Electromagnetic Wave PropagationLecture 10: Multilayer structures I
Daniel Sjoberg
Department of Electrical and Information Technology
October 4, 2012
Outline
1 Introduction
2 Multiple dielectric slabs
3 Antireflection coatings
4 Dielectric mirrors
5 Propagation bandgaps
6 Narrow-band transmission filters (Fabry-Perot resonators)
7 Conclusions
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Outline
1 Introduction
2 Multiple dielectric slabs
3 Antireflection coatings
4 Dielectric mirrors
5 Propagation bandgaps
6 Narrow-band transmission filters (Fabry-Perot resonators)
7 Conclusions
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Key questions
I How can we analyze multilayer structures?
I What can we build with them?
I How can we design the structures?
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Outline
1 Introduction
2 Multiple dielectric slabs
3 Antireflection coatings
4 Dielectric mirrors
5 Propagation bandgaps
6 Narrow-band transmission filters (Fabry-Perot resonators)
7 Conclusions
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Scattering from multilayer structure
(Fig. 6.1.1 in Orfanidis)(E1+
E1−
)=
1
2
(1 ηa1 −ηa
)P1 · · ·PM
(1 11ηb− 1ηb
)(E′M+1,+
0
)
Pi =
(cos(ki`i) jηi sin(ki`i)
jη−1i sin(ki`i) cos(ki`i)
)6 / 50
Scattering from multilayer structure
(Fig. 6.1.1 in Orfanidis)(E1+
E1−
)= P′1 · · ·P′M
1
τM+1
(1 ρM+1
ρM+1 1
)(E′M+1,+
0
)
P′i =1
τi
(ejki`i ρie
−jki`i
ρiejki`i e−jki`i
)7 / 50
Scattering parameters
The total transfer matrix relation is(E1+
E1−
)=
(T11 T12T21 T22
)(E′M+1,+
0
)=
(T11E′M+1,+
T21E′M+1,+
)which implies
T =E′M+1,+
E1+=
1
T11
Γ1 =E1−E1+
=T21E′M+1,+
E1+=T21T11
Thus, having computed the total transfer matrix, the reflectionand transmission coefficients correspond to simple rearrangements.
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Layer recursion for reflection
With propagation matrices(Ei,+Ei,−
)=
1
τi
(ejki`i ρie
−jki`i
ρiejki`i e−jki`i
)(Ei+1,+
Ei+1,−
)(EiHi
)=
(cos(ki`i) jηi sin(ki`i)
jη−1i sin(ki`i) cos(ki`i)
)(Ei+1
Hi+1
)the reflection coefficient at interface i can be found by recursion
Ei−Ei+
= Γi =ρi + Γi+1e
−2jki`i
1 + ρiΓi+1e−2jki`i, ΓM+1 = ρM+1
and the impedance at interface i in the same way
EiHi
= Zi = ηiZi+1 + jηi tan(ki`i)
ηi + jZi+1 tan(ki`i), ZM+1 = ηb
These are equivalent. Thus, the reflection properties can be foundfrom a one-pass calculation, iterating from M + 1 to 1.
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Computation of transmission
In general, to find also the transmission coefficient, the fullcascading technique must be employed. However, in the importantcase of no losses energy conservation gives us the result
1− |Γ |2
2ηa=|T |2
2ηb
Thus, for lossless structures the number 1− |Γ |2 represents thetransmittance.
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Dielectric structures
From now on, we assume the medium is non-magnetic, that is,
k = ω√εµ =
ω
c0
√εr = k0n
and
η =
õ
ε=
√µ0ε0
1√εr
=η0n
Thus, a slab is characterized solely by its physical length ` and itsrefractive index n =
√εr.
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Orfanidis’ matlab codes: multidiel.m
% multidiel.m - reflection response of isotropic or birefringent multilayer structure
%
% na | n1 | n2 | ... | nM | nb
% left medium | L1 | L2 | ... | LM | right medium
% interface 1 2 3 M M+1
%
% Usage: [Gamma,Z] = multidiel(n,L,lambda,theta,pol)
% [Gamma,Z] = multidiel(n,L,lambda,theta) (equivalent to pol=’te’)
% [Gamma,Z] = multidiel(n,L,lambda) (equivalent to theta=0)
%
% n = isotropic 1x(M+2), uniaxial 2x(M+2), or biaxial 3x(M+2), matrix of refractive indices
% L = vector of optical lengths of layers, in units of lambda_0
% lambda = vector of free-space wavelengths at which to evaluate the reflection response
% theta = incidence angle from left medium (in degrees)
% pol = for ’tm’ or ’te’, parallel or perpendicular, p or s, polarizations
%
% Gamma = reflection response at interface-1 into left medium evaluated at lambda
% Z = transverse wave impedance at interface-1 in units of eta_a (left medium)
%
% notes: M is the number of layers (M >= 0)
% n = [na, n1, n2, ..., nM, nb] = 1x(M+2) row vector of isotropic indices
%
% [ na1 n11 n12 ... n1M nb1 ] 3x(M+2) matrix of birefringent indices,
% n = [ na2 n21 n22 ... n2M nb2 ] = if 2x(M+2), it is extended to 3x(M+2)
% [ na3 n31 n32 ... n3M nb3 ] by repeating the top row
%
% optical lengths are in units of a reference free-space wavelength lambda_0:
% for i=1,2,...,M, L(i) = n(1,i) * l(i), for TM,
% L(i) = n(2,i) * l(i), for TE,
% TM and TE L(i) are the same in isotropic case. If M=0, use L=[].
%
% lambda is in units of lambda_0, that is, lambda/lambda_0 = f_0/f
%
% reflectance = |Gamma|^2, input impedance = Z = (1+Gamma)./(1-Gamma)
%
% delta(i) = 2*pi*[n(1,i) * l(i) * sqrt(1 - (Na*sin(theta))^2 ./ n(3,i).^2))]/lambda, for TM
% delta(i) = 2*pi*[n(2,i) * l(i) * sqrt(1 - (Na*sin(theta))^2 ./ n(2,i).^2))]/lambda, for TE
%
% if n(3,i)=n(3,i+1)=Na, then will get NaN’s at theta=90 because of 0/0, (see also FRESNEL)
%
% it uses SQRTE, which is a modified version of SQRT approriate for evanescent waves
%
% see also MULTIDIEL1, MULTIDIEL2
% Sophocles J. Orfanidis - 1999-2008 - www.ece.rutgers.edu/~orfanidi/ewa
function [Gamma,Z] = multidiel(n,L,lambda,theta,pol)
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Outline
1 Introduction
2 Multiple dielectric slabs
3 Antireflection coatings
4 Dielectric mirrors
5 Propagation bandgaps
6 Narrow-band transmission filters (Fabry-Perot resonators)
7 Conclusions
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Application: antireflection coating
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
An anti-reflection coating can be realized by one or many layers.
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Materials
Some materials in use for optical anti-reflection coatings are foundon page 190 in Orfanidis:
Not all refractive indices are available, meaning the design needs toeither be tolerant to deviations or adapted to the availablematerials.
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How can the parameters of the coating be chosen?
Each slab is characterized by two parameters, thickness `i andrefractive index ni.
We may fix the parameter vector a = [{`i}Ni=1, {ni}Ni=1], andcompute the response Γ (a). Setting up an optimization loop, wetypically minimize a penalty function like
p(a) =
∫ f2
f1
|Γ (a, f)− Γ0(f)|nw(f) df
where Γ0(f) is the target frequency response, and the weightfunction w(f) may emphasize different parts of the spectrum.
I This is straightforward, but as the number of layers increases,it becomes important to have a good start design, andunderstand what can and cannot be done.
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Demo
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Explicit design of a two-layer structure
We wish to match refractive index nb = 1.5 to na = 1 using twoslabs. The reflection coefficients are
Γ1 =ρ1 + Γ2e
−2jk1`1
1 + ρ1Γ2e−2jk1`1, Γ2 =
ρ2 + ρ3e−2jk2`2
1 + ρ2ρ3e−2jk2`2
Zero reflection requires Γ1 = 0, or e2jk1`1 = −Γ2/ρ1.
|Γ2|2 =ρ22 + ρ23 + 2ρ2ρ3 cos(2k2`2)
1 + ρ22ρ23 + 2ρ2ρ3 cos(2k2`2)
= ρ21
which can be transformed to
cos2(k2`2) =ρ21(1− ρ2ρ3)2 − (ρ2 − ρ3)2
4ρ2ρ3(1− ρ21)
sin2(k2`2) =(ρ2 + ρ3)
2 − ρ21(1 + ρ2ρ3)2
4ρ2ρ3(1− ρ21)
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Determining materials
Assuming n1 < n2 and n2 > nb, we have ρ2 < 0 and ρ3 > 0. Thefollowing conditions must be satisfied for the equations to have asolution: ∣∣∣∣ ρ3 + ρ2
1 + ρ2ρ3
∣∣∣∣2 < ρ21 <
∣∣∣∣ ρ3 − ρ21− ρ2ρ3
∣∣∣∣2which requires
n1 ∈ [√nb, nb] = [1.22, 1.5] and n2 6∈ [
√nb, n1
√nb] = [1.22, 1.69]︸ ︷︷ ︸
using n1 = 1.38
Materials:I Magnesium flouride: n1 = 1.38I Bismuth oxide: n2 = 2.45
Lengths (target wavelength λ0 = 550 nm)
k1`1 = 2.0696 n1`1 = 0.3294λ0 `1 = 131 nm
k2`2 = 0.2848 n2`2 = 0.0453λ0 `2 = 10.2 nm
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Results
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
Some drawbacks:
I Smaller bandwidth
I Complicated procedure
I Small dimensions
(Fig. 6.2.1 in Orfanidis)
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Special case: quarter and half wavelength slabs
For a quarter wavelength slab, ni`i = λ0/4, we have
Zi = ηiZi+1 + jηi tan(ki`i)
ηi + jZi+1 tan(ki`i)=
η2iZi+1
and for a half wavelength slab, ni`i = λ0/2, we have
Zi = ηiZi+1 + jηi tan(ki`i)
ηi + jZi+1 tan(ki`i)= Zi+1
Thus, quarter wavelength slabs inverts the impedance, whereashalf wavelength slabs preserves impedance.
I These rules can be used to make simple designs at targetwavelength.
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An impedance approach to antireflection
The goal is to transform the impedance ηb to ηa. Two possibilities:
(Fig. 6.2.2 in Orfanidis)
First possibility (quarter-quarter):
ηa =η21Z2
=η21
η22/ηb= ηb
η21η22
⇒ nanb
=n21n22
Second possibility (quarter-half-quarter):
ηa =η21Z2
=η21
η23/ηb= ηb
η21η23
⇒ nanb
=n21n23
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Resulting bandwidth
The design was made for the center wavelength λ = 550 nm, theresult for any wavelength is computed with multidiel.m:
(Fig. 6.2.3 in Orfanidis)
Note there was no requirement on the half-wavelength slab! Thefunction of this slab is to increase the bandwidth.
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How does this look in the complex plane?
-1
-0.5
0
0.5
1
-1 -0.5 0 0.5 1-0.25
-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
-0.15 -0.1 -0.05 0 0.05
Mismatched quarter-quarter, matched quarter, matchedquarter-half-quarter, mismatched quarter-half-quarter.The effect of the half-wavelength slab is to add the twist for thered curve, since Γ` = Γ0e
−2jk`.
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Outline
1 Introduction
2 Multiple dielectric slabs
3 Antireflection coatings
4 Dielectric mirrors
5 Propagation bandgaps
6 Narrow-band transmission filters (Fabry-Perot resonators)
7 Conclusions
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Dielectric mirrors
Instead of anti-reflection, we can design a multilayered structureaimed at significant reflection.
Useful to avoid losses in metal structures, which may absorb a fewpercent of power at optical frequencies. Typical design issues:thickness and bandwidth.
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Design at center wavelength
Alternating quarter wavelength slabs of high (nH) and low (nL)refractive index (short hand notation AH(LH)4G):
(Fig. 6.3.1 in Orfanidis)
The impedance at interface 2 is
Z2 =η2LZ3
=η2L
η2H/Z4=
(nHnL
)2
Z4 =
(nHnL
)4
Z6 = · · · =(nHnL
)8
ηb
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Reflection
The impedance at interface 1 after N bilayers and a final nH layeris
Z1 =η2HZ2
=η20/n
2H(
nHnL
)2Nηb
which implies the reflection coefficient
Γ1 =Z1 − ηaZ1 + ηa
= · · · =1−
(nHnL
)2N n2H
nanb
1 +(nHnL
)2N n2H
nanb
→ −1, N →∞
Thus, for many layers, we get high reflection. Without the final nHlayer, the reflection would have the limit Γ → +1.
nH nL N 1 2 4 8 16
2.32 1.38 |Γ1|2 0.7685 0.9111 0.9884 0.9998 1.0000
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Propagation on other wavelengths
(Fig. 6.3.1 in Orfanidis)(E1+
E1−
)=
1
τ1
(ejδH ρ1e
−jδH
ρ1ejδH e−jδH
)FN
1
τ2
(1 ρ2ρ2 1
)(E′M+1,+
0
)F =
1
1 + ρ
(ejδL ρe−jδL
ρejδL e−jδL
)1
1− ρ
(ejδH −ρe−jδH−ρejδH e−jδH
)δH = kH`H, δL = kL`L, ρ =
nH − nLnH + nL
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Results for different number of layers
(Fig. 6.3.2 in Orfanidis)
Computed using multidiel.m with the parameters na = 1,nb = 1.52, nH = 2.32, nL = 1.38. Bandwidth is relativelyinsensitive to the number of layers, ca 600−430
500 = 0.34.
For the infinite structure, the bandwidth is (will be shown later)
∆λ
λ0=π
2
(1
arccos(ρ)− 1
arccos(−ρ)
)= 0.336, ρ =
nH − nLnH + nL
= 0.254
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Different optical thicknesses
(Fig. 6.3.3 in Orfanidis)
Computed using multidiel.m with the parameters na = 1,nb = 1.48, nH = 4.6, nL = 1.6. The optical thicknesses arenH`H = 0.2944λ0 and nL`L = 0.2112λ0, and N = 4.
I Note that nH`H + nL`L = 0.5λ0, this is the important period.I Higher bandwidth (∆λ/λ0 = 0.73) due to higher contrast
(ρ = 0.48).
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Multiband reflectors
Short-hand notation for the dielectric mirror: AH(LH)8G, whereA denotes the air medium, H the high-index medium, L thelow-index medium, and G is the glass. Multiband reflectors can bemade by inserting extra multiples of one or both of H and L slabs:
(Fig. 6.3.4 in Orfanidis)
Note fundamental period of ∆f/f0 = 2.
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Short- and longpass reflectors
The frequency dependence can be manipulated by the outmostlayers. Original design is AH(LH)8G centered at 500nm.
Shortpass: A(0.5L)H(LH)8(0.5L)G
Longpass: A(0.5H)L(HL)8(0.5H)G
(Fig. 6.3.5 in Orfanidis)
Note how the roles of H and L are dual in the two designs. Dottedcurve is original design centered at the corresponding λ0.
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Outline
1 Introduction
2 Multiple dielectric slabs
3 Antireflection coatings
4 Dielectric mirrors
5 Propagation bandgaps
6 Narrow-band transmission filters (Fabry-Perot resonators)
7 Conclusions
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Bandwidth for an infinite structure
Consider the limit case N →∞. We look for waves propagatingwith the effective wave number K, so that there should existsolutions (where δL = kL`L, δH = kH`H, and ` = `L + `H)(
EiHi
)=
(cos(δL) jηL sin(δL)
jη−1L sin(δL) cos(δL)
)·(
cos(δH) jηH sin(δH)
jη−1H sin(δH) cos(δH)
) (Ei+2
Hi+2
)︸ ︷︷ ︸=e−jK`(Ei
Hi)
This is an eigenvalue equation, which can be put in the form
cos(K`) =cos(δH + δL)− ρ2 cos(δH − δL)
1− ρ2
where ρ = nH−nLnH+nL
. K is called the Bloch wavenumber.
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Bandgaps
With δH = δL = δ, the equation is
cos(K`) =cos(2δ)− ρ2
1− ρ2
Propagating waves correspond to real K, which implies| cos(K`)| ≤ 1.
0 1 2 3 4 5−5
−4
−3
−2
−1
0
1
δ/δ0
nH/n
L=2
nH/n
L=4
nH/n
L=8
Inside the band gap, the waves are exponentially attenuated.36 / 50
Bandwidth
The band edges are given by the condition
−1 =cos(2δ)− ρ2
1− ρ2⇒ cos2 δ = ρ2
Since δ = π2λ0/λ, where λ0 is the center wavelength, the band
edges are
λ1 =λ0π/2
arccos(−ρ)λ2 =
λ0π/2
arccos(ρ)
and the bandwidth is
∆λ
λ0=π
2
(1
arccos(ρ)− 1
arccos(−ρ)
),
∆f
f0=
4
πarcsin(ρ)
The bandwidth is zero for ρ = 0, and ∆f/f0 = 2 for ρ = 1(λ2 →∞ and f2 → 0 when ρ→ 1), where
ρ =nH − nLnH + nL
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Bandgaps generalize to 3D structures
Manufactured and measured at FOI in Linkoping.
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General periodic structures
I Propagation bandgaps may appear in any periodic structure:I CrystalsI GratingsI Periodically loaded transmission lines
I Often analyzed by solving for the Bloch waves, propagatingwith factor e−jK·r, which leads to eigenvalue problems.
Looking for solutions [E(r),H(r)]e−jK·r where E(r) and H(r)are periodic functions, turns Maxwell’s equations into(
0 −(∇− jK)× I(∇− jK)× I 0
)(E(r)
H(r)
)= −jω
(ε(r) ξ(r)ζ(r) µ(r)
)(E(r)
H(r)
)with periodic boundary conditions. For some bands of ω, there areno solutions with real-valued K-vectors.
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Multidimensional bandgap structures
See http://ab-initio.mit.edu for free text book, lecture notes andcomputational software MPB (eigensolver) and Meep (FDTD).
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MIT website
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Outline
1 Introduction
2 Multiple dielectric slabs
3 Antireflection coatings
4 Dielectric mirrors
5 Propagation bandgaps
6 Narrow-band transmission filters (Fabry-Perot resonators)
7 Conclusions
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Application: Fabry-Perot resonator
Having designed a dielectric mirror, we can use it to construct aFabry-Perot resonator, which is a narrow-band transmission filter.
Since a half wavelength slab preserves impedance, we can eliminatesuch slabs at the design frequency. By inserting a low-indexmaterial between two bilayer stacks, we obtain
(HL)NL(HL)N = (HL)N−1HL L HL(HL)N−1
→ (HL)N−1HHL(HL)N−1
→ (HL)N−1L(HL)N−1 → · · · → L
Adding another outer layer L, the final structure(HL)NL(HL)NL→ 2L allows perfect transmission at the designwavelength.
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Example of a Fabry-Perot resonator design
(Fig. 6.5.1 in Orfanidis)
Computed using multidiel.m with parameters na = nb = 1.52,nL = 1.4, and nH = 2.1.
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Two FPRs
G|(HL)N1L(HL)N1 |(HL)N2L(HL)N2 |G
(Fig. 6.5.2 in Orfanidis)
The resonance bandwidth is controlled by the number of layers.
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Three FPRs
G|(HL)N1L(HL)N1 |(HL)N2L(HL)N2 |(HL)N1L(HL)N1 |L|G
(Fig. 6.5.3 in Orfanidis)
Note an extra L layer is added. The ripple is decreased by slightincrease of middle FPR.
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Four FPRs
G|(HL)N1L(HL)N1 |(HL)N2L(HL)N2 |(HL)N2L(HL)N2 |(HL)N1L(HL)N1 |G
(Fig. 6.5.4 in Orfanidis)
No extra L layer needed. The ripple is decreased by slight increaseof the middle FPRs.
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Control of the design
The resonance bandwidth is controlled by the number of layers.
(Fig. 6.5.2 in Orfanidis)
The isolation of the resonance is controlled by the contrast.
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Outline
1 Introduction
2 Multiple dielectric slabs
3 Antireflection coatings
4 Dielectric mirrors
5 Propagation bandgaps
6 Narrow-band transmission filters (Fabry-Perot resonators)
7 Conclusions
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Conclusions
I Multilayer structures are easily analyzed by cascadingtechniques.
I Typical designs are based on quarter wavelength and halfwavelength slabs.
I A stack of high/low index slabs can form a dielectric mirror.
I Insertion of extra layers can form a Fabry-Perot resonator.
I High bandwidth and high attenuation requires many layers.
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