Electromagnetic Waves Across Interfaces

Lecture 1: Foundations of Optics Outline 1 Electromagnetic Waves 2 Material Properties 3 Electromagnetic Waves Across Interfaces 4 Fresnel Equations 5 Brewster Angle 6 Total Internal Reflection Christoph U. Keller, Utrecht University, C.U.Keller@uu.nl Lecture 1: Foundations of Optics 1

Electromagnetic Waves 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 D = 4πρ H 1 D = 4π c t c j E + 1 B = 0 c t B = 0 Symbols D electric displacement ρ electric charge density H magnetic field c speed of light in vacuum j electric current density E electric field B magnetic induction t time Christoph U. Keller, Utrecht University, C.U.Keller@uu.nl Lecture 1: Foundations of Optics 2

Linear Material Equations D = ɛe B = µ H j = σe Symbols ɛ dielectric constant µ magnetic permeability σ electrical conductivity Isotropic and Anisotropic Media isotropic media: ɛ and µ are scalars anisotropic media: ɛ and µ are tensors of rank 2 isotropy of medium broken by anisotropy of material itself (e.g. crystals) external fields (e.g. Kerr effect) Christoph U. Keller, Utrecht University, C.U.Keller@uu.nl Lecture 1: Foundations of Optics 3

Wave Equation in Matter static, homogeneous medium with no net charges: ρ = 0 for most materials: µ = 1 combine Maxwell, material equations differential equations for damped (vector) wave 2 E µɛ 2 E c 2 t 2 4πµσ c 2 E t = 0 2 H µɛ 2 H c 2 t 2 4πµσ H c 2 t = 0 damping controlled by conductivity σ E and H are equivalent sufficient to consider E interaction with matter almost always through E but: at interfaces, boundary conditions for H are crucial Christoph U. Keller, Utrecht University, C.U.Keller@uu.nl Lecture 1: Foundations of Optics 4

Plane-Wave Solutions Plane Vector Wave ansatz E = E 0 e i( k x ωt) k spatially and temporally constant wave vector k normal to surfaces of constant phase k wave number x spatial location ω angular frequency (2π frequency) t time E 0 (generally complex) vector independent of time and space could also use E = E 0 e i( k x ωt) 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: Foundations of Optics 5

Complex Index of Refraction temporal derivatives Helmholtz equation 2 E + ω 2 µ c 2 dispersion relation between k and ω k k = ω 2 µ complex index of refraction ( ɛ + i 4πσ ) E = 0 ω c 2 ( ɛ + i 4πσ ) ω ( ñ 2 = µ ɛ + i 4πσ ), k ω 2 k = ω c 2 ñ2 split into real (n: index of refraction) and imaginary parts (k: extinction coefficient) ñ = n + ik Christoph U. Keller, Utrecht University, C.U.Keller@uu.nl Lecture 1: Foundations of Optics 6

Transverse Waves plane-wave solution must also fulfill Maxwell s equations E 0 k = 0, H0 k = 0, H0 = ñ k µ k E 0 isotropic media: electric, magnetic field vectors normal to wave vector transverse waves E 0, H 0, and k orthogonal to each other, right-handed vector-triple conductive medium complex ñ, E 0 and H 0 out of phase E 0 and H 0 have constant relationship consider only E Christoph U. Keller, Utrecht University, C.U.Keller@uu.nl Lecture 1: Foundations of Optics 7

Energy Propagation in Isotropic Media Poynting vector S = c ( ) E H 4π S : energy through unit area perpendicular to S per unit time direction of S is direction of energy flow time-averaged Poynting vector given by S = c ( 8π Re E0 H ) 0 Re real part of complex expression complex conjugate. time average energy flow parallel to wave vector (in isotropic media) S = c ñ 8π µ E 0 2 k k Christoph U. Keller, Utrecht University, C.U.Keller@uu.nl Lecture 1: Foundations of Optics 8

Quasi-Monochromatic Light monochromatic light: purely theoretical concept monochromatic light wave always fully polarized real life: light includes range of wavelengths quasi-monochromatic light quasi-monochromatic: superposition of mutually incoherent monochromatic light beams whose wavelengths vary in narrow range δλ around central wavelength λ 0 δλ λ 1 measurement of quasi-monochromatic light: integral over measurement time t m amplitude, phase (slow) functions of time for given spatial location slow: variations occur on time scales much longer than the mean period of the wave Christoph U. Keller, Utrecht University, C.U.Keller@uu.nl Lecture 1: Foundations of Optics 9

Polarization of Quasi-Monochromatic Light electric field vector for quasi-monochromatic plane wave is sum of electric field vectors of all monochromatic beams E (t) = E 0 (t) e i( k x ωt) can write this way because δλ λ 0 measured intensity of quasi-monochromatic beam Ex E x + Ey E y = lim t m > 1 t m tm/2 t m/2 : averaging over measurement time t m measured intensity independent of time E x (t) E x (t) + E y (t) E y (t)dt quasi-monochromatic: frequency-dependent material properties (e.g. index of refraction) are constant within λ Christoph U. Keller, Utrecht University, C.U.Keller@uu.nl Lecture 1: Foundations of Optics 10

Polychromatic Light or White Light 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 Christoph U. Keller, Utrecht University, C.U.Keller@uu.nl Lecture 1: Foundations of Optics 11

Material Properties Index of Refraction complex index of refraction ( ñ 2 = µ ɛ + i 4πσ ), k ω 2 k = ω c 2 ñ2 no electrical conductivity real index of refraction dielectric materials: real index of refraction conducting materials (metal): complex index of refraction index of refraction depends on wavelength (dispersion) index of refraction depends on temperature index of refraction roughly proportional to density Christoph U. Keller, Utrecht University, C.U.Keller@uu.nl Lecture 1: Foundations of Optics 12

Glass Dispersion http://en.wikipedia.org/wiki/file:dispersion-curve.png Christoph U. Keller, Utrecht University, C.U.Keller@uu.nl Lecture 1: Foundations of Optics 13

Wavelength Dependence of Index of Refraction tabulated by glass manufacturer various approximations to express wavelength dependence with a few parameters typically index increases with decreasing wavelength Abbé number: ν d = n d 1 n F n C n d : index of refraction at Fraunhofer d line (587.6 nm) n F : index of refraction at Fraunhofer F line (486.1 nm) n C : index of refraction at Fraunhofer C line (656.3 nm) low dispersion materials have high values of ν d Abbe diagram: ν d vs n d Christoph U. Keller, Utrecht University, C.U.Keller@uu.nl Lecture 1: Foundations of Optics 14

Glasses ν d 2.05 95 90 85 80 75 70 65 60 55 50 45 40 35 30 25 20 2.05 n d 2.00 68 2.00 n d Abbe-Diagram n d ν d 1.95 Description of Symbols N- or P-glass 1.95 Lead containing glass 66 1.90 N-glass or lead containing glass 46A 67 1.90 Glass suitable for Precision Molding Fused Silica 31A LASF 1.85 9 57 1.85 1.80 44 21 41 33 40 51 50 47 43 45 SF 6 56A 11 1.80 1.75 1.70 1.65 1.60 1.55 1.50 52 A 51 PK PSK 34 LAF 33A 37 7 35 2 34 10 KZFS 8 8 LAK 64 14 9 35 BASF 12 10 2 7 22 5 51 KZFS5 2 21 SSK KZFS11 BAF 16 2 2 60 8 KZFS4 53A 4 2 52 14 4 5 5 SK F 58A 57 4 5 1 4 11 BALF KZFS2 LF 3 2 BAK 5 1 LLF KF 5 7 7 9 BK K 10 10 ZK7 14 4 10 69 1 15 8 5 1.75 1.70 1.65 1.60 1.55 1.50 1.45 51A FK 5 FS 1.45 August 2010 95 90 85 80 75 70 65 60 55 50 45 40 35 30 25 20 ν d Christoph U. Keller, Utrecht University, C.U.Keller@uu.nl Lecture 1: Foundations of Optics 15

Glass Ingredients http://glassproperties.com/abbe_number Christoph U. Keller, Utrecht University, C.U.Keller@uu.nl Lecture 1: Foundations of Optics 16

Internal Transmission internal transmission per cm typically strong absorption in the blue and UV almost all glass absorbs above 2 µm Christoph U. Keller, Utrecht University, C.U.Keller@uu.nl Lecture 1: Foundations of Optics 17

Metal Reflectivity http://commons.wikimedia.org/wiki/file:image-metal-reflectance.png Christoph U. Keller, Utrecht University, C.U.Keller@uu.nl Lecture 1: Foundations of Optics 18

Electromagnetic Waves Across Interfaces Introduction classical optics due to interfaces between 2 different media from Maxwell s equations in integral form at interface from medium 1 to medium 2 ( D2 D 1 ) n = 4πΣ ( B2 B 1 ) n = 0 ( E2 E 1 ) n = 0 ( H2 H 1 ) n = 4π c K n normal on interface, points from medium 1 to medium 2 Σ surface charge density on interface K surface current density on interface Christoph U. Keller, Utrecht University, C.U.Keller@uu.nl Lecture 1: Foundations of Optics 19

Fields at Interfaces Σ = 0 in general, K = 0 for dielectrics complex index of refraction includes effects of currents K = 0 requirements at interface between media 1 and 2 ( D2 D 1 ) n = 0 ( B2 B 1 ) n = 0 ( E2 E 1 ) n = 0 ( H2 H 1 ) n = 0 normal components of D and B are continuous across interface tangential components of E and H are continuous across interface Christoph U. Keller, Utrecht University, C.U.Keller@uu.nl Lecture 1: Foundations of Optics 20

Plane of Incidence plane wave onto interface incident ( i ), reflected ( r ), and transmitted ( t ) waves E i,r,t = E i,r,t 0 e i( k i,r,t x ωt) H i,r,t = c µω k i,r,t E i,r,t interface normal n 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 valid for all x in interface all 3 wave vectors in one plane, plane of incidence Christoph U. Keller, Utrecht University, C.U.Keller@uu.nl Lecture 1: Foundations of Optics 21

Snell s Law spatial, temporal behavior the same for all three waves ( k i x) z=0 = ( k r x) z=0 = ( k t x) z=0 k = ω c ñ ω, c the same for all 3 waves Snell s law ñ 1 sin θ i = ñ 1 sin θ r = ñ 2 sin θ t Christoph U. Keller, Utrecht University, C.U.Keller@uu.nl Lecture 1: Foundations of Optics 22

Monochromatic Wave at Interface H i,r,t 0 = c ωµ k i,r,t E i,r,t 0, B i,r,t 0 = c ω k i,r,t E i,r,t 0 boundary conditions for monochromatic plane wave: ( ñ1 2 E 0 i + ñ2 1E0 r ñ2 ) 2E0 t n = 0 ( k i E 0 i + k r E 0 r k t E ) 0 t n = 0 ( E i 0 + E 0 r E 0) t n = 0 ( 1 k i E µ 0 i + 1 k r E 1 µ 0 r 1 ) k t E 1 µ 0 t n = 0 2 4 equations are not independent only need to consider last two equations (tangential components of E 0 and H 0 are continuous) Christoph U. Keller, Utrecht University, C.U.Keller@uu.nl Lecture 1: Foundations of Optics 23

Two Special Cases TM, p TE, s 1 electric field parallel to plane of incidence magnetic field is transverse to plane of incidence (TM) 2 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: Foundations of Optics 24

Electric Field Perpendicular to Plane of Incidence E i E r H i H r E t H t θ r = θ i ratios of reflected and transmitted to incident wave amplitudes r s = E r 0 E i 0 t s = E t 0 E i 0 = ñ1 cos θ i µ 1 µ 2 ñ 2 cos θ t ñ 1 cos θ i + µ 1 µ 2 ñ 2 cos θ t = 2ñ 1 cos θ i ñ 1 cos θ i + µ 1 µ 2 ñ 2 cos θ t Christoph U. Keller, Utrecht University, C.U.Keller@uu.nl Lecture 1: Foundations of Optics 25

Electric Field in Plane of Incidence E i E r H i H r E t H t ratios of reflected and transmitted to incident wave amplitudes r p = E r 0 E i 0 = ñ2 cos θ i µ 1 µ 2 ñ 1 cos θ t ñ 2 cos θ i µ 1 µ 2 + ñ 1 cos θ t t p = E t 0 E i 0 = 2ñ 1 cos θ i µ ñ 2 cos θ 1 i µ 2 + ñ 1 cos θ t Christoph U. Keller, Utrecht University, C.U.Keller@uu.nl Lecture 1: Foundations of Optics 26

Summary of Fresnel Equations eliminate θ t using Snell s law ñ 2 cos θ t = for most materials µ 1 /µ 2 1 ñ 2 2 ñ2 1 sin2 θ i electric field amplitude transmission t s,p, reflection r s,p t s = t p = r s = r p = 2ñ 1 cos θ i ñ 1 cos θ i + ñ2 2 ñ2 1 sin2 θ i 2ñ 1 ñ 2 cos θ i ñ2 2 cos θ i + ñ 1 ñ2 2 ñ2 1 sin2 θ i ñ 1 cos θ i ñ2 2 ñ2 1 sin2 θ i ñ 1 cos θ i + ñ2 2 ñ2 1 sin2 θ i ñ 2 2 cos θ i ñ 1 ñ 2 2 ñ2 1 sin2 θ i ñ 2 2 cos θ i + ñ 1 ñ 2 2 ñ2 1 sin2 θ i Christoph U. Keller, Utrecht University, C.U.Keller@uu.nl Lecture 1: Foundations of Optics 27

Consequences of Fresnel Equations complex index of refraction t s, t p, r s, r p (generally) complex real indices argument of square root can still be negative complex t s, t p, r s, r p real indices, arguments of square roots positive (e.g. dielectric without total internal reflection) therefore t s,p 0, real incident and transmitted waves will have same phase therefore r s,p real, but become negative when n 2 > n 1 negative ratios indicate phase change by 180 on reflection by medium with larger index of refraction Christoph U. Keller, Utrecht University, C.U.Keller@uu.nl Lecture 1: Foundations of Optics 28

Other Form of Fresnel Equations using trigonometric identities t s = t p = 2 sin θ t cos θ i sin(θ i +θ t ) 2 sin θ t cos θ i sin(θ i +θ t ) cos(θ i θ t ) r s = sin(θ i θ t ) sin(θ i +θ t ) r p = tan(θ i θ t ) tan(θ i +θ t ) refractive indices hidden in angle of transmitted wave, θ t 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: Foundations of Optics 29

Relative Amplitudes for Arbitrary Polarization electric field vector of incident wave E 0 i, length E 0 i, at angle α to plane of incidence decompose into 2 components: parallel and perpendicular to interface E i 0,p = E i 0 cos α, E i 0,s = E i 0 sin α use Fresnel equations to obtain corresponding (complex) amplitudes of reflected and transmitted waves E r,t 0,p = (r p, t p ) E0 i r,t cos α, E0,s = (r s, t s ) E0 i sin α Christoph U. Keller, Utrecht University, C.U.Keller@uu.nl Lecture 1: Foundations of Optics 30

Reflectivity Fresnel equations apply to electric field amplitude need to determine equations for intensity of waves time-averaged Poynting vector S = c ñ 8π µ E 0 2 k k absolute value of complex index of refraction enters 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 does not matter for reflected wave ratio of reflected and incident intensities is independent of these two effects relative intensity of reflected wave (reflectivity) R = E0 r 2 E0 i 2 Christoph U. Keller, Utrecht University, C.U.Keller@uu.nl Lecture 1: Foundations of Optics 31

Transmissivity transmitted intensity: multiplying amplitude squared ratios with ratios of indices of refraction projected area on interface relative intensity of transmitted wave (transmissivity) T = ñ 2 cos θ t E0 t ñ 1 cos θ i E i 2 0 arbitrarily polarized light with E 0 i at angle α to plane of incidence R = r p 2 cos 2 α + r s 2 sin 2 α T = ñ 2 cos θ ( ) t t p 2 cos 2 α + t s 2 sin 2 α ñ 1 cos θ i R + T = 1 for dielectrics, not for conducting, absorbing materials 2 Christoph U. Keller, Utrecht University, C.U.Keller@uu.nl Lecture 1: Foundations of Optics 32

Brewster Angle r p = tan(θ i θ t ) tan(θ i +θ t ) = 0 when θ i + θ t = π 2 corresponds to Brewster angle of incidence of tan θ B = n 2 n 1 occurs when reflected wave is perpendicular to transmitted wave reflected light is completely s-polarized transmitted light is moderately polarized Christoph U. Keller, Utrecht University, C.U.Keller@uu.nl Lecture 1: Foundations of Optics 33

Total Internal Reflection (TIR) Snell s law: sin θ t = n 1 n 2 sin θ i wave from high-index medium into lower index medium (e.g. glass to air): n 1 /n 2 > 1 right-hand side > 1 for sin θ i > n 2 n 1 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: Foundations of Optics 34

Phase Change on Total Internal Reflection (TIR) TIR induces phase change that depends on polarization complex ratios: r s,p = r s,p e iδs,p phase change δ = δ s δ p ( ) 2 tan δ cos θ i sin 2 θ i n2 2 = n 1 sin 2 θ i relation valid between critical angle and grazing incidence for critical angle and grazing incidence, phase difference is zero Christoph U. Keller, Utrecht University, C.U.Keller@uu.nl Lecture 1: Foundations of Optics 35