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: When lgh ravels beween 2 pons, akes he shores pah As a smple eample, lgh ha ravels drecly from pon α o pon follows he ray pah of a sragh lne raher han some curved lne: α Bu wha ray pah does lgh follow when ravels from pon α o pon and bounces once off of a nearby mrror? Consder he 3 possble pahs shown n he dagram below: α γ δ ε mrror α Usng smlar rangles, we can see ha α δ α δ ; α γ α γ ; and α ε α ε Furhermore, snce α δ s a sragh lne beween α and, hen mus be rue ha α δ < α γ and α δ < α ε Therefore, we conclude ha α δ < α γ and α δ < α ε In oher words, he shores pah from α o ha ncludes one bounce off of he mrror s he pah hrough he pon δ ha s mdway beween α and When we alk abou rays ncden on some nerface s convenen o hnk n erms of he ray angles The ray angle s he angle beween he ray self and a lne drawn normal (perpendcular) o he plane of he nerface To see wha he above resul mples abou he angle of ncdence and angle of reflecon, θ and θ r, respecvely, consder he followng dagram: IIB-
α θ θ r ο δ α θ oher Snce rangle αοδ and rangle α οδ are congruen rangles, hen θ oher θ Snce he lne α and he nerface normal are nersecng lnes, hen θ oher θ r Therefore, we see ha θ r θ Though smple, hs mporan resul s gven a name: Law of Reflecon: When a lgh ray s refleced off of an nerface he angle of reflecon s equal o he angle of ncdence Mahemacally, he Law of Reflecon saes θ r θ Unforunaely, Hero s Prncple of Leas Dsance s no compleely correc Alhough s able o eplan propagaon of lgh rays n a homogeneous medum and he reflecon of lgh rays off of an nerface, we wll see ha s no able o eplan he refracon of lgh rays ravelng across an nerface beween wo dfferen meda 2 Reflecon off Imperfec Plane Mrrors In he real world, he descrpon of lgh refleced off of a mrror or nerface s more complcaed han we have jus assumed for 2 reasons: () he naure of he reflecon s no as smple as we jus descrbed snce no mrror s perfecly fla; () no all of he lgh s refleced some lgh s ransmed hrough he mrror and some lgh s absorbed n he maeral of he mrror self Consder () frs The Law of Reflecon always holds, bu snce praccal mrrors always have some amoun of surface roughness, no all of he lgh s refleced a eacly he same angle because he ncden angle s dfferen a dfferen pons on he surface The wo ereme cases are called perfec specular reflecon, n whch he lgh s all refleced a he same angle of reflecon, and perfec dffuse reflecon, n whch he lgh s refleced randomly no all dfferen drecons specular reflecon (smooh surface) dffuse reflecon (rough surface) IIB-2
In pracce we ge a combnaon of boh ypes of reflecon The lgh ha appears o obey he Law of Refracon when he angle of ncdence s defned relave o he average surface (gnorng he surface roughness) s sad o be specularly refleced, whle he lgh ha s refleced no random drecons s sad o be dffusely refleced For a farly smooh mrror, here wll be a dsrbuon of angles of reflecon cenered around he nomnal angle of ncdence amoun of lgh θ θ r Ne consder () For ceran ypes of mrrors, called fully reflecng mrrors, all of he lgh s eher refleced or absorbed n he maeral of he mrror no lgh s ransmed hrough he mrror A good eample of hs ype of mrror s a hck pece of meal The more reflecon and he less absorpon, he beer he mrror s Many mrrors are parally reflecng mrrors Two man ypes of parally reflecng mrrors are parally slvered mrrors, whch conss of a very hn meal flm suppored by a ransparen (usually glass) subsrae, and delecrc mrrors, whch are smply plane nerfaces beween wo maerals of farly dfferen refracve ndees parally slvered mrror delecrc mrror ncden refleced ncden refleced n glass n very hn meal flm ransmed ransmed The amoun of reflecon versus ransmsson s deermned by he hckness of he meal flm n he frs ype, and by he dfference beween refracve ndees n he second ype Mrrors ha are nenonally made o be parally reflecng are useful for splng a beam of lgh no wo drecons, or as beam-splers, and for one-way mrrors (lke n an nerrogaon room), for eample 3 Ferma s Prncple of Leas Tme Law of Refracon (Snell s Law) Perre de Ferma, a French jurs and mahemacan from he 7 h cenury, was of he opnon ha naure s economcal As a resul of hs phlosophy, he posulaed n 657 ha: Ferma s Prncple of Leas Tme : When lgh ravels beween 2 pons, akes he pah ha s raversed n he leas me Noe ha we could have used Ferma s Prncple jus as well as Hero s Prncple o derve he Law of Reflecon, snce he pahs of leas dsance and leas me are equal n a homogeneous medum (n whch he velocy of lgh s a consan) Usng Ferma s Prncple, we can now derve he Law of Refracon as well Consder a ray of lgh ravelng across an nerface ha separaes wo dfferen meda havng wo dfferen ndees of refracon, such ha he velocy of lgh s dfferen n he wo regons In IIB-3
ravelng from pon α o pon, he ray crosses he nerface a some pon ο We wsh o deermne he poson of he pon ο, and from ha poson he relaonshp beween he angle of ncdence θ and he angle of ransmsson θ α a θ ο n b θ n ( n > n ) c c Now we know lgh ravels a speed v n medum and a speed v n medum, so he oal me akes for lgh o ge from pon α o pon (usng me dsance / rae) s αο ο + v v or, n erms of dsances, a + v + 2 2 2 2 b + c v Thus he me s a funcon of he poson (whch ells us where he ray crosses he nerface) To fnd he mnmum value of he funcon (), we fnd he slope d d and se equal o zero: v a 2 2 v b c + so, d d 2 2 2 + ( + ( ) ) 2 2 a + 2 v 2 2 2 2 2 + ( + ( ) ) v 2 ( ) b c 2 2 c or, d d v a + c v b + c 2 2 2 2 se 0 so, v a + c v b + c 2 2 2 2 wll gve he value ha corresponds o mnmum Snce n pracce s easer o measure angles han posons, le s see wha hs resul ells us abou he relaonshp beween he ray angles Frs, we recognze ha c snθ and snθ ; a 2 + 2 b 2 + ( c ) 2 so he correc ray pah, accordng o Ferma s Prncple, s characerzed by he relaonshp IIB-4
snθ snθ v v Bu we know ha v c c and v, n n and hus he relaonshp beween he ray angles, called Snell s Law, s: Law of Refracon (Snell s Law): When a ray s ransmed across an nerface beween wo meda wh dfferen refracve ndees, he produc of he refracve nde and he sne of he ray angle s equal on boh sdes of an nerface Mahemacally, he Law of Refracon (Snell s Law) saes n snθ n snθ Noce ha when he nde n he ncden medum s less han he nde n he ransmed medum, he ray s ben oward he normal o he nerface plane, whereas when he nde n he ncden medum s greaer, he ray s ben away from he normal Ths resul s llusraed below θ n (ar) θ n (waer) n (waer) n (ar) θ θ n < n θ > θ n > n θ < θ How do we know ha he lgh behaves hs way? Consder he frs case n whch n < n : n Snell s Law mples snθ snθ, and we know n > ; herefore snθ > snθ n n Bu we know ha snθ s a monooncally ncreasng funcon of θ for 0 θ π / 2 Thus mus be rue ha θ > θ By he way, once you have convnced yourself ha one of he wo dagrams above s correc, you can show ha he oher s correc usng he Prncple of Reversbly 4 Prncple of Ray Reversbly Noce ha n dervng he relaonshp beween he angle of ncdence and he angle of ransmsson for a ray refraced a an nerface beween wo meda of dfferen refracve ndees, we could jus as well have nerchanged he roles of pons α and so ha he lgh ravels from pon o pon α along he eac same ray pah Ths resul s acually que general, so we gve a name: IIB-5
Prncple of Ray Reversbly: Any acual ray of lgh n an opcal sysem, f reversed n drecon, wll rerace he same pah backward Graphcally, he Reversbly Prncple can be represened as follows: 5 Toal Inernal Reflecon (TIR) and he Crcal Angle Consder he refracon (ransmsson) of a ray hrough an nerface beween wo meda of dfferen refracve ndees when he nde n he ncden medum s greaer han he nde n he ransmed medum (n > n ), and as he ncden angle θ s ncreased We see ha a some value of he angle of ncdence, called he crcal angle θ c, he ransmed angle becomes equal o π/2 (or 90 ) n θ θ θ π/2 90 n ( n > n ) θ c θ > θ c Accordng o Snell s Law, n snθ n snθ n sn( π 2 ) n c Therefore, we can wre he epresson for he crcal angle as follows: Crcal Angle: he angle of ncdence ha jus makes he angle of ransmsson equal o π/2, or n θ c arcsn for n > n n For ncden angles ha eceed he crcal angle, θ θ c, he lgh can no be ransmed, so all of he lgh s refleced Ths suaon s called Toal Inernal Reflecon, or TIR for shor Noce ha TIR assumes only specular reflecon occurs n pracce even mnue amouns of dffuse reflecon preven one from observng 00% reflecon IIB-6