Lecture 2: Thin Films. Thin Films. Calculating Thin Film Stack Properties. Jones Matrices for Thin Film Stacks. Mueller Matrices for Thin Film Stacks

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1 Lecture 2: Thin Films Outline Thin Films 2 Calculating Thin Film Stack Properties 3 Jones Matrices for Thin Film Stacks 4 Mueller Matrices for Thin Film Stacks 5 Mueller Matrix for Dielectrica 6 Mueller Matrix for Metals 7 Applications to Solar Polarimetry Introduction thin film: layer with thickness λ extends in 2 other dimensions λ 2 boundary layers to neighbouring media: reflection, refraction at both interfaces 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 : 90º n m n L, d L n 3, d 3 n 2, d 2 n s Christoph U. Keller, Utrecht University, C.U.Keller@uu.nl Lecture 2: Thin Films Christoph U. Keller, Utrecht University, C.U.Keller@uu.nl Lecture 2: Thin Films 2 Thin Films and Polarimetry Plane Waves and ThinFilm Stacks some polarizers (plate, cube) based on thinfilm coatings can dramatically reduce polarizing effects of optical components aluminum mirrors have aluminum oxide thin film Calculating ThinFilm 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 plane wave E = E 0 e i( k x ωt) layers numbered from to L with complex index of refraction ñ j, geometrical thickness d j substrate has refractive index ñ s incident medium has index ñ m angle of incidence in incident medium: θ 0 Snell s law: ñ m sin θ 0 = ñ L sin θ L =... = ñ sin θ = ñ s sin θ s θ j for every layer j : 90º n m n L, d L n 3, d 3 n 2, d 2 n s Christoph U. Keller, Utrecht University, C.U.Keller@uu.nl Lecture 2: Thin Films 3 Christoph U. Keller, Utrecht University, C.U.Keller@uu.nl Lecture 2: Thin Films 4

2 p/tm Wave: Electric Field at Interface p/tm Wave: Electric Field at Interface (continued),r,t : components parallel to interface of complex amplitudes of E 0 of incident, reflected, transmitted electric fields = cos θ 0, = cos θ 0, = cos θ,r,t : complex amplitudes of E 0 of incident, reflected, transmitted electric fields continuous electric field interface: = cos θ 0 cos θ 0 = cos θ Christoph U. Keller, Utrecht University, C.U.Keller@uu.nl Lecture 2: Thin Films 5 continuous electric field interface cos θ 0 cos θ 0 = cos θ direction of electric field vector fully determined by angle of incidence sufficient to look at complex scalar quantities instead of full 3D 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 Lecture 2: Thin Films 6 p/tm Wave: Magnetic Field at Interface nonmagnetic material (µ = ): H 0 = ñ k k E 0 (complex) magnitudes related by of E 0 and H 0,r = ñ m,r, = ñ parallel component of magnetic field continuous ñ m ñ m = ñ Christoph U. Keller, Utrecht University, C.U.Keller@uu.nl Lecture 2: Thin Films 7 Matrix Formalism: Tangential Components in one Medium single interface in thinfilm stack, combine all waves into wave that travels towards substrate ( superscript) wave that travels away from substrate ( superscript) at interface a, tangential components of complex electric and magnetic field amplitudes in medium given by E a = E a E a H a = ñ ( E cos θ a E ) a negative sign for outwards traveling electric field component a b Christoph U. Keller, Utrecht University, C.U.Keller@uu.nl Lecture 2: Thin Films 8

3 Matrix Formalism: Electric Field Propagation Plane Wave Path Length for Oblique Incidence field amplitudes in medium at (other) interface b from wave propagation with common phase factor d : geometrical thickness of layer phase factor for forward propagating wave: δ = 2π λ ñd cos θ backwards propagating wave: same phase factor with negative sign a b A D B C consider theoretical reflections in single medium need to correct for plane wave propagation path length for reflected light : AB BC AD d 2d cos θ 2d tan θ sin θ = 2d sin2 θ = 2d cos θ cos θ n Christoph U. Keller, Utrecht University, C.U.Keller@uu.nl Lecture 2: Thin Films 9 Christoph U. Keller, Utrecht University, C.U.Keller@uu.nl Lecture 2: Thin Films 0 Matrix Formalism s/te waves: Electric and Magnetic Fields at Interface at interface b in medium E b = E a (cos δ i sin δ) a E b = E a (cos δ i sin δ) n H b = H a (cos δ i sin δ) b H b = H a (cos δ i sin δ) ) from before E a,b = E a,b E a,b, H a,b = ñ cos θ (E a,b E a,b propagation of tangential components from a to b ( ) ( ) cos θ Eb cos δ ( ) i sin δ ñ = Ea H ñ b i cos θ sin δ cos δ H a parallel electric field component continuous: = parallel magnetic field component continuous cos θ 0 cos θ 0 = cos θ and using relation between H and E ñ m cos θ 0 ñ m cos θ 0 = ñ cos θ Christoph U. Keller, Utrecht University, C.U.Keller@uu.nl Lecture 2: Thin Films Christoph U. Keller, Utrecht University, C.U.Keller@uu.nl Lecture 2: Thin Films 2

4 spolarized Waves in one Medium at boundary a: 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 E a = E a E a spolarization: η = n cos θ j H a = (E a E a )ñ cos θ H t 0 ppolarization: η = n cos θ j propagation of tangential components from a a to b ( ) ( ) ( Eb cos δ i sin δ b ) = ñ cos θ Ea iñ cos θ sin δ cos δ H b H a 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 Lecture 2: Thin Films 3 Christoph U. Keller, Utrecht University, C.U.Keller@uu.nl Lecture 2: Thin Films 4 Summary of Matrix Method (continued) total characteristic matrix M is product of all characteristic matrices M = M L M L...M 2 M fields in incident medium given by ( ) ( ) Em = 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 Jones Matrices for Thin Film Stacks in a coordinate system where ppolarized light has E xaxis and spolarized light has E yaxis 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 Lecture 2: Thin Films 5 Christoph U. Keller, Utrecht University, C.U.Keller@uu.nl Lecture 2: Thin Films 6

5 Mueller Matrices for Thin Film Stacks reflection r p 2 r s 2 r p 2 r s M r = r p 2 r s 2 r p 2 r s Re ( 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 M t = η s t p 2 t s 2 t p 2 t s η 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 Mueller Matrix for Dielectric reflection M r = ( ) tan 2 α 2 sin α transmission M t = sin 2i sin 2r 2 (sin α cos α ) 2 c 2 c 2 c 2 c c 2 c 2 c 2 c c c c c c 2 c c 2 c c c 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 Lecture 2: Thin Films 7 Christoph U. Keller, Utrecht University, C.U.Keller@uu.nl Lecture 2: Thin Films 8 Applications of Thin Films in Solar Polarimetry Mueller Matrix for Reflection off Metal M r = 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 ρ 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 Instrumental Polarization of Swedish Vacuum Solar Telescope 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 Lecture 2: Thin Films 9 Wavelength: 525 nm; a) bestfit theoretical model of telescope polarization; b) same model plus empirical offsets (Courtesy Pietro Bernasconi) Christoph U. Keller, Utrecht University, C.U.Keller@uu.nl Lecture 2: Thin Films 20

6 Instrumental Polarization oblique reflections off metal surfaces introduce polarization and retardation oblique reflections off and transmission through dielectrics introduce polarization, but no retardation accurate modeling not easy because of multilayer coatings, oxide layers, and sometimes oil layers best to measure instrumental effects of optics Example: Coated Metal Mirror mirror with n =.2, k = 7.5 (aluminum at 630 nm) at nm thick dielectric layer with n =.4 on top Q V disappears, I Q reduced special mirrors minimize crosstalk (e.g. GONG turret) Christoph U. Keller, Utrecht University, C.U.Keller@uu.nl Lecture 2: Thin Films 2 Christoph U. Keller, Utrecht University, C.U.Keller@uu.nl Lecture 2: Thin Films 22 WavelengthDependence of Q V CrossTalk Oil Layers, Real Mirrors 65nm thick dielectric layer with n =.4 on top maximizes crosstalk between Q and V thin oil film on aluminum mirror can make a big change in telescope Mueller matrix aluminum is covered with thin layer of aluminum oxide that needs to be taken into account only limited wavelength range can be corrected Christoph U. Keller, Utrecht University, C.U.Keller@uu.nl Lecture 2: Thin Films 23 Christoph U. Keller, Utrecht University, C.U.Keller@uu.nl Lecture 2: Thin Films 24