OPTICS. Figure m. 1,000,000 nm m. 100,000 nm m. 10,000 nm m nm. visible m. 100 nm m. 10 nm m.

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1 PTICS ptics, oe of the oldest brches of phsics, is the stud of the properties of visible d er-visible light coverig the spectrum of wvelegths from the fr ultrviolet to the fr ifrred, rge of wvelegths from ver roughl m(m ometer 0 9 meter) to,000,000 m, s illustrted i Figure 0.. Visible light is usull cosidered to hve its spectrum from the violet of bout 400 m to the red of bout 700 m with the other colors betwee these boudries (gi, see Figure 0.). Light is prt of the electromgetic spectrum which is composed of rditio formed b combied electric d mgetic fields, trsverse wve motio becuse these fields re perpediculr to the directio of trvel of the wve. ifrred ultrviolet 0-3 m 0-4 m 0-5 m 0-6 m 0-7 m 0-8 m 0-9 m,000,000 m fr ifrred 00,000 m 0,000 m 000 m visible 00 m 0 m fr ultrviolet m Figure 0. red orge ellow gree blue violet 700 m 400 m Whe Mxwell ws formultig his theor of electricit d mgetism i the 860s, he ws ble to show tht his electric d mgetic fields obeed wve equtio; this equtio hd the sme mthemticl form s the oe for wves trvelig o strig or for soud wves. The wve equtio hs extremel useful coefficiet tht equls the squre of the speed of trvel of the wves. Whe Mxwell evluted this coefficiet i his wve equtio, he obtied vlue equl to the squre of the speed of light, speed tht ws lred resobl well kow t tht time. He mde the iferece tht light ws costructed of trvelig electric d mgetic fields, or collectivel, electromgetic fields. Now it is kow tht light is smll prt of extremel lrge spectrum, the electromgetic spectrum, of electromgetic fields tht trvel with the speed of light d rge from gmm to rdio wves from the ver short wvelegths (0 3 m) to the ver log (0 0 m), rge of 3 fctors of te (d it is possible tht the rge is lrger, sice there is o theoreticl limit t this time o how smll or lrge the wves might be). The electromgetic spectrum icludes, of course, the spectrum of wvelegths show i Figure 0.. The ture of light hs bee mtter of discussio sice the time of the Greek philosophers strtig i the 500s BC, d perhps eve erlier. However, lthough the properties of light hve bee studied for ceturies, its exct ture still remis elusive. We metioed before tht light c be described s electromgetic rditio tht stisfies wve equtio; therefore, it s temptig to coclude tht we lws c model light s wve. However, there re m experimets tht show tht light hs prticle-like properties, s well. To mke model of somethig tht is both wve d prticle is difficult problem. e w out of this qudr is to s just becuse light stisfies wve equtio, tht does ot ecessril me tht light is wve, but ol coveiet mthemticl w of describig it. As exmple of similr situtio, cosider the behvior of chrge i electroics circuit composed of resistor, cpcitor, d iductor i series; the behvior of chrge i this circuit is described b differetil equtio tht hs exctl the sme form s oe i mechics tht describes the motio of mss i viscous medium t the ed of elstic sprig. Nevertheless, we do ot s tht therefore electricit d mechics re the sme. But there is other problem: other wves, such s soud wves, require medium i which to trvel. I like mer, it ws sid tht light wves trveled i specil medium clled the ether. Also, the more rigid the medium i which soud wves trvel, the greter the speed of trvel. But o pulse of kid trvels fster th light pulse i vcuum, speed tht is mesured to be ver close to c m/s. This vlue is m times greter th the speed of soud, which mes tht the ether is extremel rigid medium. Durig the ltter prt of the 9th cetur d the erl prt of the 0th, gret effort ws expeded i serchig for the ether; however, o evidece of its existece ws foud. Thus, the cocept of the ether ws fill discrded, d light is ow thought ot to eed thig i which to trvel i fct, it trvels best through regio filled with othig, true vcuum.

2 ptics Ad there is more: It is fct tht the speed of light pulse i vcuum is lws mesured s the costt c b observer, idepedet of the motio of the source or the observer. I other words, o mtter how fst the source moves whe light pulse is emitted, observer mesures the speed of the light pulse s c; lso, o mtter how fst the observer moves, the observer still mesures c. For exmple, if the observer trvels t pproximtel the speed of light i oe directio, d pulse of light trvels i the opposite directio, the observer still mesures the speed of the pulse s it psses b s c. This cocept violtes our commo-sese ides of reltive velocities; evertheless, it is i greemet with experimet. This propert of the speed of light is oe of the bsic teets i Eistei s specil theor of reltivit. These properties of light re ver difficult to model ito somethig tht grees with our orml experiece bit of eerg tht hs both wve d prticle ture, d lws hs the speed c i vcuum. But, let s tr to mke simple model to help our thikig. We strt with picture of idel wve; mel, oe tht goes o forever from mius ifiit to plus ifiit. Now for light let s imgie tht it is just smll prt of such wve so tht it is fiite d smll eough i extet to pproximte prticle. Such piece of wve we cll photo. Wheever we mesure the speed of photo i vcuum, we obti the vlue c. Now we hve simple model of light tht remids us tht light hs both wve d prticle ture, d lthough this model is oversimplified, it c ct s strtig poit.

3 PART I GEMETRICAL PTICS 3 It is coveiet to divide the stud of optics ito prts, ech prt depedig o the complexit of the model used to represet the properties of light. Becuse the properties of geometricl optics re described b the simplest model, we begi our stud with tht topic. I this model, light trvels i stright lies through uiform medi (or more precisel, homogeeous d isotropic medi), d obes the lws of reflectio d refrctio t surfces betwee differet medi. With these simple cocepts, we c work out the bsic properties of m opticl sstems. The erliest opticl istrumets were mirrors mde of polished metl. Severl mirrors, some i excellet coditio, hve bee recovered tht were used i ciet Egpt (c. 900 BC). Mirrors re metioed i the Bible (c. 00 BC) i the buildig of the tbercle; Exodus 38:8 i the New Itertiol Versio reds: The mde the broe bsi d its broe std from the mirrors of the wome who served t the etrce to the Tet of Meetig. Leses, clled burig glsses, were kow to the Greek philosophers (c. 400 BC), d the refrctio effect of the ppret bedig of objects whe the were prtl immersed i wter ws metioed b Plto t bout the sme time. B 300 BC it ws oted tht rig t the bottom of empt vessel, whe hidde b the lip of the vessel, becme visible fter the vessel ws filled with wter. The Roms lso kew bout burig glsses b the first cetur BC, d hd observed tht glss globe filled with wter would produce mgifictio. Although other cultures i erl times hd similr socil orders, it ws i Greece tht specultive thikig tured ito rtiol thought, philosoph, d sciece b the 500s BC. This trsformtio did ot hppe quickl, but took plce over period of severl ceturies s the Greek cit-sttes itercted i trde with the civilitios of Mesopotmi d Egpt tht brought cquitce with their stroom, medicie, d techolog. The ide tht light trvels i stright lies (rectilier propgtio) ws tught b Plto bout 400 BC, d the equlit of the gles of icidece d reflectio i wht cme to be clled the lw of reflectio ppers i the works of Euclid s erl s 300 BC. Although Greek sciece lrgel cesed to develop fter pproximtel 00 BC, Hero of Alexdri d Ptolem (lso of Alexdri) were exceptios. Just whe Hero lived is ope to questio; sources give dtes tht rge betwee 50 BC d 50 AD. However, becuse Hero metios eclipse i his writigs tht is kow to hve occurred i 6 AD, curret thikig is tht he lived i the first cetur AD. Hero is importt becuse he ttempted to derive the lw of reflectio from priciple tht stted tht reflected light trveled the shortest distce betwee two poits. Although, the lw does follow from this priciple for umber of reflective surfces, lter workers showed tht it ws ot true i geerl. Ptolem (00 65 AD) is sigifict becuse of his work o refrctio. The discover of the lw of refrctio took m ceturies. But Ptolem mde importt begiig; he ot ol mesured gles of icidece d refrctio for rs of light, but lso rrged them ito tbles. Exmiig his tbles, he cocluded tht the gle of refrctio ws proportiol to the gle of icidece, which we ow kow to be true ol whe the gles re smll. Durig this time period, the Rom empire ruled the wester world. However, the Roms did little to dvce sciece; the excelled t lw, govermet, d wrfre i sciece the primril collected wht the Greeks hd lered. The Rom empire declied d bsicll cesed to exist b 475 AD; the little hppeed i sciece util the Arbi empire becme importt b pproximtel 700 AD. The Arbi scholrs trslted d bsorbed the the-kow scietific kowledge, but did little origil work, except i optics where Alhe (Al hh ZAHN) i bout 000 AD mde importt cotributios. He studied reflectio d, like those before him, cocluded tht the gles of icidece d reflectio were ideed equl, but lso dded the cocept tht these gles l i the sme ple orml to the iterfce ple tht is ow clled the ple of icidece. He studied the lw of refrctio, d repeted the mesuremets mde b Ptolem. However, he correctl cocluded tht Ptolem ws i error whe the gles were lrger, d tht the gle of refrctio ws ot i geerl proportiol to the gle of icidece. But, he filed to discover the true lw of refrctio. Not util the 00s did workers i Europe show iterest i sciece d begi to ssimilte the scietific cocepts recorded b Arbi scholrs; i prticulr, the writigs of Alhe o optics hd gret ifluece. B the 300s, pitigs showed moks werig eeglsses, d i the 500s combitios of covergig d divergig leses were described. The i the erl 600s the telescope d the microscope were iveted. Although cotested, the evidece idictes tht the first telescope ws mde b spectclemker, Hs Lippershe, of Holld i 608. The first microscope ws costructed t pproximtel the sme time, most likel b other spectcle-mker, Zchris Jsse, lso i Holld. With the ivetio of these istrumets, the serch for the lw of refrctio ws reewed. B 6, the Germ

4 4 Geometricl ptics stroomer Kepler, hd discovered totl iterl reflectio, d hd obtied the lw of refrctio for smll gles, which ws bsicll wht Ptolem d Alhe hd determied ceturies before. The true form of the lw of refrctio is usull credited to Willebrord Sell some ers lter (some sources s 6), who ws professor of mechics t the Uiversit of Lede i Holld, lthough he ever published his discover. It ppers tht he foud the lw experimetll, d expressed it i terms of cosects of the gles rther th i the moder form usig sies. I 637, Reé Descrtes of Frce, published the lw usig sies of the gles. He obtied the lw theoreticll, lthough some of his hpotheses were teuous. I fct, the Frech mthemtici Pierre de Fermt d others were quick to fid fult with the hpotheses of Descrtes, d i 657 Fermt deduced the lw of refrctio usig the ssumptio tht light trvels from poit i oe medium to poit i other medium i the lest time, priciple tht cme to be clled Fermt s priciple of lest time. Yers lter, it ws foud tht this priciple could be derived from Mxwell s equtios of electromgetism. I curret literture, the me ssiged to the lw of refrctio is uder some cotetio: I m coutries it is clled either Sell s lw or simpl the lw of refrctio; however, i Frce it is kow s Descrtes lw. We will opt o the side of clrit i this textbook, d cll it the lw of refrctio. I the mi we will stud the cosequeces of the lw of refrctio; we will lso touch o the lw of reflectio, but most of our time will be spet o refrctio. The lw of refrctio works for surfce, but becuse of the reltive ese with which sphericl surfces re mde, it is these surfces tht will be our primr cocer (ple surfces re lso icluded, sice ple surfce m be regrded s sphericl surfce of ifiite rdius). Eve though the lw of refrctio is simple to stte, whe rs re trced through sphericl surfces, m of the equtios become quite complicted. Therefore, we will follow commo prctice d obti pproximte results usig prxil rs, rs tht trvel close to the smmetr xis d mke smll gles with ormls to the surfces, the sme rs tht Ptolem, Alhe, d Kepler hd observed to hve simple properties. This procedure will llow us to obti the chrcteristics of the les sstem uder stud resobl quickl s first pproximtio; the the detils will be obtied b trcig the rs through the surfces exctl without pproximtio. R trcig, eve with prxil rs, c be ver clcultio itesive. I the ds before electroic computers, the clcultios were doe b hd with pecil d pper usig lrge tbles of logrithms. To miimie the time spet i clcultio, gret effort ws give to obtiig equtios tht would reduce the umber of rithmetic opertios, rrgig the work i tbulr form so tht umericl vlues could be used efficietl, d i developig techiques for error checkig. A skillful worker could use these techiques to trce r through tpicl les sstem i bout five miutes. The sme kid of r trcig is ow doe with the moder computer i less the secod; eve with progrmmble scietific clcultor it is doe quickl. I m optics textbooks, becuse of these computtio difficulties, results re frequetl give without derivtio. With the wide vilbilit of persol computers, the tools re ow i plce such tht most results c be obtied b everoe, eve those tht re clcultio itesive. We shll usull void givig progrms i this textbook, becuse of the wide vriet of progrmmig lguges ow vilble. Isted, we shll tr to provide guidelies i the form of flowchrts, or lgorithms, to id progrm writig i whtever progrmmig lguge is chose. These flowchrts will hve the dded dvtge of brigig together d summriig equtios derived over m pges. Most of the progrms re ot log, d to write them is extremel iformtive, istructive, d helpful i uderstdig cocepts. Becuse m of the results re best uderstood b drwig grph, softwre lguge tht hs grphicl cpbilit is dvtgeous. Smbolic lgebr softwre, such s Mthemtic, is mde to order for much of the work we shll eed to perform. Such progrms c perform clcultios to high precisio, drw grphs i two d three dimesios, d furthermore, mipulte expressios smbolicll tht rise whe the rules of lgebr or clculus re pplied. The exmples d grphs i this textbook re performed with Mthemtic o Mcitosh computer. I the umericl clcultios, the prited results will usull be rouded to pproprite umber of sigifict figures, but the clcultios will be crried out to the defult precisio of Mthemtic, which is ormll 6 deciml digits. I erlier works o r trcig i les sstems, legth uits of iches were customril used i Eglish spekig coutries. Now, metric uits or o uits t ll re used. Expressig legths with o uits is populr becuse, s we shll show i the work to follow, the dimesios of legth hve the propert of sclig lierl i les sstems. However, whe o uits re expressed, it is difficult for the begier to tell if chges i legth qutit re big or smll. Therefore, i this textbook, s id to uderstdig d comprehesio, we shll usull stte the legth uits i terms of millimeters (bbrevited s mm).

5 Chpter utlie Prxil ptics Refrctio & Trsltio 5 Chpter : Prxil ptics Refrctio & Trsltio Itroductio Bsic lws The lw of rectilier propgtio The lw of reflectio The lw of refrctio (Sell s lw) Agles Mtrices d Determits Exmple.. Performig mtrix multiplictio d illustrtig the commuttive lw Exmple.. Illustrtig the ssocitive lw Exmple..3 Multiplictio of squre mtrix d colum vector Exmple..4 The iverse of mtrix Exmple..5 Multiplictio of determits The Prxil Approximtio d Its Mtrix Represettio Prxil Mthemtics Itroductio The refrctio mtrix The trsltio mtrix The ple-to-ple mtrix Exmple.3. Trcig r through covergig les Exmple.3. Trcig r bckwrd Exmple.3.3 Trcig r through divergig les Exmple.3.4 R trcig tht eds with bckwrd extesio of the r Exmple.3.5 Trcig r through doublet A recursio reltio for M P P pticl sstems scle lierl The Gussi Costts The sstem mtrix Exmple.4. Sstem mtrix, umericl exmple The four opticl sstem cses Cse (A 0): The focl or telescopic sstem Cse (B 0): The uprimed focl poit F Cse 3 (C 0): The primed focl poit F Cse 4 (D 0): The object-imge coditio The trsverse mgifictio m T The object-imge mtrix Exmple.4. The sstem mtrix, the Gussi costts, d object, imge pir A prdox d its resolutio The logitudil mgifictio m L The Crdil Poits d Ples The focl poits, F d F The uit or pricipl poits, H d H The odl poits, N d N Derivtio of equtios A importt propert The focl legths, f d f Exmple.5. A equicovex les: the crdil poits d ples, d the focl legths Exmple.5. A equicocve les: the crdil poits d ples, d the focl legths Exmple.5.3 A opticl sstem of two leses: the crdil poits d ples, d the focl legths Exmple.5.4 A opticl sstem of oe surfce: the importt reltioships

6 6 Chpter : utlie.6 The Gussi d Newtoi Formultios The Gussi equtios The Newtoi equtios The Thi Les Approximtio Thick les lsis The thi les i ir The thi covergig les The thi divergig les Exmple.7. The telephoto les Exmple.7. The Rmsde eepiece Exmple.7.3 The Huges eepiece Exmple.7.4 A oom les Step : l les L moves Step : Both L d L 3 move Problems Aswers to Problems Refereces

7 Chpter Prxil ptics Refrctio & Trsltio 7. Itroductio.. Bsic lws ur model for light is composed of ti bits of eerg clled photos which hve both wve d prticle ture; digrm tht might represet photo is drw i Figure.. The crest-to-crest distce λ is clled the wvelegth, which our ees iterpret s color i the visible regio. A photo trvels with ver high speed: i vcuum its speed is ver close to c m/s; i other medi, the speed is smller d lso vries with wvelegth. I the visible regio, the wvelegth rges from the violet (λ 400 m) to the red (λ 700 m), where m ometer 0 9 meter. Aother uit which ws frequetl used i the pst, but is ow o loger recommeded, is the Agstrom (0 A m). λ Figure. Figure. I geometricl optics, λ<<(the dimesios of objects d opeigs), d we model photos trvelig i give rectilier directio b simpl drwig r, s show i Figure.. I this chpter, d severl of the chpters to follow, we wt to ivestigte how these rs iterct with surfces tht represet boudries betwee two differet medi where ech medium is both homogeeous d isotropic. Such medium s most importt opticl propert is its idex of refrctio which is defied to be c (.) v where c is the speed of photos i vcuum, d v is the speed i the medium. For medium tht is vcuum, this defiitio gives exctl. A medium is homogeeous whe it is the sme throughout. So crstl with its toms (or molecules) i their lttice sites is homogeeous. But crstl m hve certi directios where photos trvel t differet speeds d therefore hve differet idices of refrctio. A medium where ll the toms (or molecules) re mixed up like i liquid or glss so there re o preferred directios is clled isotropic. Thus, i medium tht is both homogeeous d isotropic, the speed of photo is costt i ll directios d the idex of refrctio is costt (for give wvelegth) everwhere i the medium. The, we hve The lw of rectilier propgtio. I homogeeous d isotropic medium, light trvels i stright lie t costt speed. I Figure.3 we illustrte wht hppes whe r the icidet r flls o surfce tht seprtes two homogeeous, isotropic medi. I geerl, the icidet r seprtes ito two rs: oe is reflected d obes the lw of reflectio, the other is refrcted d obes the lw of refrctio. Eve whe the secod medium (the medium cotiig the refrcted r) is opque, there is still refrcted r; it is simpl bsorbed fter it trvels short distce. However, i our stud we re primril iterested i trspret medi where most of the icidet r goes ito the refrcted r, ver little is reflected or bsorbed. The lw of reflectio. This lw hs two prts (see Figure.3): ) θ θ ) the reflected r lies i the ple of icidece where θ is the gle of icidece, θ is the gle of reflectio, d the ple of icidece is the ple tht cotis the icidet r d the orml to the surfce. The lw of refrctio (Sell s lw). This lw lso hs two prts (see Figure.3): ) si θ si θ (.) ) the refrcted r lies i the ple of icidece where θ is the gle of refrctio, is the idex of refrctio of the medium i which the icidet r is locted, d is the idex of refrctio of the medium i which the refrcted r is locted. Equtio. is extremel importt equtio, d m people serched for it over period of m ceturies; ol fter its discover did les desig begi. It is importt to ote tht i both lws the gles re mesured reltive to the orml, ot the surfce. reflected r orml icidet r θ ple of icidece θ surfce Figure.3 θ refrcted r

8 8 Chpter : Prxil ptics Refrctio & Trsltio For medium of ir t tm pressure d 0 C temperture, the vlue of is.0009 mesured to six sigifict figures, vlue so close to oe tht we customril s the for ir is uit, like tht for vcuum. For some other medi,.50 for commo glss,.4 for dimod, d.33 for wter. Becuse the speed of photo i medium tht is ot vcuum c vr with wvelegth, the idex of refrctio vries with wvelegth; it is usull rther smll vritio. For solids d liquids, the vritio occurs i the third or fourth sigifict figure; for gs t orml pressures, such s ir, the vritio does ot occur util the seveth sigifict figure... Agles The simplest d most effective defiitio of gle θ is θ s (.3) r where s is the rc legth, d r is the rdius, s show i Figure.4. This defiitio is cosistet with the reltioship for smll gles, si θ θ, whe si θ is give b the trigle defiitio. Becuse s d r hve commo legth uits, the gle θ is dimesioless. Thus, we might thik tht θ should hve o uits; however, gle m be defied i other ws. To prevet cofusio, the rdi (bbrevited rd) uit is used whe gles re defied b Equtio.3. But the rdi uit is ot phsicl uit like the meter, secod, or kilogrm; isted, it is clled supplemetr uit it helps to distiguish gles i rdis from gles determied i some other w, s degrees. Becuse the rdi is dimesioless, it m be dropped or ersed i equtio whe it is ot eeded. r θ r r Figure.4 The degree is the more coveiet uit for esier visulitio. To obti the coversio betwee the degree d the rdi, cosider the semicircle show i Figure.5. Becuse the circumferece of circle of rdius r is πr, the rc legth of semicircle is πr, d Equtio.3 gives θ s r πr π rd r s πr 80 deg Figure.5 r s Becuse the gle of semicircle is 80 deg, we hve the desired coversio π rd 80 deg (.4) where we hve writte the degree uit s the deg, rther th, to mke it esier to use whe uits re treted s frctios i covertig uits from oe form to other.. Mtrices d Determits Mtrices d determits re shorthd represettio of the umbers i problem writte s rectgulr rr i rows d colums. Some evidece exists which shows tht some spects of mtrices were used i the secod cetur BC d perhps eve s fr bck s the fourth cetur BC. But the mthemtici Arthur Cle i 858 gve the first bstrct defiitio of mtrix, d showed how rrs pplied to problems t tht time were specil cses of his more geerl cocept. He developed lgebr of mtrices, d i subsequet ers m pplictios were foud ot ol i mthemtics, but lso i phsics d egieerig. Mtrices were first used i optics b Smpso i 93, d their use ws lso suggested b the work of T. Smith i the 930s, but ot util rticle b Hlbck i the Americ Jourl of Phsics o mtrix represettio i optics i the erl 960s, did their use become commo i textbooks o geometricl optics. A determit is less geerl cocept th mtrix: it tkes the umbers of squre mtrix d clcultes sigle umber b followig specific procedure. I our mtrix developmet of prxil optics, determits will hve the useful propert tht their vlue is lws uit; this feture is importt both theoreticll d umericll. I geerl, mtrix A is defied to be rr of umbers, clled mtrix elemets, rrged i rows d colums; s exmple A 3 (.5) 3 is clled 3 mtrix becuse it hs rows d 3 colums. The mtrix elemets hve umericl subscripts for es referece; the first subscript refers to the row, d the secod to the colum for exmple, mes tht this elemet is i the first row d secod colum. A squre mtrix hs s m rows s colums; the followig exmple illustrtes mtrix: A (.6) Whe mtrix A is squre mtrix, the its determit A is defied quite simpl s A (.7) We s the elemets d lie log the pricipl digol; the other two elemets, d, we cll the off-digol elemets.

9 . Mtrices d Determits 9 As we metioed, etire lgebr for mtrices exists, but becuse of the spce requiremet to preset it, we will ol itroduce those prts of the lgebr tht we eed. We will lso simpl illustrte some of the lws b exmple rther th to prove them rigorousl. I prxil optics, the multiplictio of squre mtrices occurs frequetl, so we give the defiitio for multiplig two such mtrices: AB b b b b b + b b + b (.8) b + b b + b To remember the steps of mtrix multiplictio, observe tht the rows of the first mtrix re multiplied d dded ito the colums of the secod. We ext illustrte mtrix multiplictio with exmples, d show t the sme time some of the lws tht mtrix multiplictio obes. We sometimes will iclude more steps th usul to id uderstdig. Exmple.. Performig mtrix multiplictio d illustrtig the commuttive lw. 3 4 AB BA This exmple illustrtes the ver importt result, AB BA; tht is, mtrix multiplictio does ot, i geerl, obe the commuttive lw. Exmple.. Illustrtig the ssocitive lw. () A(BC) () (AB)C We observe i Exmple.. tht A(BC) (AB)C, reltioship clled the ssocitive lw; i fct, mtrix multiplictio lws obes this lw. Becuse the ssocitive lw ss tht it does ot mtter how the mtrices re grouped together for multiplictio, it is cceptble to write the multiplictio of three mtrices simpl s ABC. This lw c be exteded to the multiplictio of umber of mtrices, ot just three. However, s Exmple.. shows, the order i which the mtrices re writte must i geerl ot be chged. Aother mtrix of importce is rectgulr mtrix, deoted b ) ( b b which is lso clled colum mtrix or colum vector. Mtrix multiplictio of squre mtrix d colum vector is defied s b b + b (.9) b b + b result tht is other colum vector. However, colum vector times squre mtrix the reverse order of the multiplictio i Equtio.9 is ot defied. Exmple..3 colum vector. ( Multiplictio of squre mtrix d ) A mtrix with specil properties is the idetit or uit mtrix I ; for mtrices, it is defied s 0 I (.0) 0 e of its specil properties is tht it commutes with ll mtrices: AI IA A (.) becuse AI 0 A 0 0 IA A 0 A idetit mtrix c be defied for other squre mtrices s well, ot just mtrices, d Equtio. is lso true for these mtrices. The idetit mtrix I is used to defie iverse or reciprocl A of the mtrix A b writig AA A A I (.) tht is, A is tht mtrix which whe multiplied b A gives I. The properties of mtrix re such tht if AA I, the it c be show tht A A I must lso hold.

10 0 Chpter : Prxil ptics Refrctio & Trsltio The iverse of mtrix A hs the compct d useful form A A A where A is the determit of A, mel, A A (.3) A As we metioed before, the determits of the mtrices tht we use i prxil optics hve the vlue of oe; thus, A i Equtio.3, d we hve the simple iverse A (.4) I words, the digol elemets re iterchged, d the off-digol elemets hve their sigs chged. Methods to clculte iverses re foud i the literture. Sice the use of mtrices hs icresed, routies for the umericl clcultio of iverses re icluded o m scietific pocket clcultors. Smbolic lgebr sstems, such s Mthemtic, c determie iverses either smbolicll or umericll. Exmple..4 The iverse of mtrix. Suppose we strt with the simple mtrix A ( ) 5 The the determit of this mtrix A 7 d Equtio.3 give the iverse of A s A ( 5 ) 7 7 We c verif Equtio. with mtrix multiplictio: 7 ( 5 5 ) AA ( 5 ) A A Becuse determit represets (or equls) umber, determits obe lws similr to the lws umbers obe. We evlute the determits with Equtio.7. Exmple..5 Multiplictio of determits. Ulike mtrices, determits obe the commuttive lw; we use the mtrices i Exmple..: A B (3)( 8) 34 B A ( 8)(3) 34 Eve whe we evlute the mtrix product first before we clculte the determit, we still get the sme result: AB BA Determits obe the ssocitive lw; we use the mtrices i Exmple..: A ( B C ) 3 ( ) 8 9 (3)(( 8)( )) 468 ( 3 ( A B ) C 5 4 ) ((3)( 8)) ( ) 468 A B C (3)( 8)( ) 468 No mtter how we combie the mtrices before we clculte the determits, we get the sme result: A BC (3)(36) 468 AB C ( 34)( ) 468 ABC CBA

11 .3 The Prxil Approximtio d Its Mtrix Represettio Prxil Mthemtics.3 The Prxil Approximtio d Its Mtrix Represettio Prxil Mthemtics.3. Itroductio I this chpter, opticl sstem mes sstem of oe or more refrctig surfces composed of sectios of sphericl surfces with ll their ceters of curvture o the sme xis (the xis), d with the sectios cetered o this xis cetered opticl sstem. The outlie of sphericl sectio is show i Figure.6. Not ol do sphericl surfces hve importt phsicl properties whe the iterct with light, but mthemticll the re the simplest curved surfce to describe, requirig ol ceter d rdius (the rdius of curvture). To describe the ture of the sphericl sectio more completel, we imgie the rc show i Figure.6 to rotte bout the xis to obti surfce of revolutio. The xis hs its positive directio to the right, d becuse it is such importt referece xis, it is give specil mes such s the smmetr xis, the opticl xis, or the pricipl xis. Sice the refrctig surfces re smmetricll plced with referece to the xis, this propert suggests the smmetr xis s the most descriptive me. The ple determied b the d xes is clled the meridiol or tgetil ple. The positio of the xis mes, here d, mrks the positive ed of the xis. We will geerlie our defiitio of cetered opticl sstem to iclude reflectig surfces lter. ther shpes, such s clidricl, re permitted i opticl sstem, but we limit our discussio to sphericl sectios. r smmetr xis V C Figure.6 The rdius of curvture r is positive. As show i Figure.6, the xis is the verticl xis with positive directio up, d i lter chpters whe we eed to hve three-dimesiol referece sstem, the x xis will be drw perpediculr to the pge with positive directio poitig ito the pge. We choose this somewht wkwrd referece sstem becuse i lter chpters whe we wt to describe wht is hppeig i the x ple, it gives ver coveiet referece frme. Positive d egtive sigs re extremel importt i geometricl optics, d we will hve to p specil ttetio to them; i Figure.6, the rdius of curvture r is defied to be positive whe the ceter of curvture C lies to the right of the refrctig surfce, egtive whe it lies to the left. The poit V of the refrctig surfce is clled the vertex. A opticl sstem of two or more refrctig surfces with t lest oe surfce curved, we me les sstem; whe the sstem hs just two refrctig surfces, we cll it les, or for emphsis, simple les. The clss of simple leses is divided ito two subclsses: i the covex subclss, show i Figure.7(), icidet rs prllel to the smmetr xis bed or coverge towrd the smmetr xis fter pssig through the les; i the cocve subclss, illustrted i Figure.7(b), similr icidet rs diverge from the smmetr xis ssumig tht these leses re i ir or other medium with idex of refrctio less th tht of the les mteril. We lso cll these leses positive or egtive, s we expli lter i this chpter whe we tlk bout the focl legth of les. We observe tht covergig les is thicker i the ceter; tht is, it bulges outwrd, the bsic meig of covex. I similr w, the divergig les is thier t the ceter, it is depressed, or cocve. bicovex plocovex meiscus covex () Covergig, positive, or covex leses. bicocve plococve meiscus cocve (b) Divergig, egtive, or cocve leses. Figure.7 Cross sectios of some simple leses. Some of the leses i ech subclss re give specil mes, s we idicte i Figure.7. A bicovex les, or double-covex les, hs ech of its surfces curved outwrd; whe bicovex les hs surfces of equl rdii i mgitude, the les is clled equicovex. A plocovex les, or plr-covex les, hs oe surfce tht is ple. The meiscus-covex les is crescet-shped object. We c give similr descriptio for the cocve leses. ther mes re sometimes used for these leses, but we hve give the oes frequetl used. The first prcticl use for leses ws to ssist i redig, d ws described i writig fter the 50s. Although these erl leses were crude, the worked d ebled idividuls to perceive detil tht the could ot see with the uided ee. Becuse the leses were of the covex shpe, the remided people t tht time of letils, the Lti me for which ws les, d thereb becme the populr me for these redig ids. With time skill ws developed i mkig leses of better qulit. These leses usull hd sphericl shpe becuse tht is the turl surfce tht develops whe two mterils re rotted d rubbed gist ech other over ll sectios of both surfces.

12 Chpter : Prxil ptics Refrctio & Trsltio.3. The refrctio mtrix Cosider r i the meridiol ple tht udergoes refrctio t the poit P of sphericl refrctig surfce of rdius r, s show i Figure.8; for coveiece, we drw the rs such tht the trvel from left to right (the positive directio). The digrm i Figure.8 is referece digrm: it is used to illustrte the defiitio of qutities d to show whe the re positive, d b iferece, whe egtive. B ispectio of the digrm, we observe this rule mes tht gles drw couterclockwise from the referece lies (the stright lies touched b the tils of the rcs with rrows) re positive; whe drw clockwise, the re egtive. Becuse the rdius r (d its extesio) form the orml to the surfce, the gle θ is the gle of icidece, d the gle θ is the gle of refrctio. The gles δ d δ re the gles of iclitio (or slope gles), gles reltive to the horiotl; it is these gles tht will be the most importt i describig the directios of r i prxil optics. A quick w to obti the sig of the gle of iclitio is to ote tht whe r trvels from left to right, it is positive whe the r trvels uphill, egtive whe dowhill. θ φ δ V P δ φ Figure.8 The referece digrm for refrctio drw with ll the qutities positive. We obti severl mthemticl equtios b ispectio of the digrm i Figure.8: φ θ si φ r r θ φ + δ θ φ + δ r C (.5) (.5b) (.5c) si θ si θ (.5d) where the lst equtio is the lw of refrctio (Sell s lw), our ke equtio to be simplified. It is importt to ote tht i the digrm the coordite of the poit P before refrctio is clled, immeditel fter refrctio it is clled ; these coordites re positive whe bove the xis, egtive whe below. Clerl,. We lso observe the rther strge equlities for the idices of refrctio, d ; these ppretl redudt reltios will llow us to write prcticl itertive equtios tht is, equtios which c be performed repetedl i trcig r through opticl sstem of severl refrctig surfces. It is lso correct to write the reltio s 0, d there re occsios whe we shll use the ltter ottio. However, here we use the ottio becuse we c costruct simpler equtios b tretig the first medium (d the lst) i distictive mer. We c gretl simplif the descriptio of rs tht pss through opticl sstem b the requiremet tht the rs re prxil, where pr mes beside d xil mes, of course, xis. Thus, prxil rs re beside xis rs; tht is, the trvel close to the smmetr xis log the etire legth of the opticl sstem. The, the prxil pproximtio mes tht rs trvel close eough to the smmetr xis to mke ll gles used i the descriptio of the rs smll. Whe we mke the gles smll, we c replce the sie fuctios i Equtios.5 d.5d b simpler reltioships. I clculus it is show tht whe gle A is mesured i rdis, the the sie fuctio c be writte i terms of the ifiite series si A A A3 3! + A5 (.6) 5! d we see tht whe A is smll, we hve si A A (.6b) For exmple, si (0. rd) gives for error of 0.7% (error 00 % pproximte exct /exct). Usig Equtio.4, we fid tht 0. rd correspods to 5.73 deg. Thus, for gles tht re pproximtel 6 deg or less, the error whe Equtio.6b is used to determie the sie is bout 0.% or less. Therefore, whe φ, θ, d θ re smll, Equtios.5 d.5d become φ (.7) r r θ θ (.7b) where the gles re mesured i rdis ver importt. Substitutig Equtios.5b d.5c ito Equtio.7b, d the substitutig Equtio.7 for φ, we obti (φ + δ ) (φ + δ ) + δ + δ (.8) r r Solvig Equtio.8 for δ gives where δ δ r δ p (.9) p r ( )c (.0) is clled the power of the refrctig surfce, d c /r is the curvture of the surfce. Usig the curvture c isted

13 .3 The Prxil Approximtio d Its Mtrix Represettio Prxil Mthemtics 3 of the rdius r is especill useful whe describig ple surfce, sice i plce of writig r, we c write c 0; workig with eros is usull esier th workig with ifiities, especill i computer progrm. The power p is the ol term i Equtio.9 tht cotis the rdius r we use p isted of r becuse we obti lier reltioship betwee δ d p. This reltioship gives esier ituitive feel to the chge i δ tht occurs whe we chge p rther th thikig i terms of chge i r. Sice similr equtio to Equtio.9 c be writte for surfce of the opticl sstem, wht we hve just sid is true for surfce. Fill, we write the obvious reltio, obtied b lookig t Figure.8, (.) We ow observe b ispectio tht Equtios.9 d. re summried b the mtrix equtio ( δ ) p δ (.) 0 or more simpl s V R V (.3) where p R (.4) 0 is clled the refrctio mtrix of surfce, d ( δ ) V V ( δ re colum vectors. Becuse these colum vectors V coti iformtio o the r s it trverses the opticl sstem, we cll them r vectors. The vlues of δ d cotied i the r vectors re clled the r dt (or r prmeters). Although we hve lred used the smbol V for vertex of refrctig surfce i our digrms, the cotext tells us whe V represets vertex or r vector. Referrig to the r vectors i Equtio., we see tht the δ qutit is i the first row d the qutit is i the secod. This order could be reversed, s log s the off digol elemets i the R mtrix re lso iterchged; it is primril choice of stle. Whe we clculte the determit of the refrctio mtrix R, we obti the simple but importt result R p 0 To provide this result of oe is the reso tht the idices of refrctio d re kept with the gles of iclitio δ d δ i Equtios.9 d.. ) Ever refrctio mtrix i prxil optics hs the form of the mtrix i Equtio.4 d hs the propert tht its determit equls oe..3.3 The trsltio mtrix Now we look t the trsltio (or trsfer) of prxil r from oe poit to other i give medium, s betwee the poits P d P o the two surfces show i Figure.9 (the digrm i this figure is cotiutio of the oe i Figure.8). B ispectio of this digrm we see tht V P t t t δ δ δ δ (.5) where i the lst step we hve used the reltio to mke equtio tht leds itself more esil to itertio. From Figure.9, we hve + ( t t + t ) t δ (.6) Becuse the prxil pproximtio ss tht prxil rs trvel close to the smmetr xis (the xis), we see tht t d t re smll compred to t so Equtio.6 becomes + t t δ (.7) Thus, we c simpl use the thickess of les to describe the trsltio of r betwee its surfces. I prllel with, we write t t i Figure.9 to provide flexibilit i cretig equtios d computer progrms. I Figure.9, we drw the dimesio lies for the trsltios t t, t, d t s rrows with til d hed to mke them look like vectors. The til idictes the referece from which we mesure the qutit, d the hed gives the directio so tht we c ifer the positive or egtive sttus of the qutit. I Figure.9, the dimesio lies (or dimesio rrows) poit i the positive -xis directio; therefore, ll three of these qutities re positive. This prctice, which we shll follow i our text, helps us to write reltioships with the correct plus d mius sigs b ispectio of the digrm ver useful d time-svig method. δ δ V t Figure.9 The referece digrm for trsltio drw with ll the qutities positive. P

14 4 Chpter : Prxil ptics Refrctio & Trsltio Just s there is ifiite series represettio for the sie fuctio (see Equtio.6), there is oe for the tget fuctio: t A A + A3 3 + A5 + (.8) 5 where A is i rdis. Whe A is smll we hve t A A (.8b) B the prxil pproximtio, the gle of iclitio δ i Equtio.7 is smll; thus, Equtio.8b llows us to write Equtio.7 s + t δ (.9) Sice our prctice is to group the idex of refrctio with the gle of iclitio to mke the determit of refrctio mtrix R uit, we rewrite Equtio.9 s ( t ) + ( ) δ (.30) The we observe tht we c write both Equtios.5 d.30 i mtrix form s δ 0 δ t (.3) or more simpl s where V T V (.3) 0 T t (.33) is the trsltio mtrix (or trsfer mtrix) from surfce to. We write the subscripts of the trsltio mtrix i reverse order becuse of the subscript ptter tht emerges i Equtio.3: whe we red the subscripts from right to left o the right side of Equtio.3, we hve descriptio of the progress of the r s it trverses the sstem. The subscript i the colum mtrix V idictes tht the r strts t surfce (the prime idictes tht the r hs lred bee refrcted t tht surfce), d the subscripts of the trsltio mtrix T ss tht the r trsltes from surfce to. the left side of Equtio.3, the subscript of the colum mtrix simpl ss tht the r is ow t surfce ; sice there is o prime ttched to V, we uderstd tht the r hs ot et udergoe refrctio t tht surfce. We esil see tht the trsltio mtrix i Equtio.33 hs determit equl to oe, just like refrctio mtrix. Ever trsltio mtrix i prxil optics hs the form of the mtrix i Equtio.33, d hs the propert tht its determit is uit..3.4 The ple-to-ple mtrix We c mke complete descriptio of the r trvel so fr b substitutig Equtio.3 ito Equtio.3 to obti the exteded mtrix equtio V T R V (.34) which ow icludes the refrctio tht tkes plce t surfce. Agi we ote tht whe we red the right side of Equtio.34 from right to left, we c follow the dvce of the r through the opticl sstem. Equtio.34 keeps the sme form whe it is geerlied to describe the r trvel through sstem of oe, two, or m refrctig surfces. To brig together wht we hve lred doe, to set the ptter for trcig r through umber of refrctig surfces from begiig to ed, d to fill summrie the etire r trvel i sigle mtrix clled the ple-to-ple mtrix, we drw the referece digrm of Figure.0. This digrm illustrtes how sstem of refrctig surfces from oe to m is represeted. Followig our usul prctice, we drw the digrm such tht ll the qutities re positive. The rdii r d r re positive, but re ot show to surfce surfce P δ δ P δ δ P P δ δ t t V t t V C t t C Figure.0 The referece digrm for opticl sstem of two refrctig surfces ll qutities re positive.

15 .3 The Prxil Approximtio d Its Mtrix Represettio Prxil Mthemtics 5 keep the digrm simple; however, we do show the ceters of curvture C d C. The digrm reitertes wht we hve sid before: wheever ceter of curvture is to the right of its refrctig surfce the correspodig rdius is positive; if the ceter of curvture is to the left, the the rdius is egtive. We shll use exmples to illustrte these fetures for the rdii, s well s to illustrte the properties of the other qutities. The middle prt of this digrm, the prt betwee the surfces d, shows the ptter b which qutities would be remed if there were other refrctig surfce, s surfce 3. For exmple, the idex of refrctio reltio would become 3 betwee surfces d 3; i similr w, the trsltio reltio would become t t 3. We lso ote tht qutities tht hve ot et bee refrcted t surfce re uprimed, fter refrctio the re primed. As we hve lred metioed, this somewht mess ottio is used becuse it leds itself to coveiet itertio. Fill, it proves useful to tret the iitil d fil medi i specil w usig qutities without subscripts: for the first medium we use the uprimed qutities (, δ,, t, P), for the fil medium we use the primed qutities (,δ,, t, P ). Lstl, we write the mtrix equtio tht represets the r trvel show i Figure.0 usig smbols to represet the mtrices: V P T P R T R T P V P (.35) M P P V P (.35b) where M P P T P R T R T P (.36) is clled the ple-to-ple mtrix, squre mtrix tht llows us to clculte where r o the ple cotiig the poit P will termite o ple cotiig the poit P. Agi we ote tht b redig Equtio.35 from right to left we obti the left-to-right trvel of the r through the opticl sstem of cetered refrctig surfces show i Figure.0. Fillig out the mtrix smbols i Equtio.35 with wht ech oe represets, we obti the mtrix equtio δ 0 p 0 p t t 0 δ t (.37) 0 0 where the powers of the surfces re p r d p r (.38) No mtter how complicted the sstem of refrctig surfces is, we c represet the sstem with digrm which hs the form of Figure.0, d represet the prxil r trvel with equtios tht follow the ptter show i Equtios.35 through.38. Usig exmples, we ow trce r through severl les sstems to illustrte the cocepts itroduced so fr. Exmple.3. Trcig r through covergig les. I Figure., we show equicovex les; tht is, covergig les with rdii tht re equl i mgitude. We wish to trce r from poit P through the les to other poit P (ot show) locted o ple orml to the smmetr xis distce of mm to the right of V. Iitil r-dt vlues (the δ d ) d the vlues of the other qutities tht refer to the les sstem re give i tbulr fshio i Figure.. The heders of the colums (the r,, d t) re the geeric smbols for rdii, idices of refrctio, δ 0.00 rd 5.73 deg 5.00 mm r (mm) t (mm) Figure d trsltios, respectivel. We uderstd the tble is shorthd for the followig (refer to Figures.0 d. for the meig of the smbols): P δ C V V C r mm r mm mm 0 mm 40 mm Figure. t t mm t t 0.00 mm t t mm

16 6 Chpter : Prxil ptics Refrctio & Trsltio Usig Equtio.38, we clculte the powers of the surfces: p mm r 40 p.5 (.39) 0.05 mm r 40 d the, strtig with the iitil r vector V P, we determie the other r vectors (the colum vectors) fter ech trsltio or refrctio to trce the r through the les to the poit P. First i smbol form, we hve δ V P V T P V P V R V V T V V R V V P T P V d the substitutig ( δ ) 0.00 V P 5.00 δ 0 δ 0.00 V t.00 ( V δ ) p δ δ 0 δ V t 0.50 ( V δ ) δ V P p δ t δ It is es to red off the vlues from the bove r vectors, but we eed to ppl other step to get the δ vlues (refer to Figure.0 for the meig of the δs): δ δ 0.00 δ δ rd rd δ δ rd.000 We summrie the vlues of the gles of iclitio δ d coordites for the r the r dt s it psses through the les i the tble of Figure.3 (the δ vlues re lso expressed i degrees for coveiece). We refer to tbles s figures to mke it esier to fid them, becuse tbles re usull few i umber mkig them difficult to locte b tble umber. The umbers d refer to the first d secod surfces, respectivel, of the les i Figure.4; this digrm displs scle drwig of the pth the r follows i trvelig from P to P (the horiotl d verticl scles re ot the sme). Becuse there is ot eough spce to show the δ gles i Figure.4, we must refer to Figure.0 for tht iformtio. P δ(rd) δ(deg) (mm) P P mm C Figure.3 V 0 mm Figure.4 V C 40 mm I the bove clcultios, we hve rouded the results to pproprite umber of sigifict figures; however, it is importt to remember tht 6 deciml digits (ctull, the bir equivlet) re retied for ech of the vlues i Mthemtic s computer memor, d ll clcultios re performed with these 6 deciml digits. This prctice occsioll produces smll discrepcies; for exmple, if we use clcultor to covert the prited vlues i Figure.3 for δ (or δ )of 0.69 rd to degrees, we obti rouded vlue of 9.68 deg rther th the prited 9.67 deg vlue obtied with Mthemtic fter roudig (usig π 3.46). I r trcig it is importt to do ll clcultios with firl high precisio becuse the m subtrctios tht tke plce c cuse loss of sigifict figures, d produce iccurte results ormll, the 6 deciml digits tht Mthemtic mitis is more th eough. If we do t eed the itermedite δ d r-dt vlues, but just wt to obti the δ d vlues t the fil poit P, P

17 .3 The Prxil Approximtio d Its Mtrix Represettio Prxil Mthemtics 7 the oe of the ver useful fetures of mtrices i prxil optics is tht we c clculte just oe mtrix tht coects the fil r-vector vlues with the iitil oes. Thus, we first clculte the ple-to-ple mtrix M P P give i Equtio.36 b substitutig the tble vlues i Figure. d the power vlues of Equtio.39 ito the squre mtrices: M P P T P R T R T P (.40) where we hve used Mthemtic to clculte the product of these five mtrices, d showed the results to Mthemtic s defult deciml displ of rouded six deciml digits. We c ow use Equtio.35b to clculte V P M P P V P (.4) 3.75 or more explicitl, V P δ d quickl determie the fil r dt δ rd mm (.4) i greemet with the results we obtied before for the r dt t the poit P (see the tble i Figure.3). Exmple.3. Trcig r bckwrd. e of the remrkble properties of the mtrix pproch is tht rs c be trced bckwrd esil usig the cocept of the mtrix iverse. For exmple, if we kow the fil r vector V P d the ple-to-ple mtrix iverse M P P,we c determie the iitil r vector V P. Usig the mtrix lgebr preseted i Sectio., we strt with V P M P P V P multipl both sides b the iverse M P P of the mtrix M P P to obti M P P V P (M P P M P P)V P IV P V P d the iterchge sides to get the desired result V P M P P V P (.43) All the determits of refrctio mtrices, trsltio mtrices, d their products, re equl to oe; therefore, the determit of M P P is uit. The, ccordig to Equtio.4, the iverse of M P P is es to clculte: simpl iterchge the digol elemets, d chge the sigs of the off-digol elemets. We obti V P M P P V P ( ) (.44) which grees with the r vector V P we used iitill i the previous exmple. Thus, i oe short mtrix multiplictio, we hve trced the r bck to where it beg. Agi we should remember tht ll the umbers displed i Equtio.44 re rouded, but tht i Mthemtic s iterl memor, ll the umbers re recorded to 6 deciml digit precisio it is these high precisio umbers tht re used i the mtrix multiplictio of Equtio.44. This iformtio is importt for it explis wh the use of the displed vlues for M P P d V P i the mtrix multiplictio, s with clcultor, do ot quite give the sme result for V P. We c lso trce the r bckwrd i steps, ver much like the w we trced it forwrd i the previous exmple. To trce the r bckwrd from the fil poit P to the iitil poit P, wehve V T ( P V P 0 40 V R V V ( T ( V V R V ( V P T ( P V ) ) ) ) ) Whe we compre these results with the oes we got i the previous exmple, we see tht we hve ideed trced the r bckwrd to where it beg.