Electromagnetic Wave PropagationLecture 14: Multilayer film applications
Daniel Sjoberg
Department of Electrical and Information Technology
October 18, 2012
Outline
1 Multilayer dielectric structures at oblique incidence
2 Perfect lens in negative-index media
3 Antireflection coatings at oblique incidence
4 Omnidirectional dielectric mirrors
5 Reflection and refraction in birefringent media
6 Giant birefringent optics
7 Concluding remarks
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Outline
1 Multilayer dielectric structures at oblique incidence
2 Perfect lens in negative-index media
3 Antireflection coatings at oblique incidence
4 Omnidirectional dielectric mirrors
5 Reflection and refraction in birefringent media
6 Giant birefringent optics
7 Concluding remarks
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General geometry
(Fig. 8.1.1 in Orfanidis)
Most concepts generalize. Important steps:
I Use transverse field components
I The tangential wave vector is conserved
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Oblique propagation in bianisotropic materials
The formulation from lecture 4 can be used for numericalimplementation of completely general bianisotropic materials:
∂
∂z
(Et
Ht × z
)= −jωW(z) ·
(Et
Ht × z
)
W =
(0 −z × II 0
)·[(εtt ξttζtt µtt
)−A(kt)
]·(I 00 z × I
)
A(kt) =
[(0 −ω−1kt × z
ω−1kt × z 0
)−(εt ξtζt µt
)](εzz ξzzζzz µzz
)−1 [(0 −ω−1z × kt
ω−1z × kt 0
)−(εz ξzζz µz
)]In this lecture, we focus on isotropic media.
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Wave concepts
Tangential phase (Snel’s law):
na sin θa = ni sin θi = nb sin θb ⇒ cos θi =
√1− n2a sin
2 θan2i
Phase thickness:
δi = kz,i`i =ω
c0ni`i cos θi = 2π
f
f0
ni`iλ0
cos θi
Transverse wave impedance and refractive index:
ηTM =kzωε
=
õ
ε
kzω√εµ
= η cos θ = η0cos θ
n=
η0nTM
ηTE =ωµ
kz=
õ
ε
ω√εµ
kz=
η
cos θ= η0
1
n cos θ=
η0nTE
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Propagation of split fields
Reflection and transmission coefficients:
ρT,i =nT,i−1 − nT,inT,i−1 + nT,i
, τT,i = 1 + ρT,i
Matching and propagation matrices:(ET,i+
ET,i−
)=
1
τT,i
(1 ρT,iρT,i 1
)︸ ︷︷ ︸
matching
(ejδi 00 e−jδi
)︸ ︷︷ ︸
propagation
(ET,i+1,+
ET,i+1,−
)
Reflection coefficient recursion:
ΓT,i =ρT,i + ΓT,i+1e
−2jδi
1 + ρT,iΓT,i+1e−2jδi
initialized by ΓT,M+1 = ρT,M+1.
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Propagation of total fields
Propagation matrix:(ET,i
HT,i
)=
(cos δi jηT,i sin δi
jη−1T,i sin δi cos δi
)(ET,i+1
HT,i+1
)Transverse impedance recursion:
ZT,i = ηT,iZT,i+1 + jηT,i tan δiηT,i + jZT,i+1 tan δi
, ZT,M+1 = ηT,b
Generalization of multidiel.m:
[Gamma1, Z1] = multidiel(n, L, lambda, theta, pol)
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Single dielectric slab
(Fig. 8.3.1 in Orfanidis)
ΓT,1 =ρT,1 + ρT,2e
−2jδ1
1 + ρT,1ρT,2e−2jδ1=ρT,1(1− e−2jδ1)
1− ρ2T,1e−2jδ1
δ1 = 2πf
f0
n1`1λ0
cos θ1 = πf
f1, f1 =
f0
2n1`1λ0
cos θ1
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Frequency shift
(Fig. 8.3.2 in Orfanidis)
I Notch frequency is shifted up
I Reflection level is increased
I Bandwidth is decreased
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Frustrated total internal reflection
(Fig. 8.4.2 in Orfanidis)
kx = k0na sin θ
kza =√k20n
2a − k2x = k0na cos θ
kzb =
k0√n2b − n2a sin
2 θ if θ ≤ θc−jk0
√n2a sin
2 θ − n2b = −jαzb if θ ≥ θc11 / 50
Dependence on thickness and angle
(Figs. 8.4.3 and 8.4.4 in Orfanidis)
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Power flow
The combined field from two evanescent waves is
ET(z) =1 + ρa
1− ρ2ae−2αzbd
[e−αzbz − ρae−2αzbdeαzbz
]Ea+
HT(z) =1− ρa
1− ρ2ae−2αzbd
[e−αzbz + ρae
−2αzbdeαzbz] Ea+ηTa
The power transported in one evanescent wave is zero,
Pz =1
2Re{ET+H
∗T+} =
1
2Re
{(1 + ρa)(1− ρ∗a)|1− ρ2ae−2jαzbd|
|Ea+|2
ηTae−2αzbz
}= 0
since (1 + ρa)(1− ρ∗a) = 1− |ρa|2 + ρa − ρ∗a = 2j Im(ρa). But thecombination of two evanescent waves can carry power:
Pz =1
2Re{ETH
∗T} = · · · = (1− |Γ |2) |Ea+|
2
2ηTa
|Γ |2 = sinh2(αzbd)
sinh2(αzbd) + sin2 φa, ρa = ejφa
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Simulation
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Outline
1 Multilayer dielectric structures at oblique incidence
2 Perfect lens in negative-index media
3 Antireflection coatings at oblique incidence
4 Omnidirectional dielectric mirrors
5 Reflection and refraction in birefringent media
6 Giant birefringent optics
7 Concluding remarks
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Slab of negative index
ε = −ε0, µ = −µ0, n =√
εµε0µ0
= −1, η =√
µε = η0.
(Fig. 8.6.1 in Orfanidis)
Snel’s law:na sin θa = n sin θ ⇒ θ = −θa
Zero reflection since η = η0.
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Field solutions
The electric field is (TM polarization)
E =
E0
(x− kx
kzz)e−jkzze−jkxx, for z ≤ 0
E0
(x+ kx
kzz)ejkzze−jkxx, for 0 ≤ z ≤ d
E0
(x− kx
kzz)e−jkz(z−2d)e−jkxx, for z ≥ d
where the longitudinal wave number is
kz = sqrte(k20 − k2x) =
{√k20 − k2x, if k2x ≤ k20
−j√k2x − k20, if k20 ≤ k2x
Thus, for evanescent waves where kz = −jα, the negative mediumamplifies the transmitted wave.
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Amplification of evanescent waves
(Fig. 8.6.2 in Orfanidis)
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Influence of non-ideal parameters
Transmittance below is computed as 10 log10 |T e−jkzd|2, which isnot necessarily less than one for evanescent waves.
(Fig. 8.6.3 in Orfanidis)
Peaks correspond to plasmon resonances, where
Re(kx,res) ≈ 1d
∣∣∣ ε0−εε0+ε
∣∣∣. This implies a resolution of
∆x = 1/Re(kx,res).
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Simulation
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Outline
1 Multilayer dielectric structures at oblique incidence
2 Perfect lens in negative-index media
3 Antireflection coatings at oblique incidence
4 Omnidirectional dielectric mirrors
5 Reflection and refraction in birefringent media
6 Giant birefringent optics
7 Concluding remarks
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Antireflection coatings
nb = 1.5
na = 1 Er = 0Ei
nb = 1.5 nb = 1.5
n2 = 1.38
na = 1 ErEi na = 1 Er = 0Ei
n1 = 1.22 n3 = 2.45
Suitably designed thin layers can reduce reflection from interfaces.What happens at oblique incidence?
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Two-layer coating
(Fig. 8.7.1 in Orfanidis)
n = [1, 1.38, 2.45, 1.5]; L = [0.3294, 0.0453];
la0 = 550; la = linspace(400, 700, 101); pol= ’te’;
G0 = abs(multidiel(n, L, la/la0)).^2*100;
G20 = abs(multidiel(n, L, la/la0, 20, pol)).^2*100;
G30 = abs(multidiel(n, L, la/la0, 30, pol)).^2*100;
G40 = abs(multidiel(n, L, la/la0, 40, pol)).^2*100;
plot(la, [G0; G20; G30; G40]);
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Demo program
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Typical effects at oblique incidence
I Frequency shift
I Difference in polarization
I Possible to redesign for specific angle of incidence
I Higher reflection for higher angles
I Brewster angle for TM polarization
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Outline
1 Multilayer dielectric structures at oblique incidence
2 Perfect lens in negative-index media
3 Antireflection coatings at oblique incidence
4 Omnidirectional dielectric mirrors
5 Reflection and refraction in birefringent media
6 Giant birefringent optics
7 Concluding remarks
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Dielectric mirrors
(Fig. 8.8.1 in Orfanidis)
Increasing angle of incidence implies
I Frequency shift
I Polarization dependence
Omnidirectionality requires overlap of frequency bands for TE andTM for all angles.
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Necessary condition
The angles in all layers are related by Snel’s law
na sin θa = nH sin θH = nL sin θL = nb sin θb
and the phase thicknesses are
δH = 2πf
f0LH cos θH, δL = 2π
f
f0LL cos θL
where Li = ni`i/λ0 and cos θi =√
1− n2a sin2 θa/n2i . To prevent
a wave from the outside to access the Brewster angle inside themirror (which would imply transmission), we need
na <nHnL√n2H + n2L
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Bandgap
All procedures from normal incidence are applicable whenconsidering transverse field components. We look for wavespropagating with the effective wave number K, so that thereshould exist solutions(
Ei,THi,T
)=
(cos(δL) jηLT sin(δL)
jη−1LT sin(δL) cos(δL)
)·(
cos(δH) jηHT sin(δH)
jη−1HT sin(δH) cos(δH)
) (Ei+2,T
Hi+2,T
)︸ ︷︷ ︸=e−jK`(Ei,T
Hi,T)
This is an eigenvalue equation, which can be put in the form
cos(K`) =cos(δH + δL)− ρ2T cos(δH − δL)
1− ρ2T
where ρT = nHT−nLTnHT+nLT
depends on polarization.
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Bandgaps
| cos(K`)| ≤ 1 for propagating fields:
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0f/f0
2.5
2.0
1.5
1.0
0.5
0.0
0.5
1.0
1.5theta = 45, nH/nL = 1.68
TETM
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0f/f0
2.5
2.0
1.5
1.0
0.5
0.0
0.5
1.0
1.5theta = 45, nH/nL = 3.36
TETM
The bandgap increases with contrast, and when increasing theangle:
I The entire bandgap shifts up in frequency for bothpolarizations (the layers become optically thinner).
I The TM bandgap decreases, and the TE bandgap increases.
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Frequency response
nH = 3, nL = 1.38, N = 30
(Fig. 8.8.2 in Orfanidis)
Calculated with multidiel.m.
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Bandwidths and angle
nH = 3, nL = 1.38 nH = 2, nL = 1.38
(Fig. 8.8.6 in Orfanidis)
Omnidirectional Not omnidirectional
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Polarizing beam splitter
(Fig. 8.9.1 in Orfanidis)
The TM polarization travels at the Brewster angle if
na sin θa = na1√2=
nHnL√n2H + n2L
Hence TM is transmitted and TE reflected.33 / 50
Outline
1 Multilayer dielectric structures at oblique incidence
2 Perfect lens in negative-index media
3 Antireflection coatings at oblique incidence
4 Omnidirectional dielectric mirrors
5 Reflection and refraction in birefringent media
6 Giant birefringent optics
7 Concluding remarks
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Effective refractive index
D =
ε1 0 00 ε2 00 0 ε3
·E, B = µ0H, k = Nk0k
(Fig. 4.6.1 in Orfanidis)
N =1√
cos2 θn21
+ sin2 θn23
N = n2
ηTM = η0N cos θ
n21ηTE =
η0n2 cos θ
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Snel’s law
Snel’s law is N sin θ = N ′ sin θ′ , or, more explicitly,
n1n3 sin θ√n21 sin
2 θ + n23 cos2 θ
=n′1n
′3 sin θ
′√n′21 sin2 θ′ + n
′23 cos2 θ′
(TM)
n2 sin θ = n′2 sin θ′ (TE)
Can be solved for θ as function of θ′ or vice versa. Coded in
thb = snel(na, nb, tha, pol);
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Reflection coefficients
The reflection coefficients are
ρTM =η′TM − ηTM
η′TM + ηTM=n1n3
√n′23 −N2 sin2 θ − n′21 n
′23
√n23 −N2 sin2 θ
n1n3
√n′23 −N2 sin2 θ + n
′21 n′23
√n23 −N2 sin2 θ
ρTE =η′TE − ηTE
η′TE + ηTE=n2 cos θ −
√n′22 − n22 sin2 θ
n2 cos θ +√n′22 − n22 sin2 θ
Can be computed using routine fresnel.
I If n3 = n′3 we have ρTM =n1−n′1n1+n′1
independent of θ.
I If n = [n1, n1, n3] and n′ = [n3, n3, n1] we have
ρTE = ρTM =n1 cos θ −
√n23 − n21 sin2 θ
n1 cos θ +√n23 − n21 sin2 θ
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Examples of Fresnel coefficients
(Fig. 8.11.1 in Orfanidis)
(a) n = [1.63, 1.63, 1.5] n′ = [1.63, 1.63, 1.63]
(b) n = [1.54, 1.54, 1.63] n′ = [1.5, 1.5, 1.5]
(c) n = [1.8, 1.8, 1.5] n′ = [1.5, 1.5, 1.5]
(d) n = [1.8, 1.8, 1.5] n′ = [1.56, 1.56, 1.56]
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Differences from the isotropic case
I The Brewster angle can be zero (a)
I The Brewster angle may not exist (c)
I The Brewster angle may be imaginary (both ρTE and ρTM
increase monotonically with angle) (d)
I There may be total internal reflection for one polarization butnot for the other (b)
I The reflection coefficient can be the same for TE and TM
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Outline
1 Multilayer dielectric structures at oblique incidence
2 Perfect lens in negative-index media
3 Antireflection coatings at oblique incidence
4 Omnidirectional dielectric mirrors
5 Reflection and refraction in birefringent media
6 Giant birefringent optics
7 Concluding remarks
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Dielectric mirror
(Fig. 8.8.1 in Orfanidis)
The layers are now birefringent with
nH = [nH1, nH2, nH3], nL = [nL1, nL2, nL3]
The code multidiel.m can be used to calculate the response.
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First design
N = 50, nH = [1.8, 1.8, 1.5], nL = [1.5, 1.5, 1.5], na = nb = 1
(Fig. 8.13.1 in Orfanidis)
Relatively similar TE and TM, TM is a bit more narrowband.
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Second design
N = 30, nH = [1.8, 1.8, 1.5], nL = [1.5, 1.5, 1.8], na = nb = 1.4
(Fig. 8.13.2 in Orfanidis)
Identical TE and TM, but depends on angle.
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Slight change of second design
N = 30, nH = [1.8, 1.8, 1.5], nL = [1.5, 1.5, 1.9], na = nb = 1.4
(Fig. 8.13.3 in Orfanidis)
Perturbation at higher angles.
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GBO mirrors
M. F. Weber et al, Science, vol 287, p. 2451, 2000.
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GBO reflective polarizer
N = 80, nH = [1.86, 1.57, 1.57], nH = [1.57, 1.57, 1.57],na = nb = 1
(Fig. 8.13.4 in Orfanidis)
Mismatch only in one direction.
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Giant birefringent optics
I The “giant” refers to the relatively large birefringence
I Commercialized by 3M
I Careful matching of refractive indices may eliminatedifferences between polarizations
I Particularly, the Brewster angle can be controlled
I Materials can be manufactured from birefringent polymers
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Outline
1 Multilayer dielectric structures at oblique incidence
2 Perfect lens in negative-index media
3 Antireflection coatings at oblique incidence
4 Omnidirectional dielectric mirrors
5 Reflection and refraction in birefringent media
6 Giant birefringent optics
7 Concluding remarks
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Concluding remarks
This course has dealt with the following subjects in depth:
I Propagation of electromagnetic waves in stratified media
I Material modeling in time and frequency domain, isotropic aswell as bianisotropic
I Fundamental wave properties: polarization, wave impedance,wave number
I Tangential fields and tangential wave numbers are continuousacross interfaces, and have been used consistently
I Numerical methods and examples have appeared throughoutI Lots of codes from Orfanidis and on course home page
I Several applications of multilayer structures
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Only projects and orals remaining!
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