Physical Optics. Lecture 7: Polarization Herbert Gross.

Transcription

1 Phscal Optcs Lecture 7: Polarato 8--6 Herbert Gross

2 Phscal Optcs: Cotet No Date Subject Ref Detaled Cotet 8.. Wave optcs G Comple felds, wave equato, k-vectors, terferece, lght propagato, terferometr 5.. Dffracto G Slt, gratg, dffracto tegral, dffracto optcal sstems, pot spread fucto, aberratos 3.. Fourer optcs G Plae wave epaso, resoluto, mage formato, trasfer fucto, phase magg Qualt crtera ad Ralegh ad Marechal crtera, Strehl rato, coherece effects, two-pot G resoluto resoluto, crtera, cotrast, aal resoluto, CTF 5.. Photo optcs K erg, mometum, tme-eerg ucertat, photo statstcs, fluorescece, Jablosk dagram, lfetme, quatum eld, FRT Coherece K Temporal ad spatal coherece, Youg setup, propagato of coherece, speckle, OCT-prcple Polarato G Itroducto, Joes formalsm, Fresel formulas, brefrgece, compoets Laser K Atomc trastos, prcple, resoators, modes, laser tpes, Q-swtch, pulses, power 9.. Nolear optcs K Bascs of olear optcs, optcal susceptblt, d ad 3rd order effects, CARS mcroscop, photo magg.. PSF egeerg G Apodato, superresoluto, eteded depth of focus, partcle trappg, cofocal PSF 7.. Scatterg G Itroducto, surface scatterg sstems, volume scatterg models, calculato schemes, tssue models, Me Scatterg 4.. Gaussa beams G Basc descrpto, propagato through optcal sstems, aberratos Geeraled beams G Laguerre-Gaussa beams, phase sgulartes, Bessel beams, Ar beams, applcatos superresoluto mcroscop Mscellaeous G Coatgs, dffractve optcs, fbers K = Kempe G = Gross

3 3 Cotets Itroducto Fresel formulas Joes calculus Further descrptos Brefrgece Compoets Applcatos

4 Scalar: Helmholt equato Vectoral: Mawell equatos Scalar / vectoral Optcs ) ( r k o k H k B k D k B k j D H k J J k M H B P D r r 4

5 5 Basc Forms of Polarsato. Lear compoets phase. crcular phase dfferece of 9 betwee compoets 3. ellptcal arbtrar but costat phase dfferece

6 6 Basc Notatos of Polarato Descrpto of electromagetc felds: - Mawell equatos - vectoral ature of feld stregth Decomposto of the feld to compoets Propagato plae wave: - feld vector rotates - projecto compoets are oscllatg susodal A coste A cos( t ) e

7 7 Polarato llpse lmato of the tme depedece: llpse of the vector Dfferet states of polarato: - sese of rotato - shape of ellpse A A A cos A s

8 8 Polarato llpse Represetato of the state of polarato b a ellpse Feld compoets A (cos cos s s ) A Aes of ellpse: a, b (cos cos s s ) Rotato agle of the feld A A ta cos A A A A Agle of eccetrct b b ta a a

9 9 Crcular Polarato Spral curve of feld vector Superposto of left ad rght haded crcular polared lght: resultg lear polarato t t 3 l t t t r Ref: Maset

10 Crcular polared Lght Rotato of plae of polarato wth t / Phase agle 9 Geerato b l/4 plate out of lear polared lght b l t SA b r l LA r TA t l / 4 - plate 45 lear polarer

11 Fresel Formulas s-ad p-polarato Ref: R.Chpma

12 Fresel Formulas Schematcal llustrato of the ra refracto ( reflecto at a terface The cases of s- ad p-polarato must be dstgushed a) s-polarato b) p-polarato reflecto r B r trasmsso reflecto B r r trasmsso t B t B t t ormal to the terface ormal to the terface cdece B terface cdece B terface

13 3 Fresel Formulas lectrcal trasverse polarato T, s- or -polarato, perpedcular to cdece plae Magetcal trasverse polarato TM, p- or p-polarato, cdece plae Boudar codto of Mawell equatos at a delectrc terface: cotuous tagetal compoet of -feld t t Ampltude coeffcets for reflected feld trasmtted feld Reflectvt ad trasmsso of lght power r T r T t tt rt ttm rtm R P r P T r r TM T Pt P r TM cos t cos

14 Fresel Formulas Coeffcets of ampltude for reflected ras, s ad p Coeffcets of ampltude for trasmtted ras, s ad p t t k k k k r cos cos cos cos s cos s cos ) s( ) s( t t k k k k r cos cos cos cos s cos s cos ) ta( ) ta( t k k k t cos cos cos s cos cos cos cos cos t k k k t cos cos cos s cos cos cos cos cos 4

15 5 Fresel Formulas Tpcal behavor of the Fresel ampltude coeffcets as a fucto of the cdece agle for a fed combato of refractve dces = Trasmsso depedet o polarato Reflected p-ras wthout phase jump Reflected s-ras wth phase jump of p (correspods to r<) = 9 No trasmsso possble Reflected lght depedet o polarato r r t t Brewster agle: completel s-polared reflected lght Brewster

16 6 Fresel Formulas: erg vs. Itest Fresel formulas, dfferet represetatos:. Ampltude coeffcets, wth sg. Itest ceffcets: o addtvt due to area projecto 3. Power coeffcets: addtvt due to eerg preservato r, t R, T R, T r t t T (I) T (I) T T r.3. R (I). R (I) R R

17 Decomposto of the feld stregth to two compoets / or s/p Relatve phase agle betwee compoets Polarato ellpse Lear polared lght Crcular polared lght e A A e s cos r l Joes Vector s cos A A A A 7

18 8 Joes Calculus Joes represetato of full polared feld: decomposto to compoets Cascadg of sstem compoets: Product of matrces J, o J p J s J J ss ps J J sp pp J J op os Sstem : Sstem : 3 Sstem 3 : Sstem : J J J 3 J 4 - I prcple 3 tpes of compoets fluecg polarato:. Chage of ampltude polarer, aaler. Chage of phase retarder 3. Rotato of feld compoets rotator J J tras ret J rot t e cos ( ) s s e t p s cos

19 9 Joes Matrces Rotated compoet J ( ) D( ) J () D( ) Rotato matr Itest cos D( ) s I * J J s cos Three tpes of compoets to chage the polarato:. ampltude: polarer / aaler. phase: retarder 3. oretato: rotator Propagato:. free space. delectrc terface J J PRO TRA e ts p l OPL t p J RF rs r p 3. mrror J SP r

20 Brefrgece: Uaal Crstal Brefrgece: de of refracto depeds o feld oretato Uaal crstal: ordar de o perpedcular to crstal as etra-ordar de e alog crstal as Dfferece of dces e o Joes matr J J e p l eo p l eo (, ) e s cos cos s s cos e s cos e e e Relatve phase agle p l e o

21 Descrptos of Polarato Parameter Propertes Polarato ellpse llptct, oretato ol complete polarato Comple parameter Parameter ol complete polarato 3 Joes vectors Compoets of ol complete polarato 4 Stokes vectors Stokes parameter S o... S 4 complete or partal polarato 5 Pocare sphere Pots o or sde the Pocare sphere ol graphcal represetato 6 Coherece matr - matr C complete or partal polarato

22 Stokes Vector Descrpto of polarato from the eergetc pot of vew - S o total test S I(,) I(9,) S o - S Dfferece of test - lear S I(,) I(9,) S - S Dfferece of test lear uder 45 / 35 S I(45,) I(35,) S cos - S 3 Dfferece of crcular compoets S3 I(45,9) I(45, 9) S 3 s

23 3 Stokes Vector Descrpto of a polarato state wth Stokes parameteriterpretato: Compoets of the feld o the Pocare sphere Also partal polarato s take to accout Relato Uequal sg: partal polarato Stokes vector 4 Propagato: Müller matr M S M S S S S S3 S S S S S 3 m m m m 3 m m m m 3 m m m m 3 m m m m S

24 Lear horotal / vertcal LIear 45 Crcular clockwse / couter-clockwse S S S S S amples of Stokes Vectors 4

25 5 Partal Polarato Partal polared lght: degree of polarato p = : p = : u-polared fullpolared < p < : partal polared p p I I pol ges S S S S 3 Determato of Stokes parameter: S ( p S ps S u S p ) Upolared lght Full polared lght S S S S3 S S S3

26 6 Sphere of Pocare ver pot o a u sphere descrbes oe state of polarato I sphercal coordates: r cos s cos cos s Pots o -as: crcular polared lght Merda le: lear polarato left haded crcular polared ellptcal polared Pots sde partal polarato lear polared rght haded crcular polared

27 7 Partal Polarato o the Pocare Sphere Full polared: pot Upolared: full surface Partal polared: probablt dstrbuto, pots sde full polared u-polared partal polared P

28 8 Pocare Sphere ad Stokes Vector Stokes parameter S, S, S 3 : Compoest alog as drectos Radus of sphere, legth of vector: S o Projecto to merdoal plae: agle of polarato ellpse Projecto to merda plae: eccetrct agle S 3 S o P S S

29 9 Polarato Polarato of a doat mode the focal rego:. I focal plae. I defocussed plae Ref: F. Wrowsk

30 3 Ideal Polarato of a Sphercal Wave Ideal lear polared sphercal wavefrot Costat agle of the feld compoets wth the great crcles Ref: R.Chpma

31 3 Propagato of Polarato Joes matr chages Joes vector J Müller matr chages Stokes vector S M S Chage of coherece matr C J C J Procedure for real sstems:. Ratracg. Defto of tal polarato 3. Joes vector or coherece matr local o each ra 4. trasport of ra ad vector chages at all surfaces 5. 3D-effects of Fresel equatos o the feld compoets 6. Coatgs eed a specal treatmet 7. Problems: ra splttg case of brefrgece

32 3D calculus global coordates accordg to Chpma Ra trace defes local coordate sstem Trasform betwee local ad global coordates Polarato Ratrace L L j L L s s s s s s,,,,,,,,,,,,, s s L L L L L L L L L L L out L L L L L L s s s T s s s T,,,,,,,,,,,,,,,,,, plae of cdece terface plae comg ra outgog ra k j- k j j j j 3

33 mbedded local Joes matr Matrces of refractg surface ad reflecto Feld propagato Cascadg of operator matrces Trasfer propertes. Phscal chages. Geometrcal bedg effects Polarato Ratrace,,,,,, j j j j j j j j j p p p p p p p p p P... P P P P P M M total, s p r s p t r r J t t J, j j j j J refr,,, refr out T J T P,,, bed out T J T Q 33

34 34 Datteuato ad Retardato Chage of feld stregth: calculato wth polarato matr, trasmsso T Datteuato gevalues of Joes matr Retardato: phase dfferece of comple egevalues T P P * T D T J ret w arg ma ma / T T m m * / T w P / arg e / w / To be take to accout:. phscal retardace due to refractve de: P. geometrcal retardace due to geometrcal ra bedg: Q Retardato matr R Q total P total

35 Polarato Performace valuato Müller matr vsualato Iterpretato ot trval Retardato ad datteuato map across the pupl Polarato Zerke pupl aberrato accordg to M. Toteck (complcated)

36 36 Asotropc Meda Dfferet tpes of asotropc meda Ref: Saleh / Tech

37 37 Geometr of Ratrace at a Iterface Classcal geometr of brefrget refracto refers o terface plae Gradfathers method: Calculato teratve due to o-lear equatos prsm coordates Hstor: formulas accordg to Muchel / Schöppe Ol plae setup cosdered, crstal as plae of cdece ar crstal crstal as a s eo o e o wave ormal s cdet ra

38 38 Ra Splttg Case of Brefrgece Icdet plae wave Dfferet refractve dces for polarato drecto Importat: relatve oretato agast crstal as Arbtrar cdece: splttg of ras - ordar ra: perpedcular to optcal as - etra ordar ra: plae of cdece crstal as etraordar ra ordar ra

39 Brefrgece Ratracg for brefrgece splttg of the ra two wth dfferet polarato states at the etrace to a brefrget medum For m leses: m ras behd the sstem Lateral walkoff of the ras ffects o ampltude ad phase Re-decomposto of prcpal feld compoets possble I geeral two separated pupls for s/p, whch va ot terfere splttg.. 3. ras 4 8

40 4 Brefrgece Dfferet drectos of ad D Ra splttg ot detcal to wave splttg D ra / phase k / s H, B eerg / wave / Potg P

41 4 Asotropc Meda Relatoshp betwee electrc feld ad dsplacemet vector D r Lear relatoshp of tesor-equato D j l jl l l, m g jlm m l l, m, q jlmq m q l Frst term, coeffcet : local drecto, brefrgece Secod term, coeffcet g: gradet of feld, thrd order, optcal actvt, polarato rotated Thrd term, : forth order, trsc brefrgece, spatal dsperso of brefrgece Geeral: Due to the tesor propertes of the coeffcets, all these effects are asotropc The feld ad the dsplacemet D are o loger alged D s s s s

42 4 Fresel or Ra llpsod Vashg soluto determate of the wave equato c k c c k c c k c Value of speed of lgth depedg o the ra drecto (phase veloct) Alteratve: as / / costat eerg dest electrc feld D dsplacemet vector / /

43 Iverse matr of the delectrc tesor: de or ormal ellpsod Gves the refractve de as a fucto of the oretato Also possble: ellpsod of k-values Ide or Normal llpsod cost r k k k D electrc feld costat eerg dest D D dsplacemet vector 43

44 44 Ide llpsod for Uaal Crstals Specal case uaal crstal: ellpsod rotatoal smmetr o ordar drecto vald for two drectos e etra ordar vald for ol oe drecto Two cases: e > o : postve ( prolate, cgar ) e < o : egatv ( oblate, dsc ) Arbtrar oretato : tersecto pots a) postve brefrgece e > o b) egatve brefrgece e < o optcal as optcal as k 3 k k e-ra e-ra k () k o-ra o-ra k e k k

45 45 Wave-Optcal Costructo of the o/e-drecto Icdet plae wave Osculatg tagetal plae at the ordar de-sphere: defes ormal to o-ra drecto Osculatg tagetal plae at the de o/e-de ellpsod: ormal to e-drecto cdet ra veloct ellpsod crstal as o - ra eo-ra wavefrots

46 46 Brefrget Uaal Meda Classcal uaal meda used polarato compoets:. Quart, postve brefrget, small dfferece. Calcte, egatve brefrget, larger dfferece materal sg o eo Calcte egatve Quar, SO postve quart e o e o calcte

47 47 Refracto wth Brefrgece Optcal smmetr as of crstal materal breaks smmetr Splt of ras depeds o the as oretato Splt of feld to two orthogoal polarsato compoets erg propagato (ra, Potg) geeral ot perpedcular to wavefrot dverget ras parallel ras wth phase dfferece parallel ras phase o-ra e-ra crstal as crstal as crstal as

48 48 Polarer Polarer wth atteuato c s/p J LIN c s c p Rotated polarer J P ( cos ) s cos s cos s Polarer -drecto J P () TA

49 49 Par Polarer-Aaler Polarer ad aaler wth rotato agle Law of Malus: erg trasmsso TA TA cos I( ) I cos o I lear polarer lear polarer polarer aaler parallel perpedcular

50 5 Stress Aalss: Polarer / Aaler Crossed par of polarer - aaler plates No sample: trasmsso ero Polared sample betwee the plates: - spatal resolved rotato of polarato - aalss of stress ad stra TA TA lear polarer aaler lear polarer

51 5 Stress Ijecto modlded les wth stress brefrgece betwee crossed polarers Left: orgal oretato Rght: les rotated b 45 Ref: R.Chpma

52 5 Retarder Phase dfferece betwee feld compoets Retarder plate wth rotato agle Specal value: l / 4 - plate geerates crcular polared lght. fast as J V (, p / ) J RT e e cos s (, ) s cos e s cos e e s cos e J V SA. fast as 45 J V ( p / 4, p / ) LA

53 53 Rotator Rotate the of plae of polarato Realato wth magetc feld: Farad effect b B L V J ROT cos b s b s b cos b Verdet costat V b

54 54 Lqud Crstals Tpes of LC meda: - ematc log molecules, oreted oe drecto - smectc log molecules, oreted oe drecto several laers dfferet tpes A, B, C - dscotc plate-shapes molecules, oreted a plae - cholesterc skrew-shaped molecules Fuctoal prcple: - teracto creates domaes - oretato of the domaes b voltage - asotropc behavor, brefrgece - strog fluece of temperature Parameter to descrbe the order / etrop: S 3cos Applcato: - dsplas - spatal lght modulator

55 55 LCD Cell Oretato of the molecules a eteral feld (voltage) molecules glass plate feld : = feld : >

56 56 Tpes of Lqud Crstals ematc smectc A smectc C

57 57 Tpes of Lqud Crstals smectc C cholesterc

58 58 Lqud Crstal Dspla Trasmsso chages due to polarato Black-Whte swtch possble Reflectve or refractve polarer 9 - TN-LC cell aaler black polarer LC cell mrror whte

59 59 Mcroscopc Cotrast brght feld polarato fluorescece ep llumato tras llumato Ref: M. Kempe

60 Mcroscope Polarato of pupl

61 6 Dfferetal Iterferece Cotrast Pot spread fucto psfdc (, ) ( R) e R e psf psf (, (, ) ) aaler Wollasto prsm adjustmet phase Parameter:. Shear dstace. Phase offset 3. Splttg rato shear dstace objectve object codeser -R R splttg rato Wollasto prsm compesator polarer

62 6 Dfferetal Iterferece Cotrast Dfferetal splttg: Large cotrast at phase gradets orgal / shfted dfferetal splttg

63 63 Dfferetal Iterferece Cotrast Lateral shft : preferred drecto Vssblt depeds o oretato of detals shft 45 -shft ( ) -shft (9 )

64 64 DIC Phase Imagg Oretato of prsms ad shft se determes the asotropc mage formato Ref.: M. Kempe