Jons vctor & matrcs PY3 Colást na hollscol Corcagh, Ér Unvrst Collg Cork, Irland Dpartmnt of Phscs
Matr tratmnt of polarzaton Consdr a lght ra wth an nstantanous -vctor as shown k, t ˆ k, t ˆ k t, o o kzt kzt Colást na hollscol Corcagh, Ér Unvrst Collg Cork, Irland PY3 Dpartmnt of Phscs
Matr tratmnt of polarzaton Combnng th componnts: ˆ kzt kzt kz t ˆ ˆ ~ o o o kzt o ˆ Th trms n brackts rprsnts th compl ampltud of th plan wav o Colást na hollscol Corcagh, Ér Unvrst Collg Cork, Irland PY3 Dpartmnt of Phscs 3
Jons Vctors Th stat of polarzaton of lght s dtrmnd b th rlatv ampltuds o, o and, th rlatv phass = of ths componnts Th compl ampltud s wrttn as a two-lmnt matr, th Jons vctor ~ o ~ ~ o o o o o o Colást na hollscol Corcagh, Ér Unvrst Collg Cork, Irland PY3 Dpartmnt of Phscs 4
Horzontall polarzd lght Th lctrc fld oscllatons ar onl along th -as Th Jons vctor s thn wrttn, Th arrows ndcat th sns of movmnt as th bam approachs ou ~ o ~ ~ o o o A A whr w hav st th phas =, for convnnc Th normalzd form s Colást na hollscol Corcagh, Ér Unvrst Collg Cork, Irland PY3 Dpartmnt of Phscs 5
Vrtcall polarzd lght Th lctrc fld oscllatons ar onl along th -as Th Jons vctor s thn wrttn, ~ o ~ ~ o o o A A Whr w hav st th phas =, for convnnc Th normalzd form s Colást na hollscol Corcagh, Ér Unvrst Collg Cork, Irland PY3 Dpartmnt of Phscs 6
Arbtrar Angl For an arbtrar lnar polarzaton Thn w hav: ~ ~ o ~ o o and th Jons vctor s smpl a ln nclnd at an angl = tan ( o / o ) snc w can wrt ~ o o o o o cos sn cos A sn Th normalzd form s cos sn Colást na hollscol Corcagh, Ér Unvrst Collg Cork, Irland PY3 Dpartmnt of Phscs 7
Crcular polarzaton Th Jons vctor for ths cas whr lads s ~ o A o A o A Th normalzd form s, Ths vctor rprsnts crcularl polarzd lght, whr rotats countrclockws (from prspctv of sourc) Ths mod s calld lft-crcularl polarzd lght What s th corrspondng vctor for rght-crcularl polarzd lght? Rplac / wth / to gt Colást na hollscol Corcagh, Ér Unvrst Collg Cork, Irland PY3 Dpartmnt of Phscs 8
llptcall polarzd lght If o o,.g. f o = A and o = B Th Jons vctor can b wrttn A B Tp of rotaton? countrclockws A B Tp of rotaton? clockws What dtrmns th major or mnor as of th llps? Colást na hollscol Corcagh, Ér Unvrst Collg Cork, Irland PY3 Dpartmnt of Phscs 9
Jons vctor and polarzaton In gnral, th Jons vctor for th arbtrar cas s an llps m; m + ~ o o o B A cos sn o b tan cos o o o o a o Colást na hollscol Corcagh, Ér Unvrst Collg Cork, Irland PY3 Dpartmnt of Phscs
Lnar polarzr Slctvl rmovs all or most of th - vbratons cpt n a gvn drcton TA Lnar polarzr Colást na hollscol Corcagh, Ér Unvrst Collg Cork, Irland PY3 Dpartmnt of Phscs
Jons matr : lnar polarzr Consdr a lnar polarzr wth transmsson as along th vrtcal (). Lt a X matr rprsnt th polarzr opratng on vrtcall polarzd lght. Th transmttd lght must also b vrtcall polarzd. Thus, a b c d Thus, w know: a b c d a b c d Colást na hollscol Corcagh, Ér Unvrst Collg Cork, Irland PY3 Dpartmnt of Phscs
Jons matr : lnar polarzr Th rlatonshp must b tru for an & Thrfor a b s not a vald soluton! Instad: a b c d And: M Colást na hollscol Corcagh, Ér Unvrst Collg Cork, Irland PY3 Dpartmnt of Phscs 3
Optcal lmnts: Rotator Rotats th drcton of lnarl polarzd lght b a partcular angl SA Rotator Colást na hollscol Corcagh, Ér Unvrst Collg Cork, Irland PY3 Dpartmnt of Phscs 4
Jons matr for a rotator An -vctor oscllatng lnarl at s rotatd b an angl Thus, th lght must b convrtd to on that oscllats lnarl at ( + ) a c bcos d sn cos sn On thn fnds: acos ccos bsn d sn cos( sn( ) ) Colást na hollscol Corcagh, Ér Unvrst Collg Cork, Irland PY3 Dpartmnt of Phscs 5
Jons matr for a rotator Rmmbr Trg dntts: Thrfor: And: cos( sn( ) ) a cos cos sn cos cos sn cos b sn c sn d cos cos sn M R sn cos th Rotator Matr sn sn Colást na hollscol Corcagh, Ér Unvrst Collg Cork, Irland PY3 Dpartmnt of Phscs 6
Rotator Proprts Th rotator matr can also b usd to rotat a Matr M( ) R( ) MR( ) R( ) cos sn sn cos R( ) cos sn sn cos Start wth a lnar polarsr: M () M () R( ) cos sn sn cos cos sn Colást na hollscol Corcagh, Ér Unvrst Collg Cork, Irland PY3 Dpartmnt of Phscs 7
Matr Rotaton M( ) R( ) MR( ) Thrfor: cos sn cos sn cos cos sn cos sn cos sn cos sn cos M ( ) sn cos sn sn s th Jons matr at an arbtrar angl Colást na hollscol Corcagh, Ér Unvrst Collg Cork, Irland PY3 Dpartmnt of Phscs 8
Wav Plats Rmmbr th Optcal Path Lngth? In a brfrngnt matral, t s possbl for thr to b a dffrnt nd of rfracton for ach polarsaton. ( o) ( ) n n o d d Rmmbr that th quaton of th M fld s: So: k d n o n o p ( k t) s a masur of phas, and thrfor k d n o n nds nd For a wav plat Colást na hollscol Corcagh, Ér Unvrst Collg Cork, Irland PY3 Dpartmnt of Phscs 9
Phas Rtardr Introducs a phas dffrnc (Δφ) btwn orthogonal componnts Th fast as(fa) and slow as (SA) ar shown FA SA Rtardaton plat Colást na hollscol Corcagh, Ér Unvrst Collg Cork, Irland PY3 Dpartmnt of Phscs
PY3 Colást na hollscol Corcagh, Ér Unvrst Collg Cork, Irland Dpartmnt of Phscs Jons matr : phas rtardr W wsh to fnd a matr whch wll transform th lmnts as follows: It s as to show b nspcton that, Hr and rprsnt th advanc n phas of th componnts o o o o nto nto o o M
PY3 Colást na hollscol Corcagh, Ér Unvrst Collg Cork, Irland Dpartmnt of Phscs Quartr Wav Plat M 4 A Quartr Wav plat s charactrsd b: For ampl: 4 4 4 4 M 4 4 Th ntal phas trm s arbtrar, thus tpcall: so
PY3 Colást na hollscol Corcagh, Ér Unvrst Collg Cork, Irland Dpartmnt of Phscs 3 Jons matrcs: HWP M M HWP, SA vrtcal HWP, SA horzontal Slow as
Jons Dtals Jons Vctors and Matrcs ar Normalsd to Lnar Polarsaton: kz ˆ cos ˆ sn t cos sn cos sn Jons: * cos cos sn sn cos sn Colást na hollscol Corcagh, Ér Unvrst Collg Cork, Irland PY3 Dpartmnt of Phscs 4
PY3 Colást na hollscol Corcagh, Ér Unvrst Collg Cork, Irland Dpartmnt of Phscs 5 Jons Dtals Jons Vctors and Matrcs ar Normalsd to Crcular Polarsaton: ˆ ˆ t kz * Jons:
Jons Dtals Jons Vctors and Matrcs ar Normalsd to Lnar Polarsr: If w consdr a normalsd Jons vctor For lnar polarsd lght: X Y cos cos sn Th mamum valu s : Smlarl wth all othr matrcs! Colást na hollscol Corcagh, Ér Unvrst Collg Cork, Irland PY3 Dpartmnt of Phscs 6
Jons Dtals Whn w masur lght, w masur th ntnst or powr. Not th ampltud: Lnar Polarsaton: kz ˆ cos ˆ sn t cos I * cos sn cos sn I I S sn W can ffcntl us th Jons formalsm to calculat Th absorpton or transmsson through an optcal ntt. T * Thrfor Jons formalsm s also normalsd for Intnst c o cos B sn c o Colást na hollscol Corcagh, Ér Unvrst Collg Cork, Irland PY3 Dpartmnt of Phscs 7
Jons ampl - Consdr a lnar polarzr wth lnarl polarzd lght: cos sn cos cos cos cos sn cos sn sn cos sn sn sn cos cos sn sn Th amount absorbd s: Colást na hollscol Corcagh, Ér Unvrst Collg Cork, Irland PY3 Dpartmnt of Phscs 8
Jons ampl - Consdr a lnar polarzr wth crcularl polarzd lght: cos sn cos sn cos sn cos sn cos sn cos sn Half th lght s transmttd and half s absorbd! Colást na hollscol Corcagh, Ér Unvrst Collg Cork, Irland PY3 Dpartmnt of Phscs 9
Handlng Hardr Problms Drvng Jons matrcs,.g. for crcular polarsaton a b c d Ths gvs: a b for normalsd, c d And th soluton s not provdd. What should w do? 3 Stps: ) Appl gnral cas (normalsd) ) Appl spcfc cas (all passng) 3) Appl spcfc cas (all blockng) Colást na hollscol Corcagh, Ér Unvrst Collg Cork, Irland PY3 Dpartmnt of Phscs 3
Handlng Hardr Problms ) Appl gnral cas (normalsd) a b c d Ths gvs: a No soluton t b for normalsd, c d Colást na hollscol Corcagh, Ér Unvrst Collg Cork, Irland PY3 Dpartmnt of Phscs 3
Handlng Hardr Problms ) Appl spcfc cas (all passng) a b c d for Ths gvs: a b, c d All passng So: a b, c d Colást na hollscol Corcagh, Ér Unvrst Collg Cork, Irland PY3 Dpartmnt of Phscs 3
Handlng Hardr Problms 3) Appl spcfc cas (all blockng) a b c d for All blockng Ths gvs: a b, c d So: a b, c d Colást na hollscol Corcagh, Ér Unvrst Collg Cork, Irland PY3 Dpartmnt of Phscs 33
PY3 Colást na hollscol Corcagh, Ér Unvrst Collg Cork, Irland Dpartmnt of Phscs 34 Handlng Hardr Problms Usng ) and 3) w hav Thus: d c b a
PY3 Colást na hollscol Corcagh, Ér Unvrst Collg Cork, Irland Dpartmnt of Phscs 35 Fnal Proof ) ) 3)