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Electromagnetic Wave PropagationLecture 15: Multilayer film applications II
Daniel Sjoberg
Department of Electrical and Information Technology
October 12, 2010
Outline
1 Antireflection coatings at oblique incidence
2 Omnidirectional dielectric mirrors
3 Polarizing beam splitters
4 Reflection and refraction in birefringent media
5 Brewster and critical angles in birefringent media
6 Giant birefringent optics
7 Concluding remarks
Daniel Sjoberg, Department of Electrical and Information Technology
Outline
1 Antireflection coatings at oblique incidence
2 Omnidirectional dielectric mirrors
3 Polarizing beam splitters
4 Reflection and refraction in birefringent media
5 Brewster and critical angles in birefringent media
6 Giant birefringent optics
7 Concluding remarks
Daniel Sjoberg, Department of Electrical and Information Technology
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?
Daniel Sjoberg, Department of Electrical and Information Technology
Two-layer coating
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]);
Daniel Sjoberg, Department of Electrical and Information Technology
Demo program
Daniel Sjoberg, Department of Electrical and Information Technology
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
Daniel Sjoberg, Department of Electrical and Information Technology
Outline
1 Antireflection coatings at oblique incidence
2 Omnidirectional dielectric mirrors
3 Polarizing beam splitters
4 Reflection and refraction in birefringent media
5 Brewster and critical angles in birefringent media
6 Giant birefringent optics
7 Concluding remarks
Daniel Sjoberg, Department of Electrical and Information Technology
Dielectric mirrors
Increasing angle of incidence implies
I Frequency shift
I Polarization dependence
Omnidirectionality requires overlap of frequency bands for TE andTM for all angles.
Daniel Sjoberg, Department of Electrical and Information Technology
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
Daniel Sjoberg, Department of Electrical and Information Technology
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.
Daniel Sjoberg, Department of Electrical and Information Technology
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.Daniel Sjoberg, Department of Electrical and Information Technology
Frequency response
nH = 3, nL = 1.38, N = 30
Calculated with multidiel.m.
Daniel Sjoberg, Department of Electrical and Information Technology
Bandwidths and angle
nH = 3, nL = 1.38 nH = 2, nL = 1.38
Omnidirectional Not omnidirectional
Daniel Sjoberg, Department of Electrical and Information Technology
Outline
1 Antireflection coatings at oblique incidence
2 Omnidirectional dielectric mirrors
3 Polarizing beam splitters
4 Reflection and refraction in birefringent media
5 Brewster and critical angles in birefringent media
6 Giant birefringent optics
7 Concluding remarks
Daniel Sjoberg, Department of Electrical and Information Technology
Polarizing beam splitter
The TM polarization travels at the Brewster angle if
na sin θa = na1√2=
nHnL√n2H + n2L
Hence TM is transmitted and TE reflected.Daniel Sjoberg, Department of Electrical and Information Technology
Wavelength dependence
na = nb = 1.55 (quartz), nH = 2.3 (zinc sulfide), nL = 1.25(cryolite), N = 5
Not many material combinations exist that satisfy
na = nb =
√2nHnL√n2H + n2L
Daniel Sjoberg, Department of Electrical and Information Technology
Another solution
na = nb = 1.62 (dense flint glass), nH = 2.04 (zirconium oxide),nL = 1.385 (magnesium fluoride), N = 5
Lower reflectance for the TM polarization.
Daniel Sjoberg, Department of Electrical and Information Technology
Outline
1 Antireflection coatings at oblique incidence
2 Omnidirectional dielectric mirrors
3 Polarizing beam splitters
4 Reflection and refraction in birefringent media
5 Brewster and critical angles in birefringent media
6 Giant birefringent optics
7 Concluding remarks
Daniel Sjoberg, Department of Electrical and Information Technology
Effective refractive index
D =
ε1 0 00 ε2 00 0 ε3
·E, B = µ0H, k = Nk0k
N =1√
cos2 θn21
+ sin2 θn23
N = n2
ηTM = η0N cos θ
n21ηTE =
η0n2 cos θ
Daniel Sjoberg, Department of Electrical and Information Technology
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);
Daniel Sjoberg, Department of Electrical and Information Technology
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
Daniel Sjoberg, Department of Electrical and Information Technology
Outline
1 Antireflection coatings at oblique incidence
2 Omnidirectional dielectric mirrors
3 Polarizing beam splitters
4 Reflection and refraction in birefringent media
5 Brewster and critical angles in birefringent media
6 Giant birefringent optics
7 Concluding remarks
Daniel Sjoberg, Department of Electrical and Information Technology
Critical angles
n3 < n′3, n2 < n′2 n3 > n′3, n2 > n′2
sin θ′c =n3n
′3√
n23n′23 + n
′21 (n
′23 − n23)
sin θc =n3n
′3√
n23n′23 + n21(n
23 − n
′23 )
sin θ′c =n2n′2
sin θc =n′2n2
Daniel Sjoberg, Department of Electrical and Information Technology
Brewster angles
tan θB =n3n
′3
n21
√n21 − n
′21
n23 − n′23
tan θ′B =n3n
′3
n′21
√n21 − n
′21
n23 − n′23
Requires n1 > n′1 and n3 > n′3, or n1 < n′1 and n3 < n′3.
Daniel Sjoberg, Department of Electrical and Information Technology
Examples of Fresnel coefficients
(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]
Daniel Sjoberg, Department of Electrical and Information Technology
Normalized TM reflection
Normalized by reflectance at normal incidence.Daniel Sjoberg, Department of Electrical and Information Technology
Differences from the isotropic case
I The Brewster angle can be zero
I The Brewster angle may not exist
I The Brewster angle may be imaginary (both ρTE and ρTM
increase monotonically with angle)
I There may be total internal reflection for one polarization butnot for the other
I The reflection coefficient can be the same for TE and TM
Daniel Sjoberg, Department of Electrical and Information Technology
Multilayer birefringent structures
Everything keeps generalizing, just use transverse componentsconsistently. Snel’s law is Na sin θa = Ni sin θi = Nb sin θb, andeverything is coded in (where ni = [ni,1, ni,2, ni,3])
[Gamma1, Z1] = multidiel(n, L, lambda, theta, pol);
Daniel Sjoberg, Department of Electrical and Information Technology
Outline
1 Antireflection coatings at oblique incidence
2 Omnidirectional dielectric mirrors
3 Polarizing beam splitters
4 Reflection and refraction in birefringent media
5 Brewster and critical angles in birefringent media
6 Giant birefringent optics
7 Concluding remarks
Daniel Sjoberg, Department of Electrical and Information Technology
Dielectric mirror
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.
Daniel Sjoberg, Department of Electrical and Information Technology
First design
N = 50, nH = [1.8, 1.8, 1.5], nL = [1.5, 1.5, 1.5], na = nb = 1
Relatively similar TE and TM, TM is a bit more narrowband.
Daniel Sjoberg, Department of Electrical and Information Technology
Second design
N = 30, nH = [1.8, 1.8, 1.5], nL = [1.5, 1.5, 1.8], na = nb = 1.4
Identical TE and TM, but depends on angle.
Daniel Sjoberg, Department of Electrical and Information Technology
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
Perturbation at higher angles.
Daniel Sjoberg, Department of Electrical and Information Technology
GBO mirrors
M. F. Weber et al, Science, vol 287, p. 2451, 2000.
Daniel Sjoberg, Department of Electrical and Information Technology
GBO reflective polarizer
N = 80, nH = [1.86, 1.57, 1.57], nH = [1.57, 1.57, 1.57],na = nb = 1
Mismatch only in one direction.
Daniel Sjoberg, Department of Electrical and Information Technology
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
Daniel Sjoberg, Department of Electrical and Information Technology
Outline
1 Antireflection coatings at oblique incidence
2 Omnidirectional dielectric mirrors
3 Polarizing beam splitters
4 Reflection and refraction in birefringent media
5 Brewster and critical angles in birefringent media
6 Giant birefringent optics
7 Concluding remarks
Daniel Sjoberg, Department of Electrical and Information Technology
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
Daniel Sjoberg, Department of Electrical and Information Technology
Only projects and orals remaining!
Daniel Sjoberg, Department of Electrical and Information Technology