Handout 30. Optical Processes in Solids and the Dielectric Constant

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1 Haut Otal Sl a th Dlt Ctat I th ltu yu wll la: La ut Ka-Kg lat Dlt tat l Itba a Itaba tbut t th lt tat l C 47 Sg 9 Faha Raa Cll Uty Chag Dl, Dl Mt, a lazat Dty A hag l t a gat a a t hag aat by ta: Q Q Dl t a hag l a t Q uh that: lazat ty t a u tg hag l th ut th ub N l ut lu ( l ty) a th tgth ah l g by : N N Q C 47 Sg 9 Faha Raa Cll Uty

2 Dlt Ctat N-la Matal N-la Dlt Matal (N-la Iulat It Sut) ulu - lt lu (al ba a lt) Dlt a -l Matal gt laz wh la a lt l ( l hag l) bau th lt lu ht lat t th ul lazat ty (# l ut lu t th tgth l) D C 47 Sg 9 Faha Raa Cll Uty la Dlt Matal (la Iulat It Sut) Cat A Dlt Ctat la Matal Th h tbut t lazat Th ttal (h lt) tbut t lazat I la atal, atal lazat a -l ha tw tbut: a) Th h tbut: h h b) Th lt tbut: D h h h C 47 Sg 9 Faha Raa Cll Uty

3 Dlt Ctat Matal: h a lt Ctbut I gal, th utblt a quy t: h lt uh at tha th latt I wg at qu that a t all a t th haatt qu th ay a th axat: I w : th all qu: h Cag wth th x haut 9: g u: h M h C 47 Sg 9 Faha Raa Cll Uty h A w ha ally: M M W w th lt tbut Hgh Fquy Dlt Ctat Sl C a ual &M wa quy agatg a l: t q t, R, t q Wh th lt l ha : q, Slaly, th agt l ha : q H, q H H A th tw l a lat by th tw Maxwll quat:, H, Faaay Law H,, A Law Th tw quat tgth g th lat th &M wa: q q C 47 Sg 9 Faha Raa Cll Uty

4 Statgy t Calulat th Dlt Ctat Matal ) Stat wth th Halta bg th tat th lt wth th ltagt l: q t q t A H H H,, ) F ut hw th lt waut ( Blh ut) gt ug taa t tubat thy:, w,, t, ) F th waut, alulat th lt hag ty, a th th l ty Th ab u, althugh abl, a lttl lat a w wll u a altat aah! C 47 Sg 9 Faha Raa Cll Uty Rat Ix Sl Th at x a atal a: Th wa lat th: q A th lt l ha a b wtt a: q q, Th at x uually ha al a agay at: Th lt l ha th: q q, Th agay at th x b wa ay ( wa alat ga t) C 47 Sg 9 Faha Raa Cll Uty 4

5 5 C 47 Sg 9 Faha Raa Cll Uty Iagay at th Rat Ix a th L Ct Stulat abt W ha alay that tulat abt ult a wa t ay a u (tal l): R R q, Wh: But w al ha: q, Th a th agay at th at x : C 47 Sg 9 Faha Raa Cll Uty Hgh Fquy Dlt Ctat Sl: Iagay at Th at x a atal a: Th, ug th at that: Th l: Ug th x th agay at th at x w gt: Qut: What th al at th lt tat? a

6 La R Fut: La R Fut I a la t aat (LTI) yt, th tulu ha S() lat t th ha R() by a la ut (): R S Th la yt ut aty th llwg tw t: ) It ut b aual (yt at b th tulu al) ) A al tulu S(t) ut ult a al R(t) (wth agay t) Th t g: * a Mt l a x t la ut xal lu: Cutty: J,, Dlt Ctat: D,, C 47 Sg 9 Faha Raa Cll Uty La R Fut a Ka-Kg Rlat Th tw t, lt u l, tat that th al a agay at ay ut at b t thy ut b RLATD! R S Th lath btw th al a th agay at th ut atu by th Ka-Kg lat: 4 4 () () I w th al at all qu, th a th agay at ug Ka-Kg lat Cly, w th agay at all qu, th a th al at ug Ka-Kg lat ROOF OF KRAMRS-KRONIG RLATIONS GIVN IN ANDIX C 47 Sg 9 Faha Raa Cll Uty 6

7 7 C 47 Sg 9 Faha Raa Cll Uty Hgh Fquy Dlt Ctat Sl: Ral at W ha: A th Ka-Kg lat w w: 4 C 47 Sg 9 Faha Raa Cll Uty Hgh Fquy Dlt Ctat Sl Nt that u x lt tat ta t aut tba tat lg ly a gl al ba a a gl ut ba A alt x wul lu tba tat ag all ba th l

8 8 C 47 Sg 9 Faha Raa Cll Uty Hgh Fquy Dlt Ctat Sl,, Th ab x lu tbut tba tat ag all a ba th l Uually th t tat a at tbut at qu tt tba tat btw th hght u ba ( th al ba) a th lwt uu ba ( th ut ba) C 47 Sg 9 Faha Raa Cll Uty Fquy (h), () Fquy (h),, () g g g g g g g Baga a th Hgh Fquy Dlt Ctat,,

9 9 C 47 Sg 9 Faha Raa Cll Uty Baga a th Hgh Fquy Dlt Ctat g Ma y ugh tat: Su : g h g g g Matal wth lag baga wll ha all al at lt tat (a, th, all al at at ) Rb th lat yu hw 7: g h 4 C 47 Sg 9 Faha Raa Cll Uty ht hl lt A Httutu La (Ba Daga) N- - tal A Rg Wagu La Stutu Sut Httutu La h I ut httutu la, th w baga atal ha all at x tha th aw baga atal Th bat aw a w baga atal at l a lt tal wagu that a gu th ht Th httutu t ly th a but t al th ht!! I I

10 Dlt Ctat: Ca N-Z Cutty W ha bta a x th lt tat that at tba tal a h What th atal al ta lag t lt hl bth ( what th atal wa a ut)? G ba t Maxwll quat:, H, H, J, Faaay Law A Law Nw t (ut ty u t lt hl bth) J H,, H,, Wh: Th t th taba th -a tbut C 47 Sg 9 Faha Raa Cll Uty Dlt Ctat: N-Z Cutty W ha: U th Du l th quy t utty: Th quat th lt t lty (aug aabl/t ba D): t I ha tat (aug a ual lt l): Th ut ty : J wh: C 47 Sg 9 Faha Raa Cll Uty

11 Dlt Ctat: N-Z Cutty Aug -z t bth lt a hl th ttal utty b: W ha: h Th: h Itba tal a h Itaba tal C 47 Sg 9 Faha Raa Cll Uty Th laa Fquy Su w ha a tal a - ut whh: Th w ha: h Wh th laa quy a: g, LO F tal: ~ 4 F ut: ~ 5 Hz (UV-blu lght quy) Hz (Tahtz quy) C 47 Sg 9 Faha Raa Cll Uty

12 lt at: uttg ythg Tgth h h lt at (Itba), h h at Ft l th tba at a th l th aba th -a at S l -z ly utg atal a ha z quy lt h at: M h Cutty at (lt taba at th a at) C 47 Sg 9 Faha Raa Cll Uty lat lat t ltagt wa ul wth atal wa atal xtat It th a g t th ha wh ltagt gy b tgly ul wth atal g S xal lat a: ) h-lat ltagt wa b tgly ul wth th tal h a la u ) la-lat ltagt wa b tgly ul wth th laa wa a utg u ) xt-lat ltagt wa b tgly ul wth xt (bu lt-hl a) C 47 Sg 9 Faha Raa Cll Uty

13 C 47 Sg 9 Faha Raa Cll Uty Ta a Lgtual lat F ay u: Lgtual lat: I lgtual lat, th -l ha a -z g but th D-l ha a z g: D u u u u D D t q I th -l ha a wa-l : Th: q -l ha a -z t th t wa agat Ta lat: I ta lat, th -l a th D-l bth ha a z g: D t q I th -l ha a wa-l : Th: q -l ha t th t wa agat C 47 Sg 9 Faha Raa Cll Uty Lgtual lat t q Su th -l ha a wa-l : D D Th D-l g a: F lgtual lat w ut ha: Th ly way that bth th quat a hl th quy th lgtual lat uh that: Th ab quat g th quy th lgtual lat

14 Lgtual lat Lgtual h-lat: C a -utg la u (la ut a la ulat) wh lt tat at qu uh all tha th atal baga g axatly, Th t, M g: LO C 47 Sg 9 Faha Raa Cll Uty LO Th lgtual h-lat a jut th la lgtual tal h! Lgtual la-lat: C a utg u (l gl, l) wh lt tat at qu uh lag tha th h qu but uh all tha th atal baga g axatly, Th t, g: Th lgtual la-lat a jut th laa wa! Ta lat Su th -l ha a wa-l : q t q Th D-l g a: D F ta lat w ut ha: D Th ltagt wa quat wh : Th la wa a lut th wa quat : q Th ab quat g th th ta lat C 47 Sg 9 Faha Raa Cll Uty 4

15 Ta h-lat C a -utg la u (la ut a la ulat) wh lt tat at qu uh all tha th atal baga g axatly, M q Th lat: g th llwg quat: 4 LO q q LO Th ultg lat ltt th Fgu Nt that th a ba qu whh ltagt wa a agat th u ( agatg wa xt) Th ba all th Rtahl ba LO Rttahl ba q C 47 Sg 9 Faha Raa Cll Uty Ta la-lat C a utg u (l gl, l) wh lt tat at qu uh lag tha th h qu but uh all tha th atal baga g axatly, Th lat: q g th llwg quat: q Th ultg lat ltt th Fgu Nt that ltagt wa a agat th u wth a quy all tha th laa quy q C 47 Sg 9 Faha Raa Cll Uty 5

16 ANDIX: Ka-Kg Rlat () I a la t aat (LTI) yt, th tulu ha S() lat t th ha R() by: Th la ut (): Ralty: Caualty: R S Ral ut ut ult a al Th t g: * a I FT g: R t t t t St t t t t Caualty l that th yt at xhbt t a ut b th ut u: t t t t whh g: R t t t t St It Fquy R: N hyal yt a at t qu, : t C 47 Sg 9 Faha Raa Cll Uty Th aualty t : Ka-Kg Rlat () t t Th ut (), wh a a aalyt ut th lx la, at ha ay l th u hal th lx la th aualty t t hl C th llwg tu tgal th tu hw: S th a l th u hal la, th l tu ta l, a th tu tgal ut b z R= C 47 Sg 9 Faha Raa Cll Uty 6

17 7 C 47 Sg 9 Faha Raa Cll Uty Ka-Kg Rlat () Mathg th al a agay at bth g: 4 4 Wh th llwg lat ha b u t gt th tgal: a I a wh th al at () ay t b z at t qu, a t ha th a th lt tat, w jut at th t u th bgg wth () ( ) ta () t gt: 4 4

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