PARABOLIC AND ELLIPTIC REFLECTORS

Transcription

1 PARABOLIC AND ELLIPTIC REFLECTORS It s well known tht refletng surfes of ether prol or ellpt shpe hve the nterestng propert tht nomng lght rs wth spefed orentton n reflet off of suh surfes to produe undle of refleted rs whh onverge t one pont termed the fous. We wsh here to revew the proedure for lotng suh fol ponts. We strt the nlss lookng refl t the s refleton lw from surfe. Sne prolods nd ellpsods n n most nstnes e tken s eng smmetr, t s suffent to look t just the D verson of refleton. Ths s done frst defnng urve =f s shown n the ompnng fgure- Net we onsder n nomng lght r prllel to the s. Its unt length vetor form equls V = -j. When t hts the refletng surfe refleted r wll e estlshed. Its unt length form s denoted Vr. A thrd vetor N, not of unt length, s formed tkng the grdent to the surfe t the mpngement pont. It equls N=grd[-f. Now from opts we know tht the refleton ngle equls the ngle of ndene. Ths mens the dot produts N/ N V nd N/ N V r oth equl to os. Applng vetor ddton we then hve- V r V V N N, where N =N/ N s the unt length vetor long N. Ths equton represents essentll the Lw of Refleton for n nomng lght r nd n thus e used for oth prol nd ellpt refletors.

2 PARABOLIC REFLECTORS: Here we hve refletor whose surfe ontns the prol =. Assume n nomng lght r whose unt length vetor reds V = -j. The Inwrd norml to ths prol s- N=grd-=-+j nd the unt length vetor prllel to N wll e N j Applng the Refleton Lw we fnd- V r j But we n lso wrte the full refleton vetor s- 0-+h- j=const. V r, where the vetor lso ontns the l pont [,=[0,h. So we hve- Const.=+ / nd h=/ Ths lst result mples tht ll refleted rs produed ll lght rs omng n prllel to the s wll fous t pont =0 nd =/. Ths s the fol pont s lso shown n the followng fgure-

3 Ths fousng plt hs reeved wde pplton n the desgn of flshlghts, uto hedlghts, oust lstenng deves, nd solr trough onentrtors. Note tht lght r omng n prllel to the s nd strkng the refletor t [.=[/,/ wll produe horzontl refleted r whh lso psses through the sme fol pont t [0,/. Lrge fol length prol mrrors form the ss for ll refletor telesopes. Suh telesopes, lke the Hule spe telesope, usull do not ple ther oservton pont t the fol pont ut rther use seondr refletng mrror to onentrte the onvergng lght rs t pont ehnd the prmr mrror. Some thrt ers go we were nvolved wth retng some eght foot dmeter prol mrrors under ontrt wth NASA. We onstruted them frst mnufturng fom omposte whh ppromtes prolod nd then rotted ths form on turn tle fter ddng smll mount of lqud epo to the prolod surfe. Spnnng the form t onstnt rottonl speed untl the epo hrdened led to perfetl formed prol surfe whh ws then oted wth refletve mlr. We thus were le to produe ner perfet long fol length prol mrrors of lrge dmeter t mnml ost. Although these mrrors where not of optl qult, the were perfetl suffent to onentrte prllel rs of sunlght omng from helostt to the order of 300 suns. Ths onentrton ws enough to set on fre wooden s held t fol pont some 30 ft n front of the mrror. We were usng the onentrted sunlght to power YterumYAG glss lser. ELLIPTIC REFLECTORS: Whle prol refletors onentrte prllel lght rs to sngle fol pont, ellpt mrrors hve two fol ponts s shown n the followng grph-

4 . The ellpsod s ut the ellpse-, whose nwrd unt length norml s gven j N There re two fol ponts F nd F loted t [,=[sqrt -,0. The eentrt of the ellpse s gven -, where represents the dstne from the ellpse orgn to ts fol ponts.

5 Consder now lght r omng from fol pont F t [-,0, httng the ellpse t [,, nd then sendng refleted r towrd the s. The nomng r hs the unt vetor representton 0 j V Usng the ove stted Lw of Refleton llows us to predt tht the unt length refleted vetor wll then hve the rther length form- j V r [ [ [ [ When lookng t the spel se of [,=[0,, ths produes- j j V r } { We n lso wrte the unt vetor gong from [0, to [*,0 s- * * j V r Comprng these two forms for Vr, we n onlude tht *=. Tht s, the refleted r t [0, hts the s t the seond fol pont F loted t [,0. Ths result ontnues to hold for other refleton ponts long the ellpse for rs omng from fol pont F. Another, smple to lulte, pth s one where the ndent lght r omes from F, mpnges t = nd = /, nd then hs refleted r V r =-j pssng dretl through F. Not onl does lght soure t F lws hts F fter refleton ut lso the trnst tme s the sme nd equl to dvded the propgton speed. Ths mens,

6 mong other thngs, tht n eploson ntted t F wll onentrte ts energ t the seond fous F. Ths s essentll the seret ehnd trggerng hdrogen om gnton of n tom om t F. It s lso the prnple ehnd the ver suessful nd non-nvsve pulverzng of kdne stones. Severl dedes go one of m students Mohmmd Nsr studed the fousng plt of semellptl refletors mmersed n wter. B gntng n eletr sprk t F nd flmng the event wth hgh speed mer, he oserved the rpd growth nd then de of vtton ule t F few hundred mroseonds fter the sprk ntton t F.. Whsperng glleres re found ll over the world. The re essentll rooms wth ellpsodl elngs nd rulr wlls. St Pul s Cthedrl n London nd the Sttur Hll n the US Cptol re emples of whsperng glleres. Two people spekng ner fous F n one of these rooms n e lerl herd someone stndng long dstne w ner F. There re other tpes of refletng surfes esdes prolods nd ellpsods whh n use lght or sound wves to oth onverge nd dverge. For emple, mn ers go we nvestgted new tpe of ell shped smmetr mrror n whh lndrl rdtng lght soure pled long the mrror s ws ple of produng hgh ntenst prllel lght em. Suh mrrors would e helpful for dentsts nd phsns emnng ptents. M 3, 05