Inhomogeneous electromagnetic plane waves in crystals

Inhomogeneous Electromagnetic Plane Waves in Crystals

M I C H A E L H A Y E S

Communicated by R. A. TOUPIN

Contents

S u m m a r y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222

w 1. I n t r o d u c t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222

w 2. I n h o m o g e n e o u s P lane Waves . . . . . . . . . . . . . . . . . . . . . 225

w 3. The P r o p a g a t i o n Cond i t i on . . . . . . . . . . . . . . . . . . . . . . 227

w 4. Circular ly Polar ised Waves. D and S Paral lel . . . . . . . . . . . . . . 229

w 5. Circular ly Polar ised Waves. D a n d B Paral lel . . . . . . . . . . . . . . 231

w 5.1. Case (ii). D a n d B Paral lel . . . . . . . . . . . . . . . . . . . . 232 w 5,2. I so t ropic Eigenbivectors a n d D o u b l e R o o t s . . . . . . . . . . . . . . 233

w 5.2.1. I so t rop ic Eigenbivectors . . . . . . . . . . . . . . . . . 233 w 5.2.2. D o u b l e R o o t s . . . . . . . . . . . . . . . . . . . . . . 234

w 5.3. R e m a r k . H o m o g e n e o u s Circular ly Polar ised Waves . . . . . . . . . 234

w 6. Biaxial Crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235 w 6.1. The Secular E q u a t i o n . . . . . . . . . . . . . . . . . . . . . . 235 w 6.2. M e a n Energy F lux . . . . . . . . . . . . . . . . . . . . . . . 236 w 6.3. Universa l Re la t ions . . . . . . . . . . . . . . . . . . . . . . . 237 w 6.4. Circular ly Polar ised Waves. D and B Paral lel . . . . . . . . . . . . 240

w 6.4.1. Ci rcular ly Polar ised H o m o g e n e o u s Waves . . . . . . . . . . 240 w 6.4.2. I n h o m o g e n e o u s Circular ly Polar ised Waves . . . . . . . . . 242

w 6.5. I n h o m o g e n e o u s Circular ly Polar ised Waves. D and S parallel . . . . . 245 w 6.5,1. D e t e r m i n a t i o n o f S a n d D 245 w 6.5.2. D e t e r m i n a t i o n o f S a n d D. Geomet r i ca l M e t h o d . . . . . . . 246

w 7. Uniax ia l Crysta ls . . . . . . . . . . . . . . . . . . . . . . . . . . . 247 w 7.1. I n h o m o g e n e o u s P lane Waves . . . . . . . . . . . . . . . . . . . 248 w 7.2. Energy F lux Vector . . . . . . . . . . . . . . . . . . . . . . . 250 w 7.3. Circular ly Polar ised I n h o m o g e n e o u s Waves. D and B Paral lel . . . . . 252 w 7.4. I n h o m o g e n e o u s Circular ly Polar ised Waves. D a n d S Paral lel . . . . . 254

w 7.4.1. D e t e r m i n a t i o n o f S . . . . . �9 . ~ . . . . . . . . . . . . 254 w 7.4.2. Negat ive Crystals . . . . . . . . . . . . . . . . . . . . 255 w 7.4.3. Posi t ive Crystals . . . . . . . . . . . . . . . . . . . . . 255 w 7.4.4. Geomet r i ca l Cons t ruc t i on for the D e t e r m i n a t i o n of S+ a n d S - . 256

222 M. HAYES

w 8. Isotropic Crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . 256 w 8.1. Inhomogeneous Plane Waves . . . . . . . . . . . . . . . . . . . 257 w 8.2. Circularly Polarised Waves . . . . . . . . . . . . . . . . . . . . 258

Acknowledgement . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259

Summary

The propagation of inhomogeneous, time harmonic, eUiptically polarised, electromagnetic plane waves in non-absorbing, magnetically isotropic, but electrically anisotropic, crystals is considered. The electric displacement and the magnetic induction are assumed to have the forms D exp to(S . x - t) and B exp , o ( S . x - - t) , respectively, at the place x and time t, where D, S, B are Gibbs bivectors (complex vectors) and to is real. The implications of Maxwell's equations for the various field quantities are interpreted simply and directly through the use of bivectors and their associated ellipses.

The propagation of circularly polarised waves is considered in detail. For such waves the electric displacement bivector is isotropic: D . D = 0. In order that such waves may propagate it is found that either (i) D is parallel to the slowness bivector S, so that both D and S are isotropic and coplanar, or (ii) D is parallel to the magnetic induction bivector B, so that both D and B are isotropic and coplanar. It is shown that for type (ii) the secular equation must have a double root for the slowness and conversely if the secular equation has a double root then there exists an isotropic electric displacement right eigenbivector of the optical tensor.

Both types of waves are possible in a biaxial crystal. They complement each other in the following way. For type (i) all but two great circles on the unit sphere are possible circles of polarisation for circularly polarised waves with D and 8 parallel. Each of the exceptional great circles is such that an optic axis is normal to the plane of the circle. These two exceptional circles are the only possible circles of polarisation for circularly polarised waves of type (ii) when D and B are parallel.

The situation for uniaxial crystals is similar--the only essential difference being that for uniaxial crystals there is only one exceptional circle since there is only one optic axis.

For isotropic crystals the situation is quite different. Circularly polarised waves of type (i) are not possible. All great circles on the unit sphere are possible circles of polarisation for circularly polarised waves of type (ii) with D and B parallel.

w 1. Introduction

Elliptically polarised time harmonic inhomogeneous plane waves occur in many areas. For example, the classical Rayleigh surface wave in homogeneous isotropic linear elasticity theory is a linear combination of two such inhomogeneous waves. Similarly, gravity waves in ideal fluid flow, Love and Stoneley waves in linear elasticity, T E and T M waves in electromagnetism are all formed by combi-

Inhomogeneous Waves in Crystals 223

nations of inhomogeneous waves. This paper deals with the propagation of inhomogeneous electromagnetic time harmonic plane waves in non-absorbing crystals which are magnetically isotropic but electrically anisotropic. Inhomo- geneous plane wave solutions of the governing equations for a homogeneous crystal are sought without reference to either boundary or initial conditions. For such waves of period 2z~/o9, the electric displacement at a place x at time t has the form [D exp too(S- x -- t)] + where D and S are complex vectors, or "bivectors" as HAMILTON and GIBBS called them. Since two bivectors D and S are involved in the description of the wave train it is very useful to use the language of bivectors when formulating problems or stating results. (Basic results on bivectors may be found in the works of GIBBS [1] and SYNGE [2]. STONE [3] gives some of the results a modern setting. See also HAYES [4].) As shown by GraBS there is an ellipse associated with the bivector D. The real and imaginary parts of D, D + and D-, form a pair of conjugate semi-diameters of this ellipse. For fixed x, as time t increases, the tip of the electric displacement vector lies on an ellipse which is similar and similarly situated to the ellipse of D, namely the ellipse a pair of whose conjugate semi-diameters is D + exp (--toS- �9 x) and D- exp (--oJS-. x). The sense in which the tip of the vector moves is from D + to D-.

In a previous paper [4] it was seen for propagation in elastic homogeneous anisotropic media, that if (as is always possible) the slowness bivector S is written in the form S = Te 'r C, C ---- m + th, where T and ~b are real scalars and ri is a unit vector at right angles to m, then, on prescribing C, Tand ~ are determined from the secular equation. A similar situation holds here. Associated with the bivector C is an ellipse, the principal semi-axes of which are along m and rl, and as shown by GIBBS [1] the bivector e'r + th) has real and imaginary parts which form a pair of conjugate semi-diameters of this ellipse. This pair of conjugate semi-diameters give the directions of the planes of constant phase and the planes of constant amplitude. The phase velocity and attenuation factors are then determined when T is known. Thus, for homogeneous waves a single direction is specified, whereas for inhomogeneous waves an ellipse (m + tri) is prescribed. By taking all possible m and h with m �9 h = 0, ri �9 ri = 1, all possible inhomogeneous plane wave solutions may be obtained (w 2).

Standard results [5, 6] on wave polarisation may be given a simple direct interpretation when use is made of the fact that an ellipse may be associated with a bivector. Thus, for example, corresponding to known C there are two possible D's, say D' and D". The scalar product of these is zero: D' �9 D" ---- O.

This means for vectors that they are at right angles to each other but for bivectors D' and D" it means that the planes of the ellipses of D' and D" may not be orthogonal, in general. Also the ellipse of D' when projected upon the plane of the ellipse of D" is an ellipse which is similar to the ellipse of D" and similarly situated with respect to the ellipse of D" when rotated through a quadrant.

The implications of Maxwell's equations for the various field quantities are given a simple direct interpretation in terms of bivectors and their associated ellipses in w 3. The optical tensor is defined, and the propagation condition and secular equation are derived in the usual way.

Circularly polarised waves are next considered (w 4). If the electric displacement field is circularly polarised then there are two possibilities.

224 M. HAYES

For the first possibility (9 4) the slowness bivector S is isotropic and D and S are parallel. Such waves are not possible in an isotropic crystal, whereas for biaxial crystals all but two great circles on the unit sphere are possible circles of polarisation.

For the second possibility (9 5) the electric displacement field and also the magnetic induction field are both circularly polarised and the corresponding amplitude bivectors parallel. Analogous to a result previously given for elastic bodies [4, 7] it is shown that if the secular equation has a double root then there exists a corresponding isotropic right eigenbivector of the optical tensor and hence a circularly polarised wave may propagate. Conversely, if a circularly polarised wave may propagate then the secular equation must have a double root. In particular, homogeneous circularly polarised waves may propagate in those directions for which sheets of the real slowness surface touch or intersect, that is along the optic axes, as is well known. These are, of course, the only directions in which homogeneous circularly polarised waves may propagate.

Next (9 6) wave propagation in biaxial crystals is considered. Some general results concerning the mean energy flux (Poynting) vector are obtained (9 6.2). Also (9 6.3) universal relations are found analogous to those obtained for anisotropic elastic bodies [4]. Universal relations which are valid for triads of mutually orthogonal unit directional bivectors are also obtained. Then, (9 6.4), we consider circularly polarised waves for which the electric displacement and magnetic induction amplitude bivectors are isotropic and parallel. Included here are homogeneous circularly polarised waves. They correspond to the situation when the condition that the secular equation have a double root has a real solution. This is when the phase velocity is along an optic axis. For inhomogeneous circularly polarised waves the condition for double roots may be factored. There are two pairs of solutions. For each pair the circles of polarisation are identical but are described with opposite handedness. An optic axis is normal to the plane of the circle in each pair of cases. The other possibility for circularly polarised waves, when the electric displacement and the slowness amplitude bivectors are isotropic and parallel is considered next (9 6.5). Such waves are possible for any choice of an isotropic directional ellipse, provided an optic axis is not normal to the plane of the ellipse.

Wave propagation in uniaxial crystals is considered in w 7. The secular equation may be factored. The circularly polarised waves for which the electric displacement and magnetic induction amplitude bivectors are isotropic and parallel are examined (9 7.3). For these the optic axis is normal to the plane of the circle of polarisation and the mean energy flux vector is directed along the optic axis. Then (9 7.4) we consider the circularly polarised waves for which the electric displacement and slowness amplitude bivectors are parallel and isotropic. The mean energy flux vector is found to be in the direction of the normal to the planes ,of constant phase. It is orthogonal to the optic axis for positive crystals but is :not so for negative crystals.

The familiar case when the medium is both electically and magnetically isotropic is then considered (9 8). All inhomogeneous plane wave solutions of Maxwell's equations for homogeneous isotropic media are obtained.


Notat ion. The summation convention applies to repeated lower caselatin suffixes. Lower case bold face letters a, b . . . . represent real vectors. Unit vectors

are denoted by ti, b, ... except that unit vectors along the x, y, z axes are denoted by i, j , k respectively. Throughout the paper the vectors fi and m are orthogonal. The vector m is sometimes written rn rh. Upper case bold face letters A, B . . . .

represent bivectors. Unit bivectors are denoted by / ] ,B , ...

w 2. Inhomogeneous Plane Waves

In this section inhomogeneous plane wave solutions of Maxwell's equations for an electrically anisotropic, magnetically isotropic material are introduced. The description of the waves is in terms of bivectors.

MaxweU's equations, in the absence of currents and charges, are

V" 9 = 0, (2.1)

V �9 ~ = O, (2.2)

7^8 q- - ~ - = 0, (2.3)

0 9 VAaCg at - - O, (2.4)

where 8 is the electric intensity, ~ ' the magnetic induction, 9 the electric displacement and act' is the magnetic intensity. The constitutive equations for the crystal are taken to be

(2.5) ~3 = # a~', ~ i = pgcgi.

Here • is the permittivity tensor for the medium, assumed constant, real and symmetric, and/z is a constant, the magnetic permeability. In terms of the vacuum permittivity • and permeability/~o we have

~ij = ~oK/j, # ---- ~ o M , (2.6)

where K is the relative permittivity tensor and M is the relative permeability. It is assumed that the fields are due to the propagation of an infinite train of

inhomogeneous plane waves in the crystal. Thus

(g, acg, 9 , ~') = (E, H, D, B) exp tco(S �9 x - - t ) , (2.7)

where the bivectors (complex vectors) E, H, D, B are independent of position and time, co is the angular frequency of the waves and x is the position vector of a generic point of space. Also, using the constitutive equation (2.5),

Di = aliEn, Bi = # H i . (2.8)

The bivector S is called the "slowness" bivector and may be written

S : S + + tS- , (2.9)

226 M. HAYES

so that S + �9 x = constant are the planes of constant phase and S - , x = constant are planes of constant amplitude. The phase speed is 1/1 S+ I and the attenuation factor is I S-I. ;

The electric displacement ~ represents an infinite train of elliptically polarised waves, the plane of polarisation being that of the directional ellipse [1 ] of D. Indeed, for x = x* (say)the electric displacement 9 + lies on an ellipse which is similar and similarly situated to the ellipse of D, namely the ellipse a pair of whose conjugate semi-diameters are D + exp (--~oS- �9 x*) and D- exp (--~oS-. x*). As the time t increases the sense in which the ellipse is described is from D + to D-.

The electric displacement ~ is linearly polarised i f

D^D = 0, (2.10)

where D = D + -- tD-, is the complex conjugate of D. The wave is circularly polarised if

D �9 D ----0. ( 2 . 1 1 )

In this case the bivector D is said to be "isotropic" or "null". The slowness bivector S may also be written [4]

S=Te'r m . r i ~-0, r i . r i = 1, T, 4, real. (2.12)

The vectors m and h are along the principal axes of the ellipse of S. By fixing on a particular choice of the unit vector rl and taking m to be of arbitrary (chosen) magnitude and at right angles to h, it will be seen that T and 4, are determined by an eigenvalue problem. To determine all the possible slowness bivectors the direction of fi takes all possible directions in space. For each direction chosen for r~, the vector m has arbitrary magnitude and direction in a plane whose normal is h. Also e'r + th) is a pair of conjugate semi-diameters of the ellipse o f (m + th), so that (S + + tS-) is a pair of conjugate semi-diameters of a similar and similarly situated ellipse whose principal axes are T times the principal axes o f the ellipse of (m + th).

From equation (2.12) we have

S+ = T(cos 4,m -- sin 4,tl), (2.13)

S - = T (sin 4,rn + cos 4,h).

The angle 0 (say) between the planes of constant phase and the planes o f constant amplitude is given by

m tan 0 = (2.14)

(m 2 -- 1) cos 4, sin 4,

tan 2~o (2.15) sin 24, '

where m = tan ~p.


Since we are dealing with ellipticaUy polarised waves it is useful to have a convenient shorthand to express the fact that two ellipses are similarly situated, are similar and have the same handedness. By "similarly situated" we mean that they are in the same plane and that their major axes are parallel. By "similar" we mean that the ratio of major to minor axis is the same for each. By the "same handedness" we mean the ellipses are described in the same sense--clockwise or anti-clockwise. We say two ellipses are SASSH if they are similar and similarly situated and have the same handedness. I f they do not have the same handedness we use the term SASS.

It is also useful to have a convenient shorthand to express the fact that two ellipses are SASS or SASSH when one is rotated through a quadrant. We say two ellipses are SASSQ or SASSHQ if they are coplanar and they are SASS or SASSH when one is rotated through a quadrant in the plane.

Thus if two bivectors H a n d K a r e such that H . K = 0 and i f H a n d K are coplanar then the directional ellipses of H and of K are SASSHQ. I f the ellipses of H and K are not coplanar then H . K = 0 implies that the projection of either (H say) upon the plane of the other (K) is an ellipse SASSHQ with the ellipse of K. See [4].

w 3. The Propagation Condition

Here the well known [5] propagat ion condition is derived. However since we are dealing with bivectors the description of the results is in terms of ellipses-- the directional ellipses rather than in terms of vectors.

Inserting the expressions (2.7) into Maxwell's equations (2.1)-(2.4) gives

D . S : O ,

n . S = 0, (3.1)

SÊ : B,

SAH : - - D ,

leading in the usual way [5] to the propagation condition

[ { (S- S bij - - SiSj} ~ 1 _ # 6,q] Dq = O. (3.2)

I f we write

S : Te 'r C, C = m + th, m �9 ri = 0, ti �9 h = 1, (3.3)

then the propagat ion condition (3.2) may be written

[Q~'q - T-2e-2'~ (~iq] Dq : 0, (3.4)

where Q, the optical tensor, is given by

#Qiq : [(7. (7 (~ij - CiCj] ~r (3.5)

The condition that the waves may propagate leads to the secular equation

det [Q - T - 2 e -2~ 1] = 0. (3.6)

228 M. HAYES

This is an equation for the determination of Te '~ for given C. It is a quadratic in T -z e -z'~ since det [C. C ~o - - CiCj] = 0; and hence det q = 0.

The propagation condition for H may be derived in a way similar to the deriva- tion of equation (3.2), the propagation condition for D. It is

( empqeijkSpSj~qii I -]- ~ t~mk ) H k = O. (3.7)

The corresponding secular equation is

det (empqeijkCpCflcqi I + / z T - 2 e -2'~ r : 0 . (3.8)

On expanding this it is found that it is identical with the secular equation (3.6), as it should be [8].

If for given C the roots of equation (3.6) for T - 2 e -2'~ are denoted by 22 and 22, and the corresponding eigenbivectors are denoted by DO) and D(2 ) respectively, then

QDo) = 22D0), QD(2 ) = 222D(2). (3.9)

Now, from equation (3.5) and use of (3.1), it follows that

I.t Diaiq : C" C Dju~ 1 , (3.10)

and hence from (3.9),

f122 D(1 ) �9 D(2 ) = C " C D(2)j~jq 1 D0)a , (3.1 l)

~t222 D(2 ) �9 DO) = C" C D(1)j~ 1 D(2)q.

Thus if 2 2 =~ 2 2, it follows using the symmetry of ~r that [5]

D(I ) �9 D(2 ) = 0 . (3.12)

Also from (3.1)~

D 0 ) . C = 0, D(2 ) �9 C = 0 . ( 3 . 1 3 )

Thus, in general, for given C = m + th we have two roots T2e 2'r and TZ~e 2'~ (say). Corresponding to these we have the four possible slowness bivectors

SO) = Tle'~lC, S(2) : T2e'~'C, (3.14)

and --So) and --S(2). For simplicity we ignore --S(1) and --Sr We denote the electric displacements corresponding to So) and S(2 ) by ~(l) and ~(2)- Thus

~0) = DC1) exp/~o)(S(l ) " X - t), -'~(2) : D(2) exp t ( D ( S ( 2 ) " X - - t). (3.15)

Now

~(i)" ~(2) = o, -~(1)" so) = ~2)" s(2) = o. (3.16)

From these we conclude that in general the planes of the ellipses of ~0) and ~(2 may not be orthogonal. Neither may the planes of the ellipses of ~0) and of S(~) nor the planes of the ellipses of ~(2) and of S(2 ). Also, since the ellipses of S(l ) and S(2 ~ are SASSH, the projections of the ellipses of ~V) and ~(2) on the plane of C are SASSHQ to the ellipse of C.


Further, from equations (3.1), we have

D(I ) �9 C = 0, HO)" C = 0, EO)' / - / (1) = 0, HO)" DO) = 0 , (3.17)

D(2 ) �9 C = 0, H(2 ) �9 C = 0, E(2)" HC2 ) = 0, H~2)"/)(2) = 0 ,

where H(~), E(~), D(~) are the eigenbivectors corresponding to the two roots of equation (3.6) for given C.

In general the ellipses of DO), D(2),/-/(1) and/-/(2 ) may not be in planes which are orthogonal to the plane of the ellipse of C. The planes of the ellipses of Eo) and of H(l ) are not orthogonal, in general, nor are the planes of the ellipses of E(2 ) and H(~), those of Ho) and Do) and those of/-/(2 ) and D(2). Also, the ellipses of C, DO) and Eo) when projected upon the plane of the ellipse of/-/(1) are SASSHQ to the ellipse of H(1 ). Similarly, the ellipses of C, D(2 ) and //(2) when projected upon the plane of the ellipse of//(2) are SASSHQ to the ellipse of / / (2 ). Finally, we note that the ellipses of DO), D(2), HO) and/-/(2 ) when projected upon the plane of the ellipse of C are SASSHQ to the ellipse of C.

w 4. Circulary Po lar i sed W a v e s . D and S Para l l e l

If the ~ field is circularly polarised then the bivector D is isotropic: D �9 D ---- 0. In this section it is seen that there are only two possible ways in which this may occur, namely (i) when D and S are parallel or (ii) when D and B are parallel.

In case (i) the ellipse of S, and therefore also of (7, must be a circle. The planes of constant phase are orthogonal to the planes of constant amplitude. The normals to these planes span the plane of D. In case (ii) the ellipse of B must be a circle. Consideration of case (ii) is deferred to w 5. Here case (i) is considered in detail.

For a biaxial crystal for any circle C, it is seen that apart from the two cases when one of the optic axes is normal to the plane of the circle of C, there is a corresponding circularly polarised wave, the circle of polarisation of D being coplanar with the circle of C. For a uniaxial crystal, provided that the optic axis is not normal to the plane of the circle of C, there is a circularly polarised wave corresponding to any given circle C. For an isotropic crystal such circularly polarised waves with D and S parallel are not possible.

From equation (3.l) and the constitutive equation (2.5),

Thus

D . S = O, H . S ---- O, D = - - S ^ H , ~ H = SÊ. (4.1)

O . O = (S^H) �9 (S^H) ----- (S. S) (H. / - / ) .

Hence if D is isotropic there are three possibilities:

(i) S . S = O, H . H=~ O;

(ii) S . S =~ 0, H . H = 0;

Oii) s . s = o, n . n = O.

(4.2)

(4.3)

230 M. HAYES

In considering these we first note tha t case (iii) is no t feasible. Since H . S = 0 it follows that H and S are parallel : H---- ffS for some scalar 4~, and hence D- - - -0 by (4.1)3. I t then follows f rom the constitutive equat ion tha t E = 0. This then leads to H = 0 by (4.1)4. Thus in this case all the fields are zero.

Case (i). S . S = 0, H . H =~ 0, O �9 D = 0

Since S �9 S = 0, D �9 D = 0, D �9 S ---- 0, it follows that

D ----- ~pS,

for some scalar % Using (4.1)3 we find

~pSAE = --(SAH)Ê = - - ( S . E) H,

since H . E = 0 by (4.1)4. Using (4.1)4, equat ion (4.5) becomes

~p/zH = - - ( S . E) H .

Hence

(4.4)

(4.5)

(4.6)

Thus if S can be chosen to satisfy (4.1 I), the corresponding H, B, D, g m a y be determined to within an arbi t rary scalar multiplier. The corresponding ~ field is circularly polarised.

The planes of constant phase are o r thogona l to the planes o f constant amplitude for these waves. Also the plane of the circle o f polar isat ion o f ~ is parallel to the plane spanned by the normals to the planes o f constant phase and the normals to the planes o f constant ampli tude.

N o w letting S =- Te '~ (m rh + th) (see (3.3)), we find f rom S . S = 0 tha t

m 2 = 1. (4.12)

Hence S mus t satisfy

Uk~SkSm = --#, SjSj = 0. (4.11)

v22/~ 2 = ( S . E) 2 = - - # 2 ( H . H) , (4.7)

by (4.1)4, since S . S ----- 0. Thus

D = ~oS ---- 4 - t ( H . H)�89 S. (4.8)

Returning now to the p ropaga t ion condit ion (3.2) and using S . S = 0, we have

SiSjn~ 1 Dq = - -# Di, (4.9)

bu t since D = ~0S, it follows tha t

nEI SkSm = --F" (4.10)


Taking m ---- -l-l, without loss of generality, we then obtain from (4.11)~

T2e2'~kl (gn k -~ t~lk) (rH m "q- t~lrn ) = --[~. (4.13)

This is an equation for the determination of T and ~b for given pairs of orthogonal unit vectors rh and ri. For such rh and ri, equation (4.13) may always be solved for T and ~b provided rh and h are not such that

u~ml(rhkrhn - - hkhm) = O,

-~ ^ ^ 0, (4.14) ~r mknm : -

t h - r h : h . h = 1, r h . h = 0 ,

for then the expression on the left hand side in equation (4.13) is zero. It is now shown that for a biaxial crystal there are only two possible solutions

for equation (4.14). For each of these solutions an optic axis is orthogonal to the plane of the circle of rh -k tfi.

The tensor ~r is positive definite and symmetric and so the surface u / j l x i x j = 1 is an ellipsoid (the dielectric ellipsoid). Equation (4.14)~ expresses the fact that radii along rh and h have equal lengths. From equation (4.14)2 it follows that rh and h are along conjugate diameters of the central elliptical section by the plane th^ri �9 x = 0. However rh and h are orthogonal and hence must be along the principal axes of a central section. But, since the radii along rh and h have the same length, it follows that the central section must be a circle. There are only two such circles. Indeed the normals to them give the directions of the two optic axes of the biaxial crystals [10].

Thus we conclude for biaxial crystals that apart from the two cases when the optic axes are orthogonal to the plane of C = rh q- th , then for such C (with m 2 = 1) there is a corresponding circularly polarised wave, the circle of polarisation of ~ being coplanar with the circle of C.

For a uniaxial crystal the dielectric ellipsoid is a spheroid. In this case there is only one possible central circular section and equation (4.14) has only one possible solution. The optic axis is again orthogonal to the plane of the circle of rh -k th . Thus for any given isotropic bivector C = rh § th there is a corresponding circularly polarised wave, the circle of polarisation of ~ being coplanar with the circle of C, provided only that the optic axis is not normal to the plane of C.

For an isotropic crystal the dielectric ellipsoid is a sphere, and the equations (4.14) are satisfied for all pairs of orthogonal unit vectors rh and h. Hence these solutions with D parallel to C do not exist for isotropic media.

w 5. Circularly Polarised Waves. D and B Parallel

Here the second possible circularly polarised wave solution, namely Case (ii) S �9 S ~ 0, H . H = 0, D �9 D ---- 0 is considered. It is seen in this case that H, B and D are all parallel and isotropic. It is shown that if D is isotropic and satis-

232 M. HAYES

ties the propagation condition then the secular equation (3.6) has a double root and conversely if the secular equation has a double root then a corresponding isotropic right eigenbivector D exists. I have given similar results for the propagation of inhomogeneous plane waves in incompressible elastic materials [7]. The proofs given here are simpler and more transparent.

For a biaxial crystal it will be shown later (w 6) that if the secular equation has a double root then there are two possible polarisation circles for D, one optic axis being normal to the plane of each circle. These are precisely the polarisation circles for which a solution with D parallel to S could not be obtained (w 4). Thus, for biaxial crystals, any circle is a possible circle of polarisation for circularly polarised waves. For all but two circles D and S are parallel. For the remaining two circles D and B are parallel.

For a uniaxial crystal it will be shown later (w 7) that if the secular equation has a double root then there is only one possible polarisation circle for D, the optic axis being normal to the plane of this circle. Again, this is precisely the polarisation circle for which a circularly polarised wave with D parallel to S could not be obtained. So for uniaxial crystals any circle is a possible circle of polarisation for circularly polarised waves. For all but one circle D and S are parallel. For the remaining circle D and B are parallel

It will be shown later (w 8) that for an isotropic crystal the possibility C . C : 0 is not possible For any other C the secular equation has a double root and the amplitude bivector D may be so chosen that D is isotropic. Thus for every given non-circular directional ellipse C of the slowness bivector a circularly polarised

wave may propagate. D and B are parallel for all the possible circularly polarised waves in an isotropic medium

w 5.1. Case (ii). D and B parallel

Now for Case(ii), S . S =~0, H . H = 0. Also D �9 D = 0, and D . H = 0, since D = --S^H. It follows that D and H (or B) are parallel and hence for some scalar 0,

B : 0 D, B . B = 0. (5.1)

Thus, using equation (4.1), we have

iz D = --SAB = --OSAD,

IzSAD = O(S. S) D -- 0 2

= ----(S.S)S^D, (5.2) #

and so

B = 0 O = + ( S ~ D . (5.3)

Hence in this case H, B D, are all parallel and isotropic. D must satisfy the propagation condition (3.2) and must be isotropic.


w 5.2. Isotropie Eigenvectors and Double Roots

It is now shown that if D is an eigenbivector of Q and is isotropic, then the equation I Q - 2II = 0 has a double root. Next it is shown that if ]Q - 211 = 0 has a double root then Q possesses an isotropic right eigenbivector D.

w 5.2.1. Isotropic Eigenbivectors. Now S is not isotropic and so neither is C. For the purposes of this section there is no loss in generality in assuming that C

is a unit bivector, denoted by C, C . C = 1. Also it is assumed that /z : 1, --this does not affect the argument. Then from (3.5), Q is given by

Qiq = (6o -- C, Cj) ~jql. (5.4)

Let J, /~ be unit orthogonal bivectors which form an orthonormal triad with C.

Thus ) . C = J - L = 0 , J . ) = 1, etc. Then

o.,~ = (L~ + Z,Zj) ~,j;, . (5.5)

Now D �9 C ---- 0 and hence D is a linear combination of J and L. But also D - D : 0 , so that

D = p(J • ,L), (5.6)

for some scalar p. Also, D is a right eigenbivector of Q with eigenvalue 2 (say): Q D = 2 D. So from (5.5) and (5.6)

( J ,~ + L, Lj) u~'()q • tLq) = 2(), • tL,), (5.7)

and hence

1,,_1 ) • ,A,-, L =z , (5.8)

L,,-1 J • ,s = •

Now ~r is symmetric, and hence

2J~r -~/~ = •162 J - / a r -1/~). (5.9)

The eigenvalue 2 satisfies [ Q - 21] : 0. Using (5.5) and noting that I Q I = 0 so that one root is 2 : 0, it follows that the other roots satisfy

22 -- 21 + H = 0, (5.10)

where

211 = QiiQyj - QijQji = 12 - (J~r j)z _ (/ar i,)2 _ 2(j~r

= 2ECA, -~ Y) (/,,,-,/,) - ( A , - 1 / , ) ~ 1

: (U/2), (5.11)

on using (5.9). Thus ;t = (I/2) is a double root.

234 M. HAYES

w 5.2.2. Double Roots. Turning now to the converse, it is to be shown that if Q, given by (5.5), is such that I Q - 211 = 0 has a double root for 2, then Q possesses an isotropic right eignbivector.

Write

i .z' : ed, + cL + hi,,

where

(5.12)

a = )~r 6, b = J~r J, c = j~r d = I,~r 6, h ---- i~;z -~ i;.

(5.13)

Then from equation (5.5),

Qiq : affifq q- b)i)q § c()iZq -~ )qZi) -q- dZiCq ~- hZiLq. (5.14)

The eigenvalues of Q are the roots of

i 00 I b - - 2 c ----0. (5.15)

c h - - 2

Obviously one root is zero. The other eigenvalues are the eigenvalues of a symmetric (2 x 2) matrix. These roots coincide provided

b -- h = 4-2w. (5.16)

Then from (5.14), Q possess the isotropic right eigenbivector J q: tl, with corresponding eigenvalue (b -{- h)/2.

w 5.3. Remark. Homogeneous Circularly Polarised Waves

Homogeneous circularly polarised waves are only possible when D and B are parallel. They are not possible if D and S are parallel, for if D is isotropic so is S. But S cannot be parallel to a vector and at the same time be an isotropic bivector.

Thus, for biaxial crystals, homogeneous circularly polarised waves are only possible if an optic axis is normal to the plane of the circle of polarisation (w 6). For uniaxial crystals such waves are only possible if the optic axis is normal to the plane of the circle of polarisation (w 7). For isotropic media such waves are possible for any circle of polarisation (w 8).

lnhomogeneous Waves in Crystals 235

w 6. Biaxial Crystals

The propagation of inhomogeneous plane waves in biaxial crystals is now considered. The secular equation is derived explicitly by referring all quantities to the principal axes of the dielectric tensor. In 8 6.2 some general results concerning the mean energy flux (Poynting) vector are recalled and derived. Then (8 6.3) some universal relations connecting the wave-speeds and attenuation factors are obtained. The relations are universal in the sense that they are valid irrespective of the values of the dielectric constants and as such are independent of the dielectric constants. Their worth lies in the fact that they may be checked by the experimentist without his having first to determine the dielectric constants of the crystal.

In 8 6.4 circularly polarised waves for which D and B are parallel are examined. The condition that the secular equation have a double root is written down and the corresponding circularly polarised inhomogeneous plane wave solution is exhibited. Circularly polarised homogeneous waves are included in this class of waves (8 6.4.1). For these, explicit solutions are obtained and the various field quantities given. It is then shown how the condition for double roots for inhomogeneous waves may be factored. There are two pairs of solutions. For each pair the circles of polarisation are identical but are described with opposite handedness. An optic axis is normal to the plane of the circle in each pair of cases. Ex- plicit solutions are obtained.

Then (8 6.5) inhomogeneous circularly polarised waves for which D and S are parallel are considered. Such waves are possible for any choice of an isotropic C provided an optic axis is not normal to the plane of C. For such waves we give a simple explicit geometric method for the determination of the components S + and S - of the isotropic slowness bivectors.

let

w 6.1. The Secular Equation

Let axes be taken along the principal axes of the dielectric tensor x. Thus

Zl l --- ~1, ~22 = ~2, ~33 ----- Z3, Zii = 0 (i =~j), (6.1)

and let

z t > z2 > z3. (6.2)

Then, referred to these axes, the components of the optical tensor (Q) are given by

,uQ,q -- ( c . C O,j - CiCj) (~?' 6j,6q, q- z ; ' 6j26q2 + :r Oj3Oq3), (6.3)

/ ( C . (7 - - CI)~1-1 - -C i C2 9 21 --CiC3~31 p(Q) = ~ - - C , Czg[ l (C" C -- C~)z~ ' --C2C3x; ' ~ . (6.4)

\ - C i C 3 ~ l - C2C3~ 1 ( C . C - C~) z i l /

so that

236 M. HAYES

Note that (ntC~, x2C2, 9r T is a right eigenbivector of#(Q) with eigenvalue zero and (C~, C2 ,Ca) is a left eigenbivector also with eigenvalue zero.

The secular equation is the quadratic in T-2e-2'~:

(/tT-2e-2'*) 2 --#T-2e-2'*((C 2 + C~)x?' + (C~ + C~)n~ ~ -[- (C2~ + C~)xT' }

+ (C. C)(C2~t~s ~ + C ~ F ~ ' + C2xF'~t2 ') = O. (6.5)

This may also be written in the form of Fresnel's equation:

3

~_~ (u~,C~[C" CT2e 2'* --/tu~]-'} ----- 0. o;=1

w 6.2. Mean Energy Flux

At x the energy flux (Poynting) vector ~ and its mean ~ are defined by

2 ~ # o

= (~/2~) f ~r dt. (6.6) 0

In dealing with a wave of the form (2.7) it follows that [5]

= / ~ exp (--2o)S- �9 x) (6.7)

where

k = �89 [ e ^ ~ + .

The mean energy density -r is given [5] at a point x by

2 ---- �88 [O-/~ -+- n - / t ] exp ( - 2 ~ S - �9 x).

(6.8)

(6,9)

It has been shown [9] for a train of inhomogeneous plane waves propagating with slowness S in an electromagnetic non-dissipative system that

~ . S + : 2, ~ . S - : 0 . (6.10)

Now since S + and S- are conjugate diameters of the ellipse of S it follows that

equation (6.10) expresses the fact that the component of ~ in the plane of S is along the normal to the ellipse "a t" S +. For given C : rnrh + trl, the two slownesses $1 and S 2 have similar and similarly situated ellipses. On each of these

~

at the "points" S(~) in the plane of St~ ), ~(~) is along the outward normal l(~) (say), since _r > 0. Thus :~(~) has the form

2~1~ ~ _ _ _ _ ~(~) (l~. S2 ) + 2~S~^ SZ, or : 1, 2 (no sum), (6.11)

where 2~ are scalar multipliers.

Inhomogeneous Waves in Crystals

Finally note that the component of R along S+S -

, [e,Ûl+. ( s + s - )

t T 2 = -~- [E^H] +. (C^C)

is given by

237

t T 2 = ~ [EAH" -~ EAH]" (CAC) (6.12)

t T 2 = - T [(E" C) (H- C ) - - (E- C) (H. C)

+ (~. c) ( u . ~-) - (E. C) (U. C)].

By (3.1) B-S----0 , and hence H . C = 0 . Thus,

t T 2 [E^H]+ �9 (S~S-) = -4- ([n?- C) (H- C?) -- (E- C) (H" C)]

T 2 (6.13) = -5- fig. C) ( n . C)l-.

w 6.3. Universal Relations

As the name implies universal relations are valid for all crystals whose optical and magnetic properties are described by ~r and/x. The relations are independent of ~r and ~ and hence these do not need to be determined to check the validity of the universal relation. As such they are of use to the experimentist. Here some universal relations involving the phase speeds and the attenuation factors are obtained.

On expanding (3.5) the optical tensor may be written

pQ,q(mth + tfi) = {(m 2 - 1)t)q -- (ruth, + t?li) (mr~j q- thj)} uj~'

: t t Q i q ( - m r h - t t l ) . (6.14)

For a homogeneous wave propagating in the direction/~ (say) the corresponding optical tensor is Q (/~) given by

ktQiq([~ ) : (t)i j __/~i/~j ) ~jql. (6.15)

Now the unit vectors (th • h)/~/2 are mutually orthogonal and in terms of them

2(l~li?l j -~ l~lj?li) = (1~I i -~- Ill) (l~lj --~ ?lj) - - (1~1 i - - Hi) (r?lj - - l'ij), (6.16)

238 M. HAYES

and hence

[ t ~ + ~ , ~ - a O(mrh-kth)=m zo(rh)- O(h)+ tm Q\ t/2 ]--O(---~)J, (6.17,

where the optical tensors on the right correspond to homogeneous plane waves. It follows from (6.17) that

Q(mrh + th) + Q(mlrn - th) = 2m 2 O(rh) -- 2Q(h),

+ th) - O(mrh - th)= 2trn IO l,a

- o

+ rh fi O(m~h

I

(6.18)

The optical tensors on the left correspond to inhomogeneous waves, those on the right to homogeneous waves. 2

Now the trace of Q(/5) is equal to ~ c~2(lb), the sum of the squares of the two or

phase speeds c~(/~)of the homogeneous waves which may propagate along /~. Thus, taking the trace in equation (6.17), and using (3.6), gives the universal relation

Similarly, from (6.18),

2

cr

2 2

= m 2 ~ c ~ ( r h ) - "~ c~(h) a = l ~ = 1

, m + ~ , ~ c ~ t v 2 1 - c o ~ �9 (6.19)

2 2

~_~ TZ2e-2"~(ml~l + t{l) + ~ T;2e-2'*~(mlil -- th) ~x=l a = l

2 2

= 2m2 ~a c~(r~) -- 2 ~a c~(h), c~=l ~=1

(6.20)

2 2

c~=l or (6.21)

Now if h a is any triad of mutually orthogonal unit vectors, the sum of the 3 2

squared speeds ~ ~ c~(/~) is invariant, and equal to 2z,.i~/z -j . Thus, using f l = l c~=l

(6.20) by replacing m, h in turn by/~a,/~e and using a common value r for m it


follows that

2 3

c~=l fl ,y=l

+ T;2e-Z'*~[rfi8 -- tfiv]}

3 2

= 2(r 2 -- 1) Y, E e1(Ps) 0=1 a = l

= 4(r 2 -- 1) uffl# -1 .

(6.22)

We conclude by obtaining universal relations for inhomogeneous waves with prescribed directional ellipses corresponding to unit bivectors. If C = mrh q- th is a unit bivector then C- C = 1 and so m 2 ----- 2. Thus in these directional ellipses the major axis is i/2 times the minor axis.

If }. is a unit bivector, then from equation (3.5)

(6.23)

and so

2

Y~ i_tTgZe-2'~(i,) = ~tff' -- s �9 (6.24) ot~l

Thus, if I.a, fl = 1, 2, 3, is any orthonormal triad of bivectors, we have

~-u T~2e-Z'r = 2#-1aft l, (6.25) 8 = I a = l

and hence the universal relation

f l= l f l= l (6.26)

for any pair of orthonormal triads of bivectors {~} and {is}" In interpreting this result it must be borne in mind that the directional ellipses

of LL, L2 and I-'a do not lie on orthogonal planes. The projections of the ellipses

of any two, say L1 and L2, upon the plane of the third, La, are ellipses SASSQ to the ellipse of La.

Finally, suppose J and L is a pair of orthogonal unit bivectors. Then from (6.24)

2

Z P{T;Ze-2'~('~) q- T~-=e-2'~(/~)} or=|

= 2 ~ ' -- (JiJj -k ZiLj)ajT'. (6.27)

240 M. HAVES

Let M and N be any pair of orthogonal unit bivectors "coplanar" with J and L,, so that

/ f / = cos OJ + sin 0[,, N = --sin 0~r + cos 0~,, 0 --< 0 --< 2:r. (6.28)

Then

) i~ q- L, Lj = f , l i~l j q- )vilTj. (6.29)

Then, from (6.27) we have the universal relation

2 2 ~a (T;2e-2'*~(j) -}- T~-2e-2'*~(/')} = Y~ {T22e-2'*~(i~1) q- T~-2e-2'*~(~r)},

a=l o~=1

(6.30)

which is valid for all pairs ( J , / , ) and (M, N) of "coplanar" orthogonal unit bivectors.

w 6.4. Circularly Polarised Waves. D and B Parallel

The condition that equation (6.5) have a double root is

((C 2 .q_ C2)ui -1 _]_ (C 2 .ql_ C2)•21 ~_ (C 2 .q_ C2) u31} 2 (6.31)

: 4(C. C) (C2u2'u; ' -q- C2uFlu31 q- C2uF'u2') .

If this equation is satisfied by C for given uz then the double root is T-2e -2'~ given by

2#T-Ze -2'0 ---- (C2 z + C2)ui -1 + (C 2 + C2)•2 t + (C 2 + C2)u~ -1 . (6.32)

The corresponding isotropic right eigenbivector is D given by

D1 = 2CIC2C3(~21 -- ~ 1 ) ,

D2 = C3((C 2 + C 2) ~?1 _ (C 2 + C 2) x;~ + (C 2 _ C 2) ~-1), (6.33)

D 3 = + + - - _ +

provided C~ =~ 0, C2 =~ 0, C3 @ 0. (These cases will be considered later (w 6.4.2)). It may be checked that D �9 D = 0, onusing equation (6.31), andthus the wave is circularly polarised. The corresponding values of E are

E1 = ~i-1 D1 ' E2 _. ~21 D2 ' E3 = ~-1 D3" (6.34)

w 6.4.1. Circularly Polarised Homogeneous Waves. As remarked in w 5.3 circularly polarised homogeneous waves are only possible provided D and B are parallel. These waves arise for those directions of the phase velocity for which the sheets of the real slowness surfaces touch or intersect, that is for waves whose phase velocity is along an optic axis.


For homogeneous waves take m = 0, C = th. Then, equation (6.31), the condition for a double root, may be written

[ y 2 n 2 _~_ n 2 + 0~2n212 2 2 2 2 = 4o~ ? nln3, (6.35)

where

~ 2 , ~ 3 ( ~ L - ~2) ~,2 = 1 - - 0 ( . 2 ~1~1.(~2 - - J'g3) - - $h~2(~1~ L - - ;h[3) ' = ~2(ul -- u3)" (6.36)

Now ~1 > x2 > u3 so that ~2 > 0, 72 > 0. As is well known [10] the equation (6.35) has only two real solutions for n, namely h = c~i • y k , the directions of the optic axes.

Thus for homogeneous circularly polarised waves with D parallel to B, C = t(od • ~,k), and from (6.32)

2 # T - 2 e - 2 ' r = - - ~ . , 2 u l l - - ~r 1 - - 062~r 1

= --2~21 , (6.37)

where the identity ~ - 1 = 0~2~3 1 _]_ ~ 2 ~ ] - 1, (6.38)

follows from (6.36). Thus

T 2 = #~2, 4> = z#2 or 3z~/2. (6.39)

Before proceeding to determine the isotropic eigenbivectors we note that if D is an eigenbivector so also is -- D, and there is no essential difference between them. For that reason in writing down eigenbivectors D, both + D and - -D are to be understood.

Taking ff = ~/2 we have the following possibilities for corresponding C, S and D:

C = (to~, O, t~,),

C = - ( to~ , O, tg'),

C = (t~, O, - t y ) ,

C = - (to~, O, - t g , ) ,

s = - C u - # (~,, 0, ~),

s = ( ~ 2 ) ~ (c~, 0, ~,),

s = - ( ~ # (~,, 0, -~ , ) ,

s = ~ ) ~ (o,, o, -~ , ) ,

19 = ( - y , -4-t, o0;

o = ( - ~ , , • 0,);

o = (~,, • ~ ) ;

o = (~,, • o0 .

(6.40)

The circles of polar•177 corresponding to D ----- ( - -? , t, o~) and O = (--?, --t, ~) are identical but they have opposite handedness. The direction of one optic axis (o~, 0, ?) is at right angles to the plane of the circle. A diameter of the circle lies along the intermediate (z2) axis of the crystal. Further, the circles of polar•177 corresponding to D = (y, t, ~) and D = (y, --t, o 0 are identical, but they are described in opposite senses. Also a diameter of the circle lies along the intermediate axis of the crystal, and the direction of the other optic axis (0r 0, --y) is at right angles to the plane of the circle.

I f we take the other possibility ~b = 3z~/2 it leads to the signs of C and S being changed but there is no change in D.

We consider one example.

242 M. HAYES

Example 1. I f

s = 0,

D : b ( - - 7 , q-t, o~),

where b is some scalar, then

(6.41)

# H = B = SÊ = --tb~/u2) t (--y, q-t, o0, (6.42)

on using the identity (6.38), and hence

(~uu2)�89 [E^H]+ = --bb[o;(u~ 1 + u~I), O, 7(u{1 _}_ u~l], (6.43)

= b(--y, q-t, o~) exp [--tw(~u2) ~ (o~x q- ~,z) q- t)].

Results for all the other possibilities may be obtained from these results by replacing o~ by --o~ and 7 by --), as required.

Note that in all cases there is no component of the mean energy flux vector along the intermediate ~2 axis.

Also note that the two solutions (6.43) may be combined to give a linearly polarised wave propagating along the optic axis.

Remark. In the case of homogeneous circularly polarised waves the optical tensor Q is real, and there is an isotropic right eigenbivector D (say) with corresponding real double eigenvalue 2 (say). Thus QD = 2 D, D �9 D = 0. From this

it follows that Q . D = it D, D �9 D = 0. So two homogeneous circularly polarised waves of opposite handedness but common slowness may propagate. However, if the wave is inhomogeneous, Q is in general no longer real, and thus no inference

may be drawn relating Q and D.

w 6.4.2. Inhomogeneous Circularly Polarised Waves. It is shown here that the condition for double roots (6.31) may be factored. There are two pairs of solutions. For each pair the circles of polarisation are identical but are described in opposite senses. An optic axis is normal to the plane of the circle in each pair of cases.

We present an example in which none of the components C~ is zero. The field quantities are written down.

Then we briefly consider separately the three cases when one of the components of C is zero. One of these (C2 = 0) corresponds to the homogeneous circularly polarised waves considered in the previous section.

The condition for double roots (equation (6.31) may be written

(y2C~ + C~ § o~2C~) z = 4o~272C~C~. (6.44)

There are thus two pairs of possibilities

I: ~C1 -- o~C3 = q-tC2, (6.45)

and

II: ~C1 + ~C3 : • (6.46)


For I, from (6.33), it may be shown that

o~ D1 -k- )' Da = 0. (6.47)

Now D m u s t satisfy D - D = 0 and D . C = 0 . It follows that

O : b()', T t , --00, (6.48)

where b is a scalar and the upper and lower signs here correspond with those in equation (6.45). The optic axis (o~, 0, )') is normal to the plane of the circle of polarisation D. Using (6.38) and (6.45) it may be shown that the double root (6.32) is given by

tzT-2e -2"1' : (o~C 1 q- ),Ca) (0r -~- ) '~r (6.49)

The circles of polarisation (6.48) are identical, but they are described in opposite senses.

Similarly, for the second pair of possibilities II, it may be shown from (6.33) that o~D1--7, D 3 : 0 , and then using D .D- - - -0 , D . C = 0 , it follows that D has the form

O ---- b()', Tt , o~), (6.50)

the upper and lower signs corresponding to those in (6.46). The optic axis (o~, 0, --)') is normal to the plane of the circle of polarisation. The double root is

/zT-2e -2'§ = (o~C1 -- )'Ca) (o6~r - - ~ 1 1 C 3 ) �9 (6.51)

We conclude with some examples.

Example 2. C :~ O. Let

C ---: (1 -- 02) ~ q-- tT' 1, ) (1 -- 02) ~ t0~ , (6.52)

where 0 is a real scalar satisfying 02 < 1. For this choice of C, it is seen that )'C~ -- o~Ca = tC2, and thus it is one of the first pair of possibilities. Hence D = b(--~,, t, ~) and from (6.49)

where

#T-2e -2'4' = Re ''~, (6.53)

0 2 R cos

~2(1 - 02) '

0 o ~ ) ' - 1 R sin ~ - - ( ' ' ' ' ' ' - ' A ' ~ ( ~ 3 - t l - ) - - - ui-1)"

(6.54)

244

Thus

M. HAYES

S = Te'§ : ~ / R ) �89 e -'~/2 C,

C. C = 02/(1 -- 02), (6,55)

t t H = B = t(~uR)�89 (1 -- OZ) �89 (O/(O)) (e'e/:),

O [ E ^ ~ I § = (R /~ )~ (1 - - 0~) t bt, o~(~ -~ + ~;~)cos T ,

~,~(~31 _ ~i-I) s i n T , y ( ~ 1 1 + ~-1) cos .

Example 3. Finally we deal briefly with three special cases, when one of the components of C is zero. We consider them only in the context of the first pair of possibilities (I)--there are similar results for the second pair of possibilities II. The three cases are

(i) C1 ----- 0, (72 ~ 0, Ca =~ 0; (ii) C1 =~ 0, C2 = 0, Ca =4= 0;

(iii) C1 ~= 0, C 2 =# 0, C a = 0.

In each case for I, D is given by equation (6.48).

For case (i). from (6.45) and (6.49), and the fact that B = SÊ, we find

C2 = q:tc~C3, C- C -- 9,~C~,

S . S =/z~l , yS = (~uul) �89 (0, q:w~, 1),

(/~1) �89 H = ~tb(y, -bt, --00, (6.56)

~ , ) ~ [E^H] + = +bbt ,

k ~

where

t = 0~(u21 + u~-l) i + y(~i -~ -J- u2 l) k, (6.57)

and the upper and lower signs in each of the expressions correspond with each other. There is another pair of solutions with S replaced by --S.

For case (ii), from (6.45) and (6.49)

C 2 : 0, ~ C 1 : o~C3, ~2C" C = C 2, S " S = # h ~ 2. (6.58)

This has already been examined in w 6.4.1. The circularly polarised wave is homogeneous.


For case (iii), f rom (6.45) and (6.49),

C 3 = O, C 2 : -~-/,7C1, C " C = 0/2C12, S " S = / t u 3 ,

0r = (#U3)�89 (1, ~- t~, 0) ,

~ a ) + H = :~tb(y, i t , --~), (6.59)

(/xu3)+ [E A if]+ = --bbt,

= bb', i t , - ~ ) exp [t~o{(~.z/0,2) + x - t}] exp [=?o~(gu3y2/~2) + y].

The upper and lower signs in each o f the expressions correspond with each other.

We note that in all three cases the mean energy flux vector is along t. Corresponding results for the second set o f possibilities II may be read off

f rom (6.56), (6.58) and (6.59), by replacing y by - -y . In these cases the mean-energy flux vectors are all parallel to a vector t* (say), obtained by replacing ~, by - - y in the expression (6.57) for t.

w 6.5. Inhomogeneous Circularly Polarised Waves. D and S Parallel

Here we consider the propagat ion in biaxial crystals o f inhomogeneous circularly polarised waves for which D and S are parallel. The circle o f C is chosen and then T, if, are determined, so that S is known. D is parallel to S and thus E is known, and hence B is found, since B ---- sag. Such waves are possible for any choice o f an isotropic C provided an optic axis is not normal to the plane o f C. F o r these waves we give a simple explicit geometric method for the determinat ion of the components $+ and S - of the isotropic slowness bivector.

w 6.5.1. Determination of S and D. The slowness bivector S must satisfy the equat ions (4.11). These lead to m 2 = 1, so that the planes o f constant phase are or thogonal to the planes of constant amplitude. Then taking m ---- q- 1, without loss in generality, we obtain f rom equat ion (4.11),

# T -2 cos 2~b ----- (~-1 _ ~i-i) (m 2 _ n 2) _ (x~-~ _ ~2 ' ) ( m2 -- n2), (6.60)

/zT -2 sin 2~b = --2(~21 - - ~i -1) rain 1 -Jr- 2(~31 -- ~21) m3n3,

on using fit �9 rh ---- h �9 ri = l, h �9 th = 0. These equations enable us to determine T and ~b for almost any choice o f rh and ti satisfying r h - ~ = 0. The reason for the qualification "a lmos t any" in the choice o f rh, rl is that T and ff are not determined if the right hand sides o f (6.60) are both zero. We now consider this possibility.

On using (6.36), the condit ion that the right hand sides o f (6.60) be zero may be writ ten

o~2(m 2 - - n 2) -t- = y2(m2 -- n~), (6.61)

or = ~,2man a.

246 M. HAYES

The elimination of 0~ 2 and y2 leads either to (i)

mln3 - - m a n l = 0 ,

or to (ii)

(6.62)

mlm3 ~- nln 3 = O. (6.63)

Case (ii) is not feasible since it leads to ~2m] q- 0r = 0. In case (i) we find o~2n~ -- 72n~ = 0, 0~2m~ -- y2m~ = 0, and then using (6.61)2, we obtain

(ml q- ml, m2 q- tn2, m3 + m3)" (or 0, ~ y ) = 0, (6.64)

where either sign may be taken. Thus the plane of C has an optic axis normal to it. Hence, apart from these cases, T and 4~ may be determined from (6.60).

Here is an example.

Example 4. Let C = i + L]. Then from (6.60)

[zT -2 cos 2~ = (~1 - - ~2)/(~L~2),

# T -2 sin 24~ = 0.

Thus, taking the choice 4~ = 0, we obtain

- 2 = ( - 1 -

s ( (1, ,, 0), \x~ -- x21 .

D = b(1, t, 0),

E = b(,~i- i, t~2 l, 0),

H = ,b (O, O, - -

(6.65)

(6.66)

w 6.5.2. Determination o f S and D. Geometrical Method. Now, by using the dielectric ellipsoid, we give a simple explicit method for the determination of all the possible isotropic slowness bivectors for these inhomogeneous circularly polarised waves for which D and S are parallel. Since D is an arbitrary scalar multiple of S, knowing S is equivalent to knowing D.

Let S + and S- lie in the plane whose unit normal is/3. A central section of the dielectric ellipsoid ~ 1 xixj = 1 by the plane /3 �9 x ---- 0 is generally an ellipse ~r (say). Let its major and minor semi-axis be of lengths a and b, respectively, and be directed along the unit vectors ~ and f , respectively. Let ~ = ~ �9 x, r / = ~ �9 x. Then the equation of d is

~2 ~2 a- T + ~ - = 1. (6.67)


For S+ and S- lying in the plane of ~ and i , the condition (4.11) becomes

1 ~2 ((S+. ~)2 -- (S-" 4) 2} -[- ~ - ((S+, ~.)2 _ (S-" ~)2) = --~/A. (6.68)

Families of concentric ellipses, similar to a, r and similarly situated, have equation

~2 7~2 ~" + ff~- = ~p2, o~ = 1, 2 . . . . (6.69)

Here ~0, are constants which are used to label the ellipses. Now S + and S- are equal in length. Also they must be directed along the principal axes of ~r There are infinitely many pairs of ellipses in the family (6.69) such that the length of the major axis of one is equal to that of the minor axis of the other. Indeed this is so for any two ellipses ~0~ and ~o~ of the family (6.69) for which a~0~ -- b~o~. The condition (6.68) is also satisfied for only one pair of such ellipses ~x and ~P2 (say) provided

S+ = a~pat~, S- = b]P2 ~ , (6.70)

where

alo, = b~p2, ~p2 _ ~p22 --_ _ t z ,

# b 2 # b 2 /ta 2 __ /z (6.71) ~ 2 _ _ a 2 _ b - - - - - 5 _ e 2 a - - - ~ ' ~v2 a 2 _ b 2 e 2 .

Here e is the eccentricity of the ellipse ~r e2a 2 = a 2 - - b 2.

In the two special cases when ~ is a circle (/5 is then along an optic axis), the condition (6.68) may not be satisfied unless # = 0 since in order that Is+ I = ]S-[ we must have ~p~ = ~2.

To determine S+ and S- one may proceed as follows. Take a central section of the dielectric ellipsoid. In general this is an ellipse. Determine the lengths a, b, of the principal semi-axes and the value of the eccentricity e. Thus ~p22 = Q~/e 2) is known. Hence the ellipse ~P2 is determined--its major semi axis being a~02 and its minor semi axis is b~p2. But the minor semi-axis b~p2 of the ellipse ~02 is equal in length to the major semi-axis a~pt of the ellipse ~ot. Thus the ellipse ~p~ is determined. S + has magnitude equal to that of the major semi-axis of v2~ and is directed along it. Similarly S- has magnitude equal to that of the minor semi-axis of ~02 and is directed along it.

w 7. Uniaxial Crystals

The propagation of inhomogeneous plane waves in uniaxial crystals is considered now. The secular equation has two factors, one corresponding to the spherical wave-velocity surface and the other corresponding to the spheroidal wave- velocity surface. The eigenbivectors, energy flux vector and energy density are obtained for each mode. Then (w 7.2) it is seen that the mean energy flux vector takes simple forms for certain orientations of the planes of constant phase and the planes of constant amplitude.

248 M. HAYES

Next (w 7.3) all the possible circularly polarised plane wave solutions for which D and B are parallel are obtained. In every case the optic axis is along the normal to the circle of polarisation and the mean energy flux vector is also along the optic axis. It is well known [6] that homogeneous circularly polarised waves of opposite handedness may propagate along the optic axis. This is also found to be the case for the inhomogeneous circularly polarised waves. However there is a difference. Two homogeneous circularly polarised waves of opposite handedness which can propagate along the optic axis have the same slowness and hence may be suitably combined to give a linearly polarised wave. In the ease of inhomogeneous circularly polarised waves no two waves have a common slowness and hence combinations of such waves of opposite handedness will not give a linearly polarised wave. Also there is an infinity of possible inhomogeneous circularly polarised waves, but just two pairs of homogeneous circularly polarised waves.

Then (w 7.4) we obtain all the possible inhomogeneous circularly polarised plane wave solutions for which D and S are parallel. The mean energy flux vector is found to be in the direction of the normal to the planes of constant phase. It is orthogonal to the optic axis for positive crystals but is not so for negative crystals. Finally we give a simple geometrical construction based on the dielectric ellipsoid for the determination of S+ and S-. For a uniaxial crystal the dielectric ellipsoid is a spheroid, the axis of symmetry being along the optic axis. For circularly polarised waves, with D and S parallel, it is seen for positive crystals that the normal to a plane of constant amplitude lies along a diameter of the central circular section of the dielectric ellipsoid (spheroid). On the other hand, for negative crystals, the normal to a plane of constant phase lies along a diameter of the central circular section of the dielectric ellipsoid.

w 7.1. Inhomogeneous Plane Waves

Let

ul ----- ~2 =~ za. (7.1)

Then the optical tensor is given by

i~Q,q = (C . C ~ij - CiCj) (u~-l~jq ~ ( ~ - ~-1) t~ia~q3)" (7.2)

Now

[zQiq(C2 (~ql - - C1 (~q2) = C - C ( C 2 (~il - - C1 0i2) h~i -1 , (7.3)

tzQ,q(C~C3 Oq~ q- C2Ca hq2 -- (C 2 q- C 2) Oqa) (7.4)

= ( ( c , + + (c , c3 + c:c3 8,: - +

There are thus the two solutions. For the first, (7.3), if C is real,the slowness surface is a sphere. For the second, (7.4), if C is real, the slowness surface is a spheroid. To differentiate between the two, in conformity with standard usage, I'll refer to the solutions corresponding to (7.3) as "ordinary" and those corresponding to (7.4) as "extraordinary" waves.

Inhomogeneous Waves in Crystals

Ordinary Waves. H e r e

f f T - 2 e -2'§ : C . Cui -~,

(m 2 - - 1) T 2 cos 2~b = ffu~,

s in 2~b : 0 ,

D = aTe'~(Cz, - - C1, 0),

E ---- aTe'~xTl(C2, --C1, 0),

B = aT2e2'~(C~C3, CzC3, - - (C 2 d- C])) ~?~,

H = #-~ B ,

S = Te '~ C,

249

(7.5)

Extraordinary Waves. H e r e we have

f f T - : e -2'§ = (C~ q- C])nf' q- C]u{-', # T -2 cos 2$ = (m2fn ] - - h 2) (u f I - - u ; l) + (m 2 - - I) ~ - 1 ,

f i T - 2 sin 24, = --2mrh3h3(~i -z - - ~31) ,

a = bTe'*(C,C~, C2C3, - - ( C ~ + C~)),

i~. = b Te'§ ( Ci C3u? I, C2C3~{1, _ ( C 2 -F C 2) u ; ') ,

8 = b~,(-C~, c l , o),

H = b(--C~, Cx, 0), (7.8)

• {(m 2 - - 1) cos 2~ - - (m 2 q- 1)}.

- - a a B

r,,,~ = ~ r'e-'*[C,(Cf + ~ ) , C~(~'f + ~) , (C,E, + C~C~) C~], t , t~l

N1

T 2 + {(c~, + c2~) c3?a + (cf + c~) (gf + ~,~)}l,

w h e r e a is a sca lar . U s i n g (7.5)~ a n d

2S+ = T(e q' C -t- e -'~ C), (7.6)

2~s- = r(e '* C -- e- '* C),

we m a y s h o w t h a t (6.10) is va l id . A l s o , u s ing (7.6), we have

[~^Kq+. ( s + s - ) = aZ~ ^ ^ tt ~ TSm(m~n2 - - r~/2nl. ) (mrS/3 COS ~ - - h a sin O (7.7)

250 M. HAYES

E^H = bbTe'*[C,(C2~ q- C 2) n ; ' , C2(C~ q- C~)n~-',

c3(c, cl + c2c2).i-'l,

o . ~ = ~r2[(Cl~ + c27~) c~c3~i-' 2 --2 + (c~ + c~)(c, + ~) .7 ' 1 ,

A. i f = + c E),

where b is a scalar. Using (7.6) and (7.8)1 it may be shown that (6.10) is valid. Also, using (7.8) gives

( # m 2 + l . ) [EAH]+'(S+S-)=b~)T3m(Fnth2-- rh2h~) (rnTn3 c~ 4~-- h3 sin rb) ~-i n~ "

(7.9)

w 7.2. Energy Flux Vector

Here the form of the mean energy flux vector for the two modes is considered, and shown to have rather simple forms in certain circumstances.

For the first solution (7.5) the planes of constant phase are orthogonal to

the planes of constant amplitude and since ~i~. S- = 0, ~ " S+ -=_ Z ~, it follows

hat if ~ - . (S+S-) = 0, then ~- is given by

= 2 S§ (S+. S+ ) . (7.10)

This occurs if the right hand side of (7.7) is zero. Thus the mean energy flux vector has the form (7.10) if either

o r

o r

(i) m = 0,

(ii) rht/~2 = r n 2 n l , (7.11)

(iii) mrha cos ~ = h3 sin $.

Since 24, = 0 or ~, the possibility (m 2 -- 1) cos 24~ ----- m 2 + 1 leads to m = 0. I f m = 0 the wave is homogeneous. In case (ii) (rh^h). k = 0 so that the normals to the planes of constant

phase and the planes of constant amplitude and the optic axis are coplanar. In case (iii), by (7.6)1, S + ----- 0 so that the normal to the planes of constant

amplitude is at right angles to the optic axis and to the normal to the planes of constant phase.


E x t r a o r d i n a r y W a v e s . For the second solution (7.8), the planes of constant phase are generally not at right angles to the planes of constant amplitude. I f

then since ~ - S - = 0 , ~ - S + = - ~ , it follows that t~ is d r . = o, given by

o r

( s + s - ) ^ s - =

This occurs if the right hand side of (7.9) is zero. Thus the mean energy flux vector is given by (7.12) if either

(i) m ----- 0,

(7.12)

(ii) rh^fi, k = 0,

or (7.13)

o r

(iii) mrr/3 cos 4' = h3 sin 4',

(iv) #gl = (m 2 + l ) T 2 .

In case (i) the wave is homogeneous. In case (ii) the normals to the planes of constant phase and the planes of constant amplitude and the optic axis are coplanar. In case (iii), from (7.6), $3 + = 0, so that the normal to a plane of constant phase is at right angles to the optic axis.

Finally, in case (iv), by using (7.8)2,3 and eliminating 4, and T we find that m must satisfy the quadratic in m2:

m4(~r "1 - - ~ 3 I) (rn 2 - - 1) [rn2(~r -1 -- ~r + (~r + x31)]

+ 2m2{(~11 __ ~r ^2^2 -1 [m3n3(~r 1 _ ~tj-I) (r~t2 + /~2) , i l l _ (;~12 _~_ 1r

+ (~i -1 -- ~j-~) (h 2 -- 1) [ h 2 ( ~ { ~ - u f ~) + (~i -~ + nf~)] = 0. (7.14)

Now m z must be positive. Also ~1, u 3 > 0 , th 2 ~ 1 , h 2 ~ 1 . If the crystal is positive (g3 > ul), then (7.14) has no real solution for m 2 since

the coefficients of m 4 and m 2 are negative and so is the independent term. If the crystal is negative (u~ > u3) then (7.14) may have positive solutions

for m 2. For example if rh] = �89 h ] = �89 then (7.14) becomes

( m 4 -]- 1) (xl + 3~3) (Xl - - ~t3) = 2 m 2 ( 3 ~ 2 -F- 6~ti~r - - ;~2). (7.15)

The discriminant is

64ulu3(~J + x3) (3~3 -- ~1). (7.16)

Thus if x~ > ~3 and 3u3 > u~ there are two real positive roots for m 2. Having determined m, T is found from (7.13), and 4, is obtained from (7.8)2,3 If u ~ > u 3 and 3 u 3 < ~ there are no real roots for m 2.

Hence if the crystal is positive (~3 > ~r no solution is possible in case (iv), but if the crystal is negative two solutions may be possible.

252 M. HAYES

w 7.3. Circularly Polarised Inhomogeneous Waves. D and B Parallel

The propagation of circularly polarised waves for which D and B are parallel is considered here. All the possible solutions are obtained. It is seen in every case when such waves propagate that the optic axis is along the normal to the plane of the circle of polarisation. In each case the bivectors E, D, B, H are all parallel. Also, the circularly polarised waves are either homogeneous, or have their planes of constant phase orthogonal to their planes of constant amplitude. The mean energy flux vector lies along the optic axis for each circularly polarised wave. Also, in every case when circularly polarised waves may propagate, two waves of opposite handedness may propagate.

Turning now to the details, note that the roots (7.5)1 and (7.8)1 are equal if and only if

C 2 + C 2 ---- 0, (7.17)

the common root being given by

/ z T - 2 e - 2 ' 4 ' : C2~r . (7.18)

When (7.17) holds it follows from (7.5) or (7.8) that the mean energy flux vector is parallel to the optic axis (0, 0, 1).

Since C = mrh q- th, r h . h : 0, it follows from (7.17) that

mrn3h 3 = 0, mE(1 -- rh 2) = 1 - h 2. (7.19)

There are thus three cases to be considered:

Case (i) m ---- 0, h ] ---- 1 ;

Case (ii) rh 3 = 0, m 2 = 1 -- h 2, h 3 =~= 0;

Case (iii) m2(1 - rh 2) = 1, h3 -- 0, rh 2 # 1. (7.20)

In Case (i), C = (0, 0, tha), the wave is homogeneous, and from (7.18),

T2 = ~ g l , qb ---- Z~/2 or 3z~/2; S = 4-(/,~1) �89 (0, 0, 1). (7.21)

Taking the positive sign in S, we obtain

s = (t,~1) ~ (0, o, 1), o = ~ O ~ d ( + l , t, 0) ,

8 = _ , ( f f l~ , ) t D , O - ~ = 2a2~,, (7.22)

B . H = 2ddl~,

E^~ = 2dg(~/~,)~ (0, 0, 1),

where d is a constant. This is the well known homogeneous circularly polarised plane wave solution. The -4- signs in D correspond to waves of opposite handedness. The slowness bivector is a vector along the optic axis (0, 0, 1) and the mean energy flux vector is also along the optic axis.


Similar results are obta ined if the waves p ropaga t ing along (0, 0, - -1) are considered.

In Case (ii), rh a = 0, m 2 ---- 1 - - h j, C = (Cl , 4-tC1, tha), inhomogeneous , and f rom (7.18)

T 2 - - I~gl/h 2, (b = z~/2 or 3z~/2.

Thus

the waves are

(7.23)

( ~ tel S : + t ( p ~ l ) �89 , 4 - - ~ a , t ] , (7.24)

where any combina t ion of the 4- signs m a y be t a k e n - - t h e r e are four possible S's . We consider only one example; the others are similar. Suppose

C = (C1, ,C~, ,h3), S----,(/~ul) �89 (C_~ tC~ ) ' ha ' t , (7.25)

C 1 - - - - ( 1 - - h 2 ) �89 tn l q- th 1.

Then

where

D = d(1, t, 0),

8 = t(~/ . , )~ D ,

O " E q- B" H = 4dd/ut,

2dd E^f f ---- #�89 (0, 0, 1),

[EAH]+" S + = 2ddfir

(7.26)

Ci- S + = ~1)~ - h3 ' , 1 , S - = (/t~l) �89 ~- 0 . (7.27)

\ n3 na

The planes o f constant phase are S § �9 x : constant , where $+ is given by (7.27). Similarly the planes of constant ampli tude are S - �9 x : constant . These planes are mutual ly or thogonal .

Finally note that, if instead o f taking C given by (7.25), we take C - - - - ( C ~ , - - t C ~ , tha) then T is given as before by equat ion (7.23), S : t (~) �89 (CI, --tC1, tha)/(ha) and D has the fo rm d(1, - - t , 0). This wave has handedness opposi te to tha t o f (7.26). However the two waves have different slownesses and hence cannot be combined to give a single linearly polar ised wave solution.

In Case (iii), C = (C1, • mrh3) and we find f rom (7.18) tha t qb----0 or re, and T 2 : #u~/(mZ~h]). Again, there are four possibilities for S and two pairs o f waves m a y propagate . Within each pair the waves are o f opposi te handedness. We conclude by giving an example.

254

Let

M. HAYES

C = (C1, tCl, mrh3), C1 = -4-rhl(1 - rh•)- ~ -q- lhl, ~ ] =4 = 1 ; (7.28) (G ,G )

S = ( # g l ) t m--m3'mrh3'l , S + - s - = 0 .

Here rht, rh3, hi are arbitrary apart from the constraints rh~ =4= 1, ri �9 ri = 1. Then

D = d(1, t, 0 ) ,

B = - - t ~ D ,

2a~ EAH --/z~u~/z (0, O, 1), (7.29)

2 d a r [E AH] + �9 S+ _ ,

-- 4dd D . F . + B . H - - ,

where d is a scalar.

~ n . r h =

w 7.4. Inhomogeneous Circularly Polarised Waves. D and S Parallel

Proceeding as in w 6.5 for biaxial crystals we determine here for uniaxial crystals the inhomogeneous circularly polarised waves for which D and S are parallel. For these waves C . C = 0 and hence they are not possible among the ordinary wave solutions--they belong to the class of extraordinary wave solutions. It is seen that these circularly polarised waves are possible for any choice of the circle of C other than that one in the plane whose normal is along the optic axis. Also the mean energy flux vector is in the direction of the normal to the planes of constant phase. It is orthogonal to the optic axis for positive crystals (ul = u2 > u3) but is not so for negative crystals (ul = ~I~2 < ~1~3).

We conclude by giving a simple geometrical construction for the determination of S + and S- and hence also for the determination of the circle of D. This construction is based on the dielectric ellipsoid.

w 7.4.1. Determination o f S. The equations (4.11) may be written

(S~ 2 -~- 52 +2 )N11 -~- $3~2~ -1 _ ( 5 1 2 -]- 522 ) ~ l I - - $32931 = - - ~ ,

( s ? s F + s2+sF)~l 1 + s3+s3,~f ' = o,

S+ �9 S+ = S - �9 S - , S+ �9 S - = O. (7.30)


Proceeding as in w 6.5 we obtain

#T -z cos 2$ ~- (g31 - - g l 1) (nJ -- mJ), (7.31)

# T -2 sin 2~b = 2(~r -1 -- ~11) m3n 3 .

These equations may not be solved for T and ~b if n3 and ms are both zero. In this case the optic axis, which is along k, is normal to the circle of C. Thus, for this one choice of C, these circularly polarised waves with D and S parallel are not possible.

This case apart, we obtain from (7.31)

/.tT -2 = (g31 -- ~r I) (n j + mE), (7.32)

if • ~s (negative crystal), and

biT-2 = _ (~ -1 _ ~i-1) (n 2 q_ m]), (7.33)

for a positive crystal. We consider the two cases separately.

w 7.4.2. Negative Crystals. For a negative crystal it follows from (7.31) and (7.32) that

tan ff : (m3/n3). (7.34)

Since m = 1, it follows that S + : 0, and thus the normal to the planes of constant phase is at right angles to the optic axis. We note for these waves that T and ff are determined by the components of C along the optic axis. Having chosen C, satisfying C- C ---- 0, and such that C . k =~ 0, expressions for D, E, B, etc. may be obtained from equation (7.8). In particular the mean energy flux vector may be determined on noting that

[E^/t]+ : bbC3C3u31(S +, S +, 0). (7.35)

This follows from (7.8) by noting that C . C =- 0 and that $3 + ---- 0 is equivalent

to C3 + e 2'~ Ca = 0. Thus for these waves in negative crystals the mean energy flux vector is ortho-

gonal to the optic axis and is directed along the normal to the planes of constant phase. The mean energy flux vector has the form given by equation (7.10).

w 7.4.3. Positive Crystals. For a positive uniaxial crystal it follows from (7.31) and (7.33) that

tan ~ ---- --(na/m3). (7.36)

Since m = 1 it follows that Sj- = 0 and thus the normal to the planes of constant amplitude is at right angles to the optic axis. Now S~- = 0 is equivalent

to Ca = Cae 2'~. Using this and C- C ---- 0 it then follows from equation (7.8) that

[E^H] + : bb((--C3~S+u~ 1, - -C3C3S~-7r (2 -- C3C3) S+u{~). (7.37)

256 M. HAYES

Thus the mean energy flux vector has a component along the optic axis, unlike the situation for negative crystals.

w 7.4.4. GeometricalConstructionfor the Determination of S+ and S-. A geometrical construction for the determination of S + and S- for these waves in biaxiaI crystals has been given in w 6.5.2. This construction may also be used here. The dielectric ellipsoid is now the spheroid

)C2 .~_ y2 Z2 -}- - - = 1. (7.38)

~J ;~3

This has only one circular central section, namely z -- 0, x 2 q- y2 = ~i. Any other central section is an ellipse F (say). The central plane cuts the central circular section along a diameter which has length 2(~) ~. For a positive crystal z3 > ~ , and thus this diameter of F is along a minor axis of F. For a negative crystal za < ~r and the diameter of F in the plane of z = 0 is a major axis of F.

In order to determine S + and S-, we may draw the family of ellipses which are concentric with F and similar and similarly situated with respect to it. S + and S- are of equal lengths and are orthogonal.

For a negative crystal S+ is in the plane z = 0, is equal in length to the major semi-axis of one ellipse Fl(say) and directed along it. S ~ is equal in length to the minor semi-axis of one ellipse F2(say) and is directed along it. The minor semi-axis of F2 has length equal to that of the major semi-axis of Fz. Also the difference of the squares of the major semi-axes of F2 and F1 equals ul/~.

For a positive crystal S- is in the plane z --~ 0, and is equal in length to the minor semi-axis of one ellipse Fa(say) and directed along it. S + is equal in length to the major semi-axis of one ellipse F4 (say), and is directed along it. The minor semi-axis of F3 has length equal to that of the major semi-axis of F4. Also the difference of the squares of the minor semi-axes of F3 and F4 equals xl#.

w 8. Isotropic Crystals

The familiar case when the crystal is both electrically and magnetically isotropic is considered here. The spherical and spheroidal real slowness surfaces associated with a uniaxial crystal coalesce into a single spherical slowness surface for an isotropic crystal. The secular equation has double roots for any choice of C -- other than an isotropic C, which is not possible unless D = 0, so that there is no field. Hence homogeneous circularly polarised waves may propagate in any direction, as is well known [5].

Here we concentrate on inhomogeneous plane wave solutions and give all such solutions of Maxwell's equations for an isotropic crystal. For all of them the planes of constant phase are orthogonal to the planes of constant amplitude. It is seen that for any choice of G--other than an isotropic C--the amplitude bivector D of the electric displacement field may be so chosen that D is isotropic. Hence for such C's circularly polarised D wave may propagate.


w 8.1. Inhomogeneous Plane Waves

Since the m e d i u m is electrically as well as magnet ical ly isotropic, • = u ~ij and equat ion (3.4) becomes

( C " C - - # u T - 2 e -2'~) D = 0, (8.1)

since D - S = 0 , and thus

C " C, : # ~ T - 2 e -2~4' . (8.2)

Now, recalling tha t S = Te '~ C, C = m q- th, it follows tha t

T2e2,r _ / t~ (8.3) (m 2 - 1)"

There are two cases to be considered: Case (i) m 2 > 1 and Case (ii) m 2 < 1. The possibil i ty m 2 = 1 is not feasible since then C . C = 0 and so by (8.1), D- - - -0 . In

Case (i).

m 2 ~ 1, 2~b = 0 or 2z~,

T : 4-(#u/(m 2 - - 1)} j ,

s = T(m + th), ^

E = fl q- th q- -~- ~ , D = uE,

B = to~mT + th -t- ~ ~" , H B ,

D . E + B . H : 2ufl/~ + 2~or 2 - - 1) -1 ,

[ E A H ] + = L m 2 T q - - 7 m - - 2 T ~,

2[EAH]+. S + = 2[EAH] + - T m = O . -E q- n . i f ,

[ ~ ^ g q + . s - = [ ~ ^ t ~ + �9 r , i = 0 ,

(8.4)

the given fo rm since E . S = 0. (3.1)3: B ---- ShE.

Similarly in

Case (ii)

The expression for B then follows f rom equat ion

m 2 - ~ 1; 2~b----~ or 3~,

r = -+ ~ / ( 1 - m2))L

where o~, 13 are a rb i t ra ry constants and ~" is a unit vector parallel to rh^h. E is o f

258 M. HAYES

S = T ( , m - - fi),

(m = t y . . + + ~ , E

\ m

B = am T + th + T--------- ~ 9

D E + B f f 2 ~y~ ~ �9 . = m2 + 2 ( 1 _ m 2 ) ,

re^ ]~+ = 2 T - - T +

2I~^ff l+- s+ = 2[r^ff l+ �9 ( - 7 " , ) = o . ~ + B . ~ ,

[~^~1 + �9 s - = [~^ffl+ �9 T m = O.

(8.5)

w 8.2. Circularly Polar ised Waves

Now we consider circularly polarised ~ waves. If the possibility of isotropic D and S being parallel is considered, it may be seen from equation (8.1) that it leads to S = 0 and therefore also to D = 0 so that there is no field. Thus it is only if D and B are parallel that these circularly polarised waves are possible.

In both Cases the scalars o~,/3 and y, ~ may be chosen so that the wave is circularly polarised. Thus in Case (i) choose

1 ~2/fl2 = 1 m2 = # ~ / ( T 2 m 2 ) ,

S = • z - - 1)} �89 (m + th),

S . S =/z• (8.6)

so that D - D - - - - 0 , of the circle of polarisation being

h and {m m---~ • ( 1 _ m 2]1]�89 ~.} , (8.7)

The two solutions D+, D_ (say) given by equation (8.6) correspond to circularly polarised waves of opposite handedness. This follows since D+ �9 D_ = 0.

The mean energy flux (Poynting) vector /~ is given by

o _ - ~ + ,,~ -4- 1-~! j ,

B = • D,

the wave is circularly polarised, orthogonal semi-diameters


Note that /~ is parallel to D+^D - for each wave so that the Poynting vector is perpendicular to the plane of the circle of polarisation of 9 .

Similarly in Case (ii) take

~2 /~,2 = (~ ,O/(T2m2) ,

S : 4-t~z~/(1 -- m2)) �89 (m § ih),

S . S = --/,~, (8.9)

D = t~ 7 -}- i f i4- t - - 1 , B = 4 - t ~ / ~ ) � 8 9

D = 0, the wave is circularly polarised, orthogonal semidiameters Again, D, of the circle of polarisation being

~h and h 4- ~-~ 1 . (8.10) n'/

In this Case the mean energy flux vector is

/~ ----- ~'---~'~ (~-)�89 {-- ;" 4- (~--2 -- 1)�89 h / exp ( - 2 t ~ x)" m (8.11)

Again, the Poynting vector R is perpendicular to the plane of the circle of polarisation of 9 .

The expressions in (8.4) and (8.5) give all the inhomogeneous plane wave solutions to Maxwell's equations in isotropic media.

Note that in both cases, 4~ is an integer multiple of (z~/2) and hence the planes of constant phase are orthogonal to the planes of constant amplitude. Recall that h is a unit vector and m is at right angles to it; for fixed h the vector m takes any orientation and magnitude in the plane which has h for normal; the unit vector h may take any direction in space.

In Case (i) (see equation (8.4)) the ellipse of g has 1/31 + r and

[ ( 7 ) - ~ ' ] pair of conjugate semidiameters. Also, the ellipse of ~ 1/31 h + as a

has rm I~[ ~ + ~ and irm 10~1 h + ~ as a pair

of conjugate semi-diameters. For Case (ii) similar statements may be read off from equation (8.5).

Acknowledgement. I am greatly indebted to my colleague T. J. LAFFEY. His suggestion led to the improved proof given in w 5.2.1.

References

1. GIBBS, J. W., Elements of Vector Analysis, 1881, 1884 (privately printed) = pp. 17- 90, Vol. 2, Part 2 Scientific Papers, Dover Publications, New York 1961.

2. SVNGE, J.L., The Petrov Classification of Gravitational Fields, Comm. Dublin Inst. for Adv. Studies, A, No. 15, Dublin, 1966.

260 M. HAYES

3. STONE, J. M., Radiation and Optics, McGraw-Hill, New York, 1963. 4. HAYES, M., Inhomogeneous Plane Waves, Arch. Rational Mech. Anal. 85, 41 (1984). 5. BORN, M., & E. WOLF, Principles of Optics, sixth edition, Pergamon Press, Oxford,

1980. 6. PARTINGTON, J. R., An Advanced Treatise on Physical Chemistry, Vol. IV, Longmans,

London, 1953. 7. HAVES, M., Inhomogeneous Plane Waves in Incompressible Elastic Materials, in

Wave Phenomena: Modern Theory and Applications C. ROGERS and T. B. MOODIE (eds.), Elsevier Science Publishers B.V. (North-Holland), 1984, pp. 175-191, 1984.

8. SMITH, G. F., & R. S. RIVLIN, Photoelasticity with Finite Deformations. ZAMP 21, 101 (1970).

9. HAYES, M., Energy Flux for Trains of Inhomogeneous Plane Waves. Proc. R. Soc. Lond., A. 370, 417 (1980).

10. JEr, rKINS, F. A., & H. E. WHITE, Fundamentals of Optics, third edition, McGraw-Hill, New York, 1965.

Department of Mathematical Physics University College

Dublin

(Received April 1, 1986)

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Inhomogeneous electromagnetic plane waves in crystals