Chaptr 6: Polarization and Crystal Optics * P6-1. Cascadd Wav Rtardrs. Show that two cascadd quartr-wav rtardrs with paralll fast axs ar quivalnt to a half-wav rtardr. What is th rsult if th fast axs ar orthogonal. P6-2. Jons Matrix of a Polarizr. Show that th Jons matrix of a linar polarizr with a transmission axis making an angl θ with th x axis is 2 cos sincos T 2 (FoP 6.1-25) sincos sin Hint! Driv (FoP 6.1-25) using (FoP 6.1-18), (FoP 6.1-22) and (FoP 6.1-24) * P6-3. Thr polarizrs ar placd aftr ach othr. Th first is illuminatd by unpolarizd light with th intnsity I. Th transmission dirction for th scond and th third polarizr is rotatd 45 and 9 in rlation to th first rspctivly. S figur 6-1. Figur 6-1. Unpolarizd light with th intnsity I is passing through thr polarizrs with diffrnt transmission dirctions. a) Giv th intnsity btwn polarizr 1 and 2 in rlation to I. b) Giv th intnsity btwn polarizr 2 and 3 in rlation to I. c) Giv th intnsity aftr polarizr 3 in rlation to I. * P6-4. Two polarizrs can b usd as a continuously variabl gry filtr. What is th angl b btwn th transmission axs, so that 5. % of incoming light is transmittd? W assum that w can nglct rflctions. P6-5. Giv th propagation dirction and polarization stat for th following wavs: ikxt a) E R2 iyˆ 3zˆ ikyt b) E R xˆ izˆ i / 6 i kzt E R xˆ yˆ 3 c) Pag 26
* P6-6. Giv th polarization stat of th following wav: a) E xˆecos( kzt) yˆecos( kzt / 2) b) E xˆe ˆ cos( tkz) yecos( tkz / 2) c) E xˆecos( tkz) yˆecos( tkz) d) E xˆecos( tkz) yˆecos( tkz / 4) i t kx E R 2iyˆ 3zˆ ) ( ) * P6-7. A plan, linarly polarizd light wav, with intnsity I, is transmittd through a systm of prfct linar polarizrs (w assum that all light is transmittd in th transmission dirction but in th prpndicular dirction all light is absorbd). Giv for th following systms of polarizrs and transmission dirctions th total transmittd intnsity: (angls ar masurd in th sam dirction and rlativly to th polarization dirction of th incidnt light). a) on at 9 angl b) two at th angls 45 and 9. c) thr at th angls 3, 6 and 9. d) N polarizrs an th angls 9 / N,2 9 / N,39 / N,4 9 / N,...9 ) from d) w lt N P6-8. Brwstr window. At what angl is a TM-polarizd bam of light transmittd through a glass plat of rfractiv indx n = 1.5 placd in air (n = 1) without suffring rflction losss at ithr surfac? Such plats, known as Brwstr windows (figur 6-2), ar usd in lasrs, as dscribd in FoP Sc. 15.D. Figur 6-2. Th Brwstr window transmits TM-polarizd light with no rflction loss. * P6-9. Rflctanc of Glass. A plan wav is incidnt from air (n =1) onto a glass plat (n = 1.5) at an angl of incidnc of 45. Dtrmin th powr rflctancs of th TE and TM wavs. What is th avrag rflctanc for unpolarizd light (light carrying TE and TM wavs of qual intnsitis)? P6-1. Lft lliptically polarizd light impings from air (n = 1) to a glass surfac (n = 1.56). Th ratio of th long and short axis of th llips is 3:2 and th long axis is in th plan of incidnc. Calculat th angl of incidnc for which th rflctd light is right circularly polarizd. Pag 27
P6-11. Light impings on a glass plat with th rfractiv indx 1.5. a) Calculat th Brwstr angl. b) Calculat th angl btwn th rflctd bam and th transmittd bam. * P6-12. Whn whit light is rflctd on a glass plat on can obtain plan polarizd light if th plat is orintd at Brwstr angl. a) Calculat th fficincy of th polarizr.g. th intnsity ratio btwn rflctd plan polarizd light and incidnt unpolarizd light. Th rfractiv indx of th glass is 1.54. b) Highr fficincy can b obtaind if transmittd light is usd instad. In this cas th light will not b prfctly polarizd. It is thn bttr to us svral glass plats mountd in Brwstr angl aftr ach othr. Calculat th numbr of plats ndd so that th polarization dgr* is highr than 99%. As in a) th rfractiv indx of th glass is 1.54 I par * Polarisation dgr is givn by: Pdg I I P6-13. Start with Frsnl quations (FoP 6.2-8 and FoP 6.2-9) and show that whn light impings along th normal w hav th following formula: 2 n1 n 2 n1 n2 Utiliz that, for small angls, w can approximat sin and tan with th angl. par prp * P6-14. a) Calculat th thicknss of th thinnst possibl quartr wav plat of crystallin quartz with n = 1.5497, n =1.559 for th vacuum wavlngth 486 nm. Such a plat can b mad thickr if it consists of two plats sandwichd togthr that countract. b) A plan monochromatic, linarly polarizd wav is incidnt on two thin quartr wav rtardrs according to a). Thy ar placd aftr ach othr so that thir optical axs hav th angl α rspctivly β rlativ to th incidnt linar polarization dirction. Giv for th following combinations of α and β th final polarization stat aftr th two rtardrs. In th cas whn th rsultd light is lliptically polarizd, giv th ratio btwn th short axis and th long axis in th llips dscribing th rotation of th E-vctor. It is not ncssary to giv th rotation dirction. Angls I II III IV α 45 45 β 45 45 2 Pag 28
P6-15. Btwn two crossd polarizrs (figur 6-3) thr is a glass plat with thicknss d. This plat can b slightly birfringnt if a mchanical forc is applid at 45 rlativ to th two linar polarizrs. Giv how th transmittd irradianc I aftr th scond polarizr dpnds on th incidnt intnsity I o on th plat, th thicknss d, th vacuum wavlngth and n n n th diffrnc btwn th xtraordinary and th ordinary rfractiv indx for th plat. Figur 6-3. Adjusting th transmittd bam by a birfringnt plat. * P6-16. W hav th following st-up (figur 6-5) for th wavlngth.53 μm. Polarizr Faraday rotator Optically activ crystal Mirror d Optical axis d Vrtical polarization 45 Vrtical polarization Figur 6-4. A Faraday rotator combind with an optically activ crystal. Aftr a linar polarizr thr is a Faraday rotator that turns th polarization dirction 45. This rotator is followd by an optically activ crystal that turns th polarization dirction back to th original dirction. Answr th following qustions: a) A crtain glass typ with th Vrdt constant 5.28 Tcm 1 is usd in th Faraday rotator. It is possibl to us a magntic fild of 1. T. How long glass rod has to b usd? b) As optically activ crystal, a quartz crystal is usd. Th rfractiv indx is n - =1.54424 and n + =1.544271. How thick must th crystal b? c) Aftr th optically activ crystal thr is a mirror that rflcts th light back th sam path as it cam from. Giv th polarization stat aftr th optically activ crystal and aftr th Faraday rotator. Giv also th part (in %) of th rflx that passs back through th first linar polarizr. Pag 29
* P6-17. A Wollaston prism according to th figur 6-5 is usd. Th surrounding mdium is air and th optical axs ar markd. 3 3 Figur 6-5. A Wollaston prism. Th two parts of th prism ar in contact,.g. w can nglct any influnc of air or glu at th bordr surfac. Th matrial is calcit with n =1.6584 and n =1.4864. Unpolarizd light is impinging from lft. Draw th rays through th matrial with an accurat mark of th polarization stats. Calculat also th angl btwn th rsulting rays. * P6-18 A.9 mm thick plan paralll plat of crystallin quartz has th nd surfacs paralll with th optical axs. It is placd btwn two polarizrs which hav thir transmission dirctions paralll so that th optical axis of th quartz plat maks an angl of 45 with th transmission axs of th polarizrs. A numbr of wavlngths ar missing.g. th two adjacnt wavlngths 443 nm and 467 nm. Calculat n n n. P6-19. Th following figur (figur 6-6) shows th cross sction of a Rochon prism with th optical axs markd. Th prism is of calcit with n =1.6584 and n =1.4864. Draw th rays through th prism for an incidnt ray of unpolarizd light that impings prpndicular to th surfac on th lft. Mark th polarization stat of th bam and calculat also th angl btwn th two mrgnt bams. 3 3 Figur 6-6. A Rochon prism. Pag 3
Answrs chaptr 6: P6-1: If orthogonal thn no rtardation is obtaind. P6-3: I/ 2, I/ 4 and I /8 P6-4: 72 P6-5: a) in th positiv x-dirction, lft lliptical polarisation b) in th positiv y-dirction, lft circular polarisation c) in th positiv z-dirction, right lliptical polarization P6-6: a) right circularly polarizd b) lft circularly polarizd c) linarly polarizd d) right lliptically polarizd ) right lliptically polarizd 2 N I cos 2 ) I P6-7: a) b).25i c).42i d) N P6-8: 56.3 P6-9: TE:.92 TM:.85 For unpolarizd light.5 P6-1: 29.7 P6-11: a) 56.3 b) 9 This valu is indpndnt of n P6-12: a) 8.27 % b) at last 13 plats Pag 31
6 P6-14: a) 13 1 m b) Optical axis Optical axis I 45 Circular 45 Linar II 45 Circular Linar III Linar 45 Circular IV 9 Linar 2 Elliptic 2 For th lliptic polarization: Long axis sin 2 tan 2.36 Short axis cos2 2 P6-15: If nd thn 2 I I sin / 2 P6-16: a) 85 mm b) 2. mm c) linar at 45 and at 9, I = P6-17: S FoP Figur 6.6-3, th angl btwn th bams is 11.4 P6-18:.96 P6-19: S FoP Figur 6.6-3, th angl btwn th bams is 5.8 Pag 32