MASSACHUSETTS INSTITUTE of TECHNOLOGY Department of Mechanical Engineering 2.71/ OPTICS - - Spring Term, 2014
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1 .7/.70 Optic, Spri 04, Solutio for Quiz MASSACHUSETTS INSTITUTE of TECHNOLOGY Deprtmet of Mechicl Eieeri.7/.70 OPTICS - - Spri Term, 04 Solutio for Quiz Iued Wed. 03//04 Problem. The ive opticl ytem i mde of 3 lee d perture. The problem tte tht the compoite ytem i telecope: the ry tht eter prllel hould emere prllel. Sice prxil pproximtio i vlid (x<<z), both the ABCD mtrix method d the tep-by-tep ue of the le lw will yield the me correct olutio. () Method : ABCD Mtrix The ABCD mtrix of the ytem, trti from L d edi t L3, c be cotructed the followi. M = M M M M M ytem L3 d L d L 0 d 0 d 0 f f f d d d d d d d d d f f f f f d d d d d d d d d d d f f f3 f f f f3 f f3 f f f 3 f f 3 f f3 A perture doe ot che the ABCD mtrix, ice both the le d the lterl heiht of the ry i uffected by perture. Recll tht the ABCD mtrix relte the iput d output ry by: xout A B xi. C D out i I order for prllel ry to emere prllel, the output le hould ot be ffected by the iput heiht x. Therefore, C hould be zero. Solvi for C=0 ive: 0 dd d f d f ( d d ) f f f f f f f d d f f3 d f3 d f f 8 cm. d d f f 3 (.) (.)
2 .7/.70 Optic, Spri 04, Solutio for Quiz Method. Step-by-tep le lw For liht propti i the poitive directio, the object ditce o i defied to be poitive for object plce o the left ide of the le. The ime ditce i i defied to be poitive for ime pperi o the riht ide of the le. A prllel ry c be expreed object plced t o. Writi out the le lw for L, we ee tht, ( o, f 4 cm ) i f 4 cm. (.3) f o i After pi throuh L, prllel ry will focu t the focl poit, i f. Next, the ime from L become the object for L. The object ditce c be clculted from o d i, with repect to L. Ui the le lw for L, we obti:, ( o cm), (.4) f o i where both f d i re ukow. The ime from L become the object for L3. With repect to L3, the object ditce c be clculted from o3 d i. Sice the problem tte tht the output bem from prllel iput bem hould lo be prllel, we kow tht i 3., ( i 3, f3 3 cm ) o 3 f3 3 cm. (.5) f o3 i3 3 Plui i thi reult bck to (.4), ui o3 d i, we obti.6cm d: f 8 cm. cm.6cm f i (.6) Ideed, the me wer with (.) i obtied. (b) The ulr mifictio power of the telecope ytem i /. Ui ABCD method, we c clculte: d d d d d out i D f f out /.06 f3 f 3 Altertively, tep-by-tep le lw ive the me wer:. (.7) i i i (.8) 3 o o o3 The ulr mifictio power i i
3 .7/.70 Optic, Spri 04, Solutio for Quiz (c) Ui the ABCD mtrix pproch, we c cotruct the overll mtrix icludi the proptio from the object to L, d the proptio from L3 to the ime. M M M M A B A B 0 C D0 C D totl ytem I order for the ytem to form ime, B 0 hould be tified, which ive: B A B D C 0 A B D.06 (.9) (.0) Therefore the ime form 4.406cm to the riht of L3. Altertively, the tep-by-tep le lw yield: We et the me wer with (.0)., 4 4, i (.) f i i d (.) o i,, i 8 f 8 o i i o3 i 9.4 (.3) d (.4), (.5) f o3 3 (d) The etrce (exit) pupil i the ime of the perture top throuh ll opticl elemet before (fter) the perture top. The EP i clculted by ui the le lw for L i the bckwrd directio:,, EP 36cm. (.6) f ASL EP EP Therefore the EP i locted 36cm to the left of L. Note tht whe the object i imed i the bckwrd directio i thi ce, imply ubtituti etive vlue for o doe t work : (icorrect!). i 3
4 .7/.70 Optic, Spri 04, Solutio for Quiz For the ExP, we c write the coecutive le lw for L d L3 i the forwrd directio:, , f L AS (.7),, ExP 3.7 d f ExP 3 ExP (.8) Therefore the ExP i locted 3.7cm to the left of L3. The object i locted t the focl poit of L. Therefore, the mril ry tht leve the xil poit o the object ple become prllel to the xi fter pi throuh L. Therefore, the NA c be clculted by The dimeter of the AS i.005cm. D / t 0.5, D. 005cm. (.9) 4
5 .7/.70 Optic, Spri 04, Solutio for Quiz Problem. Silvered Gl Hemiphere A prxil ry icidet o the flt ide of the hemiphere oe throuh the followi: () Flt l urfce: refrctio, che i le if icidet ry i ot prllel. () Striht proptio iide l, + directio. (3) Reflectio t curved urfce with rdiu R: che i le, d directio. (4) Striht proptio iide l, directio. (5) Flt l urfce: refrctio, che i le if icidet ry i ot prllel. Recll tht prxil pproximtio i ecery to defie ile focu for o-prbolic urfce. Uder the prxil pproximtio, i t. () Grphiclly, tep () throuh (4) for prllel icidet ry re depicted i Fiure. Fiure The focl poit of curved reflector c be obtied from the iterectio betwee the reflectio of the prllel ry d the horizotl xi. From the equlity of the three le mrked i Fiure, we ee tht the leth fo d fp re equl. Therefore, the focl poit i the midpoit betwee O d R. Sice the mirror i cocve, R 7.5cm, d f R / 3.75cm. The focl leth of cocve mirror i poitive. 5
6 .7/.70 Optic, Spri 04, Solutio for Quiz b) We re ow ive bubble tht i iide the l hemiphere. There re two ime: () Object refrcted throuh ple, () Object reflected by hemiphericl mirror, the refrcted throuh ple. () Object refrcted throuh ple The firt ime c be obtied by trci ry tht leve the object prllel towrd the left, d other ry tht hit the xil poit of the l urfce. The ime i erect, d h the me heiht the object, how i Fiure. Thi i virtul ime ice the outoi ry from poit o the object lwy divere, rther th covere to form rel ime. Fiure The loctio of the ime c be obtied from the ell lw betwee the two le, : i i,. (.0) I the prxil limit, the two equtio of (.0) re equl. The loctio of the ime with repect to the ceter of the hemiphere O c be obtied by compri the two trile, with le, t O. The loctio of the ime c be foud from: t t, i o / / 5 cm / cm. i o o (.) Thu, the upriht, virtul ime i locted 3.33cm to the riht of the flt l urfce. 6
7 .7/.70 Optic, Spri 04, Solutio for Quiz () Object reflected by hemiphericl mirror, the refrcted throuh ple. The ecod ime c be obtied by trci ry levi the object prllel to the xi towrd the mirror, which i reflected to p throuh the focl poit. Next, ry tht hit the xil poit of the mirror c be trced. The left ide of Fiure 3 how the virtul ime formed from reflectio, i ree dotted lie. Fiure 3 Thi virtul ime i erect d i lrer th the object. The loctio of the ime c be obtied from:, i 7. 5cm f R / 7.5cm 5cm 3.75cm. (.) o i i Therefore, the virtul ime i locted 7.5cm to the riht of the vertex of the hemiphere, before refrctio by the flt l urfce. Sice the oberver i i the ir, the ree ry re oce more refrcted by the flt l urfce. The red ry i the riht ide of Fiure 3 depict thi refrctio. The olid red ry repreet the corrected ime tht the oberver i the ir c ee. The loctio of the refrcted ime c be obtied i the me wy from (.): t t, i o / / 5 cm /.5 0 cm. i o o Therefore, the virtul ime i locted 0cm to the riht of the flt l urfce. (.3) 7
8 .7/.70 Optic, Spri 04, Solutio for Quiz c) Let u coider the pth of liht to clculte the effective focl leth for the ytem. Liht i icidet from the left ide of the flt urfce, d oe throuh the followi compoet: () Refrctio. Air ito l () Proptio. d=7.5cm (3) Cocve Mirror Reflectio. f=-/r=3.75cm (4) Proptio. d=7.5cm (5) Refrctio. Gl ito ir There i o wy tht liht c rech iide the ytem without bei refrcted t the flt urfce. Therefore the ytem i defied to trt d fiih riht outide the flt l urfce i ir, rther th iide the l. The ytem ABCD mtrix c be clculted : M =M M M M M ytem (5) (4) (3) () () 0 d 0 d f 0 0 d 0 d (.4) 0 0 R d d R R d( ) 0, d 0. 4 R R where R 7.5 cm, d 7.5 cm,,.5. From (.4), we clculte the EFL to be: EFL.5 cm. (.5) C d) The priciple ple re diplced from FFP d BFP by EFL: PP+EFL=FFL, PP+EFL=BFL. Sice both FFP d BFP re locted.5cm to the riht of the flt l urfce, Both PP re locted 5cm to the riht of the flt l urfce. Altertively, ui the ABCD mtrix, we c et: PP ( ) ( ) D 5 cm, PP A 5 cm. (.6) C C Ideed, the t PP i locted 5cm to the riht of the iput ple, which i the flt urfce. The d PP i lo locted 5cm to the riht with repect to the output ple, which i i the flt urfce. (The etive i of PP reult from the etive proptio directio: -5cm to the left i the me 5cm to the riht.) The priciple ple overlp t 5cm to the riht of the flt urfce. 8
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