Fermat s Principle and Its Applications

Transcription

1 Frmat s ricipl ad Its pplicatios 3. INTRODUTION Th propagatio of light i th ralm of gomtrical optics mploys th cocpt of rays. osidr a circular aprtur i frot of a poit sourc as show i Fig. 3.. Wh th diamtr of th aprtur is quit larg (~ cm), w ca s a patch of light with wlldfid boudaris o th scr SS. s w rduc th siz of th aprtur to b vry small (<. mm), th th pattr obtaid o SS' cass to hav wll-dfid boudaris. This phomo is kow as diffractio ad is a dirct cosquc of th fiitss of th wavlgth (which is dotd by ). I haptrs 8 ad, w will show that i th limit of th diffractio ffcts will b abst. osqutly w ca obtai a ifiitsimally thi pcil of light; this is calld a ray. Th fild of optics glctig th fiitss of th wavlgth is calld gomtrical optics. Fig. 3. Th light mittd by th poit sourc is allowd to pass through a circular hol ad if th diamtr of th hol is vry larg compard to th wavlgth of light th th light patch o th scr SS has wll dfid boudaris. S S Light has a wavlgth of th ordr of 5 cm, which is small compard to th dimsios of ormal optical istrumts lik lss, mirrors, tc., o ca, i may applicatios, glct th fiitss of th wavlgth. Th path of th rays a b studid by usig Frmat's pricipl: th ray will corrspod to that ath for which th tim tak is a trmum i compariso to arby paths. Th tim tak to travrs th gomtric path ds i a mdium of rfractiv id (, y, z) is giv by ds c / ds c whr (, y, z) is th positio dpdt rfractiv id ad c is th spd of light i fr spac. If th ray to travrs th path alog th curv (Fig. 3.) th th total lapsd tim is idsi c c i ds. () Th symbol blow th itgral rprsts th fact that th itgratio is from th poit to through th curv. Whr ds i rprsts th i th arc lgth ad i th corrspodig rfractiv id. If idd rprsts th path of a ray, th will b ithr lss tha, gratr tha or qual to for all arby paths lik. Sic c is a costat, o ca altrativly dfi a ray as th path for which optical path is a trmum. ds ()

2 Mathmatically, th ray would follow th path for which ds Fig. 3. If th path rprsts th actual ray path th th tim tak i travrsig th path T will b a trmum i compariso to ay arby path. ds (3) whr th lft-had sid rprsts th chag i th valu of th itgral du to a ifiitsimal variatio of th ray path. Th actual ray path btw two poits is th o for which th optical path lgth is statioary with rspct to variatios of th path. Th Frmat pricipl tlls that i a homogous mdium, th rays will b straight lis bcaus a straight li will corrspod to a miimum valu of th optical path coctig two poits i th mdium (rfrrig to Fig. 3.3). E D Fig. 3.3 Sic th shortst distac btw two poits is alog a straight li, light rays i a homogous mdium ar straight lis; all arby paths lik E or D will tak logr tims. 3. LWS OF REFLETION ND REFRTION FROM FERM'T'S RINILE osidr a rflctig surfac as show i Figur 3.4. Light from poit is dflctd to poit by this surfac, formig th agl of icidc i (= S) ad th agl of rflctio r (= S) masurd from th ormal to th surfac. Th tim rquird for th ray of light to travl th path + is giv by =(+)/v, whr v is th vlocity of light i th mdium cotaiig th poits. W fid ( z) hi ( d z) hr z. v To trmiz to gt (z) with rspct to variatios i z, w st d (z)/dz= d z z h ( d z) h z i r i r.

3 W thus obtai th laws of rflctio from th Frmat's pricipl. Lt us ow us Frmat s pricipl to aalyz rfractio as illustratd i Figur 3.5. ad ar th agls of icidc ad trasmissio, rspctivly, masurd oc agai from th ormal to th itrfac. Th optical path lgth tak by th light to travl th distac R is L. op R R h L h (4) To trmiz this, w must hav dlop L d h L h. (5) From Fig. 3.5, w s that L si ad si. h L h Thus Eq. (5) bcoms si si. (6) which is th Sll's law of rfractio. Th laws of rflctio ad rfractio form th basic laws for tracig light rays through simpl optical systms, lik a systm of lss ad mirrors, tc. Eampl 3. (s Fig. 3.6) Eampl 3. (s Fig. 3.9) Eampl 3.3 (s Fig. 3.) Eampl 3.4 (s Fig. 3.) a S M h i R z Q h r N h M L - R N Q h Fig. 3.4 Th shortst path coctig th two poits ad via th mirror is alog th path whr th poit is such that, S ad ar i th sam pla ad S = S; S big th ormal to th pla of th mirror. Th straight li path is also a ray. L Fig. 3.5 ad ar two poits i mdia of rfractiv idics ad. Th ray path coctig ad will b such that si = si.

4 Q Q L L S Fig. 3.6 ll rays paralll to th ais of a paraboloidal rflctor pass through th focus aftr rflctio (th li is th dirctri). It is for this raso that atas (for collctig lctromagtic wavs) or solar collctors ar oft paraboloidal i shap. Fig. 3.7: paraboloidal satllit dish. hotograph courtsy McGraw Hill Digital ccss Library S Q S Fig. 3.8 Fully strabl 45m paraboloidal dishs of th Giat Mtrwav Radio Tlscop (GMRT) i u, Idia. Th GMRT cosists of 3 dishs of 45m diamtr with 4 atas i th tral rray. hotograph courtsy: rofssor Govid Swarup, GMRT, u. Fig. 3.9 ll rays maatig from o of th foci of a llipsoidal rflctor will pass through th othr focus.

5 S S O r I Q O I y y M Fig. 3. SM is a sphrical rfractig surfac sparatig two mdia of rfractiv idics ad. rprsts th ctr of th sphrical surfac. Fig. 3. : Th rfractd ray is assumd to divrg away from th pricipal ais. 3.3 RY THS IN N INHOMOGENEOUS MEDIUM I a ihomogous mdium, th rfractiv id varis i a cotiuous mar ad, i gral, th ray paths ar curvd. For ampl, o a hot day, th air ar th groud has a highr tmpratur tha th air which is much abov th surfac. Sic th dsity of air dcrass with icras i tmpratur, th rfractiv id icrass cotiuously as w go abov th groud. This lads to th phomo kow as mirag. W will us Sll's law (or Frmat s pricipl) to study th ray paths i a ihomogous mdium. W furthr assum th rfractiv id chags cotiuously alog o dirctio oly; th ais. Th ihomogous mdium ca b thought of as a limitig cas of a mdium cosistig of a cotiuous st of thi slics of mdia of diffrt rfractiv idics [s Fig. 3.(a)]. t ach itrfac, th light ray satisfis Sll s law ad o obtais si si 3 si 3. (3) Thus, w may stat that th product cos si (4) is a ivariat of th ray path; w will dot this ivariat by. If th ray iitially maks a agl (with th z ais) at a poit whr th rfractiv id is, th th valu of is cos. Th picwis straight lis show i Fig. 3.(a) form a cotiuous curv which is dtrmid from th quatio cos cos =. (5)

6 z 4 (a) (b) Fig. 3. (a) I a layrd structur, th ray bds i such a way that th product i cos i rmais costat. (b) For a mdium with cotiuously varyig rfractiv id, th ray path bds i such a way that th product () cos () rmais costat. ds dz d z 3.3. Th homo of Mirag O a hot day th rfractiv id cotiuously dcrass as w go ar th groud. Idd, th rfractiv id variatio ca b approimatly assumd to b of th form k fw mtrs (6) : th rfractiv id of air at = (just abov th groud) k : a costat. Th calculatd ray paths (s Eampl 3.8) ar show i Fig osidr a ray which touchs horizotal at =. ssum th rfractiv id is at th y positio E (= ) whr th ray maks a agl with th horizotal th cos. (7) 3 Usually << so that (m) W W R R M z (m) Fig. 3.3 Ray paths i a mdium charactrizd by a liar variatio of rfractiv id [s Eq.(6)] with k.34-5 m -. Th objct poit is at a hight of.5 m ad th curvs corrspod to +.,, -., -.8, ad.5. Th shadig shows that th rfractiv id icrass with. E cos. (8) t costat air prssur, th dpdc of o tmpratur T givs T T. (9) Eq. (9) tlls that T T T or T T T () T

7 O a typical hot day th tmpratur ar road surfac T 33 K (=5 ) ad, about.5 m abov th groud, T 33 K (= 3 ). Now, at 3,.6 givig 5.67 radias.35. I Fig. 3.3 w hav show rays maatig (at diffrt agls) from a poit which is.5 m abov th groud; ach ray has a spcifid valu of th ivariat Th figur shows that wh th objct poit ad cos. th obsrvatio poit E ar clos to th groud, th oly ray path coctig poits ad E will b alog th curv ME ad that a ray maatig horizotally from th poit will propagat i th upward dirctio as as show i figur. Th y at E will s th mirag ad ot s th objct dirctly at. Thr is a rgio R whr o of th rays (maatig from th poit ) rachs whr ithr th objct or its imag ca b s th shadow rgio. Thr is also a rgio R whr oly th objct is dirctly visibl ad th virtual imag is ot s. I Fig. 3.3, w s that th ray bdig up aftr it bcoms paralll to th z ais (th poit at which = is kow as th turig poit). Th bdig up of th caot b dirctly ifrrd from Eq. (5) bcaus at such a poit o may pct th ray to procd horizotally byod th turig poit as show by a dottd li. Howvr this bhaviour may b plaid from cosidratios of symmtry ad from th rvrsibility of ray paths; th ray path should b symmtrical about th turig poit. hysically, th bdig of th ray ca b udrstood by cosidrig a small portio of a wav frot such as W (s Fig. 3.3); th uppr dg will travl with a smallr spd i compariso with th lowr dg, ad this will caus th wav frot to tilt (s W') makig th ray to bd. Furthrmor, a straight li path lik dos ot corrspod to a trmum valu of th optical path. W t cosidr a rfractiv id variatio which saturats to a costat valu as :, () whr, ad ar costats ad rprsts th hight abov th groud. Th act ray paths ar obtaid by solvig th ray quatio (s Eampl 3.) ad ar show i Figs 3.4 ad 3.5; thy corrspod to th followig valus of various paramtrs: =.33, = ad =.33 m () Th valus of quatio () ar ot actually ralistic, vrthlss, it allows us to udrstad qualitativly th ray paths i a gradd id mdium. Figurs 3.4 ad 3.5 show th ray paths maatig from poits that ar.43 m ad.8 m abov th groud rspctivly. R E R R M z (m) Fig. 3.4 Ray paths i a mdium charactrizd by Eqs. () ad (). Th objct poit is at a hight of / (.43m) ad th curvs corrspod to = + /,, - /6, - /3, - /5 ad - /. Th shadig shows that th rfractiv id icrass with.

8 I Fig. 3.4, th poit corrspods to a valu of th rfractiv id qual to.6455 (= ) ad diffrt rays corrspod to diffrt valus of, th agl that th ray maks with th z ais at th poit. W s that wh th objct poit ad th obsrvatio poit E ar clos to th groud, th oly ray path coctig poits ad E will b alog th curv ME ad that a ray maatig horizotally from th poit will bd up i th upward dirctio, show as. Th y at E will s th mirag ad ot s th objct dirctly at. If poits ad E ar much abov th groud as i Fig. 3.5, th y will s th objct almost dirctly (bcaus of rays lik E) ad will also rciv rays apparig to maat from poits lik '. Sic diffrt rays do ot appar to com from th sam poit ad hc th rflctd imag s will hav cosidrabl abrratios RR M z (m) E E Fig. 3.5 Ray paths i a mdium charactrizd by Eqs. () ad (). Th objct poit is at a hight of.8m ad th curvs corrspod to (th iitial lauch agl) =, - /6, - /3, - /6, - /, - / ad - /8. Th shadig shows that th rfractiv id icrass with. R 5 5 R Fig. 3.6 typical mirag as s o a hot road o a warm day; photograph adaptd from ir%mirror.jpg. Th photograph was tak by rofssor iotr iraski of oza Uivrsity of Tchology i olad; usd with prmissio from rofssor iraski. Fig. 3.7 This is actually ot a rflctio i th oca, but th miragd (ivrtd) imag of th Su's lowr dg. fw scods latr (otic th motio of th bird to th lft of th Su!), th rflctio fuss with th rct imag. Th photographs wr tak by Dr. Gorg Kapla of th U. S. Naval Obsrvatory ad ar o th wbsit cratd by Dr. Youg; photographs usd with prmissios from Dr Kapla ad Dr.. Youg.

9 Thr is also a shadow rgio R whr o of th rays (from th poit ) will rach; a y i this rgio ca ithr s th objct or its imag. Th actual formatio of mirag is show i Figs. 3.6 ad 3.7. Rfr to () Eampl 3.5 () Eampl Fig. 3.8 Ray paths corrspodig to th rfractiv id distributio giv by Eq.(3) for a objct at a hight of.5 m; th valus of, ad ar giv by Eq. (). E 5 5 z (m) 3.3. Th homo of Loomig bov cold sa watr, th air ar th watr surfac is coldr tha th air abov it ad hc thr is a opposit tmpratur gradit. suitabl rfractiv id variatio for such a cas a b writt as:, (3) Th quatio dscribig th ray path is discussd i roblm 3.3. W assum th valus of, ad to b giv by Eq.(). For a objct poit at a hight of.5 m, th ray paths ar show i Fig If th y is at E, th it will rciv rays apparig to maat from '. Such a phomo i which th objct appars to b abov its actual positio is kow as loomig; it is commoly obsrvd i viwig ships ovr cold sa watrs (s Figs 3.9 ad 3.). Sic o othr rays maatig from rach E, th objct caot b obsrvd dirctly Th Gradd Id tmosphr Th ocircular shap of th sttig or th risig su (s Fig. 3. ad Fig. 8 i th prlim pags) is du to th gradd id mdium tmosphr. Th rfractiv id of th air gradually dcrass as w mov outwards. If w approimat th cotiuous rfractiv id gradit by a fiit umbr of layrs (ach layr havig a spcific rfractiv id) th th ray will bd i a way similar to that show i Fig. 3.. Thus th su (which is actually at S) appars to b i th dirctio of S. It is for this raso that th sttig su appars flattd ad also lads to th fact that th days ar usually about 5 miuts logr tha thy would hav b i th absc of th atmosphr. Obviously, if w wr o th surfac of th moo, th risig or th sttig su would ot oly look whit but also circular i shap!

10 Fig. 3.9 Th suprior mirag occurs udr rvrs atmosphric coditios from th ifrior mirag. For it to b s, th air clos to th surfac must b much coldr tha th air abov it. This coditio is commo ovr sow, ic ad cold watr surfacs. Wh vry cold air lis blow warm air, light rays ar bt dowward toward th surfac, thus trickig our ys ito thikig a objct is locatd highr or is tallr i apparac tha it actually is. Figur adaptd from Fig. 3. hous i th archiplago with a suprior mirag. Figur adaptd from Th photograph was tak by Dr. kka arviai i Turku, Filad; usd with prmissio from Dr. arviai. (b) (a) Fig. 3. Th o circular shap of th sttig su. dditioal Figur: hotographs o th moo. caus moo dos ot hav ay atmosphr, th sky ad shadows ar vry dark. I (a) ad (b) w ca also s th arth.hotographs courtsy McGraw Hill Digital ccss Library. (c)

11 Earth Fig. 3. caus of rfractio, light from S appars to com from S. S Su Su S 3.4 THE RY EQUTION ND ITS SOLUTIONS W may driv th ray quatio ow, th solutio of which will giv th prcis ray paths i a ihomogous mdium. W assum th rfractiv id chags alog th ais. It ca b cosidrd as th limitig cas of a mdium comprisig of a cotiuous st of thi slics of mdia of diffrt rfractiv idics. For a cotiuously varyig rfractiv id, th product ()cos () is a ivariat of th ray path: cos. (4) For a cotiuous variatio of rfractiv id, th picwis straight lis show i Fig. 3.(a) forms a cotiuous curv as i Fig. 3.(b). Lt ds rprsts th ifiitsimal arc lgth alog th curv, th ds d ds d dz or. (5) dz dz Rfr to Fig. 3.(b), w fid that dz ds lug Eq. (6) ito Eq. (5), w hav d dz cos. (6). (7) For a giv () variatio, Eq. (7) ca b itgratd to giv th ray path (z). W may also diffrtiat it with rspct to z to obtai: d d d d d d. (8) dz dz d dz dz d oth Eqs (7) ad (8) rprst rigorously corrct ray quatios wh th rfractiv id dpds oly o th coordiat. Rfr to Eampl 3.7 ad Eampl Ray aths i arabolic Id Mdia osidr a parabolic id mdium with th followig rfractiv id distributio: (36) Equatio (7) ca b writt as Substitutig d lads to d dz. (37) d dz. (38)

12 whr (39) ad (4) Th rgio <a is kow as th cor of th wavguid ad th rgio >a is usually rfrrd to as th claddig. Thus Writig = si w gt si z z (4) hoos th origi such that z =, w obtai si z (4) W could hav also usd Eq. (8) to obtai th ray path. I a practical optical wav-guid th rfractiv id distributio is usually writt i th form:, a cor a, a claddig (43) I a typical parabolic id fibr,. (44) a =.5, =., a= m. (45) W obtai.485, =.67 4 m Typical ray paths for diffrt valus of ar show i Fig W may writ Guidd rays Rfractig rays (46) I Fig. 3.3, th ray paths show corrspod to z = ad =4, 8.3 ad ; th corrspodig valus of ar pproimatly.496 (> ),.485 (=) ad.4 (< ) th last ray udrgos rfractio at th cor-claddig itrfac.. -. z p = = 8.3 z (mm) = 4 laddig or Fig. 3.3 Typical ray paths i a parabolic id mdium for paramtrs giv by Eq.(45) for =4, 8.3 ad. th priodical lgth z p of th siusoidal path is giv by a cos z p. (47) I Fig. 3.3, th valus of z p ar.8864 mm ad.8796 mm with =4 ad 8.3, rspctivly. I th paraial approimatio, cos ad all rays hav th sam priodic lgth. I Fig. 3.5, typical paraial ray paths ar plottd for rays lauchd alog th z ais. Diffrt rays (show i th figur) corrspod to diffrt valus of. Rmarks o th Faturs: (i) I th paraial approimatio ( ) all rays lauchd horizotally com to a focus at a particular poit. Thus th mdium acts as a covrgig ls of focal lgth giv by: f a cos a lim. (48) 4

13 (iii) Rays lauchd at diffrt agls with th ais (s, for istac, th rays mrgig from poit ) gt trappd i th mdium ad hc th mdium acts lik a 'guid'. Idd such mdia ar rfrrd to as optical wavguids ad thir study forms a subjct of grat cotmporary itrst. (iii) Ray paths would b allowd oly i th rgio whr is lss tha or qual to () [s Eq. (7)]. Furthr, d/dz would b zro (i.., th ray would bcom paralll to th z ais) wh () quals. (iv) Th rays priodically focus ad dfocus as show i Fig hysically, although th ray LQ travrss a largr path i compariso to MQ, it dos so i a mdium of lowr avrag rfractiv id thus th gratr path lgth is compsatd for by a gratr avrag spd ad hc all rays tak th sam tim to propagat through a crtai distac of th wavguid. It is for this raso that parabolic id wavguids ar tsivly usd i fibr-optic commuicatio. Gradit-Id (GRIN) lss, charactrizd by parabolic variatio of rfractiv id i th trasvrs dirctio, ar ow commrcially availabl ad fid may applicatios (s Fig. 3.4). For ampl a GRIN ls ca b usd to coupl th output of a lasr diod to a optical fibr; th lgth of such a GRIN ls would b z p /4 (s Fig. 3.5); typically z p fw cm ad th diamtr of th ls would b fw millimtrs. Such small siz lss fid may applicatios. Similarly, a GRIN ls of lgth z p / ca b usd to trasfr collimatd light from o d of th ls to th othr.. ( m) L M Q -. - z p Fig. 3.4 gradit-id ls with a parabolic variatio of rfractiv id. Th ls focuss light i a way similar to a covtioal ls; figur adaptd from z (mm) Fig. 3.5 araial ray paths i a squar law mdium. Notic th priodic focussig ad dfocussig of th bam.

14 3.4. Trasit Tim alculatios i a arabolic Id Wavguid s show i Sc. 3.4., th ray path through a parabolic id wavguid as dscribd by Eq. (36) (isid th cor) is giv by si, z (49) whr ad hav b dfid through Eqs (39) ad (4). Lt d rprst th tim tak by a ray to travrs th arc lgth ds [s Fig. 3. (b)]: ds d, (5) c / whr c is th spd of light i fr spac. Sic dz ds, w may writ Eq. (5) as d dz dz c c d si z dz (5) c Th tim tak by th ray to travrs a distac z alog th wavguid is giv as z z cos z z dz dz c c = z si z c c z z si z (5) c 4c Wh =, z z c /. (53) as pctd. For larg valus of z, th scod trm o th RHS of Eq. (5) would mak a gligibl cotributio to z(z) ad w may writ z z (54) c If a puls of light is icidt o o d of th wavguid, it would i gral cit all rays ad sic diffrt rays tak diffrt amouts of tim, th puls will gt tmporally bro add. For a parabolic id wavguid, this broadig will b giv by z z. (55) c c W hav assumd. (56) For th fibr paramtrs giv by Eq. (45), w gt.5 s/km. Rfr to Eampl 3.9 [p. 3.6] Rflctios from th Ioosphr Th ultraviolt rays i th solar radiatio rsults i th ioizatio of th costitut gass i th atmosphr rsultig i th formatio of what is kow as th ioosphr (Th ioizatio is almost gligibl blow a hight of about 6 km).

15 = 6 caus of th prsc of th fr lctros (i th ioosphr), th rfractiv id is giv by (Eq. (76), haptr 7): (m) = 3 = = 45 N q. (63) m Thus as th lctro dsity starts icrasig from th rfractiv id starts dcrasig ad th ray paths would b similar to that dscribd i Eampl 3.9. If T rprsts th rfractiv id at th turig poit (s Fig. 3.7) th - cos T. (64) z (m) Fig. 3.6 arabolic ray paths (corrspodig to =, 3,45 ad 6 ) i a mdium charactrizd by liar rfractiv id variatio i th rgio > [s Eq.(9)]. Th ray paths i th rgio < ar straight lis. Thus if a lctromagtic sigal is st from th poit (at a agl ) is rcivd at th poit, o ca dtrmi th rfractiv id (ad hc th lctro dsity) of th ioosphric layr whr th bam has udrgo th rflctio. Thrfor th short wav radio broadcasts ( m) st at a particular agl from a particular city (say Lodo) would rach aothr city (say Nw Dlhi) aftr udrgoig rflctio from th ioosphr. For ormal icidc, m N T. (65) q I a typical primt, a lctromagtic puls (of frqucy btw.5 to MHz) is st vrtically upwards ad if th cho is rcivd aftr a dlay of t scods, th h t. (66) c h is th hight at which it udrgos rflctio. Thus if lctromagtic puls is rflctd from th E layr of ioosphr (which is at a hight of about km), th cho will b rcivd aftr about 67 s. 8 km km T F rgio (N) ma 4 m -3 (N)ma.6 m -3 E rgio Fig. 3.7 Rflctio from th E rgio of th ioosphr. Th poit T rprsts th turig poit. Th shadig shows th variatio of th lctro dsity.

16 ltrativly, by masurig th dlay t, o ca dtrmi th hight (at which th puls gts rflctd) from th followig rlatio t h. (67) c I Fig. 3.8 is a plot of th frqucy dpdc of th quivalt hight of rflctio (as obtaid from th dlay tim of cho) from th E ad F rgios of th ioosphr. From th figur w fid that at =4.6 6 Hz, chos suddly disappar from th km hight. Thus, N km m q ( 4.6 ) 9 (.6 ) lctros/m. Equivalt hight (km) Frqucy of th plorig wavs i MHz Fig. 3.8 Frqucy dpdc of th quivalt hight of rflctio from th E ad F rgios of th ioosphr. If w furthr icras th frqucy, th chos appar from th F rgio of th ioosphr. Rfr to Eampl 3. [p. 3.8] 3.5 Rfractio of Rays at th Itrfac btw a Isotropic Mdium ad a isotropic Mdium I a isotropic mdium th proprtis rmai th sam i all dirctios; typical ampls ar glass, watr ad air. O th othr had, i a aisotropic mdium, som of th proprtis (such as spd of light) may b diffrt i diffrt dirctios. Wh a light ray is icidt o a crystal lik calcit, it (i gral) splits ito two rays kow as ordiary ad traordiary rays. Th vlocity of th ordiary ray is th sam i all dirctios. Thus th ordiary ray obys Sll's laws but th traordiary ray dos ot. I a uiaial homogous mdium, th rfractiv id variatio for th traordiary ray is giv by [s Eq. () of haptr ] cos + si. (77) whr o ad ar costats of th crystal ad th agl that th ray maks with th optic ais Optic is Normal to th Surfac rprsts osidr th particularly simpl cas of th optic ais big ormal to th surfac. Rfrrig to Fig. 3.9, th optical path lgth from ad is giv by Lop h L h. (78) : th rfractiv id of mdium I *Th icidt ray, th rfractd ray ad th optic ais ar coplaar.

17 h Sic W obtai h cos ad si, h h I i o Lop h L h. (79) II Optic is L - Q c / Fig. 3.9 Th dirctio of th rfractd traordiary ray wh th optic ais (of th uiaial crystal) is ormal to th surfac. h c / o For th actual ray path, w rquir thus or dl d L h L h op o ta r si i (8) ta r o, r: agl of rfractio ta r =/h. W fid o si i ta r. (8) si i This givs th agl of rfractio for a giv agl of icidc. s a simpl ampl, lt th first mdium to b air so that =. Th o si i ta r. (8) si i ssum th scod mdium to b calcit, th o =.65836, ad = Thus for i=45, w gt r 3.. If o = = th Eq. (8) simplifis to si i s i r. [ Slls's Law] (83) 3.5. Optic is i th la of Icidc osidr a mor gral cas of th optic ais makig a agl with th ormal. W furthr assum th optic ais lis i th pla of icidc as show i Fig I gral, i a aisotropic mdium, th rfractd ray dos ot li i th pla of icidc. It ca b show that if th optic ais lis i th pla of icidc th th rfractd ray also lis i th pla of icidc. Th optical path lgth from to (s Fig. 3.3) is giv by Sic =r Lop h L h. (84), w hav cos r + si r = cos cos si si + si cos cos si r r r r

18 Thus h ad h = cos + si h h cos - h h h si h cos si cos h si (85) For th actual ray path, w rquir thus L h L dl h cos si si + cos h si cos d or o cos si cos si h h cos si + si cos si i (87) cos si o op, L h L op h cos si cos h si. (86) For giv valus of th agls i ad, th abov quatio ca b solvd to giv th valus of ad hc th agl of rfractio r (= + ). h I i S II ( ) L - r Optic ais Q L Fig. 3.3 Th dirctio of th rfractd traordiary ray wh th optic ais (of th uiaial crystal) lis i th pla of icidc makig a agl with th ormal to th itrfac. h S Rmarks o Svral articular ass (i) Wh o = =, th aisotropic mdium bcoms isotropic ad Eq. (87) simplifis to sii = si( + )= sir which is othig but Sll's law. (ii) Wh =, i.., th optic ais is ormal to th surfac, Eq. (87) bcoms si si i cos si si r = (88) cos r si r o whr w hav usd th fact that r =. Th abov quatio is idtical to Eq. (8). (iii) Fially, w cosidr ormal icidc, i.., i =. Thus, Eq. (87) givs us o

19 cos si + si cos or cos r si + si r cos or cos r cos si si cos si r si cos or ta r si si cos cos Equatio (89) shows that i gral r (s Fig. 3.3). (89) I II Optic ais Fig. 3.3 For ormal icidc, i gral, th rfractd traordiary ray udrgos fiit dviatio. Howvr, th ray procds udviatd wh th optic ais is paralll or ormal to th surfac. For ormal icidc, th abov aalysis is valid for a arbitrary oritatio of th optic ais; th rfractd (traordiary) ray lis i th pla cotaiig th ormal ad th optic ais. Furthrmor, for ormal icidc, wh th crystal is rotatd about th ormal, th rfractd ray also rotats o th surfac of a co. Eq. (89) tlls that wh th optic ais is ormal to th surfac ( =) or wh th optic ais is paralll to th surfac but lyig i th pla of icidc ( = /), r= ad th ray gos u-dviatd (s Fig. 3.3). y S r Q O M Fig. 3.3 araial imag formatio by a cocav mirror.

20 S S Q y O O Q r r y M M Fig araial imag formatio by a cov mirror. Fig araial imag formatio by a cocav rfractig surfac SM. O Q R O Fig ll rays paralll to th major ais of th llipsoid of rvolutio will focus to o of th focal poits of th llips providd th cctricity = /. Fig sphrical rflctor.