5-55-7 Soluton Set #. Te varaton of refractve ndex wt wavelengt for a transarent substance (suc as glass) may be aroxmately reresented by te emrcal equaton due to Caucy: n [] A + were A and are emrcally determned constants and s te wavelengt of lgt n a vacuum. If A :, :5 ( nm), determne te ase and grou veloctes at 5 nm. n [ 5 nm] : + :5 ( nm) (5 nm) :5 v [ 5 nm] n! jk j c n :997958 8 m s v : 8 m :5 s Te grou velocty may also be called te modulaton velocty v mod, wc s te seed of te low-frequency modulaton. We need to nd an exresson for te modulaton velocty n terms of te numbers we know: d! v mod v dk! k 5 nm ) d! dk d dk )! k v k c n k c c n n dk dk + k n dc dk + kc d dk n d! v mod dk c n + k n + kc n dn dk c ck dn n n dk c n dn dn dk d d dk dn dk d dn d d dn d v mod v n dn d v + n dn c d n n [] A + ) dn d c v mod + n n B c B n n! c :5 :5 ( nm) :5 (5 nm) c :78 v :5 k dn n dk + n dn d
. or te crown and nt glasses gven n te notes wt te followng ndces measured at two vacuum wavelengts: Lne [ nm] n for Crown n for lnt C 5:8 :58 :97 8: :55 :778 Aroxmate te emrcal constants A and n Caucy s equaton (gven n #) and use tem to evaluate te refractve ndex at 589:59 nm (raunofer s D lne); comare te results to te actual values: Lne [ nm] n for Crown n for lnt D 589:59 :5 :7 Two equatons n two unknowns: n C A + C n A + Many ways to solve; I ll use matrx nverson. or te crown glass: det C C : 8 nm : 5 nm C nc ) C n (5:8 nm) (8: nm) :997 nm nc n : 8 nm : 5 nm : 5 nm : 8 nm :997 nm :5 8 :5 8 5: 5 nm 5: 5 nm :5 8 :5 8 :58 5: 5 nm 5: 5 nm :55 :5 5:8 nm at D 589:59 nm : Crown glass: n A + D 5:8 nm :5 + :5, dentcal to measurement (589:59 nm) or te nt glass: 5: 5 nm 5: 5 nm :778 :5 8 :5 8 :97 : 5: nm at D 589:59 nm : lnt Glass: n A + D 5: nm : + :7 vs. :7 (589:59 nm)
. (owles.5) Te electrc vector of a wave s gven by te real exresson: E [z; t] E ^x [k z! t] + ^y b [k z! t + ] Sow tat ts s equvalent to te comlex-valued exresson: E [z; t] E ^x + ^y b ex [ ] ex [ (k z! t)] E [z; t] E ^x [k z! t] + ^y b [k z! t + ] Re E ^x ex [ (k z! t)] + ^y b ex [ (k z! t + )] E Re ^x ex [ (k z! t)] + ^y b ex [ (k z! t)] ex [ ] E Re ^x + ^y b ex [ ] ex [ (k z! t)]. Sketc dagrams to sow te tye of olarzatons n # for te followng cases: (a) ; b E [z; t] E ^x + ^y ex [ (k z! t)] E ex [ (k z! t)] ) LP wt amltude E + E at angle tan 5 (b) ; b E [z; t] E ^x + ^y ex [ (k z! t)] E ex [ (k z! t)] ) LP wt amltude E + E at angle tan :7 radans :5 radans : (c) + ; b E [z; t] E ^x ^y ex ex [ (k z! t)] E ex ex [ (k z! t)] E ex ex ex ) LHEP wt major axs at angle tan 5 5. Wrte down te Jones vectors for te tree cases n te revous roblem. Dd t wtn te roblems (a) (b) E E (c) E ex ex
. or te followng tree Jones vectors: E E E (a) Determne te tye of olarzaton of eac wave; E [] E sn [] " # tan r E + + Lnearly olarzed wt amltude of at angle of E RHCP wt unt amltude ex ex E + ( ) " ( ) () () ( ) () # ex RHCP wt amltude (b) nd Jones vectors tat are ortogonal to eac of te tree cases and descrbe te state of olarzaton. nd a vector suc tat te scalar roduct s zero, were te de nton of te scalar roduct for vectors wt comlex-valued comonents ncludes a comlex conjugate E E? E X (E ) n E? n ) E? n ) (E ) x ceck by evaluatng scalar roduct E E? + + E? x + (E ) y or E? y E E? ex ex ex or E E? ex ex or
7. (P 5-) Intally unolarzed lgt asses n turn troug tree lnear olarzers wt transmsson axes at,, and, resectvely, relatve to te orzontal axs. Wat s te rradance of te roduct lgt exressed as a ercentage of te unolarzed lgt rradance? ts s Malus law mlemented twce, lus recognzng tat te rradance s te squared magntude of te amltude. Te rst olarzer reduces te rradance by alf and te lgt s lnearly olarzed orzontally, so te Jones vector after te rst olarzer s: r I E Te Jones matrx for te second olarzer s: M (n:b:; det M ) sn sn sn " # Te angle of te trd olarzer s : M sn sn sn " # Te outut state s te roduct of te matrces wt te nut state: " # " # r I E M M E r 8 I I E E 9 I + 9 I : I 9 I :8I tan (as t sould!) 5
8. (P 5-): Snce a seet of Polarod s not an deal olarzer, not all te energy of te E-vbratons arallel to te TA are transmtted, nor are all E-vbratons erendcular to te transmsson axs are absorbed. Suose an energy fracton us transmtted n te rst case and a fracton s transmtted n te second. (a) Extend Malus law by calculatng te rradance transmtted by a ar of suc olarzers wt angle between ter transmsson axes. Assume ntally unolarzed lgt of rradance I. Sow tat Malus law follows n te deal case. If two olarzers orented at angle : Ideal Malus Law : I I We know tat alf of rradance of unolarzed lgt s blocked by an deal olarzer, so tat f a ercentage s assed of te lgt at angle and a ercentage s assed at te ortogonal angle, ten some lgt gets troug n all cases. Assume tat te ntal olarzer s orented along x and te second at te angle relatve to x: Lgt assed by x-olarzer orented along x : (I ) x I Lgt assed by x-olarzer orented along y : (I ) y I x-axs lgt assed by olarzer # orented along : (I x ) (I ) x I y-axs lgt assed by olarzer # orented along : (I y ) (I ) y + I sn Total lgt assed: x-axs lgt assed by olarzer # n ortogonal drecton: (I x ) + (I ) x + I sn y-axs lgt assed by olarzer # n ortogonal drecton : (I x ) + (I ) y () I I I + sn + Ceck lmtng beavor of total lgt assed: set, : I I + sn + I [] dentcal to Malus law for deal olarzer: I I [] I [] (b) Let :95 and :5 for a gven seet of Polarod. Comare te rradance wt tat of an deal olarzer wen unolarzed lgt s assed troug two suc seets ave relatve angle between transmsson axes of,, 5, and 9. Ideal Malus Law: I I [] Realstc Malus Law: I I :95 + :95 :5 sn + :5 I :95 + :95 sn
) I I :95 + :95 sn :95 I :55I ) I I :5 5I ) I I I :95 + :95 ) I I :8 75I :95 + :95 sn :95 + :95 sn :95 I + :95 :5I :95 + :95 sn Just for fun, lot te deal equaton and te realstc equatons for tese values of and wt I Transmsson.5..... 5 7 8 9 Angle teta (degrees) Comarson of te realstc exresson for Malus Law (black sold lne) to te deal exresson (red dased lne) for a :95, :5. Note tat te exressons are equal at 5. Try t agan for d erent values of :75; :5 : Transmsson.5..... 5 7 8 9 Angle teta (degrees) wc sows tat te senstvty of te amount of transmtted lgt te te angle of te olarzers as become oor. 7