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
µ1 = 1
θi θr
n2 = √ε2
µ2 = 1
θt
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
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)
n1 = √ε1
µ1 = 1
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) θi = θr
From (2) kI sin θi = kR sinθr = kT sinθt
kI
θi θr k
R
y
n2 = √ε2
µ2 = 1
z θt kT
ω
ω
kI = kR = c µ1ε1 kT = c µ2ε2
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 E from Faraday’s Law ∮𝐶 E.dℓ = − ∫𝑆 B.dS
dt
∆t
dℓL
θL
EL
θR
dℓR
ER
∮𝐶 E.dℓ = EL.dℓL + ER.dℓR (as ∆t → 0)
∫𝑆 B.dS → 0
(as ∆t → 0)
-EL sinθLdℓL + ER sinθR dℓR = 0
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 = � jfree +
.dS = � H.dℓ
∂t
D, ∂D/∂t are everywhere finite, so as as ∆t → 0, ∫
For materials of finite conductivity, jfree is finite,
so ∫ jfree.dS → 0 as ∆t → 0
∂D
.dS → 0
∂t
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
∆t
dℓL
θL
HL
θR
dℓR
HR
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
H||R = H||L
∆t
dℓL
θL
HL
θR
dℓR
HR
Infinite conductivity at interface
Finite conductivity at interface
Boundary conditions on B
B field at matter/vacuum interface
∇. B = 0 ⇒ ∫ B.dS = 0
S
B1 cosθ1 dS − B 2 cosθ 2 dS = 0
⇒ B1⊥ = B 2⊥
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
1
D1
2
D2
θ2
dS1,2
∫ D1. dS1 + ∫D2.dS2 = ∫ρfree dv
D ┴1 dS - D ┴2 dS = σfree dS dS1 = dS2 = dS
D┴1 = D┴2
No interface free charges
D┴1 - D┴2 = σfree
Interface free charges
Boundary conditions summary
E||L = E||R
E|| continuous
B┴1 = B┴2
B┴ continuous
D┴1 = D┴2
D┴1 - D┴2 = σfree
D┴ continuous No interface free charges
Interface free charges
H||R = H||L
H||R = H||L + jfree,surface
H|| continuous Finite conductivity at interface
Infinite conductivity at interface
Fresnel Equations
Reflection coefficient R and Transmission coefficient T
∇2E - µoεoε ∂2E/∂t2 = 0
E(r, t) = Eo ex Re{ei(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 = ER2 √(εoε1 /µoµ1) / EI2 √(εoε1 /µoµ1)
= ER2 / EI2
T = transmitted energy / incident energy = ET2 √(εoε2 /µoµ2) / EI2 √(εoε1 /µoµ1)
= ET2 / EI2 n2 / n1 (if µ1 = µ2 = 1)
R = ER2 / EI2
T = ET2 / EI2 n2 / n1
Fresnel Equations
Fields
Normal Incidence
n1 = √ε1
µ1 = 1
kI = kR = k1
n2 = √ε2
µ2 = 1
kT = k2
x
ER BR EI k
I
kR B
I
y
ET
BT
EI = ex EoI ei(ωt - k1z)
BI = ey BoI ei(ωt - k1z)
ER = ex EoR ei(ωt + k1z)
z
kT
BR = -ey BoR ei(ωt + k1z)
ET = ex EoT ei(ωt - k2z)
Boundary conditions
BT = ey BoT ei(ωt - k2z)
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
n1 (EoI - EoR) = n2 EoT
from BoI - BoR = BoT when µ1 = µ2 = 1
BoT = n2 EoT / c
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) = 4n12 / (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
EI = ex EoI ei(ωt - k1.r)
EI
BI
n1 = √ε1
µ1 = 1
kI = kR = k1
ER
kI
kR
BR
θi θr
BT
θt ET
z
BI = (ey cosθi + ez sinθi) BoI ei(ωt - k1.r)
ER = ex EoR ei(ωt + k1.r)
y
n2 = √ε2
µ2 = 1
kT = k2
Fields
kT
Boundary conditions
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)
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
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
= 4n12 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+ n22cos2θt + 4n1n2cosθi cosθt) /(n1cosθi + n2cosθt) 2
=1
Fresnel Equations
Off-normal incidence, p-polarisation
X BI
EI
n1 = √ε1
µ1 = 1
BR
kI
X
EI = (ey cosθi + ez sinθi) EoI ei(ωt - k1.r)
kR
ER
θi θr
Fields
BI = -ex BoI ei(ωt - k1.r)
ER = (-ey cosθr + ez sinθr) EoR ei(ωt + k1.r)
y
n2 = √ε2
µ2 = 1
ET
θt BT
X
z
Boundary conditions
kT
BR = -ex BoR ei(ωt + k1.r)
ET = (ey cosθt + ez sinθt) EoT ei(ωt - k2.r)
BT = -ex BoT ei(ωt - k2.r)
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
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
= 4n12cos2θi 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
Reflectivity
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
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
EI unpolarised
n1 = 1, n2 = 1.5
θB θB
RS
θB
20
40
60
Angle of Incidence
RP
80
ER s-polarised
θt = π/2 - θB
ET partly polarised
Fresnel Equations
Fields
Normal Incidence, metal-vacuum interfaces
EI = ex EoI ei(ωt - k1z)
BI = ey BoI ei(ωt - k1z)
x
n1 = 1
µ1 = 1
α = √(ωσµo/2)
µ2 = 1
ER BR EI k
I
kR B
I
y
ET
BT
ER = ex EoR ei(ωt + k1z)
BR = -ey BoR ei(ωt + k1z)
z
kT
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
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
= −iω ey BoT ei(ωt - αz) e -αz
∂t
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
E||1 = E||2 becomes EoI + EoR = EoT
α = √(ωσµo/2)
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