Incident ray Reflected ray θ i. θ r

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Spencer Hall
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MODUL #3: BASIC OPTICS In hs module we wll learn how maer can ac on lgh, how we can use hs o manpulae lgh propagaon. We wll learn abou relecon, reracon, lenses, lens sysems and aberraons.

1 MODUL #3: BASIC OPTICS In hs module we wll learn how maer can ac on lgh, how we can use hs o manpulae lgh propagaon. We wll learn abou relecon, reracon, lenses, lens sysems and aberraons. RFLCTION AND RFRACTION: I a ray o lgh could be observed approachng and relecng o o a la mrror, hen he behavor o he lgh as relecs would ollow a predcable law known as he law o relecon. The dagram llusraes he law o relecon. Normal Incden ray Releced ray θ θ r In he dagram, he ray o lgh approachng he mrror s known as he ncden ray (see dagram). The ray o lgh leavng he mrror s known as he releced ray. A he pon o ncdence where he ray srkes he mrror, a lne can be drawn perpendcular o he surace o he mrror; hs lne s known as a normal lne. The angle beween he ncden ray and he normal s known as he angle o ncdence, θ. The angle beween he releced ray and he normal s known as he angle o relecon, θ r. The law o relecon saes ha when a ray o lgh relecs o a surace, he angle o ncdence s equal o he angle o relecon, θ θ r. Relecon o o smooh suraces leads o a ype o relecon known as specular relecon. Relecon o o rough suraces such as clohng, paper, and he asphal roadway leads o a ype o relecon known as duse III -

In oher meda (glass, or example) he velocy s less.

2 relecon. The dagram below depcs wo beams o lgh ncden upon a rough and a smooh surace. The velocy o lgh, c, n a vacuum s abou 3x0 8 meers per second. In oher meda (glass, or example) he velocy s less. The rao o c o he acual velocy s called he reracve ndex, n: n c v snce c and v εo µ o εµ, n εµ εo µ o k m k e k e ε elecrc permvy µ magnec permeably k e delecrc consan (ε/ε o ) k m relave permeably (µ/µ o ) III -

3 The color o he lgh and s requency are he same n boh meda. Thereore, he wavelengh mus shoren by he same rao as he velocy. You can hnk o hs slowng down o lgh n a ransparen medum you pcure he medum composed o ndvdual aoms or molecules ha can nerac wh he passng lgh by absorbng and reemng he lgh. Ths absorbed and re-emed lgh s added o he componen passng hrough a c n such a way ha he sum s connually slowed down wh respec o c. Ths connuous slowng down s equvalen o a phase velocy less han c. You can hnk o lke hs: The elecrons n he glass are drven o oscllae by he lgh s -eld. Ths causes he elecrons o become dpoles hemselves and hey begn o re-radae or scaer. However, only he waveles n he orward drecon are IN PHAS and nerere consrucvely. The ohers nerere desrucvely and cancel ou. ar glass 3 BASIC LAWS Three undamenal laws descrbe how a waveron o lgh neracs wh a surace ha orms he boundary beween maerals wh deren reracve ndces e.g. ar-glass nerace where ar has a reracve ndex o and glass s ypcally.5. ) Incden, releced, and ransmed waves le all n he same plane ) Angle o ncdence s equal o he angle o relecon θ θ r III - 3

4 3) SNLL S LAW: n sn(θ ) n sn(θ ) Where n s ndex o reracon o he medum and n s ndex o reracon o medum θ θ r n n Medum Medum θ Snell s Law allows us o calculae he new drecon o propagaon when lgh passes hrough an nerace beween wo maerals wh deren ndces o reracon. The angles are measured beween he normal o he surace and he lgh beam. Lgh passng rom a maeral wh a hgh ndex o reracon o a maeral wh a low ndex o reracon bends away rom he normal whereas lgh passng rom maeral wh a low o maeral wh a hgh ndex o reracon bends oward he normal. FRSNL S QUATIONS: Whle Snell's law and he law o relecon ell us somehng abou he drecon n whch releced and reraced lgh propagae, does no say anyhng abou how much lgh goes where. When lgh srkes he nerace beween wo maerals wh deren ndces o reracon, a racon o he lgh s releced (R) and a racon s ransmed (T). The values o R and T may be calculaed usng Fresnel s equaons. I s mporan o realze ha ) he sum o releced and ransmed lgh mus equal he oal ncden lgh (snce hese are racons R + T ); and ) he angle o polarzaon o he ncden M wave wh respec o he plane o he ncden maeral has III - 4

5 an eec on he respecve racons o lgh ha are ransmed or releced. a) -Feld perpendcular o he plane o ncdence B B k n k θ θ r Medum n Medum θ B k r r n n + n n sn ( θ sn ( θ + θ ) θ ) r amplude relecon coecen, rao o releced o ncden elecrc eld ampludes. n n cos θ + n cos θ sn θ cos θ sn ( θ + θ ) amplude ransmsson coecen, rao o ransmed o ncden elecrc eld ampludes. III - 5

6 b) -eld parallel o he plane o ncdence k B k θ θ r B n Medum n Medum θ B k r // // r // n n n + n an( θ an( θ θ ) + θ ) // // // n n + n sn θ sn ( θ + θ ) cos( θ θ ) Gven ha, R Relecance (W/m ) T Transmance (W/m ) When here s no absorpon, R + T, and R R T T // // r r // n n n n // III - 6

7 I θ 0, he ncden plane becomes undened and n n R R// R n + n T T// T 4n ( n + n ) n xamples:. Wha percen o lgh s releced a he nerace o ar (n.0) o glass (n.5) he angle o ncdence s 0? Answer: R n n R R // R n n R // R % Ths means ha n a lens, whch has ar-glass neraces, ransmsson hrough each nerace 96%. Ths means ha he ransmsson hrough he lens even absoluely non-absorbng s (0.96) In oher words 7.84% o he lgh s los due o relecon. Noe ha hs propery s mulplcave.. Wha percen o lgh s ransmed rom ar (n.0) o glass (n.4) he angle o ncdence s 48. Assume ha he lgh s unpolarzed. III - 7

8 Answer: The eases way o approach hs s o calculae he racon o lgh ha s releced. Assumng no absorpon, he remander s ransmed no he second medum. Frs sar wh calculang he angle o reraced lgh usng Snell's law. Gven ha θ 48 o n θ sn θ n sn sn θ.4 sn (48) 3.06 deg rees We can hus calculae he relecon coecens or parallel and perperdcularly polarzed lgh usng Fresnel equaons. Subsung hs no: R // r // and R r Nex we have o realze he lgh s unpolarzed. Praccally we can handle hs n erms o Fresnel's equaons by assumng ha here s equal quanes o parallel and perpendcularly polarzed lgh and he smply ake he average: ( r + ) R + R // R, // r // an ( ) sn ( ) R + an ( ) sn ( ) Thus R 4.0% wh T - R ollows ha: T III - 8

9 AN APPLICATION OF FRSNL QUATIONS: The Fresnel equaons descrbe he eecs o an ncomng elecromagnec plane wave on he nerace beween wo meda wh deren delecrc consans or ndces o reracon. From he deren Fresnel equaons we oban, sn ( θ R r sn ( θ + an ( θ R r // // an ( θ + θ ) θ ) θ ) θ ) Here, noe ha whle R can never be zero, R // s zero when (θ + θ ) 90. As a resul, or -eld parallel o he plane o ncdence, he relecance vanshes and he beam s compleed ransmed. Ths s Brewser's Law (see gure). III - 9

10 Anoher way o look a hs s ha or parallel polarzed lgh, here s an angle o ncdence where he relecvy 0. Ths angle, known as he Brewser's angle can be calculaed by: LNSS AND LNS SYSTMS: n θb arcan n A lens s ypcally made up o an opcally ranslucen maeral conanng wo or more reracng suraces, a leas one o whch s curved. Lenses may be used n an opcal sysem o mody a beam o lgh or o orm an mage o an objec. There are a number o acors ha need o be consdered when characerzng a lens: Dameer The dameer o a lens s ypcally chosen based on he sze o he beam and objec ha needs o be moded. Radus o Curvaure R deermnes how curved he lens s and he drecon o he curvaure. I also relaes o he ocal lengh o he lens (see lens equaons secon). Focal lengh Focal pon s dened as he pon a whch parallel rays comng no he lens converge. The dsance beween he cener o he lens a hs pon s he ocal lengh o he lens. Ths pon may be on he oppose sde o he lens as n a convex lens or he same sde as n a concave lens. Transmsson range Any gven maeral wll allow lgh o ceran wavelenghs o be ransmed whle allowng ohers o be absorbed. The lens maeral s chosen based on he wavelengh o he lgh ha s beng moded. e.g. glass ransms well rom 400 o 500 nm, however quarz needs o be used o ransm lgh n he UV whle more exoc maerals need o be used or ransmsson urher n he IR (or example: sapphre, CaF, ec.). Aberraons Aberraons are lmaons n lens behavor ha can be dermenal o s perormance. These nclude sphercal aberraons, chromac aberraons, coma, and asgmasm. III - 0

11 There are sx knds o lenses, dvded n wo man caegores; a) he posve or convex lenses and b) he negave or concave lenses. The convex lenses have n common ha hey are hcker n he cener han a he edge whle he concave lenses are hnner n he cener han a he edges: Bconvex Plano-convex Menscus convex R > 0 R R > 0 R < 0 R < 0 R > 0 _ Bconcave Plano-concave Menscus concave R < 0 R R > 0 R > 0 R > 0 R > 0 where, R Radus o Curvaure o he rs lens surace rom he le) R Radus o Curvaure o he second lens surace (rom he le) III -

12 R R Snce all rays ssung rom a source pon wll arrve a he mage pon, any wo rays wll x ha pon. There are hree rays ha are eases o apply. Two o hese make use o he ac ha a ray passng hrough he ocal pon wll emerge rom he lens parallel o he opcal axs and vce versa; he hrd s he undevaed ray hrough he cener o he lens. They are llusraed below or boh posve and negave lenses. Ray Ray Ray Ray Ray 3 Ray 3 III -

13 BASIC LNS QUATIONS: ) The ocal lengh o a lens can be calculaed by he Gaussan Lens Formula: o + where, ocal lengh, o objec dsance, mage dsance ) Anoher useul lens equaon s he Lensmaker's Formula; (n n ) R R where n l ndex o reracon o he lens n ndex o reracon o he surroundng medum ypcally ar) R Radus o Curvaure o he rs lens surace rom he le) R Radus o Curvaure o he second lens surace (rom he le) 3) Transverse Magncaon (M T ) s dened as he magncaon o he mage n he drecon perpendcular o he drecon o propagaon and s gven as: M T o 4) Longudnal Magncaon (M L ) s dened as he magncaon o he mage n he drecon o propagaon and s gven as M L M T 5) The ransverse magncaon o a wo-lens sysem ha s separaed by a dsance d ha s greaer han he sum o her ocal lenghs s gven by: III - 3

14 MT d (o ) o The magncaon n such a wo-lens sysem s smply he produc o he magncaons rom each elemen: M T MT sysem * MT Based on he locaon o he objec relave o he ocal pon he locaon, sze and ype o mage wll vary or a posve or negave lens. CONVX LNS OBJCT IMAG LOCATION TYP LOCATION ORINTATION SIZ > o > Real < < Invered Mned o Real I Invered Same Sze < o < Real > > Invered Magned o ± o < Vrual > o rec Magned CONCAV LNS OBJCT IMAG LOCATION TYP LOCATION ORINTATION SIZ Anywhere Vrual < o > rec Mned Curved Mrrors behave smlar o lenses excep ha he ormaon o he mage s reversed,.e. he concave mrror behaves lke a convex lens and a convex mrror behaves lke a concave lens. CONCAV MIRROR OBJCT IMAG LOCATION TYP LOCATION ORINTATION SIZ > o > Real < < Invered Mned o Real Invered Same Sze < o < Real > > Invered Magned o ± o < Vrual > o rec Magned III - 4

15 CONVX MIRROR OBJCT IMAG LOCATION TYP LOCATION ORINTATION SIZ Anywhere Vrual < o > rec Mned XAMPLS:. Consruc he rays o orm he mage or a posve lens gven ha he ocal lengh o he lens s m and an objec (.5 m hgh) s placed a a dsance o 3.5 m rom he lens. Answer: Gven, m o 3.5 m h(objec).5 m (Hn: To consruc he mage, draw he hree rays descrbed earler) o 3.5 m.5 m m Gven ha, III - 5

16 Thus 4.67 m + o The magncaon s gven as, 4.67 M T.33 o 3.5 Thereore, an objec.5 m hgh wll be magned.33 mes o yeld an mage.995 m hgh. Ths mage s real, nvered and magned.. Consruc he rays o orm he mage or a lens wh ocal lengh -0 cm and an objec ha s placed a a dsance o 0 cm rom he lens. Wha knd o mage do you ge? Answer: Gven, o 0 cm -0 cm, hereore s a concave (negave) lens. Draw rays o consruc he mage. o 0 cm 0 cm III - 6

17 + o Thereore - 5 cm The magncaon s gven as, ( 5) M T 0.5 o 0 Thus he mage s a vrual, erec, mned mage locaed a /. III - 7

18 ABRRATIONS: The ormulas developed earler or mage ormaon by sphercal relecng and reracng suraces are, o course, only approxmaely correc. In dervng hose ormulas was necessary o assume paraxal rays, rays boh near o he opcal axs and makng small angles wh. However, n consderng hese lens suaons wll arse when hese assumpons are no longer vald and aberraons are observed.. Sphercal Aberraons Sphercal aberraons occur due o he severe curvaure o shor ocal lengh or smaller lenses because rays ncden on he ouer regons o a lens bend more han he rays owards he cener, causng he mage o appear ou o ocus. Sphercal aberraons are correced by: usng a larger lens orenng he lens correcly usng he rgh ype o lens. Coma Coma s an o-axs aberraon ha s nonsymmercal abou he opcal axs. Ths arses rom he dependence o ransverse magncaon on he ray hegh a he lens. Because o coma, an o-axs objec pon s maged as a blurred shape ha resembles a come wh a head and a al. Ths ype o mage can be mnmzed by approprae selecon o he dameer o he lens o be used. III - 8

19 3. Asgmasm When an objec pon les ar away rom he opcal axs, he ncden cone o rays wll srke he lens asymmercally gvng rse o asgmasm. I he rays ncden on he lens n he plane o he paper (angenal plane) has a gven ocal lengh, hen he rays n he plane ha s oblquely angled wh respec o he paper (sagal plane) has a deren ocal lengh. Thus, or he ncden concal bundle o rays, he cross-secon o beam as leaves he lens s nally crcular and gradually becomes ellpcal unl mees n a lne a he ocal pon ha s angenal o he plane o he paper. 4. Feld o Curvaure In hs ype o aberraon, a gven planar objec s maged on a parabolc surace nsead o a plane as can be seen n he gure. III - 9

20 5. Dsoron Dsoron shows up as a varaon n he ransverse magncaon or pons o he objec away rom he opcal axs. In oher words, dsoron occurs because deren areas o he lens have deren ocal lenghs and deren ransverse magncaons. Objec Pncushon Dsoron Barrel Dsoron 6. Chromac Aberraon Chromac aberraon occurs or ncden rays ha conan many wavelenghs. Snce he ndex o reracon vares wh wavelengh, he ocal properes o a smple lens wll vary as well. The reracve ndex s hgher or blue lgh han red lgh. Thereore, he ocal lengh o a convex lens s shorer or blue lgh han red lgh. III - 0

21 Chromac aberraon can be correced or usng an achromac double. An achromac double consss o a convex and concave lens made o deren maerals cemened ogeher. By choosng maerals wh approprae reracve ndces, you can creae a double ha wll have he same ocal lengh a wo wavelenghs. The wo lenses correc or each oher and a ocal pon s ound somewhere n he mddle. RFRNCS:. Hech, ugene. (987) Opcs. Addson-Wesley Publshng Company, Readng, MA.. Pedro F and Pedro L. (993) Inroducon o Opcs, Prence-Hall, Upper Saddle Rver, NJ. III -

II. Light is a Ray (Geometrical Optics)

II. Light is a Ray (Geometrical Optics) II Lgh s a Ray (Geomercal Opcs) IIB Reflecon and Refracon Hero s Prncple of Leas Dsance Law of Reflecon Hero of Aleandra, who lved n he 2 nd cenury BC, posulaed he followng prncple: Prncple of Leas Dsance: