l-(n - 1) D(n) P(n)where

(17)


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P. Yeh / Op t ics of ani sotropi c la yered media 41

rev[-irl(fl)f,l 0 0 00

P(n)= exp[-ir,(n>t,l

0 00 0 exp[-ir )bJ 00i 0 0 1.xp[-ir,(~~M

(18)The matrices D(n) are called dynamical matrices because they depend only on

the direction of polarization of those four partial waves. The dynamical matricesare defined in a way such that they are block-diagonalized when the mode couplingdisappears. This requires that Ar and A2 are the amplitudes of the plane waves ofthe same mode (polarization) such that the plane wave with amplitude Ar propa-gates to the right, whereas the plane wave with amplitude Aa propagates to the left.Likewise A3 and A4 are the amplitudes of the plane waves of the same mode, prop-agating, respectively to the right or left. The matrices, P(n),re called propagationmatrices, and depend only on the phase excursion of these four partial waves. Thetransfer matrix is defined asTn- 1 ,n =ryn - l)D(n)P(n). 09)Eq. (16) can thus be written

= Tn-r,n

The matrix equation which relates A(0) nd A(s)s therefore given by

wheres=N+ 1 and tN+l 0.Eqs.16) and (21) show how systematic the matrix method is for treating elec-

tromagnetic propagation in anisotropic layered media. If eq. (21) is representedgraphically in fig. 2, two dynamical matrices can be seen to be associated with eachlayer. The overall transfer matrix is the product of all these matrices from left toright. This completes the theoretical formulation of the 4 X 4 matrix method.

The matrix method just described is an exact approach to the propagation of


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48 P. Yelr / Optics of anisotrop ic laJ ered m edia

D- 0 (1) P(1) D-l{1 ) I

tlig. 2. Diagram representation of

N

l N) P N) D-l N

he matrix method.

S

(S)

electromagnetic radiation in anisotropic layered media. Both birefringent phaseretardation and thin film interference are considered. This differs from the tradi-tional 2 X 2 Jones calculus [26] which neglects the reflection from each interface.Therefore, in calculating the transmission and reflection properties of some bire-fringent filters, the results obtained from these methods are expected to be differ-ent. In fact, they are only very different in the fine structures of the spectralresponses [27,28] when the birefringence of the material is small.

In addition, there are several interesting optical phenomena in periodic birefrin-gent layered media which have been analyzed by using this 4 X4 matrix algebra.These include the indirect optical bandgap, exchange Bragg reflection and exchangeSole-Bragg transmission. Details can be found in ref. [IO].

4. Reflection and transmissionThe matrix method just discussed is very useful in the calculation of the reflec-

tance and transmittance of an anisotropic layered medium. Because of the aniso-tropy of the medium, mode coupling appears at the interfaces. Therefore, there arefour complex amplitudes associated with the reflection and another four associatedwith the transmission. These eight complex amplitudes can be expressed in terms ofthe matrix elements of the overall transfer matrix. To illustrate this, one considers,without loss of generality, the case of an anisotropic layered medium sandwichedbetween two isotropic ambient and substrate media. Assume that the light isincident from the left side of the structure, and let A,,A,, B B and C,, C, be theincident, reflected, and transmitted electric field amplitudes, respectively. By em-ploying the matrix method described in section 3, a transfer matrix can be foundfor any given anisotropic layered structure such thatAs4 =A,Bp,

M,, MS, M33 M34 CPM A442 M43 M44_ 4 IJ0

(22)


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P. Yeh /Opt ics of ani sotropi c lay ered medi a 49

The reflection and transmission coefficients are defined and expressed in terms ofthe matrix elements as follows:rss

YSP M,,M33 -M& f,,A~=O = M ,, -M ,,M ,,Ml,43 -M,,M,,

= M,, -M,df31

tss = MS3A~=O =M ,,M ,, -M ,,M ,,tsp = 4 1/ip=o = M ,, - M ,,M ,,

-Ml,= M,, -M&f,,

ct,, = $( 1 Ml1p A,=0 =M,,M,,-M,,M,,

(23)

(24)

3 (25)

9 (26)

, (27)

(28)(29)

(30)These reflection and transmission formulas are extremely useful in the calcula-

tion of the spectral response of an anisotropic layered structure. The matrix ele-ments are obtained by carrying out the matrix multiplication in eq. (21). Thegeneral explicit forms are normally not available. For fast results, a computerprogram is in general required. Even for the special case of periodic layered medi-um, closed forms for the reflectance and transmittance are too complicated toderive. These eight complex amplitudes are spectrally correlated (see ref. [lo]).

5. Ellipsometry of anisotropic layered structuresEllipsometry has long been recognized as one of the most accurate techniques

for determining the optical properties of materials. The basic mathematical probleminvolved for the case of isotropic films has been discussed by VaCicek [29] andmany other workers [30.31]. In recent years, many mathematical techniques havebeen developed for the ellipsometric study of anisotropic layered media; many ofthese are special cases of the uniaxial system, with the optic axis either perpendi-


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5 P. Yeh / O pt ics o f nisotropic layered media

cular or parallel to the surface [32-381. The differential 4 X4 matrix methoddescribed by Berreman 1121 provides a much more general approach to the pro-blem of stratified anistropic media, including continuous variation of the refractiveindices in the media. De Smet [39] has recently presented a paper at the ThirdConference on Ellipsometry in which he discussed a 4 X4 matrix formalism. This4 X 4 matrix formalism also used the amplitudes of the electric fields and magneticfields as the elements of a column vector which is function of position. In thematrix method discussed in secion 3, a constant column vector is associated witheach layer,

The new 4 X4 matrix algebra just discussed is very useful in calculating thereflectance and transmittance amplitudes which are externally measurable via theellipsometric techniques. For example, the ellipse of polari~tion for the reflectedlight can be expressed in terms of the matrix elements as

(31)where xi represents the ellipse of polarization for the incident light. In the ellip-sometric determination of the refractive indices, crysta1 axes orientations andthicknesses of the films several independent measurements obviously must be made.Since there are so many variables involved, a computer program is generallyrequired to determine these unknowns efficiently.

6 Guided waves

Birefringent multilayer waveguides, especially titanium diffused lithium niobatewaveguides [40,41], are becoming increasingly important in integrated optics. Thewaveguiding principle is similar to that of the isotropic case. Waves are to beevanescent in the regions outside the guiding layer. The propagation characteristics,however, depend on the direction of propagation. The analytic treatment for thegeneral multilayer birefringent waveguide suffers from the serious difficulty ofsolving an eigenvalue problem involving a large number of simultaneous linear equa-tions. A systematic approach is to use the matrix method described in section 3which involves the ~nipulation of 4 X 4 matrices.

As a result of successive matrix multip~~tions, a linear relation between thefields on both sides of a finite birefringent layered medium is obtained. The reflec-tance and transmittance coefficients have been shown to be expressible in terms ofthe elements of the overall transfer matrix. It will be shown in the following thatthe poles of the reflectance coefficients play an important role in the guided-modetheory of birefringent layered media.A basic problem in particle physics is that the poles in the scattering amplitude,which are assumed to dominate the scene, correspond to exchange of particlescarrying definite angular momentum [42]. In other words, a resonance scattering


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P. Yeh /Opt ics of ani sotropi c lay ered media 51

corresponds to an eigenstate of the composite system. It was suggested by Regge[43] in 1959 that the angular momentum be treated as a complex continuousvariable. In particle scattering, the angular momentum corresponds to the impactparameter, while in the optics of birefringent layered media the direction of inci-dence (or equivalently, o. and /3) is the corresponding variable. The a and /I variablescan now be extended into complex variables, and the poles of the reflectance am-plitudes, which correspond to the scattering amplitudes, can be sought. In general,the poles occur at complex values of 01 and f3, and each of these poles correspondseither to a guided mode or to a leaky mode.

From eqs. (23) through (30) it was concluded that the poles of the reflectanceamplitudes occur atMllM,, -~13~31 =o. (32)It is important to notice that at the poles of the reflectance amplitudes, the reflec-tivities are infinite. In order to fulfill the finiteness of the electromagnetic field, thesolution of the Maxwell equations consists of outgoing waves only. Eq. (32) isactually the mode dispersion relation

for a given birefringent layered structure. Eq. (33) can also be writtenw =w3,@p>p, = ( a /3),0 n = tan- (fl/o)

(33)

(34)

(36)The subscript en in eq. (34) indicates that the dispersion relation between o and &,depends on the direction of propagation defined by 0, in the xy plane. In order tobe a confined mode, the field amplitude must decay to zero at infinity (z = +m).Therefore, the propagation constant, /In, must be big enough so that the z com-ponents of the propagation vectors (i.e., 7,) are complex. Outgoing waves withcomplex propagation constant are evanescent waves. Therefore, the optical energyis guided by the structure and propagates parallel to the layers.

Because of mode coupling, pure TE or TM waves, in general, do not exist.Except for some cases with special crystal orientations, most of the guided modesare a mixture of ordinary waves and extraordinary waves. Another distinct featureof the guided waves in birefringent layered structure is the evanescent waves in abirefringent substrate. In the case of isotropic media, the evanescent waves have apure imaginary propagation constant. This however, is no longer true in birefringentlayered media. A guided wave in birefringent layered waveguide has, in general, twocomplex ys in the birefringent substrates. This makes the evanescent wave decayexponentially with an oscillatory intensity distribution.


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52 P Yeh /Optics ofanisotropic layered media

7. ConclusionA new 4 X 4 matrix algebra has been developed for investigation of the propaga-

tion of electromagnetic radiation in anisotropic layered media. The concepts ofdynamical matrix and propagation matrix have been introduced to clarify thismethod and to make it systematic. Diagram representation is also made possible viathe use of these two matrices. Applications of this new matrix method in the ellip-sometry of anisotropic layered media, as well as the guided wave in these media,have been illustrated.

References_[l] F. Abel& Ann. Physique (Paris) 5 (1950) 596, 706.[2] P. Yeh,A. Yarivand C.S. Hong, J. Opt. Sot. Am. 67 (1977) 423, 438.[3] P. Yeh and A. Yariv, Opt. Commun. 19 (1976) 427.[4] A.Y. Cho, A. Yariv and P. Yeh, Appl. Phys. Letters 30 (1977) 471.[5] P. Yeh, A. Yarivand A.Y. Cho, Appl. Phys. Letters 32 (1978) 104.[6] W. Ng, P. Yeh, P.C. Chen and A. Yariv, Appl. Phys. Letters 32 (1978) 370.[7] J.B. Shellan, W. Ng, P. Yeh, A. Yariv and A.Y. Cho, Opt. Letters 2 (1978) 136.[8] R.D. Dupuis and P.D. Dapkus, Appl. Phys. Letters 33 (1978) 68.[9] I. Sole, Cesk. Casopis Fys. 3 (1953) 366; 10 (1960) 16; J. Opt. Soc.Am. 55 (1965) 621.

[lo] P. Yeh, J. Opt. Sot. Am. 69 (1979) 742.[ll] P. Yeh, J. Opt. Sot. Am. 68 (1978) 1423.[12] D W Berreman, J. Opt. Sot. Am. 62 (1972) 502.[13] See, for example, R.M.A. Azzam and N.M. Bashara, Ellipsometry and Polarized Light

(North-Holland, Amsterdam, 1977) section 4.7, p. 340.[14] A. Wunsche, Ann. Physik(Leipzig) 25 (1970) 201.[15] J. Schesser and G. Eichmannb, J. Opt. Sot. Am. 62 (1972) 786.[16] C.B. Curry, ElectromagneticTheory of Light (McMiBan, London, 1950) pp. 356-369.[ 17 j Ii. Schopper, 2. Physik 132 (1952) 146.[18] A.B. Winterbottom, Kgl. Norske Videnskab. Selskab, Skrifter 1 (1955) 17.[lP] A.M. Goncharenko and F.J. Federov, Opt. Spektrosk. 14 (1962) 94; Opt. Spectrosc. 14

(1963) 48.[20] J. Schesser and G. Eichmann, J. Opt. Sot. Am. 62 (1972) 786.[21] D.A. Holmes and D.L. Feucht, J. Opt. Sot. Am. 56 (1966) 1763.[22] J.J. Stamesand G.C. Sherman, J. Opt. Sot. Am. 67 (1977) 683.[23] M. Born and E. Wolf, Principles of Optics (Pergamon, Oxford, 1964).[24] See, for example, H. Goldstein, Classical Mechanics (Addison-Wesley, Reading, MA,

1965).[25] See, for example, J.M. Stone, Radiation and Optics (McGraw-Hill, New York, 1963) sec-

tion 17-5.[26] R.C. Jones, J. Opt. Sot. Am. 31 (1941) 488.[27] Ref. [lO],p. 752.[28] P. Yeh, Opt. Commun. 29 (1979) 1.[29] A. Vasicek, J. Opt. Sot. Am. 37 (1947) 145.[3O] See, for example, O.S. Heavens, Optical Properties of Thin Solid Films (Dover, New York,

1965).


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P. Yeh /Opt ics ~~a rl i soiro~ i c lay ered media 53

[ 311 See, for example, ref. [ 131, ch. 3.[32] R&W. Graves, J. Opt. Sot. Am. 59 (1969) 1225.[33] D.J. D&yam and M. Moskovits, Appl. Opt. 9 (1970) 1868.[34] D. den Engelsen, J. Opt. Sot. Am. 61 (1971) 1460.[35] D.J. De Smet, .I. Opt. Sot. Am. 63 (1973) 958.[36] D.J. De Smet, J. Opt. Sot. Am. 64 (1974) 631.1371 R.M.A. Azzam and N.M. Basbara, J. Opt. Sot. Am. 64 (1974) 128.[38] M. E~hazIy-ZaghIou~ R.M.A. Azzam and N.M. Bashara, Surface Sci. 56 (1976) 293.[39] D J De Smet, Surface Sci. 56 (1976) 293.[40] I.P. Kaminow and J.R. Carruthers, AppL Phys. Letters 22 (1973) 326.[41] R.V. Schmidt and I.P. Kaminow, Appi. Phys. Letters 25 (1974) 458.[42] S.C. Frautschi, Regge Poies and S-Matrix Theory (Benjamin, New York, 1963).[43] T. Regge, Nuovo Cimento 14 (1959) 9.51.

DiscussionJ.B. Theeten (Laboratories DElectronique et de Physique Appliquee): In your calculationon exchange Bragg scattering in those layered structures, what do you expect the effect of anonabrupt transition between two successive layers to be on the quality of the guided wave?P. Yeh: In the event when there is a non-abrupt transition between two successive layers,the phe~amena of guided waves and exchange Bragg reflection still exist. Wowever, the matrixmethod developed here becomes only an approximation to the exact solution provided thetransition region near the interface is much smaller than the wavelength. Exchange Bragg reflec-tion and guided waves normally happen under different conditions, i.e., different LYnd 0.

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(1980) Yeh, P_Optics of anisotropic layered media..A new 4x4 matrix algebra-Surface Science 96(1–3) 41-53