Accurate method to determine the eigenstates of polarization in gyrotropic media

Accurate method to determine the eigenstates ofpolarization in gyrotropic media

Theresa A. Maldonado and Thomas K. Gaylord

Many practical modulator materials include combinations of electrooptically induced birefringence, opticalactivity, and/or Faraday rotation. Thus, there is a need for a procedure to design and analyze devicesfabricated with materials exhibiting any or all of these effects. In this paper a simple procedure employing anextension of the general Jacobi method is introduced for determining the properties of the two allowedelliptical eigenpolarizations for an arbitrary direction of propagation and for the principal indices and axes ofa general lossless, electrooptic, and gyrotropic medium. The procedure uses an iterative application ofunitary transformations to diagonalize the Hermitian impermeability tensor. A complex polarizationvariable is defined from elements of the unitary transformation matrix to determine the ellipticity, azimuthangle, relative amplitude and phase, and handedness of the two orthogonal elliptical polarizations. Thephase velocity indices of refraction are readily calculated with simple derived expressions. The procedure isnumerically stable and accurate for any crystal class, external field direction, and direction of propagation.

1. Introduction

In a recent publication,' a straightforward, numeri-cally stable method for performing electrooptic effectcalculations was presented. Simple analytic proce-dures were developed for calculating the principal di-electric axes and refractive indices of an electroopticcrystal of any crystal class subject to an external elec-tric field applied in a general direction. Simple formu-las were also developed for determining the two al-lowed eigenpolarizations and their correspondingrefractive indices for a general direction of phase prop-agation. However, the optical properties of a crystalmay be affected by other influences, internal or exter-nal, such as natural optical activity, an internal orapplied magnetic field, stress, and others. The opticalproperties and the induced changes in them may bedescribed by the relative permittivity tensor [e] or itsinverse, the impermeability tensor [-q], where []

[1/n2], and n is the index of refraction.The linear and quadratic electrooptic effects include

changes in the linear birefringence of a crystal. Theseeffects may be represented as symmetric perturba-tions to the impermeability tensor. The allowed po-

The authors are with Georgia Institute of Technology, School ofElectrical Engineering, Atlanta, Georgia 30332-0250.

Received 20 September 1988.0003-6935/89/112075-12$02.00/0.© 1989 Optical Society of America.

larizations remain linear, regardless of the direction ofpropagation. However, some electrooptic crystalssuch as bismuth silicon oxide are also optically activeand, therefore, exhibit natural reciprocal circular bire-fringence. This physical effect is manifested as a rota-tion of the linear polarization of the light on transmis-sion through the crystal. Correspondingly, theeigenpolarizations are no longer linear but are, in gen-eral, elliptical. Rotation of linearly polarized lightmay also be induced by an external field. For exam-ple, an applied magnetic field may induce nonrecipro-cal circular birefringence (Faraday rotation). Fur-thermore, an external electric field may inducereciprocal circular birefringence (electrogyration ef-fect). Media exhibiting circular birefringence, wheth-er natural or induced, are referred to as gyrotropic.The mechanisms giving rise to gyrotropy may be repre-sented as imaginary antisymmetric (Hermitian) per-turbations to the impermeability tensor.

The two general questions to be addressed here areas follows: (1) Given a crystal that is linear birefrin-gent (natural or induced) and/or gyrotropic (natural orinduced), what are the principal refractive indices andprincipal dielectric axes of the crystal? (2) What arethe two eigenstates (i.e., phase velocity indices andcorresponding eigenpolarizations) for a given directionof light propagation? Assuming the crystal to be loss-less, linear, and homogeneous, the answers to theabove questions can be obtained by first determiningthe eigenvalues and eigenvectors of the perturbed im-permeability tensor. In addition, since the eigenpo-larizations are elliptical in general, three pieces of in-

1 June 1989 / Vol. 28, No. 11 / APPLIED OPTICS 2075

formation are required to describe each state ofpolarization: azimuth angle, ellipticity, and handed-ness.

Similar problems with the electromagnetic descrip-tion of gyrotropy have been addressed in the literaturein terms of macroscopic theory2 -9 and quantum me-chanics. 9 -"1 As an example of the former, a method tocalculate the eigenstates of a naturally optically active,electrooptic crystal by diagonalizing the coupled-waveequations was presented by Yariv and Lotspeich. 3 Asan example of the latter, a method using quantumelectrodynamics to determine the eigenstates in opti-cally active, linear birefringent crystals was presentedby Eimerl.10 The approach presented in this paper isbased on the macroscopic properties of the crystal.The procedure introduced employs an extension of thegeneral Jacobi method, a known, very accurate, stable,and simple numerical routine. To obtain a full de-scription of the eigenpolarizations, a complex polariza-tion variable is used to map the eigenvectors of thetransverse impermeability tensor into a complex po-larization plane. The advantages of the method arethat (1) no assumptions are required; (2) it applies toany crystal class; (3) it applies to any field direction; (4)it applies to any direction of light propagation; and (5)it is numerically stable, accurate, and straightforward.

The constitutive equation will first be used to de-scribe the optical properties of a crystal. To provide ageometric interpretation of linear and circular bire-fringence, the index ellipsoid and gyration surface willbe reviewed. Next, the procedure for solving the givenproblem will be introduced. This discussion beginswith a brief overview of the eigenvalue/eigenvectorproblem for Hermitian matrices and continues with adescription of the extended Jacobi method and thecomplex polarization variable used to obtain the eigen-states for a given direction of phase propagation. Fi-nally, bismuth silicon oxide, an optically active, elec-trooptic, electrogyratory material, is analyzed toillustrate the simplicity and accuracy of the method.Boldface is used to denote a vector and [-] to denote amatrix throughout the paper.

II. Constitutive Equation

The material properties (principal permittivities orrefractive indices) of the crystal are represented by theconstitutive relation D = [e]E, where [El is the permit-tivity tensor of the medium. Disturbances to the opti-cal properties, which are typically very small in magni-tude, may be described by this tensor. They arecommonly expressed in terms of the inverse of thepermittivity tensor, [e]-1 = 1/co[e]-1 = 1/eo[l/n 2 ], whereEo is the permittivity of free space.

For a homogeneous, lossless, and nongyrotropic me-dium the permittivity (impermeability) tensor hasonly real components. Moreover, it is symmetric forall crystal classes and for any selection of the dielectricaxes,12 -14 and can, therefore, be diagonalized. In theprincipal coordinates system, the constitutive equa-tion is

[D.,l [c 0 O1FE,-

D, 0 0 ez E,

(1)

with the principal permittivities on the diagonal.If the medium is gyrotropic, the constitutive equa-

tion may be written as13-' 5

D = []E + iG X E = ['E, (2)

where [e] is the symmetric unperturbed permittivitytensor, i is .=T, and G is the gyration vector, which isuniquely defined for the mechanism producing thecircular birefringence. The vector cross product G XE in Eq. (2) may be represented as the product of anantisymmetric tensor [G] with the vector E. Thus,

D = [e] + iO[G]E = []'E. (3)

Therefore, the permittivity tensor [' is now clearlyHermitian as indeed it must be due to thermodynamicarguments.3,14

The constitutive equation may also be written as

E = ['"D = [-q]'D = (1/co)1[n] - i[][Gl[lnjD. (4)

From Eq. (4), the antisymmetric (imaginary) part of[nl'is

O -77,,?yGz ,x?7zGy

Im[] - G] = 77yG ° -ny7zGx (5)

_nx~qzGy ?lyqzGx °

An important point is that the imaginary part of []'has no effect on the principal dielectric axes and indi-ces of the crystal. The real part of [ti]' is real andsymmetric.

For review purposes some of the different types ofinfluences, both natural and induced, on the opticalproperties are now described.

A. Dielectric Properties with No Fields Applied

1. Natural Linear BirefringenceThe optical symmetry of a crystal is represented by

the permittivity (impermeability) tensor.12'14 If alldiagonal elements are equal, the crystal is isotropic. IfEx = ey E4 Ez, the crystal is uniaxial. If Ex $4 ey # Ez z Ex,

the crystal is biaxial. Therefore, both uniaxial andbiaxial crystals exhibit natural linear birefringence;e.g., Ez -Ey E" 0.

2. Natural Optical ActivityIn general, the macroscopic properties of a medium

depend on the temporal and/or spatial variation of theelectromagnetic field. For the case of natural opticalactivity, the properties are influenced by spatial dis-persion, the dependence of [e] on the magnitude anddirection of k at fixed frequency. 4"13-'9 The macro-scopic dipole moment per unit volume of the mediumat a given point depends on the field at and near thatpoint. In the optical frequency range, the effects ofspatial dispersion, in general, are small and are charac-

2076 APPLIED OPTICS / Vol. 28, No. 11 / 1 June 1989

OPTICAL ACTIVITY

/

Only noncentrosymmetric crystals can have naturaloptical activity. Table I provides a summary of thegyration tensors for all crystal classes that exhibit opti-cal activity.13

B. Dielectric Properties with External Fields Applied

k

k

(a)

FARADAY ROTATION

i/\* \

1. Linear Electrooptic EffectAn electric field applied in an arbitrary direction to a

crystal lacking a center of symmetry produces a changein the coefficients (1/n2 )i due to the linear electroopticeffect according to

BA(1/n2 )i = rzjEj

k

(b)

Fig. 1. Sense of optical rotation relative to the direction of propaga-tion k for (a) natural optical activity and (b) Faraday rotation.

terized by the first power of a/X (<<1), a dimensionlessquantity, where a is of the order of the lattice constantand X is the wavelength of the light in the medium.Therefore, to first order, the constitutive equationisI 4 ,15 ,17 ,18

Di = eijEj + yj(,(aEj/ax,) = 1jEj, (6)

c = (,k) = jj(w) + ijj(w)kj, (7)

where eij(co) is the permittivity tensor without opticalactivity and yijl(W) is a third-rank real antisymmetrictensor in the indices i and j, resulting in E' beingHermitian. The second term can also be representedby G X E, where G is the gyration (axial) vector.'3-'5

The displacement vector D rotates in a helical fash-ion about the wavevector k, so G is parallel to k and thecomponents of G are functions of the direction cosinesof k. The sense of rotation bears a fixed relation to thedirection of propagation, as shown in Fig. 1(a). If alinearly polarized input lightwave is transmittedthrough an optically active crystal and then is reflectedback through the crystal, the net rotation of the polar-ization is zero. Therefore, natural optical activity is areciprocal effect.4,' 3

Defining the direction of k in spherical coordinates(Ok,Ok), the components and magnitude of G are13 ,17

Q = G sinOk cosok, Gy = G inOk sinfk, G = G COSk,

IGI = G = g11 sin2 Ok COS2 Ok + g2 2 sin2Ok sin2Ok + 33 COs20k

i = 1, . .,6,i = xy,z = 1,2,3, (9)

where rij is the ith element of the linear electrooptictensor in reduced-subscript notation,'3, 20 In matrixform Eq. (9) is

r01 r1 2 r13

A(I/n2 )2 r21 r22 r23 EA(l/n 2

) 3 r30 r32 r33 EA(1/ 2)4 r41 r42 r43 LEZ

A(1/n2 ) r5 l r52 r 3(1/n2) r6i r62 r63

For a noncentrosymmetric crystal, the new imperme-ability tensor in the presence of an applied electricfield is, in general, no longer diagonal in the originalaxes system. The perturbed impermeability tensor is

[/n2 + A(1/n2 )0 A(2/n2 )6 A(1/n2 )5[1/n21' = A(1/n2)6 1/n2 + A(l/n2 )2 A(1/n2 )4

A(1/n2 ), A(j/n 2)4 1/n2 + A(l/n 2 )3

(11)

However, the field-induced perturbations are real andsymmetric, so the symmetry of the tensor is not dis-

Table 1. Gyration Tensors g for All Crystal Classes Exhibiting NaturalOptical Activity13

Biaxial

Triclinic Monoclinic

1 2 (2 || y) ( y)

g1 3 1 Fg1 0 g-31 0

g-1 g2 2 g 3 Og g". O 012 0 gz3

3 1 g3 2 g33- L 1, 0 03 3 J -L 3 ° -

Uniaxial

Tetragonal

4, 422 4 42m (2 | x)

[91 ° .01 ° gl; -g': ° ] Z [g O 1

Orthorhombic

222 2mm

[0ll ° 1 0 ° g 1

0 O 0 Z~ OB

Trigonal and Hexagonal

3, 32, 622

0 g 1

° ° g3j

+ 2g12 sin2Ok sinkk CoSek

+ 2 13 sinO& cosOk cos~k

(8)

+ 2g23 sinOk CoSOk sinlk,

where gij are the components of the gyration tensor.

Isotropic (without center of symmetry)

Cubic

432, 23

0 0119ll '1


turbed. These changes to [1/n2 ] have the effect ofchanging the principal axes and indices of the crystal.The electrooptic tensor for all crystal classes is summa-rized in numerous texts.13"15 All optically active crys-tals are also electrooptic, but the converse is not true.

2. Electrogyration EffectAn applied electric field may not only induce linear

birefringence, but in many cases2' it may also inducecircular birefringence through the electrogyration ef-fect. Considering only first-order spatial dispersion,the gyration tensor is21-23

g, = j + tiikEk, (12)

where gij is the gyration tensor with no applied fieldand tijk is the third-rank electrogyration tensor, whichhas the same symmetry as the electrooptic tensor.The second-rank tensor tijkEk alters gij in the samemanner as Eq. (9) alters the impermeability tensor inEq. (11). The net result is a change in the magnitudeof the gyration vector G as found by Eq. (8) and,therefore, a change in the specific rotation (polariza-tion rotation per unit thickness) of the medium. Theelectrogyration effect may be viewed as induced opti-cal activity, a reciprocal effect. This perturbationchanges the impermeability tensor in the same manneras optical activity, that is, producing Hermitian off-diagonal elements as in Eq. (4) rather than the realsymmetric off-diagonal elements of Eq. (11).

3. Faraday RotationIn a Faraday active medium, the macroscopic prop-

erties are influenced by frequency dispersion causedby an applied magnetic field.4 '8,3 14,2 4-27 The fieldcreates a relative shift between the phase velocity indi-ces of refraction of the two allowed eigenpolarizations,inducing circular birefringence. The constitutiveequation is

D = [t]E + ieokB X E, (13)

where [El is the unperturbed permittivity tensor and ,6is the magnetogyration constant of the medium. Thereal part of the impermeability tensor is symmetric inB, and the imaginary part is antisymmetric in B.14Since only first-order effects are considered, the realpart of the tensor remains unchanged. In this case,the imaginary part is represented by the gyration vec-tor that is proportional to B; i.e., G = /'B. Thus, thecomponents of G are functions of the direction cosinesof B(0BGB), rather than those of k. That is,

G = B sinGB cosOB, Gy = VIB sinOB sinOB, G2 = kB cosOB.

(14)

The magnitude of G is merely the product of 4' and themagnitude of B. For Faraday rotation, the sense ofrotation bears a fixed relation to B, as shown in Fig.1(b). The eigenpolarizations are preserved on reflec-tion so that the net rotation is doubled. Thus, Fara-day rotation is a nonreciprocal effect.4 11"13

x

z'

Fig. 2. Index ellipsoid cross section (crosshatched) that is normalto the wavevector k and passes through the origin. The principalaxes of the crosshatched ellipse represent the directions of the al-lowed linear polarizations D0 and D2 . D1 , D 2, and k form an orthog-

onal triad.

C. Combined Effects

If, for example, a lossless, biaxial, and optically ac-tive crystal is subject to an applied electric field, theimpermeability tensor is altered by Hermitian pertur-bations and is5"13,14,24

1xx 1Ixy lXzl[1/n2 ]' = []' = 7xy 71yy 7yz|

LXZ nyz 71ZZ -

(15)

where -ij, i F j, are complex and qij are real. As statedpreviously, the imaginary parts of the off-diagonal ele-ments do not affect the principal axes or indices of thecrystal. They affect only the state of the allowedpolarizations and the phase velocity indices. The ei-genpolarizations are now elliptical, in general, ratherthan linear as with the electrooptic effect. A Hermi-tian matrix may be represented by a quadratic surfacein complex space. In real Cartesian space, however,only the real part of [7]', which is symmetric, contrib-utes to a quadratic surface (ellipsoid). Geometric sur-faces representing optical properties of the crystal arediscussed in the next section.

Ill. Geometric Approach

A. Index Ellipsoid

The index ellipsoid is a construct whose geometriccharacteristics represent the phase velocities and thedirections of electric displacement vibration for a giv-en optical wavevector direction in a crystal. The gen-eral index ellipsoid for an optically biaxial crystal isexpressed in Cartesian coordinates as12"13

(x2/n2) + ( 2/n,) + (z2/n2) = 1, (16)

where nx, np, and n, are the principal refractive indicesof the crystal. Since the permittivity (impermeabil-ity) tensor is positive definite, the surface is always anellipsoid. If nx = ny, the surface becomes an ellipsoidof revolution, representing a uniaxial crystal. An iso-tropic crystal is represented by a sphere (degenerateellipsoid) with the principal axes having equal length.These surfaces are shown in Fig. 1 of Ref. 1. Alsoshown are the optic axes for each crystal symmetry.


The eigenstates for an arbitrary direction of propa-gation in a crystal are found in the elliptical crosssection perpendicular to k that passes through theorigin of the index ellipsoid, as shown in Fig. 2. If theoptical properties are not disturbed, the major andminor axes of the cross sectional ellipse represent thetwo allowed orthogonal linear vibration directions of D(eigenpolarizations) for that particular direction ofpropagation. The lengths of these axes correspond tothe respective phase velocity indices of the allowedpolarizations. They are, therefore, referred to as thefast and slow axes. As with the principal axes andindices of the crystal, these eigenstates, in general, areaffected by real symmetric perturbations (for example,the electrooptic effect) to the impermeability tensor.However, the antisymmetric perturbations affect onlythe eigenstates (the eigenpolarizations and phase ve-locity indices). The major axes of the cross-sectionalellipse correspond to the major axes of the allowedpolarizations in this case.

B. Gyration Surface

The geometry of the index ellipsoid provides onlypartial information about the eigenstates if the al-lowed polarizations are not linear. In this case theellipsoid can be used to determine only the orientationof the allowed elliptical polarizations but nothingabout the properties of optical rotation for a givenwavevector direction. The gyration surface, however,may be used to illustrate the directional dependence ofoptical rotation in a gyrotropic crystal. For opticallyactive crystals the surface is constructed from the gy-ration tensor [g] in the same way that the index ellip-soid is constructed from the impermeability tensor.17That is, the distance from the origin to any point on thesurface is given by

G = gjkikj, (17)

where k and k are the direction cosines of the wave-vector k. Equation (17) is the directional magnitudeof the gyration vector, G = IGI, as given by Eq. (8).Since the gyration tensor is not necessarily positivedefinite, the surface may have a variety of forms.Shubnikovl7 provides a complete set of all possiblesurfaces for isotropic, uniaxial, and biaxial crystalclasses that exhibit optical activity. For example, Fig.3 is the gyration surface for right-handed quartz (class32), which is positive uniaxial and optically active.The first two diagonal elements, g1l = g2 2 , are negative,and g3 3 is positive. The surface is given by G = -glilsin 26ko + g3 3 COS20ko, where Oho is the angle between theoptic axis and k. Therefore, optical rotation along theoptic axis is right-handed and is denoted by the whitesurface. However, propagation perpendicular to theoptic axis gives optical rotation in the opposite direc-tion and is denoted by the dark surface. There is nooptical rotation (G = 0) for propagation directed 56°from the optic axis, as determined by the measuredquantity g 11/g 3 3 0.45, and the eigenpolarizations arelinear.

z, Optic AxisNoOptical

' Activity(LinearPolarization)

x ,y

Fig. 3. Gyration surface for right-handed quartz (class 32). Thewhite surface depicts right-handed optical rotation with the maxi-mum rotation occurring for propagation along the optic axis. Thedark surface depicts left-handed rotation with maximum rotationalong a direction perpendicular to the optic axis. There is no optical

rotation for propagation -56° from the optic axis.

y

B

x

Fig. 4. Gyration surface for Faraday active crystals. Maximumoptical rotation occurs for propagation parallel and antiparallel to B.The white surface depicts rotation of one sense while the dark

surface depicts rotation of the opposite sense.

For Faraday active crystals a gyration surface mayalso be constructed. As stated before, the sense ofoptical rotation bears a fixed relation to the directionof the applied magnetic field B. Maximum rotation isachieved for propagation parallel and antiparallel tothe applied magnetic field. If propagation is in adirection inclined to B, the degree of optical rotationwill decrease as cosOkB, where OkB is the angle between kand B.14 Therefore, Fig. 4 is a representation of thegyration surface for this type of crystal.

IV. Analysis

The problem is to determine the principal axes andindices of the crystal and the two allowed orthogonaleigenstates of polarization (Di and D2) and the corre-sponding phase velocity indices (n, and n 2 ) for a gener-al wavevector direction and a general external field.The solution to the problem lies in determining the


/1

eigenvalues and eigenvectors of Eq. (15). The methodchosen to address this problem involves diagonalizingthe matrix using an extension of the general Jacobimethod. For Hermitian matrices, the eigenvalues arereal. The eigenvectors, on the other hand, are com-plex in general. Thus, additional information is re-quired to describe the general state of the eigenpolari-zations (the complex eigenvectors). For linearpolarization, only the orientation (azimuth angle) inthe plane transverse to k is required. For ellipticalpolarization, the ellipticity and handedness as well asthe azimuth angle must be determined. Furthermore,for linear polarization, D oscillates in a plane in onefixed direction perpendicular to k. If D is ellipticallypolarized, it no longer oscillates in a plane but ratherpropagates in a flattened helix about k, as shown inFig. 5. A full description of the eigenpolarizations isobtained by using a complex polarization variable.

A. Principal Axes and Principal Refractive Indices

A detailed description of a procedure to determinethe new principal axes and indices of a crystal subjectto real symmetric perturbations is given in Ref. 1.That approach employs the general Jacobi method,and it is known to be an accurate, numerically stable,and simple routine for diagonalizing real symmetricmatrices. The routine is iterative, and it involves thecalculation of an elementary plane rotation angle ateach step to bring to zero the largest off-diagonal ele-ment. Also, the method allows for consistent labelingof the new axes: a global rotation axis and a globalrotation angle are calculated from the resulting cumu-lative orthogonal transformation matrix. To find theprincipal axes and indices of a crystal with Hermitianperturbations to its optical properties, as given by Eq.(15), the same procedure is performed but only on thereal part of the matrix. The authors refer the reader toRef. 1 for this part of the analysis.

B. Eigenstates of Polarization and Phase Velocity Indices

The eigenvalue problem for Hermitian matrices isaddressed by a unitary transformation, [a] [n]'[a]H =[X], where [a] is the unitary transformation matrix([a]H = [a'), [nq]' is a Hermitian matrix, [X] is the

resulting diagonal matrix of real eigenvalues, and Hdenotes complex conjugate transpose. As suggestedby Wilkinson,28 a form of the unitary matrix that canbe used is

[a] cos4' exp(iB) sin'I (18)

-exp(-iB) sinP cos J(

This matrix has the effect of transforming a systemfrom Cartesian coordinates to a complex (helical) coor-dinate system.29 There are two parameters, '1' and B,to determine. For real symmetric matrices B = 0 and4 represents the elementary Cartesian plane rotationangle. Using the expression for a unitary transforma-tion with the matrix of Eq. (18), a set of relationshipswas derived which results in a version of the generalJacobi method extended to Hermitian matrices. Thetwo unknown parameters P and B are calculated at

Fig. 5. Flattened helical contour of an elliptically polarized propa-

gating wave at an instant of time. The radial vectors to the contourrepresent the displacement vector D ( k).

Y"

x

Fig. 6. Orthogonal transformation of the (x,y,z) dielectric axes to

the (x",y",z") coordinate system of the wavevector k (z" 11 k)represented in polar coordinates (,kOk)-

each iteration step. With additional algebra, the sim-ple expressions in the Appendix were obtained forupdating the elements of [aq]' as the diagonalizationprocess proceeds. The parameter B was determinedto be the argument of the off-diagonal element A7ij, i 5dj, and is the required value for driving that element tozero with a rotation in the (ij) complex plane. Theseformulas reduce to those of the general Jacobi methodfor real symmetric matrices.' The Jacobi method us-ing these modified formulas was programmed and test-ed. The results for several test matrices were foundwith virtually zero error. These results were oftenmore accurate than those obtained with the commer-cial IMSL30 routine, EIGCH.

To find the eigenstates for a specific direction oflight propagation, a real orthogonal transformationmust first be performed on the Hermitian matrix [-q]' toplace the problem in a coordinate system of k. Thewavevector direction is specified by the spherical coor-dinates (0kkk) in the original (x,y,z) coordinate sys-tem. A new coordinate system (x",y'",z") is definedwith z" parallel to k and x" lying in the (z,z") plane.The transformation to the (x",y",z") system is pro-duced first by a rotation kk about the z-axis followed bya rotation Ok about y " as shown in Fig. 6. This trans-


ZZt .- /

formation is described by

x = x cosOk cos(k + y CoSOk sinPk - z singk,

y " = -x singk + Y CoS4k,

z = x sinOk cosok + y sinOk sinok + z cos0d.

The transformed tensor []" in the (x",y") plane,transverse to k, is

In]'"= 7 xx lxy1 ,nxy nZz_

(20)

where

XX= X c(osX k Cos + nyy sink) cos2Ok + ,, sin20,

+ 2,,y, cos2Ok CoSk sinfk

- 2 COSOk sinOk(nlxz, Cos~k + nlyzr singk),

1 yy= xx sin 2k + nyy COS 0k -

2xyr cosok sinok, (21)

nY= (nyy - nx) CosOak COSck sintk

+ COSO(7xy COS k -xy sin 0

+ sink(7ix, sinkk - n COSk) = yxI

and flij denotes the real part of qij. Therefore, theHermitian property is preserved. The third row andcolumn can be neglected, since the vibration directionof the eigenpolarizations is transverse to k. Next, thephase velocity indices of refraction are easily deter-mined using the relationships in the Appendix (with aslight modification) to diagonalize the 2 X 2 matrix ofEq. (20). The required rotation in the complex(x",y") plane is

b = /2 tan'112 sgn(7yr)InyI/(i7x - 77' (22)

where 11 denotes magnitude and sgn(n'yr) is the sign offlxyr- The phase velocity indices are

n = 77X cos2 4 + 77' sin24 + 2 sgn(727xyr)I7xy cosl' sin -"2 ,2 3

= 1n4x sin2 I + n', cos - 2 sgn(nxyr)b1xyI coso sinfl4"2 .

The corresponding states of polarization are just therows of the 2 X 2 unitary transformation matrix of Eq.(18):

D = [cosF exp(iB)sinbIH,

D2= [-exp(-iB)sin4' COS4]H, (24)

where B = arg(77xy). These states are the left eigenvec-tors of []"; the right eigenvectors are the complexconjugate transpose of Eq. (24) and are

D = [ cost ] D = F-exp(iB) sinl (25)lexp(-iB) sinj [ cos4' J

The orthogonality relation, D, - D- = 0, is satisfied.From Eq. (23) if n, < n2, as shown in Fig. 7, D, is thefast wave and D2 is the slow wave. If n1 > n2, the fastand slow waves are reversed.

Now the full description of these eigenpolarizationsmust be determined, i.e., azimuth angle, ellipticity,and handedness. The eigenpolarizations have thesame ellipticity but opposite senses of rotation with

y

x"Fig. 7. Cross-sectional ellipse of the index ellipsoid in the (x ",y")

plane. The x"' andy"' axes represent the major axes of the eigenpo-larizations oriented relative to x" andy". The two eigenstates haveorthogonal major axes, opposite handedness, and the same elliptic-ity. The wavevector k and the z " axis are normal to the plane of the

figure.

orthogonal major axes. The azimuth angle or direc-tion of the major axes for D, and D2 is easily obtainedfrom Eq. (22) as follows. The orthogonal transforma-tion performed on [7i]' can be visualized in terms of theindex ellipsoid. The cross-sectional ellipse transverseto k is found by taking only the real part of the trans-formed tensor []", giving X& T Re{[n]"}X" = 1, or

;XXX + , + 2'x'y"y" = 1 (26)where Re [1] "} is the real part of [] " and X" = [x /lyf] T.The azimuth angle A3 for D, is the angle in the realCartesian (x",y") plane required to diagonalizeRet[]'"}. It is

#I1 = 1/2 tan-'[2n1 yr/(nLx - ny)] (27)

This expression is the same as Eq. (22) if the numera-tor 2 sgn(xiyr)l1nxyl is replaced by 2 nxyr. The azimuthangle for D2 is just 02 = /1 + 7r/2. The angles /1 and /32define the orthogonal semiaxes directions (x "' and y"'in Fig. 7) of the cross-sectional ellipse of the indexellipsoid. If Im(n7xy) were zero, the lengths of thesemiaxes would correspond to the phase velocity indi-ces for the two linear polarizations. These indiceswould be calculated from Eq. (23) with /1 rather than4,.

The ellipticity and handedness of D, and D2 can befound through the use of a complex polarization vari-able (CPV) x.31 These eigenpolarizations are in theform of a two-component Cartesian Jones vector or-thogonal to k:

D - [_DJ exP(i5;)]

DYJi LIDY1 exp(iby) i (28)

The form of the CPV is then x = r exp(iWA), where r= lDy/DI and A = y - , It performs a bilinear


/(19)

y"I

imixID I

ReX

Fig. 8. Cartesian complex plane of polarization. Each point in the

plane represents a polarization state. The basis states are the

horizontal linear polarization at the origin and the vertical linearpolarization at infinity. The dashed circle represents the unit circle(unit relative amplitude). The radial line represents a contour of

constant relative phase of 7r/4.

transformation from the complex (x ",y ") plane to po-larization space, shown in Fig. 8, where points in thecomplex x plane represent polarization states. Thisplane, in fact, maps directly onto the Poincare sphere.The equator of the sphere is the horizontal real axis oflinear polarizations, and the north and south poles arethe points R and L of the right- and left-circular polar-izations, respectively. From D1, the CPV is

Xl = exp(-iB) tan@, (29)

and from the orthogonal state D2 ,

X2 = -l/xl = -exp(-iB) cot4. (30)

Relationships between the CPV xi and the azimuthangle /1 and ellipticity angle 0i are given by Azzam andBashara 3 ' as

tan2o, = 2 Re(xl)/(1 - Ixi'), (31)

sin2t 1 = 2 Im(x1 )/(l + Ix112).

In terms of the impermeability tensor elements andthe rotation angle 4,, the azimuth angle /1 was given inEq. (27), and the ellipticity angle 4, was derived to besin241 = -1Im(n')/l11xl} sin24, = -sinB sin24, or

(, = -/2 sin-'(sinB sin24). (32)

The ellipticity is given by tanki. Furthermore, therelative amplitude of the orthogonal components of DIis defined as r1 = tan@P, and the relative phase is A51 =-arg(nxY) = -B.

The handedness of the polarization is determined bythe sign of the relative phase. If Ad > 0, the ellipticalpolarization is left-handed. If Ab < 0, the polarizationis right-handed. And if sb = 0 or mr, m = 1,2,3,. . ., thepolarization is linear.

For the orthogonal polarization state, the parame-ters 02, E2, r2, and A32 for X2 are3i

02 = p1 + 7r/2, 2 = -4, r2 = cot', A62 = -7r + Able (33)

The orthogonality condition for x1 and X2 is X1X2 = -1-

k, z"', +lZ . -

Y" X

Fig. 9. Principal transverse crystal orientation of Bi12 SiO2 0 (BSO).

The external electric field is applied in the [ 1 0] direction, and the

direction of propagation is along [ 1 0]. The (xy,z) coordinate

system represents the unperturbed dielectric axes. The new coordi-nate system resulting from the electrooptic effect is represented by

(x',y',z'). The coordinate system of the wavevector k is given by(x ",y ",z ") with z" II k. Finally, the (x "',y "',z "') coordinate system

with z " I z" [[ k represents the principal axis coordinate system ofthe eigenstates for the given k.

Therefore, in the process of diagonalizing a Hermitianmatrix, the relative phase iA and ellipticity angle t arechanging iteratively. In the case of a real symmetricmatrix, the process is interpreted as a rigid body rota-tion.'

The advantages of this method include the follow-ing: (1) It is accurate and stable for all crystal classes.(2) Stable orthonormal eigenvectors are found simul-taneously with the eigenvalues. (3) The unitary trans-formation matrix is easily determined by elements ofthe perturbed impermeability tensor. (4) All descrip-tive information about the eigenpolarizations of a crys-tal for a given k is obtained from the unitary transfor-mation matrix with simple formulas. (5) Thecombined effects of real and Hermitian perturbationsto the impermeability tensor can be straightforwardlyhandled.

V. Example: the Sillenite Crystal Class

A cubic sillenite crystal of class 23 is examined toillustrate the ease and accuracy of the method justdescribed. This crystal class is being widely investi-gated for applications in dynamic real-time holograph-ic interferometry and spatial light modulation.72 3

The class includes bismuth silicon oxide (BSO), bis-muth germanium oxide (BGO), and bismuth titaniumoxide (BTO). These crystals are electrooptic, optical-ly active, electrogyratory, piezoelectric, and elastoop-tic. The electrooptic effect, optical activity, and elec-trogyration effect combined may strongly affect thepolarization in these crystals. BSO is examined toillustrate these influences on the eigenpolarizations.

Two principal configurations of BSO, both trans-verse, are used for volume holography. One of them isshown in Fig. 9.7 An external electric bias is applied inthe [ 1 0J direction, and the direction of light propaga-tion is [1 1 0] or (0k,0k) = (1350,90°). In the (x,y,z)coordinate system, the unperturbed impermeability


tensor for this cubic isotropic crystal is[/n 0 01

[1/n2] = 0 1/n 0 0 (34)

0 0 1/n'J

where no is the principal refractive index of the crystal.The index ellipsoid is a sphere. The gyration tensorfor natural optical activity is a function of the givenapplied electric field and wavevector directions and is

g11 0 t41E,

[g]' = 0 g11 A41E. ,E 4 1 E, 41Ex g1l _(35)

where 41 is the electrogyratory coefficient in contract-ed notation. The gyration surface changes from asphere to the surface [from Eqs. (8) and (17)],

G = gl - J 4 1E sin0k cosok(sink + cosk), (36)

where Ex = Ey = -1/RE, and E is the magnitude ofapplied field. If E = 0, then G' = G = gl, which is asphere. The perturbed gyration surface is shown inFig. 10. Note by Eq. (36) that for the given field ([i 10]) and wavevector ([1 1 0]) directions the effect ofelectrogyration is not present. Therefore, with theelectrooptic effect, optical activity, and the given di-rection of propagation k taken into account, the tensorbecomes

[I =

1/no

0

L/~J2I-r4lE + ig 1 /0

0

1/no

1/4-r 4,E + ig1 /no

1/2 -1/X ]Tandy = [-1/2 -1/2-1/2]T. Numericalvalues 7 for the various parameters are no = 2.53 and r4 l

= 4.41 X 10-12 mn/V at a free space wavelength of Xo =0.6328 ,um. From these numbers the lengths of theprincipal axes are nick = no - /2n3r41E = 2.52996 andny, = no + /2 nor4 lE = 2.53004 for a field magnitude of106 V/m.

The phase velocity indices are found by diagonaliz-ing the entire matrix [n" of Eq. (39). The requiredrotation angle is

D= 1/2 tan-12 sgn(7xyr)'yI(1/n2 _- /n2)} = -45°, (41)

and IqxnyI = {(r41E)2 + (gll/n4)2}1/2. The constant g,, iscalculated from the measured specific rotation of p =21.40/mm (Ref. 7) which is equal to 7rgl/Xno. There-fore, g11/n4 = Xp/7rn3 = 2.6618 X 10-4, giving InxY =2.6621 X 10-4. Optical activity dominates the magni-tude of the off-diagonal element qXY The phase veloc-ity indices are found from Eqs. (23) to be

= {1/n2 + 1,7 - 11-/2 = 2.52785 (fast wave),

n2 = {1/n - 7 1}-/2 = 2.53216 (slow wave).(42)

The circular (elliptical) birefringence is then An, = n2

-ni = 0.0043111. Without the electric field applied,the indices are n1, = 2.52785 (=nl) and n2' = 2.532158(<n2 ), giving a circular birefringence of Anct = n2' -ni

= 0.0043080. Therefore, the electrooptic effect only

1/_J2-r4 IE - ig1/n}

1/42-r4 lE -ig/no}1/n2

(37)

where G. = -GY = -1/pg,,. The new orientation ofthe index ellipsoid is obtained by applying the generalJacobi method as described in Ref. 1 to the real part ofEq. (37). The crystal is now biaxial, and the principalindices and axes are

n,,, = 2.52996 *' = [1/2 1/2 - 1C]T

ny, = 2.53 y = [-1/C2 1/ O]T (y'llk). (38)

n, = 2.53004 z' = [1/2 1/2 1 /f2]T

To determine the eigenstates of the crystal for thegiven k, the entire []' tensor of Eq. (37) must betransformed to the (x",y",z") coordinate system byEqs. (19). The resulting 2 X 2 matrix in the (x",y")plane is given by

1 /n2 (-r 4 1E + ig1 1/n4) 1, (9

[(-r4 1 E - igll/n4) 1/n2 Jandx" = -z([0 0 1]), y" = -1/V2x - 1/y([i 1 0]), andz = -1/2x + 1/Vy([1 1 0] Ilk). The cross-sectionalellipse is, by Eq. (26),

1/n0(x 2 + y ti2) - 2r4 lEx My" = 1. (40)

The azimuth angle is found by placing this cross sec-tion in principal coordinates, i.e., by diagonalizing thereal part of Eq. (39). This angle is / = /2tan-'-2r 4 E/(1/n0 - 1/n2)} =-450. Therefore, theaxes of the cross-sectional ellipse are along x"' = [1/2

(ok = 1800)

Fig. 10. Gyration surface for BSO in the (x,z) or (y,z) principalplane. The dashed circle represents the surface projection with noapplied electric field, and therefore, the optical rotation is invariantwith the wavevector k direction. The solid heart-shape contourdepicts the surface when a field is applied in the [1 0] direction.(This contour depicts actual calculations using Eq. (36) with g, = 2,t41 = 12, and E = 1.) The magnitude of radial vectors from theorigin to the surface is a measure of the optical rotation per unitlength which depends on the direction of k. Note that opticalrotation is not affected by the electric field for propagation in the

(x,y) plane.

slightly enhances circular birefringence. The corre-sponding eigenvectors are found from Eq. (25) to be


-).- X, Y ( 0k = go")

D F 1/ 1L -(0.01656 + io.99986)1/Cj (43)

L (0.01656 -iO.99986)1/V2

D2 /

and B = arg(77'y) = -89.05080.The CPV xl is -exp(-iB). From Eq. (31), the azi-

muth angle is /3 = -450, which agrees with the angleobtained before. Also from Eq. (31), the ellipticityangle is sin2t = Im(xj) = sinB, which is equal to-1m(77Y)/177xY sin24, = -sinB sin24, = sinB from Eq.(32) for 4, = -45°. Therefore, Eqs. (31) and (32) areconsistent. The ellipticity angle is then -44.52541°,and the ellipticity of the polarization is tant =-0.98357° (almost circular polarization). The rela-tive amplitude is r1 = 1, and the relative phase betweenDxj and Dy, is zAX = +89.050. The correspondingpoint on the complex x plane in Fig. 8 is on the unitcircle at the phase angle of +89.050. Since A61 > 0, D,is left-handed polarization, and n 1 = nL is the corre-sponding index. Therefore, D, corresponds to the fastwave.

For the orthogonal polarization D2, the CPV is X2 =exp(-iB). The azimuth angle is +450, and the ellip-ticity angle is -tj or +44.52541° for an ellipticity of0.98357. The relative amplitude is r2 = 1, and therelative phase between Dx 2 and Dy 2 is -90.95°. Thecorresponding point on the complex x plane in Fig. 8 ison the unit circle at the phase angle of -90.95°. SinceA32 < 0, D2 is right-handed polarization, and n2 = nR-Therefore, D2 corresponds to the slow wave.

Finally, an additional note is that the phase retarda-tion F between DI and D2 for a given crystal thicknessd may be calculated from the circular (elliptical) bire-fringence An,. It is given by r = 27r/XAncd and, ingeneral, includes the effects of both the natural opticalactivity and the external electric field (e.g., electroop-tic and electrogyration effects).

VI. Conclusion

A straightforward systematic procedure for per-forming electrooptic effect calculations was developedand presented in Ref. 1. That approach, which em-ploys the general Jacobi method, can be used to ana-lyze propagation in electrooptic materials in any crys-tal class for an arbitrary electric field direction andarbitrary wavevector direction. The properties of theimpermeability tensor were exploited to obtain simple,stable, and accurate expressions for determining theprincipal axes and indices of a crystal and the eigen-states (phase velocity indices and eigenpolarizations)for a given direction of propagation k.

In this paper that procedure has been extended togyrotropic crystals. External (or internal) influences,such as optical activity, electrogyration effect, andFaraday rotation may now be included, singly or to-gether. These circular birefringence effects are morecomplicated and, in general, produce elliptical eigen-polarizations. The extended method requires a uni-tary transformation from a Cartesian coordinate sys-

tem to a complex helical coordinate system todetermine the eigenstates for a given direction of prop-agation. Using a unitary matrix suggested by Wilkin-son,28 a set of formulas has been derived that results inan extended version of the general Jacobi method ap-plicable to Hermitian matrices. These relationshipsare given in the Appendix. In addition, a complexpolarization variable was introduced to quantify theconnection between the elements of the perturbed im-permeability tensor and the eigenpolarizations. Thenew procedure reduces easily to the less complicatedmethod of Ref. 1 where the electrooptic effect alone ispresent.

Two specific questions were posed and answered inthe present work:

(1) Given a crystal that is linear birefringent (natu-ral or induced) and/or gyrotropic (natural or induced),what are the principal refractive indices and principaldielectric axes of the crystal?

(2) What are the eigenstates for an arbitrary direc-tion of light propagation?

The step-by-step procedure introduced in this paperis summarized as follows:

(1) The general Jacobi method as described in Ref. 1is applied to the real part of the perturbed imperme-ability tensor [X]' to determine the principal indicesand axes of the crystal.

(2) A real orthogonal transformation is performedon []' to place the tensor in the coordinate system of k(x",y",z"), giving [n]".

(3) Using formulas from the Appendix, the eigen-states for the given k are obtained.

(4) From the eigenvectors of [ni]" a complex polar-ization variable (CPV) is defined. Using the CPV, thedescriptive properties of the eigenpolarizations, i.e.,azimuth angle, ellipticity and handedness, are deter-mined in terms of the elements of [n7]".

The results obtained using the derived formulaswere often more accurate than those obtained usingthe IMSL routine, EIGCH. To provide a geometricinterpretation of linear and circular birefringence, theindex ellipsoid and gyration surface were used.

Finally, the sillenite crystal class was examined toillustrate the ease and accuracy of the extended meth-od. Specifically, bismuth silicon oxide (BSO) was an-alyzed in a principal configuration to show the effectsof its natural optical activity together with the simulta-neous influences of an applied electric field (throughthe electrogyration and electrooptic effects) on theeigenstates of the crystal for a given k. Example nu-merical results were presented.

This research was sponsored in part by an Arm-Research Office Science & Technology Fellowship un-der grant DAAL03-86-G-0051.

Appendix: Complex Plane Rotations

The general Jacobi method is modified here for diag-onalizing a 3 X 3 Hermitian matrix. This iterativeprocedure involves a unitary transformation from aCartesian to a helical coordinate system. As a quick


reference the following relationships for the rotationangle, the unitary transformation matrix, and the up-dated impermeability elements are provided for rota-tion in each of the three complex planes. These ex-pressions are derived from the coordinatetransformation law for second-rank tensors. The ro-tation angle in the complex (xy) principal plane isdenoted by X, in the complex (x,z) principal plane by 0,and in the complex (y,z) principal plane by V. Theimpermeability tensor is represented as the Hermitianmatrix H.

1. Rotation in the Complex (xy) Plane

The rotation angle 0 required to zero the H12 ele-ment is found by

0 = 1/2 tan-'[21H121/(Hj, - H22)]. (Al)

This angle represents a counterclockwise rotation inthe complex (xy) plane. The transformation matrixis

cosO[a], = -exp(-iA) sino

0

where exp(iA) = H12/lH1 21.updated as follows:

exp(iA) sino 0cosO 0

0 1(A2)

The elements of H are

3. Rotation in the Complex (yz) PlaneThe rotation angle , required to zero the H23 ele-

ment is found by

xl = 1/2 tan'[21H2,1/(H,, - H33)]. (A7)

This angle represents a counterclockwise rotation inthe complex (y,z) plane. The transformation matrix is

r1 0[a]4, =10 cosV/

Lo -exp(-iC) sinp

0exp(iC) sinqj

cosO(A8)

where exp(iC) = H23/1H23 1. The elements of H areupdated as follows:

Hjj = H,,,

H,,22 = H22 cos 2, + H3 sin2'' + 2!H231 cosq/ sin{p,

H^33 = H22 sin2'4 + H33 cos24 - 2H231 cost/ sinq',

H12 = H12 cos4' + H13(H;3/IH231) sin4 = H',, (A9)

H13 = -H12(H23/!H23) sink' + H13 cos4, = Hj,

H,23 = (H33 - H22)(H23/IH23) cos4' sinik

+ H23(cos22' - sin24t) = H;2 = 0. I

H'4,, = H,, cos2 ' + H22 sin24, + 21H121 coso sino,

H', 22 = H,, sin20 + H22 cos2k - 2H121 cosk sino,

H'033 = H33,

H'12 = (H22 - H11)(H12/H12,) cosk sin A3)

+ H12(cos 20 - sin 24) = H;, = 0,

H,13 = H13 cosk + H23(H,2I/H12) sino = H',,,

H,23 = -H1(H2/IH121) sino + H23 cosO = 2-

2. Rotation in the Complex (xz) PlaneThe rotation angle 0 required to zero the H13 element

is found byO = 1/2 tan'1[21HI31/(Hl - H33)]- (A4)

This angle represents a clockwise rotation in the com-plex (x,z) plane. The transformation matrix is

coso 0 exp(iB) sinO[a]O = 0 1 0

-exp(-iB) sinO 0 coso

where exp(iB) = H13 /IH 3 1. The elements of H areupdated as follows:

Hjj = H,, cos2 0 + H,, sin20 + 21H,,I cosO sinO,

=22 H22

H33 = H,, sin20 + H33 cos2 0 - 21HI31 cosO sinO,

H'0 2 = H12 cosO + H,3(H,3/IH131) sinO = H',', (A6)

H;3= (H33 - H1,)(H3/IH131) cosO sinO

+ H13(cos2 0 - sin20) = H31; = 0,

H'023 =-H*,(H,3/IH,,) sinO + H23 cosO = H'2-

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Accurate method to determine the eigenstates of polarization in gyrotropic media

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