Calculating Thin Film Stack Properties

Transcription

1 Lecture 5: Thin Films Outline 1 Thin Films 2 Calculating Thin Film Stack Properties 3 Fabry-Perot Tunable Filter 4 Anti-Reflection Coatings 5 Interference Filters Christoph U. Keller, Leiden University, keller@strw.leidenuniv.nl ATI 2016 Lecture 5: Thin Films 1

2 Introduction thin film: layer with thickness λ extends in 2 other dimensions λ reflection, refraction at all interfaces L layers of thin films: thin film stack layer thickness d i λ interference between reflected and refracted waves infinite number of multiple reflections and refractions must be considered assumption: substrate (index n s ), incident medium (index n m ) have infinite thickness :! 0 90º n m n L, d L n 3, d 3 n 2, d 2 n 1, d 1 n s Christoph U. Keller, Leiden University, keller@strw.leidenuniv.nl ATI 2016 Lecture 5: Thin Films 2

3 Thin Films and Polarization some polarizers (plate, cube) based on thin-film coatings can dramatically reduce polarizing effects of optical components all 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, Leiden University, ATI 2016 Lecture 5: Thin Films 3

4 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, Leiden University, keller@strw.leidenuniv.nl ATI 2016 Lecture 5: Thin Films 4

5 p/tm Wave: Electric Field at Interface E i E r E i E i E r Er H i H r E i E r ñ m θ 0 θ 0 θ 0 θ 0 ñ 1 θ 1 E t θ 1 E t E t H t E t p/tm wave: electric fields are in (parallel to) plane of incidence E i,r,t : complex amplitudes of E i,r,t of incident, reflected, transmitted electric fields E i,r,t : components tangential to interface of complex amplitudes of E i,r,t 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 Christoph U. Keller, Leiden University, keller@strw.leidenuniv.nl ATI 2016 Lecture 5: Thin Films 5

6 p/tm Wave: Electric Field at Interface (continued) E i E i E r Er E i E r θ 0 θ 0 θ 1 E t E t E t continuous electric field tangential to 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, Leiden University, keller@strw.leidenuniv.nl ATI 2016 Lecture 5: Thin Films 6

7 p/tm Wave: Electric Field at Interface (continued) E i E r H i H r ñ m θ 0 θ 0 ñ 1 θ 1 E t H t continuous electric field tangential to interface E i cos θ 0 E r cos θ 0 = E t cos θ 1 electric field direction 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, Leiden University, keller@strw.leidenuniv.nl ATI 2016 Lecture 5: Thin Films 7

8 p/tm Wave: Magnetic Field at Interface E i E r H i H r ñ m θ 0 θ 0 ñ 1 θ 1 E t H t non-magnetic material (µ = 1): H = ñ k k E k, E perpendicular, cross-product becomes multiplication (complex) magnitudes of E i,r,t and H i,r,t related by H i,r = ñ m E i,r, H t = ñ 1 E t tangential component of H continuous: H i + H r = H t H i,r,t is already tangential to interface: H i,r,t = H i,r,t Christoph U. Keller, Leiden University, keller@strw.leidenuniv.nl ATI 2016 Lecture 5: Thin Films 8

9 p/tm Wave: Magnetic Field at Interface (continued) E i E r H i H r θ 0 θ 0 ñ m ñ 1 θ 1 E t H t tangential component of H continuous: H i + H r = H t expressed in electric field ñ m E i + ñ m E r = ñ 1 E t Christoph U. Keller, Leiden University, keller@strw.leidenuniv.nl ATI 2016 Lecture 5: Thin Films 9

10 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 ñ m ñ 1 θ 0 θ 0 θ 1 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, Leiden University, keller@strw.leidenuniv.nl ATI 2016 Lecture 5: Thin Films 10

11 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, Leiden University, keller@strw.leidenuniv.nl ATI 2016 Lecture 5: Thin Films 11

12 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, Leiden University, keller@strw.leidenuniv.nl ATI 2016 Lecture 5: Thin Films 12

13 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, Leiden University, keller@strw.leidenuniv.nl ATI 2016 Lecture 5: Thin Films 13

14 s/te waves: Electric and Magnetic Fields at Interface E i E r H i H r ñ m θ 0 θ 0 ñ 1 θ 1 E t H t tangential electric field component continuous: E i + E r = E t tangential 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, Leiden University, keller@strw.leidenuniv.nl ATI 2016 Lecture 5: Thin Films 14

15 s-polarized Waves in one Medium! 0 H i E i ñ m ñ 1 E r H r θ 0 θ 0 a - + n 1, d 1 θ 1 b E t 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, Leiden University, keller@strw.leidenuniv.nl ATI 2016 Lecture 5: Thin Films 15

16 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, Leiden University, keller@strw.leidenuniv.nl ATI 2016 Lecture 5: Thin Films 16

17 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, Leiden University, keller@strw.leidenuniv.nl ATI 2016 Lecture 5: Thin Films 17

18 Materials and Manufacturing Materials and Refractive Indices Christoph U. Keller, Leiden University, ATI 2016 Lecture 5: Thin Films 18

19 VLT Coating Chamber Christoph U. Keller, Leiden University, ATI 2016 Lecture 5: Thin Films 19

20 VLT Coating Chamber with Magnetogron Christoph U. Keller, Leiden University, ATI 2016 Lecture 5: Thin Films 20

21 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, Leiden University, ATI 2016 Lecture 5: Thin Films 21

22 Ion-Beam Deposition Christoph U. Keller, Leiden University, ATI 2016 Lecture 5: Thin Films 22

23 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, Leiden University, keller@strw.leidenuniv.nl ATI 2016 Lecture 5: Thin Films 23

24 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, Leiden University, keller@strw.leidenuniv.nl ATI 2016 Lecture 5: Thin Films 24

25 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 R (1 R) 2 sin 2 δ 2 Christoph U. Keller, Leiden University, keller@strw.leidenuniv.nl ATI 2016 Lecture 5: Thin Films 25

26 Fabry Perot Properties 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 I = I max R (1 R) 2 sin 2 δ 2 finesse F F = π R 1 R Christoph U. Keller, Leiden University, keller@strw.leidenuniv.nl ATI 2016 Lecture 5: Thin Films 26

27 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 = extra-dense flint glass at 600 nm: n s = 1.75 R = silicon at 600 nm: n s = 3.96 R = 0.6 loss in transparency ghost reflections Christoph U. Keller, Leiden University, keller@strw.leidenuniv.nl ATI 2016 Lecture 5: Thin Films 27

28 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 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, Leiden University, keller@strw.leidenuniv.nl ATI 2016 Lecture 5: Thin Films 28

29 Uncoated Fused Silica Christoph U. Keller, Leiden University, ATI 2016 Lecture 5: Thin Films 29

30 MgF 2 Coating on Fused Silica Christoph U. Keller, Leiden University, keller@strw.leidenuniv.nl ATI 2016 Lecture 5: Thin Films 30

31 Multi-Layer Coatings on Fused Silica Christoph U. Keller, Leiden University, ATI 2016 Lecture 5: Thin Films 31

32 AR Coating Types Christoph U. Keller, Leiden University, ATI 2016 Lecture 5: Thin Films 32

33 Highly Reflective Coatings Christoph U. Keller, Leiden University, ATI 2016 Lecture 5: Thin Films 33

34 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, Leiden University, ATI 2016 Lecture 5: Thin Films 34

35 Types of Interference Filters Christoph U. Keller, Leiden University, ATI 2016 Lecture 5: Thin Films 35

36 Bandpass Interference Filter Terminology Bandpass: range of wavelengths passed by filter Blocking: degree of attenuation outside of passband Center Wavelength (CWL): wavelength at midpoint of passband FWHM Number of cavities: Fabry-Perot-like thin-film arrangement Full-width Half-Maximum (FWHM): width of bandpass at one-half of peak transmission Peak Transmission in %: maximum transmission in passband Christoph U. Keller, Leiden University, ATI 2016 Lecture 5: Thin Films 36

37 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 Christoph U. Keller, Leiden University, ATI 2016 Lecture 5: Thin Films 37

38 Multiple Cavity Performance Christoph U. Keller, Leiden University, ATI 2016 Lecture 5: Thin Films 38