Lecture 6: Thin Films Outline 1 Thin Films 2 Calculating Thin Film Stack Properties 3 Polarization Properties of Thin Films 4 Anti-Reflection Coatings 5 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 λ extends in 2 other dimensions λ 2 boundary layers to neighbouring media: reflection, refraction at both interfaces! 0 90º n m n L, d L layer thickness d i λ interference between reflected and refracted waves L layers of thin films: thin film stack substrate (index n s ) and incident medium (index n m ) have infinite thickness : n 3, d 3 n 2, d 2 n 1, d 1 n s 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 on thin-film coatings can dramatically reduce polarizing effects of optical components aluminum mirrors have aluminum oxide thin film Calculating Thin-Film Stack Properties many layers consider all interferences between reflected and refracted rays of each interface complexity of calculation significantly reduced by matrix approach signs and conventions are not conistent in literature only 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 = E 0 e i( k x ωt) layers numbered from 1 to L with complex index of refraction ñ j, geometrical thickness d j substrate has refractive index ñ s! 0 90º n m incident medium has index ñ m angle of incidence in incident medium: θ 0 Snell s law: : n L, d L n 3, d 3 n 2, d 2 n 1, d 1 n s ñ m sin θ 0 = ñ L sin θ L =... = ñ 1 sin θ 1 = ñ s sin θ s θ j for every layer j Christoph U. Keller, Utrecht University, C.U.Keller@uu.nl ATI 2010 Lecture 6: Thin Films 4
p/tm Wave: Electric Field at Interface E i E r H i H r E t H t E i,r,t : components parallel to interface of complex amplitudes of E 0 of incident, reflected, transmitted electric fields E i = E i cos θ 0, E r = E r cos θ 0, E t = E t cos θ 1 E i,r,t : complex amplitudes of E 0 of incident, reflected, transmitted electric fields continuous electric field interface: E i E r = E t E i cos θ 0 E r cos θ 0 = E t cos θ 1 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) E i E r H i H r E t H t continuous electric field interface E i cos θ 0 E r cos θ 0 = E t cos θ 1 direction of electric field vector fully determined by angle of incidence sufficient to look at complex scalar quantities instead of full 3-D vector since the electric field is perpendicular to the wave vector and 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 E i E r H i H r E t H t non-magnetic material (µ = 1): H 0 = ñ k k E 0 (complex) magnitudes related by of E 0 and H 0 H i,r = ñ m E i,r, H t = ñ 1 E t parallel component of magnetic field continuous ñ m E i + ñ m E r = ñ 1 E t 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 towards substrate (+ superscript) wave that travels away from substrate ( superscript) H i E i E r E t H r at interface a, tangential components of complex electric and magnetic field amplitudes in medium 1 given by E a = E + 1a E 1a H a = ñ 1 ( E + cos θ 1a + E ) 1a 1 negative sign for outwards traveling electric field component a b - H t! 0 + n 1, d 1 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 wave propagation with common phase factor d 1 : geometrical thickness of layer phase factor for forward propagating wave: δ = 2π λ ñ1d 1 cos θ 1 a b -! 0 + n 1, d 1 backwards propagating wave: same phase factor with negative sign Christoph U. Keller, Utrecht University, C.U.Keller@uu.nl ATI 2010 Lecture 6: Thin Films 9
Plane Wave Path Length for Oblique Incidence! D A!! C d n B consider theoretical reflections in single medium need to correct for plane wave propagation path length for reflected light : AB + BC AD 2d cos θ 2d tan θ sin θ = 2d 1 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! 0 at interface b in medium 1 E + 1b = E + 1a (cos δ + i sin δ) a - E 1b = E 1a (cos δ i sin δ) + n H + 1b = H + 1, d 1 1a (cos δ + i sin δ) b H 1b = H 1a (cos δ i sin δ) ) from before E a,b = E + 1a,b E 1a,b, H a,b = ñ1 cos θ 1 (E + 1a,b + E 1a,b propagation of tangential components from a to b ( ) ( ) cos θ Eb cos δ 1 ( ) i sin δ ñ = 1 Ea H ñ b i 1 cos θ 1 sin δ cos δ H a 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 E i E r H i H r E t H t parallel electric field component continuous: E i + E r = E t parallel magnetic field component continuous H i cos θ 0 H r cos θ 0 = H t cos θ 1 and using relation between H and E ñ m cos θ 0 E i ñ m cos θ 0 E r = ñ 1 cos θ 1 E t Christoph U. Keller, Utrecht University, C.U.Keller@uu.nl ATI 2010 Lecture 6: Thin Films 12
s-polarized Waves in one Medium! 0 H i E i E r E t H r a - + n 1, d 1 b H t at boundary a: E a = E + 1a + E 1a H a = (E + 1a E 1a )ñ 1 cos θ 1 propagation of tangential components from a to b ( ) ( ) 1 ( Eb cos δ i sin δ = ñ 1 cos θ Ea 1 iñ 1 cos θ 1 sin δ cos δ H b H a ) Christoph U. Keller, Utrecht University, C.U.Keller@uu.nl ATI 2010 Lecture 6: Thin Films 13
Summary of Matrix Method for each layer j calculate: θ j using Snell s law: n m sin θ 0 = n j sin θ j s-polarization: η j = n j cos θ j p-polarization: η j = n j cos θ j phase delays: δ j = 2π λ n jd j cos θ j characteristic matrix: M j = ( cos i δj η j sin δ j iη j sin δ j cos δ j ) 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 characteristic matrices M = M L M L 1...M 2 M 1 fields in incident medium given by ( ) ( ) Em 1 = M H m η s complex reflection and transmission coefficients r = η me m H m η m E m + H m, t = 2η m η m E m + H m Christoph U. Keller, Utrecht University, C.U.Keller@uu.nl ATI 2010 Lecture 6: Thin Films 15
Materials and Manufacturing Materials and Refractive Indices dielectric materials: Material n(550 nm) transparency range in nm MgF 2 1.38 210-10000 TiO 2 2.2-2.7 350-12000 HfO 2 2.0 220-12000 SiO 2.0 500-8000 SiO 2 1.46 190-8000 ZrO 2 2.1 340-12000 metals: Al Ag Au Christoph U. Keller, Utrecht University, C.U.Keller@uu.nl ATI 2010 Lecture 6: Thin Films 16
VLT Coating Chamber Christoph U. Keller, Utrecht University, C.U.Keller@uu.nl ATI 2010 Lecture 6: Thin Films 17
VLT Coating Chamber with Magnetogron Christoph U. Keller, Utrecht University, C.U.Keller@uu.nl ATI 2010 Lecture 6: Thin Films 18
Evaporation evaporation of solid material through high electric current (e.g. classic Al mirror coatings) sputtering: plasma glow discharge ejects material from solid substance Deposition uncontrolled ballistic flights, mechanical shields to homogenize beam ion-assisted deposition Christoph U. Keller, Utrecht University, C.U.Keller@uu.nl ATI 2010 Lecture 6: Thin Films 19
Ion-Beam Deposition Christoph 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 and s-polarized light has E y-axis reflection transmission ( ) rp 0 J r = 0 r s ( ) tp 0 J r = 0 t s 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 Stacks reflection r p 2 + r s 2 r p 2 r s 2 0 0 M r = 1 r p 2 r s 2 r p 2 + r s 2 0 0 2 0 0 2Re ( rp ) r s 2Im ( rp ) r s 0 0 2Im ( rp ) ( ) r s 2Re r p r s transmission t p 2 + t s 2 t p 2 t s 2 0 0 M t = η s t p 2 t s 2 t p 2 + t s 2 0 0 2η m 0 0 2Re ( tpt ) s 2Im ( tpt ) s 0 0 2Im ( tpt ) ( ) s 2Re t p t s M t includes correction for projection Christoph U. Keller, Utrecht University, C.U.Keller@uu.nl ATI 2010 Lecture 6: Thin Films 22
Mueller Matrix for Dielectric reflection M r = 1 ( ) tan 2 α 2 sin α + transmission M t = 1 sin 2i sin 2r 2 (sin α + cos α ) 2 c 2 + c 2 + c 2 c 2 + 0 0 c 2 c 2 + c 2 + c 2 + 0 0 0 0 2c + c 0 0 0 0 2c + c c 2 + 1 c 2 1 0 0 c 2 1 c 2 + 1 0 0 0 0 2c 0 0 0 0 2c α ± = θ i ± θ t, c ± = cos α ±, and Snell: sin θ i = n sin θ t Christoph U. Keller, Utrecht University, C.U.Keller@uu.nl ATI 2010 Lecture 6: Thin Films 23
Mueller Matrix for Reflection off Metal M r = 1 2 ρ 2 s + ρ 2 p ρ 2 s ρ 2 p 0 0 ρ 2 s ρ 2 p ρ 2 s + ρ 2 p 0 0 0 0 2ρ s ρ p cos δ 2ρ s ρ p sin δ 0 0 2ρ s ρ p sin δ 2ρ s ρ p cos δ real quantities ρ s, ρ p, δ = φ s φ p defined by ρ p e iφp = sin (i r) cos i ñ cos r ρ s e iφs = = sin (i + r) cos i + ñ cos r tan (i r) tan (i + r) = (ñ cos r cos i cos i + ñ cos r ) ( ) ñ cos r cos i sin 2 i ñ cos r cos i + sin 2 i Christoph U. Keller, Utrecht University, C.U.Keller@uu.nl ATI 2010 Lecture 6: Thin Films 24
Example: Coated Metal Mirror mirror with n = 1.2, k = 7.5 (aluminum at 630 nm) at 45 1.000 0.028 0.000 0.000 0.028 1.000 0.000 0.000 0.000 0.000 0.983 0.180 0.000 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 0.000 0.000 0.000 1.000 0.000 0.000 0.000 0.000 1.000 Q V disappears, I Q reduced special 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 1899 interference between partially transmitting plates containing medium with index of refraction n angle of incidence in material θ, distance d path 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 continued path difference between successive beams: = 2nd cos θ phase difference: δ = 2π /λ = 4πnd cos θ/λ incoming wave: e iωt intensity transmission at surface: T intensity reflectivity at surface: R outgoing wave is the coherent sum of all beams write this as Ae iωt = Te iωt + TRe i(ωt+δ) + TR 2 e i(ωt+2δ) +... A = T (1 + Re iδ + R 2 e i2δ +... = T 1 Re iδ Christoph U. Keller, Utrecht University, C.U.Keller@uu.nl ATI 2010 Lecture 6: Thin Films 28
Fabry Perot continued emerging amplitude A = emerging intensity is therefore I = AA = with I max = T 2 /(1 R) 2 T 1 Re iδ T 2 1 2R cos δ + R 2 = T 2 (1 R) 2 + 4R sin 2 δ 2 I = I max 1 1 + 4R (1 R) 2 sin 2 δ 2 Christoph U. Keller, Utrecht University, C.U.Keller@uu.nl ATI 2010 Lecture 6: Thin Films 29
Fabry Perot Properties I = I max 1 1 + 4R (1 R) 2 sin 2 δ 2 transmission is periodic distance between transmission peaks, free spectral range FSR = λ 2 0 2nd cos θ Full-Width at Half Maximum (FWHM) λ λ = FSR/F finesse F F = π R 1 R Christoph U. Keller, Utrecht University, C.U.Keller@uu.nl ATI 2010 Lecture 6: Thin Films 30
Anti-Reflection Coatings Reflection Off Uncoated Substrate reflectivity of bare substrate: ( nm n s R = n m + n s ) 2 fused silica at 600 nm: n s = 1.46 R = 0.035 extra-dense flint glass at 600 nm: n s = 1.75 R = 0.074 silicon at 600 nm: n s = 3.96 R = 0.6 loss in transparency ghost reflections Christoph U. Keller, Utrecht University, C.U.Keller@uu.nl ATI 2010 Lecture 6: Thin Films 31
Analytical treatment of single layer assume single-layer coating determine optimum thickness and index of material amplitude reflection from coating/air interface A 1 = n 1 1 n 1 + 1 amplitude reflection from coating/substrate interface A 1 = n s n 1 n s + n 1 amplitudes subtract for 180 degree phase difference coating should have optical path length λ/4 thick best cancellation for n 1 = n s 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
MgF 2 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 things band-pass, long-pass, short-pass, dichroic filters colored glass substrates often used in addition to coatings sensitivie to angle of incidence evaporated 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 filter Bandpass Interference Filter: interference filter designed to transmit specific wavelengths band Blocking: degree of attenuation outside of passband Center Wavelength (CWL): wavelength at midpoint of passband FWHM Filter Cavity: Fabry-Perot-like thin-film arrangement Full-width Half-Maximum (FWHM): width of bandpass at one-half of maximum transmission Peak Transmission: maximum percentage transmission within passband Christoph U. Keller, Utrecht University, C.U.Keller@uu.nl ATI 2010 Lecture 6: Thin Films 38
Cavities basic design element: Fabry-Perot cavity cavity: λ/2 spacer and reflective multi-layer coatings on either side stacks of cavities provide much better performance www.fluorescence.com/tutorial/intfilt.htm Christoph U. Keller, Utrecht University, C.U.Keller@uu.nl ATI 2010 Lecture 6: Thin Films 39