Lecture 1: Basic Concepts of Polarized Light
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Lecture 1: Basic Concepts of Polarized Light The Sun in Polarized Light Magnetic Field Maps from Longitudinal Zeeman Effect Outline 1 2 3 4 5 6 The Sun in Polarized Light Fundamentals of Polarized Light Electromagnetic Waves Across Interfaces Fresnel Equations Brewster Angle Total Internal Reflection Christoph U. Keller, Utrecht University, C.U.Keller@uu.nl Lecture 1: Basic Concepts of Polarized Light 1 Christoph U. Keller, Utrecht University, C.U.Keller@uu.nl Lecture 1: Basic Concepts of Polarized Light 2 Fundamentals of Polarized Light Second Solar Spectrum from Scattering Polarization Electromagnetic Waves in Matter Maxwell’s equations ⇒ electromagnetic waves optics: interaction of electromagnetic waves with matter as described by material equations polarization of electromagnetic waves are integral part of optics Maxwell’s Equations in Matter ~ = 4πρ ∇·D ~ ~ − 1 ∂ D = 4π~j ∇×H c ∂t c ~ ~ + 1 ∂B = 0 ∇×E c ∂t ~ =0 ∇·B Christoph U. Keller, Utrecht University, C.U.Keller@uu.nl Lecture 1: Basic Concepts of Polarized Light 3 Christoph U. Keller, Utrecht University, C.U.Keller@uu.nl Symbols ~ D ρ ~ H c ~j ~ E ~ B t electric displacement electric charge density magnetic field speed of light in vacuum electric current density electric field magnetic induction time Lecture 1: Basic Concepts of Polarized Light 4 Linear Material Equations Wave Equation in Matter Symbols static, homogeneous medium with no net charges: ρ = 0 dielectric constant ~ = E ~ D ~ = µH ~ B for most materials: µ = 1 µ magnetic permeability combine Maxwell, material equations ⇒ differential equations for damped (vector) wave σ electrical conductivity ~j = σ E ~ Isotropic and Anisotropic Media isotropic media: and µ are scalars ~ − ∇2 H ~ ~ µ ∂ 2 H 4πµσ ∂ H − =0 c 2 ∂t 2 c 2 ∂t ~ interaction with matter almost always through E ~ are crucial but: at interfaces, boundary conditions for H anisotropy of material itself (e.g. crystals) external fields (e.g. Kerr effect) Lecture 1: Basic Concepts of Polarized Light ~ ~ µ ∂ 2 E 4πµσ ∂ E =0 − 2 2 2 ∂t c ∂t c damping controlled by conductivity σ ~ and H ~ are equivalent ⇒ sufficient to consider E ~ E anisotropic media: and µ are tensors of rank 2 isotropy of medium broken by Christoph U. Keller, Utrecht University, C.U.Keller@uu.nl ~ − ∇2 E 5 Christoph U. Keller, Utrecht University, C.U.Keller@uu.nl Lecture 1: Basic Concepts of Polarized Light 6 Complex Index of Refraction temporal derivatives ⇒ Helmholtz equation ω2µ 4πσ ~ 2~ ∇ E + 2 +i E =0 ω c dispersion relation between ~k and ω Plane-Wave Solutions ~ =E ~ 0 ei (~k ·~x −ωt ) Plane Vector Wave ansatz E ~k ~k |~k | ~x ω t ~0 E spatially and temporally constant wave vector normal to surfaces of constant phase wave number spatial location angular frequency (2π× frequency) time (generally complex) vector independent of time and space 2 ~k · ~k = ω µ + i 4πσ ω c2 complex index of refraction ω2 4πσ 2 ñ = µ + i , ~k · ~k = 2 ñ2 ω c split into real (n: index of refraction) and imaginary parts (k : extinction coefficient) ñ = n + ik ~ =E ~ 0 e−i (~k ·~x −ωt ) could also use E damping if ~k is complex ~ real electric field vector given by real part of E Christoph U. Keller, Utrecht University, C.U.Keller@uu.nl Lecture 1: Basic Concepts of Polarized Light 7 Christoph U. Keller, Utrecht University, C.U.Keller@uu.nl Lecture 1: Basic Concepts of Polarized Light 8 Energy Propagation in Isotropic Media Transverse Waves Poynting vector ~ energy through unit area perpendicular to S ~ per unit time |S|: ~ is direction of energy flow direction of S time-averaged Poynting vector given by D E ~ = c Re E ~0 × H ~∗ S 0 8π plane-wave solution must also fulfill Maxwell’s equations ~ ~ 0 · ~k = 0, H ~ 0 · ~k = 0, H ~ 0 = ñ k × E ~0 E µ |~k | Re real part of complex expression ∗ complex conjugate h.i time average isotropic media: electric, magnetic field vectors normal to wave vector ⇒ transverse waves ~ 0, H ~ 0 , and ~k orthogonal to each other, right-handed vector-triple E ~ 0 and H ~ 0 out of phase conductive medium ⇒ complex ñ, E ~ 0 have constant relationship ⇒ consider only E ~ ~ 0 and H E Christoph U. Keller, Utrecht University, C.U.Keller@uu.nl Lecture 1: Basic Concepts of Polarized Light ~ = c E ~ ×H ~ S 4π energy flow parallel to wave vector (in isotropic media) D E ~ = c |ñ| |E0 |2 S 8π µ 9 Quasi-Monochromatic Light Christoph U. Keller, Utrecht University, C.U.Keller@uu.nl ~k |~k | Lecture 1: Basic Concepts of Polarized Light Polarization of Quasi-Monochromatic Light monochromatic light: purely theoretical concept monochromatic light wave always fully polarized electric field vector for quasi-monochromatic plane wave is sum of electric field vectors of all monochromatic beams real life: light includes range of wavelengths ⇒ quasi-monochromatic light ~ (t) = E ~ 0 (t) ei (~k ·~x −ωt ) E quasi-monochromatic: superposition of mutually incoherent monochromatic light beams whose wavelengths vary in narrow range δλ around central wavelength λ0 can write this way because δλ λ0 measured intensity of quasi-monochromatic beam 1 tm −>∞ tm δλ 1 λ D E D E ~xE ~∗ + E ~yE ~ ∗ = lim E x y measurement of quasi-monochromatic light: integral over measurement time tm tm /2 −tm /2 ~ x (t)E ~ ∗ (t) + E ~ y (t)E ~ ∗ (t)dt E x y measured intensity independent of time quasi-monochromatic: frequency-dependent material properties (e.g. index of refraction) are constant within ∆λ slow: variations occur on time scales much longer than the mean period of the wave Lecture 1: Basic Concepts of Polarized Light Z h· · · i: averaging over measurement time tm amplitude, phase (slow) functions of time for given spatial location Christoph U. Keller, Utrecht University, C.U.Keller@uu.nl 10 11 Christoph U. Keller, Utrecht University, C.U.Keller@uu.nl Lecture 1: Basic Concepts of Polarized Light 12 Electromagnetic Waves Across Interfaces Introduction classical optics due to interfaces between 2 different media Polychromatic Light or White Light from Maxwell’s equations in integral form at interface from medium 1 to medium 2 ~2 − D ~ 1 · ~n = 4πΣ D ~2 − B ~ 1 · ~n = 0 B ~2 − E ~ 1 × ~n = 0 E ~2 − H ~ 1 × ~n = − 4π K ~ H c wavelength range comparable wavelength ( δλ λ ∼ 1) incoherent sum of quasi-monochromatic beams that have large variations in wavelength cannot write electric field vector in a plane-wave form must take into account frequency-dependent material characteristics intensity of polychromatic light is given by sum of intensities of constituting quasi-monochromatic beams ~n normal on interface, points from medium 1 to medium 2 Σ surface charge density on interface ~ surface current density on interface K Christoph U. Keller, Utrecht University, C.U.Keller@uu.nl Lecture 1: Basic Concepts of Polarized Light 13 Christoph U. Keller, Utrecht University, C.U.Keller@uu.nl Lecture 1: Basic Concepts of Polarized Light 14 Plane of Incidence Fields at Interfaces ~ = 0 for dielectrics Σ = 0 in general, K plane wave onto interface ~ =0 complex index of refraction includes effects of currents ⇒ K incident (i ), reflected (r ), and transmitted (t ) waves requirements at interface between media 1 and 2 ~2 − D ~ 1 · ~n = 0 D ~2 − B ~ 1 · ~n = 0 B ~2 − E ~ 1 × ~n = 0 E ~2 − H ~ 1 × ~n = 0 H ~ i,r ,t E ~ i,r ,t H interface normal ~n k z-axis spatial, temporal behavior at interface the same for all 3 waves (~k i · ~x )z=0 = (~k r · ~x )z=0 = (~k t · ~x )z=0 ~ and B ~ are continuous across interface normal components of D ~ and H ~ are continuous across interface tangential components of E Christoph U. Keller, Utrecht University, C.U.Keller@uu.nl Lecture 1: Basic Concepts of Polarized Light ~ i,r ,t ei (~k i,r ,t ·~x −ωt ) = E 0 c ~ i,r ,t ~ i,r ,t k ×E = µω valid for all ~x in interface ⇒ all 3 wave vectors in one plane, plane of incidence 15 Christoph U. Keller, Utrecht University, C.U.Keller@uu.nl Lecture 1: Basic Concepts of Polarized Light 16 Monochromatic Wave at Interface Snell’s Law ~ i,r ,t , B ~ i,r ,t = c ~k i,r ,t × E ~ i,r ,t ~ i,r ,t = c ~k i,r ,t × E H 0 0 0 0 ωµ ω boundary conditions for monochromatic plane wave: ~ i + ñ2 E ~ r − ñ2 E ~ t · ~n = 0 ñ12 E 0 1 0 2 0 ~k i × E ~ i + ~k r × E ~ r − ~k t × E ~ t · ~n = 0 0 0 0 ~i +E ~r −E ~ t × ~n = 0 E 0 0 0 1 1 ~i ~ i ~ r − 1 ~k t × E ~ t × ~n = 0 k × E0 + ~k r × E 0 0 µ1 µ1 µ2 spatial, temporal behavior the same for all three waves (~k i ·~x )z=0 = (~k r ·~x )z=0 = (~k t ·~x )z=0 ~ ω k = c ñ ω, c the same for all 3 waves Snell’s law ñ1 sin θi = ñ1 sin θr = ñ2 sin θt 4 equations are not independent only need to consider last two equations (tangential components ~ 0 and H ~ 0 are continuous) of E Christoph U. Keller, Utrecht University, C.U.Keller@uu.nl Lecture 1: Basic Concepts of Polarized Light 17 Christoph U. Keller, Utrecht University, C.U.Keller@uu.nl Lecture 1: Basic Concepts of Polarized Light 18 Electric Field Perpendicular to Plane of Incidence Two Special Cases Hi Er Ei Hr Et TM, p 1 2 TE, s Ht electric field parallel to plane of incidence ⇒ magnetic field is transverse to plane of incidence (TM) θ r = θi ratios of reflected and transmitted to incident wave amplitudes electric field particular (German: senkrecht) or transverse to plane of incidence (TE) general solution as (coherent) superposition of two cases choose direction of magnetic field vector such that Poynting vector parallel, same direction as corresponding wave vector Christoph U. Keller, Utrecht University, C.U.Keller@uu.nl Lecture 1: Basic Concepts of Polarized Light 19 rs = E0r E0i = ts = E0t E0i = Christoph U. Keller, Utrecht University, C.U.Keller@uu.nl ñ1 cos θi − ñ1 cos θi + µ1 µ2 ñ2 cos θt µ1 µ2 ñ2 cos θt 2ñ1 cos θi ñ1 cos θi + µµ12 ñ2 cos θt Lecture 1: Basic Concepts of Polarized Light 20 Summary of Fresnel Equations Electric Field in Plane of Incidence eliminate θt using Snell’s law ñ2 cos θt = Ei Er Hi q ñ22 − ñ12 sin2 θi for most materials µ1 /µ2 ≈ 1 Hr electric field amplitude transmission ts,p , reflection rs,p Et ts = Ht 2ñ1 cos θi q ñ1 cos θi + ñ22 − ñ12 sin2 θi tp = ñ22 cos θi ratios of reflected and transmitted to incident wave amplitudes rp = E0r E0i = tp = E0t E0i = ñ2 cos θi µµ12 − ñ1 cos θt rs = ñ2 cos θi µµ12 + ñ1 cos θt ñ1 cos θi 2ñ1 cos θi ñ2 cos θi µµ12 + ñ1 cos θt Christoph U. Keller, Utrecht University, C.U.Keller@uu.nl Lecture 1: Basic Concepts of Polarized Light ñ1 cos θi rp = ñ22 cos θi ñ22 cos θi 21 Christoph U. Keller, Utrecht University, C.U.Keller@uu.nl 2ñ1 ñ2 cos θi q + ñ1 ñ22 − ñ12 sin2 θi q − ñ22 − ñ12 sin2 θi q + ñ22 − ñ12 sin2 θi q − ñ1 ñ22 − ñ12 sin2 θi q + ñ1 ñ22 − ñ12 sin2 θi Lecture 1: Basic Concepts of Polarized Light 22 Other Form of Fresnel Equations using trigonometric identities Consequences of Fresnel Equations complex index of refraction ⇒ ts , tp , rs , rp (generally) complex ts = real indices ⇒ argument of square root can still be negative ⇒ complex ts , tp , rs , rp real indices, arguments of square roots positive (e.g. dielectric without total internal reflection) tp = rs = rp = therefore ts,p ≥ 0, real ⇒ incident and transmitted waves will have same phase therefore rs,p real, but become negative when n2 > n1 ⇒ negative ratios indicate phase change by 180◦ on reflection by medium with larger index of refraction 2 sin θt cos θi sin(θi +θt ) 2 sin θt cos θi sin(θi +θt ) cos(θi −θt ) sin(θi −θt ) − sin(θ i +θt ) tan(θi −θt ) tan(θi +θt ) refractive indices “hidden” in angle of transmitted wave, θt can always rework Fresnel equations such that only ratio of refractive indices appears ⇒ Fresnel equations do not depend on absolute values of indices can arbitrarily set index of air to 1; then only use indices of media measured relative to air Christoph U. Keller, Utrecht University, C.U.Keller@uu.nl Lecture 1: Basic Concepts of Polarized Light 23 Christoph U. Keller, Utrecht University, C.U.Keller@uu.nl Lecture 1: Basic Concepts of Polarized Light 24 Reflectivity Fresnel equations apply to electric field amplitude Relative Amplitudes for Arbitrary Polarization need to determine equations for intensity of waves D E ~ = c |ñ| |E0 |2 ~k time-averaged Poynting vector S ~ i , length E i , at angle α to electric field vector of incident wave E 0 0 plane of incidence 8π µ |~k | absolute value of complex index of refraction enters decompose into 2 components: parallel and perpendicular to interface i i E0,p = E0i cos α , E0,s = E0i sin α energy along wave vector and not along interface normal each wave propagates in different direction ⇒ consider energy of each wave passing through unit surface area on interface use Fresnel equations to obtain corresponding (complex) amplitudes of reflected and transmitted waves does not matter for reflected wave ⇒ ratio of reflected and incident intensities is independent of these two effects r ,t r ,t E0,p = (rp , tp ) E0i cos α , E0,s = (rs , ts ) E0i sin α relative intensity of reflected wave (reflectivity) r 2 E R = 0 2 E i 0 Christoph U. Keller, Utrecht University, C.U.Keller@uu.nl Lecture 1: Basic Concepts of Polarized Light 25 Christoph U. Keller, Utrecht University, C.U.Keller@uu.nl Lecture 1: Basic Concepts of Polarized Light 26 Transmissivity Brewster Angle intensity of transmitted wave by multiplying ratios of amplitudes squared with ratios of indices of refraction projected area on interface relative intensity of transmitted wave (transmissivity) 2 |ñ2 | cos θt E0t T = 2 |ñ1 | cos θi E i 0 ~ i at angle α to plane of for arbitrarily polarized light with E 0 incidence 2 2 2 rp = tan(θi −θt ) tan(θi +θt ) = 0 when θi + θt = π 2 corresponds to Brewster angle of incidence of tan θB = 2 R = |rp | cos α + |rs | sin α |ñ2 | cos θt 2 T = |tp | cos2 α + |ts |2 sin2 α |ñ1 | cos θi n2 n1 occurs when reflected wave is perpendicular to transmitted wave reflected light is completely s-polarized transmitted light is moderately polarized R + T = 1 for dielectrics, not for conducting, absorbing materials Christoph U. Keller, Utrecht University, C.U.Keller@uu.nl Lecture 1: Basic Concepts of Polarized Light 27 Christoph U. Keller, Utrecht University, C.U.Keller@uu.nl Lecture 1: Basic Concepts of Polarized Light 28 Total Internal Reflection (TIR) Phase Change on Total Internal Reflection TIR induces phase change that depends on polarization complex ratios: rs,p = |rs,p |eiδs,p Snell’s law: sin θt = n1 n2 phase change δ = δs − δp r 2 cos θi sin2 θi − nn21 δ tan = 2 sin2 θi sin θi wave from high-index medium into lower index medium (e.g. glass to air): n1 /n2 > 1 right-hand side > 1 for sin θi > relation valid between critical angle and grazing incidence n2 n1 for critical angle and grazing incidence, phase difference is zero all light is reflected in high-index medium ⇒ total internal reflection transmitted wave has complex phase angle ⇒ damped wave along interface Christoph U. Keller, Utrecht University, C.U.Keller@uu.nl Lecture 1: Basic Concepts of Polarized Light 29 Christoph U. Keller, Utrecht University, C.U.Keller@uu.nl Lecture 1: Basic Concepts of Polarized Light 30
