Outline. Classes of polarizing devices Polarization states. Eigen-polarization of crystals. Momentum matching at boundaries Polarization calculations

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1 Cstal optics lectue C 566 Adv. Optics Lab Outline Classes of polaizing devices Polaization states igen-polaization of cstals Momentum matching at boundaies Polaization calculations Muelle matices Jones matices Anisotopic powe walk-off

2 Polaizing devices C 566 Adv. Optics Lab Thee tpes of phsics. Diattenuation (polaization dependent loss) Tansmission is polaization dependent Polaizes A.k.a.: polaization dependent loss (PDL), dichoism. Retadance Optical path length is polaization dependent. Wave plates, optical activit, electo-optic A.k.a.: polaization mode dispesion (PMD) Poincae sphee geomet 3. Depolaization The degee of polaization ma decease depending on input polaization A.k.a.: polaization scambling R.A. Chipman, Polaizes, Retades, and Depolaizes, PI Annual Mtg,

3 tates of polaization C 566 Adv. Optics Lab O hea.6, aleh & Teich 6 Polaization states Linea states of polaization Linea polaized, Linea polaized, Linea -45 o polaized, φ φ π Linea θ polaized, φ φ, π 3

4 tates of polaization C 566 Adv. Optics Lab Polaization states lliptical states of polaization Right-hand cicula, φ φ π Left-hand cicula, φ φ π lliptical, and/o φ φ ± π 4

5 5 C 566 Adv. Optics Lab tokes vectos Complete desciption of polaization state Pefom 6 iadiance measuements with ideal polaizes: Hoizontal linea Vetical linea o linea o linea R Right cicula L Left cicula + L R V H V H H V R L tates of polaization

6 tates of polaization C 566 Adv. Optics Lab Poincaè sphee Useful visualization of tokes paametes R 3 V H L i j i j + i j + i3 j3 uface is polaized, cente is unpolaized quato is linea polaized Noth hemisphee is ight elliptical, south is left elliptical Othogonal polaizations ae on opposite points of sphee 6

7 7 C 566 Adv. Optics Lab tokes vectos Degee of polaization 3 DoP + + Degee of polaization 3 DoCP Degee of cicula polaization Degee of linea polaization DoLP + tokes vectos ae polchomatic on addition: H + V at diffeent fequencies give Lissagous figue with no aveage polaization state H + V H polaized DoP /3 R.A. Chipman, Polaizes, Retades, and Depolaizes, PI Annual Mtg, tates of polaization

8 tates of polaization igenpolaizations of cstals C 566 Adv. Optics Lab Uniaial suface Positive uniaial n e >n o n Cos ϕ in ± n ± o, + no n e ϕ Fou solutions, odina etaodina 8

9 tates of polaization igenpolaizations of cstals C 566 Adv. Optics Lab Biaial k-suface igenpolaizations ae shown as lines on sufaces lices though aes ae dawn on bounding cube 9

10 Momentum matching C 566 Adv. Optics Lab Momentum-matching at half-space Genealization of nell s Law nˆ ε k n θ tans k tans tˆ k inc θ inc k t k k t t tans t k n eˆ e k k inc tans nˆ + k tˆ k tˆ k t tˆ j( ωtk ) inc t tans inθ Beak tansmitted wave vecto into nomal and tansvese components. inc inθ tans citation on bounda. Tansvese wave vecto conseved. nˆ, tˆ Poblem: Given oientation of bounda ( ), mateial ( ), and bounda ecitation ( t), how does plane wave popagate into mateial ( )? k n ε

11 Momentum matching C 566 Adv. Optics Lab Isotopic efaction a.k.a. nell s law gives a diections at bounda Real space ε ε I k n nˆ θ tans k tans tˆ k inc θ inc k t ε I Fouie space n n Glass Ai.5.5 θ inc n tans n inc n t n n ε n t.5 θ efl n efl Real pat Imag. pat.5 n t n n n inc tans efl inθ inθ inc inθ tans efl

12 Momentum matching C 566 Adv. Optics Lab Total intenal eflection a.k.a. evanescent waves Real space ε I nˆ α n tans e k ( jn tˆ t α n nˆ) k t tˆ k inc θ inc θ efl k efl ε ε I Momentum space n n Glass Ai.5.5 n inc n t θ inc α n n t ε.5 θ efl n efl Real pat Imag. pat.5 n t n n inc efl inθ inθ inc efl

13 Momentum matching C 566 Adv. Optics Lab aleh & Teich 6.3 taodina efaction and othe fun with cstals Real space k n nˆ k k e o tˆ ε o ε ε o ε e k inc k t ε I Momentum space ĉ taodina Odina Ai n t.5 n n n n e o n inc Real pat Imag. pat.5 n t 3

14 Momentum matching C 566 Adv. Optics Lab ample Non-cstalogaphic uniaial slab Note that biefingence has been highl eaggeated. 4

15 Momentum matching C 566 Adv. Optics Lab Conoscop Used to measue ais and biefingence Real space Fouie space Conoscopic patten fo LiNbO 3 o how would we calculate this patten??? 5

16 6 C 566 Adv. Optics Lab Muelle Matics Complete polaization modeling.g. Linea polaize 44 mati of eal values descibing tansfomation tokes paams. Can descibe de-polaizing elements. igenvectos ae eigenpolaization of sstem (unchanged). Polaization calculations Muelle matices Must use Muelle matices in sstems with finite coheence (since can be patiall polaized) Muelle matices & tokes paametes have nice popet that the can be measued BUT, Muelle matices have 6 elements. Is thee something simple?

17 7 C 566 Adv. Optics Lab Jones vectos implified fo full polaized sstems A A J whee A and A ae the comple amplitudes of the and polaized electic fields. * * A A A A J J + Inne poduct H V j R j L Polaization calculations Jones vectos

18 Polaization calculations Muelle & Jones matices C 566 Adv. Optics Lab Table of Jones and Muelle matices 8

19 Polaization calculations Jones vectos C 566 Adv. Optics Lab P J Out P L θ Polaizes Jones mati of linea polaize passing H Powe tansmission of analize and abita linea polaization: T J Out JOut cosθ cosθ sinθ ( ) cos θ Malus Law Dichoism: polaization dependent absoption P e α z Cstal polaizes e β z.g.: heet polaize, polaizing sunglasses, Polaco TM, wie-gid Reflective polaizes: Bewste s angle, thin-film coatings.g.: Wollaston, Rochon, énamont.g.: Glan-Focault 9

20 Polaization calculations Jones vectos C 566 Adv. Optics Lab Rotation matices Components not aligned with o J R( θ )J Rotate a Jones vecto b an angle θ R cosθ sinθ sinθ ( θ ) cosθ whee R(θ) is given b ( θ ) T ( θ ) T R R A otated component, T

21 Polaization calculations Jones vectos C 566 Adv. Optics Lab Jones mati eample Cascaded polaizes Cossed polaizes: z -pol -pol P P so no light leaks though. Rotation toleance otated -pol -pol R ( θ ) P R( θ ) sin θ sinθ cosθ P sin θ sinθ cosθ sinθ sinθ cosθ cos θ so component of leaks into component of Geoage Tech Cente fo Ultafast optics

22 Polaization calculations Jones vectos C 566 Adv. Optics Lab R Retades Common use of uniaials jγ e Jones mati of etade with fast ais in Γ π π 4 Quate-wave plate. Convets linea into cicula. π Γ π n Half-wave plate. Convets linea into linea. Cos θ in θ, + o o e ( θ ) n n n.g. Half-wave plate. Γ L π λ o λo n ( n n ) e o L π Quatz QWP is onl 48 micons thick at 55 nm. Cossing plates of diffeing dispesion can educe λ, θ dependence

23 Polaization calculations Jones vectos C 566 Adv. Optics Lab Poincaè and etades Retades otate the polaization state on the Poincaè sphee. Ais of otation connects eigenpolaizations of the etade. R 3 V H L ample: QWP with hoizontal ais convets 45 o linea into RHC ample: Opticall active (o Faada) otato convets H to 45 o 3

24 Powe walk-off C 566 Adv. Optics Lab Powe walk-off Conside a Fouie popagation poblem in which the k-suface is a tilted line: Fouie space Real space k z z θ k (, z) F F { (,) } dk dk z z, e dk j dk.5 n n z k k s z Fouie popagato hift theoem n t.5.5 4

Waves and Polarization in General

Waves and Polarization in General Waves and Polaization in Geneal Wave means a distubance in a medium that tavels. Fo light, the medium is the electomagnetic field, which can exist in vacuum. The tavel pat defines a diection. The distubance