5 04 Interference and Diffrac tion Sec. 9.7 505

Co ncave spheri cal mirror. A sphere is said to be " nestled" at the apex ofa par aboloid if it is tangen t to th e paraboloid th ere and has the sameradi us as the radius of curvatu re of th e para boloid there. It is not difficultto show tha t th e radius of suc h a nestled sphe re is 2[. Sec Fig. 9.24.

far to th e right, the ellipsoid degenerates into a pa raboloid. Rays emit tedfrom F th en for m a para llel bea m (because they still focus at F ' , infini te lyfar away). This is sho wn in Fig. 9.23.

If the parabolic mir ror ape rture has a dia met er D, th en a point sourceat F doe s not form a perfec tly parall el bea m. Th e angular wid th of theinterfere nce maxim um is liB ::: 'A/ D. If D is " in fini te:' we get a perfec tpla ne wav e from th e point source.

Conv ersely, an incident plan e wave (per fectly well defined in angle)focuses to an imag e at F tha t is not a poin t unless D is infinite. The imagehas a wid th liz z. J 1I8 z. JX/ D.

Fig. 9.24 Concave spherical mirro r ("incontact" with an imagined nestled para­bolic mirTor) . The sphere's center is atC; its ra d ius is 2f Ray a reflected fromthe sphere is not parallel to . the am;ray a' reflected f rom the paraboloid u.Th il illuetrates $pherical aberration:

Fig. 9.23 Concave parabolic mi'rror.

Ellip soidal mirror. I n Fig. 9.22 we see a hollow ellipsoid of revolutionwith a spec uJar ly refiecung inn er surface and with a point source of lightlocat ed a t F, one of th e two p rin cipal foci. Fr om th e de finition of anellipse , the d istances from F to th e other focus F ' ar e th e same for allpa ths (excep t for th e dir ect path not involving a reflecti on]. Th erefore thefocus F ' is a region of co mplete co nstruc tive interference for radia tione mitted by elect rons in th e surface that are dri ven by radiati on from F.We say th at th e sour ce at F is imaged at th e poin t F ' .

Th e image at F ' is not a poin t ; the phase of the resultan t field at a pointnear F ' is within ab out ± 11 of th e ph ase at F ' pr ovided th e poin t lieswithin a sphere with rad ius ab out >"/ 4 ce ntered at F '. Therefore th at isroughly th e size of th e ima ge at F ' .

Conca ve parabolic mirror. I magine th at th e focal poin t F an d the focalle ngth Jof th e ellipsoid in Fig. 9.22 are held fixed, but tha t th e focal point F 'is moved to the righ t; th e ellipse is "s tr etched.' If F ' is moved infini tely

III.,III "- 1f..- f -+lI · -,I ·

Fig. 9.22 Ellipsoida l m irror.

The de viation a is a co nstant, independent of the angle of in ciden ce, aslong as w e stay near normal incidence. Equation (89) is easily derived asfollow s (See Fig. 9.25 ): At the base of the prism, the wave front transversesthe distan ce I at velocity er n. At the apex, the veloc ity is n times larger(sinc e the prism thickness is zero there), and the same wavefron t thereforetravels a distance nl in the same time. Thus the wa vefront is ahead by adistance (11 - 1)1at the top . Thi s distan ce divided by the width IV of thepri sm is (for small angles) the angle of deviation 8 = (11 - 1)(l/ IV) =(11 - 1)" , which is Eq. (89).

506 Interference and Diffracti on

Spheri cal abe rra tio n. For a small aperture diameter D -<2[, a sphericalmirror is essentially "in co ntact" with an imagined nestled parabolic mirror.Th en a point source at F forms an almost parallel beam . For large aper­tures, the deviation of the sphe rical surface from that of a paraboloid pro­duces "spherical aberration." (See Fig. 9.24 .)

For a discussion of image formation by co ncave mirrors, see PSSC,Physics, 2nd ed., Cbap. 12 (D . C. Heat h and Company, Boston, 1965).You can obtain experience with co ncave mirrors by ge tting a cheap "shav­ing mirror" and formin g images of (for example) a candle flame or yourface. (The bowl of a shiny new spoo n works almost as well.) For experi­ence with convex mirrors, we recommend playing with a silve ry sphericalChristmas tree ornament. (Or, turn the spoon ove r.)

Sec. 9.7

is de viated "toward the base of the prism" by an angle 8 give n by

8 = (11 - 1)" .

507

(89)

Fig. 9.25 Deviation by a thi n prism.

D evi ation of a light ray at near -normal in cidence o n a thin glass pri sm.A "thin" prism is one for which the wedge angle a is so small that we canuse the small-angle approximations sin a :::::; ex, cos a :::::: 1. For near-normalincidence , we can also use small-angle approximations for the angle ofincid e nce. Then a monochromat ic plane wave at near-normal incidence

r- n1-j

I[

+1 zl-

Colo r dispersion of pri sm. As an example of a thin prism , suppose a is30 deg (for whic h th e small-angle app roximation is still not too bad, for ourpurpose) an d 11 is 1.50. Th en the deviati on is 15 deg , according to Eq. (89).That is actually the ave rage deviation, because for typical glass with anaverage index of refrac tion of 1.5, blue light of wave length 0.45 )J. actuallyhas i ndex ah out 0.0 1 larger tha n red light of wave lengt h 0.65)J.. Th ere­fore th e blue light is devia ted mor e th an is the red light by abo ut 0.01" .For c of 30 deg. blue is devia ted abou t 0.3 deg more than red. In radia ns,since 30 deg is about half a radian (1 rad = 57.3 °), blue is deviat ed byabout ~ radian more than red. On a screen one meter beyond the30-deg prism, blue is therefore separated from red by abo ut t em. Aprism spec trome ter makes use of this dispe rsive e ffec t of a glass prism toanalyze spec tra. In optical instrument s involvin g glass lenses, dispersionleads to chromatic abe rra t ionc--i.e. , rays of different colors do not focus atthe same places . One can avoid chromatic abe rration in a telescop e byusing a parabolic mirror to gathe r the light to a focus rather than a refract­ing lens. (The law of specular reflecti on holds for all colors.) On e canalso eliminate chromatic aberration by using two kinds of glass with differ­ent dispersio n. See Prob . 9.5.3.

Focusing of paranal light rays by a thin lens. Suppose we have a glassle ns in air with two co nvex spherical surfaces normal to a common axis ofsym metry z. A light ray is incident from th e left, tra veling parallel to theaxis of symmetry of the lens at distan ce y = h from th e axis. If the lensis " thin," we (by definition) neglect the variatio n of y as the ray passesth rough the len s; we also neglect the thickn ess co mpared with th e focallength . To consider only "paraxial" rays means that we keep h small co m­pared to the radii of curvature of the two surfaces, so that we ca n usesmall-angle approximations for all angles of interest.

5 08 Interference and Diff raction Sec. 9. 7 509

~----- /8

Foca l plane

(90)

Let us find the focal point F where a ray which is incident parallel tothe syrnmetry axis crosses the symmetry axis after deviation by the lens, asshow n in Fig. 9.26. W e see that if the incident ray is focussed.at F it musthave been deviated by the small angle

Necessa ry co ndition fo r a foc us. Thus we see that the necessary condi­tion for the existence of a common focal poin t for all paralle l paraxial inci­dent rays is that the de t:riation be linearly proportional to the disp lacement hof the ray from th e axis. Thu s if Eq. (90) is satisfied for all h (but alwaysassu ming small-ang le deviations), then all p arallel rays will be focus ed atthe same distan ce f behin d the lens. This condit ion holds f or any similarf ocusing problem, for example for the focu sing of a beam of charged parti­cles by a magnetic lens.

It re mains to be seen whether a thin lens with spherical surfaces satis­fies Eq. (90) wi th f independ ent of h. This is seen as follows: As far as theray in Fig. 9.26 is co nce rned, it could just as well have been deviated byan equivalent thin prism. The first surface is at angle h/ R1 to the vertical(where the ray strikes it). The second surface is at an angle hlR, to thevertic al in the oppos ite sens e. The equivalent pri sm angle a is thereforehR, - l + hR , - 1 The devia tion 8 by the equivalent thin pri sm is (n - I)«,so that we have

Thin ·lens f ormu la. But

8, = ~ , 8, = ~ , S hP q and =l '(The deviat ion is always hl f , independ ent of angle of incidenc e.) Th ere­fore Eq. (93) gives

Fig. 9.2 7 Foca l plan e.

( ~3)8, - S + 8, = O.

Foca l plane. Now co nsider a bundle of parallel rays which are not paral­lel to the symmetry a;xis but rather make angle 0 with the axis, Thedeviation of a thin prism is Indep ende nt of the angle of incide nce (for smallangles). Therefore a ray that s trikes the lens at distance h [rom its centeris de Diated by 8 = hl f, independent af the angle of incidence. Thatmeans that any parallel bundle focuses at a point in a plane, called thef oca l plane, a distance f behi nd th e lens, and the lateral displace men t inthe plan e of the point is f8 from th e axis, as show n in Fig. 9.27.

Real p oint im ogc of a point object . We have found the point image of aparalle l bea m, i .e ., a bea m from an objec t poin t (source) infinitely far tothe left. Le t us now co nsider an objec t point 0 at distance p to the left ofour converging lens and find its image I at distance q to the right. We leto be on the symmetry axis; then I will be on the axis. Now consider Fig.9.28. It is obvious from the figure that if we start with a vector pointingfrom 0 in the + :i direction and the n perform the rotations +01, - 8, and+ O2, we are back to the + :i axis: .

(91)8 = (n - l )h(R, - ' + R , - ' ).

Fig. 9.26 Thi n le ns. Inci de nt raypar allel to the (Iris.

Lens-mu ker 's fo rm ula . We see that Eq. (91) satisfies the condition for afocus, nam ely that S be proportional to h ; the focal length f is given by[see Eq. (90)]

i.e.,

l = (/1 - 1 ) (~ + ~)f R , R,

Equati on (92) is called the lens-maker's f ormula.

(92)

~ - ~ + ~f - p q '

[*+*=71Equ ati on (94) is called the thin-lens form ula.

(94)

510 Int erference aOODiffraction Sec. 9.7 511

Fig. 9.28 Reol point im age of a pointobjec t.

\0

t ~2

.:q

Lateral magnifi cat ion. Th e an gles of de viati on of rays by a thi n lens areun ch an ged if th e le ns is given a slight rota t ion ab ou t an axis through itsce nter perp endi cular to th e plane of F ig. 9.28. Th us the ray from th e ob­ject point through th e cen ter of the lens is still un de viated , and the raystriking th e lens at distance h from the center is deviate d by hlf. Th ere­fore th e object and ima ge point s in F ig. 9.2 8 are unchanged if th e lens isgive n a slight rot ati on abou t its ce nter. (On th e oth er hand, if th e lens isgive n a slight tr anslati on perpendi cul ar to its axis, th en th e image pointwill b e tr anslat ed . Th e ne w location is obtained by th e observat ion thatth e ray thr ough th e cente r of th e lens is und eviated .) Instead of making aslight rot ation of th e lens abo ut its cente r, suppose we hold the lens fixedand give the objec t poin t a slight up ward translation perpendicular to theaxis of th e lens. The entire ray diagram can then be rot ated about thece nte r of th e lens (b ecause th e deviation s are ind ependent of th e angle ofincid enc e, for near-normal incid en ce). Thus we see th at if th e objectpoint is translated up an amo un t y, th en th e image point will be translateddow n an am ount th at is greate r th an y by th e rati o of th e "lever arm s" qand p. O ne expresses thi s by saying th at th e lat eral magnificati on is - ql p:

Virtu al ima ge. If th e objec t point in Fig. 9.28 is at distan ce f to the leftof th e thin co nve rging len s sh own, then th e de via tion hl f of rays at dis­tan ce h from th e len s ce nter is just suc h as to fo rm a parallel beam to theright of the lens. If th e objec t point is closer than [ , then th e devi ati onhI ! is insufficien t to dir ect th e ray ba ck towar d th e axis. Th erefo re theray never crosses th e axis aga in. Thus th er e is no real image. Thi s raysee ms to come from a " v ir t ua l" point to th e left of the lens. O ne says thatther e is a virt ual image. See F ig. 9.29. It is ea sy to show (we shall letyou do it ) th at the location of th e virtual image is still give n by th e thin­lens formul a, Eq . (94), provide d we int erpret a negative value of 4 tomean a distan ce measured to the left of th e lens.

Di vergin g lens . If a lens is thinner at the cente r th an at th e edges , it is adiverging lens (assuming it is a glass lens in air ). If we thin k of the lens asconsisting of thin pr isms (as we did for the conve rging lens), then the apexof each pri sm is closer to th e axis th an the base is. Rays ar e deflect edaw ay from th e lens axis (rather than tow ard it as in a converging lens). Apa rallel beam incident from th e left gives a dive rgin g beam that diver gesfrom a o-i-rtua l f ocus to th e left of the lens, as shown in Fig. 9.30 . It is

.,'

F

Fig. 9.29 Virtual point image oj pointobject. The ob ject distance p is lessthan the Joca l length f.

La te ral magni ficati on = - !l. (95)P

The minus sign te lls u s that if th e objec t poin t goes up th e ima ge po intgoes down. If th e objec t is not a Single point but an extende d object, likea littl e arrow with head and tail, we see th at th e image is i-nverted.

Converging lens . The lens show n in Fig. 9.28 is a converg ing lens . Theima ge of an object that is at a distance grea te r th an the focal length f froma conve rging thin lens is a real in verted image. Th e adjec tive ureal"means th at the re really is light at th e ima ge. By contras t, an ima ge in anordina ry plane mir ror is " virtu al" - there is no light behind the mirrorsurface.

Fig. 9.30 Diverging lens.

5 12 Interference and Diffract ion Sec. 9.7 513

easy to show (we sha ll let )'ou do it) that all the formulas obtained for thinco nverging lenses can be used for thin diverging lenses if we give suitableinterpretation to the meaning of negative quantities. 111U S if we say thata div erging lens has a nega tive focal length , f = - II I, we can use thethin-lens formula to relate object and image distances. For example, Fig.9.30 corres ponds to p = + 00 , q = - If I, and f = -I I I in th e formula

Lens power in diop ters. The inverse focal length in units of inversemeters is called the lens pow er in diopiers. Thus a co nverging lens offocal le ngth 50 c m has pow er +2 diopt ers ( + 2 D). A diverging lens offocal le ngth - 50 e m has power - 2 D. Th e inverse focal length (thepower) has the nice feature that it is linear in the follow ing sense: If onethin lens is followed imm ediate ly hy a nother, the total power of the twothin lenses in contact is the su m of their individ ual powers. That is easilysee n as follows. The first lens devi ates a ray toward the axis by an anglehilt , where j, is positive for a convergi ng lens and negati ve for a dive rg­ing lens. U the second lens is located at the exit end of the first lens, thenthe ray does not have a chance to c hange its transverse distance 11 from thecommon axis of the two lenses. Therefore it is incident on the second lensat the same distance h as it was on the first lens. Therefore the deviationproduced by the second lens is hl f,. Th e total deviation prod uced by thetwo len ses is hl /t + hIt.. Th at is the devia tion that would be producedby an equivalent single lens of focal length I suc h that III = 11/1 + 1/ fz.Thus the total power, or total equivalent inverse focal length , is the sumof the individu al pow ers. Of course if there is a space betwee n the twolenses, then the ray does not st rike the seco nd lens at the same distance hfrom the axis as it did the first. The powers of lenses in series thereforeadd linearly only if we can neglect the sepa ration between the lenses.

If you wea r glasses , you should take th em off and measur e (roughly) thepower of eac h lens in both the horizontal and vert ical planes. Use a dis­tant point source (or the sun). If the lens is a positive lens yo u ean fonnan image of the source on a wall or piece of paper. Are th e focal lengthsof eac h lens the same in both planes? (U th ey are different the lens is saidto be "astigmatic," and yo u are said to have astigmatism in the eye thatdiffers from the norm .)

The distance q from the eye lens to the retina is about 3 em. In inversemeters (rnr ' ) th is gives q: » = (0.03 m)-I = 33 m- I, th at is, qr ' is about33 inverse meters. An eye focused on a very distant objec t at distancep = 00 has lens power r: give n by t : = p- I + q: » = 0 + 33 m- I =33 D . To focus on an object at distance p = 25 em from your eye the ac­co mmodation muscles of the eye must increase the lens power by an

a moun t p-I = (0.25 m)-1 = 4 m " ! = 4 0 , giving a total of ab ou t 37 D.If you have sufficiently goo d accommodation muscles, you can instead in­crease your lens power by about 10 D and can then focus on an objec t atdistan ce p = {10 D)- I = 0.1 m = 10 c m. Th en the objec t looks largerand rou ca n see its details better. If you could b ring it to wit hin 1 ern ofyo ur eye and still focus an image on the retina, it would look 25 timeslarger than at 25 c m; corres pondin gly, you could resolve detai ls 25 timessmaller. No one has that much accommod ation.

Simple magn ifier. You can hold a small objec t at abo ut 25 cm from yourunaided eye and examine it with out fatigue, if you have normal vision. Ifthe heigh t of the objec t is h (in cm), it subte nds an angle " / 25 (radians) atyo W" eye, and this determines the size of the image on the retin a. If youca n bring the objec t closer, it will give a larger image on the retina . Tomaintain a clear [i.e., focused) image, the accommodati on muscles must in­crease th e le ns pow er. Th at is difficult aod tiring. Now use a lens of focalle ngth I (cm). Hold the lens just in front of the eye. Brin g the objectcloser. When th e object is at th e focal plan e of the len s, eac h point onthe objec t \ViU give a pa rallel bundle nf rays out of the lens an d into youreye . Thi s is easy for you to focus- your eye lens is rela:xed. We shall letyou show that th e ang ular size of th e objec t increases by a fact or 2511(assuming small angles so that you can use small-angle approximations).See Fig. 9.3 1. You ca n make a cheap magni fier (for ahout 50 ce nts) byge tting a lens of focal length 2 or 3 cm and tap ing it on a microscope slide.(You ca n also buy one for about $1.00 . See . for exa mple, th e catalog ofEdmund Scientific Co., Barrin gton. N.J. 08007 ).

Fig. 9.31 Simp l€ magnifier. 1'he powe rof the eye lens is supp lemented by thatof the magnifier. The objec t can bemove d closer to the eye and com cquentlygives a larger im age.

514 Interf erence and Diffraction Sec. 9.7 515

Fig. 9.32 Telescope.

Pin hole magnifi er. Take a piece of aluminum foil and make in it a pin­hole of about t- mm dia meter or less. Hold it close in front of yo ur eye.Look at a light source. The "Heaters" y Oll see are the diffraction patternsof chains of ce lls in you r eye. (They are not on the surface, as you cansee by trying to wipe them off by blinkin g.) Now look at a well-illuminatedprinte d page thr ough the pinh ole . (If you wea r glasses, take th em off.You don 't need them a nd th ey do n' t do any good.) Bring the page upcloser and closer to your eye. Notice that the word you are looking atstays "in focus" and is magni fied as it is hrought closer! (It finally getsfuzz)' because your pin hole is not small en ough.) The magni fication iseasily calculated by a sketch like that of F ig. 9.3 1 with the lens re placedby a pinh ole.

Do you really see things upside down? Here is a way you can convinceyourself that the image on your retina is inverted. Look through your pin­hole at a broad light source. Hold a pe ncil poin t in front of the pinholeand look at its shadow on yo ur reti na. Everything behaves as ex pected.Now reverse the order and put the pencil point betwe en the pinhole andyour eye. Move the pencil and notice the direction of motion of theshadow! Now make a sketch and explain what's happening .

Exercising the pup ils. Wh en you look at a broad source (like the sky)through your pinhole. yo u see a bright circle. That circle is the projectionof your pupil on your re tina. You can study the dilation and contractionof your pupil by co vering and uncovering yo ur other eye. the eye that isnot looking th rough the pinhole! When you uncover the other eye so thatlight enters it , its p up il contracts. So does the pupil of the eye lookingthrough the pinhole! You can easily see these "sy mpathe tic" pu pil con­tractions. Notice tha t it takes a time of the ord er of t sec for the pupil tocontract or dilate when the light intensity is suddenly changed.

/ EyeplCce

:' t .

Telescope. A telescope cons ists of two lenses. The first is the "objective"lens, which forms a real image of a distant object. To a good approxima­tion, the image is in the focal plane of the objec tive lens. If 80 is the an­gula r size of the dista nt object and [v is th e focal le ngth of the objectivelens, then th e height h , of the image form ed by the objec tive lens ish, = f ,00. Th e seco nd lens of a telescope is called the eyepiece. It ise ffectively a simple magnifier that is used to examine the real imagefa nn ed hy the objecti ve lens. If the eyepiece is adjusted so that the imageform ed by the objec tive lens is in the focal plane of the eye piece, then apoint on the image gives a parallel beam into the eye. Then the eye is re­laxed , just as if it were lookin g at the distant object wit bout the telescope.The an gular size subte nded at th e eyepiece by the image of height h, ishd f" where f, is the focal length of the eyepiece lens. This is larger thanthe an gular size 00 by th e rat io (hdf,) / Oo = (1, 00 /12)/ 00 = f df,. Th usthe angular magnificati on is it / f,. See Fig. 9.32.

ft"icrm~cnlJe. A microscope is like a telescope in having an objective lensto form a real image of the object and an eyepiece to examine this image.The bug to he examined is nearly (but not exactly) at the focal plane of theobjective. The image is formed a long distance L from the objective- sayL "" 20 ern. Thi s distan ce is esse ntially the length nf the barr el of themicroscope . A bug of width x located at distance approximately 11 fromth e objecti ve gives a real image of wid th h , = (L/h)x at the image point.This image is a distance h from the eyepiece and subtends an angle h1/hthere. If the bug were examined with the naked eye at distance 25 cm, itwould subte nd angle x/ 25 em. Thus the magnification is (hdf, )/ (x/ 25) '=25L/!tf, . See Fig . 9.33.

Fig. 9.33 Microscope.

516 Interference and Diffra ct ion Sec. 9. 7 517

You can use Eq . (100) and a jar to measure th e ind ex of refraction of wateror of (for example) min eral oil. [E quation (100) holds for eith er a cylinde ror a sphe re.) See Home Exp. 9.4 2.

Now follow th e ray to the rear surface. It gets closer to th e axis by anamount 2R ti mes th e deviation 6. Thus it rea ches th e rear surface at dis­tance h' that is

h' = h - 2R8 = h - 2h (1- ~) = hH-1). (98)

At the rear surface the ray is again deviat ed toward the axis. By th e s}m­metry of a circle about a cho rd, the deviation upon eme rging is the sameas th at upon enterin g. Thu s th e ray em erges at an angle 25 to the axisand at lateral distan ce h', It will th erefore hit the axis at distance f 'beyond the last surface , whe re

(99)

(100)R (2 - n)"2 (n - 1) '

h'28=7Equa tions (97), (98), and (99) give

h (~ - 1)f ' hI =.:.n:-_-:--:-=28= ~ (1_ ~)

L eeuwenhoek 's mi croscop e. The world 's first microscop e was merely atin y glass sphere. You ca n make one. (You can obtain tin y clear glassspheres by th e pound from a chemical supply house. Make sure th ey areclear inst ead of tr anslucent. ) Here is how it works. Pu t the sphere rightin fron t of your eye. Put the bu g (to be looked at) at the focal point F of Fig.9.34. A point on th e b ug gives a pa ralle l b eam of light ente ring th e eye .Since it is a parallel beam, you ca n relax you r acc ommod ation mu scles,and th e bea m will focu s to a point on the re tina. Anothe r point on th ebug will focus at anot her poin t on th e ret ina. Let us calculate the magm­fication of th is lens. Suppose th e bug has lateral extension rbug . Raysfrom th e extremes of the bu g thr ough th e cen te r of the sphe re are notdeviated . Tha t means th at th e angul ar size of th e bug is Xb Ug divided byth e distance fro m F to th e ce nte r of the sphe re :

D eviation at a single spherica l surf ace. Le t us trace parallel rays thr oughthi s lens. Th e ray thr ough th e cente r of th e sphere or cir cle is no t deviated.The ray at tr an sver se distance h from the cen ter lin e makes an angle ofinci dence OJ given by 0, = h/R, for hlR <: 1. Th e devialion 8 of this rayat th e first surface is eq ual to th e angle of inc ide nce ()i minu s the an gle ofrefraction Or. For small an gles, Snell's law, n l sin () j = na sin Oz, beco mesnl()l = n z()z. Th en th e deviation toward the llorma l at one surf ace isgiven by

Index = 1

/l = 0, - 0,

=01 ( 1 -~)

---.----F

Thi ck spherical or cylindrical lens. A sma ll glass bab y-food jar makes agood cylindri cal lens. (We recommend chocolate pudding. Eat the pud ­din g, scrape off th e label, and fill th e clean jar with wa ter or an y othercl ear liqui d.) In Fig. 9.34 we show th e form ation of an ima ge of a parallelheam of light by such a lens.

Fig. 9.34 Example of "thick" lens.Th e fo ca l poi nt F is locat ed a di stance f'bey ond the last surface . Ind ices of re­fraction: a ir = 1, lens = n.

Equatio n (96) is general (for small an gles) and is useful for tracing raysthr ough complicate d syste ms. In th e present example, we find for th edeviation at th e first surface

(101)(96)

(97)

e -~bUg - R + f"

T his is the angle bet ween th e paral lel beams th at corr espond to the imag esof the ext remes of th e bug on YOill reti na and is th erefore th e angu lar sizeyou " see" using th e microscope. When you look at th e bug witho ut themicroscope, you must hold th e bug abo ut 25 cm away to focus on it com­for tably. Th e angular size of th e bug is th en xbug/25 em. Th e angular

518 Interference and Diffra ction

magnificat ion AI is therefore

Problems and Home Experime nts

Problems and Home Experimen ts

519

Fig. 9.35 Retrodirection of light by anideal Scotchlite refl ector having index1) = 2.

(10-2)

Th us, for example, if R = 1 mm and n = 1 (glass), we get M = 167.

Scot chli te Tetrodir eclive refi ect or. If n = 2, then according to Eq . (98) ap araxial ray entering at transverse dista nce h strikes the rear surface of thesp he re of Fig. 9.32 at distan ce h' = O. Thus a parallel beam is focusedexactly at the rear surface. The bea m is partly reflected and partl y trans­mi tted the re. Th e reflected part is eve ntually directed back at 180 deg totb e original directi on, as is seen by insp ection of Fig. 9.35. Th e transmittedligb t at (be rear sur face can be largely reflected back into th e glass byco vering th e rear surface with a silvery reflector.

A reflecting mater ial called Scotchlite, whic h uses th is principle. can beobtain ed at any hardware store . It is used for bright road signs, a mongot her thin gs. Examin e it with a magnif yin g glass. You will see that itco nsists of man y tiny glass sphe res e mbedde d on a sticky silvery surfaceand th en paint ed wi th clea r red sbe llac (for Ted Scotchlite) or somethi ngelse f OT othe r e ffects . It turns ou t t hat the largest index th at one caneasily ge t witb glass is abo ut n = 1.9. This is close enougb to 2 so tbat itwo rks fairly well.

The " next genera tion" of the world's largest liquid hydrogen bubblechambe rs, now (1968) being designed, will (at least some of them will) useScotchlite on the bott om of th e cha mbe r to retr odir ect light rays towardtheir source. You can easily measu re th e ret rcdlr ectt ve properties ofScotchlit e. See Home Exp. 9.35.

9. 1 Nea r fiel d a nd far field . How far away should you be from a double slit ofslit spacing 0. 1 mrn irradiat ed w ith visible light in order to use the far-field app roxi­mation wi thout making use of a le ns? How far should you be from two microwa veantennas having spacing 10 em and emitt ing 3-c m mic rowaves to use the far-fieldap proximation?

9.2 A doub le slit of slit separ at ion 0.5 mm is illumina ted by a paralle l beam from aheliu m-neon laser tha t emits monochr omatic light of wavelength 6328 A. Fivemeters beyond the slits is a screen . Wh at is the separation of the interfer ence fringesall the screen?

9.3 Wh at is the "mean length" of the classical wave train (wave pac ket) correspond ­ing to the ligh t emitte d by an atom wi th mea n decay tim e 10- 8 sec? In an ordinarygas-discharge source the atoms do not decay freely but rather have an effect ivecoherence time :::: 10- 9 sec due to Doppler broade ning and collision broadening.\ Vhat is the length of the correspon ding classical wavetr ain?

9 .4 If a " line" sou rce of visib le light is not real lya line bu t has width 1 mm , howfar must it be from a doub le slit whic h it illu minates in order for the two slits to be

reasonably cohe rent? Assum e the slit separa tion is t mm .

9 .5 How far away is an aut omob ile when you can barely resolve the two headlightswith your eye?

9 .6 Ven us has a diamete r of abo ut 8000 miles. \Vhen it is visible as a " morni ngstar" (or "evening sta r" ), it is abo ut as far away as the su n, i.e ., ab out 93 millionmiles . I t looks " larger th an a poin t" to th e unaided eye. Are you seeing the truesize of Venus?

9.7 Resolut ion of the eye . Tak e two light bulbs of the same power (say 150 watt),one with a clear glass envelope and reas on ably small filam ent ( :::::: 1 in . by t in.), theother with a frosted bulb zz 3 in. in diameter. Find out by experim ent bow far awayyou must walk before the two light s have th e same apparent size. (It will be a blockor two. ) At this same large dista nce , compa re the appare nt sizes of two frosted bulbshaving the same actual size but differin g in power hy a factor of two or three. Howdo you explain the resul t? why does Venus look larger th an a point? (See Prob. 9.6.)

Home experiment

520 Interference and Diffraction Problems and Home Experiments 521

Home experiment 9 .8 Diffract ion gr ating mo ire patt ern. You need a whi te line source and twoidentical gratings. [Th e best line source (which you will need for many of these ex­

peri ments) is a "dis play lamp." For example, one of abou t 40 watt s with a 3-in . longstraigh t filament in a clea r glass envelope is availa ble for abou t 40 cen ts in groce ryand hardware sto res. Your optics kit has only one grating. More gralillgs are availa ­ble for abo ut 25 cents each (or about JOcents eac h in boxes of 100) from , e.g. , Edmun dScientifi c Company, Barrin gton, N.J. 08007 .] With the line sourc e oriente d vertically,look thr ough one grating (hold it up d ose to one eye) and orient the grating so thatthe colors are sp read ou t ho rizontally. Now superpose the seco nd grating on the first.Carefully rotate it so as to superpose exactl y the first-ord er images from the two grat ­ings. With car e, you will succeed (in a minute) ill obtaining " black stripes " acrossthe colored first -ord er image. Here is part of the expla nation. Th e line spacing onthe gra ting is d . Suppose the space betw een the plan es of th e two gra tings is s.Think of them as tw o pick et fences supe rposed with a slight space be twe en them oras two identi cal sc reens parallel to one another. At some ang les, the grating scratcheswill l ie one behind the othe r. At other angles, the scratches of one grating will lie (inprojecti on) halfwa y between the scratches of the othe r. At thes e angles, the effectivenumber of scratc hes pe .. unit length (f.e ., d- 1) is doubl ed . Now co mes the physics :\Vhy do you get the black stripes? Do they correspond to angles where the effectivenum be r of lines is " sing le" or " dou ble"? Given the number of tines per em, d:», foreac h gra ting, how ca n you dete rmine the spaci ng s? Give n s, how can you deter­

mine d ?

Home experiment 9 .9 Si lk-stocking diffra c t ion patt ern. You nee d a shee r silk (or nylon) stoc kingand a poin t source of white light . Although a reasonably distan t street light will per ­haps do for th e poin t source, th e bes t point source for this an d othe r expe rime nts ismade from a &-volt flashligh t , for example a " cam per" flashlight , with a bulb that hasa filamen t abo ut! mm lon g. To get a goo d point source , remove th e glass lens andcover th e par abolic reflect or with a piece of da rk 'cloth or pap er (with a hole cut forthe bulb ). Or Simply look a t the bulb from the side, out of the beam from th e reflector.(Note: A "sealed be am" flashlight will not workl)

Look thr ough th e stoc king at the poin t sourc e. From the pattern you see , youcould d etermine the avera ge thr ead spaci ng and the num ber of se ts of thr eads atdifferent an gles. Fold ove r man y layers and look at th e sour ce again . Th e patt ernof concen tric circl es you see is similar to an x-ray "powder diffraction patte rn ."

Home experiment 9 .10 Long -playing diffraction grati ng. Look at a white point source re flected atnear-glan cing inciden ce on a 33- rp rn record. Th e record grooves make a good reflec­tion gra ting. Measure crud ely the wavelengt h of red an d of green light using therecord. Describe your me thod . How can )'ou easily determine the loca tion of thezero th-o rde r "spec ular" maximum?

Home experiment 9 .1 1 Which side ha s th e sc ratches? On e side of the plasti c of your diffractiongrating is smooth; th e oth e r side has the scra tches. You can find out which side has

the scra tches by looking thr ough it at a white source afte r rubbing one side of thegrating wit h an oily finger ; th en clean it and t ry the other side . Wha t is the explanation?

9 . 12 Consider nestled spherical and para bolic mirr ors as shown in Fig. 9.24. Takethe + i dir ection to the right (along the axis of sym metry ) and x tra nsverse to z: take.r :;;:; z :;;:; 0 at the apex of the mirr ors.

(a) Show that the parab olic surfa ce is given by

"Z = 4[ .

(b) Show that the spherical surface is given (for %<.f ) by

(c) Comp are a spheric al mirror with aperture of diameter 0 and with focal length Jto a par ab olic mir ror with the same 0 and f For the sphe rical mirr or, consider theangular dev iation M of the "worst" rays (near the rim of the ape rture ) due to sphe ri­cal abe rrati on. (68 is the deviation from the i direc tion for rays from a point source ).Show that 68 is less than the diffra ction angular width ~8 =:: AI D provided that

Th us (for example) for visible light and for focal length f ~ 50 iu., a sphe rical mirroris abo ut as good as a parabolic mirror prov ided the mirror diameter D is less than-3.5 in .

9 .13 A plane slab of glass of thi ckn ess t and ind ex n is insert ed be tween an observer'sey e and a point source. Show that the point source app ear s to he displaced to apoint closer to the observe r by approximatel y [(n - l )/ n ]t. Use small-angleapproximations.

9 .1 4 A "co rne r reflect or" co nsists of three plan e mirrors joined so as to form an in ­

side comer of a rectangular box. Show that a light be am that stri kes a corne r reflectoris directed back at 180 deg to its original dir ection, indepe nde nt of the angle of inci­dence, as long as it hits all thr ee surfaces.

9.15 Show that a plane wave normally incident on one face of a wedge-shapedprism of an gle A is deviated by an amout 8dey , where

n sin A :;;:; sin (A + 8du .) .

9 .16 A diffracti on-limit ed laser beam of diamet er 1 em is p ointed at the moon.Wha t is the diame ter of the area illum inated on the moon? (Th e moon is 240,000 miawa y.) Take the light wave length to be 6328 A. Neglect sca ttering in the earth'satmosphere.

522 lnterj erence and Diffraction Problems and Home Experim ents 523

Home experiment 9 .17 Single-slit diffr acti on patte rn . Tape a piece of aluminum foil by its edges toa microscop e slide. (Th e most convenie nt tape is Scotch translucen t " magic mend ­ing tape .") Cut a single slit with a razor blade or sharp knife. Hold the slit close infront of one eye an d look at a white line source. Estimate the angula r full width ofth e cen tra l maximum by (for exam ple) makin g marks on a piece of pap er that is be­

hind the line source to give a scale. Estima te the ratio of the wavelength of red lightto tha t of green ligh t , where these colors are given by your gelatin filte rs. Using thered filter, estimate the width of the razor cut, i.e., the width of your slit, using themeasur ed angul ar width of the diffraction pattern and assuming A -... 6500 A. If youhave a magnifying glass, you can lay your slit on a millimeter scale and estima te theslit width directly. H ow do the two results for the width co mpare?

Herne experiment 9 .18 Double-s lit d iff racti on and - interfe re nce pa tt e rn. Make tw-o par allel slitsse parated by t rom or less, using the technique of Home Exp. 9.17 . Make one slitabou t i cm longer than the othe r, so that you can go quickly from the double-slit pat­tern to the sing le-slit pa tte rn by displacing the slits slightly. You can th us see what

part of the double-slit pa ttern is the "Single-slit modula tion : ' due to the nonzerowidth of the single slit. To see the effect of variab le slit spacing d easily, cut one slitat a slight angle to th e othe r, so th at they c ross in a "vee" shape. You should mak e

many slits (it takes 10 sec to make one pair of slits-the ten th ti me )'ou do it); somewill be be tte r than others. (Hold the slit up to a light and examine it to see wby it is

a bad one , if it is.)

Herne experiment 9 .19 Three -sl it pattern, Try this only aft er you have made a num ber of gooddoub le slits by the techniqu e of Home Exps. 9. 17 and 9. 18. Cut a th ird slit parallelto the first two slits. Make th e thi rd slit not as long as the first two, so that you canqui ckly go from two to thr ee slits by a slight translation. TIle important thin g to try

to see is the narrowi ng of th e inten sity maxima when the third slit is adde d. [Youca n ge t a very bea u tifu l set of single, doub le , tripl e , and qu adruple slits, alon g wi thslits of var iable width and an a rray of up to 80 slits, al] mounted on a single conve ni­cot slide called th e Co rnell Slit£ lm Dem onstr ator, which is obtainable from Th eNational Press, B50 H ansen Way, Palo Alto, Calif. List price is $1.50.]

Home experiment 9 .20 Coherence -size of a " point " source or line source, Use a Single slit ofknown (estim ated) w idth. Place the red gelatin filter over the source. Stand farenough from the sou rce so tha t you obtain a sha rp single-s lit patt ern . Now movecloser to the source . Find the distan ce L at which the single-slit pattern "washes out: '[It washes out at a distance at whic h differ ent parts of the filament o f the flashlightbulb (if tha t is your point source) become independent ligbt sources an d thus are in­

coherent for the resolution tim e of your eye, as discussed in Sec . 9.4.J Use your esti­mates of th e sizes of the source and the slit and your meas ure ment of the distance Lat which th e patt ern washes out to estimate the wave length of the light . using the re­

lation derived in Sec . 9.4, d(source) D(slit) '" LA.

9 .2 1 Coherence -Lloyd ' s mirror , th e " gua ranteed coh e re nt doubl e s lit ." if

you hold au ordinary doubl e slit in front of one eye and look at a broad source likethe sky or a h osted light bulb, you will see no interfer ence pa tte rn. \Vby is tha t?'w e shall now design a double slit that will give a two-slit interferen ce patt ern evenw hen you look at a frosted light bulb . First make a single slit by the techniqu e ofHome Exp. 9. 17. Now take a seco nd microscope slide an d place it edgewi se to thefirst slide (the one wi th the slit) and parallel to the slit , so tha t th e mirr or ima ge of theslit in the sec ond slide is parallel to th e first slit. Stick the seco nd slide to th e firstwith a big glob of putt y or mod eling clay (nondryi ng putt y, e.g. , Iu-Cl aze glazingcompound works well), SO tha t you can easily \\i ggle the second slide to adjust it, butit will st ay in place when you don ' t push on it. Adjust the mirror (the second slide)so as to get as narr ow a separ atio n betvv·een the slit and its " image slit" as you canma nage-sa)' t mm. Do this by holding the assem bly a foo t or so from )'our bead,so tha t you can eas ily focu s your eyes on the "double" slit while )'ou hold it in frontof a bright bac kground and ad just the mirror. Wh en you have a good double slit,bring the assembly up close to c ue eye and focus the eye at a large distance [i.e., onth e light source). Loo k for th ree or four "b lack stre aks" parallel to the "coh erentdoub le slit ." Th ese are interfere nce ze ros due to destructi ve inte rfe rence be tweenthe light comi ng from th e real slit and tha t coming from th e image slit . Th e imageslit is of course always co mplet ely coherent with the real slit. (' Vh y?) Because ofth e phase rever sal in reflect io n , the slit curre nts and the " image slit curre nts " are180 deg out of phase. Th erefore the fringe at the plane of th e mirr or is "b lack"-aninterfe rence zero. Here is a qu estion , to be ans wered bot h b y experiment and by" theory" : Are th e " bright" lin es between th e " black" lines exactly as brigh t as thebright backgroun d th at one sees with just the single slit? Bright er? Dimmer?

9.22 Paper-clip Lloyd' s mirr or . (Sec Hnme Exp. 9.21.) A pape r clip illumi natedby a light bulb gives a shiny narro w line source. Hold the clip against (and parallelto) the edge of a microsco pe slide used as a mirr or. when you get a decen t looking"c ohere nt doubl e slit" of separa tion less than t mm, brin g it up close to one eye andlook for the dark interference ba nds discussed in Home Exp . 9.2 1. It takes a littl emore practice tha n the metho d of Hom e Exp . 9.2 1. Th e light must be at nearlygr azi ng incidence on the mirr or . Also the illumin ation should be arranged so thatthe light sourc e doesn 't blind you.

9 .23 Two -dlmenslonal diffr action pa tte rns. (a) Look at a distan t str eetlightthr ough a piece of ordin ary window scree n, Turn the screen sideways so tha t thepr ojecte d wire separa tio n is as small as you please. Problem: How far away must ast ree tlight of 2O-cm diam eter (frosted bulb ) be in orde r to give coherent illumin ation

over two neighboring wires of the scree n?(b) Look at a st ree t ligh t or your 8ashligbt point source through vario us kinds of

cloth-a silk handkerchief , nylon p anties, an umb rella. et c.

Home ex.periment

Home experiment

Home experiment

524 Interf erence and Diffm ctionProblems Gild Home Exp eriment s 525

Home experi ment

Home experi ment

(e) Look at a point source through tw o diffracti on grat ings of the typ e tha t you havein your kit. Rotate one gra ting so that its lines are perpendicular to those of the first.. otice that one gets some (rather faint ) bright spots at 45 deg to the two sets of lines.These spots are something new, not obtained by superposing int ensities from the twogratings. Of course th ey mus t be due to superposition of amplitu des from the twosets of lines . Make a sketch and explain the origin of these "extra spots." The dlf­fraction pattern produ ced by two crossed gratings is similar to the pattern producedby diffraction from a single crys tal. You may have seen the movie made for EducationDevelopment Cen te r (EDC, formerly ESI) by L Germer, showing diffracti on of amonoenerge tic electro n beam reflect ed from th e surface of a single crystal. (Thetechnique is easier for reflected than for transmitt ed waves, if one wants to look at asingle crystal. Similarly, you ca n ge t reflection gratings from Edmu nd Scien tific Co.Th ey are like your transmission gra ting. excep t that the sur face is lightly silvered toenhance reflection.)

9 .2 4 Diffr action gr atin g-gel at in filt er passb and s . Use your diffraction gratingas follows to measur e the wavelength s of the red and green passed by your filters.Put a line (or point) source right next to a wall or door. Make a mark on the wallabout a foot to the side of th e source. Look at the source thr ough the grating, hold­ing th e filter over your grating (or put the filte r over th e source - but don't melt it!)Move closer and farth er from th e source until th e color of interest app ear s to besuperposed with your mar k on the wall. Measure th e appropriate distan ces and cal­culate A. Th us calibrate the wavelengths tra nsmitted by your red, green, and purplefilters. 'te morize the results. (Th en you can use your filte rs and the gra ting to findthe wavelengths of oth er colors when you wish to, without repeatin g the geometricmeasureme nt of this experimen t.)

9 .25 Spectral lines. Pour some table salt on a wet knife or spoon (one that youdon 't mind m ining). Set the knife in the Harne of a gas stove. Look at the yellowflame throu gh your diffraction gra ting (thi s is easiest at night in a darkened room).Notice tha t the first-orde r (and higher-orde r) images of the yellow sodiu m flame areas sharp and clear as the zeroth-order " dir ect " image. Th at is because th e yellowlight is a "spec tral line " havin g narrow bandwi dth . (Actually the yellow light fromsodium is a "doublet " of two lines with wavelength s 5890 and 5896 A.) Now look ata ca ndle . In zeroth order, it does not look terri bly diHerent from the sod ium Same;they are both yellow. But in the first-order di ffract ion image, th e ca ndle is verymuch spread out in color, whereas th e sodium remain s sharp. Th e "yellow" of thecandle, which is du e to hot part icles of carbon, has a wavelength spec trum extendingover (and beyond) the entire visible ran ge.

Here are other convenient sources of sharp spec tral lines: look at them thr ough yourgra ting:

Mercury vapor: Fluorescent lamps, mercury-vapor street lights, sunlamps. (A sun­lamp is convenient in that it screws directly into an ordinary l Hl-volt AC socket. Itis probably the cheapes t source of mercury -vapor spec tr al lines; the cost is abou t $10.)

.Veo ll: Many adve rtising signs. Neon has a profusion of lines ; you see "man}'signs." A cheap b road monochromatic source is a G.E . bulb NE -3--l which screwsdirec tly into a llO -volt AC socket (the cost is about 81.60). Others are a "c ircuitcontinuity teste r," which plugs into any wall receptacle and which costs about 81 (ata hardware store), and a neon " night light."

Strontiu m. Stront ium chloride salt (available at a chemical supply house for abou t25 cents/ ox). dissolve a littl e in a few dr ops of water and put it in the gas flame onyour ruined spoon. Th e wavelength of the red line is a famous length standard.

Copper: Coppe r sulfate: availabilit y and technique as for st rontium chloride. Itgives a bea utiful green color.

Hy drocarbon : Look at your gas Bame in the first-ord er spect rum. Th ere are ashar p, clear blue image a nd a sharp, clea r green image. Th e "b lue" color of th e flameis therefore due to one or more almost monochrom atic spect ral lines.

9 .26 Mono chromat ic toi let pap er . Burn a piece of toilet pap er and look at it

thr ough your diffracti on grating (held, as always, close in front of one eye). Noticeth e bea utifully clea r "first-order flame," This shows that th e soft yello w light is al­

most monochromatic, wi th very litt le "whi te light" color spectnun du e to hot carbon.Th e yellow that you see is the by now familia r (we hope) sodium doublet of wave­lengths 5890 and 5896 "-

Now that you recognize "sodium yellow," light an ordinary pape r ma tch and lookat it with your grating . Most of the light is "hot carbon yellow," which is not reallyyellow but a complete " white" color spectrum. But look closely! In th e yellow partof the hot carbon spec t ru m, down low next to the cardboard , where the Harne is"blue" looking- below the blindi ngly brigh t hot carbon spectrum -c-dc )'ou see acrisp, clear littl e monochromat ic match flame? If you don 't, try aga in ! Now burnothe r things and look. You may well conclude that everythi ng is made of salt or is atleast contaminated by it.

9 .27 Fabry -Perot sodium fringes . The world's cheapes t broad. almost monochro­matic light source is obtained by burning a wad of toilet paper. You can use this sourceto see Fabry- Pe rot fringes. Bum the paper. (Th e room should be dark- perhapsalso you shouJd have so me water hand y!) Look through the flame at the image ofth e flame at near -normal incidence in a piece of glass- a microscope slide or pictur e­fra me glass. You will see f inger-pr lnt -llke fringes. If the glass is optic ally 6at , thefr inges will be circles ce ntered on your eyeballs; in any case, you can see them easily.U you have a gas stove or bunsen burner, you can get a brighter mon ochr omaticsodium source by sprinkling salt on a wet knife and immersing it in the Bame. Thenyou ca n see the Fabry-Perot fringes even in the dayt ime. For a nice, st eady, broadmonochromatic source wi th which to look at the fringes, use the G.E. neon bulb NE.J4.

Home exper iment

Home experim ent

52 6 Interference and D ijJractiollProblems and Home Experiments 527

Home experiment 9 .2 8 Mail ing-tube spec tromete r- Fra unh ofe r lines . Usc a mailing tube l ! to2 ft long. Moun t your diffracti on gratin g on one end . Moun t a single slit 0 11 theother en d. Th e slit is best mad e with two single-edged ra zor blades . Glue or tapeone blade per mane ntly in place; stick the othe r on wi th nonharde ning putty (glazingco mpound, obtainable in any hard ware store), so tha t you can easily adjust it (narrowerfor better resolution, wide r for more light ). Look at the specl ra menti oned in HomeExp. 9.25.

Prob lem : Should you be able to resolve the sodium doublet (wavelengths 5890 and5896 A) with this spec tro meter?

Am. No-the line separa tion give n by this grating is just abou t eq ual to the imagewidth due to diffraction in th e pu pil of your eye.

Can )'ou resolve it by using a longer mailing tube?'Ans . No. Th ere are two ways to improve the resolution . On e is to get a grating

with smalle r line spac ing d. The other is to increase the nu mbe r of lines that areused , i.e ., to increase the width D of gra ting used. With the desig n above, D is thewidth of yOUT pup il, abo ut 2 mm. If you add a telescope with an objec tive lens ofdiameter 2 em, and if all th e rays that en te r the objecti ve lens ge t thr ough the pup ilof your eye , then w'tth the diffraction gra ting at the objec tive lens, your angularresolution, 'A/ D. is improved tenfold.

With this Simple spec tro mete r you can see th e Fraunh ofer lines in the spec trum ofthe sun. C o outside on a sun ny day. La y a pile of half a dozen sheets of whitepaper on the ground (more tha n one so that it is as "while as possible"). Look at thesunlit paper with your spect rometer. Use a coat or blanket to cover your head tokeep out stray light ; otherwise you will have difficulty seeing the first-order spectnun.Also, use the edge of the tube to "hi de" the blindingly bright zeroth-order light . Ad­just the slit to about j mm. Look for three or four or five dark lines crossing the con­tinuous spect ru m of th e su n. If you don' t see anything, keep trying- adjust the slitwidth for comfortable intensity. Another technique is to cover th e slit with severallayers of waxed paper , use a very narrow slit. and look at the sky near th e sun , vary­ing the intensity by how close you come to pointing the spec trometer at the sun.

Th e dark Fraunh ofer lines you see are absor ption lines. Atoms in the relativelycool oute r gas mantle of the sun ar e driven by the continuous spect ru m e mitted bythe hot sun. Those frequ en cies that correspond to natur al resonances of th e atomsexcite the atoms. Thi s takes energy out of the contin uous spec tru m at th e resonan tfrequency. Th e outer gas is actuall y opaque at those freq uencies, so th at the spec­trum has corresponding "b lack lines" at colors where the sunlight has been co mpletelyabsorbed. The easiest lines to see are some closely spaced lines in th e yellow-greendue to iron, calcium, and magnesium ; the H line in the blue-green due to hydrogen;and several closely spaced lines in the blue du e to hydr ocarb ons-similar to theemission lines you see with a gas flame. Th e sodium D line is also presen t , bu t hardto see (for me at least). To see where to look for it, look at the sodium emission lineby throwin g salt on a gas Dame. That is the color th at is "missing" in the F raunhoferspec trum. (See the color plate foUowin g p. 528.)

9 .29 Diffra ct ion of wat er waves. Illuminate a bathtub from above with anincand escen t lamp that has a small fila ment in a clear enve lope, so as to get sharpshadows. Ge ner ate traveling wave s tha t are "s traight waves" -the two-dimensionalanalog of plan e waves-by jiggling a floating stick or board placed across the end ofth e tub . F10at a coffee cup as an opaque obstacle. Estima te the distance downstreamat which the "shadow" of the cup is " healed:' Supp ose t ha t you did n't know thediame ter of the cup. Determine this diame ter (approximately) experimen tally bymultiplying th e " healing length : ' Lo, by the wavelength of the wate r waves, A, andtaking the squar e root. ( \Ve assume you know where tha t formul a comes from. SeeSec . 9.6.) This is one way of 6ndin g the diameter of nucl ei-by measuring their dif­fraction "c ross section: ' (Note: It is rathe r difficult to measur e the wavelength of thewater wa ves with th e cru de technique we suggest It is easier to shake the stick at arep rodu cible tempo (as fast as you can ) and then measure th e frequency. The wave­len gth can then be obtained from th e dispe rsion relation for water waves, as tabulatedin Sec. 4.2.) How does yOUT cross-section measurement of th e cup diame ter comparewith a direct measur ement of it?

9 .30 How wide is a " pla ne wave " fro m a distant poin t source ? we have oftensaid that the traveling wave from a distant point source is 'like" a plane wave over a" limited region " tran sverse to the line of Sight from the point source to th e field point.How limited is th e region? Suppose the source is at distance L and we wish to con­sider a circular plane region of radius R transverse to the line of sight from the source.How large can R be so that the phase at the cen te r of the circle and that at the edgeof the circle differ by less th an 6.cp radians?

r1ns . The phase a t the ce nter of the circle is ahead of tha t at the edge (the ce nte r iscloser to the sour ce) by an am oun t acp = 11R2/ U . Thu s th e phase is " the same"over the entire plane of the circle to the extent that the ar ea of the circle is smallcompared with U .

9 .31 The world' s largest pa rabolic radio antenna at presen t, at the National RadioAstronomy Observatory, Gree n Bank, west Virginia, is a para boloid dish 300 ft in

diameter . 'what is its angula r resolution in radians and in minutes of arc (the uni ts

used by astro nomers) for the famous z l -cm radiation of hydr ogen?Ans. A point source will look like a volley ball at a distance of 300 ft .

9 ,32 Telescope " e xit pupil. " Suppose you have a Simple telescope consisting ofan objective lens an d an eyepiece. Th e angul ar magnification is It /h , where h and[z are the focal lengths of the objective and eyepiece lenses, respecti vely. Show thatnot all of th e rays from a distan t objec t which stri ke a very large diameter objectivelens ge t into your eye and that in fact th e "useful diam eter" of th e objec tive lens is

about It /h times the diam eter of your eye pup il. Thus, in an eight-power telesco pe,if the exit beam is a parall el beam of width 4 mm (twice as wide as the pupil of youreye, so that your eye need not be perf ectly Lined up and also so that off-axis poin ts in

the field of view deliver all th eir light), the objective lens should be 32 mm in

diam eter . A larger dia meter is a waste of objective lens.

Home experiment

Home experiment

Home experiment

528 l nteiierence alld Diffracl ian

9.33 Eye-pup il s ize and me nta l ac tivity . If someone shows you a picture of agood-loo king individual of the opposite sex, yom eye-pup il dia meter may increase by

as much as 30%, according to Eckhard H. Hess, SCientific Am erican p. 46 (April, 1965).This large a change is very easy to dete ct in your own pu pil by using a pinhole in apiece of aluminum foil that covers one eye, wi th a brigh t source illuminating the pin­bole, as discussed in Sec. 9.7. Perh aps by just thinking, you can vary your pupil size,dep ending on wha t you thin k ab out. Have someone read to you. (Concentra te onlistening. not on the pupil size .}

9.34 Diffraction by an opaque obstacle. This expe riment works well with a whitepoint source consisting of a 6-volt " campe r" flashlight with lens removed and reflectorcovered by dark cloth. (The filament size is about t mm.) Th e source should be atleast thr ee meters from the obstacle, so tha t you get a decent "coherent plane wave"over an obstacle the size of a pin . TIle detecting " screen" is a microscope slide onwhich is stuc k a layer of Scotch translucent magic mending tape. Make the shadowof the object fall on the screen with the screen held a foot in front of your face (orwhatever distan ce you find comfor tab le for looking at the screen). Your eye shouldbe almost in line with the light source an d the image on the screen so as to takeadv antage of the lar ge inten sity scattered at small angles (about the forward direction)from the translucent screen. Aside from looking at the bea utiful fringes, one putpJseof th e experiment is to explore (crudely) the concept of the "length of the shadow,"Lo, given by LoA :::::: D2, where D is the width of the obstacle . Among othe r thin gs,look at a pin (if the pin width is -t mm, then Lo :::: 50 ern for visible light) and ahum an hair (yours). [Mine has width about -itr mm. This gives Lo:::::. t cm.]

F irst conside r the pin. Place the screen 5 or 6 meters downstrea m from the pin.Th e diffraction image should then be large enough so that you do not need a magni­fying glass. It may belp to jiggle the screen slightly, so as to wash out the effect ofthe irregularities of the magic mendin g tape. Notice the famous bright spo t at thecen ter of the "shadow" of the pinhead and the bright line at the center of the pinshaft. Is the bright spot or line brighter or dimmer than the bright screen Itself (at apoint well outside the image)? Next examine the image of the pin with the screen ata distance of only 5 ern downstream from the pin. (You will need a magnifying glass,unless you have very good eyes.) Notice that the shadow is a nice solid black, withno brigh t spot in th e center. Th at is because you are much closer than Lo. At theedges it shows fringes, as is expected from our discussion in Sec. 9.6.

Next conside r the human hair. Put the scree n immedia tely behin d the hair (Le.,about 1 mm downstr eam). Look at the shadow wi th a magnifying glass. It shouldbe nice and black, since L is small compared with 4. Now go to a distance of a fewcentimeters. You should see nice fringes. Go to 5 or 6 meters downstream. This is

several hun dred times 1..0. According to our discussion, the shadow should be practi­cally "healed" and the image of the hair very difficult to see against the backgronndof light from the source . Your eyes are very sensitive detectors of constrast, and youwill see something. Look at other things, knife-edges, holes in aluminum foil, etc.

OPTICAL SPECTRA