Final solutions - FINAL EXAM Course ECE 716 Name g d L...

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Unformatted text preview: FINAL EXAM Course# ECE 716 Name g d L gg‘F / 0 4/9 Quarter Autumn 2006 This exam is being administered under the Electrical Engineering Honor Code. You may use THREE 8.5” x 11” pieces of paper with anything you want written anywhere on them, a ruler, and your calculator. If you are lacking a physical constant you may ask for it. If there is an equation you already know exists, you can ask for that too as long as you describe it well enough so I know you're not fishing. If you don't get an answer, but need it to complete the next problem, then write down that you didn't get the result, that you're assuming (choose a reasonable value), and proceed. Make sure ou make this clear, however. Show all your work. mil- If you get a ridiculous answer, write down that you recognize it's ridiculous and speculate as to why (if you don't have time to find your mistake). If part (b) relies /" on having part (a) correct, and you couldn't do (a), then write down "assume the answer to "a" is ...", and continue. There are __total points on this exam; all questions are not weighted evenly. You havel hour and 48 minutes to finish the exam. page 1 of 11 1‘” f WU 3 {flu K () 1. A plane wave whose phase fronts form an angle of 10° with respect to the x-axis and are parallel to the y axis is incident on a lens/’for focal length f=20 cm as shown below. () a) Find the (x, y, z) coordinates of the point image formed. Explain your reasoning. ear) W) (DMI‘YVLL’C\ rmwci'rd V0») J [06‘ (JV-£3? ova/HA “firth/(x H! Um}, AVA vvfi [0ch 1+ " “k V“ at] I ‘ {)ch CU}, M {mar (WLU tL A.“ LA (c lac‘amf). I if) 2 CL“ {00 X‘ = 3r} (AM WW If 304' Vla (Lula/hex {yd} :0 );o. Com/l (ILA/v2 rSI} 2:10ij (fir?) :- (3,1‘3/ O, 9.0) () b) If the wavelength is k20.5 um, what is the spatial frequency of this plane wave in the plane 220? ) l m 9 = mm 12, 1x .1. K jirlua‘w ‘ 7/ fix' mic ‘: ;’ ~18?“ Qw )7):th A7LrivLL] .. I‘()( ' 2;— : 32.7éutm“ Lifflc‘ié page 2 of 11 1. Continued. () C) Write the complete mathematical expression for the wave before and after the lens. 0(W 0W f ([96th “4 UV”) L/-ch/,¢L “3(K’sz}? > "/kY‘-cvf) (I? 4. E 50 e a g 1““ “ i if; k:-§;,/Oxe2«/o m / “ kc, [lewulmJ Swo “4/1 3 to" "z * . (of; b ’71—? {1'}ka Ana JIOOALQ/ $9M =l 2 M lg”! -)(l Law J.) (3<V~o 03%) t 71 ”if" 1 C 249.2%) ‘ (2’0.lm\ Q Val/‘61!” \l £3k2 )(3 1&7/0 4.6“) ‘ 6 page 3 of 12 1. Continued. ‘1» 0 k (I—[)2~—y2 d) A wave described by E = Bog-flu» __.’P_ 6- [ Ill—6°) "—7 I» «.0 L'W’e’” is incident on the same lens above. The lens causes the beam to collapse to a point source. Find the location of the point source created by the lens. If the second point source is am image of the source of the original beam, What is the magnification? Is the image real or imaginary? (“)7 wév)‘. fil/wri WW l W 47‘9““ all) ‘Wéfl" £10m ac; ,9 AMT 5 cvvlf H9,— ' ‘ \ ~- 0“ » CVAUA V0: ‘/ «7;»0/ $.76” 56) L7 (“j/”(L/‘,6k 7L 07‘“ (I ,1, fjttvxiarl 7[mm {jiKV‘ dV‘ /3"((( ”j C‘A‘J ( (iv/)4 6W1 Jon fill‘vs we LIAM/z l, — ’ (Qt/WNW M‘— 0:60 10:75) in; —L J J- HL ‘L. J— 4 — 50 Hex} 15 z“, 7’ 60° ‘+ ”(J ’ F a 6‘0 LIL; : 10 3‘) o/L' 30 > ’07“ ”k ‘ ’%L' 533’ “of 5'3 X|2 '0.V(m)“ul(t):~of , M u ‘1 “A. 60 \E’ a A' “A I Watch/v» * ’ crypt»? IV \ 30:7: ” 0 J‘V‘t“ \j W /) ‘ ~ * ) (x! (7‘ 2’) v 0 SI 0’ 30 Magma) t‘ r #r , L“, (M {g ”fivm‘ (thm : ‘Or 3’ g 3591””? L( d'lLtvf :\ r, 1 “‘4 'W‘)‘ " “3M “7’ m ) J 2&1 (WM {hr dt w) flag," WC 3 R CV) ’N/ (“' Z J “>5 rcwtw ("m4 4 (3“ l, 4 :1 MW V57] page 4 of 12 O 2. For each of the optical situations below, indicate which of the suggested matrices, if any, describes it. Explain your reasoning. dashed lines is an unspecifiled optical system, and th out of (right) the system are shown. The region between the e rays going into (left) and [0 B] C Y) “57¢ 0 if and? (Vb (on. (Ct Flam waw Hewlett} cm a 0 3. A hologram is constructed using a plane wave reference and an off- axis point source object as shown below. This is similar to the system on the second midterm except for the location of the object. () a) The hologram is then reconstructed using the same plane wave used to make it. Sketch on the figure above the phasefronts for all the beams you 5L expect from reconstruction, and label them. ‘ l 1276‘ rsrwutr/Lk trim: [Dc/if)" id“ Jicifidf' l" 52"“ (((RC'M “7 gay '. (oak; ltlze “(anglwucl‘iw Lew \ M (’0 (ML “"7 > “L2 (04k [xL-L comju'j‘v‘j' l‘o grkeu‘ta/L WM. W’t (CMJJ) ,- M ELM” ‘— [le M Libk‘l— (Cl/\pw «‘2sz fxaelj KL; (TOjvj‘L M‘a’k M\‘ (yr [‘1 5‘ UKK-(s ) va \CVLCL" rye , l’\' l. glow F or,“ LC Ck () b) Can this hologram be thought of as two superimposed lenses? Explain. You score for this problem will depend on the quality of your explanation, not just whether you have some general idea. Use the back of this sheet if needed. (On the midterm I received some very sketchy answers). VCSwfikatl/(v) ”A 54“" ‘11 Writ-“W 2/ 0%) dL/K‘n} LUCJ‘ MAO/(£9 71/ (7‘7“Sn A lfm)‘ (cmufvl’f Cx fkdtm Mat <67}? HA l’fi CK fur-«l ft'uvtc, TlA‘I lnclaj‘rw (W (rtccl1/ +WQ j/lriu‘t cup mph/(J v' 0'“ ply" (at/f: ' a K j VI rdmk 5°V‘V" t 0‘43 erbFLa-l Uncut/t \J‘ ‘o‘llv «7 «“7 I 56 Ed! ("‘7 muggy-f (S 1/}1 31/01 (Cur [quit/5 I?» L“ H ((4% I 07‘ AJ C/jybLM). O‘+L{r Fa‘K‘F-‘J‘CVK( —‘ A LE CM)» Arc/{l J» 6m [calcivy AV chleh7 (ILL 7/ L') ed ltwskwlu.)jl A¢W»«_+%‘:§?c50 13th} LL “glut/l. r ‘ ' W» i " Vl“\l ijllv hf {ncdxwuy (x [1611 l‘ // L) I ‘ 3. Continued () C) Suppose the same hologram is reconstructed with the object beam (the point source) instead. Sketch and label below the principle beam and the E00 beam (we never had a good name for it):__pon’t worry about the conjugate beam. “ Lao ,Vc.-_v‘.,_____._,c - page 6 of 11 0 4. A GaAs vertical-cavity surface—emitting diode laser (A4520 nm) produces a round Gaussian beam waist of wo=25 um. , 3’59W4V J F' . , . () a) 1nd the Raylelgh range (I n W01 : fl (”(1%“) “ML 2 , at. (J (A («AM 1 ‘ / ..__ . 7 5 ) A0 {/Loflo‘ ‘f... :3}, MM () b) Find the far-field divergence angle and indicate your units (degrees or radians) av 11de Lgiv‘trircwt ($3 ’) baffffiz‘uta/V #:H 61% - *6} 3 - ~, o 9’ 71% L W Local. will“, “"‘W 'le (Znaf‘ual [l t l fly as? of 3;“; ‘1 810%; ”UV/V ”7 () c) We wish to couple as much as possible of this light into a fiber, Q using a lens to reimage the beam waist on the fiber tip. If the lens will be 59911-31 S m m away from the waist, what should the diameter of the lens be? Explain your choice. WM {VIA/(’1’ *LL K41), (171" mmgx/ {o M (OM W G alacvt (V 1 ' " i I) lxw} l/LL Sr} 1174: a" 12::5‘M“ ‘ 1 _ x "6 ‘7' in 1 6L @/? W(l’)~(ll¥(d m\ [l ‘l’é‘s‘ Wt) 1 @l/ xix/(Vlzng'flm ‘MJU Vl/‘(rs ‘5 k (“4“; ”ppm A\¢Vrgifiv’d':'j lib «11) {mm I) .1..———- W “37“ (\A ruckus H ”ll—4 V6 {(e’kwol dlnl' or ellX mbhsilg‘ Epolf did I) I ;~ A; {9 k” y_ - 7 ‘ & \YU/jr‘vl‘w E l : 013 So wlr we W M Kym; gutsy/um m} L; (um) . {iv/villi» («45" c, lvjvvl’ Vlfi{‘l’h. \1’ CLEWLL‘ W H71“ ll] 647 vaj Jubu< / fl ( \‘sf wwkxlx L4 lnavc’x l1) mwlue Sual,‘ 0x l«W) but)" ll‘l r) liv‘r’ik LI.’ 6 y H s M 3* mwflrw Mm!“ ( :‘wx WW“t (filial—ELL} ”47ch r‘taJ‘CV‘W‘j 'l/Lxcu. \ow' avatar, Mn? «N / page80f12 “a “lime, :4: ”iv ‘qéyfi (,{o Vu’fiC/Jii’x{j I /{ 0 6. We said in class that the Gaussian beam diverges the least of all possible beams. There is a type of beam called the Bessel beam, however, that doesn’t diffract at all— its phasefront is planar everywhere. The Gaussian beam is a solution to the paraxial wave equation, but the Bessel beam is an exact solution of the full-up wave equation. Furthermore, the Gaussian beam has a finite RMS width, but the Bessel beam’s RMS width is infinite. The drawing below compares the two profiles. Gaussian beam Explain why it is not possible to generate a true non-diffracting Bessel beam in the laboratory. Why will any physically realizable beam necessarily diffract? Sly“: \“l/L @(SSdI Liam (S tW,+lV\l+-€ (\A w€‘(l(\/ ‘14- wIll 1’le Fob l2); HVWC aft/(J i"\ Aux] Pky$i(u,Q 57Il-(m’ F‘L‘C ‘FrvALLJVVV‘ Wulkrw" page 9 of 11 O 7. Sometimes reflections that come back from an optical system have to be defeated. For example, in launching light into a fiber, the Fresnel reflection, although small, can mess up the oscillation of the laser if the reflections get back into the laser. Below is a trick commonly used to stop them. . . i reflection mcndent linear polarizer QWP FA/lx @ 45° () a) When light reflects from a material of higher refractive index (such as the face of the fiber), one polarization (call it x) is phase-shifted by 7: radians; the other polarization is not phase—shifted. When the LCP light is reflected from the face of the fiber, will it come back as RCP or LCP? Lcr: [f l a ‘j “5 {5 tam: y m flan. 31cm ) W* wW my 77, W7 7 vull {HH-(Cifivbk) L! (ll/monk i471: C 0347(ch A) 1 his palavficuiwv‘ WU Le (up [i] , ) ; Y, a v» 315%” “term “1 " page 10 of 11 7. Continued b) Show that the system shown will prevent any reflected light. we Mjw (w (oi/La AW) «M HA “1“ Warsaw, He A4,, L I i (' Wi . ,1 ,r. .5 LM, an M mm! 4%} Wm «+- mflfb t H La law [A r , ,,, ac \ rc‘ K. m WV f Q ~ 0. TL!“ 3‘ frufc) L» (c “A V) Q L [-0 81?] A»)!!! LkL Thmjk MA [Ari/1741' ’E m X a“ (I rcWVflJ (Riflki N4 merr (COWELWKJ > 5) WM 1”" 5%ng E [J A) «VJ-c: / '7 1 W x - Q i ”,J _ LN“ rbrnfr; [W3 9 Mug (.4) :l: [/2 Z t ‘3: I I g “3’ 9 W (964/3 SWIG j: , \ I «J1: v\ so we 50+ 0v“F £7“ air/W ”a L i " I O \ { (E g f A ~,: ' «if / l’ '\ l i) L + f6 ”r 0"“ c) e ) «m c) Does is matter how the quarter wave plate is oriented (e.g, where its fast axis is?) ‘4’?) wt Wt‘fikrsx Fol Exow’JL/ (fly “I Fist runa- wtx 01"“? (fl-«L‘rt‘ be L‘ Q “0 (wish: (VOW—QYCJWh’g (”m/J ‘{/LL( wkh f/Luv Ova/(a PLEDGE: N0 aid received, given, or observed. 5' d L LA I Q N’S page 12 of 12 ...
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