DERIVATION OF THE BASIC LAWS OF GEOMETRIC OPTICS

Transcription

1 DERIVATION OF THE BASIC LAWS OF GEOMETRIC OPTICS It s well kow that a lght ay eflectg off of a suface has ts agle of eflecto equal to ts agle of cdece ad that f ths ay passes fom oe medum to aothe that the teface codto s that the poduct of the dex of efacto tmes the se to the teface omal emas costat both meda. Oe fds umeous dscussos o UTube coceg the devato of these laws. Ufotuately these ae mostly peseted a dffcult to udestad heavly acceted mae ad ofte much too log fo what s beg coveyed. We emedy ths stuato hee by e-devg the basc laws of geometcal optcs statg wth the Femat Pcple that a lght ays wll always choose a path whch mmzes(actually extemzes the tast tme gog betwee two pots. We beg ou devatos ad dscussos wth the followg smple dagam- It dcates a comg lght ay emaatg at pot P[x,], httg a flat suface at R[x,0], ad the beg patally eflected towad pot Q[x,] ad patally efacted gog towad pot S[x,-] the secod medum. The thee agles assocated wth these ays elatve to the suface omal N ae gve as dcated. LAW OF REFLECTION: Hee we stat at pot P[x,y], ht the teface at pot R[x,0], ad the sed ts eflected poto to Q[x ]. The tme t takes to go fom pot P to Q alog ths two pat path s just-

2 T= ( ( x x x x c Hee c=3x0 0 cm/sec s the speed of lght vacuum ad the dex of efacto the fst medum. Typcal values fo ae fo vacuum,.0009 fo a,.33 fo wate,.5 fo cowt glass, ad.4 fo damod. Takg the devatve of T wth espect to x ad settg the esult to zeo poduces the Femat codto - ( x x ( x x ( x ( x x x Ths s equvalet to sayg s( =s (, whee the agles ae measued elatve to the suface omal N=j at the eflecto pot R[x,0]. Thus we have the Law of Reflecto- Whe ths esult s used fo ay-tacg though complcated optcal systems such as ceta sola mo aays, oe usually pefes the vecto fom of ths esult. Ths vecto fom ca be deved as follows. Stat wth the vecto epesetatos of the comg ad eflected ay- V [ a bj]/ a b ad V [ a bj]/ a b Next take the dffeece to get- V V bj / a b But N=j ad b/sqt(a +b =cos(=-(nv. Hece the esultat vecto fom of the eflecto law becomes- V V ( V N N A athe volved applcato of ths fomula ca be foud oe of the techcal atcles I publshed back 980 Sola Eegy 5, pp-3. CORNER REFLECTORS: Aothe mpotat applcato of the efecto law apples to coe eflectos. I ts smplest fom a coe eflecto cossts of two pefectly eflectg mos mouted at ght agles to each othe as show-

3 A comg ay hts the eflectg suface wth a cdet agle elatve to thye omal N. O eflecto =. A secod eflecto at omal N has cdet ad eflected agles equal to (/-. Hece addg up all fou agles we get +(- =. Thus vectos V ad Vout ae 80 deg out of phase wth each othe meag what goes s eflected back the same decto egadless of the coe eflecto oetato. Bllad playes wll be vey famla wth ths effect. Addg a thd eflecto at ght agles to the fst two does ot chage the esults. It s ths popety of coe eflectos whch makes them useful measug dstaces usg lase pulses. Ideed, dug oe of the ealest tps to the moo, astoauts left a 3D coe eflecto lyg o the moo suface allowg obseves fom eath to make vey accuate detematos of the eath-moo dstace. LAW OF REFRACTION: To geeate the Law of Refacto also kow as Sell s Law, we cosde pots P[x,],R[x,] ad S[xt,] the above dagam. Ths tme the tme fo the lght ay to go fom P to S s gve by- T ( x x ( x x c c O dffeetatg oce wth espect to x ad settg the esult to zeo, poduces the tme mmum codto-

4 ( x x ( x x ( x t ( x x x t Fom the dagam we ecogze the quotets ths expesso to be just the se of ad t. Hece we have the Law of Refacto, also kow as Sell s law, - s( s( t Ths mpotat law foms the bass fo most les calculatos employg ay tacg. Notce that whe < a codto may exst whee a lght ay comg fom S wll be etely teally eflected at the teface. Ths codto s kow a total teal eflecto ad plays a majo ole lossless lght pulse tasmsso though optcal fbes. If you ae lyg at the bottom of a swmmg pool ad lookg up you wll oly be able to see the a sde wth a ccle fomg a coe wth you eyes at the vetex. Ths coe has the half-agle - half coe acs( 48.75deg.33 SIMPLE EXAMPLE OF RAY TRACING: To demostate how the Refacto Law s used ay tacg, we cosde the followg optcal set-up-

5 ad ty to fd the deflecto agle t betwee the extg ay V t ad the secod omal N. We have hee a glass wedge of vetex agle w =acta(/ ad dex of efacto =.5. A lght ay V =-j comes paallel to the y axs ad hts the top of the wedge at pot [,0.5].The suface omal at the top of the wedge s detemed by a gadet opeato ad just equals- N ( j / 5, whe adjusted to have ut legth. The agle of cdece wth espect to ths omal s just- accos( j accos( / 5 N By Sell s Law ths yelds the efacto agle t of- acs( 3 5 t deg deg The efacted ay ow moves a staght le alog the t decto utl t hts the bottom suface. Thee t s aga efacted extg the bottom of the wedge at agle t elatve to the bottom omal N =-j. A lttle geomety volvg the oblque tagles exstg wth the wedge poduces a cdet agle at the lowe suface of - w t Fom ths last esult t follows that the deflecto agle s- t 3 acs s[acta(/ acs( / 3 5] =3.904 deg Note that ths ay wll evetually coss the y axs. It gves a dcato of how leses wok to poduce a focal pot. REFLECTION COEFFICIENT AT AN INTERFACE: A emag fomula eeded geometcal optcs calculatos s oe elatg the eflected ay testy compaed to the cdet testy. Fo ths calculato t becomes ecessay to cosde lght as a tasvese wave wth te depedet electc ad magetc felds at ght agles to the popagato decto. The fst calculato usg such a model was caed out by the Fech physcst ad egee August-Jea Fesel ( Although bette kow to the publc fo hs dscovey of Fesel leses fo

6 lght houses, hs fou equatos goveg lght tasmsso ad eflecto a teface,motvated by eale wok of Youg ad pedatg Maxwell by some foty yeas, was tuly a emakable achevemet. Let us show how to deve oe of these equatos based upo a polazed lght wave whee ts electc feld E les pepedcula to the page the above dagam. Ude these codtos the magetc feld B les the plae of the pape. The bouday codtos at the teface ae that- E +E =E t ad B cos( -B cos( =B t cos( t Also we have the electomaget waves elato- B=E/c applcable fo cdet, eflected ad tasmtted waves. Elmatg the Bs ad E t fom these equatos yelds- { E cos( E cos( } {[ E E Solvg fo the ato E /E we get- ]} E E cos( cos( t cos( cos( t Oe ca smplfy ths esult futhe by usg the eflecto law = ad the Sell s Refacto Law that t acs{ s( } Now we kow that the ato of eflected lght testy to the cdet testy ca be cosdeed as the Reflecto Coeffcet R ad s epeseted by the squae of E/E. A case of specal teest occus whe the cdet lght hts the teface alog the omal decto. I that case = = t ad the eflecto coeffcet educes to the smple fom- ( R ( Fo a a-glass teface we fd appoxmately R=(0.5/.5 =0.04. That s oly about 4% of the comg cdet othogoal ay s eflected. The est s tasmtted. That s the easo that oe looses vey lttle testy of the comg lght whe lookg though a pae of glass.

7 Febuay 6, 07