Electromagnetic Wave Propagation Lecture 8: Propagation in birefringent media

Electromagnetic Wave Propagation Lecture 8: Propagation in birefringent media Daniel Sjöberg Department of Electrical and Information Technology September 27, 2012

Outline 1 Introduction 2 Maxwell s equations for linear and circular polarization 3 Uniaxial and biaxial media 4 Chiral media (optical activity) 5 Gyrotropic media 6 Oblique propagation in biaxial media 2 / 41

Key questions I I I 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? 4 / 41

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 5 / 41

The wave equation Eliminating the magnetic field implies ( 2 z 2 1 2 ) 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. 6 / 41

Wave propagation in bianisotropic media In a previous lecture we showed that in general bianisotropic media ( ) ( ) Et Et = jωw z H t ẑ H t ẑ where W = ( ) 0 ẑ I I 0 [( ) ] ( ) ɛtt ξ tt I 0 A(k ζ tt µ t ) tt 0 ẑ I Looking for solutions ( E t H t ẑ) e jk zz implies the eigenvalue problem ( ) ( ) k z Et Et = W ω H t ẑ H t ẑ where the eigenvalue k z /ω corresponds to the phase velocity and the eigenvector corresponds to the polarization of the wave. 7 / 41

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). 8 / 41

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 at this point no connection to left or right) We typically write ê + = ˆx jŷ and ê = ˆx + jŷ E = E x ˆx + E y ŷ = E + ê + + E ê where { E + = 1 2 (E x + je y ) E = 1 2 (E x je y ) { Ex = E + + E E y = 1 j (E + E ) 10 / 41

Maxwell s equations in one dimension Maxwell s equations in one dimension are { ẑ z E = jωb ẑ z H = jωd 11 / 41

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 11 / 41

Anisotropic media Biaxial media are described by D x ɛ 1 0 0 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 1.309 1.313 Calcite CaCO 3 1.658 1.486 Rutile TiO 2 2.616 2.903 Sapphire Al 2 O 3 1.768 1.760 Biaxial material n 1 n 2 n 3 Perovskite CaTiO 3 2.300 2.340 2.380 Topaz Al 2 SiO 4 (F, OH) 2 1.618 1.620 1.627 13 / 41

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 14 / 41

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 14 / 41

Propagated field The total propagated field at distance z = l is (with E(0) = Aˆx + Bŷ, A and B real) E(l) = 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. 15 / 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). 16 / 41

General polarization change 17 / 41

Polarizers A polarizer is a spatial filter which discriminates one of the polarizations. In the linear polarization case, we can see this as complex propagation constants k 1 = β 1 jα 1 k 2 = β 2 jα 2 so that the linear polarization E(0) = ˆxA + ŷb is propagated as E(l) = ˆxAe jk 1l + ŷbe jk 2l = (ˆxAe α 1l + ŷe α 2l e jφ )e jβ 1l In addition to the phase shift, the polarizations may suffer different attenuation. If α 2 α 1, then the y-polarization is virtually zero at z = l. 18 / 41

Polarizer 19 / 41

Chirality Some structures do not possess mirror symmetry: These are called chiral structures, and leads to birefringence in circular polarization. 21 / 41

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, that is, there are no chiral effects for very low frequencies. 22 / 41

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 (implies a < 1) this is ( ) ( ) ( ) E± jka k E± = z ηh ± k jka ηh ± 23 / 41

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 24 / 41

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 24 / 41

Natural rotation Start with a linearly polarized field at z = 0 E(0) = 2Aˆx = Aê + + Aê 25 / 41

Natural rotation Start with a linearly polarized field at z = 0 E(0) = 2Aˆx = Aê + + Aê Propagating the circular polarizations individually implies E(l) = Aê + e jk +l + Aê e 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) 25 / 41

Natural rotation, continued Collecting the results, we see that a linearly polarized field propagates to become E(0) = 2Aˆx 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. 26 / 41

Rotation of polarization plane 27 / 41

Quarter-wavelength retarder (Fig. 4.2.1 in Orfanidis) 28 / 41

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. 30 / 41

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 { { 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± ) ɛ ± 31 / 41

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 32 / 41

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. 32 / 41

Faraday rotation Calculations corresponding to the natural rotation now leads to that the initial linear polarization is propagated to E(0) = 2Aˆx 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. 33 / 41

Conclusions for propagation in birefringent media Different polarizations may generate different responses 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. 34 / 41

Biaxial, non-magnetic media We consider the material relation (with B = µ 0 H) D x ɛ 1 0 0 E x n 2 1 0 0 E x D y = 0 ɛ 2 0 E y = ɛ 0 0 n 2 2 0 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 36 / 41

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. 37 / 41

Special geometry, ˆk = ˆx sin θ + ẑ cos θ (Fig. 4.6.1 in Orfanidis) ( ) 1 n2 1 N 2 ( ) 1 n2 3 N 2 E = E x ˆx + E z ẑ E x = (E x sin θ + E z cos θ) sin }{{}}{{} θ ˆk E ˆx ˆk E z = (E x sin θ + E z cos θ) cos }{{}}{{} θ ˆk E ẑ ˆk E = E y ŷ ( ) 1 n2 2 N 2 E y = 0 38 / 41

k-surfaces For each propagation direction ˆk, there exist two solutions, corresponding to two level surfaces in k-space for fixed ω: The explicit solutions are (extra-ordinary and ordinary wave) 1 N 2 eo = cos2 θ n 2 1 + sin2 θ n 2, 3 1 N 2 o = 1 n 2 2 The group velocity ω k = P/w is orthogonal to the k-surfaces. Power is not propagating along k for the extraordinary wave!! 39 / 41

Birefringence in optics Calcite: no = 1.658, ne = 1.557. 40 / 41

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). 41 / 41