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, C.U.Keller@uu.nl 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, C.U.Keller@uu.nl 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, C.U.Keller@uu.nl 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, C.U.Keller@uu.nl 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, C.U.Keller@uu.nl 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, C.U.Keller@uu.nl 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, C.U.Keller@uu.nl 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, C.U.Keller@uu.nl 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

+-

Christoph U. Keller, Utrecht University, C.U.Keller@uu.nl ATI 2010 Lecture 6: Thin Films 9

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

Christoph U. Keller, Utrecht University, C.U.Keller@uu.nl ATI 2010 Lecture 6: Thin Films 10

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

)

Christoph U. Keller, Utrecht University, C.U.Keller@uu.nl ATI 2010 Lecture 6: Thin Films 11

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

Christoph U. Keller, Utrecht University, C.U.Keller@uu.nl ATI 2010 Lecture 6: Thin Films 12

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, C.U.Keller@uu.nl 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

)

Christoph U. Keller, Utrecht University, C.U.Keller@uu.nl ATI 2010 Lecture 6: Thin Films 14

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

Christoph U. Keller, Utrecht University, C.U.Keller@uu.nl ATI 2010 Lecture 6: Thin Films 15

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

Christoph U. Keller, Utrecht University, C.U.Keller@uu.nl ATI 2010 Lecture 6: Thin Films 16

VLT Coating Chamber

www.eso.org/public/images/eso9711aChristoph U. Keller, Utrecht University, C.U.Keller@uu.nl ATI 2010 Lecture 6: Thin Films 17

VLT Coating Chamber with Magnetogron

www.eso.org/public/images/eso9950jChristoph U. Keller, Utrecht University, C.U.Keller@uu.nl ATI 2010 Lecture 6: Thin Films 18

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

Christoph U. Keller, Utrecht University, C.U.Keller@uu.nl ATI 2010 Lecture 6: Thin Films 19

Ion-Beam Deposition

www.4waveinc.com/ibd.htmlChristoph U. Keller, Utrecht University, C.U.Keller@uu.nl ATI 2010 Lecture 6: Thin Films 20

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, C.U.Keller@uu.nl 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

Christoph U. Keller, Utrecht University, C.U.Keller@uu.nl ATI 2010 Lecture 6: Thin Films 22

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

Christoph U. Keller, Utrecht University, C.U.Keller@uu.nl ATI 2010 Lecture 6: Thin Films 23

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

)

Christoph U. Keller, Utrecht University, C.U.Keller@uu.nl ATI 2010 Lecture 6: Thin Films 24

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)

Christoph U. Keller, Utrecht University, C.U.Keller@uu.nl ATI 2010 Lecture 6: Thin Films 25

Wavelength-Dependence of Q ⇒ V Cross-Talk

only limited wavelength range can be corrected

Christoph U. Keller, Utrecht University, C.U.Keller@uu.nl ATI 2010 Lecture 6: Thin Films 26

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

Christoph U. Keller, Utrecht University, C.U.Keller@uu.nl ATI 2010 Lecture 6: Thin Films 27

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

Christoph U. Keller, Utrecht University, C.U.Keller@uu.nl ATI 2010 Lecture 6: Thin Films 28

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

Christoph U. Keller, Utrecht University, C.U.Keller@uu.nl ATI 2010 Lecture 6: Thin Films 29

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

Christoph U. Keller, Utrecht University, C.U.Keller@uu.nl ATI 2010 Lecture 6: Thin Films 30

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

Christoph U. Keller, Utrecht University, C.U.Keller@uu.nl ATI 2010 Lecture 6: Thin Films 31

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

Christoph U. Keller, Utrecht University, C.U.Keller@uu.nl ATI 2010 Lecture 6: Thin Films 32

Uncoated Fused Silica

Christoph U. Keller, Utrecht University, C.U.Keller@uu.nl ATI 2010 Lecture 6: Thin Films 33

MgF2 Coating on Fused Silica

Christoph U. Keller, Utrecht University, C.U.Keller@uu.nl ATI 2010 Lecture 6: Thin Films 34

Multi-Layer Coatings on Fused Silica

Christoph U. Keller, Utrecht University, C.U.Keller@uu.nl ATI 2010 Lecture 6: Thin Films 35

Highly Reflective Coatings

Christoph U. Keller, Utrecht University, C.U.Keller@uu.nl ATI 2010 Lecture 6: Thin Films 36

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

Christoph U. Keller, Utrecht University, C.U.Keller@uu.nl ATI 2010 Lecture 6: Thin Films 37

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

Christoph U. Keller, Utrecht University, C.U.Keller@uu.nl ATI 2010 Lecture 6: Thin Films 38

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, C.U.Keller@uu.nl ATI 2010 Lecture 6: Thin Films 39