Optics and Optical design Problems

Transcription

1 Optics nd Opticl design 0 Problems Sven-Görn Pettersson / Cord Arnold :3 This mteril is tken from severl sources. Some problems re from the book Våglär och Optik by Görn Jönsson nd Elisbeth Nilsson. Some re from the book: Fundmentl of Photonics, second edition by Bh Sleh nd Mlvin Crl Teich. Another source is Exempelsmling i våglär by Lrs Engström. Exmples nd figures tht re referenced to in the textbook Fundmentl of Photonics re lbelled with FoP. The exercise mrked with * re the most relevnt for the course. The other exercises re ment s repetition nd/or s preprtion for the lbortory exercises. Pge

2 Chpter : Ry Optics P-. Show tht the distnce p tht ry is displced prllel to its initil direction when trnsmitted through plne prllel glss plte of thickness d, is given by d sin( ) p cos where is the ngle of incidence nd is the ngle of the trnsmitted ry. See the following figure (figure -). Figure -. A plne prllel plte does not chnge the direction of light ry. However, the ry is trnslted distnce p tht is dependent on the thickness d of the plte. P-. At the bottom of.0 m deep swimming pool there re illuminting lmps. Due to totl reflection, the light from single lmp cn only be seen within limited region. Clculte the re of this region. Assume tht the refrctive index for wter is.33. * P-3. The high index of refrction of dimonds is utilised to obtin repeted totl reflections which cuses beutiful scttering of the light. The most common type of dimond polishing is clled brillint. In figure - the refrction nd reflection of red light ry is shown s it is propgting through dimond. The refrctive index for dimond nd red light is.407. ) Show tht the ry is totlly reflected t the point P b) Clculte the ngle of incidence t the upper surfce of the dimond. Figure -. Totl reflection in dimond. Pge

3 * P-7. Proof of the Snell s Lw. The proof of Snell s lw is n exercise in the ppliction of Fermt s principle. Referring to figure -4, we seek to minimize the opticl pth length nab nbc between points A nd C. We therefore hve the following optimiztion problem: Minimize nd sec nd secwith respect to the ngles nd, subject to the condition dtndtn d. Show tht the solution of this constrined minimiztion problem yields Snell s lw. Figure -4. Construction to prove Snell s lw. * P-8. Lens in Wter. Determine the focl length f of biconvex lens with rdii 0 cm nd 30 cm nd refrctive index n =. 5. Wht is the focl length when the lens is immersed in wter (n =.33)? * P-9. In n otoscope (medicl instrument for the exmintion of the er), the imging is schemticlly performed with cylindricl glss rod of the type shown in figure -5. The rod hs refrctive index of.49 nd the end surfces hve the rdii of curvture R = 5 mm nd R = 35 mm. ) Clculte the finl imge of n object plced 0.0 mm to the left of the surfce to the left. b) Is the imge virtul or rel? Figur -5. A simplified drwing of n otoscope. * P-. An object is plced 0 cm to the left of lens L mounted on n opticl bench. The lens hs the focl length cm. Slightly to the right of the lens L, nother lens L, is plced. This lens cretes finl imge 0 cm to the right of L. This imge is 6.0 times lrger thn the object nd upside down. ) Determine the distnce between the lenses L nd L. b) Wht is the focl length of the second lens L? Pge 3

4 * P-8. Numericl Aperture nd Angle of Acceptnce of n Opticl Fiber. An opticl fiber is illuminted by light from source (e.g., light-emitting diode, LED). The refrctive indices of the core nd cldding of the fiber re n nd n, respectively, nd the refrctive index of ir is (figure -9). Show tht the ngle of the cone of rys ccepted by the Figure -9. Acceptnce ngle of n opticl fiber. fiber (trnsmitted through the fiber without undergoing refrction t the cldding) is given by / NA sin ( n n) (FoP.-5) The prmeter NA sin is known s the numericl perture of the fiber. Clculte the numericl perture nd cceptnce ngle for silic glss fiber with n =.475 nd n =.460. P-9. Numericl perture of Clddless Fiber Determine the numericl perture nd the cceptnce ngle of n opticl fiber if the refrctive index of the core is n =.46 nd the cldding is stripped out (replced with ir n ). * P-0. Fiber Coupling Spheres. Tiny glss blls re often used s lenses to couple light into nd out of opticl fibers. The fiber end is locted t distnce f from the sphere. For sphere of rdius = mm nd refrctive index n =.8, determine f such tht ry prllel to the opticl xis t distnce y = 0.7 mm is focused onto the fiber, s illustrted in figure -0. Figure -0. Focusing light into n opticl fiber with sphericl glss bll. * P-. The Grin Slb s Lens. Show tht SELFOC slb (see figure -) of length d / nd refrctive index given by n ( y) n0 ( y ) cts s cylindricl lens ( lens with focusing power in the y-z plne) of focl length Pge 4

5 f n 0 sin( d) (FoP.3-3) Show tht the principl point (defined in the figure) lies t distnce from the slb edge AH (/ n ) tn( d / ). Sketch the ry trjectories in the specil cses d / nd /. 0 Figure -. The SELFOC slb used s lens; F is the focl point nd H is the principl point. * P-. For grdient-index lens, with dimeter of.0 mm, the refrctive index on the xis is.608 nd t the edge.534. Clculte its numericl perture nd period. Wht length is needed in order to imge the surfce of the first end on the other end? Note tht for grdientindex lens the profile cn be pproximted with n( y) n0( y ). * P-3. Numericl Aperture of the Grded-Index Fiber. Consider grded-index fiber with the index profile given by n n0 x y nd rdius. A ry is incident from ir into the fiber t its center, which then mkes n ngle 0 with the fiber xis in the medium (see FoP Figure.3-8). Show, in the prxil pproximtion, tht the numericl perture is NA sin n0 (FoP.3-6) where is the mximum cceptnce ngle for which the ry trjectory is confined within the fiber. Compre this to the numericl perture of step-index fiber such s the one discussed in FoP Ex.-5. To mke the comprison fir, tke the refrctive indices of the core nd cldding of the step-index fiber to be n n0nd n n0 n0( ), respectively. * P-4. A Set of Prllel Trnsprent Pltes. Consider set of N prllel plnr trnsprent pltes of refrctive indices n, n,..., nn nd thicknesses d, d,..., dn plced in ir (n = ) norml to the z xis. Show tht the ry-trnsfer mtrix is Pge 5

6 Note tht the order of plcing the pltes does not ffect the overll ry-trnsfer mtrix. Wht is the ry-trnsfer mtrix of n inhomogeneous trnsprent plte of thickness d 0 nd refrctive index n(z)? * P-5. A Gp Followed by Thin Lens. Show tht the ry-trnsfer mtrix of distnce d of free spce followed by lens of focl length f is * P-6. Imging with Thin Lens. Derive n expression for the ry-trnfer mtrix of system comprised of free spce/thin lens/free spce, s shown in Figure -. Show tht if the imging condition (/ d/ d / f) is stisfied, ll rys originting from single point in the input plne rech the output plne t the single point y, regrdless of their ngles. Also show tht if d f, ll prllel incident rys re focused by the lens onto single point in the output plne. Figure -. Single lens imging system. * P-7. A Periodic Set of Pirs of Different Lenses. Exmine the trjectories of prxil rys through periodic system comprising sequence of lens pirs with lternting focl lengths f nd f, s shown in Figure -3. Show tht the ry trjectory is bounded (stble) if 0 ( d d )( ) f f (FoP.4-35) Figure -3. A periodic set of lenses * P-8. The ry trnsfer mtrix for curved boundry with curvture R nd refrctive index n before the surfce nd refrctive index n fter the surfce is given by: Pge 6

7 A B 0 M n n C D nr ) Clculte the ABCD mtrix of thin sphericl lens, mde up of two closely spced dielectric interfces, of rdii R nd R enclosing mteril of refrctive index n. The lens is immersed in medium of refrctive index n. b) From the ABCD mtrix it is esy to find the focl length f of the lens. Give n expression for f. * P-30. Ry-Trnsfer Mtrix of GRIN Plte. Determine the ry-trnsfer mtrix of SELFOC plte [i.e. grded-index mteril with prbolic refrctive index n( y) n0( y )] of thickness d. Pge 7

8 Answers to the problems: P-: 6 m P-3: b) 0 P-4: 4.8 P-6: F b = -.0 cm, yb 3.0 cm P-8: f = 4 cm in ir nd f = 94 cm if immersed in wter. P-9: ) 8 mm to the left of the first surfce b) virtul P-0: f = 4 cm P-: f = -0 cm, R = 5 cm, y(finl imge) -.6 cm, upside down P- : ) 5 cm b) -0 cm P-3 : ) 7 cm b) 80 cm to the right of the lens. The imge is upside down nd enlrged x. P-4 : 750 m P-5 : ) -33 b) 80 cm x. m P-6 : +,5 cm Focl plne +,5 cm -,5 cm -,5 cm P-7 : 5 cm P-8 : NA = 0., = P-9 : NA = P-0 : 0.03 mm P- : See FoP Figure.3-5 P- : NA = 0.49, period = 0,7 mm, length = 0,4 mm P-3. Grded-index fiber: NA = 0.04, Step-index fiber: NA = d0 dz P-4: M 0 nz ( ) 0 Pge 8

9 d d dd( ) f f P-6: M d f f 0 A B P-8: ) M n n C D n R R For lens with n =n nd n =we hve n r r P-9: R ; R ' ' r r n n f n R R b) (FoP.-) f R R P-30: M sind cosd sind cosd Pge 9