Lecture 6: Thin Films
Outline
1 Thin Films2 Calculating Thin Film Stack Properties3 Polarization Properties of Thin Films4 Anti-Reflection Coatings5 Interference Filters
Christoph U. Keller, Utrecht University, [email protected] ATI 2010 Lecture 6: Thin Films 1
Introduction
thin film:layer with thickness . λextends in 2 other dimensions� λ
2 boundary layers to neighbouringmedia: reflection, refraction at bothinterfaceslayer thickness di . λ⇒interference between reflected andrefracted wavesL layers of thin films: thin film stacksubstrate (index ns) and incidentmedium (index nm) have infinitethickness
:
ns
n1, d1
n2, d2
nL, dL
n3, d3
nm
90º
!0
Christoph U. Keller, Utrecht University, [email protected] ATI 2010 Lecture 6: Thin Films 2
Thin Films and Polarization
some polarizers (plate, cube) based onthin-film coatingscan dramatically reduce polarizing effectsof optical componentsaluminum mirrors have aluminum oxidethin film
Calculating Thin-Film Stack Properties
many layers⇒ consider all interferences between reflected andrefracted rays of each interfacecomplexity of calculation significantly reduced by matrix approachsigns and conventions are not conistent in literatureonly isotropic materials here
Christoph U. Keller, Utrecht University, [email protected] ATI 2010 Lecture 6: Thin Films 3
Plane Waves and Thin-Film Stacks
plane wave ~E = ~E0ei(~k ·~x−ωt)
layers numbered from 1 to L withcomplex index of refraction nj ,geometrical thickness dj
substrate has refractive index ns
incident medium has index nm
angle of incidence in incidentmedium: θ0
Snell’s law:
nm sin θ0 = nL sin θL = ... = n1 sin θ1 = ns sin θs
⇒ θj for every layer j
:
ns
n1, d1
n2, d2
nL, dL
n3, d3
nm
90º
!0
Christoph U. Keller, Utrecht University, [email protected] ATI 2010 Lecture 6: Thin Films 4
p/TM Wave: Electric Field at Interface
Ei E
r
Et
Hi H
r
Ht
Ei,r ,t : components parallel to interface of complex amplitudes of~E0 of incident, reflected, transmitted electric fields
Ei = Ei cos θ0, Er = Er cos θ0, Et = Et cos θ1
Ei,r ,t : complex amplitudes of ~E0 of incident, reflected, transmittedelectric fieldscontinuous electric field ‖ interface: Ei − Er = Et
Ei cos θ0 − Er cos θ0 = Et cos θ1
Christoph U. Keller, Utrecht University, [email protected] ATI 2010 Lecture 6: Thin Films 5
p/TM Wave: Electric Field at Interface (continued)
Ei E
r
Et
Hi H
r
Ht
continuous electric field ‖ interface
Ei cos θ0 − Er cos θ0 = Et cos θ1
direction of electric field vector fully determined by angle ofincidencesufficient to look at complex scalar quantities instead of full 3-Dvector since the electric field is perpendicular to the wave vectorand in the plane of incidence
Christoph U. Keller, Utrecht University, [email protected] ATI 2010 Lecture 6: Thin Films 6
p/TM Wave: Magnetic Field at Interface
Ei E
r
Et
Hi H
r
Ht
non-magnetic material (µ = 1): ~H0 = n ~k|~k |× ~E0
(complex) magnitudes related by of ~E0 and ~H0
Hi,r = nmEi,r , Ht = n1Et
parallel component of magnetic field continuous
nmEi + nmEr = n1Et
Christoph U. Keller, Utrecht University, [email protected] ATI 2010 Lecture 6: Thin Films 7
Matrix Formalism: Tangential Components in one Medium
single interface in thin-film stack,combine all waves into
wave that travels towardssubstrate (+ superscript)wave that travels away fromsubstrate (− superscript)
at interface a, tangentialcomponents of complex electric andmagnetic field amplitudes inmedium 1 given by
Ea = E+1a − E−1a
Ha =n1
cos θ1
(E+
1a + E−1a
)negative sign for outwards travelingelectric field component
Ei E
r
Et
Hi H
r
Ht
n1, d1
!0
a
bb
+-
Christoph U. Keller, Utrecht University, [email protected] ATI 2010 Lecture 6: Thin Films 8
Matrix Formalism: Electric Field Propagation
field amplitudes in medium 1 at(other) interface b from wavepropagation with common phasefactord1: geometrical thickness of layerphase factor for forward propagatingwave:
δ =2πλ
n1d1 cos θ1
backwards propagating wave: samephase factor with negative sign
n1, d1
!0
a
bb
+-
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Plane Wave Path Length for Oblique Incidence
n
!
!
!
A
B
C
D
d
consider theoretical reflections in single mediumneed to correct for plane wave propagationpath length for “reflected light”: AB + BC − AD
2dcos θ
− 2d tan θ · sin θ = 2d1− sin2 θ
cos θ= 2d cos θ
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Matrix Formalism
at interface b in medium 1
E+1b = E+
1a(cos δ + i sin δ)
E−1b = E−1a(cos δ − i sin δ)
H+1b = H+
1a(cos δ + i sin δ)
H−1b = H−1a(cos δ − i sin δ)
n1, d1
!0
a
bb
+-
from before Ea,b = E+1a,b − E−1a,b, Ha,b = n1
cos θ1
(E+
1a,b + E−1a,b
)propagation of tangential components from a to b(
EbHb
)=
(cos δ cos θ1
n1i sin δ
i n1cos θ1
sin δ cos δ
)(EaHa
)
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s/TE waves: Electric and Magnetic Fields at Interface
Ei
Er
Et
Hi
Hr
Ht
parallel electric field component continuous: Ei + Er = Et
parallel magnetic field component continuous
Hi cos θ0 − Hr cos θ0 = Ht cos θ1
and using relation between H and E
nm cos θ0Ei − nm cos θ0Er = n1 cos θ1Et
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s-Polarized Waves in one Medium
Ei
Er
Et
Hi
Hr
Ht
n1, d1
!0
a
bb
+-
at boundary a:
Ea = E+1a + E−1a
Ha = (E+1a − E−1a)n1 cos θ1
propagation of tangential components from a to b(EbHb
)=
(cos δ 1
n1 cos θ1i sin δ
i n1 cos θ1 sin δ cos δ
)(EaHa
)Christoph U. Keller, Utrecht University, [email protected] ATI 2010 Lecture 6: Thin Films 13
Summary of Matrix Methodfor each layer j calculate:
θj using Snell’s law: nm sin θ0 = nj sin θj
s-polarization: ηj = nj cos θj
p-polarization: ηj =nj
cos θj
phase delays: δj = 2πλ njdj cos θj
characteristic matrix:
Mj =
(cos δj
iηj
sin δj
iηj sin δj cos δj
)
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Summary of Matrix Method (continued)total characteristic matrix M is product of all characteristicmatrices
M = MLML−1...M2M1
fields in incident medium given by(EmHm
)= M
(1ηs
)complex reflection and transmission coefficients
r =ηmEm − Hm
ηmEm + Hm, t =
2ηm
ηmEm + Hm
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Materials and Manufacturing
Materials and Refractive Indicesdielectric materials:Material n(550 nm) transparency range in nmMgF2 1.38 210-10000TiO2 2.2-2.7 350-12000HfO2 2.0 220-12000SiO 2.0 500-8000SiO2 1.46 190-8000ZrO2 2.1 340-12000
metals:AlAgAu
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VLT Coating Chamber
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VLT Coating Chamber with Magnetogron
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Evaporationevaporation of solid material through high electric current (e.g.classic Al mirror coatings)sputtering: plasma glow discharge ejects material from solidsubstance
Depositionuncontrolled ballistic flights, mechanical shields to homogenizebeamion-assisted deposition
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Ion-Beam Deposition
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Polarization Properties of Thin Films
Jones Matrices for Thin Film Stacks
in a coordinate system where p-polarized light has ~E ‖ x-axis ands-polarized light has ~E ‖ y -axisreflection
Jr =
(rp 00 rs
)transmission
Jr =
(tp 00 ts
)other orientations through rotation of Jones matrices
Christoph U. Keller, Utrecht University, [email protected] ATI 2010 Lecture 6: Thin Films 21
Mueller Matrices for Thin Film Stacksreflection
Mr =12
|rp|2 + |rs|2 |rp|2 − |rs|2 0 0|rp|2 − |rs|2 |rp|2 + |rs|2 0 0
0 0 2Re(r∗p rs
)2Im
(r∗p rs
)0 0 −2Im
(r∗p rs
)2Re
(r∗p rs
)
transmission
Mt =ηs
2ηm
|tp|2 + |ts|2 |tp|2 − |ts|2 0 0|tp|2 − |ts|2 |tp|2 + |ts|2 0 0
0 0 2Re(t∗p ts)
2Im(t∗p ts)
0 0 −2Im(t∗p ts)
2Re(t∗p ts)
Mt includes correction for projection
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Mueller Matrix for Dielectricreflection
Mr =12
(tanα−sinα+
)2
c2− + c2
+ c2− − c2
+ 0 0c2− − c2
+ c2− + c2
+ 0 00 0 −2c+c− 00 0 0 −2c+c−
transmission
Mt =12
sin 2i sin 2r
(sinα+ cosα−)2
c2− + 1 c2
− − 1 0 0c2− − 1 c2
− + 1 0 00 0 2c− 00 0 0 2c−
α± = θi ± θt , c± = cosα±, and Snell: sin θi = n sin θt
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Mueller Matrix for Reflection off Metal
Mr =12
ρ2
s + ρ2p ρ2
s − ρ2p 0 0
ρ2s − ρ2
p ρ2s + ρ2
p 0 00 0 2ρsρp cos δ 2ρsρp sin δ0 0 −2ρsρp sin δ 2ρsρp cos δ
real quantities ρs, ρp, δ = φs − φp defined by
ρseiφs = −sin (i − r)
sin (i + r)=
cos i − n cos rcos i + n cos r
ρpeiφp =tan (i − r)
tan (i + r)=
(n cos r − cos icos i + n cos r
)(n cos r cos i − sin2 in cos r cos i + sin2 i
)
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Example: Coated Metal Mirrormirror with n = 1.2, k = 7.5 (aluminum at 630 nm) at 45◦
1.000 0.028 0.000 0.0000.028 1.000 0.000 0.0000.000 0.000 0.983 0.1800.000 0.000 −0.180 0.983
126-nm thick dielectric layer with n = 1.4 on top
1.000 −0.009 0.000 0.000−0.009 1.000 0.000 0.0000.000 0.000 1.000 0.0000.000 0.000 0.000 1.000
Q ⇒ V disappears, I ⇒ Q reducedspecial mirrors minimize cross-talk (e.g. GONG turret)
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Wavelength-Dependence of Q ⇒ V Cross-Talk
only limited wavelength range can be corrected
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Fabry-Perot Filter
Tunable Filter
invented by Fabry and Perot in 1899interference between partially transmitting plates containingmedium with index of refraction nangle of incidence in material θ, distance dpath difference between successive beams: ∆ = 2nd cos θ
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Fabry Perot continuedpath difference between successive beams: ∆ = 2nd cos θphase difference: δ = 2π∆/λ = 4πnd cos θ/λincoming wave: eiωt
intensity transmission at surface: Tintensity reflectivity at surface: Routgoing wave is the coherent sum of all beams
Aeiωt = Teiωt + TRei(ωt+δ) + TR2ei(ωt+2δ) + ...
write this as
A = T (1 + Reiδ + R2ei2δ + ... =T
1− Reiδ
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Fabry Perot continuedemerging amplitude
A =T
1− Reiδ
emerging intensity is therefore
I = AA∗ =T 2
1− 2R cos δ + R2 =T 2
(1− R)2 + 4R sin2 δ2
with Imax = T 2/(1− R)2
I = Imax1
1 + 4R(1−R)2 sin2 δ
2
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Fabry Perot Properties
I = Imax1
1 + 4R(1−R)2 sin2 δ
2
transmission is periodicdistance between transmissionpeaks, free spectral range
FSR =λ2
02nd cos θ
Full-Width at Half Maximum (FWHM)∆λ
∆λ = FSR/F
finesse F
F =π√
R1− R
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Anti-Reflection Coatings
Reflection Off Uncoated Substratereflectivity of bare substrate:
R =
(nm − ns
nm + ns
)2
fused silica at 600 nm: ns = 1.46⇒ R = 0.035extra-dense flint glass at 600 nm: ns = 1.75⇒ R = 0.074silicon at 600 nm: ns = 3.96⇒ R = 0.6
loss in transparencyghost reflections
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Analytical treatment of single layerassume single-layer coatingdetermine optimum thickness and index of materialamplitude reflection from coating/air interface
A1 =n1 − 1n1 + 1
amplitude reflection from coating/substrate interface
A1 =ns − n1
ns + n1
amplitudes subtract for 180 degree phase difference⇒ coatingshould have optical path length λ/4 thickbest cancellation for n1 =
√ns
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Uncoated Fused Silica
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MgF2 Coating on Fused Silica
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Multi-Layer Coatings on Fused Silica
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Highly Reflective Coatings
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Interference Filters
Overview
many layers can achieve many thingsband-pass, long-pass, short-pass, dichroic filterscolored glass substrates often used in addition to coatingssensitivie to angle of incidenceevaporated coatings are very temperature sensitive
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Terminology (adapted from www.fluorescence.com/tutorial/int-filt.htm)
Bandpass: range of wavelengths passed by filterBandpass Interference Filter: interference filter designed totransmit specific wavelengths bandBlocking: degree of attenuation outside of passbandCenter Wavelength (CWL): wavelength at midpoint of passbandFWHMFilter Cavity: Fabry-Perot-like thin-film arrangementFull-width Half-Maximum (FWHM): width of bandpass atone-half of maximum transmissionPeak Transmission: maximum percentage transmission withinpassband
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Cavities
www.fluorescence.com/tutorial/int-
filt.htm
basic design element: Fabry-Perot cavitycavity: λ/2 spacer and reflectivemulti-layer coatings on either sidestacks of cavities provide much betterperformance
Christoph U. Keller, Utrecht University, [email protected] ATI 2010 Lecture 6: Thin Films 39