; we re considering the shape of the interface between medium 1 and medium 2. Consider n 2. , so that the rays from S will be refracted inwards.

Transcription

1 Prf. Raghuveer Parthaarath Uvert f Oreg Phc 35 Wter L ENSES We fte wh t cllect ad re-hape electrmagetc wavefrt t create mage f bject. Lee are pwerful tl fr achevg thee gal ad are bvul ver ueful, frmg the eetal magg elemet f telecpe, mcrcpe, camera, ur ee, ad ma ther devce. The deal hape f a le urface geerall me -phercal cc ect (hperbla, parabla, etc.), but practce phercal lee are tpcall ued, ce the are vatl eaer t make tha apherc (-phercal) lee. Tpcall, e ue phercal lee ad the crrect fr ther aberrat (-deal behavr), e.g. b ug cmbat f lee. 4. A phercal terface Cder a pt urce emttg phercal wave frm pt S, a medum f dex f refract (ee Fgure 4.). Ca we ctruct a phercal terface f radu R that fcue the emtted lght t pt P, regardle f where t ht the terface? What huld R be? Pt P embedded a medum f dex f refract ; we re cderg the hape f the terface betwee medum ad medum. Cder >, that the ra frm S wll be refracted ward. Fgure 4.. A phercal terface. C, S, A,ad P refer t partcular pt the ceter f the phercal terface, the bject pt, the pt at whch the ra draw ht the terface, ad the mage pt, repectvel. Italczed letter refer t dtace. Greek letter refer t agle te that α ASC ad β CPA. Nte Optc R. Parthaarath 008 Page 7

2 Pa attet t the tat Fgure 4.. Pt C the ceter f the phere f radu R. The dtace betwee the bject pt, S, ad the terface, ad the dtace betwee the mage pt, P, ad the terface. The agle that the cdet ad reflected ra make wth repect t the rmal t the terface are θ ad θ. A uual θ ad θ are related b Sell Law: β θ θ. We ca relate θ t β va the law f e :. Relatg θ t α R θ R t qute a traparet; frt te that SAC π θ, ( SAC) ( π θ) θ, ad α θ the appl the law f e t SAC t get:. Iertg all th t Sell Law: R R + R+ R α β,.e. ( R+ ) α ( R) β. R R Mre gemetr: α ad β. l + l Therefre: ( R+ ) ( R) + + ( R+ ) + ( R) + We ve derved a relat that mut hld fr fcug at P t ccur. I ther wrd, we kw what R we eed the R that atfe the abve expre. Ufrtuatel, t deped, the pt at whch ur ra ht the terface! Therefre dfferet ra wll t fcu t the ame mage pt. 4. A phercal terface the paraxal regme What we ve hw, fact, that a trul phercal terface wll t erve a a deal le. There a wa ut f th, hwever, whch t lmt urelve t the paraxal regme, meag that we cder l lght that earl parallel wth the ptcal ax, SP. I ther wrd, we cder A B C Recall frm gemetr the Law f Se: Fr a tragle, ; ee the fgure fr tat. a b c Nte Optc R. Parthaarath 008 Page 8

3 mall α ad β. Therefre, equat: ad fall: ( ) ( ) ad R+ R, r R + R ( ) + ( ) ( + ) ( ) R R R are mall, allwg u t eglect them the bxed +. A mple, clea, ueful relat! See Fgure 4. fr a R llutrat f the path f ma dfferet ra. (B the wa, we culd al have derved th drectl frm Fermat prcple, b determg the R fr whch SAP a extremal path fr a A.) Shuld we be bthered b lmtg urelve t the paraxal cae? Ye ad. I practce e de tr t deg ptcal tem uch that beam are cle t the ceter f phercal le elemet r, equvaletl, t have e mage ad bject dtace be large cmpared t the ze f the le. If e de th, the abve relat wrd ver well. I practce, e wrk the paraxal regme, ad apple addtal crrect f ecear. We wll ctue thkg abut the paraxal regme. Fgure 4.. Fcug b a phercal terface the paraxal regme, whch all ra frm S are refracted t P. 4.3 Fcal Pt If R, ad are fxed, decreag mea that creae (ad vce vera), frm the abve bxed relat. Let creae utl, ther wrd parallel ra emerge frm the Nte Optc R. Parthaarath 008 Page 9

4 terface; what? Frm abve: ( ) +, therefre R ( ) th dtace fcue t ft. We ll call th dtace the bject fcal legth, The phercal wave frm the pt urce tur t plae wave ee Fgure 4.3. R -- a bject at R. f Fgure 4.3. Lght emaatg frm the bject fcal dtace fcued t a mage dtace f ft (.e. ra becme parallel). The ame hld f we d t cder a em-fte medum the rght, but rather a fte le wth a phercal urface at the left ad a flat urface at the rght a pla-cvex le (See Fgure 4.4.) Nte that ce the rght edge flat, all ra are rmal t t, ad there bedg f the ra due t refract. Pla-cvex lee are ver ueful. Fgure 4.4. Parallel ra geerated b a pla-cvex le frm a urce lcated at the bject fcal legth. We ca f cure cder the ppte tuat, whch plae wave (parallel ra frm ) are fcued t a mage at me. Th partcular deted f, the mage fcal legth. 4.4 Real ad Vrtual Image Slvg the bxed le equat Sect 4. fr, we have. If > f, the > 0, ad pt P t the rght f R f the terface. The ra frm S cverge at P. T a berver at the rght, t lk a f lght Nte Optc R. Parthaarath 008 Page 30

5 emaatg frm pt P. We have what called a real mage at P (ee Fgure 4.5). If, fr example, we put a pwer meter at P, we detect a hgh degree f pwer due t the fcued lght If < f, the < 0, ad pt P t the left f the terface. The ra d t actuall ht pt P, but the appear t a berver at the rght a f the are emaatg frm P (ee Fgure 4.5). We have what called a vrtual mage at P. If, fr example, we put a pwer meter at P, we d t detect a hgh tet fcued pt, ce there pt there. Fgure 4.5. Real ad vrtual mage. Left: lght emaate frm P. Rght: Lght lk t a berver lke t emaatg frm pt P lcated t the left f the terface. 4.5 Ccave lee The ame aal wrk fr ccave lee, but we treat R a egatve ( R < 0 ). Sce +, f > the < 0 we have a vrtual mage (ee Fgure 4.6). R Fgure 4.6. A cvex le. Nte the vrtual mage (f > ). 4.6 Th lee Let glue e le f radu f curvature R t ather f R. (See Fgure 4.7.) We ll cder th lee, ad eglect the le thcke d (.e. we re aumg d maller tha ther legth vlved). Nte Optc R. Parthaarath 008 Page 3

6 Fgure 4.7. Fcug lght wth a th le (mage d mall). The bject ad mage legth fr le (the left half f the le) are related b +. The mage f le prvde the bject fr le. Therefre R 0 + d, where the egatve g are becaue, a defed abve, a ptve mage legth ad a ptve bject legth le ppte drect. Cderg le : +, where we keep track f whch dex f refract whch. R We eed t adpt a ctet et f g cvet fr the rad. A ted abve, a cvex left le ha R > 0, ad a ccave left le ha R < 0. Fr the rght de le, thee are wtched. Here are me llutrat f thee rule: Bcvex, R >0, R <0 Plaar cvex, R, R <0 Plaar cvex, R >0, R Mecu cvex, R >0, R >0 Bccave, R <0, R >0 Returg t ur th le, addg the tw expre abve: Nte Optc R. Parthaarath 008 Page 3

7 + ( ) R R Fr a th le ar, ; le, gvg u the Th Le Equat, r Lemaker Frmula: + ( ) le R R The fcal legth, f, gve ether b r (t de t matter whch): f ( ) le R R f.. We ca the wrte the th le equat a +, al kw a the Gaua Le Frmula. Th e the mt mprtat relat fr the deg f ptcal tem. Fr example: Cder parallel ra cdet a gla (.5), R, R 50 mm pla-cvex le. (See the fgure, rght, ad te the relat betwee the hape f the le urface ad the g f the R.) Where wll thee ra be fcued t? Awer: (.5 ), f 00 mm, 00 mm. f 50mm 4.7 Magfcat Lee magf bject. The magfcat ca be > r < (whch Hecht call mfcat ). See the fgure f the th le, belw, whch magfg a exteded bject (.e. t a pt urce) th cae, a apple. Fgure 4.8. Lee magf mage chematc. Nte Optc R. Parthaarath 008 Page 33

8 Pt F ad F are each a dtace f, the fcal legth, frm the le. Cder lght emaatg frm the tp f the apple. The ra that ge thrugh F wll emerge frm the le parallel t the ax (thk: wh?). The ra the leave the apple parallel t the ax wll g thrugh F (thk: wh?). The ra that ge thrugh the ceter f the le wll be udeflected the th le lmt ee Hecht fr a dcu. The magfcat, M T, defed t be the heght f the mage relatve t the heght f the bject.e. M T. Tragle S S O mlar t tragle P P O,, M T ; the egatve g hw that the mage verted. Tragle AOF mlar t tragle P P F,. f ( f ) Tragle BOF mlar t tragle S S F,. Cmbg thee, ( f ) f f f. Nte that -f x (ee fgure). Ug the lat mlar tragle relat aga, f f f x f. Ad : M T f. We culd al have wrtte: M T. f x A the bject dtace x lwered, the magfcat creae. (Thk abut what happe f x < 0,.e. the bject cler tha the fcal pt. Drawg ra, u huld be able t cvce urelf that the le cat frm a mage f the bject.) 4.8 Relut Whe cderg gle-lt dffract, we realzed that θ m λ/a, where λ the wavelegth f lght ad a the dameter f the magg devce, the agular relut f the devce. Tw bject mut have a agular eparat f at leat θ m f the are t be relved a eparate bject. Ug lee t magf bject, th agular relut crter tll hld. Mrever, the fact that the bject dtace ca t be cler tha the fcal legth tur ur relut relat t a dtace crter, a we ll (prbabl) ee Prblem Set 4. T be relvable b a magfg le, tw bject mut be eparated b at leat x m λ. A befre, we re grg factr f, etc., th expre. The true expre, f u re tereted, x m λ/( θ m ), where the dex f refract f the medum ad θ m the maxmal agle f the ce f lght the le cllect. Fr ptcal wavelegth (λ 500 m), magg water (.3), th expre et a fudametal lmt f at bet 00 m fr the relut f mcrcpe. A we ll ee later cla, ma peple are wrkg clever trck t get arud th dffract lmt. Nte Optc R. Parthaarath 008 Page 34