Fresnel Equations

Consider reflection and transmission of light at dielectric/dielectric boundary

Calculate reflection and transmission coefficients, R and T, as a function of incident light polarisation and angle of incidence using EM boundary conditions s-polarisation p-polarisation

n1 = √ε1 θi θr µ1 = 1

n = √ε 2 2 θ µ = 1 t 2 s-polarisation E perpendicular to plane of incidence p-polarisation E parallel to plane of incidence

Fresnel Equations

Snell’s Law

Boundary conditions apply across the entire, flat interface (z = 0) Incident, reflected and transmitted waves are like

i(ωt - kI.r) EI = (ey cosθi + ez sinθi) EoI e i(ωt - kR.r) ε kI ER = (-ey cosθr + ez sinθr) EoR e n1 = √ 1 θ θ i r kR i(ωt - kT.r) µ = 1 ET = (ey cosθt + ez sinθt) EoT e 1

y To satisfy BC (k . r) = (k . r) = (k . r) I z=0 R z=0 T z=0 ε n2 = √ 2 k (1) wave vectors lie in single plane z θt T µ2 = 1 (2) projection of wave vectors on xy plane is same

From (1) θi = θr ω ω θ θ θ µ ε µ ε From (2) kI sin i = kR sin r = kT sin t kI = kR = c 1 1 kT = c 2 2 kI sin θi = kT sinθt becomes sin θi / sinθt = µ2ε2 / µ1ε1

Boundary conditions on E

E fields at matter/vacuum interface d Boundary conditions on from Faraday’s Law d = .d E E. ℓ dt B S ∮퐶 − ∫푆

∆t E.dℓ = EL.dℓL + ER.dℓR (as ∆t 0)

ER ∮퐶 B.dS 0 (as ∆t → 0) dℓL θR ∫푆 → → θL θ θ dℓR -EL sin LdℓL + ER sin R dℓR = 0 EL EL sinθL = ER sinθR

E||L = E||R E|| continuous Boundary conditions on H

H fields at matter/vacuum interface

Boundary conditions on H from Ampère’s Law ∇ x H = jfree + ∂D/∂t

∂D ∇ x H .dS = j + .dS = H.dℓ free ∂t � � � ∂D , ∂ /∂t are everywhere finite, so as as ∆t 0, .d 0 D D ∂t S → ∫ → ∆t For materials of finite conductivity, jfree is finite, so j .dS 0 as ∆t 0 HR free dℓL θ R ∫ → → θL For materials of infinite conductivity, jfree is infinite, dℓR HL so j .dS j dℓ as ∆t 0 free free,surface j is surface current per unit length free∫,surface → →

Boundary conditions on H

.d = j dℓ H ℓ free,surface � -HL sinθLdℓL + HR sinθR dℓR = j dℓ free,surface HR sinθR dℓ = HL sinθL dℓ + j dℓ free,surface

H||R = H||L + jfree,surface Infinite conductivity at interface

H||R = H||L Finite conductivity at interface

∆ t

HR dℓL θR θL dℓR HL Boundary conditions on B

B field at matter/vacuum interface

∇. B = 0 ⇒ ∫B.dS = 0 S

B1 cosθ1 dS −B2 cosθ2 dS = 0

⇒ B1⊥ = B2⊥

θ1

1 B1

2 B2 θ2 dS1,2

Boundary conditions on D

D field at matter/vacuum interface

∇.D = ρfree

∫D.dS = ∫ρfree dv

∫ D.dS = 0 No free charges at interface

∫ D.dS = ∫ ∇.D dv = σfree dS Free charge density σfree at interface

θ1 ∫ . d + ∫ .d = ∫ρ dv D1 S1 D2 S2 free D dS - D dS = σ dS dS = dS = dS 1 D1 ┴1 ┴2 free 1 2 D┴1 = D┴2 No interface free charges 2 D2 D┴1 - D┴2 = σfree Interface free charges θ2

dS1,2

Boundary conditions summary

E||L = E||R E|| continuous

B┴1 = B┴2 B┴ continuous

D┴1 = D┴2 D┴ continuous No interface free charges

D┴1 - D┴2 = σfree Interface free charges

H||R = H||L H|| continuous Finite conductivity at interface

H||R = H||L + jfree,surface Infinite conductivity at interface

Fresnel Equations Reflection coefficient R and Transmission coefficient T

2 2 2 ∇ E - µoεoε ∂ E/∂t = 0

i(ωt - k.r) E(r, t) = Eo ex Re{e } k = ω√(µoµεoε)

∇ x E = -i k x E take curl of plane wave E

∇ x E = - ∂B/∂t Faraday’s law

- ∂B/∂t = -iω B time harmonic, plane wave B

-iω B = -i k x E

B = k x E / ω = k ek x E / ω = ω√(µoµεoε) ek x E / ω = √(µε) ek x E / c

Fresnel Equations

B = k x E / ω = k ek x E / ω = ω√(µoµεoε) ek x E / ω = √(µε) ek x E / c

N = E x H = E x B / µoµ = E x (√(µoµεoε) ek x E) / µoµ

2 N = E √(εoε /µoµ)

2 2 R = reflected energy / incident energy = ER √(εoε1 /µoµ1) / EI √(εoε1 /µoµ1) 2 2 = ER / EI

2 2 T = transmitted energy / incident energy = ET √(εoε2 /µoµ2) / EI √(εoε1 /µoµ1) 2 2 = ET / EI n2 / n1 (if µ1 = µ2 = 1)

2 2 R = ER / EI 2 2 T = ET / EI n2 / n1 Fresnel Equations Fields Normal Incidence i(ωt - k1z) n = √ε n = √ε EI = ex EoI e 1 1 x 2 2 µ1 = 1 µ2 = 1 i(ωt - k1z) kI = kR = k1 kT = k2 BI = ey BoI e

i(ωt + k1z) ER = ex EoR e ER BR EI ET kI z ω k kT i( t + k1z) R BI BT BR = -ey BoR e y i(ωt - k2z) ET = ex EoT e

Boundary conditions i(ωt - k2z) BT = ey BoT e

E||1 = E||2 EoI + EoR = EoT B┴ = D┴ = 0 (normal incidence)

H||1 = H||2 (BoI - BoR) / µ1µo = BoT / µ2µo B = µµo H µ1 = µ2 = 1

Fresnel Equations

BoI = n1 EoI / c BoR = n1 EoR / c BoT = n2 EoT / c

n1 (EoI - EoR) = n2 EoT from BoI - BoR = BoT when µ1 = µ2 = 1

EoI + EoR = EoT

EoT = EoI + EoR = n1(EoI - EoR) / n2 Eliminate EoT

EoR (n1 + n2) = EoI (n1 - n2)

EoR / EoI = (n1 - n2) / (n1 + n2) EoR / EoI < 0 if n1 < n2) => π change of phase

Fresnel Equations

n1 (EoI - EoR) = n2 EoT Eliminate EoR

EoI + EoR = EoT

EoR = EoT - EoI = EoI - n2 EoT / n1

EoT (n1 + n2) = 2n1 EoI

EoT / EoI = 2n1 / (n1 + n2)

2 2 2 2 2 2 (EoR / EoI) + (EoT / EoI) = (n1 - n2) / (n1 + n2) + 4n1 / (n1 + n2) ≠ 1!

Fresnel Equations

Reflectivity

2 2 2 R┴ = (EoR / EoI) = (n1 - n2) / (n1 + n2)

Transmittivity

2 2 2 2 T┴ = (EoT / EoI) √(µ2ε2) / √(µ1ε1) = 4n1 / (n1 + n2) (n2 / n1) = 4n1n2 / (n1 + n2)

Energy conservation

2 2 2 R┴ + T┴ = (n1 - n2) / (n1 + n2) + 4n1n2 / (n1 + n2) = 1

Fresnel Equations

Fields Off-normal incidence, s-polarisation

i(ωt - k1.r) EI EI = ex EoI e ER kR BI kI θ θ i(ωt - k1.r) n1 = √ε1 BI = (ey cos i + ez sin i) BoI e BR µ1 = 1 θi θr = E ei(ωt + k1.r) kI = kR = k1 ER ex oR y = (- cosθ + sinθ ) B ei(ωt + k1.r) n = √ε BR ey r ez r oR 2 2 θ µ = 1 t ET 2 E = e E ei(ωt - k2.r) k = k BT T x oT T 2 z kT i(ωt - k2.r) Boundary conditions BT = (ey cosθt + ez sinθt) BoT e

E||1 = E||2 EoI + EoR = EoT

H||1 = H||2 (BoI - BoR) cosθi / µ1µo = BoT cosθt / µ2µo µ1 = µ2 = 1 Fresnel Equations

B = √(µε) k x E / ck = n / (ck) k x E in uniform dielectric

BoI = n1 EoI / c BoR = n1 EoR / c BoT = n2 EoT / c

n1 (EoI - EoR) cosθi = n2 EoT cosθt from (BoI - BoR) cosθi / µ1µo = BoT cosθt / µ2µo with µ1 = µ2 = 1

EoI + EoR = EoT Eliminate EoT

EoT = EoI + EoR = n1(EoI - EoR) cosθi / (n2 cosθt )

EoR (n1 cosθi + n2 cosθt) = EoI (n1 cosθi - n2 cosθt)

EoR / EoI = (n1 cosθi - n2 cosθt) / (n1 cosθi + n2 cosθt)

Fresnel Equations

n1 cosθi (EoI - EoR) = n2 cosθt EoT Eliminate EoR

EoI + EoR = EoT

EoR = EoT - EoI = EoI - n2 cosθt EoT / (n1 cosθi)

EoT (n1 cosθi + n2 cosθt) = 2n1 cosθi EoI

EoT / EoI = 2n1 cosθi / (n1 cosθi + n2 cosθt)

Reflectivity

2 2 2 RS = (EoR / EoI) = (n1 cosθi - n2 cosθt) / (n1 cosθi + n2 cosθt)

Fresnel Equations

Transmittivity

2 TS = (EoT / EoI) √(µ2ε2) cosθt / √(µ1ε1) cosθi 2 2 2 = 4n1 cos θi / (n1cosθi + n2cosθt) (n2cosθt / n1cosθi) 2 = 4n1n2 cosθi cosθt / (n1cosθi + n2cosθt)

Energy conservation

2 2 2 2 R+T =(n1cosθi - n2cosθt) /(n1cosθi + n2cosθt) + 4n1n2cos θi /(n1cosθi + n2cosθt)

2 2 2 2 2 = (n1 cos θi - 2n1n2cosθi cosθt+ n2 cos θt + 4n1n2cosθi cosθt) /(n1cosθi + n2cosθt)

= 1

Fresnel Equations

Off-normal incidence, p-polarisation Fields

i(ωt - k1.r) X BI EI = (ey cosθi + ez sinθi) EoI e BR kR EI kI X i(ωt - k1.r) n1 = √ε1 BI = -ex BoI e ER µ1 = 1 θi θr i(ωt + k1.r) ER = (-ey cosθr + ez sinθr) EoR e y B = -e B ei(ωt + k1.r) n2 = √ε2 R x oR θt µ = 1 X BT 2 ω ET E = (e cosθ + e sinθ ) E ei( t - k2.r) z kT T y t z t oT

Boundary conditions i(ωt - k2.r) BT = -ex BoT e

E||1 = E||2 (EoI - EoR) cosθi = EoT cosθt

H||1 = H||2 (BoI + BoR) / µ1µo = BoT / µ2µo µ1 = µ2 = 1 Fresnel Equations

B = √(µε) k x E / ck = n / (ck) k x E in uniform dielectric

BoI = n1 EoI / c BoR = n1 EoR / c BoT = n2 EoT / c

n1 (EoI + EoR) = n2 EoT from (BoI + BoR) / µ1µo = BoT / µ2µo with µ1 = µ2 = 1

(EoI - EoR) cosθi = EoT cosθt

EoT = (EoI + EoR) n1 / n2 = (EoI - EoR) cosθi / cosθt Eliminate EoT

EoR (n1 / n2 + cosθi / cosθt) = EoI (- n1 / n2 + cosθi / cosθt)

EoR / EoI = (- n1 / n2 + cosθi / cosθt) / (n1 / n2 + cosθi / cosθt)

= (n2cosθi - n1cosθt) / (n2cosθi + n1cosθt)

Fresnel Equations Reflectivity

2 2 2 RP = (EoR / EoI) = (n2cosθi - n1cosθt) / (n2cosθi + n1cosθt)

EoR = EoT n2 / n1 - EoI = EoI - EoT cosθt / cosθi Eliminate EoR

EoT (n2 / n1 + cosθt / cosθi) = 2EoI

EoT / EoI = 2EoI / (n2 / n1 + cosθt / cosθi)

EoT / EoI = 2 / (n2 / n1 + cosθt / cosθi) = 2n1cosθi / (n1cosθt +n2cosθi)

Fresnel Equations Transmittivity

2 TP = (EoT / EoI) √(µ2ε2) cosθt / √(µ1ε1) cosθi 2 2 2 = 4n1 cos θi n2cosθt / (n1cosθt + n2cosθi) n1cosθi 2 = 4n1cosθi n2cosθt / (n1cosθt + n2cosθi)

Energy conservation

2 2 RP + TP = ((n2cosθi - n1cosθt) + 4n1cosθi n2cosθt ) / (n2cosθi + n1cosθt) = 1

Fresnel Equations Normal incidence 2 2 R┴ = (n1 - n2) / (n1 + n2) 2 T┴ = 4 n1n2 / (n1 + n2)

S-polarisation 2 2 RS = (n1cosθi - n2cosθt) / (n1cosθi + n2cosθt) 2 TS = 4n1n2 cosθi cosθt / (n1cosθi + n2cosθt)

P-polarisation 2 2 RP = (n2cosθi - n1cosθt) / (n2cosθi + n1cosθt) 2 TP = 4n1n2 cosθi cosθt / (n1cosθt + n2cosθi)

Energy conservation R + T = 1 in each case Fresnel Equations Light polarisation by reflection - the Brewster angle 2 2 RS = (n1cosθi - n2cosθt) / (n1cosθi + n2cosθt) 2 2 RP = (n2cosθi - n1cosθt) / (n2cosθi + n1cosθt)

If n1 < n2 (e.g. n1 = 1, n2 > 1), θi > θt then n1cosθi < n2cosθt

Consequently RS ≠ 0 for any θi

If n1 < n2, n2cosθi = n1cosθt then RP = 0 for θi = θB Brewster angle

0.7 EI unpolarised 0.6 0.5 n = 1, n = 1.5 0.4 1 2 θB θB ER s-polarised 0.3 RS 0.2 Reflectivity 0.1 θ RP B θt = π/2 - θB 0 20 40 60 80 Angle of Incidence ET partly polarised Fresnel Equations

Fields Normal Incidence, metal-vacuum interfaces

i(ωt - k1z) EI = ex EoI e

i(ωt - k1z) x BI = ey BoI e

n1 = 1 α = √(ωσµo/2) E = e E ei(ωt + k1z) µ1 = 1 µ2 = 1 R x oR

i(ωt + k z) ER BR EI ET = - B e 1 kI z BR ey oR

k kT R BI BT i(ωt - αz) -αz ET = ex EoT e e y i(ωt - αz) -αz Boundary conditions BT = ey BoT e e

E||1 = E||2 EoI + EoR = EoT B┴ = D┴ = 0 (normal incidence)

H||1 = H||2 (BoI - BoR) / µ1µo = BoT / µ2µo B = µµo H µ1 = µ2 = 1 Fresnel Equations

∂B ≠ √(µε) x / ck in lossy matter, use Faraday’s law instead ∇ x = T B k E ET ∂t − i(ωt - αz) -αz ∇ x ET = ey EoT e e α(1 + i) ∂B T = iω B ei(ωt - αz) e -αz ∂t − ey oT α = ω −EoT (1 +− i) i BoT

BoI = n1 EoI / c BoR = n1 EoR / c BoT = α(1 − i) EoT / ω

H||1 = H||2 becomes n1 (EoI - EoR) / c = α(1 − i) EoT / ω α = √(ωσµo/2)

E||1 = E||2 becomes EoI + EoR = EoT set n1 = µ1 = µ2 = 1

Fresnel Equations

(EoI - EoR) / c = α(1 − i) EoT / ω

EoI + EoR = EoT Eliminate EoT

E = E + E = (E - E ) / a(1 − i) a = ω / αc = √(σ/2ε ω) oT oI oR oI oR o

EoR (a(1 − i) + 1) = EoI (1 - a(1 − i))

EoR / EoI = (1 - a(1 − i)) / (1 + a(1 − i))

2 R┴ = |EoR / EoI| ≈ 1 - 2 / a = 1 - 2 √(2εoω/σ)

7 -1 14 -2 For Cu metal, σ = 6.7 x 10 (Ωm) For ω = 7 x 10 R ┴ ≈ 1 - 2.8 x 10 = 0.97