Electromagnetic Wave Propagation Lecture 5: Propagation in birefringent media

Transcription

1 Electromagnetic Wave Propagation Lecture 5: Propagation in birefringent media Daniel Sjöberg Department of Electrical and Information Technology April 15, 2010

2 Outline 1 Introduction 2 Wave propagation in isotropic media 3 Wave propagation in bianisotropic media 4 Interpretation of the fundamental equation 5 Maxwell s equations for linear and circular polarization 6 Uniaxial and biaxial media 7 Chiral media 8 Gyrotropic media 9 Oblique propagation in biaxial media

3 Outline 1 Introduction 2 Wave propagation in isotropic media 3 Wave propagation in bianisotropic media 4 Interpretation of the fundamental equation 5 Maxwell s equations for linear and circular polarization 6 Uniaxial and biaxial media 7 Chiral media 8 Gyrotropic media 9 Oblique propagation in biaxial media

4 Key questions How can we compute the wave number and wave impedance in bianisotropic materials? What is the effect if a material has different properties for different polarizations? How can we convert between linear and circular polarization? What are the effects when a wave is not propagating along an optical axis?

5 Outline 1 Introduction 2 Wave propagation in isotropic media 3 Wave propagation in bianisotropic media 4 Interpretation of the fundamental equation 5 Maxwell s equations for linear and circular polarization 6 Uniaxial and biaxial media 7 Chiral media 8 Gyrotropic media 9 Oblique propagation in biaxial media

6 Propagation in isotropic media Maxwell s equations for nondispersive, isotropic media are E = µ H t H = ɛ E t Assuming the fields do not depend on x or y implies E = ẑ E z ẑ E z = µ H t ẑ H z = ɛ E t z ( ) E = H ẑ t The cross products with ẑ make the z-components vanish. ( ) ( ) 0 µ E ɛ 0 H ẑ

7 The wave equation Eliminating the magnetic field implies ( 2 z ) c 2 t 2 E(z, t) = 0 with the solution E(z, t) = F + (z ct) + F (z + ct) The corresponding magnetic field is H(z, t) = 1 η ẑ F +(z ct) 1 η ẑ F (z + ct) The minus sign for the wave in negative z-direction is due to that both (E +, H +, ẑ) and (E, H, ẑ) must be right-handed triples. The wave speed is c = 1/ ɛµ and the wave impedance is η = µ/ɛ.

8 Observations Some fundamental properties are observed: The wave speed c = 1/ ɛµ and wave impedance η = µ/ɛ depend on the material properties. Waves can propagate in the positive or negative z-direction. For each propagation direction, there are two possible polarizations (ˆx and ŷ, or RCP and LCP etc).

9 Outline 1 Introduction 2 Wave propagation in isotropic media 3 Wave propagation in bianisotropic media 4 Interpretation of the fundamental equation 5 Maxwell s equations for linear and circular polarization 6 Uniaxial and biaxial media 7 Chiral media 8 Gyrotropic media 9 Oblique propagation in biaxial media

10 Wave propagation in general media We now generalize to arbitrary materials in the frequency domain. Our plan is the following: 1. Write up the full Maxwell s equations in the frequency domain. 2. Assume the dependence on x and y appear at most through a factor e j(kxx+kyy). 3. Separate the transverse components (x and y) from the z components of the fields. 4. Eliminate the z components. 5. Identify the resulting differential equation as a dynamic system for the transverse components, ( ) ( ) ( ) Et W11 W = jω 12 Et z H t ẑ W 21 W 22 H t ẑ

11 1) Maxwell s equations in the frequency domain Maxwell s equations in the frequency domain are [ ] H = jωd = jω ɛ(ω) E + ξ(ω) H [ ] E = jωb = jω ζ(ω) E + µ(ω) H The bianisotropic material is described by the dyadics ɛ(ω), ξ(ω), ζ(ω), and µ(ω). The frequency dependence is suppressed in the following.

12 2) Transverse behavior Assume that the fields depend on x and y only through a factor e j(kxx+kyy) (corresponding to a Fourier transform in x and y) E(x, y, z) = E(z)e jkt r where k t = k x ˆx + k y ŷ Since x e jkxx = jk x e jkxx, the action of the curl operator is then E(x, y, z) = e ( jk jkt r t + ) z ẑ E(z)

13 3) Separate the components, curls Split the fields according to E = E t + ẑe z where E t is in the xy-plane, and ẑe z is the z-component. In the curls, we have ( jk t + ) z ẑ E(z) = jk t E }{{} t jk t ẑe z + }{{} z ẑ E t }{{} parallel to ẑ orthogonal to ẑ orthogonal to ẑ There is no term with Ez z, since ẑ ẑ = 0.

14 3) Separate the components, fluxes Split the material dyadics as ( ) ( ) ( ) ( ) ( ) Dt ɛtt ɛ = t ẑ Et ξtt ξ + t ẑ Ht ẑd z ẑɛ z ɛ zz ẑẑ ẑe z ẑξ z ξ zz ẑẑ ẑh z ɛ tt can be represented as a 2 2 matrix operating on xy components, ɛ t and ɛ z are vectors in the xy-plane, and ɛ zz is a scalar. The transverse components of the electric flux are (vector equation) D t = ɛ tt E t + ɛ t E z + ξ tt H t + ξ t H z and the z component is (scalar equation) D z = ɛ z E t + ɛ zz E z + ξ z H t + ξ zz H z

15 4) Eliminate the z components We first write Maxwell s equations using matrices ( ) ( ) ( ) ( ) 0 ẑ I E 0 jkt I E = z ẑ I 0 H jk t I 0 H ( ) ( ) ɛ ξ E jω ζ µ H The z components of these equations are (using ẑ (k t ẑe z ) = 0 and ẑ (k t E t ) = (ẑ k t ) E t ) ( ) ( ) ( ) 0 0 jẑ kt Et = 0 jẑ k t 0 H t ( ) ( ) ( ) ( ) ɛz ξ jω z Et ɛzz ξ jω zz Ez ζ z µ z H t ζ zz µ zz H z from which we solve for the z components of the fields: Ez «ɛzz ξ» 1 = zz 0 ω 1ẑ ««k t ɛz ξ H z ζ zz µ zz ω 1 z Et ẑ k t 0 ζ z µ z H t

16 4) Insert the z components The transverse part of Maxwell s equations are z ( ) 0 ẑ I ẑ I 0 ( Et H t ( ɛtt ξ jω tt ζ tt µ tt ) ( ) ( ) 0 jkt ẑ Ez = jk t ẑ 0 H z ) ( ) ) ( ) Et Ez H t ( ɛt ξ jω t ζ t µ t Inserting the expression for [E z, η 0 H z ] previously derived implies z ( ) 0 ẑ I ẑ I 0 ( Et H t ) ( ) ɛtt ξ = jω tt ζ tt µ tt where the matrix A is due to the z components. ( Et + jωa ) H z H t ( Et H t )

17 4) The A matrix The matrix A is a dyadic product [( 0 ω A = 1 ) ( )] k t ẑ ɛt ξ ω 1 t k t ẑ 0 ζ t µ t ( ) 1 [( ɛzz ξ zz 0 ω 1 ) ( )] ẑ k t ɛz ξ ζ zz µ zz ω 1 z ẑ k t 0 ζ z µ z In particular, when k t = 0 we have ( ) ( ) 1 ( ) ɛt ξ A = t ɛzz ξ zz ɛz ξ z ζ t µ t ζ zz µ zz ζ z µ z and for a uniaxial material with the axis of symmetry along the ẑ direction, where ɛ t = ξ t = ζ t = µ t = 0 and ɛ z = ξ z = ζ z = µ z = 0, we have A = 0.

18 4) The A matrix, oblique propagation Assuming k t 0 and isotropic media, implies (writing a = kt k 0 ẑ = c 0 ω 1 k t ẑ) [( 0 ω A = 1 ) ( )] k t ẑ ɛt ξ ω 1 t k t ẑ 0 ζ t µ t ( ) 1 [( ɛzz ξ zz 0 ω 1 ) ( )] ẑ k t ɛz ξ ζ zz µ zz ω 1 z ẑ k t 0 ζ z µ z ( 0 c 1 = 0 a ) ( ) ( ɛ c 1 c 1 0 a 0 0 µ 1 0 a ) c 1 0 a 0 = 1 ( µ 1 ) aa 0 c 2 0 ɛ 1 aa 0 The vector a has amplitude a = k t /k 0 = sin θ, and indicates the direction of a TE-polarized electric field.

19 5) Dynamical system Using ẑ (ẑ E t ) = E t for any transverse vector E t, it is seen that (0 ) ( ) ( ) ( ) ẑ I 0 ẑ I Et Et = I 0 ẑ I 0 H t ẑ H t The final form of Maxwell s equations is then! ««E t 0 ẑ I ɛtt ξ = jω tt I 0 + jωa z H t ẑ I 0 ζ tt µ tt 0 ẑ I or ( ) ( ) Et Et = jωw z H t ẑ H t ẑ This can also be written as (just a matter of convention) ( ) ( ) [ ( ) ] Et 0 ẑ I ɛtt ξ = jω tt + jωa z H t ẑ I 0 ζ tt µ tt ( Et H t E t H t ẑ )!

20 Outline 1 Introduction 2 Wave propagation in isotropic media 3 Wave propagation in bianisotropic media 4 Interpretation of the fundamental equation 5 Maxwell s equations for linear and circular polarization 6 Uniaxial and biaxial media 7 Chiral media 8 Gyrotropic media 9 Oblique propagation in biaxial media

21 Propagator If the material parameters are constant, the dynamical system ( ) ( ) Et Et = jωw z H t ẑ H t ẑ has the formal solution ( Et (z 2 ) H t (z 2 ) ẑ ) ( = exp ) jω(z 2 z 1 )W } {{ } P(z 1,z 2 ) ( ) Et (z 1 ) H t (z 1 ) ẑ The matrix P(z 1, z 2 ) is called a propagator. It maps the transverse fields at the plane z 1 to the plane z 2. The propagator exists even if W does depend on z, due to the linearity of the problem.

22 Eigenvalue problem For media with infinite extension in z, it is natural to look for solutions on the form ( ) ( ) Et (z) Et (0) = e jβz H t (z) ẑ H t (0) ẑ This implies z [E t(z), H t (z) ẑ] = jβ[e t (0), H t (0) ẑ]e jβz. We then get the algebraic eigenvalue problem ( ) β Et (0) ω H t (0) ẑ ( ) 0 ẑ I I 0 ( Et (0) = W H t (0) ẑ ] [( ɛtt ξ tt ζ tt µ tt ) A ) = ( I 0 0 ẑ I ) ( ) Et (0) H t (0) ẑ

23 Interpretation of the eigenvalue problem Since the wave is multiplied by the exponential factor e j(ωt βz) = e jω(t zβ/ω) = e jω(t z/c), we identify the eigenvalue as the inverse of the phase velocity β ω = 1 c = n c 0 which defines the refractive index of the wave. The transverse wave impedance is defined through the relation E t (0) = Z (H t (0) ẑ) where [E t (0), H t (0) ẑ] is the corresponding eigenvector.

24 Example: standard media An isotropic medium is described by ( ) ( ) ɛ ξ ɛi 0 = ζ µ 0 µi For k t = 0, we have A = 0 and W = ( ) 0 ẑ I I 0 [( ) ] ( ɛtt ξ tt I 0 A ζ tt µ tt 0 ẑ I = ( ) 0 µi = ɛi 0 ) 0 0 µ µ ɛ ɛ 0 0

25 Eigenvalues The eigenvalues are found from the characteristic equation det(λi W) = 0 λ 0 µ 0 λ µ 0 0 λ µ = 0 λ 0 µ ɛ 0 λ 0 = 0 0 λ µ ɛ λ 0 0 = ɛ λ λ µ = (λ 2 ɛµ) 2 0 ɛ 0 λ 0 0 ɛ λ 0 0 ɛ λ Thus, the eigenvalues are (with double multiplicity) β ω = λ = ± ɛµ

26 Eigenvectors Using that H t ẑ = ˆxH y ŷh x, the eigenvectors are (top row corresponds to β/ω = ɛµ, bottom row to β/ω = ɛµ): E x 0, H y 0 E x 0 H y, 0 0 E y 0 H x 0 E y 0 H x,, Z = E x H y = Z = E x H y = E y µ = H x ɛ = η E y µ = H x ɛ = η

27 For oblique propagation we have z ( Et 0 ẑ I I 0 ) = H t ẑ [ «jω ( = = jω ɛi c 2 Oblique propagation «ɛi 0 +jω 1 µ 1 aa 0 0 µi c 2 0 ɛ 1 aa 0 «{z } =A 0 µi c 2 0 µ 1 aa 0 0 ɛ 1 bb ] I 0 0 ẑ I «Et H t ẑ ) ( ) Et H t ẑ where a = kt k 0 ẑ is the direction of TE polarized electric field and b = k t /k 0 is the direction of TM polarized electric field, with amplitudes a = b = sin θ. Wave number and wave impedance are β = ω ɛµ c 2 0 sin 2 θ, Z = ωµ aa β a 2 + β bb ωɛ b 2

28 Propagator The propagator matrix for isotropic media can be represented as (after some calculations) ( ) Et (z 2 ) = H t (z 2 ) ẑ ( cos(βl)i j sin(βl)z j sin(βl)z 1 cos(βl)i ) ( Et (z 1 ) ) H t (z 1 ) ẑ which is recognized as the ABCD matrix for a homogeneous transmission line.

29 Conclusions for propagation in bianisotropic media We have found a way of computing time harmonic waves in any bianisotropic material. The transverse components (x and y) are crucial. There are four fundamental waves: two propagation directions and two polarizations. Each wave is described by The wave number β (eigenvalue of W) The polarization and wave impedance (eigenvector of W) The propagator P(z 1, z 2 ) = exp( jk 0 (z 2 z 1 )W) maps fields at z 1 to fields at z 2. A simple computer program is available on the course web for calculations of W.

30 Outline 1 Introduction 2 Wave propagation in isotropic media 3 Wave propagation in bianisotropic media 4 Interpretation of the fundamental equation 5 Maxwell s equations for linear and circular polarization 6 Uniaxial and biaxial media 7 Chiral media 8 Gyrotropic media 9 Oblique propagation in biaxial media

31 Linear and circular polarization To describe fields in the xy-plane, we can either use linear polarization ˆx and ŷ or circular polarization (since there is no propagation direction there is no connection to left or right) We typically write where ê + = ˆx jŷ and ê = ˆx + jŷ E = E x ˆx + E y ŷ = E + ê + + E ê E + = 1 2 (E x + je y ) E x = E + + E E = 1 2 (E x je y ) E y = 1 j (E + E )

32 Maxwell s equations in one dimension Maxwell s equations in one dimension are { ẑ z E = jωb ẑ z H = jωd

33 Maxwell s equations in one dimension Maxwell s equations in one dimension are { ẑ z E = jωb ẑ z H = jωd With linear polarization ( ) ( E H = Ex ) ( ˆx H yŷ or Eyŷ ) H x ˆx { z E x = jωb y { z E y = jωb x z H y = jωd x z H x = jωd y

34 Maxwell s equations in one dimension Maxwell s equations in one dimension are { ẑ z E = jωb ẑ z H = jωd With linear polarization ( ) ( E H = Ex ) ( ˆx H yŷ or Eyŷ ) H x ˆx { z E x = jωb y { z E y = jωb x z H y = jωd x With circular polarization ( E H (using ẑ (ˆx jŷ) = ±j(ˆx jŷ)) { z E + = ωb + ) = ( E+ (ˆx jŷ) H + (ˆx jŷ) z H x = jωd y ) ( or E (ˆx+jŷ)) { z E = ωb H (ˆx+jŷ) z H + = ωd + z H = ωd

35 Outline 1 Introduction 2 Wave propagation in isotropic media 3 Wave propagation in bianisotropic media 4 Interpretation of the fundamental equation 5 Maxwell s equations for linear and circular polarization 6 Uniaxial and biaxial media 7 Chiral media 8 Gyrotropic media 9 Oblique propagation in biaxial media

36 Anisotropic media Biaxial media are described by D x ɛ E x D y = 0 ɛ 2 0 E y D z 0 0 ɛ 3 E z If two parameters are equal, ɛ 2 = ɛ 3 = ɛ o, the material is uniaxial. Often the refractive index n = ɛ/ɛ 0 is tabulated. Examples (at λ = 590 nm, stronger anisotropy possible for composite materials): Uniaxial material n o n e Ice H 2 O Calcite CaCO Rutile TiO Sapphire Al 2 O Biaxial material n 1 n 2 n 3 Perovskite CaTiO Topaz Al 2 SiO 4 (F, OH)

37 Solutions in linear polarization Maxwell s equations for linear polarization are { z E x = jωµ 0 H y { z E y = jωµ 0 H x z H y = jωɛ 1 E x z H x = jωɛ 2 E y

38 Solutions in linear polarization Maxwell s equations for linear polarization are { z E x = jωµ 0 H y { z E y = jωµ 0 H x z H y = jωɛ 1 E x z H x = jωɛ 2 E y There is no coupling between the equations, and eliminating H x and H y implies 2 z E x = ω 2 µ 0 ɛ 1 E x 2 z E y = ω 2 µ 0 ɛ 2 E y

39 Solutions in linear polarization Maxwell s equations for linear polarization are { z E x = jωµ 0 H y { z E y = jωµ 0 H x z H y = jωɛ 1 E x z H x = jωɛ 2 E y There is no coupling between the equations, and eliminating H x and H y implies 2 z E x = ω 2 µ 0 ɛ 1 E x 2 z E y = ω 2 µ 0 ɛ 2 E y with the forward moving solutions E x (z) = Ae jk 1z E y (z) = Be jk 2z k 1 = ω µ 0 ɛ 1 = k 0 n 1 k 2 = ω µ 0 ɛ 2 = k 0 n 2

40 Propagated field The total propagated field at distance z = l is (with E(0) = Aˆx + Bŷ) E(z) = ˆxAe jk1l + ŷbe jk2l = [ˆxA ] + ŷbe j(k 1 k 2 )l } {{ } polarization The phase difference due to propagation is φ = (k 1 k 2 )l = (n 1 n 2 )k 0 l = (n 1 n 2 ) 2πl λ 0 e jk 1l where λ 0 is the vacuum wavelength. This changes the polarization state as the wave is propagating.

41 Example: linear to circular polarization Send in a wave with equal linear polarization in x and y E(0) = A(ˆx + ŷ) The propagated field becomes circularly polarized at z = l E(l) = A(ˆx + ŷe jφ )e jk1l = A(ˆx + jŷ)e jk 1l provided that φ = (n 1 n 2 ) 2πl λ 0 = π 2 (n 1 n 2 )l = λ 0 4 This is called a quarter-wavelength retarder (or quarter-wavelength plate).

42 General polarization change

43 Outline 1 Introduction 2 Wave propagation in isotropic media 3 Wave propagation in bianisotropic media 4 Interpretation of the fundamental equation 5 Maxwell s equations for linear and circular polarization 6 Uniaxial and biaxial media 7 Chiral media 8 Gyrotropic media 9 Oblique propagation in biaxial media

44 Chirality Some structures do not possess mirror symmetry: These are called chiral structures, and leads to birefringence in circular polarization.

45 Chiral media Chiral media are modeled by D = ɛe jχh B = µh + jχe This is a bi-isotropic material, with material matrix ( ɛ ) jχ jχ µ which is hermitian symmetric if ɛ, µ, and χ are real, and positive definite if ɛµ χ 2 > 0. However, usually all parameters are complex and depend on frequency. Typically, χ(ω) 0 as ω 0.

46 Solutions in circular polarization Maxwell s equations for circular polarization are (where ± refers to the polarization ˆx jŷ) { { z E + = ω(µh + + jχe + ) z E = ω(µh + jχe ) z H + = ω(ɛe + jχh + ) z H = ω(ɛe jχh )

47 Solutions in circular polarization Maxwell s equations for circular polarization are (where ± refers to the polarization ˆx jŷ) { { z E + = ω(µh + + jχe + ) z E = ω(µh + jχe ) z H + = ω(ɛe + jχh + ) z H = ω(ɛe jχh ) Defining c = 1/ µɛ, η = µ/ɛ, k = ω/c = ω µɛ, and the dimensionless parameter a = cχ = χ/ µɛ (so that a < 1) this is ( ) ( ) ( ) E± jka k E± = z ηh ± k jka ηh ±

48 Solutions in circular polarization Maxwell s equations for circular polarization are (where ± refers to the polarization ˆx jŷ) { { z E + = ω(µh + + jχe + ) z E = ω(µh + jχe ) z H + = ω(ɛe + jχh + ) z H = ω(ɛe jχh ) Defining c = 1/ µɛ, η = µ/ɛ, k = ω/c = ω µɛ, and the dimensionless parameter a = cχ = χ/ µɛ (so that a < 1) this is ( ) ( ) ( ) E± jka k E± = z ηh ± k jka ηh ± This is diagonalized by the change of variables E R± = 1 2 [E ± jηh ± ] E ± = E R± + E L± E L± = 1 2 [E ± + jηh ± ] H ± = 1 jη [E R± E L± ]

49 Solution, continued In the new variables we have (see Orfanidis for details) ( ) ( ) ( ) ER+ jk+ 0 ER+ = z E L+ 0 jk E L+ ( ) ( ) ( ) EL jk 0 EL = z 0 jk + where E R E R k ± = k(1 ± a) = ω( µɛ ± χ) = n ± k 0

50 Solution, continued In the new variables we have (see Orfanidis for details) ( ) ( ) ( ) ER+ jk+ 0 ER+ = z E L+ 0 jk E L+ ( ) ( ) ( ) EL jk 0 EL = z 0 jk + where E R This has the solutions E R k ± = k(1 ± a) = ω( µɛ ± χ) = n ± k 0 E + (z) = E R+ (z) + E L+ (z) = A + e jk +z }{{} forward RCP E (z) = E L (z) + E R (z) = A e jk z }{{} forward LCP RCP travel with k +, LCP with k, in either direction. + B + e jk z }{{} backward LCP + B e jk +z }{{} backward RCP

51 Natural rotation Start with a linearly polarized field at z = 0 E(0) = ˆx2A = ê + A + ê A

52 Natural rotation Start with a linearly polarized field at z = 0 E(0) = ˆx2A = ê + A + ê A Propagating the circular polarizations individually implies E(l) = ê + Ae jk +l + ê Ae jk l ( ) = ê + e j(k + k )l/2 + ê e j(k + k )l/2 Ae j(k ++k )l/2

53 Natural rotation Start with a linearly polarized field at z = 0 E(0) = ˆx2A = ê + A + ê A Propagating the circular polarizations individually implies E(l) = ê + Ae jk +l + ê Ae jk l ( ) = ê + e j(k + k )l/2 + ê e j(k + k )l/2 Ae j(k ++k )l/2 Denoting (k + k )l/2 = φ, we go back to the linear basis (remember that ê ± = ˆx jŷ) ê + e jφ + ê e jφ = ˆx(e jφ + e jφ ) + ŷ( je jφ + je jφ ) = 2(ˆx cos φ ŷ sin φ)

54 Natural rotation, continued Collecting the results, we see that a linearly polarized field propagates to become E(0) = ˆx2A E(l) = (ˆx cos φ ŷ sin φ)2ae j(k ++k )l/2 where φ = (k + k )l/2. Thus, the polarization is always linear, but the plane of polarization is rotating.

55 Natural rotation, continued Collecting the results, we see that a linearly polarized field propagates to become E(0) = ˆx2A E(l) = (ˆx cos φ ŷ sin φ)2ae j(k ++k )l/2 where φ = (k + k )l/2. Thus, the polarization is always linear, but the plane of polarization is rotating. If the propagation is in the negative z-direction, change the roles of k + and k.

56 Natural rotation, continued Collecting the results, we see that a linearly polarized field propagates to become E(0) = ˆx2A E(l) = (ˆx cos φ ŷ sin φ)2ae j(k ++k )l/2 where φ = (k + k )l/2. Thus, the polarization is always linear, but the plane of polarization is rotating. If the propagation is in the negative z-direction, change the roles of k + and k. This means a rotation by φ is canceled by a rotation φ if the wave is reflected and travels back the same distance.

57 Natural rotation, continued Collecting the results, we see that a linearly polarized field propagates to become E(0) = ˆx2A E(l) = (ˆx cos φ ŷ sin φ)2ae j(k ++k )l/2 where φ = (k + k )l/2. Thus, the polarization is always linear, but the plane of polarization is rotating. If the propagation is in the negative z-direction, change the roles of k + and k. This means a rotation by φ is canceled by a rotation φ if the wave is reflected and travels back the same distance. Optical activity is detected by rotation of transmission polarization, not reflection.

58 Quarter-wavelength retarder

59 Outline 1 Introduction 2 Wave propagation in isotropic media 3 Wave propagation in bianisotropic media 4 Interpretation of the fundamental equation 5 Maxwell s equations for linear and circular polarization 6 Uniaxial and biaxial media 7 Chiral media 8 Gyrotropic media 9 Oblique propagation in biaxial media

60 Gyrotropic media A gyroelectric medium (plasma) is modeled by D x ɛ 1 jɛ 2 0 E x D y = jɛ 2 ɛ 1 0 E y, B = µh D z 0 0 ɛ 3 E z and a gyromagnetic medium (ferromagnetic) is modeled by B x µ 1 jµ 2 0 H x B y = jµ 2 µ 1 0 H y, B z 0 0 µ 3 H z D = ɛe The medium is lossless and passive if ɛ 1,2,3 are real, and ɛ 1 > 0, ɛ 2 < ɛ 1, and ɛ 3 > 0 (correspondingly for µ 1,2,3 ). Usually complex and frequency dependent parameters.

61 Circularly polarized fields The gyrotropic medium is diagonalized by circularly polarized waves: D ± = (ɛ 1 ± ɛ 2 )E ± B ± = µh ± (gyroelectric) B ± = (µ 1 ± µ 2 )H ± D ± = ɛe ± (gyromagnetic) and we introduce (keeping only gyroelectric case) ɛ ± = ɛ 1 ± ɛ 2, k ± = ω µ µɛ ±, η ± = Use the corresponding change of variables as in the chiral case ɛ ± E R± = 1 2 [E ± jη ± H ± ] E ± = E R± + E L± E L± = 1 2 [E ± + jη ± H ± ] H ± = 1 jη ± [E R± E L± ]

62 Solution In the new variables we have (see Orfanidis for details) ( ) ( ) ( ) ER+ jk+ 0 ER+ = z 0 jk + z E L+ ( EL E R E L+ ) ( ) ( ) jk 0 EL = 0 jk E R

63 Solution In the new variables we have (see Orfanidis for details) ( ) ( ) ( ) ER+ jk+ 0 ER+ = z 0 jk + z E L+ ( EL E R This has the solutions E L+ ) ( ) ( ) jk 0 EL = 0 jk E + (z) = E R+ (z) + E L+ (z) = A + e jk +z }{{} forward RCP E (z) = E L (z) + E R (z) = A e jk z }{{} forward LCP E R + B + e jk +z }{{} backward LCP + B e jk z }{{} backward RCP Forward RCP and backward LCP travel with k +, forward LCP and backward RCP with k.

64 Faraday rotation Calculations corresponding to the natural rotation now leads to that the initial linear polarization is propagated to E(0) = ˆx2A E(l) = (ˆx cos φ ŷ sin φ)2ae j(k ++k )l/2 where φ = (k + k )l/2. Thus, we still have rotation of the polarization plane,

65 Faraday rotation Calculations corresponding to the natural rotation now leads to that the initial linear polarization is propagated to E(0) = ˆx2A E(l) = (ˆx cos φ ŷ sin φ)2ae j(k ++k )l/2 where φ = (k + k )l/2. Thus, we still have rotation of the polarization plane, but the roles of k + and k remain the same when propagating in the negative z direction.

66 Faraday rotation Calculations corresponding to the natural rotation now leads to that the initial linear polarization is propagated to E(0) = ˆx2A E(l) = (ˆx cos φ ŷ sin φ)2ae j(k ++k )l/2 where φ = (k + k )l/2. Thus, we still have rotation of the polarization plane, but the roles of k + and k remain the same when propagating in the negative z direction. Thus, a reflected wave experiences twice the rotation of the polarization plane.

67 Faraday rotation Calculations corresponding to the natural rotation now leads to that the initial linear polarization is propagated to E(0) = ˆx2A E(l) = (ˆx cos φ ŷ sin φ)2ae j(k ++k )l/2 where φ = (k + k )l/2. Thus, we still have rotation of the polarization plane, but the roles of k + and k remain the same when propagating in the negative z direction. Thus, a reflected wave experiences twice the rotation of the polarization plane. Faraday rotation is a non-reciprocal phenomenon, caused by the break in symmetry by an applied magnetic field.

68 Conclusions for propagation in birefringent media Different polarizations may generate different response from the material. Linear polarization is natural for uniaxial and biaxial media. Circular polarization is natural for chiral and gyrotropic media. In uniaxial and biaxial materials, an originally linear polarization may undergo transformations during propagation: linear circular linear circular etc. Natural rotation (chiral media) is restored upon reflection. Faraday rotation (gyrotropic media) is doubled upon reflection.

69 Outline 1 Introduction 2 Wave propagation in isotropic media 3 Wave propagation in bianisotropic media 4 Interpretation of the fundamental equation 5 Maxwell s equations for linear and circular polarization 6 Uniaxial and biaxial media 7 Chiral media 8 Gyrotropic media 9 Oblique propagation in biaxial media

70 Biaxial, non-magnetic media We consider the material relation (with B = µ 0 H) D x ɛ E x n E x D y = 0 ɛ 2 0 E y = ɛ 0 0 n E y D z 0 0 ɛ 3 E z 0 0 n 2 3 E z and look for plane wave solutions E(r) = Ee jk r, H(r) = He jk r where k = ˆkk. The effective refractive index N is defined from where k 0 = ω/c 0. k = Nk 0

71 Maxwell s equations for biaxial media Looking for plane wave solutions Maxwell s equations become k E = ωµ 0 H k H = ωɛ E Eliminating the magnetic field implies Using k = ˆkk = ˆkNk 0 implies ˆk (ˆk E) = }{{} =E t k (k E) = ω 2 µ 0 ɛ E 1 ɛ 0 N 2 ɛ E E 1 ɛ 0 N 2 ɛ E = ˆk(ˆk E) For a given propagation direction ˆk this is a homogeneous system, having non-trivial solutions E only if the determinant is zero.

72 Special geometry, ˆk = ˆx sin θ + ẑ cos θ ( 1 n2 1 N 2 E = E x ˆx + E z ẑ ) E x = (E x sin θ + E z cos θ) sin θ ( ) 1 n2 3 N 2 E z = (E x sin θ + E z cos θ) cos θ E = E y ŷ ( ) 1 n2 2 N 2 E y = 0

73 k-surfaces For each propagation direction ˆk, there exist two solutions. This corresponds to two k-surfaces: The explicit solutions are (extra-ordinary and ordinary wave) 1 N 2 eo = cos2 θ n sin2 θ n 2, 3 1 N 2 o = 1 n 2 2 Using k = N ω c 0 (cos θẑ + sin θˆx) the group velocity v g = ω/ k can be calculated. We show the energy propagates in this direction.

74 Energy propagation in the group velocity direction Orfanidis makes the detailed calculations, here a quick argument is given. Poynting s theorem for the plane wave is k 1 2 Re{E H } = ω 2 Re(D E + B H ) = ωw where w = 1 2 Re(D E + B H ) is the energy density.

75 Energy propagation in the group velocity direction Orfanidis makes the detailed calculations, here a quick argument is given. Poynting s theorem for the plane wave is k 1 2 Re{E H } = ω 2 Re(D E + B H ) = ωw where w = 1 2 Re(D E + B H ) is the energy density. Differentiating with respect to k x implies ˆx 1 2 Re{E H } = w ω k x = wv g,x where v g,x is the x-komponent of the group velocity

76 Energy propagation in the group velocity direction Orfanidis makes the detailed calculations, here a quick argument is given. Poynting s theorem for the plane wave is k 1 2 Re{E H } = ω 2 Re(D E + B H ) = ωw where w = 1 2 Re(D E + B H ) is the energy density. Differentiating with respect to k x implies ˆx 1 2 Re{E H } = w ω k x = wv g,x where v g,x is the x-komponent of the group velocity v g = ω 1 k = 2 Re{E H } w This demonstrates that the group velocity is the ratio of the Poynting vector over the energy density.

77 k-surfaces revisited The k-surfaces can be interpreted as level surfaces for a fixed ω: The group velocity v g = ω/ k is then orthogonal to this surface. Power is not propagating along k for the extraordinary wave!!

78 Conclusions for oblique propagation Biaxial media supports two typical waves, ordinary and extra-ordinary. An effective refractive index can be defined, as the solution to a matrix equation. The ordinary wave is polarized along an optical axis, and propagates as if in an isotropic medium. The extra-ordinary wave is polarized between two optical axes. The power of the extra-ordinary wave does not travel in the k-direction (direction of the phase).