2.710 Optics Spring 09 Solutions to Problem Set #2 Due Wednesday, Feb. 25, 2009

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1 MASSACHUSETTS INSTITUTE OF TECHNOLOGY.70 Optics Sprig 09 Solutios to Prolem Set # Due Weesay, Fe. 5, 009 Prolem : Wiper spee cotrol Figure shows a example o a optical system esige to etect the amout o water preset o the wishiel o a car to ajust the wiper spee. As show i this gure, we ca use the wishiel as a waveguie to guie the light rom a source locate at oe e (ottom o the wishiel) to a etector locate i the opposite e. The light su ers total-iteral re ectio (TIR) at the glass-air iterace. However, whe rai rops are preset, some o the light will su er rustrate TIR escapig outsie the waveguie. Sice we kow the power o the light source, a give rop i power ca e correlate to the amout o water preset a use to ajust the wiper spee. To quatiy the easiility o this esig we start y computig the critical agles o the glass-air a glass-water iteraces, air cg a ar csi 4:8 ; glass water cg w arcsi 6:5 : glass The iciece agle o a give ray propagatig isie this waveguie is restricte to the ollowig cases:. For < cg a : The light will su er rustrate TIR a escape out o the waveguie regarless o whether the iterace is glass-air or glass-water.. For cg a < < cg w : The light will su er TIR at the glass-air iterace a rustrate TIR at the glass-water iterace. 3. For cg w < : The light will su er TIR at oth iteraces. Prolem : Telescope with i ite cojugates a) The geometry o this prolem is show i Figure. By ispectio, we see that the rst les (L) will ocus the light at a istace o at its ack ocal plae. This poit ow acts as a poit source oject or the seco les (L) a sice it is locate at its rot ocal plae (provie that + ), the light eam gets collimate a thus exits the optical system as a parallel ray ule (plae wave). ) To aswer this questio we use the matrix ormulatio. The output agle a height are relate to the iput agle a height accorig to,

2 Figure : Wiper spee cotrol prolem. Figure : Telescope with i ite cojugates.

3 x x : x () So the optical power o the composite optical system is, P ( + ) ; () a the e ective ocal legth is, EF L P : (3) ( + ) Note that or the case o +, equatios a 3 ecome: P 0; a EF L. These type o optical systems are calle aocal. c) From equatio, we the equatios or output agle, ; a height, x, + P x (4) + ; x + x (5) ( + ) x : To the eam with, a, we cosier the case o ormal iciece (i.e. 0), a x x (6) a : 3

4 ) From Figure, we ca see that a real image is orme o the o -axis oject at i ity a is locate at a istace at the ack ocal plae o les L. The elevatio o the image poit respect to the optical axis is, x i : (7) The seco les collimates this image poit a the output agle is, x i (8) The relatios show i equatios 7 a 8 ca also e use as a simple way to compute the agular a lateral magi catios. To compute the agular a lateral magi - catios, we comie the previous relatios a solve or the ratio etwee output a iput agles, M A M L ; (9) ; M A which are cosistet with the results we ou i part (c). e) Figure 3 shows the ew geometry o the system i we assume that L is egative ( < 0). We see that i orer to have a collimate output eam, the separatio etwee leses is still + ; however, to esure a positive separatio istace (i.e. > 0) we require that > j j. The rst les ow orms a virtual image at a istace to the let o L. The elevatio o the virtual image respect to the optical axis agai x i, ut sice < 0 we see that ow the image height is egative. For the seco les, L, the light seems to e comig rom a poit source locate a istace at its rot ocal plae a thus it collimates the eam with the same output agle,, as that give y equatio 4, ut ow > 0. The size o the output eam, a, will also e the same as that give y equatio 6 ut sice > j j, we see that the eam gets expae a ot iverte. This type o telescope is kow as Galilea telescope. Prolem 3: Telescope with ite cojugates a) The geometry o this prolem is show i Figure 4. By ispectio we see that sice the o -axis poit source is locate i the rot ocal plae o the rst les, L, it gets collimate. The seco les, L, receives a o -axis plae wave that the ocuses to orm a real image o its ack ocal plae (i.e. a istace to the right o L). ) To solve this part we use the matrix approach. The image height a agle are relate to the oject height a agle accorig to, 4

5 Figure 3: Galilea telescope. Figure 4: Telescope with ite cojugates. 5

6 x P x 0 3 P 5 P + x (0) ; x where, P ( + ) ; () a or the case o +, the optical power P 0, so equatio 0 ecomes, x 0 : ( ) 0 We ow compute the elevatio x a the lateral magii catio M T, x x M T x x : x x ; (3) c) Agai rom equatio, the arrival agle,, a the agular magi catio, M A, are give y, M A : M T ; (4) ) We ow cosier the case i which L is egative (i.e. < 0). The geometry o this case is show i Figure 5. Similar to the previous prolem, i orer to have a positive separatio istace etwee oth leses ( > 0), we require that > j j. The same equatios hol as or the case o a positive les; however, we otice that the image orme y the seco les is ow virtual a erect. The agular magi catio is also the same as that o equatio 4 except or the sig. 6

7 Figure 5: Galilea telescope. Prolem 4: Immersio les a) We egi y computig the composite matrix o the optical system, i xi " # " 0 0 R 0 s i R 0 # 0 0 s o o x o " + so P P + s i P + s i s i + so P # o x o ; (5) where, P 0 R R R R ; (6) a we see that or the case o ; equatio 6 reuces to the "Les Maker s " equatio that we saw i class. Equatio 6 is a more geeralize versio o this equatio. To i, we cosier a plae wave comig rom the let such that o x i 0 (i.e. or a oject at i ity, s o ), 0; a 0 si + P x o (7) ) i : 7

8 We ow compare i with the ocal legth o the same les immerse i air, a. Tale summarizes some o the possile coitios that the les ca have give the our variales, ; ; R ; a R : Tale R > 0 R R < 0 > > < 0 R > 0 > > B C B A B C B A > C D > A D > C D > A D R > 0 R R > 0 > > < 0 R < 0 > > B A B C B C B A > C D > A D > A D > C D Where, Note: This hol or R > R ; or R < R, i ( ; ; R ; R ) ** Note: This hol or R < R ; or R > R, i ( ; ; R ; R ) A : i < a B : i a C : i > a D : i ( ; ; R ; R ): ) To erive the imagig coitio, we ote that all the rays, regarless o their origiatig agle o, arrive at the image poit x i. From equatio 5, we wat to the coitio o 0, si + s o si + P 0 ) (8) + : s o s i c) The agular magi catio is give y, 8

9 @ i M A o so s o : From the imagig coitio o equatio 8, we solve or s i, s o : (0) s i s o Usig equatio 0 i equatio 9 we get, M A s o : () s i Similarly, we erive the lateral magi catio usig the equatio or x i or a o-axis ray (i.e. o 0), si x i xo () x i si s i M L : (3) x o s o ) We cosier the triagle orme y a ray that origiates a the oject s tip a aims at the optical ceter o the les. The agle o the ray respect to the optical axis is, x o ta o ~ o : (4) s o From equatio 5 we see that the oject a image agles are relate accorig to, i so o so x o s o x o o ; s o xo x o (5) which is Sell s law! 9

10 Prolem 5: Hyperoloial reractor a) The geometry or this prolem is show i Figure 6. Fermat s priciple requires that the propagatio legth o ray, l 0, is equal to that o ray, l, where, l 0 + s; a (6) l c + : By similar triagles we see that, c a Sice x a +, a c + a, s sc ) ; (7) as s ) : x a + s x ) a ; (8) + s 3 c x 6 x s ) c s 5 : Comiig equatio 8 with equatio 6 we get, s l 0 l (9) # s + s " + + x ) ( + s) s + + x ) ( + s) ( + s) + x ) + (s) + s + s + s + x ) s x + s ) x + s + + ) x s + : + + 0

11 Figure 6: Hyperoloial reractor prolem. ) We egi y rewritig equatio 9 to match the orm o a shite hyperola, where, (s h) x ; (30) a From equatio 30 we solve or s; a h ; a (3) + r : + s(x) a r + x We expa equatio 3 ito a Taylor series evaluate at x 0, where, 0 00 x + h: (3) s(x)~ s(0) + s (0)x + s (0) + high orer terms, (33)

12 0 ax s (0) h i 0; (34) + x 00 a s (0) h ax i a + x h i + x 3 4 : So equatio 33 ecomes, a s(x)~ (a + h) + x + high orer terms, a we see that or small jxj compare to the raii o curvature, (i.e. i the paraxial regio), the hyperoloial surace is approximate as a paraoloial surace, s cx (the costat term cacels out as a h or our case):

13 MIT OpeCourseWare /.70 Optics Sprig 009 For iormatio aout citig these materials or our Terms o Use, visit:

2.710 Optics Spring 09 Solutions to Problem Set #3 Due Wednesday, March 4, 2009

2.710 Optics Spring 09 Solutions to Problem Set #3 Due Wednesday, March 4, 2009 MASSACHUSETTS INSTITUTE OF TECHNOLOGY.70 Optics Sprig 09 Solutios to Problem Set #3 Due Weesay, March 4, 009 Problem : Waa s worl a) The geometry or this problem is show i Figure. For part (a), the object