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Polarization and Crystal Optics * - Polarization ↔ Time Course of the Direction of E (R,T) - Polarization Affects:  Amount of Light Reflected at Material Interfaces

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Polarization and Crystal Optics * - Polarization ↔ Time Course of the Direction of E (R,T) - Polarization Affects:  Amount of Light Reflected at Material Interfaces EE 485, Winter 2004, Lih Y. Lin Chapter 6 Polarization and Crystal Optics * - Polarization ↔ Time course of the direction of E (r,t) - Polarization affects: Amount of light reflected at material interfaces. Absorption in some materials. Scattering. Refractive index (thus velocity) of anisotropic materials. Optically active materials to rotate polarization. 6.1 Polarization of Light Consider (z,t) = Re{A exp(jω(t − z )} (6.1-1) E c = + With complex envelope A Axxˆ Ay yˆ (6.1-2) Trace the endpoint of E (z,t) at each position z as a function of time. The polarization ellipse = ϕ = ϕ Ax ax exp( j x ), Ay ay exp( j y ) = + E (z,t) Exxˆ Ey yˆ (6.1-3) = a cos[ω(t − z )+ ϕ ] (6.1-4a) Ex x c x = a cos[ω(t − z )+ ϕ ] (6.1-4b) Ey y c y 2 E 2 EE Ex + y − ϕ x y = 2 ϕ 2 2 2cos sin (6.1-5) ax ay axay ay → An ellipse. The shape of the ellipse depends on and ϕ . The size ax a2 + a2 of the ellipse determines intensity I = x y (η: impedance of the medium). 2η 1 EE 485, Winter 2004, Lih Y. Lin Rotation direction (viewed from the direction towards which the wave is approaching): ϕ > ϕ y x : Clockwise rotation ϕ < ϕ y x : Counter-clockwise rotation • Linearly-polarized light = → ax 0 Ey only = → ay 0 Ex only Or ϕ = 0 or π a = y for ϕ = 0 Ey a Ex → x = − ay ϕ = π Ey E for ax = → ± ax ay 45 polarization • Circularly-polarized light 2 EE 485, Winter 2004, Lih Y. Lin = = ax ay a0 ϕ = π → Right circularly-polarized 2 ϕ = − π → Left circularly-polarized 2 B. Matrix Representation The Jones Vector * Complex envelopes for E (r,t): = ϕ = ϕ Ax ax exp( j x ), Ay ay exp( j y ) A J ≡ x Ay • Orthogonal polarizations ()= * + * = J1, J 2 A1x A2x A1y A2 y 0 (6.1-7) • Expansion of arbitrary polarization If J1 and J2 are normalized and orthogonal to each other, then an arbitrary = α + α polarization J 1J1 2J 2 α = ()α = () 1 J,J1 , 2 J, J 2 3 EE 485, Winter 2004, Lih Y. Lin 1 1 1 1 1 Example: = + 0 2 j 2 − j Matrix representation of polarization devices A A Input: 1x , Output: 2 x A1y A2 y A2x T T A1x A1x = 11 12 ≡ T (6.1-9) A2 y T21 T22 A1y A1y T: Jones matrix ⇒ = J 2 TJ1 (6.1-10) • Linear polarizers 1 0 T = (for xˆ polarizer) (6.1-11) 0 0 ⇒ = = A2x A1x , A2 y 0 • Wave retarders 1 0 T = (6.1-12) 0 exp(− jΓ) A A 2x = 1x − Γ A2 y exp( j )A1y → y component is delayed by a phase Γ. x: fast axis, y: slow axis. Examples: (1) Γ = π / 2 (quarter-wave retarder) 1 1 → 1 − j 4 EE 485, Winter 2004, Lih Y. Lin 1 1 → j 1 (2) Γ = π (half-wave retarder) 1 1 → 1 −1 1 1 → j − j • Polarization rotators cos θ − sin θ T = (6.1-13) sin θ cos θ cos θ cos(θ + θ) 1 → 1 θ θ + θ sin 1 sin( 1 ) • Cascaded polarization devices = ⋅ ⋅ ⋅ T TM ... T2 T1 Coordinate transformation J'= R(θ)J (6.1-14) cos θ sin θ R(θ) = (6.1-15) − sin θ cosθ T'= R(θ)TR(−θ) (6.1-16) 5 EE 485, Winter 2004, Lih Y. Lin T = R(−θ)T'R(θ) (6.1-17) Normal modes (of a polarization system) States of polarization that are not changed when the wave is transmitted through the optical system. TJ = µJ (6.1-19) µ: eigenvalue J: eigenvector If T is Hermitian, i.e., T12 = T21*, the normal modes are orthogonal to each other, and can be used as an expansion basis. The response to the system can be evaluated more easily if the input wave is decomposed into the two normal modes: = α + α J 1J1 2J 2 = α + α = α µ + α µ TJ T( 1J1 2J 2 ) 1 1J1 2 2J 2 Examples (Exercise 6.1-4): (a) The normal modes of the linear polarizer are linearly polarized waves. (b) The normal modes of the wave retarder are linearly polarized waves. (c) The normal modes of the polarization rotator are right and left circularly polarized waves. 6.2 Reflection and Refraction E E E = 1x = 2x = 3x J1 , J 2 , J3 E1y E2 y E3 y = J 2 tJ1, t : 2×2 Jones matrix for transmission = J3 rJ1, r : 2×2 Jones matrix for reflection t 0 r 0 t = x , r = x 0 t y 0 ry 6 EE 485, Winter 2004, Lih Y. Lin = = E2x tx E1x , E2 y t y E1y (6.2-2) E = r E , E = r E (6.2-3) 3x x 1x 3 y y 1y For transverse electric (TE) polarization, E = Exˆ, H = Hy n cosθ − n cosθ r = 1 1 2 2 (6.2-4) x θ + θ n1 cos 1 n2 cos 2 t = 1+ r (6.2-5) x x For transverse magnetic (TM) polarization, E = Ey, H = Hxˆ n cos θ − n cos θ r = 2 1 1 2 (6.2-6) y θ + θ n2 cos 1 n1 cos 2 = n1 + t y (1 ry ) (6.2-7) n2 rx, tx, ry, ty can be complex numbers → = ϕ = ϕ rx | rx | exp( j x ), ry | ry | exp( j y ) TE polarization • < External reflection (n1 n2 ) ϕ = π x n − n | r |= 2 1 at θ = 0º (θ = 0º) x + 1 2 n2 n1 7 EE 485, Winter 2004, Lih Y. Lin • > Internal reflection (n1 n2 ) θ θ ϕ = For 1 < c, x 0 n − n At θ = 0º (θ = 0º), r = 1 2 1 2 x + n1 n2 θ θ = At 1 = c, rx 1 θ θ = For 1 > c, total internal reflection, | rx | 1, ϕ sin 2 θ − sin 2 θ tan x = 1 c (6.2-9) θ 2 cos 1 TM polarization • < External reflection (n1 n2 ) ry is real n − n At θ = 0º (θ = 0º), r = 2 1 is positive 1 2 y + n2 n1 θ θ = −1 n2 = θ At 1 = B tan , ry 0 ( B: Brewster angle) n1 θ θ ϕ = π For 1 > B, ry becomes negative, y θ ϕ = π At 1 = 90º, ry = -1, y 8 EE 485, Winter 2004, Lih Y. Lin • > Internal reflection (n1 n2 ) n − n At θ = 0º (θ = 0º), r is negative, | r |= 2 1 , ϕ = π 1 2 y y + y n2 n1 θ θ θ ϕ = π For 1 < B, |ry| decreases with , y θ θ = At 1 = B, ry 0 θ θ θ ϕ = For 1 > B, ry becomes positive and increases with , y 0 θ θ = At 1 = c, ry 1 θ θ = For 1 > c, total internal reflection, | ry | 1, ϕ sin 2 θ − sin 2 θ tan x = 1 c (6.2-11) θ 2 θ 2 cos 1 sin c 9 EE 485, Winter 2004, Lih Y. Lin Power reflectance and transmittance R =| r 2 | (6.2-12) T = 1− R (6.2-13) At normal incidence, for both TE and TM, 2 n − n R = 1 2 + n1 n2 Examples: (a) Glass (n = 1.5) and Air (n = 1) interface R = 0.04 for normal incidence (b) GaAs (n = 3.6) and Air (n = 1) interface R = 0.32 for normal incidence 10 EE 485, Winter 2004, Lih Y. Lin 6.3 Optics of Anisotropic Media A. Refractive Indices Permittivity tensor = ε Di ∑ ij E j , i, j = 1, 2, 3 (6.3-1) j {ε }≡ ij ε: Electric permittivity tensor (3 × 3) D = εE ε ε = ε is symmetrical, ij ji → only 6 independent numbers. Principal axes and principal refractive indices Choose a coordinate system such that ε 1 0 0 = ε ε 0 2 0 ε 0 0 3 = ε = ε = ε D1 1E1, D2 2 E2 , D3 3E3 (6.3-2) → This coordinate system defines the “principal axes” of the crystal. Principal refractive indices: ε ε ε n = 1 , n = 2 , n = 3 (6.3-3) 1 ε 2 ε 3 ε 0 0 0 Isotropic, uniaxial, and biaxial crystals = = Isotropic: n1 n2 n3 = = Uniaxial: n1 n2 no (ordinary index, ordinary axes) = n3 ne (extraordinary index, extraordinary axis) > < Positive uniaxial: ne no Negative uniaxial: ne no ≠ ≠ Biaxial: n1 n2 n3 The index ellipsoid x2 x2 x2 1 + 2 + 3 = 2 2 2 1 n1 n2 n3 (6.3-7) Direction of D determines the refractive index from the index ellipsoid. 11 EE 485, Winter 2004, Lih Y. Lin B. Propagation Along a Principal Axis Consider a plane wave propagating along z-direction. Normal modes = = ε For E E1xˆ, D 1E1xˆ , = → = c0 k n1k0 c n1 = = ε For E E2yˆ, D 1E2yˆ , = → = c0 k n2k0 c n2 Polarization along an arbitrary direction Decompose the electric field: = + E E1xˆ E2yˆ The x- and y-components travel with different speeds. The phase retardation ϕ = − (n2 n1)k0d → Linearly-polarized wave becomes an elliptically-polarized wave. The crystal acts like a wave retarder. C. Propagation in an Arbitrary Direction 12 EE 485, Winter 2004, Lih Y. Lin Determine the polarizations and refractive indices na and nb of the normal modes of a wave traveling in uˆ direction. (1) Draw a plane passing thru the origin of the index ellipsoid, normal to uˆ . The intersection of the plane with the ellipsoid is an ellipse, called the index ellipse.
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