Diffraction Patterns and Polarization

Transcription

1 chpter 38 Diffrction Ptterns nd Polriztion 38.1 Introduction to Diffrction Ptterns 38.2 Diffrction Ptterns from Nrrow Slits 38.3 Resolution of Single-Slit nd Circulr Apertures 38.4 The Diffrction Grting 38.5 Diffrction of X-Rys by Crystls 38.6 Polriztion of Light Wves When plne light wves pss through smll perture in n opque brrier, the perture cts s if it were point source of light, with wves entering the shdow region behind the brrier. This phenomenon, known s diffrction, cn be described only with wve model for light s discussed in Section In this chpter, we investigte The Hubble Spce Telescope does its viewing bove the tmosphere nd does the fetures of the diffrction pttern tht not suffer from the tmospheric blurring, cused by ir turbulence, tht plgues ground-bsed telescopes. Despite this dvntge, it does hve limittions due to occurs when the light from the perture is diffrction effects. In this chpter, we show how the wve nture of light limits llowed to fll upon screen. the bility of ny opticl system to distinguish between closely spced objects. (NASA Hubble Spce Telescope Collection) In Chpter 34, we lerned tht electromgnetic wves re trnsverse. Tht is, the electric nd mgnetic field vectors ssocited with electromgnetic wves re perpendiculr to the direction of wve propgtion. In this chpter, we show tht under certin conditions these trnsverse wves with electric field vectors in ll possible trnsverse directions cn be polrized in vrious wys. In other words, only certin directions of the electric field vectors re present in the polrized wve. 1111

2 1112 CHAPTER 38 Diffrction Ptterns nd Polriztion Viewing screen Dougls C. Johnson/Cliforni Stte Polytechnic University, Pomon Source Opque object From M. Cgnet, M. Frnçon, nd J. C. Thrierr, Atls of Opticl Phenomen, Berlin, Springer-Verlg, 1962, plte 32 Figure 38.1 The diffrction pttern tht ppers on screen when light psses through nrrow verticl slit. The pttern consists of brod centrl fringe nd series of less intense nd nrrower side fringes. Figure 38.2 Light from smll source psses by the edge of n opque object nd continues on to screen. A diffrction pttern consisting of bright nd drk fringes ppers on the screen in the region bove the edge of the object Introduction to Diffrction Ptterns P. M. Rinrd, Am. J. Phys. 44: Notice the bright spot t the center. Figure 38.3 Diffrction pttern creted by the illumintion of penny, with the penny positioned midwy between the screen nd light source. In Sections 35.3 nd 37.1, we discussed tht light of wvelength comprble to or lrger thn the width of slit spreds out in ll forwrd directions upon pssing through the slit. This phenomenon is clled diffrction. When light psses through nrrow slit, it spreds beyond the nrrow pth defined by the slit into regions tht would be in shdow if light trveled in stright lines. Other wves, such s sound wves nd wter wves, lso hve this property of spreding when pssing through pertures or by shrp edges. You might expect tht the light pssing through smll opening would simply result in brod region of light on screen due to the spreding of the light s it psses through the opening. We find something more interesting, however. A diffrction pttern consisting of light nd drk res is observed, somewht similr to the interference ptterns discussed erlier. For exmple, when nrrow slit is plced between distnt light source (or lser bem) nd screen, the light produces diffrction pttern like tht shown in Figure The pttern consists of brod, intense centrl bnd (clled the centrl mximum) flnked by series of nrrower, less intense dditionl bnds (clled side mxim or secondry mxim) nd series of intervening drk bnds (or minim). Figure 38.2 shows diffrction pttern ssocited with light pssing by the edge of n object. Agin we see bright nd drk fringes, which is reminiscent of n interference pttern. Figure 38.3 shows diffrction pttern ssocited with the shdow of penny. A bright spot occurs t the center, nd circulr fringes extend outwrd from the shdow s edge. We cn explin the centrl bright spot by using the wve theory of light, which predicts constructive interference t this point. From the viewpoint of ry optics (in which light is viewed s rys trveling in stright lines), we expect the center of the shdow to be drk becuse tht prt of the viewing screen is completely shielded by the penny. Shortly before the centrl bright spot ws first observed, one of the supporters of ry optics, Simeon Poisson, rgued tht if Augustin Fresnel s wve theory of light were vlid, centrl bright spot should be observed in the shdow of circulr object illuminted by point source of light. To Poisson s stonishment, the spot ws observed by Dominique Argo shortly therefter. Therefore, Poisson s prediction reinforced the wve theory rther thn disproving it Diffrction Ptterns from Nrrow Slits Let s consider common sitution, tht of light pssing through nrrow opening modeled s slit nd projected onto screen. To simplify our nlysis, we ssume the observing screen is fr from the slit nd the rys reching the screen

3 38.2 Diffrction Ptterns from Nrrow Slits 1113 Incoming wve Slit L u The pttern consists of centrl bright fringe flnked by much weker mxim lternting with drk fringes. min min mx min min Viewing screen From M. Cgnet, M. Frnçon, nd J. C. Thrierr, Atls of Opticl Phenomen, Berlin, Springer-Verlg, 1962, plte 18 ACTIVE FIGURE 38.4 () Geometry for nlyzing the Frunhofer diffrction pttern of single slit. (Drwing not to scle.) (b) Photogrph of single-slit Frunhofer diffrction pttern. re pproximtely prllel. (This sitution cn lso be chieved experimentlly by using converging lens to focus the prllel rys on nerby screen.) In this model, the pttern on the screen is clled Frunhofer diffrction pttern. 1 Active Figure 38.4 shows light entering single slit from the left nd diffrcting s it propgtes towrd screen. Active Figure 38.4b is photogrph of single-slit Frunhofer diffrction pttern. A bright fringe is observed long the xis t u 5 0, with lternting drk nd bright fringes on ech side of the centrl bright fringe. Until now, we hve ssumed slits re point sources of light. In this section, we bndon tht ssumption nd see how the finite width of slits is the bsis for understnding Frunhofer diffrction. We cn explin some importnt fetures of this phenomenon by exmining wves coming from vrious portions of the slit s shown in Figure According to Huygens s principle, ech portion of the slit cts s source of light wves. Hence, light from one portion of the slit cn interfere with light from nother portion, nd the resultnt light intensity on viewing screen depends on the direction u. Bsed on this nlysis, we recognize tht diffrction pttern is ctully n interference pttern in which the different sources of light re different portions of the single slit! To nlyze the diffrction pttern, let s divide the slit into two hlves s shown in Figure Keeping in mind tht ll the wves re in phse s they leve the slit, consider rys 1 nd 3. As these two rys trvel towrd viewing screen fr to the right of the figure, ry 1 trvels frther thn ry 3 by n mount equl to the pth difference (/2) sin u, where is the width of the slit. Similrly, the pth difference between rys 2 nd 4 is lso (/2) sin u, s is tht between rys 3 nd 5. If this pth difference is exctly hlf wvelength (corresponding to phse difference of 180 ), the pirs of wves cncel ech other nd destructive interference results. This cncelltion occurs for ny two rys tht originte t points seprted by hlf the slit width becuse the phse difference between two such points is 180. Therefore, wves from the upper hlf of the slit interfere destructively with wves from the lower hlf when or when 2 sin u 56l 2 sin u 56 l 1 If the screen is brought close to the slit (nd no lens is used), the pttern is Fresnel diffrction pttern. The Fresnel pttern is more difficult to nlyze, so we shll restrict our discussion to Frunhofer diffrction. b Pitfll Prevention 38.1 Diffrction Versus Diffrction Pttern Diffrction refers to the generl behvior of wves spreding out s they pss through slit. We used diffrction in explining the existence of n interference pttern in Chpter 37. A diffrction pttern is ctully misnomer, but is deeply entrenched in the lnguge of physics. The diffrction pttern seen on screen when single slit is illuminted is ctully nother interference pttern. The interference is between prts of the incident light illuminting different regions of the slit. Ech portion of the slit cts s point source of light wves. /2 /2 sin u 2 The pth difference between rys 1 nd 3, rys 2 nd 4, or rys 3 nd 5 is (/ 2) sin u. Figure 38.5 Pths of light rys tht encounter nrrow slit of width nd diffrct towrd screen in the direction described by ngle u (not to scle). u

4 1114 CHAPTER 38 Diffrction Ptterns nd Polriztion Dividing the slit into four equl prts nd using similr resoning, we find tht the viewing screen is lso drk when sin u562 l Likewise, dividing the slit into six equl prts shows tht drkness occurs on the screen when sin u563 l Therefore, the generl condition for destructive interference is Condition for destructive interference for single slit Pitfll Prevention 38.2 Similr Eqution Wrning! Eqution 38.1 hs exctly the sme form s Eqution 37.2, with d, the slit seprtion, used in Eqution 37.2 nd, the slit width, used in Eqution Eqution 37.2, however, describes the bright regions in twoslit interference pttern, wheres Eqution 38.1 describes the drk regions in single-slit diffrction pttern. sin u drk 5 m l m 561, 62, 63, c (38.1) This eqution gives the vlues of u drk for which the diffrction pttern hs zero light intensity, tht is, when drk fringe is formed. It tells us nothing, however, bout the vrition in light intensity long the screen. The generl fetures of the intensity distribution re shown in Active Figure A brod, centrl bright fringe is observed; this fringe is flnked by much weker bright fringes lternting with drk fringes. The vrious drk fringes occur t the vlues of u drk tht stisfy Eqution Ech bright-fringe pek lies pproximtely hlfwy between its bordering drk-fringe minim. Notice tht the centrl bright mximum is twice s wide s the secondry mxim. There is no centrl drk fringe, represented by the bsence of m 5 0 in Eqution Quick Quiz 38.1 Suppose the slit width in Active Figure 38.4 is mde hlf s wide. Does the centrl bright fringe () become wider, (b) remin the sme, or (c) become nrrower? Exmple 38.1 Where Are the Drk Fringes? Light of wvelength 580 nm is incident on slit hving width of mm. The viewing screen is 2.00 m from the slit. Find the positions of the first drk fringes nd the width of the centrl bright fringe. SOLUTION Conceptulize Bsed on the problem sttement, we imgine single-slit diffrction pttern similr to tht in Active Figure Ctegorize We ctegorize this exmple s strightforwrd ppliction of our discussion of single-slit diffrction ptterns. Anlyze Evlute Eqution 38.1 for the two drk fringes tht flnk the centrl bright fringe, which correspond to m 5 61: sin u drk 56 l Let y represent the verticl position long the viewing screen in Active Figure 38.4, mesured from the point on the screen directly behind the slit. Then, tn u drk 5 y 1 /L, where the subscript 1 refers to the first drk fringe. Becuse u drk is very smll, we cn use the pproximtion sin u drk < tn u drk ; therefore, y 1 5 L sin u drk. The width of the centrl bright fringe is twice the bsolute vlue of y 1 : 2 0 y L sin u drk `6L l ` 5 2L l m m m m mm Finlize Notice tht this vlue is much greter thn the width of the slit. Let s explore below wht hppens if we chnge the slit width.

5 38.2 Diffrction Ptterns from Nrrow Slits cont. WHAT IF? pttern? Wht if the slit width is incresed by n order of mgnitude to 3.00 mm? Wht hppens to the diffrction Answer Bsed on Eqution 38.1, we expect tht the ngles t which the drk bnds pper will decrese s increses. Therefore, the diffrction pttern nrrows. Repet the clcultion with the lrger slit width: 2 0 y L l m m m m mm Notice tht this result is smller thn the width of the slit. In generl, for lrge vlues of, the vrious mxim nd minim re so closely spced tht only lrge, centrl bright re resembling the geometric imge of the slit is observed. This concept is very importnt in the performnce of opticl instruments such s telescopes. Intensity of Single-Slit Diffrction Ptterns Anlysis of the intensity vrition in diffrction pttern from single slit of width shows tht the intensity is given by sin 1p 2 sin u/l2 I 5 I mx c d (38.2) p sin u/l where I mx is the intensity t u 5 0 (the centrl mximum) nd l is the wvelength of light used to illuminte the slit. This expression shows tht minim occur when Intensity of single-slit Frunhofer diffrction pttern or p sin u drk l 5 mp sin u drk 5 m l m 561, 62, 63, c in greement with Eqution Figure 38.6 represents plot of Eqution 38.2, nd Figure 38.6b is photogrph of single-slit Frunhofer diffrction pttern. Notice tht most of the light intensity is concentrted in the centrl bright fringe. Intensity of Two-Slit Diffrction Ptterns When more thn one slit is present, we must consider not only diffrction ptterns due to the individul slits but lso the interference ptterns due to the wves coming Condition for intensity minim for single slit I I mx p 3p 2p p p 2p 3p l sin u From M. Cgnet, M. Frnçon, nd J. C. Thrierr, Atls of Opticl Phenomen, Berlin, Springer-Verlg, 1962, plte 18 b A minimum in the curve in corresponds to drk fringe in b. Figure 38.6 () A plot of light intensity I versus (p/l) sin u for the single-slit Frunhofer diffrction pttern. (b) Photogrph of singleslit Frunhofer diffrction pttern.

6 1116 CHAPTER 38 Diffrction Ptterns nd Polriztion ACTIVE FIGURE 38.7 The combined effects of two-slit nd single-slit interference. This pttern is produced when 650-nm light wves pss through two 3.0-mm slits tht re 18 mm prt. Interference fringes I The diffrction pttern cts s n envelope (the blue dshed curve) tht controls the intensity of the regulrly spced interference mxim. Diffrction minim 3p 2p p p 2p 3p p sin u l from different slits. Notice the curved dshed lines in Figure 37.7 in Chpter 37, which indicte decrese in intensity of the interference mxim s u increses. This decrese is due to diffrction pttern. The interference ptterns in tht figure re locted entirely within the centrl bright fringe of the diffrction pttern, so the only hint of the diffrction pttern we see is the flloff in intensity towrd the outside of the pttern. To determine the effects of both two-slit interference nd single-slit diffrction pttern from ech slit from wider viewpoint thn tht in Figure 37.7, we combine Equtions nd 38.2: pd sin u sin 1p 2 sin u/l2 I 5 I mx cos 2 bc d (38.3) l p sin u/l Although this expression looks complicted, it merely represents the single-slit diffrction pttern (the fctor in squre brckets) cting s n envelope for twoslit interference pttern (the cosine-squred fctor) s shown in Active Figure The broken blue curve in Active Figure 38.7 represents the fctor in squre brckets in Eqution The cosine-squred fctor by itself would give series of peks ll with the sme height s the highest pek of the red-brown curve in Active Figure Becuse of the effect of the squre-brcket fctor, however, these peks vry in height s shown. Eqution 37.2 indictes the conditions for interference mxim s d sin u 5 ml, where d is the distnce between the two slits. Eqution 38.1 specifies tht the first diffrction minimum occurs when sin u 5 l, where is the slit width. Dividing Eqution 37.2 by Eqution 38.1 (with m 5 1) llows us to determine which interference mximum coincides with the first diffrction minimum: d sin u sin u 5 ml l d 5 m (38.4) In Active Figure 38.7, d/ 5 18 mm/3.0 mm 5 6. Therefore, the sixth interference mximum (if we count the centrl mximum s m 5 0) is ligned with the first diffrction minimum nd is drk.

7 38.3 Resolution of Single-Slit nd Circulr Apertures 1117 Quick Quiz 38.2 Consider the centrl pek in the diffrction envelope in Active Figure Suppose the wvelength of the light is chnged to 450 nm. Wht hppens to this centrl pek? () The width of the pek decreses, nd the number of interference fringes it encloses decreses. (b) The width of the pek decreses, nd the number of interference fringes it encloses increses. (c) The width of the pek decreses, nd the number of interference fringes it encloses remins the sme. (d) The width of the pek increses, nd the number of interference fringes it encloses decreses. (e) The width of the pek increses, nd the number of interference fringes it encloses increses. (f) The width of the pek increses, nd the number of interference fringes it encloses remins the sme. (g) The width of the pek remins the sme, nd the number of interference fringes it encloses decreses. (h) The width of the pek remins the sme, nd the number of interference fringes it encloses increses. (i) The width of the pek remins the sme, nd the number of interference fringes it encloses remins the sme Resolution of Single-Slit nd Circulr Apertures The bility of opticl systems to distinguish between closely spced objects is limited becuse of the wve nture of light. To understnd this limittion, consider Figure 38.8, which shows two light sources fr from nrrow slit of width. The sources cn be two noncoherent point sources S 1 nd S 2 ; for exmple, they could be two distnt strs. If no interference occurred between light pssing through different prts of the slit, two distinct bright spots (or imges) would be observed on the viewing screen. Becuse of such interference, however, ech source is imged s bright centrl region flnked by weker bright nd drk fringes, diffrction pttern. Wht is observed on the screen is the sum of two diffrction ptterns: one from S 1 nd the other from S 2. If the two sources re fr enough prt to keep their centrl mxim from overlpping s in Figure 38.8, their imges cn be distinguished nd re sid to be resolved. If the sources re close together s in Figure 38.8b, however, the two centrl mxim overlp nd the imges re not resolved. To determine whether two imges re resolved, the following condition is often used: When the centrl mximum of one imge flls on the first minimum of nother imge, the imges re sid to be just resolved. This limiting condition of resolution is known s Ryleigh s criterion. The ngle subtended by the sources t the slit is lrge enough for the diffrction ptterns to be distinguishble. The ngle subtended by the sources is so smll tht their diffrction ptterns overlp, nd the imges re not well resolved. S 1 S 1 S 2 u S 2 u Slit Viewing screen Slit Viewing screen b Figure 38.8 Two point sources fr from nrrow slit ech produce diffrction pttern. () The sources re seprted by lrge ngle. (b) The sources re seprted by smll ngle. (Notice tht the ngles re gretly exggerted. The drwing is not to scle.)

8 1118 CHAPTER 38 Diffrction Ptterns nd Polriztion Figure 38.9 Individul diffrction ptterns of two point sources (solid curves) nd the resultnt ptterns (dshed curves) for vrious ngulr seprtions of the sources s the light psses through circulr perture. In ech cse, the dshed curve is the sum of the two solid curves. The sources re closer together such tht the ngulr seprtion stisfies Ryleigh s criterion, nd the ptterns re just resolved. The sources re fr prt, nd the ptterns re well resolved. The sources re so close together tht the ptterns re not resolved. From M. Cgnet, M. Frnçon, nd J. C. Thrierr, Atls of Opticl Phenomen, Berlin, Springer-Verlg, 1962, plte 16 b c From Ryleigh s criterion, we cn determine the minimum ngulr seprtion u min subtended by the sources t the slit in Figure 38.8 for which the imges re just resolved. Eqution 38.1 indictes tht the first minimum in single-slit diffrction pttern occurs t the ngle for which sin u5 l where is the width of the slit. According to Ryleigh s criterion, this expression gives the smllest ngulr seprtion for which the two imges re resolved. Becuse l,, in most situtions, sin u is smll nd we cn use the pproximtion sin u < u. Therefore, the limiting ngle of resolution for slit of width is u min 5 l (38.5) where u min is expressed in rdins. Hence, the ngle subtended by the two sources t the slit must be greter thn l/ if the imges re to be resolved. Mny opticl systems use circulr pertures rther thn slits. The diffrction pttern of circulr perture s shown in the photogrphs of Figure 38.9 consists of centrl circulr bright disk surrounded by progressively finter bright nd drk rings. Figure 38.9 shows diffrction ptterns for three situtions in which light from two point sources psses through circulr perture. When the sources re fr prt, their imges re well resolved (Fig. 38.9). When the ngulr seprtion of the sources stisfies Ryleigh s criterion, the imges re just resolved (Fig. 38.9b). Finlly, when the sources re close together, the imges re sid to be unresolved (Fig. 38.9c) nd the pttern looks like tht of single source. Anlysis shows tht the limiting ngle of resolution of the circulr perture is Limiting ngle of resolution for circulr perture u min l (38.6) D where D is the dimeter of the perture. This expression is similr to Eqution 38.5 except for the fctor 1.22, which rises from mthemticl nlysis of diffrction from the circulr perture. Quick Quiz 38.3 Ct s eyes hve pupils tht cn be modeled s verticl slits. At night, would cts be more successful in resolving () hedlights on distnt cr or (b) verticlly seprted lights on the mst of distnt bot?

9 38.3 Resolution of Single-Slit nd Circulr Apertures 1119 Quick Quiz 38.4 Suppose you re observing binry str with telescope nd re hving difficulty resolving the two strs. You decide to use colored filter to mximize the resolution. (A filter of given color trnsmits only tht color of light.) Wht color filter should you choose? () blue (b) green (c) yellow (d) red Exmple 38.2 Resolution of the Eye Light of wvelength 500 nm, ner the center of the visible spectrum, enters humn eye. Although pupil dimeter vries from person to person, let s estimte dytime dimeter of 2 mm. (A) Estimte the limiting ngle of resolution for this eye, ssuming its resolution is limited only by diffrction. SOLUTION Conceptulize In Figure 38.9, identify the perture through which the light trvels s the pupil of the eye. Light pssing through this smll perture cuses diffrction ptterns to occur on the retin. Ctegorize We determine the result using equtions developed in this section, so we ctegorize this exmple s substitution problem. Use Eqution 38.6, tking l nm nd D 5 2 mm: u min l D m m b rd < 1 min of rc (B) Determine the minimum seprtion distnce d between two point sources tht the eye cn distinguish if the point sources re distnce L 5 25 cm from the observer (Fig ). SOLUTION Figure (Exmple 38.2) Two point sources seprted by distnce d s observed by the eye. d S 1 S 2 u min L Noting tht u min is smll, find d: sin u min < u min < d L S d 5 Lu min Substitute numericl vlues: This result is pproximtely equl to the thickness of humn hir. d 5 (25 cm)( rd) cm Exmple 38.3 Resolution of Telescope Ech of the two telescopes t the Keck Observtory on the dormnt Mun Ke volcno in Hwii hs n effective dimeter of 10 m. Wht is its limiting ngle of resolution for 600-nm light? SOLUTION Conceptulize In Figure 38.9, identify the perture through which the light trvels s the opening of the telescope. Light pssing through this perture cuses diffrction ptterns to occur in the finl imge. Ctegorize We determine the result using equtions developed in this section, so we ctegorize this exmple s substitution problem. Use Eqution 38.6, tking l m nd D 5 10 m: u min l D m b 10 m rd < s of rc Any two strs tht subtend n ngle greter thn or equl to this vlue re resolved (if tmospheric conditions re idel). continued

10 1120 CHAPTER 38 Diffrction Ptterns nd Polriztion 38.3 cont. WHAT IF? Wht if we consider rdio telescopes? They re much lrger in dimeter thn opticl telescopes, but do they hve better ngulr resolutions thn opticl telescopes? For exmple, the rdio telescope t Arecibo, Puerto Rico, hs dimeter of 305 m nd is designed to detect rdio wves of 0.75-m wvelength. How does its resolution compre with tht of one of the Keck telescopes? Answer The increse in dimeter might suggest tht rdio telescopes would hve better resolution thn Keck telescope, but Eqution 38.6 shows tht u min depends on both dimeter nd wvelength. Clculting the minimum ngle of resolution for the rdio telescope, we find u min l D m 305 m b rd < 10 min of rc This limiting ngle of resolution is mesured in minutes of rc rther thn the seconds of rc for the opticl telescope. Therefore, the chnge in wvelength more thn compenstes for the increse in dimeter. The limiting ngle of resolution for the Arecibo rdio telescope is more thn times lrger (tht is, worse) thn the Keck minimum. A telescope such s the one discussed in Exmple 38.3 cn never rech its diffrction limit becuse the limiting ngle of resolution is lwys set by tmospheric blurring t opticl wvelengths. This seeing limit is usully bout 1 s of rc nd is never smller thn bout 0.1 s of rc. The tmospheric blurring is cused by vritions in index of refrction with temperture vritions in the ir. This blurring is one reson for the superiority of photogrphs from the Hubble Spce Telescope, which views celestil objects from n orbitl position bove the tmosphere. As n exmple of the effects of tmospheric blurring, consider telescopic imges of Pluto nd its moon, Chron. Figure 38.11, n imge tken in 1978, represents the discovery of Chron. In this photogrph, tken from n Erth-bsed telescope, tmospheric turbulence cuses the imge of Chron to pper only s bump on the edge of Pluto. In comprison, Figure 38.11b shows photogrph tken from the Hubble Spce Telescope. Without the problems of tmospheric turbulence, Pluto nd its moon re clerly resolved The Diffrction Grting The diffrction grting, useful device for nlyzing light sources, consists of lrge number of eqully spced prllel slits. A trnsmission grting cn be mde by cutting prllel grooves on glss plte with precision ruling mchine. The spces between the grooves re trnsprent to the light nd hence ct s seprte slits. A reflection grting cn be mde by cutting prllel grooves on the surfce of Figure () The photogrph on which Chron, the moon of Pluto, ws discovered in From n Erth-bsed telescope, tmospheric blurring results in Chron ppering only s subtle bump on the edge of Pluto. (b) A Hubble Spce Telescope photo of Pluto nd Chron, clerly resolving the two objects. Courtesy U.S. Nvl Observtory/Jmes W. Christy Chron Pluto Dr. R. Albrecht, ESA/ESO Spce Telescope Europen Coordinting Fcility; NASA b

11 38.4 The Diffrction Grting 1121 Incoming plne wve of light P First-order mximum (m 1) Figure Side view of diffrction grting. The slit seprtion is d, nd the pth difference between djcent slits is d sin u. Centrl or zeroth-order mximum (m 0) Diffrction grting P First-order mximum (m 1) u d u d d sin u reflective mteril. The reflection of light from the spces between the grooves is speculr, nd the reflection from the grooves cut into the mteril is diffuse. Therefore, the spces between the grooves ct s prllel sources of reflected light like the slits in trnsmission grting. Current technology cn produce grtings tht hve very smll slit spcings. For exmple, typicl grting ruled with grooves/cm hs slit spcing d 5 (1/5 000) cm cm. A section of diffrction grting is illustrted in Figure A plne wve is incident from the left, norml to the plne of the grting. The pttern observed on the screen fr to the right of the grting is the result of the combined effects of interference nd diffrction. Ech slit produces diffrction, nd the diffrcted bems interfere with one nother to produce the finl pttern. The wves from ll slits re in phse s they leve the slits. For n rbitrry direction u mesured from the horizontl, however, the wves must trvel different pth lengths before reching the screen. Notice in Figure tht the pth difference d between rys from ny two djcent slits is equl to d sin u. If this pth difference equls one wvelength or some integrl multiple of wvelength, wves from ll slits re in phse t the screen nd bright fringe is observed. Therefore, the condition for mxim in the interference pttern t the ngle u bright is d sin u bright 5 ml m 5 0, 61, 62, 63,... (38.7) We cn use this expression to clculte the wvelength if we know the grting spcing d nd the ngle u bright. If the incident rdition contins severl wvelengths, the mth-order mximum for ech wvelength occurs t specific ngle. All wvelengths re seen t u 5 0, corresponding to m 5 0, the zeroth-order mximum. The first-order mximum (m 5 1) is observed t n ngle tht stisfies the reltionship sin u bright 5 l/d, the second-order mximum (m 5 2) is observed t lrger ngle u bright, nd so on. For the smll vlues of d typicl in diffrction grting, the ngles u bright re lrge, s we see in Exmple The intensity distribution for diffrction grting obtined with the use of monochromtic source is shown in Active Figure Notice the shrpness of the principl mxim nd the brodness of the drk res compred with the brod bright fringes chrcteristic of the two-slit interference pttern (see Fig. 37.6). You should lso review Figure 37.7, which shows tht the width of the intensity mxim decreses s the number of slits increses. Becuse the principl mxim re so shrp, they re much brighter thn two-slit interference mxim. Quick Quiz 38.5 Ultrviolet light of wvelength 350 nm is incident on diffrction grting with slit spcing d nd forms n interference pttern on screen distnce L wy. The ngulr positions u bright of the interference Pitfll Prevention 38.3 A Diffrction Grting Is n Interference Grting As with diffrction pttern, diffrction grting is misnomer, but is deeply entrenched in the lnguge of physics. The diffrction grting depends on diffrction in the sme wy s the double slit, spreding the light so tht light from different slits cn interfere. It would be more correct to cll it n interference grting, but diffrction grting is the nme in use. Condition for interference mxim for grting m l d l d 0 l d 2l d ACTIVE FIGURE sin u Intensity versus sin u for diffrction grting. The zeroth-, first-, nd second-order mxim re shown.

12 1122 CHAPTER 38 Diffrction Ptterns nd Polriztion mxim re lrge. The loctions of the bright fringes re mrked on the screen. Now red light of wvelength 700 nm is used with diffrction grting to form nother diffrction pttern on the screen. Will the bright fringes of this pttern be locted t the mrks on the screen if () the screen is moved to distnce 2L from the grting, (b) the screen is moved to distnce L/2 from the grting, (c) the grting is replced with one of slit spcing 2d, (d) the grting is replced with one of slit spcing d/2, or (e) nothing is chnged? Conceptul Exmple 38.4 A Compct Disc Is Diffrction Grting Light reflected from the surfce of compct disc is multicolored s shown in Figure The colors nd their intensities depend on the orienttion of the CD reltive to the eye nd reltive to the light source. Explin how tht works. SOLUTION The surfce of CD hs spirl grooved trck (with djcent grooves hving seprtion on the order of 1 mm). Therefore, the surfce cts s reflection grting. The light reflecting Figure (Conceptul Exmple 38.4) A compct disc observed under white light. The colors observed in the reflected light nd their intensities depend on the orienttion of the CD reltive to the eye nd reltive to the light source. from the regions between these closely spced grooves interferes constructively only in certin directions tht depend on the wvelength nd the direction of the incident light. Any section of the CD serves s diffrction grting for white light, sending different colors in different directions. The different colors you see upon viewing one section chnge when the light source, the CD, or you chnge position. This chnge in position cuses the ngle of incidence or the ngle of the diffrcted light to be ltered. Kristen Brochmnn/Fundmentl Photogrphs, NYC Exmple 38.5 The Orders of Diffrction Grting Monochromtic light from helium neon lser (l nm) is incident normlly on diffrction grting contining grooves per centimeter. Find the ngles t which the first- nd second-order mxim re observed. SOLUTION Conceptulize Study Figure nd imgine tht the light coming from the left origintes from the helium neon lser. Let s evlute the possible vlues of the ngle u. Ctegorize We determine results using equtions developed in this section, so we ctegorize this exmple s substitution problem. Clculte the slit seprtion s the inverse of the number of grooves per centimeter: d cm cm nm Solve Eqution 38.7 for sin u nd substitute numericl vlues for the first-order mximum (m 5 1) to find u 1 : Repet for the second-order mximum (m 5 2): sin u l d sin u l d 5 u u nm nm nm nm WHAT IF? Wht if you looked for the third-order mximum? Would you find it?

13 38.4 The Diffrction Grting cont. Answer For m 5 3, we find sin u Becuse sin u cnnot exceed unity, this result does not represent relistic solution. Hence, only zeroth-, first-, nd second-order mxim cn be observed for this sitution. Applictions of Diffrction Grtings A schemtic drwing of simple pprtus used to mesure ngles in diffrction pttern is shown in Active Figure This pprtus is diffrction grting spectrometer. The light to be nlyzed psses through slit, nd collimted bem of light is incident on the grting. The diffrcted light leves the grting t ngles tht stisfy Eqution 38.7, nd telescope is used to view the imge of the slit. The wvelength cn be determined by mesuring the precise ngles t which the imges of the slit pper for the vrious orders. The spectrometer is useful tool in tomic spectroscopy, in which the light from n tom is nlyzed to find the wvelength components. These wvelength components cn be used to identify the tom. We shll investigte tomic spectr in Chpter 42 of the extended version of this text. Another ppliction of diffrction grtings is the grting light vlve (GLV), which competes in some video disply pplictions with the digitl micromirror devices (DMDs) discussed in Section A GLV is silicon microchip fitted with n rry of prllel silicon nitride ribbons coted with thin lyer of luminum (Fig ). Ech ribbon is pproximtely 20 mm long nd 5 mm wide nd is seprted from the silicon substrte by n ir gp on the order of 100 nm. With no voltge pplied, ll ribbons re t the sme level. In this sitution, the rry of ribbons cts s flt surfce, speculrly reflecting incident light. When voltge is pplied between ribbon nd the electrode on the silicon substrte, n electric force pulls the ribbon downwrd, closer to the substrte. Alternte ribbons cn be pulled down, while those in between remin in n elevted configurtion. As result, the rry of ribbons cts s diffrction grting such tht the constructive interference for prticulr wvelength of light cn be directed towrd screen or other opticl disply system. If one uses three such devices one ech for red, blue, nd green light full-color disply is possible. In ddition to its use in video disply, the GLV hs found pplictions in lser opticl nvigtion sensor technology, computer-to-plte commercil printing, nd other types of imging. Another interesting ppliction of diffrction grtings is hologrphy, the production of three-dimensionl imges of objects. The physics of hologrphy ws Source Collimtor Slit u Telescope Courtesy Silicon Light Mchines ACTIVE FIGURE Grting Digrm of diffrction grting spectrometer. The collimted bem incident on the grting is spred into its vrious wvelength components with constructive interference for prticulr wvelength occurring t the ngles u bright tht stisfy the eqution d sin u bright 5 ml, where m 5 0, 61, 62,.... Figure A smll portion of grting light vlve. The lternting reflective ribbons t different levels ct s diffrction grting, offering very high-speed control of the direction of light towrd digitl disply device.

14 1124 CHAPTER 38 Diffrction Ptterns nd Polriztion Figure In this hologrm, circuit bord is shown from two different views. Notice the difference in the ppernce of the mesuring tpe nd the view through the mgnifying lens in () nd (b). Photo by Ronld R. Erickson; hologrm by Nicklus Phillips b developed by Dennis Gbor ( ) in 1948 nd resulted in the Nobel Prize in Physics for Gbor in The requirement of coherent light for hologrphy delyed the reliztion of hologrphic imges from Gbor s work until the development of lsers in the 1960s. Figure shows single hologrm viewed from two different positions nd the three-dimensionl chrcter of its imge. Notice in prticulr the difference in the view through the mgnifying glss in Figures nd 38.17b. Figure shows how hologrm is mde. Light from the lser is split into two prts by hlf-silvered mirror t B. One prt of the bem reflects off the object to be photogrphed nd strikes n ordinry photogrphic film. The other hlf of the bem is diverged by lens L 2, reflects from mirrors M 1 nd M 2, nd finlly strikes the film. The two bems overlp to form n extremely complicted interference pttern on the film. Such n interference pttern cn be produced only if the phse reltionship of the two wves is constnt throughout the exposure of the film. This condition is met by illuminting the scene with light coming through pinhole or with coherent lser rdition. The hologrm records not only the intensity of the light scttered from the object (s in conventionl photogrph), but lso the phse difference between the reference bem nd the bem scttered from the object. Becuse of this phse difference, n interference pttern is formed tht produces n imge in which ll three-dimensionl informtion vilble from the perspective of ny point on the hologrm is preserved. In norml photogrphic imge, lens is used to focus the imge so tht ech point on the object corresponds to single point on the film. Notice tht there is no lens used in Figure to focus the light onto the film. Therefore, light from ech point on the object reches ll points on the film. As result, ech region of the photogrphic film on which the hologrm is recorded contins informtion bout ll illuminted points on the object, which leds to remrkble result: if smll section of the hologrm is cut from the film, the complete imge cn be formed from the smll piece! (The qulity of the imge is reduced, but the entire imge is present.) These light rys trvel to the film without striking the object. These light rys strike the object nd then trvel to the film. M 2 M 1 L 2 Figure Experimentl rrngement for producing hologrm. Lser B L 1 Film

15 38.5 Diffrction of X-Rys by Crystls 1125 If the m 1 rys re extended bckwrd, virtul imge of the object photogrphed in the hologrm exists on the front side of the hologrm. Figure Two light rys strike hologrm t norml incidence. For ech ry, outgoing rys corresponding to m 5 0 nd m 5 61 re shown. Virtul imge Incoming light ry Hologrm m 0 m 1 m 1 m 1 Rel imge Incoming light ry m 0 m 1 A hologrm is best viewed by llowing coherent light to pss through the developed film s one looks bck long the direction from which the bem comes. The interference pttern on the film cts s diffrction grting. Figure shows two rys of light striking nd pssing through the film. For ech ry, the m 5 0 nd m 5 61 rys in the diffrction pttern re shown emerging from the right side of the film. The m 5 11 rys converge to form rel imge of the scene, which is not the imge tht is normlly viewed. By extending the light rys corresponding to m 5 21 behind the film, we see tht there is virtul imge locted there, with light coming from it in exctly the sme wy tht light cme from the ctul object when the film ws exposed. This imge is wht one sees when looking through the hologrphic film. Hologrms re finding number of pplictions. You my hve hologrm on your credit crd. This specil type of hologrm is clled rinbow hologrm nd is designed to be viewed in reflected white light Diffrction of X-Rys by Crystls In principle, the wvelength of ny electromgnetic wve cn be determined if grting of the proper spcing (on the order of l) is vilble. X-rys, discovered by Wilhelm Roentgen ( ) in 1895, re electromgnetic wves of very short wvelength (on the order of 0.1 nm). It would be impossible to construct grting hving such smll spcing by the cutting process described t the beginning of Section The tomic spcing in solid is known to be bout 0.1 nm, however. In 1913, Mx von Lue ( ) suggested tht the regulr rry of toms in crystl could ct s three-dimensionl diffrction grting for x-rys. Subsequent experiments confirmed this prediction. The diffrction ptterns from crystls re complex becuse of the three-dimensionl nture of the crystl structure. Nevertheless, x-ry diffrction hs proved to be n invluble technique for elucidting these structures nd for understnding the structure of mtter. Figure shows one experimentl rrngement for observing x-ry diffrction from crystl. A collimted bem of monochromtic x-rys is incident on Photogrphic film Crystl X-ry source X-ry bem Figure Schemtic digrm of the technique used to observe the diffrction of x-rys by crystl. The rry of spots formed on the film is clled Lue pttern.

16 1126 CHAPTER 38 Diffrction Ptterns nd Polriztion Figure () A Lue pttern of single crystl of the minerl beryl (beryllium luminum silicte). Ech dot represents point of constructive interference. (b) A Lue pttern of the enzyme Rubisco, produced with widebnd x-ry spectrum. This enzyme is present in plnts nd tkes prt in the process of photosynthesis. The Lue pttern is used to determine the crystl structure of Rubisco. Used with permission of Estmn Kodk Compny I. Andersson Oxford Moleculr Biophysics Lbortory/ Science Photo Librry/Photo Reserchers, Inc. b Pitfll Prevention 38.4 Different Angles Notice in Figure tht the ngle u is mesured from the reflecting surfce rther thn from the norml s in the cse of the lw of reflection in Chpter 35. With slits nd diffrction grtings, we lso mesured the ngle u from the norml to the rry of slits. Becuse of historicl trdition, the ngle is mesured differently in Brgg diffrction, so interpret Eqution 38.8 with cre. Brgg s lw The blue spheres represent Cl ions, nd the red spheres represent N ions. crystl. The diffrcted bems re very intense in certin directions, corresponding to constructive interference from wves reflected from lyers of toms in the crystl. The diffrcted bems, which cn be detected by photogrphic film, form n rry of spots known s Lue pttern s in Figure One cn deduce the crystlline structure by nlyzing the positions nd intensities of the vrious spots in the pttern. Figure 38.21b shows Lue pttern from crystlline enzyme, using wide rnge of wvelengths so tht swirling pttern results. The rrngement of toms in crystl of sodium chloride (NCl) is shown in Figure Ech unit cell (the geometric solid tht repets throughout the crystl) is cube hving n edge length. A creful exmintion of the NCl structure shows tht the ions lie in discrete plnes (the shded res in Fig ). Now suppose n incident x-ry bem mkes n ngle u with one of the plnes s in Figure The bem cn be reflected from both the upper plne nd the lower one, but the bem reflected from the lower plne trvels frther thn the bem reflected from the upper plne. The effective pth difference is 2d sin u. The two bems reinforce ech other (constructive interference) when this pth difference equls some integer multiple of l. The sme is true for reflection from the entire fmily of prllel plnes. Hence, the condition for constructive interference (mxim in the reflected bem) is 2d sin u 5 ml m 5 1, 2, 3,... (38.8) This condition is known s Brgg s lw, fter W. L. Brgg ( ), who first derived the reltionship. If the wvelength nd diffrction ngle re mesured, Eqution 38.8 cn be used to clculte the spcing between tomic plnes. Incident bem The incident bem cn reflect from different plnes of toms. Reflected bem Figure Crystlline structure of sodium chloride (NCl). The length of the cube edge is nm. Upper plne Lower plne u d sin u u Figure A two-dimensionl description of the reflection of n x-ry bem from two prllel crystlline plnes seprted by distnce d. The bem reflected from the lower plne trvels frther thn the bem reflected from the upper plne by distnce 2d sin u. u d

17 38.6 Polriztion of Light Wves Polriztion of Light Wves In Chpter 34, we described the trnsverse nture of light nd ll other electromgnetic wves. Polriztion, discussed in this section, is firm evidence of this trnsverse nture. An ordinry bem of light consists of lrge number of wves emitted by the toms of the light source. Ech tom produces wve hving some prticulr orient tion of the electric field vector E S, corresponding to the direction of tomic vibrtion. The direction of polriztion of ech individul wve is defined to be the direction in which the electric field is vibrting. In Figure 38.24, this direction hppens to lie long the y xis. All individul electromgnetic wves trveling in the x direction hve n E S vector prllel to the yz plne, but this vector could be t ny possible ngle with respect to the y xis. Becuse ll directions of vibrtion from wve source re possible, the resultnt electromgnetic wve is superposition of wves vibrting in mny different directions. The result is n unpolrized light bem, represented in Figure The direction of wve propgtion in this figure is perpendiculr to the pge. The rrows show few possible directions of the electric field vectors for the individul wves mking up the resultnt bem. At ny given point nd t some instnt of time, ll these individul electric field vectors dd to give one resultnt electric field vector. As noted in Section 34.3, wve is sid to be linerly polrized if the resultnt electric field E S vibrtes in the sme direction t ll times t prticulr point s shown in Figure 38.25b. (Sometimes, such wve is described s plne-polrized, or simply polrized.) The plne formed by E S nd the direction of propgtion is clled the plne of polriztion of the wve. If the wve in Figure represents the resultnt of ll individul wves, the plne of polriztion is the xy plne. A linerly polrized bem cn be obtined from n unpolrized bem by removing ll wves from the bem except those whose electric field vectors oscillte in single plne. We now discuss four processes for producing polrized light from unpolrized light. Polriztion by Selective Absorption The most common technique for producing polrized light is to use mteril tht trnsmits wves whose electric fields vibrte in plne prllel to certin direction nd tht bsorbs wves whose electric fields vibrte in ll other directions. In 1938, E. H. Lnd ( ) discovered mteril, which he clled Polroid, tht polrizes light through selective bsorption. This mteril is fbricted in thin sheets of long-chin hydrocrbons. The sheets re stretched during mnufcture so tht the long-chin molecules lign. After sheet is dipped into solution contining iodine, the molecules become good electricl conductors. Conduction tkes plce primrily long the hydrocrbon chins becuse electrons cn move esily only long the chins. If light whose electric field vector is prllel to the chins is incident on the mteril, the electric field ccelertes electrons long the chins nd energy is bsorbed from the rdition. Therefore, the light does not pss through the mteril. Light whose electric field vector is perpendiculr to the chins psses through the mteril becuse electrons cnnot move from one molecule to the next. As result, when unpolrized light is incident on the mteril, the exiting light is polrized perpendiculr to the moleculr chins. It is common to refer to the direction perpendiculr to the moleculr chins s the trnsmission xis. In n idel polrizer, ll light with E S prllel to the trnsmission xis is trnsmitted nd ll light with E S perpendiculr to the trnsmission xis is bsorbed. Active Figure (pge 1128) represents n unpolrized light bem incident on first polrizing sheet, clled the polrizer. Becuse the trnsmission xis is oriented verticlly in the figure, the light trnsmitted through this sheet is polrized verticlly. A second polrizing sheet, clled the nlyzer, intercepts the bem. In z S B y S E S c Figure Schemtic digrm of n electromgnetic wve propgting t velocity c S in the x direction. The electric field vibrtes in the xy plne, nd the mgnetic field vibrtes in the xz plne. The red dot signifies the velocity vector for the wve coming out of the pge. S E S E Figure () A representtion of n unpolrized light bem viewed long the direction of propgtion. The trnsverse electric field cn vibrte in ny direction in the plne of the pge with equl probbility. (b) A linerly polrized light bem with the electric field vibrting in the verticl direction. b x

18 1128 CHAPTER 38 Diffrction Ptterns nd Polriztion ACTIVE FIGURE Two polrizing sheets whose trnsmission xes mke n ngle u with ech other. Only frction of the polrized light incident on the nlyzer is trnsmitted through it. Unpolrized light The polrizer polrizes the incident light long its trnsmission xis. S E 0 The nlyzer llows the component of the light prllel to its xis to pss through. u Trnsmission xis Polrized light Mlus s lw Active Figure 38.26, the nlyzer trnsmission xis is set t n ngle u to the polrizer xis. We cll the electric field vector of the first trnsmitted bem S E0. The component of S E0 perpendiculr to the nlyzer xis is completely bsorbed. The component of S E0 prllel to the nlyzer xis, which is trnsmitted through the nlyzer, is E 0 cos u. Becuse the intensity of the trnsmitted bem vries s the squre of its mgnitude, we conclude tht the intensity I of the (polrized) bem trnsmitted through the nlyzer vries s I 5 I mx cos 2 u (38.9) where I mx is the intensity of the polrized bem incident on the nlyzer. This expression, known s Mlus s lw, 2 pplies to ny two polrizing mterils whose trnsmission xes re t n ngle u to ech other. This expression shows tht the intensity of the trnsmitted bem is mximum when the trnsmission xes re prllel (u 5 0 or 180 ) nd is zero (complete bsorption by the nlyzer) when the trnsmission xes re perpendiculr to ech other. This vrition in trnsmitted intensity through pir of polrizing sheets is illustrted in Figure Becuse the verge vlue of cos 2 u is 1 2, the intensity of initilly unpolrized light is reduced by fctor of one-hlf s the light psses through single idel polrizer. Polriztion by Reflection When n unpolrized light bem is reflected from surfce, the polriztion of the reflected light depends on the ngle of incidence. If the ngle of incidence is 0, the reflected bem is unpolrized. For other ngles of incidence, the reflected light is Figure The intensity of light trnsmitted through two polrizers depends on the reltive orienttion of their trnsmission xes. The red rrows indicte the trnsmission xes of the polrizers. The trnsmitted light hs mximum intensity when the trnsmission xes re ligned with ech other. The trnsmitted light hs lesser intensity when the trnsmission xes re t n ngle of 45 with ech other. The trnsmitted light intensity is minimum when the trnsmission xes re perpendiculr to ech other. Henry Lep nd Jim Lehmn b c 2 Nmed fter its discoverer, E. L. Mlus ( ). Mlus discovered tht reflected light ws polrized by viewing it through clcite (CCO 3 ) crystl.

19 38.6 Polriztion of Light Wves 1129 The dots represent electric field oscilltions prllel to the reflecting surfce nd perpendiculr to the pge. Incident bem u 1 u 1 The rrows represent electric field oscilltions perpendiculr to those represented by the dots. n 1 n 2 Reflected bem Electrons t the surfce oscillting in the direction of the reflected ry (perpendiculr to the dots nd prllel to the blue rrow) send no energy in this direction. Incident bem u p u p 90 n 1 n 2 Reflected bem Figure () When unpolrized light is incident on reflecting surfce, the reflected nd refrcted bems re prtilly polrized. (b) The reflected bem is completely polrized when the ngle of incidence equls the polrizing ngle u p, which stisfies the eqution n 2 /n 1 5 tn u p. At this incident ngle, the reflected nd refrcted rys re perpendiculr to ech other. u 2 u 2 Refrcted bem b Refrcted bem polrized to some extent, nd for one prticulr ngle of incidence, the reflected light is completely polrized. Let s now investigte reflection t tht specil ngle. Suppose n unpolrized light bem is incident on surfce s in Figure Ech individul electric field vector cn be resolved into two components: one prllel to the surfce (nd perpendiculr to the pge in Fig , represented by the dots) nd the other (represented by the ornge rrows) perpendiculr both to the first component nd to the direction of propgtion. Therefore, the polriztion of the entire bem cn be described by two electric field components in these directions. It is found tht the prllel component represented by the dots reflects more strongly thn the other component represented by the rrows, resulting in prtilly polrized reflected bem. Furthermore, the refrcted bem is lso prtilly polrized. Now suppose the ngle of incidence u 1 is vried until the ngle between the reflected nd refrcted bems is 90 s in Figure 38.28b. At this prticulr ngle of incidence, the reflected bem is completely polrized (with its electric field vector prllel to the surfce) nd the refrcted bem is still only prtilly polrized. The ngle of incidence t which this polriztion occurs is clled the polrizing ngle u p. We cn obtin n expression relting the polrizing ngle to the index of refrction of the reflecting substnce by using Figure 38.28b. From this figure, we see tht u p u ; therefore, u u p. Using Snell s lw of refrction (Eq. 35.8) gives n 2 5 sin u 1 5 sin u p n 1 sin u 2 sin u 2 Becuse sin u 2 5 sin (90 2 u p ) 5 cos u p, we cn write this expression s n 2 /n 1 5 sin u p /cos u p, which mens tht tn u p 5 n 2 (38.10) n 1 This expression is clled Brewster s lw, nd the polrizing ngle u p is sometimes clled Brewster s ngle, fter its discoverer, Dvid Brewster ( ). Becuse n vries with wvelength for given substnce, Brewster s ngle is lso function of wvelength. We cn understnd polriztion by reflection by imgining tht the electric field in the incident light sets electrons t the surfce of the mteril in Figure 38.28b Brewster s lw

20 1130 CHAPTER 38 Diffrction Ptterns nd Polriztion into oscilltion. The component directions of oscilltion re (1) prllel to the rrows shown on the refrcted bem of light nd therefore prllel to the reflected bem nd (2) perpendiculr to the pge. The oscillting electrons ct s dipole ntenns rditing light with polriztion prllel to the direction of oscilltion. Consult Figure 34.12, which shows the pttern of rdition from dipole ntenn. Notice tht there is no rdition t n ngle of u 5 0, tht is, long the oscilltion direction of the ntenn. Therefore, for the oscilltions in direction 1, there is no rdition in the direction long the reflected ry. For oscilltions in direction 2, the electrons rdite light with polriztion perpendiculr to the pge. Therefore, the light reflected from the surfce t this ngle is completely polrized prllel to the surfce. Polriztion by reflection is common phenomenon. Sunlight reflected from wter, glss, nd snow is prtilly polrized. If the surfce is horizontl, the electric field vector of the reflected light hs strong horizontl component. Sunglsses mde of polrizing mteril reduce the glre of reflected light. The trnsmission xes of such lenses re oriented verticlly so tht they bsorb the strong horizontl component of the reflected light. If you rotte sunglsses through 90, they re not s effective t blocking the glre from shiny horizontl surfces. These two rys re polrized in mutully perpendiculr directions. Unpolrized light Clcite E ry O ry Figure Unpolrized light incident t n ngle to the optic xis in clcite crystl splits into n ordinry (O) ry nd n extrordinry (E) ry (not to scle). The E nd O rys propgte with the sme velocity long the optic xis. Optic xis Figure A point source S inside double-refrcting crystl produces sphericl wve front corresponding to the ordinry (O) ry nd n ellipticl wve front corresponding to the extrordinry (E) ry. S E O Polriztion by Double Refrction Solids cn be clssified on the bsis of internl structure. Those in which the toms re rrnged in specific order re clled crystlline; the NCl structure of Figure is one exmple of crystlline solid. Those solids in which the toms re distributed rndomly re clled morphous. When light trvels through n morphous mteril such s glss, it trvels with speed tht is the sme in ll directions. Tht is, glss hs single index of refrction. In certin crystlline mterils such s clcite nd qurtz, however, the speed of light is not the sme in ll directions. In these mterils, the speed of light depends on the direction of propgtion nd on the plne of polriztion of the light. Such mterils re chrcterized by two indices of refrction. Hence, they re often referred to s double-refrcting or birefringent mterils. When unpolrized light enters birefringent mteril, it my split into n ordinry (O) ry nd n extrordinry (E) ry. These two rys hve mutully perpendiculr polriztions nd trvel t different speeds through the mteril. The two speeds correspond to two indices of refrction, n O for the ordinry ry nd n E for the extrordinry ry. There is one direction, clled the optic xis, long which the ordinry nd extrordinry rys hve the sme speed. If light enters birefringent mteril t n ngle to the optic xis, however, the different indices of refrction will cuse the two polrized rys to split nd trvel in different directions s shown in Figure The index of refrction n O for the ordinry ry is the sme in ll directions. If one could plce point source of light inside the crystl s in Figure 38.30, the ordinry wves would spred out from the source s spheres. The index of refrction n E vries with the direction of propgtion. A point source sends out n extrordinry wve hving wve fronts tht re ellipticl in cross section. The difference in speed for the two rys is mximum in the direction perpendiculr to the optic xis. For exmple, in clcite, n O t wvelength of nm nd n E vries from long the optic xis to perpendiculr to the optic xis. Vlues for n O nd the extreme vlue of n E for vrious double-refrcting crystls re given in Tble If you plce clcite crystl on sheet of pper nd then look through the crystl t ny writing on the pper, you would see two imges s shown in Figure As cn be seen from Figure 38.29, these two imges correspond to one formed by the ordinry ry nd one formed by the extrordinry ry. If the two imges re viewed through sheet of rotting polrizing glss, they lterntely pper nd dispper becuse the ordinry nd extrordinry rys re plne-polrized long mutully perpendiculr directions.

21 38.6 Polriztion of Light Wves 1131 TABLE 38.1 Indices of Refrction for Some Double- Refrcting Crystls t Wvelength of nm Crystl n O n E n O /n E Clcite (CCO 3 ) Qurtz (SiO 2 ) Sodium nitrte (NNO 3 ) Sodium sulfite (NSO 3 ) Zinc chloride (ZnCl 2 ) Zinc sulfide (ZnS) Some mterils such s glss nd plstic become birefringent when stressed. Suppose n unstressed piece of plstic is plced between polrizer nd n nlyzer so tht light psses from polrizer to plstic to nlyzer. When the plstic is unstressed nd the nlyzer xis is perpendiculr to the polrizer xis, none of the polrized light psses through the nlyzer. In other words, the unstressed plstic hs no effect on the light pssing through it. If the plstic is stressed, however, regions of gretest stress become birefringent nd the polriztion of the light pssing through the plstic chnges. Hence, series of bright nd drk bnds is observed in the trnsmitted light, with the bright bnds corresponding to regions of gretest stress. Engineers often use this technique, clled opticl stress nlysis, in designing structures rnging from bridges to smll tools. They build plstic model nd nlyze it under different lod conditions to determine regions of potentil wekness nd filure under stress. An exmple of plstic model under stress is shown in Figure Figure A clcite crystl produces double imge becuse it is birefringent (double-refrcting) mteril. Henry Lep nd Jim Lehmn Polriztion by Scttering When light is incident on ny mteril, the electrons in the mteril cn bsorb nd rerdite prt of the light. Such bsorption nd rerdition of light by electrons in the gs molecules tht mke up ir is wht cuses sunlight reching n observer on the Erth to be prtilly polrized. You cn observe this effect clled scttering by looking directly up t the sky through pir of sunglsses whose lenses re mde of polrizing mteril. Less light psses through t certin orienttions of the lenses thn t others. Figure (pge 1132) illustrtes how sunlight becomes polrized when it is scttered. The phenomenon is similr to tht creting completely polrized light upon reflection from surfce t Brewster s ngle. An unpolrized bem of sunlight trveling in the horizontl direction (prllel to the ground) strikes molecule of one of the gses tht mke up ir, setting the electrons of the molecule into vibrtion. These vibrting chrges ct like the vibrting chrges in n ntenn. The horizontl component of the electric field vector in the incident wve results in horizontl component of the vibrtion of the chrges, nd the verticl component Peter Aprhmin/Science Photo Librry, Photo Reserchers, Inc. Figure A plstic model of n rch structure under lod conditions. The pttern is produced when the plstic model is viewed between polrizer nd nlyzer oriented perpendiculr to ech other. Such ptterns re useful in the optiml design of rchitecturl components.

22 1132 CHAPTER 38 Diffrction Ptterns nd Polriztion The scttered light trveling perpendiculr to the incident light is plne-polrized becuse the verticl vibrtions of the chrges in the ir molecule send no light in this direction. Unpolrized light Air molecule Figure The scttering of unpolrized sunlight by ir molecules. of the vector results in verticl component of vibrtion. If the observer in Figure is looking stright up (perpendiculr to the originl direction of propgtion of the light), the verticl oscilltions of the chrges send no rdition towrd the observer. Therefore, the observer sees light tht is completely polrized in the horizontl direction s indicted by the ornge rrows. If the observer looks in other directions, the light is prtilly polrized in the horizontl direction. Vritions in the color of scttered light in the tmosphere cn be understood s follows. When light of vrious wvelengths l is incident on gs molecules of dimeter d, where d,, l, the reltive intensity of the scttered light vries s 1/l 4. The condition d,, l is stisfied for scttering from oxygen (O 2 ) nd nitrogen (N 2 ) molecules in the tmosphere, whose dimeters re bout 0.2 nm. Hence, short wvelengths (violet light) re scttered more efficiently thn long wvelengths (red light). Therefore, when sunlight is scttered by gs molecules in the ir, the shortwvelength rdition (violet) is scttered more intensely thn the long-wvelength rdition (red). When you look up into the sky in direction tht is not towrd the Sun, you see the scttered light, which is predominntly violet. Your eyes, however, re not very sensitive to violet light. Light of the next color in the spectrum, blue, is scttered with less intensity thn violet, but your eyes re fr more sensitive to blue light thn to violet light. Hence, you see blue sky. If you look towrd the west t sunset (or towrd the est t sunrise), you re looking in direction towrd the Sun nd re seeing light tht hs pssed through lrge distnce of ir. Most of the blue light hs been scttered by the ir between you nd the Sun. The light tht survives this trip through the ir to you hs hd much of its blue component scttered nd is therefore hevily weighted towrd the red end of the spectrum; s result, you see the red nd ornge colors of sunset (or sunrise). Opticl Activity Mny importnt pplictions of polrized light involve mterils tht disply opticl ctivity. A mteril is sid to be opticlly ctive if it rottes the plne of polriztion of ny light trnsmitted through the mteril. The ngle through which the light is rotted by specific mteril depends on the length of the pth through the mteril nd on concentrtion if the mteril is in solution. One opticlly ctive mteril is solution of the common sugr dextrose. A stndrd method for determining the concentrtion of sugr solutions is to mesure the rottion produced by fixed length of the solution. Moleculr symmetry determines whether mteril is opticlly ctive. For exmple, some proteins re opticlly ctive becuse of their spirl shpe. The liquid crystl displys found in most clcultors hve their opticl ctivity chnged by the ppliction of electric potentil cross different prts of the disply. Try using pir of polrizing sunglsses to investigte the polriztion used in the disply of your clcultor. Quick Quiz 38.6 A polrizer for microwves cn be mde s grid of prllel metl wires pproximtely 1 cm prt. Is the electric field vector for microwves trnsmitted through this polrizer () prllel or (b) perpendiculr to the metl wires? Quick Quiz 38.7 You re wlking down long hllwy tht hs mny light fixtures in the ceiling nd very shiny, newly wxed floor. When looking t the floor, you see reflections of every light fixture. Now you put on sunglsses tht re polrized. Some of the reflections of the light fixtures cn no longer be seen. (Try it!) Are the reflections tht dispper those () nerest to you, (b) frthest from you, or (c) t n intermedite distnce from you?

23 Objective Questions 1133 Concepts nd Principles Summry Diffrction is the devition of light from stright-line pth when the light psses through n perture or round n obstcle. Diffrction is due to the wve nture of light. The Frunhofer diffrction pttern produced by single slit of width on distnt screen consists of centrl bright fringe nd lternting bright nd drk fringes of much lower intensities. The ngles u drk t which the diffrction pttern hs zero intensity, corresponding to destructive interference, re given by sin u drk 5 m l m 561, 62, 63, c (38.1) Ryleigh s criterion, which is limiting condition of resolution, sttes tht two imges formed by n perture re just distinguishble if the centrl mximum of the diffrction pttern for one imge flls on the first minimum of the diffrction pttern for the other imge. The limiting ngle of resolution for slit of width is u min 5 l/, nd the limiting ngle of resolution for circulr perture of dimeter D is given by u min l/D. A diffrction grting consists of lrge number of eqully spced, identicl slits. The condition for intensity mxim in the interference pttern of diffrction grting for norml incidence is d sin u bright 5 ml m 5 0, 61, 62, 63,... (38.7) where d is the spcing between djcent slits nd m is the order number of the intensity mximum. When polrized light of intensity I mx is emitted by polrizer nd then is incident on n nlyzer, the light trnsmitted through the nlyzer hs n intensity equl to I mx cos 2 u, where u is the ngle between the polrizer nd nlyzer trnsmission xes. In generl, reflected light is prtilly polrized. Reflected light, however, is completely polrized when the ngle of incidence is such tht the ngle between the reflected nd refrcted bems is 90. This ngle of incidence, clled the polrizing ngle u p, stisfies Brewster s lw: tn u p 5 n 2 n 1 (38.10) where n 1 is the index of refrction of the medium in which the light initilly trvels nd n 2 is the index of refrction of the reflecting medium. Objective Questions denotes nswer vilble in Student Solutions Mnul/Study Guide 1. Wht combintion of opticl phenomen cuses the bright colored ptterns sometimes seen on wet streets covered with lyer of oil? Choose the best nswer. () diffrction nd polriztion (b) interference nd diffrction (c) polriztion nd reflection (d) refrction nd diffrction (e) reflection nd interference 2. Wht is most likely to hppen to bem of light when it reflects from shiny metllic surfce t n rbitrry ngle? Choose the best nswer. () It is totlly bsorbed by the surfce. (b) It is totlly polrized. (c) It is unpolrized. (d) It is prtilly polrized. (e) More informtion is required. 3. If plne polrized light is sent through two polrizers, the first t 45 to the originl plne of polriztion nd the second t 90 to the originl plne of polriztion, wht frction of the originl polrized intensity psses through the lst polrizer? () 0 (b) 1 4 (c) 1 2 (d) 1 8 (e) A Frunhofer diffrction pttern is produced on screen locted 1.00 m from single slit. If light source of wvelength m is used nd the distnce from the center of the centrl bright fringe to the first drk fringe is m, wht is the slit width? () mm (b) mm (c) mm (d) 1.00 mm (e) mm 5. Consider wve pssing through single slit. Wht hppens to the width of the centrl mximum of its diffrction pttern s the slit is mde hlf s wide? () It becomes one-fourth s wide. (b) It becomes one-hlf s wide. (c) Its width does not chnge. (d) It becomes twice s wide. (e) It becomes four times s wide.

24 1134 CHAPTER 38 Diffrction Ptterns nd Polriztion 6. Assume Figure 38.1 ws photogrphed with red light of single wvelength l 0. The light pssed through single slit of width nd trveled distnce L to the screen where the photogrph ws mde. Consider the width of the centrl bright fringe, mesured between the centers of the drk fringes on both sides of it. Rnk from lrgest to smllest the widths of the centrl fringe in the following situtions nd note ny cses of equlity. () The experiment is performed s photogrphed. (b) The experiment is performed with light whose frequency is incresed by 50%. (c) The experiment is performed with light whose wvelength is incresed by 50%. (d) The experiment is performed with the originl light nd with slit of width 2. (e) The experiment is performed with the originl light nd slit nd with distnce 2L to the screen. 7. In Active Figure 38.4, ssume the slit is in brrier tht is opque to x-rys s well s to visible light. The photogrph in Active Figure 38.4b shows the diffrction pttern produced with visible light. Wht will hppen if the experiment is repeted with x-rys s the incoming wve nd with no other chnges? () The diffrction pttern is similr. (b) There is no noticeble diffrction pttern but rther projected shdow of high intensity on the screen, hving the sme width s the slit. (c) The centrl mximum is much wider, nd the minim occur t lrger ngles thn with visible light. (d) No x-rys rech the screen. 8. Off in the distnce, you see the hedlights of cr, but they re indistinguishble from the single hedlight of motorcycle. Assume the cr s hedlights re now switched from low bem to high bem so tht the light intensity you receive becomes three times greter. Wht then hppens to your bility to resolve the two light sources? () It increses by fctor of 9. (b) It increses by fctor of 3. (c) It remins the sme. (d) It becomes one-third s good. (e) It becomes one-ninth s good. 9. Certin sunglsses use polrizing mteril to reduce the intensity of light reflected s glre from wter or utomobile windshields. Wht orienttion should the polrizing filters hve to be most effective? () The polrizers should bsorb light with its electric field horizontl. (b) The polrizers should bsorb light with its electric field verticl. (c) The polrizers should bsorb both horizontl nd verticl electric fields. (d) The polrizers should not bsorb either horizontl or verticl electric fields. 10. When you receive chest x-ry t hospitl, the x-rys pss through set of prllel ribs in your chest. Do your ribs ct s diffrction grting for x-rys? () Yes. They produce diffrcted bems tht cn be observed seprtely. (b) Not to mesurble extent. The ribs re too fr prt. (c) Essentilly not. The ribs re too close together. (d) Essentilly not. The ribs re too few in number. (e) Absolutely not. X-rys cnnot diffrct. 11. When unpolrized light psses through diffrction grting, does it become polrized? () No, it does not. (b) Yes, it does, with the trnsmission xis prllel to the slits or grooves in the grting. (c) Yes, it does, with the trnsmission xis perpendiculr to the slits or grooves in the grting. (d) It possibly does becuse n electric field bove some threshold is blocked out by the grting if the field is perpendiculr to the slits. 12. Why is it dvntgeous to use lrge-dimeter objective lens in telescope? () It diffrcts the light more effectively thn smller-dimeter objective lenses. (b) It increses its mgnifiction. (c) It enbles you to see more objects in the field of view. (d) It reflects unwnted wvelengths. (e) It increses its resolution. Conceptul Questions denotes nswer vilble in Student Solutions Mnul/Study Guide 1. Why cn you her round corners, but not see round corners? 2. Holding your hnd t rm s length, you cn redily block sunlight from reching your eyes. Why cn you not block sound from reching your ers this wy? 3. Fingerprints left on piece of glss such s windowpne often show colored spectr like tht from diffrction grting. Why? 4. () Is light from the sky polrized? (b) Why is it tht clouds seen through Polroid glsses stnd out in bold contrst to the sky? 5. A lser bem is incident t shllow ngle on horizontl mchinist s ruler tht hs finely clibrted scle. The engrved rulings on the scle give rise to diffrction pttern on verticl screen. Discuss how you cn use this technique to obtin mesure of the wvelength of the lser light. 6. If coin is glued to glss sheet nd this rrngement is held in front of lser bem, the projected shdow hs diffrction rings round its edge nd bright spot in the center. How re these effects possible? 7. How could the index of refrction of flt piece of opque obsidin glss be determined? 8. A lser produces bem few millimeters wide, with uniform intensity cross its width. A hir is stretched verticlly cross the front of the lser to cross the bem. () How is the diffrction pttern it produces on distnt screen relted to tht of verticl slit equl in width to the hir? (b) How could you determine the width of the hir from mesurements of its diffrction pttern? 9. A rdio sttion serves listeners in city to the northest of its brodcst site. It brodcsts from three djcent towers on mountin ridge, long line running est to west, in wht s clled phsed rry. Show tht by introducing time delys mong the signls the individul towers rdite, the sttion cn mximize net intensity in the direction towrd the city (nd in the opposite direction) nd minimize the signl trnsmitted in other directions.

25 Problems John Willim Strutt, Lord Ryleigh ( ), invented n improved foghorn. To wrn ships of costline, foghorn should rdite sound in wide horizontl sheet over the ocen s surfce. It should not wste energy by brodcsting sound upwrd or downwrd. Ryleigh s foghorn trumpet is shown in two possible configurtions, horizontl nd verticl, in Figure CQ Which is the correct orienttion? Decide whether the long dimension of the rectngulr opening should be horizontl or verticl nd rgue for your decision. 11. The toms in crystl lie in plnes seprted by few tenths of nnometer. Cn they produce diffrction pttern for visible light s they do for x-rys? Explin your nswer with reference to Brgg s lw. 12. Figure CQ38.12 shows megphone in use. Construct theoreticl description of how megphone works. You my ssume the sound of your voice rdites just through the opening of your mouth. Most of the informtion in speech is Figure CQ38.12 crried not in signl t the fundmentl frequency, but in noises nd in hrmonics, with frequencies of few thousnd hertz. Does your theory llow ny prediction tht is simple to test? Doug Pensigner/Getty Imges Figure CQ38.10 Problems The problems found in this chpter my be ssigned online in Enhnced WebAssign 1. denotes strightforwrd problem; 2. denotes intermedite problem; 3. denotes chllenging problem 1. full solution vilble in the Student Solutions Mnul/Study Guide 1. denotes problems most often ssigned in Enhnced WebAssign; these provide students with trgeted feedbck nd either Mster It tutoril or Wtch It solution video. Section 38.2 Diffrction Ptterns from Nrrow Slits 1. Light of wvelength 540 nm psses through slit of width mm. () The width of the centrl mximum on screen is 8.10 mm. How fr is the screen from the slit? (b) Determine the width of the first bright fringe to the side of the centrl mximum. 2. Helium neon lser light (l nm) is sent through mm-wide single slit. Wht is the width of the centrl mximum on screen 1.00 m from the slit? 3. Sound with frequency 650 Hz from distnt source psses through doorwy 1.10 m wide in sound-bsorbing wll. Find () the number nd (b) the ngulr directions of the diffrction minim t listening positions long line prllel to the wll. denotes sking for quntittive nd conceptul resoning denotes symbolic resoning problem denotes Mster It tutoril vilble in Enhnced WebAssign denotes guided problem shded denotes pired problems tht develop resoning with symbols nd numericl vlues 4. A horizontl lser bem of wvelength nm hs circulr cross section 2.00 mm in dimeter. A rectngulr perture is to be plced in the center of the bem so tht when the light flls perpendiculrly on wll 4.50 m wy, the centrl mximum fills rectngle 110 mm wide nd 6.00 mm high. The dimensions re mesured between the minim brcketing the centrl mximum. Find the required () width nd (b) height of the perture. (c) Is the longer dimension of the centrl bright ptch in the diffrction pttern horizontl or verticl? (d) Is the longer dimension of the perture horizontl or verticl? (e) Explin the reltionship between these two rectngles, using digrm. 5. Coherent microwves of wvelength 5.00 cm enter tll, nrrow window in building otherwise essentilly opque to the microwves. If the window is 36.0 cm wide, wht is the distnce from the centrl mximum to the first-order minimum long wll 6.50 m from the window? 6. A bem of monochromtic light is incident on single slit of width mm. A diffrction pttern forms on wll 1.30 m beyond the slit. The distnce between the positions of zero intensity on both sides of the centrl mximum is 2.00 mm. Clculte the wvelength of the light. 7. A screen is plced 50.0 cm from single slit, which is illuminted with light of wvelength 690 nm. If the distnce between the first nd third minim in the diffrction pttern is 3.00 mm, wht is the width of the slit? 8. A screen is plced distnce L from single slit of width, which is illuminted with light of wvelength l. Assume L... If the distnce between the minim for m 5 m 1 nd m 5 m 2 in the diffrction pttern is Dy, wht is the width of the slit?

26 1136 CHAPTER 38 Diffrction Ptterns nd Polriztion 9. Assume light of wvelength 650 nm psses through two slits 3.00 mm wide, with their centers 9.00 mm prt. Mke sketch of the combined diffrction nd interference pttern in the form of grph of intensity versus f 5 (p sin u)/l. You my use Active Figure 38.7 s strting point. 10. Coherent light of wvelength nm is sent through two prllel slits in n opque mteril. Ech slit is mm wide. Their centers re 2.80 mm prt. The light then flls on semicylindricl screen, with its xis t the midline between the slits. We would like to describe the ppernce of the pttern of light visible on the screen. () Find the direction for ech two-slit interference mximum on the screen s n ngle wy from the bisector of the line joining the slits. (b) How mny ngles re there tht represent two-slit interference mxim? (c) Find the direction for ech single-slit interference minimum on the screen s n ngle wy from the bisector of the line joining the slits. (d) How mny ngles re there tht represent single-slit interference minim? (e) How mny of the ngles in prt (d) re identicl to those in prt ()? (f) How mny bright fringes re visible on the screen? (g) If the intensity of the centrl fringe is I mx, wht is the intensity of the lst fringe visible on the screen? 11. A diffrction pttern is formed on screen 120 cm wy from mm-wide slit. Monochromtic nm light is used. Clculte the frctionl intensity I/I mx t point on the screen 4.10 mm from the center of the principl mximum. b u 12. Wht If? Suppose light strikes single slit of width t n ngle b from the perpendiculr direction s shown in Figure P Show tht Eqution 38.1, the condition for destructive interference, must be modified to red Figure P38.12 sin u drk 5 m l 2 sin b m 5 61, 62, 63,... Section 38.3 Resolution of Single-Slit nd Circulr Apertures In Problems 14, 17, 20, 21, nd 61, you my use the Ryleigh criterion for the limiting ngle of resolution of n eye. The stndrd my be overly optimistic for humn vision. 13. The ngulr resolution of rdio telescope is to be when the incident wves hve wvelength of 3.00 mm. Wht minimum dimeter is required for the telescope s receiving dish? 14. The pupil of ct s eye nrrows to verticl slit of width mm in dylight. Assume the verge wvelength of the light is 500 nm. Wht is the ngulr resolution for horizontlly seprted mice? 15. The objective lens of certin refrcting telescope hs dimeter of 58.0 cm. The telescope is mounted in stellite tht orbits the Erth t n ltitude of 270 km to view objects on the Erth s surfce. Assuming n verge wvelength of 500 nm, find the minimum distnce between two objects on the ground if their imges re to be resolved by this lens. 16. A pinhole cmer hs smll circulr perture of dimeter D. Light from distnt objects psses through the perture into n otherwise drk box, flling on screen t the other end of the box. The perture in pinhole cmer hs dimeter D mm. Two point sources of light of wvelength 550 nm re t distnce L from the hole. The seprtion between the sources is 2.80 cm, nd they re just resolved by the cmer. Wht is L? 17. Wht is the pproximte size of the smllest object on the Erth tht stronuts cn resolve by eye when they re orbiting 250 km bove the Erth? Assume l nm nd pupil dimeter of 5.00 mm. 18. Yellow light of wvelength 589 nm is used to view n object under microscope. The objective lens dimeter is 9.00 mm. () Wht is the limiting ngle of resolution? (b) Suppose it is possible to use visible light of ny wvelength. Wht color should you choose to give the smllest possible ngle of resolution, nd wht is this ngle? (c) Suppose wter fills the spce between the object nd the objective. Wht effect does this chnge hve on the resolving power when 589-nm light is used? 19. A helium neon lser emits light tht hs wvelength of nm. The circulr perture through which the bem emerges hs dimeter of cm. Estimte the dimeter of the bem 10.0 km from the lser. 20. Nrrow, prllel, glowing gs-filled tubes in vriety of colors form block letters to spell out the nme of nightclub. Adjcent tubes re ll 2.80 cm prt. The tubes forming one letter re filled with neon nd rdite predominntly red light with wvelength of 640 nm. For nother letter, the tubes emit predominntly blue light t 440 nm. The pupil of drk-dpted viewer s eye is 5.20 mm in dimeter. () Which color is esier to resolve? Stte how you decide. (b) If she is in certin rnge of distnces wy, the viewer cn resolve the seprte tubes of one color but not the other. The viewer s distnce must be in wht rnge for her to resolve the tubes of only one of these two colors? 21. Impressionist pinter Georges Seurt creted pintings with n enormous number of dots of pure pigment, ech of which ws pproximtely 2.00 mm in dimeter. The ide ws to hve colors such s red nd green next to ech other to form scintillting cnvs, such s in his msterpiece, A Sundy Afternoon on the Islnd of L Grnde Figure P38.21 SuperStock/SuperStock

27 Problems 1137 Jtte (Fig. P38.21). Assume l nm nd pupil dimeter of 5.00 mm. Beyond wht distnce would viewer be unble to discern individul dots on the cnvs? 22. A circulr rdr ntenn on Cost Gurd ship hs dimeter of 2.10 m nd rdites t frequency of 15.0 GHz. Two smll bots re locted 9.00 km wy from the ship. How close together could the bots be nd still be detected s two objects? Section 38.4 The Diffrction Grting Note: In the following problems, ssume the light is incident normlly on the grtings. 23. A helium neon lser (l nm) is used to clibrte diffrction grting. If the first-order mximum occurs t 20.5, wht is the spcing between djcent grooves in the grting? 24. White light is spred out into its spectrl components by diffrction grting. If the grting hs grooves per centimeter, t wht ngle does red light of wvelength 640 nm pper in first order? 25. Consider n rry of prllel wires with uniform spcing of 1.30 cm between centers. In ir t 20.0 C, ultrsound with frequency of 37.2 khz from distnt source is incident perpendiculr to the rry. () Find the number of directions on the other side of the rry in which there is mximum of intensity. (b) Find the ngle for ech of these directions reltive to the direction of the incident bem. 26. Three discrete spectrl lines occur t ngles of 10.1, 13.7, nd 14.8 in the first-order spectrum of grting spectrometer. () If the grting hs slits/cm, wht re the wvelengths of the light? (b) At wht ngles re these lines found in the second-order spectrum? 27. The lser in compct disc plyer must precisely follow the spirl trck on the CD, long which the distnce between one loop of the spirl nd the next is only bout 1.25 mm. Figure P38.27 shows how diffrction grting is used to provide informtion to keep the bem on trck. The lser light psses through diffrction grting before it reches the CD. The strong centrl mximum of the diffrction pttern is used to red the informtion in the trck of pits. The two first-order side mxim re designed to fll on the flt surfces on both sides of the informtion trck nd re used for steering. As long s both bems re reflecting Compct disc First-order mxim Lser Figure P38.27 Centrl mximum Diffrction grting from smooth, nonpitted surfces, they re detected with constnt high intensity. If the min bem wnders off the trck, however, one of the side bems begins to strike pits on the informtion trck nd the reflected light diminishes. This chnge is used with n electronic circuit to guide the bem bck to the desired loction. Assume the lser light hs wvelength of 780 nm nd the diffrction grting is positioned 6.90 mm from the disk. Assume the first-order bems re to fll on the CD mm on either side of the informtion trck. Wht should be the number of grooves per millimeter in the grting? 28. A grting with 250 grooves/mm is used with n incndescent light source. Assume the visible spectrum to rnge in wvelength from 400 nm to 700 nm. In how mny orders cn one see () the entire visible spectrum nd (b) the short-wvelength region of the visible spectrum? 29. A diffrction grting hs rulings/cm. On screen 2.00 m from the grting, it is found tht for prticulr order m, the mxim corresponding to two closely spced wvelengths of sodium (589.0 nm nd nm) re seprted by 1.54 mm. Determine the vlue of m. 30. The hydrogen spectrum includes red line t 656 nm nd blue-violet line t 434 nm. Wht re the ngulr seprtions between these two spectrl lines for ll visible orders obtined with diffrction grting tht hs grooves/cm? 31. Light from n rgon lser strikes diffrction grting tht hs grooves per centimeter. The centrl nd firstorder principl mxim re seprted by m on wll 1.72 m from the grting. Determine the wvelength of the lser light. 32. Show tht whenever white light is pssed through diffrction grting of ny spcing size, the violet end of the spectrum in the third order on screen lwys overlps the red end of the spectrum in the second order. 33. Light of wvelength 500 nm is incident normlly on diffrction grting. If the third-order mximum of the diffrction pttern is observed t 32.0, () wht is the number of rulings per centimeter for the grting? (b) Determine the totl number of primry mxim tht cn be observed in this sitution. 34. A wide bem of lser light with wvelength of nm is directed through severl nrrow prllel slits, seprted by 1.20 mm, nd flls on sheet of photogrphic film 1.40 m wy. The exposure time is chosen so tht the film stys unexposed everywhere except t the centrl region of ech bright fringe. () Find the distnce between these interference mxim. The film is printed s trnsprency; it is opque everywhere except t the exposed lines. Next, the sme bem of lser light is directed through the trnsprency nd llowed to fll on screen 1.40 m beyond. (b) Argue tht severl nrrow, prllel, bright regions, seprted by 1.20 mm, pper on the screen s rel imges of the originl slits. (A similr trin of thought, t soccer gme, led Dennis Gbor to invent hologrphy.) 35. A bem of bright red light of wvelength 654 nm psses through diffrction grting. Enclosing the spce beyond

28 1138 CHAPTER 38 Diffrction Ptterns nd Polriztion the grting is lrge semicylindricl screen centered on the grting, with its xis prllel to the slits in the grting. Fifteen bright spots pper on the screen. Find () the mximum nd (b) the minimum possible vlues for the slit seprtion in the diffrction grting. Section 38.5 Diffrction of X-Rys by Crystls 36. If the spcing between plnes of toms in NCl crystl is nm, wht is the predicted ngle t which nm x-rys re diffrcted in first-order mximum? 37. Potssium iodide (KI) hs the sme crystlline structure s NCl, with tomic plnes seprted by nm. A monochromtic x-ry bem shows first-order diffrction mximum when the grzing ngle is Clculte the x-ry wvelength. 38. Monochromtic x-rys (l nm) from nickel trget re incident on potssium chloride (KCl) crystl surfce. The spcing between plnes of toms in KCl is nm. At wht ngle (reltive to the surfce) should the bem be directed for second-order mximum to be observed? 39. The first-order diffrction mximum is observed t 12.6 for crystl hving spcing between plnes of toms of nm. () Wht wvelength x-ry is used to observe this first-order pttern? (b) How mny orders cn be observed for this crystl t this wvelength? trnsmission xis t 15.0 reltive to the preceding filter. (d) Comment on compring the nswers to prts (), (b), nd (c). 45. The criticl ngle for totl internl reflection for spphire surrounded by ir is Clculte the polrizing ngle for spphire. 46. For prticulr trnsprent medium surrounded by ir, find the polrizing ngle u p in terms of the criticl ngle for totl internl reflection u c. 47. You use sequence of idel polrizing filters, ech with its xis mking the sme ngle with the xis of the previous filter, to rotte the plne of polriztion of polrized light bem by totl of You wish to hve n intensity reduction no lrger thn 10.0%. () How mny polrizers do you need to chieve your gol? (b) Wht is the ngle between djcent polrizers? Additionl Problems 48. Lser light with wvelength of nm is directed through one slit or two slits nd llowed to fll on screen 2.60 m beyond. Figure P38.48 shows the pttern on the screen, with centimeter ruler below it. () Did the light pss through one slit or two slits? Explin how you cn determine the nswer. (b) If one slit, find its width. If two slits, find the distnce between their centers. Section 38.6 Polriztion of Light Wves Problem 52 in Chpter 34 cn be ssigned with this section. 40. The ngle of incidence of light bem onto reflecting surfce is continuously vrible. The reflected ry in ir is completely polrized when the ngle of incidence is Wht is the index of refrction of the reflecting mteril? 41. Unpolrized light psses through two idel Polroid sheets. The xis of the first is verticl, nd the xis of the second is t 30.0 to the verticl. Wht frction of the incident light is trnsmitted? 42. Why is the following sitution impossible? A technicin is mesuring the index of refrction of solid mteril by observing the polriztion of light reflected from its surfce. She notices tht when light bem is projected from ir onto the mteril surfce, the reflected light is totlly polrized prllel to the surfce when the incident ngle is Plne-polrized light is incident on single polrizing disk with the direction of E S 0 prllel to the direction of the trnsmission xis. Through wht ngle should the disk be rotted so tht the intensity in the trnsmitted bem is reduced by fctor of () 3.00, (b) 5.00, nd (c) 10.0? 44. An unpolrized bem of light is incident on stck of idel polrizing filters. The xis of the first filter is perpendiculr to the xis of the lst filter in the stck. Find the frction by which the trnsmitted bem s intensity is reduced in the three following cses. () Three filters re in the stck, ech with its trnsmission xis t 45.0 reltive to the preceding filter. (b) Four filters re in the stck, ech with its trnsmission xis t 30.0 reltive to the preceding filter. (c) Seven filters re in the stck, ech with its Figure P In single-slit diffrction pttern, ssuming ech side mximum is hlfwy between the djcent minim, find the rtio of the intensity of () the first-order side mximum nd (b) the second-order side mximum to the intensity of the centrl mximum. 50. The second-order drk fringe in single-slit diffrction pttern is 1.40 mm from the center of the centrl mximum. Assuming the screen is 85.0 cm from slit of width mm nd ssuming monochromtic incident light, clculte the wvelength of the incident light. 51. In wter of uniform depth, wide pier is supported on pilings in severl prllel rows 2.80 m prt. Ocen wves of uniform wvelength roll in, moving in direction tht mkes n ngle of 80.0 with the rows of pilings. Find the three longest wvelengths of wves tht re strongly reflected by the pilings. 52. Two motorcycles seprted lterlly by 2.30 m re pproching n observer wering night-vision goggles sensitive to infrred light of wvelength 885 nm. () Assume the light propgtes through perfectly stedy nd uniform ir. Wht perture dimeter is required if the motorcycles hedlights re to be resolved t distnce of 12.0 km?

29 Problems 1139 (b) Comment on how relistic the ssumption in prt () is. 53. Light from helium neon lser (l nm) is incident on single slit. Wht is the mximum width of the slit for which no diffrction minim re observed? 54. The Very Lrge Arry (VLA) is set of 27 rdio telescope dishes in Ctron nd Socorro counties, New Mexico (Fig. P38.54). The ntenns cn be moved prt on rilrod trcks, nd their combined signls give the resolving power of synthetic perture 36.0 km in dimeter. () If the detectors re tuned to frequency of 1.40 GHz, wht is the ngulr resolution of the VLA? (b) Clouds of interstellr hydrogen rdite t the frequency used in prt (). Wht must be the seprtion distnce of two clouds t the center of the glxy, light-yers wy, if they re to be resolved? (c) Wht If? As the telescope looks up, circling hwk looks down. Assume the hwk is most sensitive to green light hving wvelength of 500 nm nd hs pupil of dimeter 12.0 mm. Find the ngulr resolution of the hwk s eye. (d) A mouse is on the ground 30.0 m below. By wht distnce must the mouse s whiskers be seprted if the hwk cn resolve them? 58. Iridescent pecock fethers re shown in Figure P The surfce of one microscopic brbule is composed of trnsprent kertin tht supports rods of drk brown melnin in regulr lttice, represented in Figure P38.58b. (Your fingernils re mde of kertin, nd melnin is the drk pigment giving color to humn skin.) In portion of the fether tht cn pper turquoise (bluegreen), ssume the melnin rods re uniformly seprted by 0.25 mm, with ir between them. () Explin how this structure cn pper turquoise when it contins no blue or green pigment. (b) Explin how it cn lso pper violet if light flls on it in different direction. (c) Explin how it cn present different colors to your two eyes simultneously, which is chrcteristic of iridescence. (d) A compct disc cn pper to be ny color of the rinbow. Explin why the portion of the fether in Figure P38.58b cnnot pper yellow or red. (e) Wht could be different bout the rry of melnin rods in portion of the fether tht does pper to be red? Dine Hirsch/Fundmentl Photogrphs, NYC b Figure P38.58 Figure P Review. A bem of 541-nm light is incident on diffrction grting tht hs 400 grooves/mm. () Determine the ngle of the second-order ry. (b) Wht If? If the entire pprtus is immersed in wter, wht is the new secondorder ngle of diffrction? (c) Show tht the two diffrcted rys of prts () nd (b) re relted through the lw of refrction. 56. Why is the following sitution impossible? A technicin is sending lser light of wvelength nm through pir of slits seprted by 30.0 mm. Ech slit is of width 2.00 mm. The screen on which he projects the pttern is not wide enough, so light from the m 5 15 interference mximum misses the edge of the screen nd psses into the next lb sttion, strtling coworker. 57. A 750-nm light bem in ir hits the flt surfce of certin liquid, nd the bem is split into reflected ry nd refrcted ry. If the reflected ry is completely polrized when it is t 36.0 with respect to the surfce, wht is the wvelength of the refrcted ry? istockphoto.com/cbpix 59. Light in ir strikes wter surfce t the polrizing ngle. The prt of the bem refrcted into the wter strikes submerged slb of mteril with refrctive index n s shown in Figure P The light reflected from the upper surfce of the slb is completely polrized. Find the ngle u between the wter surfce nd the surfce of the slb. 60. Light in ir (ssume n 5 1) strikes the surfce of liquid of index of refrction n, t the polrizing ngle. The prt of the bem refrcted into the liquid strikes submerged slb of mteril with refrctive index n s shown in Figure P The light reflected from the upper surfce of the slb is completely polrized. Find the ngle u between the wter surfce nd the surfce of the slb s function of n nd n,. 61. An Americn stndrd nlog television picture (non- HDTV), lso known s NTSC, is composed of pproximtely 485 visible horizontl lines of vrying light intensity. Assume your bility to resolve the lines is limited only by the Ryleigh criterion, the pupils of your eyes re 5.00 mm in dimeter, nd the verge wvelength of the light coming from the screen is 550 nm. Clculte the rtio of the up u u p Air Wter Figure P38.59 Problems 59 nd 60.