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 E perpendicular to plane of incidence p-polarisation E parallel to plane of incidence

n1 = √ε1 µ1 = 1

n2 = √ε2 µ2 = 1

θt

θi θr

s-polarisation p-polarisation

Fresnel Equations Snell’s Law

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

EI = (ey cosθi + ez sinθi) EoI ei(ωt - kI.r)

ER = (-ey cosθr + ez sinθr) EoR ei(ωt - kR.r)

ET = (ey cosθt + ez sinθt) EoT ei(ωt - kT.r)

To satisfy BC (kI . r)z=0 = (kR . r) z=0 = (kT . r) z=0 (1) wave vectors lie in single plane (2) projection of wave vectors on xy plane is same From (1) From (2) kI sin θi = kR sinθr = kT sinθt kI = kR = ωc µ1ε1 kT = ωc µ2ε2

kI sin θi = kT sinθt becomes sin θi / sinθt = µ2ε2 / µ1ε1

n1 = √ε1 µ1 = 1

n2 = √ε2 µ2 = 1

θt

θi θr kI

kR

kT z

y

θi = θr

Boundary conditions on E E fields at matter/vacuum interface

Boundary conditions on E from Faraday’s Law ∮ E.dℓ = −𝐶ddt∫ B.dS𝑆

∮ E.dℓ = 𝐶 EL.dℓL + ER.dℓR (as ∆t → 0)

∫ B.dS𝑆 → 0 (as ∆t → 0) -EL sinθLdℓL + ER sinθR dℓR = 0

EL sinθL = ER sinθR E||L = E||R E|| continuous

EL

ER

θL θR dℓL

dℓR

∆t

Boundary conditions on H H fields at matter/vacuum interface Boundary conditions on H from Ampère’s Law ∇ x H = jfree + ∂D/∂t

∇ x H .dS = jfree +∂D∂t

.dS =H.dℓ

D, ∂D/∂t are everywhere finite, so as as ∆t → 0, ∫ ∂D∂t .dS → 0

For materials of finite conductivity, jfree is finite, so ∫ jfree.dS → 0 as ∆t → 0 For materials of infinite conductivity, jfree is infinite, so ∫ jfree.dS → jfree,surface

dℓ as ∆t → 0 jfree,surface is surface current per unit length

HL

HR

θL θR dℓL

dℓR

∆t

Boundary conditions on H H.dℓ = jfree,surface

dℓ

-HL sinθLdℓL + HR sinθR dℓR = jfree,surface dℓ

HR sinθR dℓ = HL sinθL dℓ + jfree,surface dℓ

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

HL

HR

θL θR dℓL

dℓR

∆t

Boundary conditions on B

⊥⊥ =⇒=−

=⇒=∇ ∫

21

2211

BB0dS cos BdS cos B

0.d0 .

θθ

S

SBB

1

2

B1 B2

θ2

θ1

dS1,2

B field at matter/vacuum interface

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

∫ D1. dS1 + ∫D2.dS2 = ∫ρfree dv

D 1 dS - D 2 dS = σfree dS dS1 = dS2 = dS

D1 = D2 No interface free charges D1 - D2 = σfree Interface free charges

1

2

D1 D2

θ2

θ1

dS1,2

Boundary conditions summary E||L = E||R E|| continuous B1 = B2 B continuous D1 = D2 D continuous No interface free charges D1 - D2 = 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 ∇2E - µoεoε ∂2E/∂t2 = 0

E(r, t) = Eo ex Reei(ωt - k.r) 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 N = E2 √(εoε /µoµ) R = reflected energy / incident energy = ER

2 √(εoε1 /µoµ1) / EI2 √(εoε1 /µoµ1)

= ER2 / EI

2 T = transmitted energy / incident energy = ET

2 √(εoε2 /µoµ2) / EI2 √(εoε1 /µoµ1)

= ET2 / EI

2 n2 / n1 (if µ1 = µ2 = 1) R = ER

2 / EI2

T = ET2 / EI

2 n2 / n1

Fresnel Equations Normal Incidence

x

z

y

ER EI ET

BT BI

BR kI

kR kT

Fields EI = ex EoI ei(ωt - k1z)

BI = ey BoI ei(ωt - k1z)

ER = ex EoR ei(ωt + k1z)

BR = -ey BoR ei(ωt + k1z)

ET = ex EoT ei(ωt - k2z)

BT = ey BoT ei(ωt - k2z) Boundary conditions

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

n1 = √ε1 µ1 = 1 kI = kR = k1

n2 = √ε2 µ2 = 1 kT = k2

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) (EoR / EoI)2 + (EoT / EoI)2

= (n1 - n2)2 / (n1 + n2)2 + 4n12 / (n1 + n2)2 ≠ 1!

Fresnel Equations Reflectivity R = (EoR / EoI)2 = (n1 - n2)2 / (n1 + n2)2

Transmittivity T = (EoT / EoI)2 √(µ2ε2) / √(µ1ε1) = 4n1

2 / (n1 + n2)2 (n2 / n1) = 4n1n2 / (n1 + n2)2

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

Fresnel Equations Off-normal incidence, s-polarisation

Fields EI = ex EoI ei(ωt - k1.r)

BI = (ey cosθi + ez sinθi) BoI ei(ωt - k1.r)

ER = ex EoR ei(ωt + k1.r)

BR = (-ey cosθr + ez sinθr) BoR ei(ωt + k1.r)

ET = ex EoT ei(ωt - k2.r)

BT = (ey cosθt + ez sinθt) BoT ei(ωt - k2.r) Boundary conditions

E||1 = E||2 EoI + EoR = EoT H||1 = H||2 (BoI - BoR) cosθi / µ1µo = BoT cosθt / µ2µo µ1 = µ2 = 1

θt

θi θr

EI kI BI ER kR

BR

BT kT ET

z

y

n1 = √ε1 µ1 = 1 kI = kR = k1

n2 = √ε2 µ2 = 1 kT = k2

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

Fresnel Equations Transmittivity TS = (EoT / EoI)2 √(µ2ε2) cosθt / √(µ1ε1) cosθi = 4n1

2 cos2θi / (n1cosθi + n2cosθt) 2 (n2cosθt / n1cosθi) = 4n1n2 cosθi cosθt / (n1cosθi + n2cosθt) 2 Energy conservation R+T =(n1cosθi - n2cosθt)2 /(n1cosθi + n2cosθt)2 + 4n1n2cos2θi /(n1cosθi + n2cosθt) 2

= (n12cos2θi - 2n1n2cosθi cosθt+ n2

2cos2θt + 4n1n2cosθi cosθt) /(n1cosθi + n2cosθt) 2

= 1

Fresnel Equations Off-normal incidence, p-polarisation

Fields

EI = (ey cosθi + ez sinθi) EoI ei(ωt - k1.r)

BI = -ex BoI ei(ωt - k1.r)

ER = (-ey cosθr + ez sinθr) EoR ei(ωt + k1.r)

BR = -ex BoR ei(ωt + k1.r)

ET = (ey cosθt + ez sinθt) EoT ei(ωt - k2.r)

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

E||1 = E||2 (EoI - EoR) cosθi = EoT cosθt H||1 = H||2 (BoI + BoR) / µ1µo = BoT / µ2µo µ1 = µ2 = 1

n1 = √ε1 µ1 = 1

n2 = √ε2 µ2 = 1

θt

θi θr

BI kI EI BR kR

ER

ET kT BT

z

y

X

X

X

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 RP = (EoR / EoI)2 = (n2cosθi - n1cosθt)2 / (n2cosθi + n1cosθt)2 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 TP = (EoT / EoI)2√(µ2ε2) cosθt / √(µ1ε1) cosθi = 4n1

2cos2θi n2cosθt / (n1cosθt + n2cosθi) 2 n1cosθi = 4n1cosθi n2cosθt / (n1cosθt + n2cosθi) 2 Energy conservation RP + TP = ((n2cosθi - n1cosθt)2 + 4n1cosθi n2cosθt ) / (n2cosθi + n1cosθt)2 = 1

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

S-polarisation RS = (n1cosθi - n2cosθt)2 / (n1cosθi + n2cosθt)2

TS = 4n1n2 cosθi cosθt / (n1cosθi + n2cosθt) 2 P-polarisation RP = (n2cosθi - n1cosθt)2 / (n2cosθi + n1cosθt)2 TP = 4n1n2 cosθi cosθt / (n1cosθt + n2cosθi) 2 Energy conservation R + T = 1 in each case

Fresnel Equations Light polarisation by reflection - the Brewster angle RS = (n1cosθi - n2cosθt)2 / (n1cosθi + n2cosθt)2

RP = (n2cosθi - n1cosθt)2 / (n2cosθi + n1cosθt)2 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 20 40 60 80

0.10.20.30.40.50.60.7

Ref

lect

ivity

Angle of Incidence

n1 = 1, n2 = 1.5 RS

RP θB

θB θB

EI unpolarised

ER s-polarised

ET partly polarised

θt = π/2 - θB

Fresnel Equations Normal Incidence, metal-vacuum interfaces

x

z

y

ER EI ET

BT BI

BR kI

kR kT

Fields EI = ex EoI ei(ωt - k1z)

BI = ey BoI ei(ωt - k1z)

ER = ex EoR ei(ωt + k1z)

BR = -ey BoR ei(ωt + k1z)

ET = ex EoT ei(ωt - αz) e -αz

BT = ey BoT ei(ωt - αz) e -αz Boundary conditions

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

n1 = 1 µ1 = 1

α = √(ωσµo/2) µ2 = 1

Fresnel Equations B ≠ √(µε) k x E / ck in lossy matter, use Faraday’s law instead ∇ x ET = − ∂BT

∂t

∇ x ET = − ey EoT ei(ωt - αz) e -αz α(1 + i)

− ∂BT∂t = −iω ey BoT ei(ωt - αz) e -αz

EoT α(1 + i) = iω 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

EoT

= EoI +

EoR = (EoI - EoR) / a(1 − i) a = ω / αc = √(σ/2εoω)

EoR (a(1 − i) + 1) = EoI (1 - a(1 − i)) EoR / EoI = (1 - a(1 − i)) / (1 + a(1 − i)) R = |EoR / EoI|2 ≈ 1 - 2 / a = 1 - 2 √(2εoω/σ) For Cu metal, σ = 6.7 x 107 (Ωm)-1 For ω = 7 x 1014 R ≈ 1 - 2.8 x 10-2 = 0.97