6.6. PolarizationPolarization andand crystalcrystal OpticsOptics Spatial evolution of a plane wave vector: helicoidal trajectory http://sar.kangwon.ac.kr/polsar/Tutorial/Part1_RadarPolarimetry/1_What_Is_Polarization.pdf The electric field may be represented in an orthonormal basis (x, y, z) defined so that the direction of propagation in z-axis. 증명!!! http://sar.kangwon.ac.kr/polsar/Tutorial/Part1_RadarPolarimetry/ Polarization ellipse Polarization ellipse The polarization ellipse shape may be characterized using 3 parameters : - A is called the ellipse amplitude and is determined from the ellipse axis as is the ellipse orientation and is defined as the angle between the ellipse major axis and x. Is the ellipse aperture, also called ellipticity, defined as τ Sense of rotation : Time-dependent rotation of The sense of rotation may then be related to the sign of the variable τ By convention, the sense of rotation is determined while looking in the direction of propagation. Right hand rotation : Left hand rotation : Right hand rotation : Left hand rotation : Quick estimation of a wave polarization state A wave polarization is completely defined by two parameters derived from the polarization ellipse - its orientation, - its ellipticity with sign(τ) indicating the sense of rotation Three cases may be discriminated from the knowledge of Æ the polarization is linear since τ = 0 Æ the orientation angle is given by Æ the polarization is circular, since τ = ±π/4 Æ the sense of rotation is given by sign(δ). Æ If δ<0, the polarization is right circular, whereas for δ>0 the polarization is left circular. Æ If δ<0, the polarization is right elliptic, whereas for δ>0 the polarization is left elliptic. Jones vector τ Jones vector τ A Jones vector can be formulated as a two-dimensional complex vector function of the polarization ellipse characteristics : This expression may be further developed Jones vectors for linear polarizations Jones vectors for circular/elliptical polarizations Jones vectors Æ Jones matrix Coordinate transform of Jones vector/matrix y’ y x’ θ x The Jones vector is given by ⎡ cosθθ sin ⎤ JRJ'()==θ ⎢ ⎥ J ⎣−sinθ cosθ ⎦ The Jones matrix T is similarly transformed into T’ TR'()= θ TR ()−θ (6.1-23) 증명!! TR=−()θ TR' ()θ Poincare sphere and Stokes parameters Æ A characterization method of the wave polarization by power measurements Given the Jones vector E of a given wave, we can create the hermitian product as follows if we consider the Pauli group of matrices where the parameters {g0, g1, g2, g3} receive the name of Stokes parameters. http://sar.kangwon.ac.kr/polsar/Tutorial/Part1_RadarPolarimetry/ Representation of Stokes vectors: The Poincaré sphere g3 2τ 2φ g1 g2 The Stokes vectors for the canonical polarization states 6.26.2 ReflectionReflection andand RefractionRefraction TE pol. TME pol. Development of the Fresnel Equations From Maxwell', s EM field theory we have the boundary conditions at the interface TE-case for the TEcase: EEir+= E t BBiirrttcosθθ−= cosco Bs θ The above conditions imply that the tangential GG components of both E and B are equal on both sides oftheinterface. We have also TM-case assumed thatμμit≅≅ μ0 , as is true for most dielectric materials. For the TM mode : −+EEiirrcosθθ cos =− E ttcos θ BBir+= B t Development of the Fresnel Equations ⎛⎞c nE Recall that E==v B⎜ ⎟ B ⇒B = TE-case ⎝⎠n c n2 Let n1 = refractive index of incident medium n2 = refractive index of refracting medium n1 For the TE mode : EEir+ = E t TM-case nE11i cosθθθirrtt− nE cos = n2 E cos n2 For the TM mode : n1 −Eiircosθθ+ EE cosrtt= − cos θ n1Eni + 1EnErt= 2 Development of the Fresnel Equations Eliminating Et from each set of equations TE-case and solving for the reflection coefficient we obtain : n2 E cosθθ− n cos TE case: r ==r it Eniicosθθ+ cos t n1 E ncosθθ− cos TM case: r = r = it Ei ncoscosθθi + t n where n = 2 n1 TM-case n2 We know that sinθθi = n sin t sin2 θ n nncosθ = 1− sin222θ =−nn 1i =− sin θ 1 t tin2 Development of the Fresnel Equations Substituting we obtain the Fresnel equations for reflection coefficients r : E cosθθ−−n22 sin TE-case TE case: r ==r ii E 22 n i cosθθii+−n sin 2 E nn222cosθθ−− sin TM case: r ==r ii E 222 i n cosθθii+−n sin n1 For the transmission coefficient t : E 2cosθ TE case : t ==ti n2 E 22 n ≡ i cosθθii+−n sin n1 TM-case En2cosθ TM case: t ==ti n E 222 2 i n cosθθii+ n − sin TE: t= r +1 These mean just the boundary conditions TntrM : = +1 n1 TIR TIR TIR TIR PowerPower :: Reflectance(R)Reflectance(R) andand Transmittance(T)Transmittance(T) The quantities r and t are ratios of electric field amplitudes. The ratios R and T are the ratios of reflected and transmitted powers, respectively, to the incident power : P P RT==r t PPii From conservation of energy : PPPirt=+ ⇒1 =+RT We can express the power in each of the fields in terms of the product of an irradiance and area : PIAPIAiiirrr== PIA ttt = ⇒ IAii= I rrA + IA tt IAiirrcosθθθ= IA cos + IAttcos Iiirrtcosθθθ= II cos+ cos t 1 1 11 ButInc= ε E 2 ⇒ nε cE 2 cosθε=+ncE22 cos θ ncE ε cos θ 2 0 0 2 10 0i i 2210 0rr 20 0 tt EnE22cosθθ E 2⎛⎞ cos E 2 ⇒=1 020rttr + = 0 +nRT⎜⎟ tt0 =+ E 22nEcosθθ E 2cos E 2 2 010iiii 0⎝⎠ii 0 R = rr* = r 22 EE2 ⎛⎞cosθθ ⎛⎞ cos 2 ⇒ RrTn= 00rttt== =n t ⎛ cosθ ⎞ ⎛ cosθ ⎞ 2 22⎜⎟ ⎜⎟ T = ⎜n t ⎟tt* = ⎜n t ⎟t E00ii⎝⎝cosθ ⎠⎠E i cosθi ⎜ ⎟ ⎜ ⎟ ⎝ cosθi ⎠ ⎝ cosθi ⎠ 6.36.3 OpticsOptics ofof anisotropicanisotropic mediamedia 6.36.3 OpticsOptics ofof anisotropicanisotropic mediamedia ε 0 1 Impermeability tensor : for principal axes ηi ≡=2 εiin *Note, impedance , for example, Determination of two normal modes (with refractive indices na and nb) An index ellipse is defined. Let’s start with E S For uniaxial case k-surface obtained from dispersion relation k3 k3 k3 k1 k1 k1 Optic axis Optic axis Optic axis k3 k3 k3 k2 k2 k2 k2 k2 k2 k1 k1 k1 Determine the wavenumbers k and indices of two normal modes u Determine the direction of polarization of two normal modes Z k θ θ D.D. Rays, Rays, wavefronts, wavefronts, and and energy energy transport transport k k surface Equi-frequency surface E.E. DoubleDouble refractionrefraction == BirefringenceBirefringence AIR 6.46.4 OpticalOptical activityactivity andand faradayfaraday effecteffect 6.56.5 OpticsOptics ofof liquidliquid crystalscrystals Principles of LCD Optics Operation of TN LCD 90 75 60 a (0V) a (5V) [deg.] 45 a (8V) b (0V) 30 director director b (5V) VLc = 0V (off) VLc = 5V (on) 15 b (8V) 0 0 0.2 0.4 0.6 0.8 1 normalized depth TNLC as a polarization rotator (6.1-23) 6.66.6 PolarizationPolarization devicesdevices .