Wave Propagation in General Anisotropic Media

!WAVE PROPAGATION IN GENERAL ANISOTROPIC MEDIA,

A ~hesisPresented to the Faculty of the College of Engineering and Technology Ohio University

In Partial Fulfillment Of the Requirements for the Degree Master of Science

Habib Taouk -- 1 August, 1986 iii

TABLE OF CONTENTS Page Acknowledgements ...... v Abstract ...... vi CHAPTER 1 BASIC EQUATIONS IN GENERAL ANISOTROPIC MEDIA 1.1 Introduction ...... 1 1.2 Maxwell's Equations ...... 1 1.3 Constitutive Relations, Dielectric and Permeability Tensors ...... 2 1.4 Plane-Wave and Homogemeous Equations ...... 3 1.5 Dispersion Equation ...... 6 1.6 Directions of the Field Vectors .....,...... 11 1.7 Isonormal Waves ...... 16 1.8 Group and Energy Velocities ...... 18 1.9 Summary ...... 23 CHAPTER 2 REFLECTION FROM GENERAL ANISOTROPIC MEDIA 2.1 Introduction ...... 24 2.2 Phase Matching ...... 25 2.2.1 Law of Reflection and Transmission .... 25 2.2.2 Booker Quartic ...... 27 2.3 Transmission and Reflection Coefficients ... 32 2.4 Reflected and Transmitted Waves for Normal Incidence ...... 36 2.5 Summary ...... 39 CHAPTER 3 WAVES PROPAGATION BIUNIAXIAL MEDIA Introduction, Optical Classification of Crystals ...... 41 Wave Vector Surfaces ...... 42 Directions of the Field Vectors ...... 47 Energy ~ensities,Poynting Vectors, Ray Vector Surfaces, and Velocities in Biuniaxial Media 53 Reflected and Transmitted Wave Vectors at an Isotropic-Biuniaxial Interface ...... 56 Conditions for the Transmitted Waves to Exist 59 ~eflectionand Transmission Coefficients ... 61 Reflected and Transmitted Field Vectors for Normal Incidence ...... 66 3.9 Special Cases ...... 71 3.9.1 optic Axis Parallel to the Plane of Incidence ...... 71 3.9.2 optic Axis Parallel to the Interface .. 74 3.9.3 optic Axis is perpendicular to the ~ntersectionof the Interface and the Plane of Incidence ...... 76 3.10 Rotation of the Plane of polarization upon Reflection; BrewsterlsAngle ...... 78 3.10.1 optic Axis Parallel to the Plane of Incidence ...... 80 3.10.2 Optic Axis Perpendicular to b, the Intersection of the Plane of Incidence and the Interface ...... 83 3.11 Special Case: Brewster's Angle at Isotropic- Isotropic Interface When €, f €, and u ,$ P, . . . 86 3.12 Energy Relations ...... 89 3.13 Total Reflection: Special Cases ...... 92 3.14 Summary ...... 94 CHAPTER 4 BIUNIAXIAL MEDIA: NUMERICAL APPLICATION 4.1 Introduction ...... 98 4.2 General Equations: Arithmetical Form ...... 98 4.3 Optic Axis Parallel to the Plane of Incidence ...... 102 4.4 Optic Axis Perpendicular to the Plane of Incidence ...... 107 4.5 General Case: Optic Axis Has Any Direction . 109 REFERENCES ...... 134 APPENDIX A VECTOR AND TENSOR ANALYSIS A.l Introduction ...... A-1 A.2 Vector Analysis ...... A-1 A.3 Tensor Analysis ...... A-3 A.4 Operator V I1Del" and Tensors ...... A-8 A.5 Dyadic Decomposition of a Matrix; Solution of Homogeneous Equations; Eigenvalue Problem A-9 A.6 Hermitian Matrices ...... A-12 APPENDIX B COMPUTER PROGRAM ...... B-1 ACKNOWLEDGEMENTS

I would like to acknowledge my gratitude to The Lebanese University for their support in my study in the U.S.A., and I would like to express my sincere thanks to Dr. Chen for his kind advise and good guidance throughout the course of my work in this thesis. I also wish to thank "the one who shakes the cradle with her right hand, shakes the world with her left handg1 for her unlimited love and support, as well as my relatives who provided me with the encouragement and the financial means to continue my study. Never to forget the one who stands, her feet above the dust, her head higher than the clouds, waiting for the lightest breeze to blow toward the west to send the ever smallest letter by the purest mean repeating in the distances of the space,

ItI love you!" ABSTRACT

The use of anisotrpic material, especially in the area of optical fibers and integrated optics, has increased significantly in the past few years; consequently, more theoretical research is needed to find the characteristics of the wave propagation in such materials. Most of the previous studies on this subject considered materials with either electric or magnetic anisotropy [3,8], using one or more coordinate systems. The subject of this thesis is to present a profound study of the wave propagation in general-anisotropic and lossless media, that are electrically and magnetically anisotropic making the permittivities and permeabilities hermitian matrices. These media are considered source-free and homogeneous. We adopt the coordinate-free approach, introduced by Chen 161, as a mathematical means that greatly facilitates the solutions of our problems. his method is based on direct manipulation of vectors, dyadics, and their invariants, eliminating the use of coordinate systems. In chapter one, we derive the various equations that characterize the wave propagation in unbounded general- anisotropic media, such as the dispersion equation, the directions of the field vectors, and the energy and group velocities. In chapter two, we consider semi-bounded media. vii We determine Booker quartic equation and the transmission and reflection coefficients at the interface of isotropic- general-anisotropic media where the incident wave is traveling in the isotropic medium. The third chapter is a special case of the first two chapters, considering real diagonal permittivity and permeability matrices with two repeated elements. In this case, the medium is called "Biuniaxialll. A more detailed study is conducted in this medium due to its importance in the practical applications. We determine the wave vector surfaces, the directions of the field vectors, the energy densities, the wave vectors, the reflection and transmission coefficients, the rotation of the incident plane of polarization upon reflection and Brewsterls angles, and the transmittivities and reflectivities. Chapter four is a direct numerical application of the third chapter. The reflectivities and transmittivities are ploted versus the angle of incidence and compared to those at isotropic-isotropic interfaces. Results and concepts that are difficult to be derived from the mathematical expressions are obtained graphically. Finally, we shall mention that the notations used throughout the text of this thesis are defined in appendix A

§A. 1. Moreover, if the reader wishes to follow the deriva- tions of the equations, we recommend that he has a certain degree of knowledge of tensors and vectors analysis; if not, he can read appendix A as a quick reference. CHAPTER ONE BASIC EOUATIONS IN GENERAL ANISOTROPIC MEDIA

1.1 Introduction Extensive studies have been conducted on anisotropic materials whose optical properties depend on the direction of propagation of the light waves [3,6,12,14,15,17].However, most of these studies considered the material to be either electrically or magnetically anisotropic; in other words, only one of the permittivity and permeability can be a tensor (3x3 matrix), the other a scalar. Few papers we-?, presented considering the electric and magnetic anisotropies in the same time [6,8,9,13], but for specific purposes (e.g., dispersion equation) or special cases (e.g. gyro- electromagnetic biaxial media). This chapter will present the different equations that characterize the uniform plane wave propagation of monochromatic light in a general anisotropic, lossless, homogeneous, and source free media. These equations will be derived using the coordinate free approach invented by Chen [6].

1.2 Maxwellls Emations For handy reference, we will review in this paragraph Maxwell's equations in time domain and in frequency domain for any frequency and for monochromatic fields, in a source- free ( p=O and J=O) region. a) Time domain [3,6,7]

where 8,Aft 3, and 9 are real vectors that represent the electric and magnetic field intensities and the electric and magnetic flux densities respectively. b)Freuuencv Domain [3,6,7,14,17] In general, we are dealing with steady state sinusoidal time-varying fields. It is convenient to represent each field vector as a complex phasor by applying Fourier transform. Assuming that we have monochromatic wave, we can write -iwt 8 =Re (E e 1.2 where E is a complex vector and function of r, and w is the angular frequency of the wave. The other field vectors can be written under the same notation of eq.1.2 . Thus, Maxwell s equations (1.1) become VxE = iwB VxH =-ioD

V*B = 0

V0D = 0

The complex wave fields E, H, B, and D can also be con-

sidered to vary sinusoidally in space [6,14,17] :

A where K=Kk is the "wave vector^, the propagation vector, or

simply the K vector, K is the "wave number1', and the unit vector k is the "wave normal". Substitution for El given in eq. 1.4, and similarely for H, B, and D into Maxwell's equations 1.3 gives

(a) KxE, = wB.

(b) XxH, = -wD, 1.5

(C) K'D, = K'B, = 0 Note that K is perpendicular to both B, and Do. However, B, is not orthogonal to E, , nor D, is orthogonal to H, since B,, E., D,,and H. are complex.

1.3 constitutive Relations. Dielectric and Permeability Tensors [3,6,14,17] Maxwell's equations alone do not allow a unique determination of the field vectors. We introduce the Constitutive equations that describe the effect of the material media on the electromagnetic fields. where E, and p are the permittivity and permeability of vacuum, given respectively by

and p, = 4~ x lo-' H/m - and p are, respectively, the relative dielectric and permeability tensors. They are considered to be nonsingular.

Note that in a General Anisotropic media, neither E, is parallel to D, nor H, is parallel to B,. - - In order to find the characteristics of E and p , we need to define the complex ~oyntingvector P :

P 1 1.7 = -2 EXH* In a source-free and lossless media, the divergence of the time average of the real Poyntingfs vector must be zero, after the energy conservation law; then [6]

V ' = Re(V0P) = 0 1.8 Substituting eq.1.7 into eq.1.8, making use of eq.A.9- (g) and eq.1.3, and solving the obtained relation for nonzero electric and magnetic fields, we obtain [6] - ;+ = & 1.9 - and ;+ = ,, 1.10 where the superscript if+ff denotes the hermitian conjugate. Therefore, the matrices and ; must be hermitians. We can also prove that E and ; must be definite positive matrices 5 by using the fact that the time-averaged electric and magnetic energy densities must always be positive. They are given as

and where We and Wm are respectively the instantaneous electric and magnetic energy densities, and T is the period.

After substituting eq. 1.2 (for 8 and ) into eq. 1.11, we integrate. Knowing then that the hermitian form is real, and using eq.1.9, we get [6]

In the same way,

Since () and () must be positive, then the tensors - e E and must be definite positive. For real wave vector K , we can easily prove, using eq.1.5, that

(a) E;D: =E;D,* =H,-B,* =H;B, *

(b) E;D, = H;B, Therefore, the total time average energy density is 1.4 - Plane-Wave and Homoseneous ~quations [3,6] i(K*r wt) We consider a plane wave of the type e - where the magnitude of K is real and depends on its direction, as will be shown later. It is important to mention that the components of K, in general, may not be real. In such case, K may be given by K=K +iK2 where K1 and KZ are real vectors. The wave then behaves as e -Kzore i(Kl *r - wt) It decays in K2 direction and propagates in K1 direction.

Since the constant phase front given by K Or - wt = constant has no constant amplitude, the plan wave is called IVNonuni- form Plane Wave. IV However, from now on, we will only consider the uniform plan waves where the wave vector K has real components. Here, the amplitude of the wave is constant since K is real (K, = 0). At a given time, the phase is constant if K*r is constant. The plane that is formed by the tip of the position vector r which satisfies K*r = constant at a given time and for a given wave vector direction is called the VVSurfaceof Constant PhasetV9or simply the "Plane Wave," and is perpendicular to 2. It travels with the "Phase Velocityo

The wavelength is given by

We define the "Reflective index vectorM as

where c is the light velocity in free space: -1 C = (E. Po) and K, is the wave number in free space:

KO = U(E. PO) 1.21 The Maxwell's equations (eq.1.5) are a set of coupled equations. They can be manipulated to give equations in which each of the field vectors satisfies separately. Eq. 1.5, to which we introduce the antisymmetrical matrix (Kxf), and use eq.A.27, can be written as - (a) (KX?) *E, = UB, = u V,U-H, - (b) (KX~)OH, = -UD, = -UE,E-E, 1.22

[K2(kxT)*;-lo (kx?) + K:;]*E. = 0 1.23 Dot premultiplying eq.1.23 by E", we get

A - [K2;-'* (kxi)*;-I (kx1) + ~ii]*E. = 0 1.24 We introduce the matrices k and % as: - - M = E-lo (SX?) 1.25 - - and N = p-l. (;xi 1.26 Then eq.1.24 becomes - - (K~M-N + K!F) -E, = o Using eq.1.6, eq.1.23 may be expressed as [5]

[K2(CxfxI) u-'- (k^xf) E-' + K:?] -Do= 0 1.28 or [K2(GbN)+ + K:I]*D, = 0 1.29 Here, eqs. 1.9, 1.10, A. 13 (f), A. 23 (d) , and A. 62 (d) were used. In the same way we solve Maxwell's equations for H, and B, and obtain

[K2(k~?) (kxi) + 3 *Ho= 0 and [K (k~i)*E-le (;Xi) -;-I + K~TI~B~= o or and

We notice from eqs.1.27, 1.29,1.32, and 1.33 that the field and flux vectors E, , D,, H, , and B. are respectively the eigenvectors of the matrices (), (k*fi)+, (5-k) , and -K~ (i*%)+corresponding to the the same eigenvalue m= 3 = -$, where n is the reflective index vector. Now we will show that n2 (and hence K2) is always real and positive for

A any given wave vector direction k, even complex. Dot premultiplying eq. 1.23 by E: and solving for n2= we get ''K: , * - n2 = E, 'E *E, 1.34

(kx~,)*-i-'* 0 - Since E , 11 , and hence are positive definite hermitian matrices, as shown in § 1.3, the hermitian forms ET*E -E, and - (GXE,) ** p-l (kx~,) are real and positive. Therefore, n is always real and positive, and hence K is always real and positive in a lossless media.

1.5 Dispersion Equation [4,5] The dispersion equation characterizes the behavior a plane wave in the medium. It relates the components of the - wave vector K to the dielectric and permeability tensors, E and i,and the angular frequency, w, of the wave. 9 According to §A.4, the homogeneous equation 1.27 has nontrivial solutions if and only if

lIC2 a.1 + K? TI = 0 1.35 First, eq. 1.35 is expended according to eq.A. 36 (b) , then, using eqs.A.24(c) and A.29(a), we find that

IR*RI = 0 1.36 Therefore, replacing eq.1.36 into the expended expression of eq.1.35, we get

[adj (tgfi)],K4 + (t-t) K~K' + K: = 0 1.37 Using eqs.A.24[(j) and (g)], and A.25(e) we obtain

= (k*i*k)(k9i.k) [adj(P-H) ] lF*jil Using eqs. A.35(f), A.22, and A25(b) we obtain - - k0;*{(adj E) *u [(adj T)-i],i}*k (BsG), = IE*;l - Substituting eqs. 1.38 and 1.39 into eq. 1.37, we obtain the dispersion equation:

where, A = (k*i0k)(k* E*k) 1.42 =: - - A B = k*u*{ [ (adj E) *GI i - (adj z) -6)-k - - and C = IE*~I 1.44 The solutions of the other homogeneous equations 1.29, 1.32, and 1.33, lead to the same dispersion equation 1.40. To show these results, we, first, can easily prove, using eqs.A.13 (d), A.24(j), A.28, 1.9, and 1.10, that (8.1) ,= [(M*N)+], = (i*z)= [(G*fi)+lt 1.45 10 and - - [adj(8-fi) It =[adj (8*I)+lt =[adj (I-fi)It =[adj (N*M)+]~ 1.46 Knowing that a scalar is equal to its transpose, we can find another expression for the term B of the dispersion equation by taking the transpose of eq.1.43 and using eq.A.l3(e): - - B = k*;*{[(adj - (adj ;)*;}*k 1.47 Or, either following the same procedures starting from eq.1.30 or directly applying the identity A.17 on eq.1.47, we get h - - B = k*e*{[(adjE)*z],T - (adj ;)*;}*k- 1.48 A z - - - - or B = k*e*{[(adju)*;lti - (adj P)-E}*~ 1.49 - - Thus, E and F are interchangeable as well as and E, and - and 1-1 . We can find other forms of the dispersion equation if we solve for eq.1.23:

I (~xi)*;-l=(KX~) + K:E~ = o 1.50 Using eq.A.35(f), we get - - Kau*K Z 1 I (KZE P -j$1-1*KK*II)- 0 - ICl + I = If we expand eq.1.51 according to eq.A.37(b) we find K'ii'K 1 1 - K.U.K - - IK5 - 1-11 + K*;*adj[Ki; - i.l]*1-1*K= 0 1.52 ICI - 11-11 If the matix [K;: - (K*;*K); /]ill is nonsingular, we devide eq.1.52 by its determinant, we obtain

Using eq.A.23, eq.1.53 can be written as - - or, interchanging ; and u, and if [K2.i - (K*~*K)i/lF I] is nonsingular, then - 1 + K0F*[K:(adj E)*i - (K*E*K)?]'~*K= 0 1.56 - - - -1 - or 1 + K*[K~i-adj - (Ke€*K)I] *E*K= 0 1.57 -., - We can also interchange E into and 11 into 11 in eqs.1.54- 1.57, and obtain another series of equations. The dispersion equations show that for a given direction of propagation k, there are, in general, two K values, and hence two phase velocity values and two reflective index values. The directions of the wave normals that give only one value for K, are called the "Optic Axis."

1.6 Directions of the Field Vectors [5,61 Given the wave vector K, we will proceed to find the directions of the field vectors, E,, Do,B,, and H,. Introducing eqs .A.22 and A. 35 (f) to eq. 1.23, and substituting K 2/~? by n , we get - - A - {lilg + n2[l*kk*,i- (k*~-k)i]}*~,= 0 1.58 Then if we dot premultiply eq.1.58 by i-'and expend, it becomes - - [(adj ) - n2(kgc*k)?]*E.= -n2(k*c*EO)k We can see two possibilities from eq.1.59:

- - 2"- a) The matrix = r t adi u *E - n tk*u*k)11 is nonsinsular If we dot premultiply eq.1.59 by A", we get

E, =-n2 (k*c*~,) {adj [ (adj- ) - n2 (k*c*k)~]}*k 1.60 I (adj i)*s - n (k*c*k)~~ . - Since [-n2(k*p*~,)/~~l]A - is a scalar, then the electric field vector direction, e, is- given by A-A- e = {adj[ (adj i) - n2 (k*~*k) I] ) *k 1.61 The directions of the other field vectors Do, B,, and H, can be found by using Maxwellls equations 1.5, the constitutive relation 1.6, and eq.1.61:

b=- nxe C

- If the matrix [(adj E)*i - n2 (kg€A--k)I] A- is nonsingular, we can find, starting from eq.1.30, similar forms of the directions of the field vectors: - ht= {adj [(adj G) *; - nZ(k*u*k)f] *k) 1.65 bl= poi*ht 1.66

dl= - - nxht C 1.67 el= LC-'- *dt o 1.68 Note that eq.1.65 can be obtained directly from eq.1.61 by - - interchanging e, E, and into hl,P , and respetively.

Since KwD0= 0 and K*B, = 0, we can find other forms of the dispersion equation. By dot multiplying eqs.1.62 and

1.66 by k, and replacing e and h1 by eqs.l.61 and 1.65 respectively we get - k*~-[ (adj ii) -2 - n2 (2*ii*k)z1-l*k = 0 if, I (adj u)*E - n2(k*G*k)?l # 0 and, - - - A - k*i0[(adj E)*W - n2 (k*~*k)i]-'-k= 0 - if, I (adj ;) *; - n2(k*z0k)il # 0 - - - h - b) The matrix A =T(adi U)E - n2(k-v*$)?l is sinsular Then

Therefore, the method used above to find the fields directions can not be applied here. - Let ui be the ith eigenvector of (adj ;) *Z correspon- - ding to the ith eigenvalue (n*F*n) i or n *p*ni (no summation over i) since is a given matrix. Then, using eq.A.56, we

Moreover, since 1%1 = 0, then, using eq.A.22, we get - A*(adj i)= 0

If A is planar, we can write A-[(adj A)*v] = o for any vector v. Therefore, for the specific case where v = n if and substituting A by its expression, we get - [(adj G)9:-(niG *ni)?] *{adj[(adj F) *Z - (nioE*ni)f] *nil = 0 Hence, comparing eqs.1.72 and 1.73, we deduct - - ui = a adj[(adj u) *E - (ni*;*ni)?l0ni 1.74 where a is an arbitrary constant since eq.1.72 does not have a unique solution. On the other hand, substituting eq. 1.71 into eq. 1.52 and making use of eq.A.24[(c), (g), and (1)1, we get for - the ith eigenvalue (n; u n; )

Substituting eq.1.74 into eq.1.75, we find - Therefore, ni is perpendicular to the complex vector (uSui) since ni is real. Now we will find the direction of the field vector H,. To do so, we expend the first term of eq.1.30 according to eq.A.35(f). After dot premultiplying the result by ;-land substituting n by nit we get

which becomes after we dot premultiply by ui0 and make use

- On the other side, dot premultiplying eq.1.72 by [(adj c) *El -I, using eq.A.23(c) and A.24(g), taking its - transpose, and solving for uia;*(adj 51, we get

- I E'U 1 u i *;* (adj E) = ui (n,*;an, ) Substituting eq.1.79 into eq.1.78 and rearranging, we get

Again, two possibilities exist:

In other words, since H. and (u) are complex, H. * - - * is orthogonal to (ui *U ) * = ui *u = 1.1 *ui . In the same time, since K0B,= 0,

- and hence, He is orthogonal to (no;) * = Won since H, and (ni* ; ) are complex vectors, and n is real vector. Having - - Ho(*ni)= 0 and HOB(u*ui)= 0, therefore

Then using eq.A.35(b), the direction hi of H. corresponding to ni is given by - h = p-I* i (nixui) Hence,

To prove eq. 1.86 (a), we substitute eq. 1.84 into Maxwell's equation, di= -(Xi xhj / u , expend it according to eq.A. 4 (b), and use eq.- 1.76. To prove eq. 1.86(b), we dot premultiply eq.1.72 by 11 and substitute the obtained equation,

into eq.1.86(a). - ii) ui*p*~,# o In this case we must have, after eq.1.80,

On the other hand, the solution of the dispersion equation 1.41 is given by

where A, B, and C are given by eqs.1.42, 1.43, and 1.44 respectively. Since, using eqs.l.41, 1.42, 1.44, and 1.89, 16 then, according to eqs. 1.90 and 1.91, the two values of n are equal and hence the direction of propagation is along the optic axis. In this case, the wave field vectors are only governed by Maxwellls equations.

1.7 Isonormal Waves [S] According to the dispersion equation 1.40, two different values of the reflective indices,n, and n,, and two corresponding different phase velocities exist for any given direction of the wave normal k not parallel to an optic axis. Therefore, two different plane waves and hence two different polarizations exist. Having the same wave . . normal, k, but two different phase velocities, the waves are called lfisonormal.llLet EL1) and E,(~), and the same for the other field vectors, be the electric field vectors corresponding to n, and n, respectively. Rewriting eqs.1.27 and 1.29 for EJ1' and D,"'* and substituting K/K, by n, we get - - (M-N)-ELI) = -T1 E,(" 1.92 nl (M.N)+.Db2J* = -T1 ~(2)* 1.93 n2 Dot premultiplying eqs.1.92 and 1.93 by DJ2'* and E~"' respectively, taking the transpose of the second obtained equation, and subtructing,we get

Since n:#nz aside from the optic axis then E'?) *D.?'* = 0, and, in the same way we prove that

(1) Hi2, E0(1'.D(2)* = Do(l,.E.(2'* = H,'l?. B:2'* = B, . * = 0 From eq.1.95 and Maxwell's equations 1.5, we get

E~LD~)~' E.(~'LBY* B,il) I E;:

.'2',- 621 * 12' E~'~'-LD~/~' B, L k

!21 f H,u)L oU* ,and (1) A H,~)LB. D, Ik

H;~'LBJ~) Hj2\ID(2)* D,(2) ~j; where is equivalent to the statement 'Iperpendicular to."

Using eq. 1.96, we notice that DL2' is perpendicular to k and

Eil! ;and B:"* is perpendicular to k and E:" also. Therefore, since k and E:" are not parallel, the vectors B!"* and Di2' are parallel. In the same way, since D:" and B,'~'* are both pependicular to the vectors k and E" ,then D:" and B.'2'* are parallel (see Fig. 1.1) . ,f ,f D b2'

.r ,DL1' B b2' *

Figure 1.1 Directions of the field vectors of the two isonormal waves.

Note that DJ2' and B!"* are perpendicular to the plane formed by the three coplanar vectors k, Hi2'*, and E:~' ; and D, l! and

B:~'* are pependicular to the plane formed by the coplanar vectors k, H:"*, and E o'2). 18

1.8 Group and Enersv Velocities

W We have already defined V = k as the phase velocity P x with which the monochromatic plane wave propagates along the direction of the wave normal k. However, a monochromatic wave has never been achieved in practice, although very narrow bandwidthes have been obtained using the advanced technology of laser. Therefore, we will be more realistic if we speak of a wave group or a wave pocket which can be viewed as a linear superposition of many monochromatic plane waves, each with definite frequency w and wave vector K. The wave pocket propagates with the group velocity [3,4,12,15]

Moreover, the energy does not travel in the same direction of k, and it has different velocity value, as it will be shown. We introduce the velocity of energy of transport (or energy velocity, or ray velocity) as

where <9>is the time average of the real Poyntingls vector given by

and is the time average of the energy density given by

In addition, in a lossless, frequency-dispersive medium

[w= w(K) 1, it can be shown that the group velocity is equal to the velocity of energy of transport [4,15]. Now we will find the velocity of energy of transport and the group velocity and we will show that they are equal by comparision. a) Enerqv Velocity We need first to find and < g>.substitution of eq. 1.60 into eq. 1.100 after making use of of the constitutive equation 1.6 yields:

If we expand the adjoint according to eq.A.36 (c) and make use of the dispersion equations 1.69 and 1.54, eq.l.101 becomes

Introducing the identity

to eq.1.102 and again making use'of the dispersive equation 1.69, we get

The Poynting vector can be determined if the term E~XH,is determined. Using Maxwell's equations 1.5 and the constitutive relations 1.6, we can write 20 which becomes after using the identities A.34[ (c) and (b)] and A.32 (a):

which we expand according to eq.A.35 and get

Now, dot premultiplying eq. 1.60 by i, developing the adjoint according to eq.A.36(c), and rearranging, we get

w - % -(x*iwE.){(K-i*~) [K*~*K - K: (E0adj- i)t]i *K w - K: (K-u-K)1; 1;-K K: 1;. (adj 2) *; OK) P'E. = + + li lK?(adj c)*E - (K*~*K)?/

Again, dot multiplying eq. 1.105 by E?, and substituting E? by the conjugate of eq. 1.60, and using the dispersion equations 1.54 and 1.69, we get - - IK-P*E.l2 {K: (K*;-K)(E-adj i), - (~011*K)~ + K: ~iiI KO;* (adja)*c* [K: (adjii) ge-(~gii-~)~l-l-~) E,*i*Et= 106 IK' (adj t)*; - (K*~*K)I/

But, using eqs.A.24 (g and h) and eq.A. 17, we establish the following relations: -w - - - (ad] e)-u = {adj[(adj u)*~l)/l;l - 1 - - - - - = --{[(ad] u)*;l2 - [(adj u)*~],(adj ;)* E 11-1 1 - + li I[ (ad1 :) -;lti} and, using the dispersion equations 1.54 and 1.69, we get

~*i*[ (adje) *i] [K: (adj ;) *E - (X-;*K)?]-~ *K - - - - = K*;*- (adj i) *>-[K: (adj i) *T - (K*;*K)~]-'OK - [(adj 2)*Glt - K*E*KK - [ (ad] ;) gilt Therefore, since we can write

eq.1.106 becomes

Now, substituting eqs. 1.105 and 1.107 into eq. 1.104 and simplifying, we get - - - - we. ~K*;*E. 1 2{(~*u*~);.~+ K:;- (adj E) *U*K

Finally, substituting eq. 1.108 and its conjugate and eq.1.103 into eq.1.98, making use of eqs.A.l3[(d) and (e)], and simplifying, we obtain the expression of the velocity of energy transport:

We can prove, using eqs.l.5, 1.16, 1.17, 1.98, and 1.99, that the relation

is still valid in the general anisotropic media [4]. Therefore, the phase velocity is the projection of the energy velocity in the direction of the wave normal. Grow Velocity The group velocity given in eq.1.97 may also be given

where F = F(x,w) is the wave surface given by the dispersion equation 1.40 which can be rewritten as w ' - F(x,m) = * (x)- F(~*c OX)[(adj z)gilt 2 2 - - - - *U + %X*VC (adj E) OX + c2 -P 1 = 0 1.112

Using eq.A.44(h), we get - - - - -8F - ax - (X0jl*K)(E+?) + (x-E*K)(i?+E) + Kt<*{ (adj e) *G - [(adj E) *El I) + K: {~l*adj - [i~adj ;lti}*x 1.113 and

- - 3 -aF - -2~ X0F*((adj- S)*O [(adj E)-PltP}*x+ qw - - am c2 - ~IE-PI which, using the dispersion equation 1.40, becomes

Finally, replacing eqs. 1.113 and 1.114 into eq. 1.111, we obtain the expression of the group velocity

By comparing eqs.1.123 and 1.117, we confirm that

VE = Vg 1.116 In other words, in the general anisotropic and lossless media, the energy velocity and the group velocity are always equal. 23

1.9 Summary The main purpose of this chapter was to establish formulas that describe the behavior of the wave propagation in unbounded general-anisotropic, homogeneous, lossless, and source free media. The permeability and permittivity were considered hermitian matrices, the most general case in such media. Although the wave number had to be real, the media imposed no restriction on the type of the propagating plane waves. However, we only considered uniform plane waves where the unit wave vectors were real. We began our study by deriving the homogeneous equations in tensorial forms. The solution of any of these equations, using the tensor algebra, led to the establish- ment of the dispersion equation in a quadratic form. We noted from the dispersion equation that two wave number values exist in any direction of propagation away from the optic axis, and they depend on this direction. Then we determined the directions of the field vectors considering all possible situations that might occur. Along the optic axis, and similar to the isotropic media, these directions are only controlled by Maxwell's equations. After that, we examined the isonormal waves, which have one direction of propagation but different wave number values, and we showed the various relations that relate between their field vectors. At the end, we derived the group and energy transport velocities showing that they are equal. CHAPTER TWO

REFLECTION FROM A GENERAL ANISOTROPIC MEDIA

2.1 Introduction In the previous chapter, we derived the equations that characterize the wave propagation in unbounded media. In this chapter, we will investigate the effect of a boundary on the wave propagation. We consider two homogeneous and source free media ,

(I) and (11), separated by a plane boundary ( ) , also called interface (see Fig. 2.1) . We consider a uniform plane wave that is incident on the separating boundary from the isotropic region (I) to the general anisotropic region (11). Reflected and transmitted waves are generated in regions (I) and (11). In studying the reflection and transmission of electromagnetic waves, we need to distinguish between the dispersion analysis and the amplitude analysis. Phase matching is used in the dispersion analysis to study the types of waves excited by the incident waves. In amplitude analysis, we calculate the amplitudes of the reflected and transmitted waves in terms of the amplitude of the incident wave at a plane boundary. 2.2 Phase Matchinq 2.2.1 Law of Reflection and ~ransmission[6,14] In this section, we will briefly review the basic theorems and phenomena that we will need in our study. The space-time dependencies of the generated waves at the interface are given by:

i(Ki'r wt) the incident wave Ei = E,i e - the 'reflected waves E:n) = Ear(n) (~2""r - ut) the tansmitted waves where n = lt2,3,. . '. and p = 1,2, 3, , . . : Ki, K:~), and K~P)are the incident, the nth reflected, and the pth transmitted wave vectors; and r is the position vector. ~ccordingto the boundary conditions, the tangential components of the field intensities E and H must be continuous, and the normal components of the field flux densities B and D must be continuous too. In frequency domain, the boundary conditions are given by:

where the unit vector normal the plane interface (1)directed from the first media (I) to the second media

Using eqs.2.2, we get the general laws of reflection and refraction

= a where

is a constant vector perpendicular to the plane of incidence formed by the two non-parallel vectors q and Ki. I k;" KP'

region (11)

interface ( 1) region (I) K:" K;~'

K (3' 1.r Figure 2.1 Reflected and transmitted wave vectors. e;, 9Lm' I L and 8:) are the incident, mth reflected and nth transmitted angles. From eq.2.3, we derive the following important phenomena: a) All the incident, reflected, and transmitted wave vectors, Kil K(,"), and xin) lie in one plane called "the Plane of In~idence.~~

b) Ki(ei) sin(ei) = sin(e$!ml) 2.5 C) Ki( ei) sin( ei) = K?)( 0:"') sin ( 0:")) (snellls law. ) where m = 1,2,..., and n = 1,2, ... 2.2.2 Booker Quartic [4,6] As we notice, the wave vectors can not be determined from the dispersion equation alone. In this section, we will find the equations that determine the directions and magnitudes of the wave vectors for a given incident wave Xi. The incident wave travels from the isotropic medium (I) toward the anisotropic medium (11). Using eq.2.3, we can see that all the wave vectors, (n) xi , Kn, and Kt , obey the relation

Ka = b + qaq

where b = *a

and 4, = xa'q

Eqs. 2.6 to 2.8 show that b is constant for a given incident wave and all tips of the incident, reflected, and transmitted waves lie on one straight line parallel to q, the unit vector normal to the interface. To determine any wave vector K we only need to find a Geometrically, the tip of K are the intersections of 4, . a the straight line that is parallel to q and touches the tip of b with the wave vector surface given by the dispersion equation in the corresponding medium. substituting into the dispersion equation expending, and rearranging according the power the variable q,, we get Booker quartic equation under the form where

- - - Ff= (b0;*b)(b*Z0b) + K:bg;* [(adj E) *; - (u-adjElti] *b - - + K,41 U* &I

Note that Booker Quartic equation reduces to [qi - (~~*q)] in the isotropic medium (I). It has two solutions:

q, = K:q for the incident wave 1

h and q,=-K:q for the reflected wave. 1 For a given incident wave, only one reflected wave exists. Next, we will derive the conditions for which Booker Quartic, given in equation 2.9, can be reduced to a biquadratic form which is easy to solve: a) Normal------Incidence

h In this case, Ka = qaq since b = 0. Then eq.2.9 reduces to - - - 2" - (q*E*q)(q=C*q)q; K.q0u*[(adj E) (;*adj E)tI~*;I + - - - $ + K~IE-uJ = o 2.11

In general, eq.2.11 has two different solutions q, and q2 in the general anisotropic media. Therefore, two transmitted

A wave vectors exist: K, = q,q and K2 = qZq. These transmitted 29 waves are isonomal since they have the same direction q but two different magnitudes ql and q,. b) B.! ------= D' =O Since rr is definite (u-rr*u*#~for any vector u), then we can see from eq.2.10 that B'=D1=O is satisfied under the following conditions:

h We find from eq.2.12 that all the vectors ii-q, E*q, i2q, -- - - % .E .; .G, .z 2 Goq, and obviously q, must be coplanar since they are perpendicular to the same vector b. Therefore, the cross product of any non-parallel two of these vectors must be parallel to the real vector b. Taking the cross product of ()and ()and using eq.A.35(a), we get

TWO cases arise:

In other words, Goq and q are parallel. Then we can write

where h is a constant scalar. Therefore, q is an eigenvector of corresponding to the eigenvalue A. Using eq.2.14, we can also obtain the following: Note that A *= A since the eigenvalues of a hermitian matrix are real. Now, using eqs.2.15 and A.35(a), we get - - (~oE-;-e)x(;*~2-;*~)= h2 [adj (Em;) 1 [&(E*q) 1 2.16 Again, we can see that two possibilities exist:

h (i) ex(E.4) = 0, then the vectors q and ;*q are parallel, and we can write: - €94 = m4 where m is a real constant eigenvalue for the matrix .

A Therefore, q is an eigenvector of Z corresponding to m. In this case, we can also prove that and Therefore,- - all the vectors and !J*E'*LI*~, and their complex conjugates are parallel to q and therefore, they are perpendicular to the interface.

( ii) In this case,

h -* must be parallel to b, after eq.2.16. Moreover, qx(r0q) is parallel to b too. Therefore, the vectors &x(E and - *q) A -A [adj (E* ;) 1 [qx(~*q)] are parallel, and we can write:

[adj(go;) ] [qx(zeq)I = a[b(~*q)I

- - Therefore, b must be an eigenvector of adj(~*p). A Then the- vector b(p*q) is parallel to b, and after eq. 2.13, (adj ii ) [&(c *i)] is parallel to b too. In this case, we can write:

., (adj G) *b = ab ." Therefore, b is an eigenvector of adj Since I l#O, we can . - easily prove from eq.2.21 that b is also an eigenvector of ; corresponding to the eigenvalue B1=IF(/B . Then we have - b*v = B'b 2.22 using eqs.2.22 and 2.12, we find that - --A -A Then the vectors E u *q and F2 *v *q are perpendicular to b, and we can write - - -. A --A (E*P)x(E*~*~)= (adj E) *[&(;*G)

h - A ZA At the end, since the vectors [ (E - q) x (; ioq)] and [qx( p*q) ] are parallel to b, eq.2.24 may be written as - (adj E)*b = Yb - Therefore, b is an eigenvector of adj (or adj ) corresponding to the eigenvalue y (or y *) . In summary, Booker Quartic equation reduces to a biquadratic form if: is an eigenvector of 11 and E , A - or q is an eigenvector of p, and b is an eigenvector of

-. - or b is an eigenvector of adj and adj E . or, in a simpler form, since 1: l#O and 1; l#O: q is an eigenvector of and E , or 4 is an eigenvector of , and b is an eigenvector of or. b is an eigenvector of E and i. otherwise, the Booker ~uarticequation (eq.2.9) yields four roots. We choose those two such that soq>O since the ray vector s must be in side for an existing transmitted wave. Once we find q,, we can find the corresponding Ka using eq.2.6, and therefore the directions of the field vectors.

2.3 Transmission and Reflection coefficients [4,6] In this section, we will determine the reflection and transmission coefficients at the interface of isotrpic- general-anisotropic medium. As before, we consider a uniform plane wave propagating from the isotropic medium toward the general-anisotropic medium.

Let u, and E, be the relative permeability and permittivity that characterize the isotropic medium 1. In this medium, all the field vectors are perpendicular to their corresponding wave vectors. Therefore, the incident

and reflected field vectors Eoi,Hoi, EOr,and ?Ior can be decomposed into two components: one is perpendicular to the plane of incidence and the other is parallel to the plane of incidence. Then, using Maxwell's equations, as given in

eq.1.5, we can write (see Chen, [6]): for the incident wave,

and for the reflected wave,

and

In medium 2, however, the transmitted field vectors are not orthogonal to their corresponding wave vectors since - E and are general hermitian matrices. The field directions of the transmitted wave are given by eqs.1.61-1.64 or eqs .l.- 84 -1.8 7, depending on the singularity of the matrix [ (adj i) *; - 11 (k*; *k) i] . The transmitted wave vectors are found from Booker Quartic given in eq.2.9: we choose only those two, K+ and K-, which satisfy s*q > 0. Let e+ and e- be the directions of the electric fields, E+ and E-, corresponding respectively to the wave vectors K+ and K- . Therefore, with the help of Maxwell's equations, we can write: Eo+ - C+e+

Ha+ = C+h+ where h =- ;-la(~+xe+) + W~o and

where medium 2 interface medium 1

Figure 2.2 Orientations of the incident, reflected, and transmitted wave vectors in media 1 and 2.

Now, we will proceed to determine the relations that connect the reflected and transmitted wave fields components BI, , C+,and C-, with those of the incident field vector

AL and A,,. Using the boundary conditions, as given in eqs .2.2 [ (a) and (b)] where E==Ei+Err EII=E++E-~ H1=Hi+Hrr and HI1=H++H-, and eqs.2.26 to 2.29, we get [4]

- [xi(bee-) + uueVlqi(a*h-)]C- B,, 2.33 21Ziqia = To have more compact and symmetrical form for these equations, we introduce the vectors N+, N -, F+ , and F- defined by N* (qii bb)*e, uu.ulqi(bht) 2.34 = + - - - h = (q.i + bb)*e, + wuaulqi(qxht) 2.35 F* 1 - - Dot multiplying eqs.2.34 and 2.35 simultaneously by a and b, we get

a'Nt- = qilqi(aOet)- + uu.~,(b*h,)l- "OF,- = qi[qi(a0e*) - uuall,(b*h 11 b*N+ Kr(boet) uu.ulqi(a*ht) - = - - - b*Ft- = Kf (beet)- + wuaplqi(a*ht)- Substituting eqs.2.36 into eqs.2.30 and 2.32, arranging into a tensor form, and solving for C+ and C-, we get

where Tij are the transmission coefficients given by

and a = (a*N+)(b9N-) - (a*~-)(b*~,) 2.39

Similarly, substituting eqs.2.36 into eqs.2.31 and 2.33, and using eq.2.37, we get

are the reflection coefficients given by where 'ij and A is again given by eq. 2.39. Since eqs. 2.36 in general are complex, the reflected and transmitted coefficients are complex too. Therefore, the reflected wave can have any polarization even if the incident wave is linearely polarized. In the special case where the incident wave is normal to the interface, eqs. 2.30 to 2.41 are not valid since ax0 and hence, the plane of incidence does not exist. We will analyze this problem in the next paragraph.

2.4 Reflected and Transmitted Waves for Normal Incidence In this case, all the wave normals coincide. Then

The transmitted wave numbers K+ and X- can be easily determined from the biquadratic form of Booker Quartic given in eq.2.11. These wave numbers are equal to their components along the vector 6, but they are not equal to each other. Therefore, the tansmitted waves are isonormal. Their wave vectors become

Using the boundary conditions as given in eq.2.2, we can write [6] :

h qx(EOi + Ear - Eo+ - E,-) = 0

A and qx(Hoi + Her - He+ - H,-) = 0 or Hoi + Her - Ho+ - Ha- = aq 2.45 where a is an arbitrary constant. Now, substituting eq.2.42 into eq. 1.5 (c), we get

A A h h qoBoi = qwBor IqgBO+ = qgB0- = 0 2.46

h h and q0Hei = q*Hor = 0 2.47 Dot multiplying eq. 2.45 by , using the constitutive relations (eq.1.6), and solving for a, we obtain

a=--1 e.11-1. 1-I. (B.+ + B.-1 Now, let

where b+ and b - are the magnetic density directions found in 81.6. Then eq.2.48 becomes

substituting eq. 2.50 into eq.2.45 and using eqs.2.49 and 1.6, we get

Bei + Bar + lJl[(q* pl*b+)q -C-l* b-IC,

+ p,[ (q*;-'*b-)q -;-'*b-]C- = 0 2.51 Using eqs.2.43 and 2.49 and Maxwell's equation 1.5, eq.2.44 becomes

BOi BOr -Kib~7 ++-7?-Kib-C - =O 2.52 - - Adding and Subtracting eqs.2.51 and 2.52, we obtain Since the transmitted waves are isonormal, then, from eq.1.95, we have

If we dot multiply eq. 2.53 by b: and b? simultaneously and make use of eqs.2.46 and 2.55, we get

Then the transmitted wave coefficients can be found by putting eqs.2.56 and 2.57 under a tensor form and solving for C+ and C-:

where

The reflected magnetic flux vector B. can be easily r determined by substituting C+ and C - as given in eq.2.58 into eq.2.54: The other reflected field vectors Hart E. r, and Dar can be determined by substituting eq.2.60 into Maxwell's equations and using the constitutive relations.

2.5 Summary In this chapter, we studied the effect of a semi- bounded media on propagating wave and field vectors. We examined the reflected and transmitted waves generated by a given wave traveling from an isotropic media toward a general-anisotropic media where both of the permeability and permittivity were hermitian matrices. ~irst, we found a general expression for Booker quartic equation from which we can determine the components of the reflected and trans-

. mitted wave vectors along the normal to the interface. The directions and magnitudes of these wave vectors can be then easily determined. After that, we examined the conditions which reduce Booker quartic to a biquadratic equation that is easy to solve. Second, we determined the the reflection and transmission coefficients which are the ratios of the amplitudes of the corresponding field vectors over those of the incident wave. We studied both cases of oblique and 40 normal incidence. These coefficients, however required the use of the directions of the field vectors of the transmitted waves found in chapter one. CHAPTER THREE WAVE PROPAGATION IN BIUNIAXIAL MEDIA

3.1 Introduction: Optical Classification of Crystals In the last two chapters, we derived many useful equations that characterize the uniform plane wave propagation of monochromatic light in general-anisotropic, lossless, homogeneous, source free, unbounded and semi-bounded media. - The permittivity and permeability tensors, € and c, were taken to be hermitians. Since a hermitian matrix can always be transformed to a diagonal form, crystals are described, in general, by the symmetrical matrices ;and ;. By a suitable choice of axis, these matrices can be reduced to a diagonal form. In general, the chosen normal axis for ; and u do not have to coincide. However, to simplify, we will assume that they do coincide. In this coordinate system, called the principal system, we have

In orthorhombic, monoclinic, and triclinic crystals,

all crystallographic axis are unequal: E~#E~#E~and/or ~d Eyfcz, and the medium is optically biaxial. In tetragonal, hexagonal, and trigonal crystals, two crystallographically equivalent directions may be chosen in one plane perpendicular to the third, which is taken to be the axis of symmetry

of the crystal. In this case, where we take E~=E~=EI$~= E,, and ux=uyuifu ,=u,, , the medium is said to be optically biuniaxial. If either E, = E,, or p, = p,,, the medium becomes

uniaxial. If both E, = E~,and pL = u,,, the medium is isotropic. In this chapter, we will assume that the medium is biuniaxial. Therefore,

where E~,E/ , ul , and w, are considered to be real.

3.2 Wave Vector Surfaces [6] In studying the wave propagation in biuniaxial media and its properties, it is very convenient to decompose the matrices into a dyadic form of multipication of vectors. - Under this form, E and can be written as

where is a unit eigenvector of ; and corresponding to

the nonrepeated eigenvalues E,, and p,,, respectively. As will be shown later, 6 is parallel to the optic axis of the

media. E, and u, are the repeated eigenvalues of ; and , respectively. 43 Using eq.A.39, we calculate the determinants and adjoints of and and get - (a) IEI = E?EN

(b I ill = pip,, 3.4 Ah (C) adj E = [E//?+(E~-E,,)CC]

(dl adj ii = P, [~,,i+(u,-u,,) Gc] The wave vector surfaces given by the dispersion equation 1.40 can be written for symmetrical tensors and as

Using eqs.3.3 and 3.4, we find

- - Ah (ad] E *P = E, [E,,U~I+ (E~P,, -E/ P~)CC~ - - and [ (adj €1 *uit = (2€,,~, +E,U/)

Then we derive the useful expression - - - - - uW{[(adj ~)*~]~i-(adj) = U~U~IE~E+ \E,,IILU 3.7 substituting eqs.3.4[(a) and (a)], 3.6, and 3.7 into eq.3.5, and factorizing, we get

Setting the two terms of eq.3.8 to zero, we obtain two wave vector surfaces:

and

TWO values of the wave number K are obtained from eqs.3.9 and 3.10. Substituting K by I& into eqs.3.9 and 3.10 and solving for K, we obtain and

Since both of the wave numbers K- and K+ depend on their

h respective unit wave vectors k- and k+, the corresponding waves are both extraordinary. To illustrate the form of the

wave vectors ' surfaces, we substitute eqs. 3.3 [ (a) and (b)]

respectively into eqs.3.9 and 3.10 and get after some manipulation

(K-4)= " 2 surface (1-) + (K-xc) =1 3.13 K?EL i-lL Kf ~,/i-l~

surface (1,)

Therefore, the wave vectors' surfaces, ( ) and ( +) ,

given by eqs.3.13 and 3.14, are two ellipsoids of revolution around the axis i, and represent extraordinary waves. We also notice that the principal vectors of the two ellipsoids along axis are equal. In other words, the two wave vectors' surfaces are tangent to each other at their intersections with the optic axis. Then

Along the optic axis, the wave vectors X- and K+

must be equal. In this case, making K-=+=Kt into eqs.3.11

and 3.12, we get which becomes after using eq.3.3

We notice from eq.3.15 that two possibilities exist:

1) (&L/E// f (k/U//) In this case, aq.3.15 is valid only if ktx^c=O. Then we have kt= k-=A Ak+=c. Therefore, c is an optic axis of the medium and the only one that exists. In illustrating graphically eqs .3.13 and 3.14, four situations are possible: E,,, >(or <) cI and i-111 >(or <) UL . A crystal is said to be electrically (or magnetically) positive if ex > E, (or,, ) , negative otherwise. For convenience, we introduce the notation ()to indicate a (electrically positive or negative, magnetically positive or negative) biuniaxial medium. Now, we will illustrate graphically all possible cases: a) (+,+) medium: E,,, > E, and v,, > u, (see Fig.3.1)

~igure3.1 Wave vector surfaces, x- and + in a (+,+) medium. b) (-medium: E,, < E, and u,/ >ul, then &,!jl, E,v// (see Fig. 3.2) c) + , - medium: &,,, > EL and u,,

Figure 3.2 Figure 3.3 Wave vector surface in a Wave vector surface in a (-,+) medium (+,-) medium d) (-, -) medium: E,, < and u,,

Figure 3.4 Wave vector surfaces, Z - and I+,in a (- ,-) medium.

In this special case, the two wave surfaces 1- and I+overlap (see Fiq. 3.5) . The resulting surface is still an A ellipsoid of revolution around the axis c. According to eqs.3.11 and 3.12, only one wave number exists but depends on its wave vector. Therefore, like an isotropic medium, only one wave may propagate in any direction; however, unlike an isotropic medium, the propagating wave is always extraordinary. The wave vector surface, in this case, is given by

It is worth noting that this case only occurs in either

(+,+) or (-, -) crystals.

Figure 3.5 Wave vector surface in a (-,-) medium, and r. *C where EL PI, = EN VL.

3.3 ~irectionsof the Field Vectors [6] For the real symmetrical matrices E and 11, eq.1.59 can be written as

[~f(adj c) *E - (K*i*K)I]*E,= -(K*~*E,)K

Using eqs.3.3 and 3.4, we have

whose determinant, using eq.A.39(b), is given by where K- and K+ are given by eqs.3.11 and 3.12 respectively. In both cases, the field vectors corresponding to K- and K+ were determined in general form in 31.6. a) Field Vectors Correspondins to K, - Here, we have 1 K (ad ) E - (KO; K) 11 $0, and eqs. 1.61 to 1.64 are valid. Using eq. 1.61, the electric field direction is given by

substituting eq. 3.18 into eq. 3.20, expending the obtained ad joint according to eq. A. 3 9 (c), and dropping the scalar factor, we get

Simpler forms of eq.3.21 can be obtained using an alternative method.

ternative Method This method requires heavy calculations, but it leads to different forms of the field directions. Their components, however, are still proportional. For simplicity, we rewrite eq.1.58 as

- - - where A = ~:lPI E + (U*K)(K-U) - (K=U*K)U Substituting eqs .3.3 and 3.4 (b) into eq. 3.23 and rearranging, we get

Expending adjz according to eq.A.SO(c), we obtain

Eq.3.22 has a nontrivial solution only if lxl=0. Therefore, ii*adj%= 1 X 1 i=a. On the other hand, as seen from eq. 3.25, adjlifa for the wave corresponding to K,. Then we can write for any vector u

comparing eqs.3.22 and 3.26, we get

In this case where K=K, and

since u is an arbitrary vector if we make: i) u=K, and substitute eq.3.25 into eq.3.27, we get

and the electric field direction is given by ii) u=6 and proceed the same way, we get

e, = -(K,*&)K, + K:E~P~C 3.30

Now, using eqs. 3.28 and 3.3, we can easily show the simila- rity of eqs.3.21, 3.29, and 3.30 by proving their components along $ and K, are proportional:

To find the other field directions, we substitute eq. 3.30 into eqs.1.62 to 1.64 and get

We notice from eqs.3.31 and 3.32 that the field vectors b, and h, are parallel to each other and they are perpendicular

A to the plane formed by the coplanar vectors K,, c, e,, and d, (see Fig.3.6).

Directions of the field vectors of the extraordinary wave corresponding to K,. b) Field Vectors Corres~ondinqto K+ FO; the extraordinary wave corresponding to Kc, we have, after eq.3.10, and IK~(adj FI) *E - (K+*~*K+)?I=0 3.34 according to eq.3.19. substituting eq.3.32 into eq.3.18, we

get - - - - Ah Ki(adj ;) *E - (K+*p0K+)T= KiLl (tuL-iIJ,,)cc 3.35 - and adj [K: (adj. ;) *E - (K+*;*K+)I]= 0

According to 81.6(b), we need to determine the eigenvector U+ of K: (adj i); corresponding to the eigenvalue (K+*; K+) . Using the solution .of eq.A.60 and eq.3.35, we find that the

A eigenvector u+ must be perpendicular to the optic axis c. Moreover, substituting eq.3.3(b) into eq.1.76, and using the identity u+*c=O,A we get

u+-(U*K+) = u+*K+ = 0 Therefore, since u+ is perpendicular to both c and K+, we can write u+ = K+XC 3.36

The directions of the field vectors, given by eqs.1.84 through 1.87, become

and

Here again, we notice that the field vectors e+ and d+ are parallel to each other and perpendicular to the plane formed

A by the coplanar vectors K+, c, b+, and h+ (see Fig.3.7).

Figure 3.7 Directions of the field vectors of the extraordinary wave corresponding to K+. C

ernative Method 1 The electric field direction can be obtained in a more direct method. Substituting eq. 3.33 into eq. 3.25, we

get A h adjx = u: u,, K?( E,, uL -cl ii,, (K+XC)(K+XC) 3.38 According to eq.3.26 and 3.27, E.=(adji) *u where u is any vector such that (K+xc)*u#O. If we choose u=K+x& and dot multiply with eq.3.38, we get the electric field direction

A as e+ = K+xc which is the same as that given in eq.3.37.

Alternative Method 2 Note that all the equations derived in 03.2 can be classified into two categories: the first group of formulas by the electrical anisotropy (e.g., K, and its related equations), and the second affected by the magnetic anisotropy (e.g., K+ and its related equations). These two groups - can be obtained from each other by interchanging 2 and u , 53 and K, and K+. Moreover, comparing eqs.1.27 and 1.32, where the first was used to determine the directions of the field vectors, we note that they can be derived from each other by interchanging 2 and , and E. and H.. Therefore, we can obtain the directions of the field vectors corresponding to K+ once we have those corresponding to K- and vice-versa, simply by applying the following duality equation:

K+ -K- 3.39 e+, h+, b+, and d+ -h- , -e-, -d- , and b, respectively Note that in the case where ELU,/=E//IIL, the directions of the field vectors given by eqs.3.29 to 3.31, or eqs.3.37 are not valid. They are only controlled by Maxwell's equations, but the propagating wave is extraordinary where the wave number K depends on the direction of propagation.

3.4 Enersv Densities, Poyntinff Vectors, Ray Vector Surfaces and Velocities in Biuniaxial Media [6] Regardless of their magnitudes, the field vectors will be used according to their directions given in eqs. 3.29, 3.31, and 3.37. According to eq.1.16, the time averaged energy densities are given by

and ~ccordingto eq.1.99, the time averaged ~oyntingvectors are given by 1 WE .E~(x-x6) (E-K-) 3.42 < g-> = I e-xh- = 2 and

According to eq.3.40 through 3.43, the ray vectors are given by

and

When the matrices and are real, the ray vector surfaces can be obtained from the wave vector surfaces by applying the duality principle [6] given by

~pplyingeq. 3.46 to eqs.3.13 and 3.14, we get the ray vector surfaces given by

and (s+*C)~+- (s+xc)~_ -1 KS E, lJL K: EL u,,

A which again are ellipsoids of revolution around the axis c. Using eq.1.17, the phase velocities are given by The energy transport velocities are given by

and

The group velocities are derived using eqs. 1,111, 3.28,

3.33, and A.44(h) :

and

Therefore, the energy transport velocities V,, and Ve+ are equal to the group velocities Vg, and Vg+, respectively. However, they are different from their corresponding phase velocities Vp- and Vp+ In other words, the energies are not traveling in the same directions of their corresponding plane waves. If we dot multiply eqs. 3.49 and 3.53 with eqs 3.50 and 3.54 respectively, we get

As a result, the phase velocities are the projections of the group velocities on the directions of the wave vectors. All these vectors are illustrated in Fig.3.8, below. Figure 3.8 Wave vectors, wave vector surfaces, ray vectors, field vectors, Poynting vectors, and velocities in a (+, -) medium: E,, >E, and LI,,

3.5 Reflected and Transmitted Wave Vectors at Isotro~ic- Biuniaxial Interface [6] We can find the reflected and transmitted wave vectors, K,=b+q, q where q, is given in Booker Quartic equation 2 .9, by following the same procedures of 82.2.2. However, since we were able to factorize the dispersion equation into two terms, it is much easier to determine q+ and q- of the transmitted waves directly from eqs. 3.9 and 3.10 . Using eq.2.6, we have

A K- = b + q-q 3.55 and K+ = b + q+q 3.56

Substituting eqs.3.55 and 3.56 into eqs.3.9 and 3.10 respectively, we get

A-A - A 2 (q0&*q)qi+ 2(b0&.q)q-+ [(b*;*b)-cL~,,lliKe]= 0

A and (e*i*q)q$+ 2(b*u0q)q+ + [(b-i*b)+~~~,,&,~:]= 0 whose solutions are given by

and where the valid values of q- and q+ must obey the conditions

,. s-*q>0 and S+*q>O because the ray vectors and the normal to the interface must always be in the same quarter of space. Therefore, using eqs.3.44 and 3.45, we must have:

and

Thus,

and

Therefore, the only valid valuer of q- and q+ are

and where A, and A+ are given by [6]

A - A A- = K:E~E,~u,(q*~*q) - a*(adj E) *a and

The transmitted wave vectors are then given by

In putting K- and K+ in such form, we used the relations [6]

A-A h -A - A b(q0 ~*q)- q(be~*q) = (E0q)xa

- A - A h and b(q*l.*q)- q(b0u-q)= (u0q)xa The reflected wave vectors can be found by substituting ; and u- in eq. 2.9 by f and u, f respectively, since the incident and reflected waves propagate in the isotropic medium. Then we get where plus sign and minus sign attribute to the incident and reflected wave, respectively. Therefore,

A Xi = Kl ki = b + qiq

A and Kr = K, kr = b - qiq where 9i = JK?E, u, - a2 Moreover, K$ = = K: = K~E,u,

The incident, reflected, and transmitted wave vectors are illustrated in Fig.3.9 as an example. From eqs. 3.61 and 3.62, we notice that a real transmitting wave does not always exist. There are different restrictions for different permeability and permittivity 59 - matrices, E and 11, that impose certain conditions on the existence of the wave vectors K, and/or K+. We will investigate these conditions in the next section.

Biuniaxial- medium Z , p

Isotropic medium ,111

Figure 3.9 Geometrical construction of the wave vectors at the interface of an isotropic-biuniaxial (+, +) medium where ~~p~

3.6 Conditions for the Transmitted Waves to Exist Real transmitting wave vectors X- and/or K+ exist only if A ->O and/or A+>O where A- and A+ are given by eqs.3.63 and 3.64. Let a, 3, and 9i be the angles formed by the axis =,

h the projection of c on the interface, and Ki with the normal to the interface 4, a, and ;I respectively (see ~ig.3.10) .

Figure 3.10 Vector orientations in the orthonormal system formed by the axis a, b, and q. substituting eqs.3.3(a), 3.4(c), 2.4, and the identities a*c = la1 sinacos a and = K:sin26i = K:E ~1 sin26i a2 = (Kixq) 11 into eq.3.62, we get

Since L, must be positive in order for a real K, to exist, then we have

From eq.3.73, we observe the following: a) If a medium is electrically negative (E,, ) and €1 i-11 c ~//p~,the transmitted wave K, always exists. b) If a medium is electrically positive (E, >E, ) and c, ul> c//uL , then a maximum angle of incidence always exists,

after which the wave vector X, does not exist. This angle is called the lvcriticalangleIn given by

sin 2 ec- = ELu/,[pL + (1.~/i-p- )cos2a1 3.75 E1ul[~+(p//-~L)(cos2u+sin2a sin 2 5) I c) In any other case, the existence of the transmitted wave K, depends on the direction of the optic axis. If the term of the right hand side of eq. 3.74 is greater than one, this transmitted wave exists for any angle of incidence. Otherwise, the critical angle is given by eq.3.75. The same procedures are followed to determine the conditions of existence of the transmitted wave K+. We must have sin2gi < EL&/ [!JL f (P,,-k )c0s2aI E (p,/ 11!J [k+ -pl ) (cos2a+sin2asin2B ) ] and the corresponding critical angle Bc+ is given by

2 sin ec+ = ELW [!JL + (v/,-u~ )c0s2~~ 3.76 E, u, [k+(!J,/ --11, ) (cos2a+sin2asin2B ) 1

Here, K+ always exists if the medium is magnetically negative (p,,

3.7 Reflection and Transmission Coefficients [4,6] The reflection and transmission coefficients determine the ratios of the amplitudes of the reflected and transmitted field vectors to those of the incident wave. We already derived these coefficients at an isotropic-general anisotropic interface in 62.3. We will use those results to obtain the reflection and transmission coefficients at an isotropic-biuniaxial interface as a special case. As an easy and handy reference, we will list below all the related field and wave vectors. The incident field vector, as given in eq.2.26:

h Eoi = A,a + A,, (kjxa) and

The reflected field vector as given in eq.2.27:

Eor = B,a + ~/(k,xa) and

The transmitted field vectors, as given by eqs.2.28, 2.29, 3.30, 3.32, and 3.37 away from the optic axis:

h h 2 E,, = C-e- where e, = -(K-*c)K- + KOE,Kc 3.79 h H,, = C-h, where h- = uEoE,(K-xc)

A E = C+e+ where e+ = K+xc

A 3.80 1 2 and H. + = C+h+ where h+ = r (K+'C) K+ - KO5jll Cl ulJ 0 'Jl

The incident and reflected wave vectors as given by eqs. 3.68 through 3.71. ~ccordingto eqs.3.11, 3.12, 3.55, 3.56, and 3.61 throgh 3.66, the transmitted wave vectors are:

and

Dot multiplying eqs.3.79 and 3.80 by a and b simultaneously, and using eqs.3.81, we get

A A A b*e- = (K:E~II~-~) (b*c) - q-a (q0c)

h A bob+ = UP .P,) [ (K~E~II,-a2 ) (bee) - q+a2(q0c) I

A b-h- = UE, ~,q-(age) Substituting eq.3.82 into eq.2.36, we get

A 3.83 A A aF+= -qi{ (U,/P, [ (a2-K:cL uL ) (bee)+q+a2 (q0c) ]+qi~+* (axc) )

where N+- and F+- are given by eqs.3.35 and 3.36. The same as in eqs.2.38 and 2.41, the transmission and reflection coefficients are, respectively, given by

and -. '11 = (l/A) [ (aaF+)(b0N-) - (a0F-)(b8N+) ] r12 = (xi/qiA) [ (a*F+)(a'N-1 - (a*F-1(a0N+) I - 3.85 r2, = - (qi/KiA) [ (b'F+) (b0N-1 - (b'F-1 (b'N+) I T~ = (l/A) [ (b0F+)(a*N-) - (b*F-)(a0N+) ] where A = (agN+)(b0N-) - (a9N,) (b*~+) 3.86

We note that, as in the uniaxial media 161, ii we reverse the orientation of the optic axis, only the transmission coefficients change signs: however, the reflection coefficients and the reflected and transmitted field vectors 64 do not change signs. We will show now that the reflection and transmission coefficients given in eqs. 3.85 and 3.84 reduce to those of uniaxial media when ;=~,f. In this case, eqs.3.62, 3.64, and 3.66 give

q: q: = K.~E,P, - a 2

h A and K+* (ax=) = q+ (be.) - a2(p*~) substituting eq.3.87 into eq.3.83, we get

. . a*N- = qi~i&L(P2qi+"ig-) (a-c)

A. b0N+ = K~P~(E~~~+E~~+)(a-C) bgN- = ~:[~$(b*C^)-~-a~(~*c)]+K~E~plqi[K-* (axc) ]

a*F+ = -(qi/u2) (~,qi-u,q+)[K+* (ax;) I

a*F- = qiK: E ( ~1 2qi-Plq-) (a-C)

A 2 b°F+ = -KOvl( ELqi- &lq+) (a*c) bWF- = K :[q;(b-o^) - q-a2(q0c) ] + K?cL ulqi[K-*(ax& 1

which are the same equations as those of the reflection and transmission coefficients at the isotropic-uniaxial interface [see Chen, 96.51. Now, it is interesting to note that the magnitudes of the transmission and reflection coefficients are not symmetric with respect to in the plane of incidence. In other words, if we rotate the incident wave vector Ki so that the angle of incidence changes from 0i to -0i (see Fig. 3.11) , the magnitudes of the reflected and transmitted 65 coefficients change. To prove this, we write eqs.3.61 and 3.62 as A AA q- - (EL ) (b*C) (q*~)+ , - S*Z*~

and q+ - (~I-UH (q*A c) + q 4-899

A When the angle of incidence changes sign, (b-c) changes sign, butlaccording to eqs. 3.63 and 3.64, A- and A+ do not change. Therefore, q- and q+ still exist but with different values. Thus, the magnitudes of the reflection and transmission coefficients given in eqs.3.84 and 3.85 change values. As a result, we must consider the sign of the angle of incidence in the presence of biuniaxial media.

Figure 3.11 Ki symmetrical rotation of the incident wave vector from one quadrant to another in the interface plane of incidence. wa

Along the optic axis, however, the directions of the field vectors are only controlled by Maxwell's equations. In this case, only one transmitted wave propagates along the optic axis. The transmitted wave vector is given by

Kt = K- = K+ = Kt; = K.- c 3.89 substituting eq.3.3 into eq.1.5 and simplifying, we get

Then the electric and magnetic fields are perpendicular to the optic axis 6, and hence to Kt. Therefore, the reflection and transmission coefficients have the same forms as those at an isotropic-isotropic interface. They are given by [4]

TZ; - -C// = 2nt (kiwq) - 2r\tc0~0 i A// (k)+ ktQiCOS ei + 'I~COS st and where A,, A,, , BLf B,, , C,, and C/, are the perpendicular and parallel components to the plane of incidence of the incident, reflected, and transmitted electric fields respectively. r\i anc r\t are respectively, the impedances of the incident and transmitted waves given by

and

3.8 Reflected and Transmitted Field Vectors for Normal

Incidence [6] This case needs a special study since the concept of the incident plane is not valid anymore. The incident, reflected, and transmitted wave vectors given in eqs.3.68,

3.69, and 3.81 become where q+ - and q, = KO Q'lJ'Q

Using eq.3.92, the directions of the field vectors given in eqs.3.79 and 3.80 become

and

Using eqs.3.93 and 3.94, we get

4 ,A b: b: = 3 (qxc)

and

NOW substituting eqs.3.95 and 3.96 into eqs.2.58 and 2.59, we get and the transmitted wave coefficients

After extensive mathematical manipulations, the reflected magnetic flux given by eq.2.60 becomes

Therefore, using Maxwell's equations, the constitutive relations, and eq. 3.96, we derive the reflected electric field

We notice from eqs. 3.99 and 3.100 that the reflected field vectors can be decomposed into two components: one

A A h hA parallel to (qxc) and the other to [qx(qxc)1. If we introduce the plane formed by the vectors q and c as a substitute to the plane of incidence, we can decompose the incident and reflected field vectors into components parallel and perpendicular to that plane as follows:

and A Ah Ear = Bl(Wc) + B,/(&(Pc) I

Now, making E. +=C+e+ and E,,=C,e,, substituting eqs. 3.101 and 3.102 into eqs.3.98 and 3.100, using eq.3.92, and arranging, we get

--c- - -2Ki A// K?EI (hKi + plK-) and = KiK+ - K?EIIJ~= - &LKi AL KiK+ + K~.E~u~clK+ + ~lKi B/, = uLKi - u,K- A, FI~K~+ u,K-

ternative Mew In this specific case of normal incidence, we can directly obtain eqs.3.103 and 3.104 by substituting E.+=C+e+ and Eo,=C,e,, where e+- are given in eqs.3.93 and 3.94, into the boundary conditions

and A Qx(8.i + Ear - H,+ - no,) = whose solutions are obtained by making their components along and (qxc)A A equal to zero: The solutions of these equations lead to the same results as those given in eqs.3.103 and 3.104. During our study above, we considered an oblique

h A optic axis c. However, a more special case might occur if c is perpendicular to the interface beside normal incidence. In this case, neither the concept of the plane of incidence nor that of the plane formed by and & is valid. Then eqs. 3.101 and 3.102 are not valid either. Again the field vector directions are only controlled by Maxwell's equations since the field vectors are perpendicular to their wave vector Kt. The transmitted wave behaves as that in an isotropic media. Therefore, since q=c=kt,AAA the transmitted and reflected field vectors are given by [6]

Note that E~ and p,, have no effect in this particular case.

Figure 3.12 - - Wave vectors for E ,U interface normal incidence El and normal c. 3.9 S~ecialCases We will consider in this section some special cases that are important in practical applications. 3.9.1 Optic Axis Parallel to the Plane of Incidence

h In this case, is perpendicular to a (c0a=O) but can have any direction parallel to the plane of incidence (see Fig.3.13). Then using eq.3.4, we can write:

a*(adj i)*a = PIp// aZ

am(adj E)*a = EL€// a2 and, therefore, eqs.3.63 and 3.64 become

In order to simplify the form of the reflection and transmission coefficients, we need to establish the following identities: K+* (ax;?) = [&(boo) -a2~//(Go=) I/ (6.6 *e)

*-A K-• (axe) = [&(be&) -a2€/./(;Iec) I/ (q*~*q) and, using eqs.3.108 to 3.110,

plane of incidence Figure 3.13 A Optic axis c of biuniaxial media is parallel to the plane of incidence. 72 Substituting eq. 3.111 into eq. 3.83 and having a-G=o, we get a*N+ = -qi[qi + (P1/ULU,y)61 [q9 (ax;) 1 b*N- = (1/~,,) (K~K+ KIEL~//P1qi) [K-'(~xc)] a*F+ = -qi[qi - (P1/PL1~//)&I I%*(axe) I b*F- = (1/~//)(K~K - K:E~CP,~~) [K-• (ax;)] a*N- = b*N+ = agF-= b0F+ = 0

Therefore, the transmission coefficients given by eq. 3.84 become -2qia2 T11= [qi + (P l/~Lill/ &I [4*(axb) I

T12=Tzl= 0 and the reflection coefficients given by eq.3.85 become

r,, = rzl= 0 In this case where the optic axis is parallel to the plane of incidence, two more specific cases may occur: the optic axis c^ is either parallel or perpendicular to the interface. a) The__--_ o~tic------axis is earallel ...... to the interface l-l-b~, NOW, we have 2 2 a*; = 0 and (b)= b = a

A A-A q*Z*q= €1 and q0u*q= i?, b.jioqA = 0 Substituting eq.3.115 into eqs.3.109, 3.61, 3.62, and 3.110, we get

Therefore, using eq. 3.116, the transmission and reflection coefficients given by eqs.3.113 and 3.114 become

Figure 3.14 Optic axis c is parallel to b. b) The-o~tjs-qxis-js-~$~~e!!~is~1qr~Sothe_jntergqse- (c=q) Two possible situations may arise: normal incidence and oblique incidence. Since we have already considered the case of normal incidence and normal optic axis in 93.8, we will only consider here the case of oblique incidence. Here, we have A = b*ch = b*&*e- = b*u*q- A = 0 - q*~*q= EN and q*ioq = U// axc = -b and K+*(axc)- = -a2 and, from eq.3.116, we get

Therefore, the transmission and reflection coefficients given in eqs.3.113 and 3.114 respectively become

Figure 3.15 optic axis is perpendicular to the interface.

3.9.2 O~ticAxis Parallel to the Interface (Fig.3.16) In this case, q*A 6-0, but 2 can have any orientation parallel to the interface. Now we have

* - A A - A E and qo&*q= 1 q0u=q= u,

q+ = -DL and q- = K/E~

Using eq.3.121, eq.3.83 becomes Figure 3.16 I I I Optic axis c is parallel to the interface.

The transmission and reflection coefficients are then obtained by substituting eq.3.122 into eqs.3.84 and 3.85. We note in this particular case that the magnitudes of the reflection and transmission coefficients are symmetrical with respect to q in the plane of incidence since, according to eq.3.121, q+ and q, do not change values if b change to -b. Therefore, we need to consider the angle of incidence as a variable only between zero and the critical angle. Again in this special case, two specific situations may occur: optic axis is either parallel or perpendicular to the plane of incidence. The former has been already studied in §3.9.l(a). We will consider now the latter case. optic Axis Perpendicular to the Plane of Incidence

h Here, c=a=a/lal and, hence, a0c=lal and b=c=O. Then

q+ = JK:E~~//- a2 and K$ = K?E~LI// and K? = K:?,~ Using eq.3.123, eq.3.122 becomes

substituting eq. 3.124 into eqs.3.84 and 3.85, the transmission and reflection coefficients become

h " Q ~igure3.17 ,b optic axis 2 is perpendicular to the plane of incidence. a

3.9.3 0~ticAxis is Perpendicular to the Intersection of

the Interface and the Plane of Incidence (b0&=0) Since the optic axis is parallel, but arbitrarily oriented, to the plane perpendicular to b, then we have the following identities: axe squar~?~~ao=,uoTq3aTjax pue uoysspusuexq aqq 'ATT~U?~ In this case also, the magnitudes of the reflection and transmission coefficients retain their values when the angle of incidence changes sign.

3.10 Rotation of the Plane of Polarization won ~eflection,

Brewsterns Anqle [6] The plane formed by the electric field E, and the wave vector K is called the Ivplane of p~larization.'~The angle a formed between the plane of incidence and the plane of polarization is called the "azimuthal anglevf(Fig.3.19).

~igure3.19 Plane of polarization and azimuthal angle of either the incident or reflected wave where K is perpendicular to a, (Kxa), and E,.

Since the incident and reflected waves propagate in an isotropic medium, then the wave vectors are perpendicular to the corresponding field vectors in this medium. There- fore, E., a, and (Kxa) are coplanar, and E, can be written as: the incident wave E,i = Ala + A,/ (kxa)

A the reflected wave EOr = B,a + B,, (kxa) The incident and reflected azimuthal angles are given by

and respectively. Using eq. 2.40, we estabilsh the following relation between ai and a=:

tan a, - rl2+rI1 tan ai r22+r21 tan ai

It is of special interest that under the conditions

the reflected azimuthal angle a, is independent of the incident azimuthal angle ai. Substituting eq.3.85 into eq.3.130, the condition for the reflected wave to be linearly polarized, independent

of the polarization of the incident wave, becomes, for A#0,

Under the condition given in eq.3.130, and using eq.3.85, the reflected azimuthal angle may be given by

As a result, the reflected wave is linearly polarized if the condition given in eq.3.131 is satisfied. The angle of incidence Bg that fulfills this condition is called tvBrewsterfsanglew or the Ifpolarization angle." We will proceed next to apply these results to some of the special cases we saw in 93.9. 3.10.1 Optic Axis Parallel to the Plane of Incidence Substituting eq. 3.114 into eq. 3.129, the reflected azimuthal angle becomes tan a, =-tanrll Cli r 2 2 [qiJu,u, - P,JE~K~(i*C*i)-a2] - ~4i.G+ EIJvAK: (BoE*B)-a2I tan ai [qiJlliu, + 'JIJ~L~i(a*~*e)-a'] [qiJEIE/I - E~J~~K~(~*E*~)-~~I Having a*F-=boF+=O,the condition of the linearly polarized reflected wave given in eq.1.31 becomes

According to eq.3.134, two possibilities exist:

The reflected azimuthal angle given in eq.3.132 becomes tan ar = -Ki(a0F-) = O qi (b*F-1 since a*F-=O ; Thus, ar=O. Therefore, using eq. 3.112 and having a0F+=O, we get the condition

under which the reflected plane of polarization is parallel to the plane of incidence. To derive Brewster's angle, we first substitute eq.3.109 into eq.3.135, solve for a2, and get Then,since a 2=~i~in202 i where Bi is the angle of incidence, Brewstergs angle 8~+is given by

A-A sin2eB+ = '~'J-L'I - 'I,~L(q*~*q) El (PLP// - P;) and Brewstergs law is given by

where the conditions for eB+ to exist are given by

A-A or ~,V~>L~~E~((I*U*~)>E,P*PI In this case, 2 2 qi - KeP1[EI(q*~*q)- El&] , - u:)

and q- is given in eq.3.61.

The reflected azimuthal angle is given by

tan ,-Ki(a0F+) = qi (b0F+) since b0F-=O; thus, ar= ~/2. Therefore, under the condition ~*F,=O which, using eq.3.112, can be written as

the reflected plane of polarization is perpendicular to the plane of incidence. Substituting eq.3.109 into eq.3.141 and solving for a2 , and again using a2 =Ki~in2 2 e it Brewsterrs angle is given by and Brewsterls law is given by

where the conditions for @B- to exist are given by

In this case.

and q+ is given in eq.3.61.

As a result, two Brewster's angles eB+ and Bg, might exist if the conditions given in eqs.3.139 and 3.144 are both satisfied. At these angles, the reflected planes of polarization are either parallel or perpendicular to the plane of incidence. A very interesting situation occurs if either

E P = u and €11 ul = u// 3.146 or E :UIU~ = JI:E~ E~ and c is parallel to b 3.147 where eq.3.133 is simplified to

In other words, if the optic axis is parallel to the plane of incidence and if any of the conditions given in eqs.3.146 83 and 3.147 is satisfied, then the incident and reflected azimuthal angles ai and a, are always equal. 3.10.2 O~ticAxis Pemendicular to b, the Intersection of the Plane of Incidence and the Interface In this case, b=&=o. Substituting eq. 3.127 into eq. 3.131, the condition for a linearly polarized reflected wave becomes

A a2(V1qi-Ulq+) (~~qi-~~q-) (q*~) 2 AA2 + K~~~u~(\qi'll~q-)(€~qi-~~q+)(~~c)= 0 3.149 Under this condition, the reflected azimuthal angle given by eq.3.132 becomes

Ah tan ar = a2~~(~~qi-u,q+)(q*c)A = -Ki~~(~~lii-~~q-)(a*c) 3.150

Eq.3.150 can be given in another form by substituting r12and I',,as given in eq.3.127 into tan ar =I',,/T,,:

Note that, in general, neither the numerator nor the denominator of eq.3.150 or eq.3.151 vanishes. Therefore, the reflected plane of polarization does not have to be parallel or perpendicular to the plane of incidence. Next, we will examine an important subcase where the optic axis is perpendicular to the plane of incidence. O~ticAxis Pemendicular to the Plane of Incidence In this case, we have qgc=OAA and aWF+=b*F,=O. Substi- tuting eq.3.125 into eq.3.129, we get r tan ar =~rtanai = (hqi-lq-) (~~qi+E19+)tan a i 3.152 r2 2 (~~qi+~~q-)(ELqi-E1q+) 84 and the condition for linearly polarized reflected wave given in eq.3.131 or eq.3.149 becomes

We see that two possibilities exist: awF,=O or b8F+=O.

Here, the reflected azimuthal angle given by eq. 3.132 becomes tan a, - Ki(a0F+) = O qi (b' F+ since a0F+=O; Hence, a r=O. In other words, the reflected plane of polarization is parallel to the plane of incidence. NOW, using eq. 3.123 and having vLqi-v lq-=O and qi=I?-i-a22 , we

which, after beeing solved for a2, leads to

a2 = Kiu,uL (u~E//-c1k )/ (11:-11:) and, therefore, 2 Si = (KOP,)~(E~u~-~//IJ~)/(u:-u~)

2 q- = (K-UJ (E~U~-E//U~)/(U;-U:) 3.156 q+ -22- K0[u1 (~Lv//-~//uL)-v:(~*U//-€lul 1 l/(u;-!J:) Brewster's law is obtained from eq.3.155 as

2 tan eB+ = "L (~~E//-E,PL)/U~(c1vl-c/,uL) 3.157 where 8~+exists only under the conditions

From eq.3.154, we notice the following: 85 i) if E =El, and LI,= uLf then uLqi-u1 q-=O and, hence, a*F-=O are valid for any angle of incidence. Therefore, the reflected plane of polarization is always parallel to the plane of incidence (ar-0) for all angles of incidence. ii) if u =!J~but E #E/ , then eq. 1.154 can never be valid and hence, agF-#O always. Therefore, cr r#O and eB+ does not exist.

Then, since boF-=Or

and, hence, crr=~/2.In other words, the reflected plane of polarization is perpendicular to the plane of incidence. Now, using eq.3.123, and noting that ELqi-Elq+=Of we get

2 qi = (E:/E~) (K:E~U~-~~) and

a2 = K~E~E~(E~u//-u~E~)/(E~-E:) Therefore, 2 qi ' K~E~(E~U,-E~U/,)/(E:-&~)

Brewsterls law is obtained from eq.3.160:

The conditions for 9g- to exist become 86 Here also, we notice from eq.3.159 the following: i) if E, = and p,/=u , then E, qi- &,q+=O and, hence, b0F+=O are valid for any angle of incidence. Therefore, the reflected plane of polarization is always perpendicular to the plane of incidence (mr=~/2)for all angles of incidence. ii) if E~ = cl but p,, # u then eq. 1.159 can never be valid and, hence, b*F+#O always. Therefore, ar#.rr/2 and eg- does not exist. Next, we will derive Brewsterts angle at an isotropic- isotropic interface as a special case of that at an isotropic-biuniaxial interface.

3.11 ~DecialCase: Brewsteras Anule at Isotro~ic-Isotro~ic

Interface when E 1 2 E and LI 1 2 u 2 In this case, any direction of propagation can be considered an optic axis, since only one transmitted wave exists. Then we must be able to obtain Brewsterts law as a special case of the previous section ( 5 3.10.1) . Here, a*F,= b0F+=O and the condition for the linearly polarized reflected wave is (b-F-)(a0F+) =O. Making E, =€/=c2 and u, =u/,=pl into eqs.3.138, 3.139, 3.143, and 3.144 produces two solutions: a) if a0F+ = 0, then the reflected wave is linearly polarized parallel to the plane of incidence. Brewsterts law given in eq.3.138 becomes

tan2eB+ = U2 (EI~J~-E~FII) u1 (E2u2-E1Ul 87 where the conditions for eB+ to exist are given by

b) if b*F- = 0, then the reflected wave is linearly polarized perpendicular to the plane of incidence. Brewsterrs law given in eq.3.143 becomes

where the conditions for 0~-to exist are given by

According to eqs.3.164 and 3.166, we have

which is always negative. Therefore, one and only one Brewster's angle always exists, BB+ or eB-, at the interface of any given isotropic-isotropic media. Moreover, if either

= p or E = E,then either eB+ or 8~-never exists, but the other always exists and is given by tan24 B-= (E, /E, ) or tan2 Og+= (u2/p ,) respectively.

A] ternative Me- At an isotropic-isotropic interface, Brewsterts angle can also be obtained in a direct way by examining the relation between the azimuthal angles at the interface. In this case, the reflection coefficients are given by (see Chen, 44.6) where 81 and 6t are the incident and the transmitted angles, respectively. The relation between the reflected and incident azimuthal angles, ar and ai, given in eq.3.129 becomes

tan ar = tan CLi since r12= r2,= 0. The reflected azimuthal angle is independent of the incident azimuthal angle if either r,,=O or r2,=0. a) If rll =0, then 8i=eB is Brewster's angle and, according to eq.3.168, we get

COS eB - /E 2v / E p2COs 8t = 0 3.170 Snellls law at this angle is given by

sin eB = sin et solving eqs.3.170 and 3.171 for eB, we get Brewsterls law as

which is the same as that given by eq.3.164. Here again, the reflected plane of polarization is parallel to the plane of incidence (c+=O), after eq.3.169. b) If r2, =0, in the same way as above, using eqs. 3.168 and 3.171, Brewsterls law is given by

which is the same as that given by eq.3.166. Here, again, 89 the reflected plane of polarization is perpendicular to the plane of incidence (ar=v2), after eq.3.169.

3.12 Eneray Relations [6,12,16,171 At the interface, the law of conservation of energy flow must be applied. The power flux of one side of the interface, given by the normal components of the instantaneous Poyntingts vectors of the incident wave and reflected waves di and 9rIis equal to the power flux of the other side, given by the normal components of the instantaneous Poyntingts vectors of the transmitted waves, 8+ and 8-. Hence, the following relation

where the instantaneous Poyntingtsvectors are given by

Eq.3.174 can be proved by using eq. 3.175, the boundary conditions given by

and Maxwell's equations for the incident, reflected, and transmitted waves, and the constitutive relation (61. Taking the time averageof eq.3.174, we get

We define the reflectivity r and the transmittivity t as the energy reflection and transmission coefficients, respectively, and they are given by

and where Yt=9++9_. Ngw, using Maxwellls equations, we get

We can easly prove, using eqs.3.178 and 3.180, that r+t=l. substituting eq.3.180 into eq.3.178 and using eqs.3.77 and 3.78, we get

where a. = r;l+ ri1

ba = r11r12+ r21r22 and ca = r;2+ r;z where ai is the incident azimuthal angle. We define the parallel and the perpendicular reflectivity as

A and ry=- <'C>'q - 1~~1~= (r2,+T2,tan ail2 <9l>*S IA,,I 2

Using eq. 3.181, we can prove that the reflectivity never vanishes unless the necessary condition 1. - r r =0 91 is satisfied. In other words, the reflectivity may not vanish unless the reflected wave is linearly polarized. However, this is only a necessary but not sufficient condition. The transmittivities of the extraordinary waves are given by 3.184 and

The time averaged Poynting vectors < p+>and < 9,> can be obtained from eqs.3.79 and 3.80 after normalizing the

electric field directions e+ and e, as

Ah where 1 e+ 1 = (K+x;) = (bxC + q+qxc)

and

Dot multiplying eqs.3.186 and 3.187 by 4, we get The normalized and time averaged Poynting vector of the incident wave is given by

and its normal component,

Substituting eqs. 3.190 and 3.193 into eq. 3.184, and eqs. 3.191 and 3.193 into eq.3.185, we get

where h Ah A 2 [K~&LPLLI-(K~*C)lqk (K~EIU~-K$)(K+*c) (qWc) y+ - - 3. 196 qi/e+l2

The total transmittivity is given by t=t++t,.

3.13 Total Reflection: Special Cases We studied in section 3.6 the conditions for real transmitted waves to exist. The critical angles given in eqs. 3.75 and 3.76 are the angles where a total reflection for the corresponding wave occurs. Next we will determine the simple forms of these critical angles in some special cases. 93 a) If the optic axis is parallel to the plane of incidence, then B=r/2, and eqs.3.75 and 3.76 become

and sin 2 ec- - ~[EL+(E//'~L)~OS~~I E. u. Two subcases exist: 1) c parallel to b, then u IT/^, and eqs. 3.197 and 3.198 become sin2ec+ = sin 2 8,- = E~I+/E,P, 3.199 2) 6 parallel to q, then a =0, and

b) If the optic axis is parallel to the interface, then a=1~/2;therefore, the critical angles are given by

EL UL P// sin 2 ec+ = E1ul[uL+ (v//-ll,) sin2~l

In the subcase where the optic axis is perpendicular to the plane of incidence (B=O), eqs.3.201 and 3.202 become

2 sin2%+ = ELP///~,~, and sin ec- = ~/,p~/~~p~ 3.203 c) If the optic axis is perpendicular to the intersection of the plane of incidence and the interface (B=O), we get sin2ec+ = €l.k//~,l-'~and sin2 8.- = E~, h/~,pl 3.204

Eq.204 shows that the critical angles 0,+ and 0 c- are 94 independent of the direction of the optic axis, which is parallel to the plane perpendicular to b. ~oticethat eqs. 3.200, 3.203, and 3.204 are the same, since the cases where the optic axis is perpendicular to the interface or to the plane of incidence are specific cases of where the optic axis is parallel to the plane perpendicular to b.

3.14 summary In this chapter, we used the results of the first two chapters to study in more detail the effects of biuniaxial media on the wave propagation. The permittivity and the permeability were considered real diagonal matrices where two of their diagonal elements were equal. First, using the dispersion equation, we found that both of the wave vector surfaces were ellipsoids of revolution around the optic axis. They represented two extraordinary waves since both of the wave numbers depended on the directions of the waves propagation. In the special case where the permittivity and permeability matrices commute, the two ellipsoids overlap: however, the resulting ellipsoid can never degenerate to a circle unless the biuniaxial media also degenerate to a uniaxial or isotrpic media. Second, after deriving the directions of the field vectors, we observed that one of the propagating waves possesses a magnetic isotropy while the other possesses an electric isotropy. In the former, the magnetic field 95 intensity and the magnetic flux density are parallel to each another, and they are perpendicular to the corresponding wave vector, the optic axis, the electric field intensity, and the electric flux density simultaneously. In the latter, the electric field intensity and the electric flux density are parallel to each another, and they are perpendicular to the corresponding wave vector, the optic axis, the magnetic field intensity, and the magnetic flux density simultaneously. We then determined the energy densities, the tinie averaged Poynting vectors, the ray vectors, the ray vector surfaces, and the phase, group, and energy velocities associated with each propagating wave in biuniaxial media, showing that the phase velocity is the projection of the group velocity along the direction of the wave vector. In the second part of this chapter, we moved on from unbounded media to semi-bounded media. ~irst,we determined the magnitudes and directions of the reflected and transmitted wave vectors. Then we studied the conditions for the transmitted waves to exist, establishing the equations giving the critical angles. We noticed that in certain cases, transmitted waves always exist while, in other cases, critical angles less than 90' always exist, independant of the orientation of the optic axis. Second, we derived the reflection and transmission coefficients at the interface of isotropic-biuniaxial media, showing that a study on both sides of the normal to the 96 interface is needed due to the non-symmetrical properties involved, making the angle of incidence varies in a range of 180" rather than 90'. We also considered the specific cases of normal incidence and incidence along the optic axis, and each of the following special cases: optic axis parallel to the plane of incidence, parallel to the interface, and perpendicular to the intersection of the interface and the plane of incidence. In the last two special cases, we noted that the reflection and transmission coefficients are symmetrical with respect to the normal to the interface; hence, the angle of incidence needs to vary only in one quadrant for a complete study. After that, we studied the effect of the biuniaxial media on the plane of polarization of the reflected wave. The reflected azimuthal angle is a function of the incident azimuthal angle. Brewstergs angle is the incident angle that makes the reflected azimuthal angle independent of the incident azimuthal angle, resulting in a linearly polarized reflected wave. In general, more than one Brewsterts angle may exist with the reflected azimuthal angles that might have any value between -90' and 90". In the special cases where the optic axis is either parallel or perpendicular to the plane of incidence, we showed that two Brewstergs angles may exist, but one of the associated reflected planes of polarization has to be parallel to the plane of incidence 97 while the other is perpendicular the plane of incidence. Finally, we used these results to derive Brewsterfs law at the interface of isotropic-isotropic media where both of the permittivity and permeability of one medium were different from those of the other medium. We proved that one and only one Brewster's angle always exists, but the plane of polarization of the reflected wave could be either parallel or perpendicular to the plane of incidence, depending on the given permittivities and permeabilities of the media. We also achieved the same results using a direct method by analyzing the polarization of the reflected wave directly at the interface of isotropic-isotropic media. Note that, with all its simplicity, this study has not been attempted before. CHAPTER FOUR BIUNIAXIAL MEDIA: NUMERICAL APPLICATION

In the last chapter, we established the mathematical expressions of the reflected and transmitted coefficients at the interface of isotropic-biuniaxial media. However, it is so complex to visualize the shape of these coefficients as functions of the angles of incidence. With the help of the computer, we can plot these functions, reach approximate solutions that are impossible to reach mathematically, and even draw some results that can not be observed directly or easily from the mathematical expression. In this chapter, we will examine graphically the reflected and transmitted power coefficients at the interface of isotropic-biuniaxial media. We will plot these coefficients versus the angles of incidence and compare them with those at the interface of isotropic-isotropic media. We will assume that the parameters used in the last chapter are still valid here.

4.2 General Equations: Arithmetical Form Most of the equations in chapter 3 were given in vectorial form. However, in order to be able to feed these expressions to the computer program, we need to express 99 them in arithmetical forms in terms of the angles formed between the involved vectors. Let the notation (u,v) be the algebraic angle oriented from the vector u toward the vector v in the counterclockwise direction. Therefore, the following angles are defined as (see Fig.4.l): = (q,Ahc) , it varies between 0' and 180' :

9 = (a,cf), it varies in a 360' range:

and di = (k)it varies between -90' and 90': where 2.1 is the projection of the optic axis c on the interface. Note that if the angle of incidence Bi changes

sign, the angle 4 changes to 4-n. We assume, as a reference, that the vectors a, b, and are aligned with the axis of

the orthonormal system when ei>0.

Figure 4.1 Orientations of the angles Q , 4, and 8 i.

In algebraic form, we can write the following factors which are repeated in most of the equations as: KO=JJE,LI, KO qi=Ki cos 5 i la1 = lbI = Kilsin ail b*c = lalsin$ sin 6

- A b*~*q= la1 [(El/-EL)/2] sin(24') sin 4 b-iiWq= (a1[(u//-PL)/2] sin(2Q) sin 6 q*Eq = E~+(E//-E~) cos2 + dou*B= u,+(u/,-u,) c0s2 1U a*(adj B) *a = ELa2 [E~- (E)cos2 4 sin2$] a*(adj G) *a = uLa2 [u, - (-) cos2 4 sin2$1 Using these equations, we can calculate A+ = u,u, K? (q*i*q)- a*(adj G) *a A- = Kf (q-;*q) - a* (ad] ;) *a

h A-A q+ - [-(b*u*q)+ &+l/(q*lJ*q)

A q- = [-(bo€*q)+ /-EI/(~-Z*~)

h K+*c- = q+ cos++ la( sin$sin 4 ~+*(ax;)- = -a2cos++ q+lalsin$ sin 6 and lK+I = (a2 + Therefore, substituting eq.4.1 into eq.4.2 and both into eqs.3.83 through 3.86, we can calculate the transmission and reflection coefficients which are then used to calculate and plot the power reflection and transmission coefficients given in eqs.3.181 through 3.183 and eqs.3.194 through 3.196. The normalizing factors given in eqs.3.188 and 3.189 can be written as: le+l = a2 (1-sin2+sin2@) + q:lalsin(2$)sin $+ q:sin2 $ 4.3 le-I2 = (q-cos + +I a[sin $ sin @) (K~-~K~E~~~)+(K~E~P~) According to eqs.3.75 and 3.76, the critical angles are given by

2 sin Bc+ = ELP// [UL+ (PY -UL) c0s2$I EIP~[pL+(v,/-~) (cos2$+sin2qsin26)] 4.4 and sin2 gC- = ,I [E~+(E,~-cl c0s2$I 2 E 1-l [E~+ (E// -E* ) (cos2$+sin2@sin 4 ) ] For easy comparison, we will plot the power reflection and transmission coefficients at both of the interfaces of isotropic-biuniaxial and isotropic-isotropic media on the same graph. coefficents_-_-______-___-____-- at ~sotro~ic-Isotro~ic_-_------Interface [6]

At the isotropic-isotropic interface, the reflection and transmission coefficients are given by:

1 where nj =(p .'JI /€.El)* and nt =(lioP2/E0E2)~,and Bt is the angle of transmission given by Snellls law as:

sin8t = sinei 4.6 The power reflection and transmission coefficients are then given by: If it exists, the critical angle is given by

sin 2 Oc = E2U,/E,U, Brewsterls angle is given by eq.3.172 or 3.173 Now, using the computer program given in appendix B, we will plot the power coefficients versus the angles of incidence at both of the interfaces of isotropic-biuniaxial and isotropic-isotropic media, starting with special cases. Note that the symbols rL, r, , r, t+, t-, t, c, q;, rl,t., ti, alid t1 used in the text, were substituted by R1, R2, R, T(+), T-) TI RI1, RI2, RI, TI1, T12, and TI, respectively on the graphs. In addition, continuous lines are used to plot the power coefficients at the biuniaxial-isotropic interface, while non-continuous lines are used to plot those coefficients at the isotropic-isotropic interface.

4.3 Optic Axis Parallel to the Plane of Incidence

According to 03.9, TI, = ~,,=T,, =T2, =O. The power reflection coefficients rL = T and r, = Ti2 are independent of the incident polarization angle ai. However, the power transmission and the total power reflection and transmission coefficients vary as ai varies. In this case, the numerical results showed the folllowing: a) All the reflection and the transmission coefficients are symmetrical with respect to q, the normal to the interface, in the plane of incidence. Figures 4.2 (a) and (b) (and many 103 others that are not included here) support this statement. Note that this symmetry did not show up during the theoretical study. b) Figures 4.3 to 4.5 show that two Brewsterls angles may exist in one quadrant of the plane of incidence. The values of these angles increase when the pennittivities and permeabilities of the second medium (biuniaxial) increase relatively to those of the first medium (isotropic). c) Comparing Figs. 4.3 and 4.4, we notice that (or LI/, ) has very little effect on the shape of the power coefficients at the biuniaxial-isotropic interface, compared to a considerable change at the isotropic-isotropic interface. In fact, the coefficients rL and t, and Brewsterls angle eB+ (or r,, , t+, and 9 B,) remain constant when E/, (or , ) changes its value. However, Brewster s angle 8 B, (or 8 B+) shifts with a comparable amount to the change of eB (compare fig.4.3 with fig. 4.4 and fig. 4.5) . ~rewster's angles 4B- and/or e B+ may appear as the values of E,, and/or P, are increased but they never disappear.

d) On the other hand, a little change of E, and/or wL reflects a considerable change in the power coefficients, which is comparable to the change in the coefficients at the isotropic-isotropic interface (compare figs.4.3 and 4.6). An

increase of EL and/or vL results in a much faster increase of Brewsterts angles, affecting their existence.

2 e) In the special case where E: vL = u EL E, and E, vL = EL '+I, we 104 have tanar=tanai, according to eq.3.148. Looking at the data listed in fig. 4.7, we see that eB=O', ai=O' and a i=9 0 " that is impossible. Also, e~+=eB-=27' and ar+=O0 anda r-=90°, which is impossible, indicates that a polarization did not take place. Moreover, we see that the curves of r, and r,, overlap, indicating that ar+=ai- at any angle of incidence. f) Fig.4.8 shows that an abnormality took place at the angle of incidence ai=20°. Investigating the matter, we found that the transmitted angles 0t+=f3t-=15.01', which is almost equal to the angle that the optic axis makes with the normal to the interface. We know from the last chapter that when the transmitted waves propagate in the direction of the optic axis, they are not linearly polarized, and therefore the equations used to calculate the transmittivities t+ and t- are not valid at this angle of incidence. g) In general, when the optic axis is parallel to the plane of incidence, the power reflection coefficients r and r,, are not equal at normal incidence, compared to the isotrpic case where the curves of r,' and r;; are always tangent to each other at 9 i=Oe (see fig. 4.9) . In this special case, two subcases are worth looking at: the optic axis is perpendicular or parallel to the interface. In these cases, we need to vary the angle of incidence only in one quadrant since the power coefficients

are symmetrical with respect to q. 105

h A A) O~ticAxis Per~endicularto the Interface (c=q) In this case, figures 4.10 through 4.12 illustrate some examples of the power coefficients when the pemittivi- ties and permeabilities of the second (transmitting) medium are much higher, in the same brder, or much smaller than those of the first (isotropic) medium. We notice from these figures that the reflection coefficents rl and 5, are tangent to each other at normal incidence, always the case at an isotropic-isotropic interface. Also, the power and total power reflection and total power transmission coefficients have almost the same shape as those of the isotropic case, especially when the permittivities and/or the permeabilities of the second medium are much higher than those of the first medium. B) O~ticAxis Parallel to the Interface (G Jb) Here, the critical angles 8,+ and 9 c- are equal and given by a,+=@ ,-= E~ vL/ E~ pl, independent of E, or uy . Also, comparing figs.4.13 and 4.10, we notice that Brewster's angles shift down when we rotate the optic axis away from q in the plane of incidence. We will now examine the two specific cases where the incident plane of polarization is parallel or perpendicular to the plane of incidence. 4.3.1 Incident Plane of Polarization Parallel to the Plane of Incidence (a i=Oo)

~ccordingto 03.12, we have r, = rr= , t+=0,

and ~=~-=K:IJ,E,Y-T:~. These coefficients are functions of q- but not q+. Since AI=O, rLhas no physical presence in this case, and therefore will not be plotted.

h Since r,/ is independent of ail while c is parallel to the plane of incidence, it has the same shape for any ai, including ai=Oo. Therefore, making ai=O' causes the reflectivity r to overlap with r,,. It also forces the power of one of the transmitted wave t+ to zero, making t=t,. At the isotropic-isotropic interface, r1 and t1 overlap, respectively, with r,' and t,) which do not change with ai either. These results are illustrated in figs.4.14 and 4.15, which are compared to figs.4.5 and 4.9. In the specific cases where c=~or 6 is parallel to b, nothing more notable would happen. We can easily figure out their shapes by applying these results to figures 4.10 through 4.13. 4.3.2 Incident Plane of Polarization Pemendicular to the Plane of Incidence. (aj=90°) Again, according to 43.12, we have r=r, = r:, , t-=O, and t=t+=u,~+q~,, but r// is undetermined. These coefficients are functions of q+ but not q-. Here, the reflectivity r overlaps with r1 which is independent of Qi. The power of one of the transmitted waves vanishes, t,=O, making t=t+. At the interface of isotropic-isotropic media, r1 overlaps with r; , and t overlaps with ti. Comparing figs. 4.16 and 4.17 with figs.4.5 and 4.9, respectively, we confirm these

results. Again, in the specific cases where q or c is parallel can obtain the curves the involved 107 power coefficients by applying these results to figs.4.10 through 4.13. Combining these results, we see that when the incident azimuthal angle ai varies between 0' and 90a, i) the reflectivity varies between r, and r,, and it is always located between them. ii) the transmitted power coefficient t+ varies between 0 and t, which is a function of ail while t, varies in the opposite direction, between t and 0. his is an interesting phenomenon since we can control the power intensity in either direction of the two transmitted waves only by changing the incident plane of polarization.

4.4 Optic Axis Perpendicular to the Plane of Incidence

In this case, we have r =I' 2 1 T T , 0 and the power reflection coefficients r,= r :,and r,, = r :, are independent of ai. However, r, is now a function of q, but not q+, while r,, is a function of q+ but not q,. Also t+ and t,, which vary with ail are functions of only q+ and q,, respectively. Therefore, the existence of the transmitted wave vectors K, (or K+) does not affect the existence of r,/ and t+ (or r, and t,) . Since these coefficients are symnet- rical functions of the angle of incidence 8i, we will plot them against 8i in only one quadrant. The graphic study showed the following: a) As in the previous case, more than one Brewsteris angle 108 may exist and the values of these angles increase when and/or vL increase, relative to El and/orv, (compare figs. 4.18 and 4.19). Note that eB+ is not a function of or u,, and neither is BB- a function of E// or uL . Moreover, the existence of BB+ and eB, depends greatly on the direction of the optic axis. Comparing fig.4.20 with fig.4.5, we notice that the two Brewster's angles in the previous case do not exist here. b) The critical angles of both cases, where the optic axis is perpendicular to the plane of incidence and perpendicular to the interface, are equal. However, while the coefficients r, and t+ (or r, and t,) were terminated at the critical angle ec+ ( or BC,) in the former case, the coefficients r, and t+ (or r1 and t,) were terminated at the critical angle ec+ (or 8 c,) in the latter case (compare figs.4.21 and 4.10). c) When the incident wave is linearly polarized parallel to the plane of incidence, we have r=r,, = T: ,, t-=O, and t=t+ (fig.4.22). However, when it is polarized perpendicular to the plane of incidence, we have r=rL = rl,,2 t+=O, and t-t, (fig.4.23) . Therefore, similar to when c is parallel to the plane of incidence, we can control the power and the presence of the transmitted waves simply by controlling the polarization of the incident wave. For instance, increasing ai from 0' to 90' increases t, from zero to a maximum and decreases t+ from a maximum to zero, and the reflected azimuthal angle changes from 0" to 90". However, changing the optic axis direction from perpendicular to parallel to the plane of incidence, while ai=Oo (or 90 ") , it changes t+ from a zero to a maximum (or from a maximum to a zero) and t, from a maximum to a zero (or from a zero to a maximum), while the reflected plane of polarization is kept parallel (or perpendicular) to the plane of incidence. d) Changing the values of the permittivities and permeabilities of the second (biuniaxial) medium relative to the first medium has the same effect as when the optic axis is parallel to the plane of incidence.

4.5 General Case: Optic Axis Has Any Direction To draw and generalize some results, we will start by considering a linearly polarized incident wave that is parallel and perpendicular to the plane of incidence. A) If the incident plane of polarization is parallel to the

2 plane of incidence, we have r, = T2 , r=~.:~+ ri2 , t+=u 1~:2 Y+, 2 and t-=Kop,~,T: 2Y-. Figures 4.24 and 4.25 showed that r, is a symmetrical function of the angle of incidence. This indicates that r:, is symmetrical but T:, is not. B) When the incident plane of polarization is perpendicular to the plane of incidence, we have r,= r: ,, r=r21+2 rll, 2 t+= T and t-=~ju,~,Ti, Y-. Here, figures 4.26 and 4.27 show that r, is a symmetrical function of the angle of 110 incidence indicating that T:, is symmetrical but F:, is not. Therefore, in general, we perceive the following: a) The magnitudes of the diagonal elements of the matrix giving the reflection coefficients are symmetrical functions of the angle of incidence ei: however, the non-diagonal elements are not. b) All the partial and total power reflection coefficients are non-symmetrical functions of hence the necessity to vary 0i between - 0 and + Bc in order to obtain a complete study for the reflectivities and transmittivities (figs.4.28 through 4.3 0) . c) If a Brewsterts angle 8 exists, then the angle -BB is also a Brewsterls angle, despite the non-synunetricity of the reflectivities and transmittivities. However, the corresponding reflected azimuthal angles are different. Also, more than one Brewsterls angle may exist in one quadrant (see fig.4.31). The existence of these angles depends on the values of the pennittivities and permeabilities and the direction of the optic axis. For example, changing the optic axis direction of fig.4.31 may causes these Brewsterls angles to disappear (see fig. 4.32) , indicating that the reflected wave can never be polarized. d) In general, the reflected azimuthal angle may have any value between -90' and 90'. Therefore, neither rL nor r,, has to go to zero at Brewster1s angle. Moreover, if r or 5, goes to zero at a certain angle of incidence, it does not 111 imply that this angle is a Brewsterls angle. In fact, changing the incident azimuthal angle ai shifts the position of this zero while Brewster's angles are independent of ai (compare figs.4.29 and 4.30). e) On the other hand, if the total power reflection coefficient r goes to zero, then the corresponding angle of incidence is Brewster's angle [fig.4.26(b)]. f) When the incident azimuthal angle varies between 0' and 90°, the powers of the transmitted waves t+ and t, almost interchange. Examining figs.4.25, 4.30, 4.29, and 4.27, we notice that a zero of t, at cli=O0 shifted to a maximu? at cli=9O0 and a maximum of t+ shifted ta zero. Therefore, an elliptically polarized incident wave will cause the powers of the transmitted waves t+ and t, to have sinusoidal patterns with a frequency twice as much as that of the incident wave polarization (fig.4.33). We also notice that whenever t+ (or t,) passes through a zero, then t, (or t+) passes through a maximum.

- Remark on the following figures, the power and total power reflection and transmission coefficients at the isotropic- isotropic interface are represented by the lines: ------. *.. -,,: for r/; and t/:, , . .. '-.. : for ri, rl,ti, and t'. Figure 4 -2

Figure 4.4 Figure 4.5 (b)

Figure 4.6 Figure 4.7

Figure 4 .a Figure 4.9

Figure 4.10 Figure 4.11 Figure 4.12 Figure 4.13 Figure 4.14

Figure 4.15 Figure 4.16

Figure 4.17 Figure 4.18

Figure 4.19 Figure 4.20

Figure 4.21

(b)

Figure 4.24 Figure 4.25

Figure 4.27 (a)

Figure 4.27

Figure 4.26 (b)

Figure 4.28 Figure 4.29

Figure 4.30 Figure 4.31

Figure 4.32

REFERENCES

1) Adams, Roy N. and Eugene D. Denman, "Wave propagation and Turbulant Media," American Elsevier Publishing Company, Inc. New York, 1966. 2) Bell, W. W., !#Matricesfor Scientists and Engineers," Van Nostrand Reinhold Company, New York, 1975. 3) Born, Max, and Emil Wolf, llPrinciples of optic^,^^ The MacMillan Company, New York, 1964. 4) Chen, Hollis C., "A Coordinate-Free Approach to Wave ~eflection From an ~nisotropic Medium,ll Radio Science /Vo1.16, No.6/ Nov.-Dec., 1981, pp. 1213-1215. 5) Chen, Hollis C., "A coordinate-Free Approach to Wave propagation in Anisotropic Media,I* J. ~pplied physics /Vo1.53/ 1982, pp. 4606-4609. 6) Chen, Hollis C., llTheory of ~lectromagnetic Waves, A coordinate-Free Approach,' McGrow- ill Book Company, 1983. 7) Collin, Robert E., "Field Theory of Guided Waves," MacGrow Hill Book Company, Inc., New York, 1960. 8) Damaskos, N. J., A.L. Maffet, and P.L.E. ~slenghi, llDispersion Relation For General ~nisotropicMedia," IEEE Trans. Antennas Propag. AP-30, p. 991-993, 1982. 9) Damaskos, N. J., A.L. Maffet, and P.L.E. Uslenghi, "Reflection and Transmission for ~yroelectromagnetic Biaxial Layered ~edia,"J. Opt. Soc. /Vol.2, No.3/ March, 1985, p. 454-461. 10) Gourlay, A.R. and G.A. Watson, w~omputationalMethods for Matrix EigenproblemstN John Wiley & Sons, New York, 1973. 11) Hague, B., llAn Introduction to Vector Analysis for Physicists and Engineers, Methuen & Ltd, 6th e., London, 1970. 12) Hewson, A.C., llAn Introduction to the Theory of Electro- magnetic Waves,11Longman, London, 1970. 13) Hlawiczka, Paul, "Gyrotropic WaveguidesIN Academic Press, Inc., New York, 1981. 14) Kong, Jin Au, "Theory of Electromagnetic Waves," Jhon Wiley & Sons, New York, 1975.

15) Okashi, T. , "Optical Fibers,It ~cademicPress, New York, 1982. 16) Owyang, ~ilfertH., "Foundation of Optical ~aveguides,~ ~lsevierNorth Holland, Inc., New York, 1981.

17) Yariv, Amnon and Pochi Yeh, ltOpticalWaves in Crystal," John Wiley & Sons, Inc., New York, 1984. APPENDIX A

VECTOR AND TENSOR ANALYSIS

New mathematical techniques were developped [4] to meet the need of the Coordinate-Free Approach method where vector analysis and matrix algebra are used. Two kinds of notations are used: index notation and direct notation. We use the first to define an operator and to do operation and the second to list the results in easy usable forms. In index notation, a repeated index, also known as a dummy index, indicates a summation over that it. 3 For example, the notation UiVi is equivalent to 1UiVi. A i=l non-repeated index,also called free index, indicates the

with a bar at its top (e.g., i)indicates a 3x3 matrix; a bold letter (e.g., u) is a vector; and a non-bold letter

(e.g. , h ) is a scalar. A """ over a bold letter indicates a - unit vector, and a " " over a letter with bar indicate the transpose of the matrix represented by that letter.

A.2 Vector Analysis [4,5,9] There are three types of products between two vectors: A-2 i) scalar product (or dot product): the result is a scalar.

u-v = uivi ii) cross product (or vector product): the result is a vector.

+l if i,],k, is an even permutation of 1,2,3 where Eijk = -1 if i,~,k,is an odd permutation of 1,2,3 o if any two of i, j, k are equal iii) dyadic product; the result is a matrix.

Some related relations between vectors a, b, c, and d are listed below:.

(a) a*(bxc) = b*(cxa) = c*(axb) (b) ax(bxc) = (a0c)b - (a0b)c

The differential operator V (also called Ifdelffor "nablaw) is of central importance in electromagnetics and is defined as:

Let u and v be two arbitrary vectors functions of r and 4 and d two arbitrary scalar functions of r. Then we define the gradiant of a scalar function , the divergence of a vector function, and the curl of a vector function as and vxu auk - ~ijkajuk -'ijk -ax j - respectively and , hence, the following identities:

A (a) Vr = r and (cor) = c where c is a constant vector

(b) VxV@ = 0

(e) V* (4 u) = @(Vau) + u*V$

(f) ox(@ U) = @Vxu+ V@XU

A.3 Tensor Analysis We will be studying 3x3 matrices and 3-dimentional vectors, though some of the rules and formulas can be used

- with different dimentional tensors as well. We denote the transpose of a matrix by either a tilde << - >> or the superscprit "T", the trace by the subscript "t'>, and the complex conjugate by the superscript "*". A.3.1 ~atrixAluebra The following operations and their definitions are given as 1) suml E = x+S - Cij = aij + bij A. 10 - 2) Product, c = A9B - Cij = aij bij 3) the matrix A as a linear Operator, where aij , bij , and Cij are the elements of the matrices a, g, and C, respectively.

A.3.2 Promities of Matrix Transpose and Trace - - - - (a) (%+BIT = i+B (b) (A+B)t = At+& - - - - (c) (A*B)T = ?j*i (a) (A*B)t = (B*A)~ --. - - (e) ~t = st (f) A=A - - is a' symmetrical matrix if A=A, and antisymmerical if ------i=-i. If A=A and B=-5, then (A*B)~=O.

A.3 .3 Determinant and Adi oint of a Matrix The determinant of a matrixlis defined as

It can be written in terms of trace of A as

The adjoint of A, adj A, is the transpose of the coefactor of A. In index form, it is defined as adj - (ad] A)ns = (1/2) Eijs Elmn ail ajm A.16 In direct notation, we have - - adj = i2 - AtA + (adj Alt'r

(adj i)t = [(It)'- (A2)t]/2 and %*(adj 5) = (adj %)*A = lilt where is the unit matrix defined as - and (b) u*? = IOU = u

A matrix 5 is said to be singular if li1=0,and nonsingular otherwise. A nonsingular matrix A has a unique inverse A so that - K.K-l= i-1 *A = -I - where A = (adj %)/lxl A. 22

Some of the proprities of the inverses of the matrices A and are given as follow: - (b). . (A-l)-I = A

(d) (i-I ) T = (i)

The following useful identities of the determinant ant adjoint of matrices are established:

(a) 121 = 1x1 (b) lail = a3 1x1

(e) ladj = 1x1' (f) adj (ai) =a2 (adj A) A.24 (g) adj i-'=(adj ii)-l=i/lii1 (h) adj (adj %) = I%IA - (i) (adj = adj (j) adj (i-5)= (adj i)*adj

A.3.4 Dvad and Antismmetric Matrix uxr We already defined the dyadic product in OA.1. We will now see some of its proprities:

(el (uv*A)~= v*A-u

Sometimes it is very useful to express the cross product of two vectors into a different form to faciliate the calcula- tion. This is why we introduce the matrix uxi, which may be given in index form as

Then the cross product of vectors u and v can be written as - UIW = (ux~)*V = (In).V = U*(vx?) = U*(IIW) ~.27

since (ux?)~= -(uxI) A. 28

According to eq.A.28, the matrix uxi is antisymmetric and, hence, we find that (a) (uxilt= luxil =O A. 29 and (b) adj (uxf) = uv Any antisymmetric matrix can always be represented as uxf such that u - ui ' -21 'ijk alk A. 30 The cross product of vector u and matrix A is defined as - in index farm, US - (uxA)i j = Eilm UL a,j A. 31 in direct form, (a) uxi = (uxf)*ii = (?XU)'A A. 32 and, similarly, (b) &.a = i*(uxf) = A*(?xu)

If ii = ab, then (a) ux(ab) = (uxf) ab = (uxa)b

and (b) (ab)xu = ab*(?xu) = a(bxu)

The following identities are also useful: - - (a) (&u)~ = -ux~ and (us) = -Am

(b) uw(kuv) = (u*A)w and (us)*v= ux(Aev) - (c) U*(vz) = (uxv) *: and (%xu) *v = A*(uxv) A-7

A.3.5 More Identities between Tensors and their Functions - (a) (adj g).*(UW) = (P*U)X(XXV) = (ud)x(v0x)

(b) i-l . = [ (Agu) (Av) I AI - (c) Po(urn) = III [ (:-I *u)x(I-l *v)] (d) A- (uxz)*A = (u-adji) xi

(e) (vxT)*x* (uxi) = [ (adj 5)*u] (rl*v) - (u-kl*v) adj

-w * .. (f) (vxz) (adj x) (uxT) = x*uv*x- (u*k*v)% A. 35 (g) (axb)(cxd) = (axb) (cxd)I + (a*d) cb + (b0c)da - (a0c)db - (b0d)ca (h) (axb) (cxd) = (a'c) (bod)- (aod)(b0c) ( i) [ (cxd)xf ] [ (axb)xi] = (axb)(cxd) - (axb) (cxd)f (j) (ji*a)x(i*b) (i*c)= (A((axb0c)

Now we will present the traces, determinants, and adjoints and their traces of matrices eqs.

If c = 5 f AT, then - (a) Ct=& + 31 (b) 1e1 = 2 X3 + ith2 -+ (adj x)t~+ 131 A.36 (c) adj C = A~Ff X(xt?-x) + adj x

(d) (ad] Elt = 3h2 f 2dtu (adj ii)t

If 6 = A + uv, then

(b) 151 = 1A1 +v0(adjg)*u

(d) (adj E)t = (adj'a)t + (umv)Xt - v*X0u - If 5 = I + uv0B, then (a) Et = 3~ + v-B*u (b) 151 =X2(X +v*~*u) (c) adj 5 = X [(A + v*B0u)I - uv-i] (d) (adj C)t = X (3h + 2v-%u)

If E = + uv + mn, then (a) Et = 3X + u*v + men (b) IZ'I = h[A2 + h(u*v + mgn) + (vxn)*(uxm)]

(d) (adj E) = A [ (3X+2(u*v+men) ]+ (vxn) (uxm)

If C = I + a1 + bm + cn, then - (a) Ct = 3X + a01 + b*m + con

(b) ICl= ~~+(a*l+b*m+c*n)~~+[(mxn).(bxc)+(nxl) (cxa) + (lxm) (axb)]A + (a0bxc)(l0mxn) (c) adj = [ (X+ael+ born + con)? - (al+bm+cn)] A. 40 + (mxn) (bxc)+ (nxl) (cxa)+ (lxm)(axb) (d) (adj E)t = x[~X + 2(a*l+b*m+c*n)]+ (mxn) (bxc)+ (nxl) (cxa)+ (lxm) (axb)

A.4 Overator v "DelW and Tensors

In §A. 1, we applied the differential operator V on scalar and vector functions only. NOW, we will apply this operator on tensors. We will define the gradient of a vector function g(r) as the divergence of a matrix A as L'*A - aiaij the curl of a matrix A as Ox% - Eijk 3 jakl Now we will list a number of useful formulas:

(a) 7 (UV) = (Veu)v+ (u-V)V

(b) V * ($x) = v$*X + 4v.X (c) v '(VU - uv) = vx(u X v) (d) v (U x X) = (V x u) *X - us(V x 8) v *(ix U) = vx u

(e) V *ox X = o

(f) V ($u) = (V$ )u + $7 u (9) v (u x v) = (Vu)x- v - (vv)x u (h) v (r*C0r)= (C+C)*r where C is a constant matrix.

A.5 Dyadic ~ecompositionof a Matrix; Solutions of Homoseneous Equations: Eisenvalue Problems A matrix A that can not be reduced to a sum of less than three dyads is said to be ltcomplete.w - A = ai'ci = 5, 9 + 9C2 + a3 C3 A. 45 ~eometrically,the vectors a, , a2 , and a3 are non-coplanar, and the vectors cl, c,, and C? are non-coplanar either. A matrix 8 that can be reduced to a sum of only two dyads is said to be "planar." A-10 Here the vectors a,, a ,, and a,(or c, , c, , and c?) are coplanar, but not collinear. A matrix x that can be reduced to one dyad is called

"1inear. "

A = ac A. 47

Here, alla2, and a3 ( or c,, c2, and c*) are collinear. To summerize, we have the following necessary and sufficient conditions: 1) if ls/#0, i=al+bm+cn is complete. Then

1x1 = (aebxc)(lwmxn) and adj 5 = (mxn)(bxc) + (nxl)(cxa) + (lxm)(axb) 2) if lsl=0 but adj A # 5, A is planar. Then - A = a1 + bm and adj A = (lxm)(axb) 3) if IAJ=Oand adj 5 = 0, but jif0, then A is linear: - A = ab

A homogeneous equation is defined by

i*u = 0 A.53

If U=O is a solution, we call it a tttrivialMsolution. However, we are interested in nnontrivialw solutions where ufO. A nontrivial solution exists if and only if =0, and therefore, two possibilities occur: a) adj % # 6, then

u = (adj g)*c = (bac)a A. 54 where c is an arbitrary constant vector and adj % = ab. - b) adj A = 0, then % is linear (&-ab) and A*u = (b0u)a= 0 Therefore, u is any vector perpendicular to b. We define the "eigenvalue problem" by

where h is an eigenvalue of and u is the corresponding eigenvector. It has nontrivial solution if and only if

which is called the characteristic equation. In this case, three possibilities are possible. a) If the matrix (A- Xi) is planar, then

and the corresponding eigenvector u is given by

u = [adj (A - XI) ] -c = (b0c)a A.59 where c is defined above. Note that u* is also an eigenvector corresponding to X *. b) If (i- Xi) is linear, then % - Xf = ab A. 60 and any vector u perpendicular to b is an eigenvector of 5 corresponding to A. The vector a is also an eigenvector of corresponding to the eigenvalue (X+a0b). - C) If (x - AT) = 0, then B-u = Xu A. 61 and therefore, any vector is an eigenvector corresponding to the eigenvalue X. A-12

A.6 Hermitian Matrices The complex conjugate of the transpose of a matrix A is called I1Hermitian1lconjugate of the matrix A, written as - if= i*.It has the following properties:

(a) (%+I + = li

(b) (% + 5)+ = A+ + ii+

A matrix A is "hermitian" if - A+ = A A. 63 A matrix B is I1antihermitian1l(or skew-hermitian) if - - B+=-B A. 64 A hermitian matrix can always be expressed as - - ~=s+iN A. 65 - (or i) = N - ig for an antihermitian matrix) where 8 and k are real symmetrical and antisymmetrical matrices respectively. The "hermitian formM is expressed as: - H = u**~*~= UirC aij uj A. 66 It is always real. If the matrix A and the vector u are real, the hermitian form is called Itquadratic form.ll The eigenvalues of a hermitian matrix are real, and the eigenvectors corresponding to different eigenvalues are orthogonal. Let Ui and Uj be two unit vectors of these orthonormal eigenvectors. Then, A hermitian matrix is said to be definite if, for any vector u,

It is definite positive if and definite negative if u**ii0u< 0 A. 70 Also, using eqs.A. 62 (e) and A. 63, we can easily prove that the determinant of a hermitian matrix 1 is always real, and, if R is definite nonsingular hermitian matrix, so are its inverse f-I and its adjoint adj B. APPENDIX B

COMPUTER PROGRAM

This program calculates the normal components of the incident and transmitted wave vectors, the reflection and transmission coefficients, and the power reflection and transmission coefficients at both the isotropic-biuniaxial and the isotropic-isotropic interfaces. It also determines the type of the crystal, the critical angles, Brewster's angles and the corresponding reflected azimuthal angles . The following parameters used in the program are defined as: PT1 and PB1 are, respectively, the permittivity and the permeability of the first medium. PT21, PT22, PB21, and PB22 are, respectively, the repeated and nonrepeated diagonal elements of the permittivity and permeability matrices of the second medium.

AD and BD are the angles (4,$) and (a,61) respectively (see fiq.4.1). ALID is the incident azimuthal angle . At the Isotro~ic-Biuniaxial Interface QI, QP, and QN are the normal components of the wave vectors Ki , K+, and K, respectively. T11, T12, T21, and T22 are the transmission coefficients given in eq.3.84. R11, R12, R21, and R22 are the reflection coefficients given in eq.3.85. RD, RL, and RT are the perpendicular, parallel, and total power reflection coefficients. TP, TN, and TT are,respectively, the transmitted power coefficients of the wave vectors K+ and K, and the total transmitted power coefficient. TBA and ALR are, respectively, Brewsterfs angles and the corresponding reflected azimuthal angles. TCP and TCN are the critical angles of the Waves K+ and K,. X is the angle of incidence. At the Isotro~ic-Isotropic Interface

TPE, TPA, and TPT are the perpendicular, parallel, and total power transmission coefficients respectively. RPE, RPA, and RPT are the perpendicular, parallel, and total power reflection coefficients respectively.

TB and AR are Brewsterls angle and the corresponding reflected azimuthal angle. the critical angle. the angle of incidence. DIMENSION X(183)~QI(1R3)~QP(1R3)~QN(183~~T11~183)~Tl2(l83)~TAOOOO I0 8 T21(183)vT22(183)vRll(l83)~RI2(IR3)~R21(183)~R22(l83)~ TA000020 & RD(1R3).RL(183),TP~l83),TN(lR3),RT(le3),TT(l3,TBA(8)ALR8 TA000030 DIMENSIONXU(183),TPE(183).TPA(I83).TPT(183),RPE(I83RPAl83, TA000040 & RPT(I83) TA000050 READ(5,II) PT1~PBt~PT2l~PT22~PB21tPB22vAD~BO~ALID TA000060 FORMAT(IX.FR.3) TA000070 LL=O TAOOOOEO MT=O TA000090 W=Z.OE+ 15 TAOOO 100 PI-3.34159265358979 TAOOO I10 PT018.854E-12 TA000 120 PBO=Pl04.E-7 TA000 130 EXM=O. 008 TA000 140 TAOOO 1 SO WRITE(6.7)PT1.PBl,PT21.PT22.PB2ltPt322,W TA000 160 FORMAT('1',3X,'PTI =',F8.5,2X,'PBI =',FR.5.2X.'PT2l=',F8.5, TA000 170 & 2X.'PT22=',F8.5/4X,'PB21=',F8.51ZX,'P822=',F852X'W'E9.4 TA000180 WRITE(6.9)AD.BD.ALID TA000 190 FORMAT('O',3X.'THE OPTIC AXIS DIRECTION IS GIVEN BY THE ANGLES' TA000200 8 ,': AD=',F6.2/53X,'BDa',F6.2/4X,'AND THE INCIDENT AZIMUTHAL ', TA000210 8 'AN(;LE'.lSX.'ALID=',F6.2) TA000220 TA000230 ********oo*******DETERMINATION OF THE CRYSTAL TYPE *0****00*000**TA000240 TA000250 IF((PT21.NE.PT22).OR.(P821.NE.P822))GO TO 12 TA000260 WRITE(6.13) TA000270 FORMAT('OO,'THE MEDIUM IS ISOTROPIC') TA000280 GO TO 1000 TA000290 IF((PT2l0PB22).NE.(PT22*P1321)) GO TO 15 TA000300 WRlTE(6.14) TA0003 10 FORMAT('OO,'THE PERMITTIVITIES AND PERMEABILITIES COMMUTE') TA000320 GO TO 1000 TA000330 TA000340 IF((PT21.EQ.PT22).0R.(PB21.EQQPE)22)) GO TO 25 TA000350 DATA POS/'POSI0/,ANE/'NEGA'/ TA000360 S l =PO5 TA000370 S2-I.;I TA000380 IFtPT21 .(;T.PT22) SI=ANE TA000390 IF (PB21. (iT.PB22) SZ=ANE TA000400 WRITE(6.20) SI.S2 TA0004 10 FORMAT('0',3X.'THE BIUNIAXIAL MEDIUM IS ELECTRICALLY '.A4.'TIVE0 TA000420 (L /25X.'AND MACiNETICALLY '.A4.'TIVE .'I TA000430 GO TO 28 TA000440 WRITE(6.27) TA000450 FORMAT(5X,'THE SECOND MEDIUM IS UNIAXIAL') TA000460 s~o*~~o~ooooo~~~oo~o~oooo*CRYSTAL TYPE END ~0*000*0*******00*****~~~0047~ TA000480 AKOS=(W*"2)"PTO*PBO TA000490 AKO=SORT( AKOS) TA000500 AI