Physics Optics vol 1

7/28/2019 Physics Optics vol 1

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" W h e n t w o U n d u l a t i o n s [ w a v e s ], f r o m D i f f e r e n t O r i g i ns , c o i n c id e e i t h e r p e r f e ct l y o r v e r yn e a r l y i n d i re c t io n , t h e i r j o i n t e f f o rt is a C o m b i n a t i o n o f th e M o t i o n s b e l o n g in g t o ea c h . "

~Thom as Young,stating the p rinciple of interference in "Theory of Light and Colours," 1801

C h a p t e r 1 6OpticsC h a p te r O v e r v i e wSect ion 16.1 in troduces the chapter . Sect ion 16.2 in troduces in ter ference and d i f f rac-t ion, largely in the c ontex t of wa ter w aves . (However , these ideas apply to a l l types ofwaves, including sound waves and l ight waves.) Having in troduced a l l these phenom-ena, the rema inder o f the chap te r d iscusses l igh t f rom a some wha t h is to rical v iewpo in t .

Sect ions 16 .3 -16 .5 d iscuss the ear ly v iews o f l igh t , wh ich tended to th ink o f l igh tas a part ic le emi t ted by a source . A m ore comple te d iscus s ion~ not the purpose o fthe p resent wo rk ~ w ou ld a lso inc lude the s tudy o f focus ing by mi rro rs and lenses,and appl icat ion of these focusing pr incip les to opt ica l instruments l ike the te lescopeand the microscope. Sect ion 16.3 presents a br ie f h istory of opt ics to the end ofthe 17th century. I t emphasizes tw o of the successes of the par t ic le emission theory:exp lana t ions o f the laws o f re f lec t ion and re f ract ion , and the fo rma t ion o f ra inbows.Sect ion 16 .4 enumera tes a number o f 17 th -cen tu ry exper iments on l igh t , many o fwh ich are expla inable only by the wave theory. Se ct ion 16.5 summa r izes the som etimescon t rad ic to ry theore t ica l deve lopments o f the 17 th cen tu ry .

Sect ions 16 .6 -16 .8 cons ider the m ore modern v iew o f l igh t , wh ich is based upona wave v iew po in t . (Bo th the par t ic le and wave v iewpo in ts w ere specu la ted on by theanc ien t Greeks, some 2 000 years ago . ) Sec t ion 16 .6 in t roduces Young, app lies h isanalysis of cons truct ive and destruct ive in ter ference in num erous con texts, and ana-lyzes h is fam ous tw o-sl i t in ter ference ex per imen t. S ect ion 16.7 in troduces F resnel andgives his analysis of the observed intensi ty patterns both for Young's two -sl i t in ter -ference expe r imen t and for single-sli t d i f f ract ion. Sec t ion 16.8 d iscusses b irefr inge ntcrysta ls, by whic h the ph eno m eno n of polar izat ion was d iscovered.Sect ions 16.9 and 16.10 d iscuss the appl icat ions of d i f f ract ion , by wh ich w e analyzeboth the very large and the very smal l . Sect ion 16.9 considers the d i f f ract ion grat ing,by wh ich we de te rmine the n a tu re o f the very d is tan t s ta rs . Sect ion 16 .10 cons idersdi f f ract ion by crysta ls of x- rays ( like l ight, a form of e lectrom agn et ic radiat ion) , bywh ich we de te rmine the pos i t ions o f ind iv idua l a toms w i th in a crysta l, a

16ol

6 7 8

I n t r o d u c t i o nM a n h a s b e c o m e a c e l e s t i a l d e t e c t i v e ~ h i s f o r e n s i c t o o l s t h e t e l e s c o p e a n d t h ed i f fr a c t io n g r a t i n g ~ i n v e s t i g a t i n g t h e b i r t h , e v o l u t i o n , a n d d e a t h o f t h e s ta rs .1 . Th e t e l e s cope , w i th a w id e ap er tu re , l e t s l igh t o f a l l pu re co lo r s pas s th ro ug hin nea r ly a s t r a igh t line , in tens i f ied by focus ing w i th a lens . Th i s i s an ap p l i ca t ion


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16 .2 Interference and D iffraction 679

o f geometrical optics, w h i c h i n c lu d e s t h e f u n d a m e n t a l p h e n o m e n a o f r e fl e c ti o nand r e f r ac t ion a t s u r faces .2 . The d i f f r ac t ion g ra t ing , w i th e i the r many na r row l ines fo r s ca t t e r ing o rm a n y n a r r o w a p e r t u r e s f o r t r a n s m i s s i o n , a n a l y z e s t h e l i g h t i n t o i t s c o m p o n e n tco lo r s . Th i s i s an app l i ca t ion o f physical optics, w h i c h i n c l u d e s polariza t ion (dis-c u s s e d i n C h a p t e r 1 5 ) a n d t h e v e r y g e n e r a l a n d c l o se l y r e l a t e d w a v e p h e n o m e n ak n o w n a s interference a n d diffraction ( to b e i n t r o d u c e d i n S e c t i o n 1 6 .2 ) .C h a p t e r 1 5 's d is c u s si o n o f l i g h t a s a w a v e p h e n o m e n o n d e p e n d e d h e a v -i ly u p o n m a t h e m a t i c a l r e a s o n in g . T h e s o l u t io n o f M a x w e l l ' s e q u a t i o n s y i e ld e dp l a n e w a v e s o l u t i o n s o f a ll f re q u e n c i e s a n d w a v e l e n g t h s , c o r r e s p o n d i n g t o a c o u -p l e d o s c il la t io n o f t h e e l e c t r ic a n d m a g n e t i c f i e l d s ~ e l e c t r o m a g n e t i c r a d i a t i o n ~t r ave l ing a t the s peed o f l igh t in vacuum. The iden t i f i ca t ion o f l igh t a s a s mal lp a r t o f t h i s s p e c t r u m w a s c le ar . H o w e v e r , e v e n w i t h o u t k n o w i n g t h a t l i g h t is a ne l e c t r o m a g n e t i c w a v e , b y 1 8 01 T h o m a s Y o u n g h a d e s t a b l i sh e d t h a t l i g h t w a s atyp e o f w ave , w i th a spec i fi c r ange o f w ave leng ths .

16o2 I n t e r f e r e n c e a n d D i f f r a c t i o nA l t h o u g h t h i s c h a p t e r p r e s e n t s b a c k g r o u n d m a t e r i a l a b o u t o p t i c s in g e n e ra l , t h i ss e c t i o n p r e s e n t s w a t e r w a v e e x p e r i m e n t s t o i l lu s t r a te t h e p h e n o m e n a o f i n t er f e r-e n c e a n d d i ff r ac t io n . T h e s e p h e n o m e n a g e n e r a li z e to o t h e r t y p e s o f w a v e s, s u c has s ound and l igh t .W e c a n d i r e c t l y o b s e r v e t h e m e d i u m - - t h e w a t e r s u r f a c e ~ a s s o c i a t e d w i t ha w a t e r w a v e . A s Y o u n g d e m o n s t r a t e d i n h i s l e c t u r e s t o t h e R o y a l I n s t i t u t i o n( 1 8 0 2 - 1 8 0 3 ) , w h e n a p lu n g e r i n a ripple tank con ta in ing w a te r o s c i l l a t e s up andd o w n a t a fi x e d f r e q u e n c y f , t h e r e i s a n e x p a n d i n g w a v e o f w a v e l e n g t h ~ t h a th a s c i r cu l a r s y m m e t r y . A t a n y i n s t a n t o f t i m e t h e r e a r e m a x i m a ( pe a ks , o r c r e st s )a n d m i n i m a ( t ro u g h s ) . F i g u r e 1 6. 1 p r e s e n t s a s c h e m a t i c o f t h e t o p v i e w o f a

Maximum am plitude(wave peak)

Figure 16.1 Schematic of the top view of a r ipple tan kcontaining a plunger that oscillates vertically at a fixedfrequency f . The dark circles indicate, at a given instantof time , the m axima, or wave peaks (crests). The distancebetw een the peaks is the wavelength X. In water, Xf = v,whe re v is the wa ve velocity.


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680 Chapte r 16 ~ Opt ics

r i p p l e t a n k . T h e c i rc le s d e n o t e t h e i n s t a n t a n e o u s l o c a ti o n s o f t h e w a v e p e ak s ,w h i c h m o v e r a d i al ly o u t w a r d w i t h v e l o c it y v = f )~ . B e c a u s e w a t e r w a v e s a r ed i spe r s ive , v va r ie s wi t h )~, r a t he r t h a n be i ng a c ons t a n t , a s i s t he c a se fo r sounda nd l i gh t .

16,2.1 I n te r f e renceW h e n t w o i d e n t i c a l p l u n g e r s w i t h i n a r i p p l e t a n k o p e r a t e a t th e s a m e f r e q u e n c yf , a n d t h e r e f o r e a t t h e s a m e w a v e l e n g t h X , t h e y m a y p r o d u c e a p a t t e r n o f m a x -i m a a n d m i n i m a o n t h e w a t e r s u r fa c e. I f tw o m a x i m a ( o r t w o m i n i m a ) a r r iv ea t a po i n t P a t t he s a m e t i m e ( i. e. , i n pha se ) , w e s a y t ha t t he re i s constructiveinterference. S e e F i g u r e 1 6 . 2 (a ) . I f a m a x i m u m a n d a m i n i m u m a r r iv e a t a p o i n tP a t t he s a m e t i m e ( i . e. , ou t o f pha se by 180~ we s a y t ha t t he re i s destructiveinterference. S e e F i gu re 16 .2 (b ) .L e t t h e t w o p l u n g e r s b e s e p a r a t e d b y t h e d i s t an c e d a n d l e t t h e w a v e l e n g t h b e),. T h e o v e r al l p a t t e r n o f m a x i m a a n d m i n i m a d e p e n d s o n t h e r e l at i v e p h a s e o ft he p l unge r s ; fo r s i m p l i c i t y , c ons i de r on l y t he c a se whe re a t t i m e t t he p l unge r s 've r t i c a l d i sp l a c e m e n t s A ( t ) ha ve t he s a m e a m pl i t ude a a nd pha se ~ b : A ( t ) =a co s( co t + ~ ) . A s w i ll b e s h o w n , t h e p a t t e r n o f m a x i m a a n d m i n i m a d e p e n d s o nt h e r a t io o f ~ t o d an d is i n d e p e n d e n t o f t h e c o m m o n a m p l i t u d e a a n d p h a s e ~ .Con s i de r t h a t t he f r e qu e nc y f c a n va ry, a nd t ha t i n i t ia l l y it i s ve ry l ow, wh i c hm e a ns t ha t ~ i s ve ry l a rge . In t h i s c a se , t he wa ve s p roduc e d by t he t wo p l unge r sa re ve ry ne a r l y i n pha se e ve rywhe re , g i v i ng a pa t t e rn m uc h l i ke t ha t fo r a s i ng l ep l un ge r (F i gu re 16 .1 ) . No w l e t f i nc rea se , wh i c h de c re a se s ;v. Th i s c a use s de -p h a s i n g b e t w e e n t h e w a v e s p r o d u c e d b y t h e t w o p l u n g e r s. F i g u r e 1 6 .3 d e p i c t st he c a se wh e re d ~ 0 .7 )~, t he o pe n c i r cl e s de no t i ng t he p l unge r s . The re , t h e t wowa ve s a re i n pha se (de no t e d by c l o se d do t s , w i t h a n i m a g i na ry l i ne C c onne c t -i n g t h e d o t s ) o n l y a l o n g t h e p e r p e n d i c u l a r b i s e c t o r o f t h e l in e c o n n e c t i n g t h ep l unge r s . A s t i m e pa s se s , t he m a x i m a m ove ou t w a rd , a s do t he do t s. (T he l ine s

A dd

Mi n i ma

CancelM a x i m u m pSl

l mu m

f$2

(a) (b)Figure 16.2 (a) Construct ive interference between waves produced by two sourceswith th e same am pli tude, wavelength, and phase. M inima (and a half-period la ter,maxim a) arrive a t the same t ime, giving a doubling o f the net wave am pli tude.(b) Destruct ive interference between two waves of the same wavelength and fromsources that are in phase with one another. M inima of one and maxim a of the otherarrive at the same time, giving zero net wave amplitude.


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16 .2 In ter ference and D if fract ion 681

C o n s t r u c t i v e ~ x ~ D 'inter fe rence . /7 " , , / /

structiveinterference ~ ~' ~ ~ D

Figure 16.3 I n t e rf e r e n c e b e t w e e n w a v e s f r o m t w o s o u r c e s( the open c i rc l es ) wi th the same ampl i tude , f requency,and phase , s epara ted by d = 0 .7)~ . Con s t ruc t ive in t e r -fe rence occurs only a long the l ine C . Des t ruc t ivein te r fe rence occurs only a long the curves D and D ' .

D a n d D ' c o r r e s p o n d t o m i n i m a , o r d e s t r u c ti v e i n t e rf e r e n c e, t o b e d i s c u s se dshort ly . )

I f f i s f ur t h er i n crea s ed , s o t h a t d > z , t h ere a re a d d i t i o n a l curv es a l o n gw h i ch co n s t ruc t i v e i n t er f eren ce o ccurs . F i g ure 1 6 . 4 d ep i c t s t h e re s p ec t i v e ca s e sw h ere d = 1 . 5 ) ~ a n d d = 4 .5 ) ~ . F o r d = 1 . 5 Z , in a d d i t i o n t o t h e p erp en d i cu la r

I n d i v i d u a l C o n s t r u c t i v emaximum interference

1

0

Constructiveinterference

1 3210

-1-2

2 -3

a = 4.~x -4(a) (b)

Figure 16.4 In te r fe rence be tween waves f rom two sources ( the open c i rc l es ) wi ththe sam e ampl i tude , f requency, and phase . ( a) Separa t ion d = 1 .5z , wher e the re a rethree curves a long wh ich cons t ru c t ive in t e r fe rence occurs . (b) Separa t ion d = 4 .5Z,wh ere the re a re n ine curves a long wh ich con s t ruc t ive in t e r fe rence occurs .


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6 8 2 Ch ap t e r 1 6 ~ O p t i c s

b i s e c t o r t h e r e is an a d d i t i o n a l p a i r o f c u r ve s , o n e t o e a c h s id e o f t h e p e r p e n d i c u l a rb i s e c t o r, w h e r e c o n s t r u c t i v e i n t e r f e r e n c e o c c u r s . F o r d = 4 .5 ~ , t h e r e a r e f o u ra d d i t i o n a l p a i r s o f c u rv e s .

L e t t h e d i s t a n c e s f r o m a n y p o i n t P to t h e p l u n g e r s b e r l a n d r 2. C o n s t r u c -t i v e i n t e r f e r e n c e o c c u r s w h e n t h e s e p a r a t i o n r 2 - r l i s a n i n t e g r a l n u m b e r o fw a v e l e n g t h s , o r

r2 r l = m ~ , m = O , + 2 , . . .B y d e f in i ti o n , w h e n t h e d i s ta n c e s b e t w e e n a p o i n t P a n d t w o f o ci d if f er b y ac o n s t a n t , t h e c u r v e t r a c e d o u t is a h y p e r b o l a . T h u s , t h e l o c u s o f p o i n t s w h e r et h e m a x i m a o c c u r t r a c e o u t a s er ie s o f h y p e r b o l a e , d e f i n e d b y t h e i n t e g e r m a n dt h e r a t i o d / ) ~ . T h e l a rg e r t h e w a v e l e n g t h X, o r t h e s m a l l e r t h e s o u r c e s e p a r a t i o nd , t h e l a r g e r t h e s e p a r a t i o n b e t w e e n m a x i m a . T h i s is s e e n in F i g u re s 1 6 . 4 ( a ) a n d1 6 . 4 ( b ) , w h e r e t h e n u m b e r t o t h e r i g h t o f e a c h c u r v e is t h e c o r r e s p o n d i n g v a l u eof m.

D e s t r u c t i v e i n t e r f e r e n c e o c c u r s w h e n t h e s e p a r a t i o n r2 - r l i s a h a l f -i n t e g ra ln u m b e r o f w a v e l en g t hs , o r

( m 1 ) 1 1 3 ( d es t ru c t iv e i n t e ~ e ~ e )2 - r l = + ~ , m + -~ = + - ~ , 4 - ~ ,

D e s t r u c t i v e i n t e r f e r e n c e c o r r e s p o n d s t o t h e l o ca l h e i g h t o f t h e w a t e r s u r f a c eb e i n g u n d i s t u r b e d ~ s t i l l w a t e r . I n F i g u r e 1 6 . 3, w h e r e d ~ 0 .7 )~ , t h e p a i r o f c u r v e sD a n d D ' c o r r e s p o n d t o m + 1 _ 4-89 i n ( 1 6 .2 ) , w h e r e t h e r e a r e m i n i m a . T h el a r g e r t h e w a v e l e n g t h , t h e l a r g e r t h e s e p a r a t i o n b e t w e e n m i n i m a . I n F i g u re s1 6 . 4 ( a ) a n d 1 6 . 4 ( b ) , b e t w e e n t h e m a x i m a l ie m i n i m a , w h i c h a r e n o t d r a w n .

~ Interference max ima and minim aT w o p l u n g e r s a r e s ep a r a t ed b y d = 2 5 cm . ( a ) W h a t i s t h e l a r g e s t )~ f o r w h i c ht h e r e i s a m ax i m u m co r r e s p o n d i n g t o n r 0 ? ( b ) W h a t i s t h e l a r g e s t ;~ f o rw h i ch t h e r e i s d e s t r u c t i v e i n t e r f e r en ce?Solution: (a) I f the wavelen g th ;~ exceeds the la rges t va lue o f r ? - r l , wh ic h he reis d = 25 cm, the n (16.1) can be sat isf ied only for m - 0. Hen ce ~ _< 25 cmfor a ma xima w i th n r 0 . When equal i ty ho lds , on ly on the l ine def ined by thetwo sources (and no t in the r eg ion be tween the sources ) i s there cons t ruc t ivein ter f erence . (b ) I f one-ha l f the wavelen g th )~ exceeds th e la rges t va lue o f r2 -r l , wh ich here i s d = 25 cm, then (16 .2) fo r des t ruc t ive in ter f er ence canno t besa ti sf ied , even fo r m - 0 . Hence , fo r des t ruc t ive in ter f er ence )~ < 50 cm. W henequ ality holds, so r2 - r l - ( 89 on ly on the l ine def ined by the two sources (andno t in the r eg ion be tween the sources ) i s there des t ruc t ive in ter f er ence .

~ Some details of interferencen patternT w o p l u n g e r s a r e s ep a r a t ed b y d = 2 5 cm . A l o n g t h e i r p e r p en d i cu l a r b is ec t o r,a d i s ta n c e D = 5 2 c m a w ay , t h e r e i s a m a x i m u m a t y - 0 . N o r m a l t o t h e


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1 6 . 2 I n t e rf e r e n ce a n d D i f f r a c t i o n 683

IiIIII Imaginary line ~

PFigure 16.5 Schema t i c fo r ana lyz ing t hein t e r fe rence p a t t e rns o f F igure s 16 .3 and16.4. Point P l ies a long a l ine para l le l to thel i ne o f c en t e r s o f t he two p lunge r s , bu t ad i s t ance D away .

b i s e c t o r a t t h i s d is t a n c e , t h e r e i s a s e c o n d m a x i m u m a t y - 1 2 c m . S e e F i g u r e1 6 . 5 . ( a) F i n d t h e w a v e l e n g t h ~ . ( b ) F i n d t h e p o s i t i o n y o f t h e f i r st m i n i m u m .S o l u t i o n : (a) Use r2 = v / D e + ( y + d / 2 ) 2 and r l = v / D e + ( y - d / 2 ) 2. Placed in(16.1) , w i th m = 1 and y - 12 cm, this yie lds )~ = 5.48 m. (b ) Using Z = 5.48 min (16.2) wi th m = 0 gives

v /D2 + (y + d /2 ) 2 - v /D 2 + (y - d /2 ) 2 = 1Z = 2 .74 m .The so lu t i on , found n umer i ca l l y , is y = 5 .90 cm. Th i s li e s abou t ha l fway be twe enthe m ax im a a t y = 0 and y = 12 cm.

16.2,2 Dif f ract ionA r i p p l e t a n k a l s o c a n b e u s e d t o d e m o n s t r a t e d i f f r a c t i o n . C o n s i d e r a w a t e r w a v ew i t h p l a n a r w a v e f r o n t o f w a v e l e n g t h z i n c i d e n t o n a b a r r ie r w i t h a n o p e n i n g o fw i d t h d . F o r d >> )~ ( i. e. , f o r s h o r t w a v e l e n g t h s ) , a s i n F i g u r e 1 6 . 6 ( a ) , t h e w a v et h a t p a s s es t h r o u g h t h e b a r r i e r g o e s n e a r l y i n a s t r a i g h t li ne , a s i f t h e w a v e w e r ea p a r t i c l e o n a s t r a ig h t - l in e p a t h . T h e r e l a t e d c a s e o f s c a t te r i n g b y a n o b s t a c l e o fl e n g t h d i s g i v e n i n F i g u r e 1 6 . 6 ( b ) .

T h e c a s e w h e r e d ~ )~ i s d e p i c t e d i n F i g u r e 1 6 . 7 b o t h f o r a b a r r i e r w i t h a no p e n i n g a n d f o r a n o b s t a c l e .

Figure 16.6 ( a ) Sca t t e r i ng by a shor t -wave l eng th wave i nc iden t on aba r r i e r . ( b ) Sca t t e r i ng by a shor t -wave l eng th wave i nc iden t on t hecor re spond ing obs t ac l e . The da rk l i ne s r ep re sen t i ns t an t aneousm a x i m a .


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68 4 Ch ap ter 16 ~:~ Op tics

Figure 16.7 (a) Scat tering by an interm ediate-w avele ngth waveincident on a barrier . (b) Scat tering by an intermediate-wavelengthwave incident on the corresponding obstacle. The dark l ines representins tan taneous maxima.

T h e ca s e w h e re d < < 5~ ( i. e ., f o r lo n g w av e l en g t h s ) i s d e p i c t ed i n F i g u re 1 6 . 8 ( a )f o r a b a r r i e r w i t h a n o p e n i n g . T h e w a v e t h a t p a s s e s th r o u g h t h e b a r r i e r s p r e a d so u t u n i f o r m l y , a s i f t h e o p e n i n g w e r e a p o i n t s o u r ce . I n th e c o r r e s p o n d i n g c a s ef o r a n o b s t ac l e , j u s t b e h i n d i t t h e w a v e a m p l i t u d e b u i l d s u p . S e e F ig u r e 1 6 . 8 ( b ) .A t l a r g e d i s ta n c e s f r o m t h e o b s t a c l e, a n d o u t o f t h e i n c i d e n t b e a m , t h e d i f f r a c te dp a t t e r n is t h e s a m e a s fo r t h e c o r r e s p o n d i n g b a r r ie r . T h i s i d e n t i c a l d i f f ra c t i o n p a t -t e r n f o r b o t h c a se s is an e x a m p l e o f w h a t i s k n o w n i n o p t i c s a s Babinet's principle.T h i s s ay s t h a t , f o r a n y p a ir o f s c a t t e r e r s w i t h c o m p l e m e n t a r y g e o m e t r i e s ( e. g. , a no p a q u e s c r e e n w i t h a s t a r c u t o u t , a n d t h e c o r r e s p o n d i n g o p a q u e s t a r ), o u t s i d et h e g e o m e t r i c a l s h a d o w o f e i th e r , t h e p a t t e r n s o f s c a t t e r e d l i g h t a re t h e s a m e .T o s u m m a r i z e , s h o r t w a v e l e n g t h s a r e a s s o c i a t e d w i t h s t r a i g h t - l i n e p r o p a g a -t i o n , a n d l o n g w a v e l e n g t h s a r e a s s o c i a te d w i t h a m o r e s p r e a d o u t p r o p a g a t i o n .R e l a t e d t o t h i s a r e t w o p h e n o m e n a a s s o c i a t e d w i t h s o u n d : ( 1 ) a c r i c k e t , w i t hi ts h i g h - p i t c h e d ( s h o r t - w a v e l e n g t h ) c h i rp , c a n b e lo c a l i z e d m o r e r e a d i l y t h a n ab u l l f r o g , w i t h i t s l o w - p i t c h e d ( l o n g - w a v e l e n g t h ) c r o a k . ( 2 ) W h e n t h e c o r n e r o fa b u i l d in g s e p a r a te s t h e m f r o m u s, t h e l o w - p i t c h e d b u l lf r o g c a n b e h e a r d m o r er e a d i l y t h a n t h e h i g h - p i t c h e d c r i c k e t .

Figure 16.8 (a) Scat tering by a long-wavelength wave incident on abarrier . (b) Scat tering by a long-wav elength wave inciden t on thecorresponding obstacle. The dark l ines represent instantaneous maxima.


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16 .3 Op t ics to the End of the 1 7 th Century 685

~ W a v e spreading diffraction)R ad io wav es o f f r eq u en cy 9 1 .1 M H z an d so u n d wav es o f f r eq u en cy 4 4 0 H zare inc iden t on a meta l wal l wi th a c ircu lar ho le o f d iam eter 1 .2 m. Take thespeed of sound to be v = 340 m /s . Descr ibe the ex ten t o f the spread ing ofth e wav es tr an sm i t t ed t h r o u g h th e h o le .Solution: Th e re lation f;~ = v im plies tha t ;~ = v / f . For the radio wave, withv = c = 3.0 x 108 m/ s, this gives )~ = 3.29 m. For the soun d w ave this gives)~ = 0.77 m . The radio wave , wit h larger wa veleng th (3.29 m ) tha n the 1.2 mdiameter circular hole (as in Figure 16.8a), should exhibit mostly spreading, withsome localization in the forward direction. T he s ound wave, with shorter wave-length (0.79 m) than the 1.2 m diameter circular hole, should exhibit mostlypropagation in the forward direction, with some spreading (as in Figure 16.6a).Sections 16.7.2 and 16.7.3 consider diffraction in more detail.

16.3 O p ti c s to t h e E n d o f t h e 1 7 t h C e n t u r ySu r e ly p r im i t iv e m an k n ew th a t l i g h t t r av e l s (n ea r ly ) i n s t r a ig h t l i n e s , an d th a tb eh in d an i l l u m in a t ed o b jec t i s a sh ad o w. E u c l id k n ew th e l aw o f r e f l e c ti o n ,a s d id t h e R o m an s , wh o u sed a m e ta l l i c a l l o y ca l l ed s p e c u l u m f o r t h e i r m i r r o r s .[ T h i s i s t h e o r ig in o f t h e t e r m specu lar re f lec t ion (e .g . , o f f a mir ror , o r a t a g lanc ingan g le f o r a l e ss p e r f ec t ly sm o o th su r face , su ch a s a p i ece o f wr i t i n g p ap e r ) , a so p p o s e d t o diffuse reflection (e.g. , o f f a d u l ly p a in t e d w a l l) . ] P to l em y , s o m e 1 8 0 0y ea r s ag o, s tu d i ed r e f r ac t io n , b u t h e ch a r ac t e r i z e d i t q u an t i t a t i v e ly o n ly fo r sm a l lan g le s . L ik ewise , t h e an c i en t s k n e w o f t h e r a in b o w, an d p e r h ap s o f t h e p r i sm ,b u t t h e y h a d l i t tl e u n d e r s t a n d i n g o f t h e s e p h e n o m e n a . S o m e v i e w e d l i g h t asb e in g g en e r a t ed b y th e ey e , r a th e r t h an t ak in g th e co r r ec t v i ew , f i r s t s t a t ed b yA 1 -H asan (c. 1 0 0 0 ) , o f t h e ey e a s a r e ce iv e r o f l ig h t wh o se so u r ce i s e l sewh e r e .L ig h t h e r e was c o n s id e r e d to b e p a r t i cl e - l ik e , t r av e l in g in a s t r a ig h t li n e.De Do m in u s , a r o u n d 1 5 9 0 , u sed a l a r g e wa te r - f i ll ed sp h e r i ca l g l as s v e sse l t or e p r o d u c e t h e h a l o s h a p e o f a r a in b o w , w i t h a b r i g h t b o w a t 4 2 ~ t o t h e i n c i d e n ts u n l i g h t . H e s h o w e d t h a t t h e p r i m a r y ( a n d s e c o n d a r y ) r a i n b o w s i n v o l v e d o n e(an d two ) i n t e r n a l r e f l e c t i o n s i n ad d i t i o n to t h e r e f r ac t io n s o n en t e r in g an d ex i t -i n g . F ig u r e 1 6 .9 (a ) p r e sen t s a g eo m e t r y wh e r e su n l ig h t i s i n c id en t f r o m th e l e f t ,d i r ec t e d a lo n g th e x -ax i s. (T h e an g le o f i n c id en c e r e l a t i v e t o t h e n o r m a l i s 0 , an dth e o b se r v a t io n an g le r e l a t i v e t o t h e b ac k w ar d d i r ec t io n i s ~b.) De Do m in u s co u ldn o t e x p l a i n , h o w e v e r , h o w t h e h a l o s h a p e o c c u r r e d , o r w h y t h e c o l o r s s e p a r a t et o h a v e a c o n t i n u o u s d i s t r ib u t i o n o f b o w s .

t6.3.1 The L aw of R e f ract ion (S ne ll 's L aw)M easu r ed r e l a t i v e t o t h e n o r m a l , l e t 0 b e t h e an g le o f i n c id en c e in a i r, an d l e t 0 'b e t h e a n g le o f r e f r ac t i o n i n t o a n o t h e r m e d i u m . T h e l a w o f r e fr a c ti o n , i n a f o rme q u i v a l e n t t o

sin 0 = n sin 0', (16.3)


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6 8 6 Chapte r 16 ~ Opt ics

In

5 04 0

30

20100 0 2b 4'0 60 8'0

0 - - - - P -0 , )

1()0

Figure 1 6 .9 (a) Path o f a ray incid ent on a water- f il led spherical glass vessel . I tundergoes refraction on enter ing, one ref lection within, and refraction on leaving.(b) Exit angle ~b as a func tion of incide nt angle O. N ot draw n are th e ray ref lectedon ente r ing , nor the t r ansm it ted ray assoc ia ted wi th in te rna l r e f lec t ion , no r theref lected ray on leaving.

w h e r e t h e i n d e x o f r e f ra c t i o n n , w a s k n o w n t o S n e l ( o n e ' T ' ) b y 1 6 2 1 . S i n c en > 1 , w e ha ve O ' < 0 . T h e l a r g e r t h e n , t h e l a rg e r t h e m a g n i t u d e o f t h e a n g u l a rd e v i a t i o n 0 - 0 ' . For smal l angles , 0 - n O ' , a s k n o w n b y P t o le m y .

(a)

Theory o f the R ainbow A ng leF r o m n u m e r i c a l r a y t r a c i n g u s i n g S n e l l ' s l a w , a n d f r o m e x p e r i m e n t s , D e s c a r t e s( 1 6 3 7 ) s t u d i e d t h e r e l a t i o n s h i p b e t w e e n 0 a n d ~b i n F i g u r e 1 6 . 9 ( a ) . T h e s e r e s u l t sc a n b e s u m m a r i z e d b y F i g u r e 1 6 . 9 ( b ) . T h i s i s c a l c u l a t e d o n t h e b a s i s o f a r a ys u b j e c t t o r e f r a c t i o n o n e n t e r i n g , a n i n t e r n a l r e f l e c ti o n , a n d a r e fr a c t i o n o n l e av -i n g, a s i n F ig u r e 1 6 . 9 ( a ). ( N o t d r a w n a r e t h e r a y r e f l e c t e d o n e n t e r i n g , n o r t h et r a n s m i t t e d r a y a s s o c i a t e d w i t h t h e i n t e r n a l r e f l e c t i o n , n o r t h e r e f l e c t e d r a y o nl e a v in g . ) F i g u re 1 6 . 1 0 p r e s e n t s a d e t a i l e d v e r s i o n o f F i g u r e 1 6 . 9 ( a ) , w i t h l i g h tr a y s i n c i d e n t f r o m t h e l e f t , d i r e c t e d a l o n g t h e x - a x i s .

bb O ~J \. . . . . . \

a " = a " + a = zc + 2 0 - 4 0 "= z c - a " = 4 0 " - 2 0

Figure 16 .10 Details of ref lection and refraction for Figure 16.9.


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16 . 3 Opt ic s to the En d o f the 17 th C entury 6 8 7

In F igu re 16 .10 , a r a y i s i nc i de n t on t h e r a i nd ro p a t a n a ng l e 0 r e l a ti ve t o t heno rm a l , a nd r e f ra c t s a t a n a ng l e 0 '. T h e r e f r a c t e d r a y d e f le c ts f r o m a st r a i g h t p a t ha l ong t he x -a x i s by a c l oc kwi se a ng l e= 0 - 0 ' . ( 16 .4 a )

Th e re f ra c t e d r a y t r a ve l s a long a s t r a i gh t pa t h wi t h i n t he r a i nd ro p u n t i l i t i si n t e rna l l y r e f l e c t e d . The r e d i re c t i on on r e f l e c t i on c a n be de sc r i be d a s a n a dd i -t i ona l c l oc kwi se ro t a t i on by Jr - 20 ' . Th us t he i n t e rna l l y r e f l e c t e d r a y m a ke s ac l oc kwi se a ng l e r e l a t i ve t o t he x -a x i s o foe ' = oe + z r - 2 0 ' = J r + 0 - 30 ' . (16.4b)

Th i s i n t e rna l l y r e f l e c t e d r a y t he n t r a ve l s in a s t r a i gh t l i ne un t i l i t h i t s t he i n t e rna lsu r fa c e o f t he r a i nd ro p a t an a ng l e O ' r e l a t i ve t o t he no rm a l . The re i t r e f r a c t st o a n a ng l e 0 r e l a t i ve t o t he no rm a l . Th i s c o r re sponds t o a ne t de f l e c t i on by a na dd i t i ona l a ng l e ~ - 0 - O ' c l oc kwi se r e l a ti ve t o t he x -a x i s. Th e ne t e f f e c t i s ac l oc kwi se a ng l e o f ro t a t i o n by

oe" = oe ' + o l = z r + 20 -4 0 ' ( ] 6 . 4 c )

r e l a t i ve t o t he x -a x i s . V i e we d by a n obse rve r r e l a t i ve t o t he r a i nbow, t h i s c o r re -sponds t o a c l oc kwi se a ng l e r e l a t i ve t o t he x -a x i s o f= J r - o ~ " = 4 0 ' - 2 0 . (16.5)

Th e ~b ve r sus 0 c u rve o f F i gu re 16 .9 (b ) i s c o m pu t e d fo r n = 4 / 3 , a s a pp rop r i a t ef o r w a te r . T h e m a x i m u m is n e a r 4 2 ~ N o r ay s a re d e f l e c te d b y m o r e t h a n 4 2 ~c o r r e s p o n d i n g t o t h e d a r k r e g io n o b s e r v e d o u t s i d e a r a in b o w . T h u s t h e p r i m a r yr a i n b o w is c a u s e d b y a b u n c h i n g u p o f r a y s a t t h e m a x i m u m a n g l e n e a r 4 2 ~ W h e nr a i n d r o p s a r e p r e s e n t , a b o w c a n b e s e e n in t h e p a r t o f t h e s k y t h a t c o r r e s p o n d st o t h i s a ng l e . S e e P l a t e 2 fo r a p r i m a ry a nd a s e c onda ry r a i nbow.I fw e use (16 .5 ) fo r ~b a nd (16 .3 ) r e l a t i ng O ' t o O , t he n s e t t i ng d ~ / d O = 0 givest h a t ~b h a s a m a x i m u m f o r

~ 4 - n 2 ( r a i n b o w a n g l e )sin 0 - 3 "F o r n = 4 / 3 t h i s y ie l ds s i nO = 0 . 8 6 1 , s o 0 = 5 9 . 4 ~ a n d t h e n ( 1 6 . 1 ) g iv e ss in O' = 0 .64 5, so O' = 40.2 ~ F inal ly , (16.5) g ives ~b = 42. 0 ~ in ag ree m en t w i tht h e m e a s u r e d r e s u l t an d w i t h t h e g r a p h o f F i g u re 1 6 . 9 ( b ) .R ef lec t ion , R e f ract ion , and the P r inc ip le of Least T imeDe sc a r t e s ga ve a "de r i va t i on" o f S ne l l ' s l a w. A l t hough e l s e whe re De sc a r t e sa r g u e d t h a t l i g h t p r o p a g a t e d i n s t an t a n e o u s ly , h i s m e c h a n i c a l l y b a s e d d e r iv a -t i o n a s s u m e d t h a t l i g h t h as a fi n it e ve lo c it y , w h o s e c o m p o n e n t p a r a ll e l to t h esu r fa c e i s c onse rve d on r e f ra c t i on . Th i s r e a son i ng y i e l de d v~ s in01 = v2 s in02,r a t h e r t h a n w h a t w e n o w k n o w t o b e t h e c o r r e c t r e s u l t , g i v e n b y ( 1 5 . 6 1 ) :


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688 Chap ter 16 ~ Optics

BA i

S P ~ ~ P o 2 S

I ~ B "

a l

A

2 S

(a) (b)Figure 16.11 Reflection and refraction of light from a source at A, incident on thesurface S. (a) Reflection to point B, where point B' is the image of B. The path ofleast time is APoB'. (b) Refraction to p oint B. Th e general pat h APB is shown.sin 01/vl = s in 02 /v2 . To ob ta in ag re em en t w i th ex per im en t , D es ca r t e s had toas s ume tha t l igh t t r ave l s more qu ick ly in w a te r and g las s than in a i r .U n s a t i s f i e d w i t h D e s c a r t e s ' s d e r i v a t i o n , F e r m a t u n d e r t o o k t o d e t e r m i n e f o ra pa r t i c l e o f l igh t the s t r a igh t - line pa th f ro m A to B tha t t akes the l eas t t ime . I n1657 , he s how ed tha t th i s r ay sa t is fi e s a l aw o f r e f rac t ion equ iva len t to ( 15 .61) .( H i s d e r i v a t i o n u s e d a m e t h o d h e h a d d e v e l o p e d e a r l i e r t o f i n d t h e t a n g e n t s ~i .e ., t h e s l o p e s m t o c u rv e s. N e w t o n g e n e r a l iz e d F e r m a t ' s m e t h o d t o y i el d w h a tw e now know as the d i f f e r en t i a l ca lcu lus . ) S ince the ve loc i ty o f ligh t w as no tmeas u red accu ra te ly un t i l F izeau ( 1850) , i t w as no t pos s ib le a t tha t t ime tover i fy Ferm at ' s fo rm o f the l aw o f r e f r ac t ion . Fo l low ing Fermat , w e n ow app lyth e p r i n c i p l e o f le a s t t i m e to s tudy r e f l ec t ion and r e f r ac t ion .T h e l aw o f re f l ec t i on . C o n s i d e r a p o i n t A i n m e d i u m 1 w h e r e a r a y o f l i g h ti s em i t t ed , and a po in t B w h ere l igh t r e f l ec ted f rom s u r f ace S i s obs e rved . SeeFigure 16.1 1 (a). D raw the ima ge p oin t B ' associated w ith B, and a s t ra igh t l inef rom A to B ' , i n t e r s ec t ing S a t P 0 . Cons ide r l igh t tha t t r ave l s s t r a igh t f rom A toa genera l po in t P on S, and the n s t r a igh t to B . Th e to ta l d i s t ance th a t i t t r ave ls i sg iven by AP + P B, w h ic h equa l s AP + P B ' . F igu re 16 .11 ( a) s how s th a t th e p o in tP g iv ing the s ho r tes t to ta l d i s t ance i s P0 . Th i s co r r es ponds to a s t r a igh t l ine f romA to B ' , and an ang le of ref lect ion 0~ tha t equa ls the angle of incidenc e 01"

0~ -- O1 ( la w o f r e f le c t io n ) { 1 ! ! )S ince the ve loc i ty is un i fo rm in m ed ium 1 , th i s pa th i s bo t h the s ho r tes t pa thand the pa th o f l eas t t ime .T h e l aw o f re frac t i on . L e t p o i n t A b e i n m e d i u m 1 , w i t h l i g h t v e l o c i t y v l ,and l e t po in t B be in me d iu m 2 , w i th l igh t ve loc i ty v2 . We f ir s t cons ide r a r ayo f l igh t tha t goes s tr a igh t f rom A in med ium 1 to a po in t P on the s u r f ace S,a n d t h e n s t ra i g h t i n m e d i u m 2 f ro m S t o t h e p o i n t B . W e t h e n f i n d t h e p o i n t f o rwh ich t he t rave l t im e T is leas t . See Figure 16.11 (b) .Le t the ne a res t d i s t ance f rom A to the s u r f ace S be a l , the nea res t d i s t ancef rom B to S be a 2 , and l e t the d i s t ance a long the s u r f ace f rom A to P be x . I f the


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1 6 . 4 L a t e 1 7 t h - C e n t u r y D i s c o ve r ie s a b o u t L i g h t 689

to ta l d i s t ance a long x f rom A to B i s l , then the d i s t ance a long x f rom P toB is l - x . T h u s t h e t o t a l t i m e T t o g o fr o m A t o B is t h e s u m o f t h e t i m eA P / v l - v / a2 + x 2 / v l t o g o f r o m A t o P a n d t h e t i m e B P / v 2 - v / a 2 + (l - x ) 2 / v 2ito go f rom P to B . Thus

~ _ v / ~ 2 + ~ =721

/ _~/a~ 2r- ( l - x ) 2+ . (16.7a)V2T h e p o s i t io n x w h e r e T i s m i n i m i z e d is t h e s o l u ti o n o f d T / d x - O . N o t i n gt h a t

d ~/a~ + x2= x and d v/a2 + ( l - x ) 2 = - ( l - x )dx , / 4 + x= m , / 4 + c~-x~=(16.7a) y ie lds

d T 1 x 1 l - x 1 1= = ~ sin O1 sin 02,d x V l v /a 2 1 + x 2 v 2 ~ / a ~ + ( l - x ) 2 v l v 2 (16.7b)

w h ere F igu re 16 .11 ( b ) has been u s ed to ob ta in the s ines. Se t t ing ( 16 .7b) to ze ro,t h e m i n i m u m t i m e o c c u r s f o rsin 01 sin 02= ~ . (16.8)Vl V2

This i s p r ec i s e ly the s am e as ( 15 .61) and l eads to the l aw o f r e f r ac t ion in thefo rm

i i i i i i i i i i i i i ii i i i i i i i i i i i ii ! i i i i i i ii i i i i! ~ i i ! ~ i l i i i ! i i i i i i i i ! i i i !

Like D es ca r tes ' s exp lana t ion o f the r a inbow , Fe rma t ' s de r iva t ion o f the l aw s o fr e f l ec t ion and r e f r ac t ion w as a g rea t t r iu m ph o f the pa r t i c l e v iew po in t o f l igh t .

16~4 L a te 1 7 t h - C e n t u r y D i s c o v e ri es a b o u t L i g h tC o n s i d e r a b l e e x p e r i m e n t a l p r o g re s s w a s m a d e i n t h e s e c o n d h a l f o f t h e 1 7 t hc e n tu r y , m u c h o f i t i n c o m p r e h e n s i b l e f r o m t h e p a r t i c l e v i e w p o i n t .1 . G r im ald i ( 161 6- 1 673 ) d i s covered and na m ed d i f f r ac t ion ( 1665) . U singa s m a l l h o l e t h r o u g h w h i c h s u n l i g h t p a s s e d , h e n o t i c e d t h a t ( 1 ) t h e s h a d o w o fob jec t s on a nea rby s c reen i s s l igh t ly l a rge r than w o u ld b e ex pec ted geome t r i ca l ly ;( 2) o n m o v i n g f a r t h e r o u t o f t h e g e o m e t r i c a l s h a do w , t h e r e is a s e q u e n c e o f b r i g h tand da rk f r inges , ge t t ing na r row er the f a r the r they a r e f rom the s hadow ( w i th ineach b r igh t f r inge the cen te r i s w h i te , the s ide nea res t the s hadow i s b lu i s h ,and the s ide f a r thes t f rom the s hado w i s r edd i s h ) ; ( 3 ) the cen te r o f the f ir s tb r igh t f r inge i s no t i ceab ly b r igh te r than the un i fo rm i l lumina t ion f a r ou t s ide theshadow. See Figure 16.12.


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6 9 0 Chapter 16 = Optics

Figure 1 6 . 1 2 (a) Diffraction of a distant light around an edge , for an observer no tso close that the geometrical shadow is completely obscured. Although firs tstudied b y Grimaldi, this is called Fresnel diffraction, after the ma n wh o firstexplained the phenom enon. (b) Intensity is plotted a long the vertical and positionalong the horizontal.

2 . Hoo ke ( 163 5- 17 03) , u s ing glas s p la tes p res s ed toge the r , in h i s Micrographia( 1665) gave the f i rs t pub l i s h ed s tud ies o f the co lo rs s een in l igh t r e f l ec ted o f fth in f i lms and p la tes . He no t i ced tha t the co lo r s w ere r e la ted to the th icknes sd o f th e g a p b e t w e e n t h e p l a t e s; t h e h a r d e r t o g e t h e r h e p r e s s e d t h e p l at e s,the f a r the r apa r t the co lo red f r inges . He s ugges ted tha t pe rhaps r e f l ec t ion o f fthe f ron t and back s u r f aces cou ld exp la in the e f f ec t , bu t he d idn ' t deve lop theidea quan t i t a t ive ly . See F igu re 16 .13 . His geom et r i e s inc lude d p la tes o f va r i ab leth icknes s , and l ens es p laced one upon ano the r . How ever , he cou ld no t meas u re

Figure 1 6 . 1 3 Interference p atterns on reflection of monochrom atic light from twolenses in near con tact, first observed b y Hooke. (a) Unpolished surfaces. (b) Surfacespolished flat to w ithin a fraction of a wavelength, and slightly tilled relative to oneanother.


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1 6 . 4 L a t e 1 7 t h - C e n t u r y D i s co v e r ie s a b o u t L i g h t 69 1

Figure 1 6 . 1 4 W hite light incident upon a prism, w hich separates the light into its purecolors (defined by their wavelengths) , f irs t s tudied b y N ewton.

t h e t h i c k n e s s o f t h e f il m s o r p l a t e s t h a t c o r r e s p o n d e d t o e a c h c o l o r e d f r in g e ,l eav ing tha t a s a cha l l enge fo r fu tu re w orker s .3 . N e w t o n ( 1 6 4 3 - 1 7 2 7 ) p e r f o r m e d e x p e r i m e n t s o n t h e p r i sm ( 1 6 66 ) , es -t a b l is h i n g t h a t l i g h t w a s c o m p o s e d o f a c o n t i n u u m o f m a n y p u r e c o l o rs , a n dt h a t o n c e a b e a m o f p u r e c o l o r w a s i s o la t e d f r o m o t h e r c o lo r s, i t re t a i n e d i t sin teg ri ty . See F igu re 1 6 .14 a nd P la te 1 . He r ea l i zed tha t , in man y cases, l igh t tha ta p p e a r e d t o b e o f a c e rt a i n c o l o r (e .g ., o r a n g e ) w o u l d , w h e n a n a l y z e d w i t h t h ep r i s m, s epara te in to va r ious pu re co lo r s ( e .g . , r ed and ye l low ) . Wi th th i s idea ,


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692 Chapter 16 R Optics

Figure 16.15 Unpolarized light refracted by a calcite crystal,which splits i t into tw o beam s, one satisfying Snell 's law (theordinary, or O, bea m), and one not satisfying Snell 's law (theextraordinary, or E, beam ). A lso given are the electric f ieldpolarizations of these beams (the op en circles denote fieldnormal to the page).N e w t o n e x p l a in e d t h e c o lo r s e p ar a ti o n o f th e r a i n b o w a s t h e c o n s e q u e n c e o fthe d i f f e r en t co lo r s hav ing d i f f e r en t ind ices o f r e f r ac t ion in w a te r . N e w ton a l soe x t e n d e d H o o k e ' s w o r k o n t h e c o l or s o f t h i n f i lm s a n d p l a t es , u s i n g a l e n s g e o m -e t r y t h a t p e r m i t t e d h i m t o m e a s u r e t h e f i lm t h i c k n e s s d. ( W e s h a ll d is c u ss t h i sin more de ta i l . )4 . B a r t h o l i n u s ( 1 6 2 5 - 1 6 9 8 ) d i s c o v e r e d double refraction ( 1 6 6 9 ) , w h i c h o c -cu r s on ly fo r c rys ta l s w i th a p re fe r r ed ax i s ( s uch as w ha t w as know n to h im asI c e l and spar , n o w k n o w n a s calci te , o r C a C O 3 ) . S e e F ig u r e 1 6 . 15 . W h e n t h e r e i so n l y o n e b e a m i n c i d e n t o n t h e c r y s t a l - - e v e n i f i t i s n o r m a l l y i n c i d e n t ~ t y p i c a l l ythe re a r e tw o r e f r ac ted beams . O ne o f thes e ( the o rd inary , o r O , bea m ) s a t is fi e sSne l l ' s l aw ; the o the r ( the ex t r ao rd ina ry , o r E , beam) does no t . ( F igu re 16 .15a l so ind ica tes the d i r ec t ions o f po la r iza t ion , a s now und er s too d ; a t the t im e , s c i -e n t i st s w e r e n o t a w a r e t h a t t h e t w o e x i t in g b e a m s h a d d i f f e r e n t p r o p e r ti e s . ) F o ro n e p a r t i c u l a r d i r e c t i o n o f p r o p a g a t i o n i n t h e c r y s t al ( t h e opt ic ax is ) , t h e O a n dE b e a m s c o i n c i d e.5 . R o e m e r ( 1 6 7 5 ) s h o w e d t h a t t h e t i m e s o f e c l ip s e s o f J u p i t e r ' s f ir st m o o n ,Io , d e p e n d o n t h e m o t i o n o f J u p i t e r r e l a t iv e t o t h e e a r t h ( th i s is a f o r m o f t h eD o p p l e r e f fe c t, w h e r e t h e f r e q u e n c y is d e t e r m i n e d b y t h e p e r i o d o f I o 's m o t i o naround J up i t e r ) . See F igu re 16 .16 , w here in mov ing f rom A to B , I o ' s pe r ioddecreas es , and in mov ing f rom C to D , I o ' s pe r iod inc reas es . Th i s ind ica ted tha tt h e s p e e d o f l i g h t i s f i n i t e . F r o m R o e m e r ' s d a t a a n d t h e t h e n b e s t k n o w n v a l u ef o r t h e d i a m e t e r o f t h e e a r t h ' s o r b i t a b o u t t h e s u n ( a b o u t t w o - t h i r d s o f it s c o r r e c tv a l u e ), H u y g e n s l a t e r e s t i m a t e d t h e s p e e d o f l i g h t c to b e a b o u t 2 x 1 08 m / s.

Figure 16.16 Schematic of the m otion of theear th around the sun, and the m otion of Jupi ter ' smo on Io around Jupiter , with pe riod Tio. W h e nthe ear th moves toward ( f rom) Jupi ter, the per ioddecreases (increases) , in proportion to th e ratio o fthe earth 's velocity to the velocity of l ight.


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1 6 . 5 L a t e 1 7 t h - C e n t u r y V i e w s o f L i g h t 6 9 3

6 . One more f igu re o f t h i s t ime pe r iod mus t be men t ioned : Kep le r , morewel l -kno wn fo r e s tab l ish ing , by 1618 , t he t h r ee l aws o f p l ane t a ry m ot ion . H i s1604 book A s t r o n o m i a P a rs O p t i c a i nves t iga t ed image fo rmat ion by the p inho lecamera , exp la ined v i s ion as due to imag ing by the l ens o f t he eye on the r e t ina ,co r r ec t ly desc r ibed the causes o f l ong-s igh tedness and shor t -s igh tedness , andexp la ined how bo th eyes a r e used fo r dep th pe r cep t ion . In 1608 Lipper sheymade a t e l e scope wi th a converg ing ob jec t ive l ens and a d ive rg ing eye l ens ,wh ich Ga l i l eo a lmos t immedia t e ly improved upon and app l i ed to t e r r es t r i a l andast ronomical s tudies , in 1610 discover ing four sa te l l i tes orbi t ing about Jupi ter ,and se t t i ng the s t age fo r t he Copern ica n r evo lu t ion in ou r v i ews o f t he un iver se .Kepler ' s 1611 book D i o p t r i c e descr ibed to ta l in ternal ref lect ion for large angles ,and founded modern geomet r i ca l op t i c s us ing P to l emy ' s smal l ang le r e su l t 0 =n 0' . I t descr ibed real , v i r tual , upr ight and inver ted images and magnif icat ion. I texp la ined the p r inc ip l es o f L ipper shey ' s t e l e scope , and a l so p roposed wha t hasbeco me the m ode rn t e l e scope , w i th a converg ing ob jec t ive and a converg ing eyelens. To Kepler , who had spent years t ry ing to explain p lanetary orbi t s as c i rc lesupon ci rc les , and then as e l l ipses, the s t ra ight l ines of geometr ical opt ics musthave bee n a p iece o f cake .H ence , by 1675 , nea r ly al l t he bas i c phen om ena o f phys i ca l op ti c s had beend i scovered . W ha t l acked was a un i fy ing exp lana t ion .

L a t e 1 7 t h - C e n t u r y V i e w s o f L i g h tR o b e r t H o o k e a n d W a v e P u l s e sH oo ke a rgued tha t l i gh t is a ssoc ia t ed wi th r ap id , sm al l -ampl i tude v ib r a t ions o f anunspec i f i ed med ium ca l led the " e ther ." H e e mp loyed an ana logy to sou nd in a ir ,wh ere the m ed i um fo r t he v ib r a t ions is a ir . H e a lso deve loped a t heo ry o f l i gh tp ropaga t ion where the source p roduces pu l ses a t r egu la r t ime in t e rva l s ( r a the rt h a n t h e m o r e m o d e r n v i e w w h e r e t h e s o u rc e p r o d u c e s a c o n ti n u o u s o s c i ll a ti o n

vefront

Rays

Figure 16 .17 Ray s and wa vefront for a sphericalsource.

whose pe r iod equa l s t ha t r egu-lar t ime in terval ) . Such pulsesp ropaga te ou tward , by ana logyto water waves, " l ike Rays f romt h e c e n t e r o f a S p h e r e . . . a ll t h epar t s o f t hese spheres cu t t heRays a t r ight angles ." He stud iedref lect ion and ref ract ion using ar ay-and-wavef ron t cons t ruc t ion .(A w a v e f r o n t o r - - t o u s e H o o k e 'sword , a p u l s e - - g i v e s t he s imul -t a n e o u s p o s i t io n o f m a n y o u t g o -ing par t ic les associa ted wi th d i f -f e r en t r ays emi t t ed a t t he samet ime. See Figure 16.17.) Accord-ing to H ooke, th e or ig in of colorsl ay in the pu l se shape , and w hi t el ight was a color u nto i tse lf . Thisv i e w w a s n o t c o n s i s te n t w i t h N e w t o n ' s w o r k w i t h t h e p r is m , w h i c h s h o w e d t h a tthe r e is more than one w ay to ob ta in a g iven color .


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6 9 4 Chapte r 16 9 Opt ics

W i t h t w o a s s u m p t i o n s ~ ( 1 ) w h e n a w a v e f r o n t im p i n g e s o n a n i n te r fa c e, i ti n i ti a te s a n e w w a v e f r o n t i n t h e n e w m e d i u m ; a n d ( 2 ) t h e n e w w a v e f r o n t a n dt h e i n c i d e n t w a v e f r o n t h a v e t h e s a m e v e l o c i t i e s p a r a l l e l t o t h e i n t e r f a c e ~ H o o k ede r i ve d t he s a m e e r ron e ou s l a w o f r e f r a c t ion a s De sc a r t e s : V l s i n 01 - v2 sin 02.

1 6 . 5 . 2 Chr is t ian Huygens and Huygens 's Pr inc ip le forthe Addi t ion of WaveletsH uyge ns , i n h i s T r ea t is e o n L i g h t ( 1 6 7 8 ) , d e v e l o p e d a w a v e t h e o r y o f l ig h t . " I f , in -d e e d , o n e l o o ks fo r s o m e o t h e r m o d e o f a c c o u n t i n g f o r t h e [ u n i f o r m s p e e d ]o f l i gh t, he wi l l ha ve d i f f i c u lt y i n fi nd i ng one be t t e r a da p t e d t ha n e l a s t ic i t y[o f t he e t he r ] . " T h i nk i ng o f l i gh t wa ve s i n t he s e nse o f pu l s e s i n t i m e ( j u s t a s d i dH o o k e ) , h e d e v e lo p e d w h a t w e n o w k n o w a s Hu yg en s ' s p r i n c i p l e : I f a wa ve f r o n t i sk n o w n a t a n y i n s t a n t o f t im e , t h e n a t a n y f u t u r e t i m e t h e n e w w a v e f r o n t i s d e t e r m i n e db y t h e e n v e lo p e o f se c o n d a r y w a v e l e t s t h a t a r e p r o d u c e d a t e a c h p o i n t o f th e i n i t ia lwa ve f r o n t . ( T h e s e c o n d a r y w a v e l e ts a r e t a k e n t o t r a v el a t t h e s a m e s p e e d a s t h ep r i m a r y w a v e . )F i g u re 1 6 . 1 8 ( a ) d e p i c t s t h e p r o p a g a t i o n o f l i g h t e m i t t e d a s s p h e r i c al w a v e sa t t i m e t = 0 f r o m p o s i t i o n A . T h e d a s h e d l i n e s r e p r e s e n t r a y s p r o p a g a t i n g r a -d i a l l y o u t w a r d f r o m A , y i e l d i n g t h e o u t g o i n g s p h e r i c a l p r i m a r y w a v e f r o n t s H H 'a t t i m e t, a nd J J ' a t t i m e 2 t . (The v a l ue o f t i s i m m a t e r i a l ; H u yg e ns d i d no tt h i n k o f l i gh t as pe r i od i c e i t he r i n spa c e o r in t i m e . ) A t p o i n t s C , D , a nd E onH H ' a t ti m e t, s e c o n d a r y w a v e l e t s a r e g e n e r a t e d t h a t t r a v e l a t t h e s a m e s p e e da s t he p r i m a ry wa ve , y i e l d i ng s e c onda ry wa ve l e t s cc ' c" , d d 'd " , a n d ee ' e" on J J 'a t t i m e 2 t . T h e e n v e l o p e o f t h e s e s e c o n d a r y w a v e l e ts i s a s p h e r e t h a t c o i n c id e sw i t h t h e p r i m a r y w a v e f r o n t . S im i la r ly , F ig u r e 1 6 . 1 8 ( b ) d e p i c t s a p r i m a r y w a v e -f r o n t t h a t i s a d o w n w a r d - d i r e c t e d p l a n e a t t = 0 . It g e n e r a t e s s e c o n d a r y w a v e l e t sw h o s e e n v e l o p e a t t i m e t c o i n c id e s w i t h t h e p r i m a r y w a v e f r o n t a t t im e t . H e n c e ,H u y g e n s ' s c o n s t r u c t i o n p e r m i t s s p h e r i c al , p l a n ar , a n d e v e n c y l i n d r ic a l w a v e f r o n t st o m a i n t a i n t h e i r s h a p e . C o n s i d e r i n g o n l y a n e n v e l o p e f r o m t h e f o r w a r d - m o v i n g

A/ \

/ \/ \

/ \C / \

/ \c , ' \ e "

J ' ~ c/ / ~ ' d "" \ " j-

d .

A a 1 a 2 a 3 a 4 a 5 a6 A"t = 0 I I I I I I

b2 b3 b4 bs b6 ~t

B B "

(a) (b)Figure 1 6 . 1 8 Huyg ens's principle . (a) The p oint source a t A produces a wavefrontHH '. C, D, and E on HH ' are sources of secondary wavelets cc'c", dd'd", ee'e",whose envelope forms the new spherical wavefront JJ ' . (b) A planar wavefront AA'has sources a l , . . . , p roduc ing the secondary wave let s b l . . . . . . w hose enve lope formsthe new planar wavefront BB' .


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w a v e l e t s , H u y g e n s ' s c o n s t r u c t i o n i m p l i c i t l y i n c l u d e s i n f o r m a t i o n a b o u t b o t h t h ewa ve pos i t i on a nd t he wa ve ve l oc i t y .B e c a u s e H u y g e n s ' s p r i n c i p l e w a s a w a v e th e o r y , b y a n a l o g y to s o u n d h e c o u l di m m e d i a t e l y j u s ti f y h o w l i g h t w a v e s c an p a ss t h r o u g h o n e a n o t h e r i n d e p e n d e n t l y( i ne xp l i c a b l e i n a pa r t i c l e the o ry , e xc e p t on a s sum i n g e x t ra o rd i na r i l y l ow pa r t i c l ede ns i t i e s ). H e c ou l d e xp l a i n sha do ws , qua l i t a t ive l y , by a rgu i ng t ha t wi t h i n t h es h a d o w t h e s e c o n d a r y w a v e l e t s " d o n o t c o m b i n e a t t h e s a m e i n s t a n t [ i. e. , i npha se ] t o p rod uc e one s i ng l e wa ve . "

Huygens's Theory of Ref lect ion and Refract ionM o s t i m p o r t a n t , f o r a p l a n a r w a v e f r o n t i n c i d e n t o n a p l a n a r i n t er f a c e b e t w e e nt w o t r a n s p a r e n t m e d i a , H u y g e n s d e v e l o p e d H o o k e ' s a r g u m e n t s t o o b t a i n t h el a w s o f r e f l ec t io n a n d r e f r a c ti o n . L i k e F e rm a t , H u y g e n s t o o k t h e v e l o c i ty o fl i gh t i n g la s s t o be l es s t ha n i n a i r , r a t he r t h a n g re a t e r t h a n i n a i r , a s e r ron e ous l ya s s u m e d b y H o o k e a n d N e w t o n .C o n s i d e r F i g u r e 1 6 . 1 9 . T h i s d e p i c t s t w o m e d i a w i t h d i f f e r e n t i n d i c e s o fre f ra c t i on ( in f a c t, w i t h n l > n 2 , so 01 < 02 , a s oc c u rs whe n l i gh t i s i nc i de n t f romw a t e r t o a ir ). I n c i d e n t r a ys i n m e d i u m 1 y i e ld b o t h r e f l e c t e d r ay s i n m e d i u m 1a nd re f ra c t e d r a ys i n m e d i um 2 . The i nc i de n t r a ys m a ke a n a ng l e 01 t o t heno r m a l N , so t he i r w a ve f ro n t s ( e .g . , t he da sh e d l i ne A A ' ) m a ke a n a ng l e 01 t o t hei n t e r fa c e S S . A t t = 0 , t he w a ve f ron t A A ' i s p l a na r , w i t h i t s e dge a t A ju s t m a k i ngc on t a c t wi t h t he i n t e r fa c e . S phe r i c a l wa ve l e t s , de p i c t e d a t t i m e t , a r e ge n e ra t e da t t -0 i n bo t h m e d i a . A l so a t t i m e t a s e c on d i nc i de n t r a y h i t s t he i n t e r fa c ea t B , wh i c h be g i ns t o i n i t i a t e a no t he r s e c onda ry wa ve l e t . To f i nd t he t i m e te n v e l o p e o f w a v e l e t s i n e a c h m e d i u m , d r a w t h e c o m m o n t a n g e n t s t o a ll t h e t i m et wa ve l e t s . F i gu re 16 .19 show s on l y t he i n f i n i t e si m a l r a d i us w a ve l e t ( i. e. , a po i n t )

Incident rays\ / N ~ \ Reflected rays, \ ~ . . . . IncidentReflected ~ " .1 ~ ,~ ] ,~. ) ( " ~ .. " wavefront at t= 0wavefront ~ V ' ' /t t " ~ / ~ ? / - "[ V - " o ,/ o 9 " V / '

t

" ~-- Refracted rays~NRefracted----- "fwavefront at tFigure 16 .19 Applicat ion of Huygens's principle to reflect ion andrefraction. Incident rays from 1 intersect surface S at A an d B,produ cing secondary wavelets in both 1 and 2. The secondarywavelets originat ing at A are shown; a t time t the w aveffont has justreached B, so the secondary w avelets originat ing at B have zero radius.


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1 6 . 5 . 4

a t B , a n d t h e w a v e l e t p r o d u c e d a t A . I n m e d i u m 1 , t h e t a n g e n t t o t h e w a v e l e tp r o d u c e d a t A is a t B'. In m e d i u m 2 , t h e t a n g e n t t o t h e w a v e l e t p r o d u c e d a t Ais at B". Let u s s ee how th i s l eads to the l aw s o f r e f l ec t ion and r e f r ac t ion .Ref lec t ion . I n F igu re 16 .19 , co ns ide r the r igh t t r iang les AA'B and BB 'A, w i thc o m m o n s id e AB n o r m a l t o t h e i r c o m m o n a n g le o f 9 0 ~ B e c a u se s i de s A B ' a n dB A ' c o r r e s p o n d t o t h e s a m e t i m e i n te r v al , A B ' = B A '. H e n c e A A ' B a n d B B ' A a res imi la r t r i ang les . There fo re the ang le 0~ made by the ou tgo ing w avef ron t to thein te r f ace equa l s the ang le 01 m ade by the incom ing w av ef ron t to the in te r f ace , o r0 { - - 0 1 . ( 1 6 . 1 0 )

B e c a u s e t h e a n g l e b e t w e e n t h e n o r m a l a n d a r a y e q u a l s t h e a n g l e b e t w e e n t h ein te r f ace and a w avef ron t , w e have thus es tab l i s hed the taw of reflection: theangle of reflection equa ls the angle o f incidence. T h i s h a s t h e s a m e f o r m f o u n d b yFermat , de r ived ea r l i e r a s ( 16 .6 ) .Ref rac t ion . I n F igu re 16 .19 , con s ide r the r igh t t r i ang les BB 'A and BB"A. T h e s es a t is fy the r e la t ions h ipsAB" - AB sin 02 an d AB ' = AB sin 01. (16.11 a)

Tak ing the r a t io s o f the l e f t- and r igh t - h and s ides o f thes e equa t ion s the n y ie ldsAB ' s in 01= . (16.11b)AB" sin 02

The d i s t ances AB ' and AB" a re p ropor t iona l to the ve loc i t i e s v l and v2 , s oAB' v l

~ - ~ oAB" v2Equ a t ing th e r igh t - ha nd s ides o f ( 16 .11 b ) and ( 16 .11 c ) y ie lds

sin 01 Vl,, , !sin 0z v2

o r

(16.1 lc)

(16.12)

nl sin 01 = n2 sin 0 2 , rnl,2 = 9 (16.13)V l , 2This i s the s am e fo rm fo und by Ferm at , de r ived ea r l i e r as (16 .9 ) . Hen ce , bo ththe p a r t i c l e and the w ave v iew po in t s y ie ld the l aw s o f r e f l ec t ion and r e f r ac t ion .

I saac Newton ' s Exper iments and V iews o f L i gh tN e w t o n b e g a n h i s r e s e a r c h e s o n l i g h t i n t h e m i d - 1 6 6 0 s a n d p u b l i s h e d p a p e r son th i s w ork in the 1670s . How ever , he d id no t pub l i s h h i s Opticks un t i l 1704 ,a f t e r the d ea ths o f h is r iva ls Hooke and Huygens . Som e h i s to r i ans a t t r ibu teth i s to h i s desi r e to avo id d i s ag reem en ts w i th thes e s c ien t i st s , ne i the r o f w ho mi n i t i a l l y a c c e p t e d t h a t N e w t o n ' s p r i s m e x p e r i m e n t s e s t a b l i s h e d t h a t s u n l i g h t i sa s uperpo s i t ion o f m any p u re co lo r s.N e w t o n r e j e c t e d t h e v i e w t h a t l i g h t i s a w a v e b e c a u s e h e c o u l d n o t s e eh o w i t c o u ld l e a d t o n e a r ly s t r ai g h t -l i n e p r o p a g a t i o n w i t h p r o n o u n c e d s h ad o w s .


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To avo id hav ing h is com prehens ive wo rk sub ject to m is in terpre ta t ion, Ne wton carefu l l yt r ied to d is t ingu ish exper im enta l fac ts f rom theoret i ca l hypotheses. Book l p resents h isexper ime nts on the co lor separat ion o f sun l ight by a pr ism, and h is experimentum crucis(c ruc ia l exper iment) showing that these co lors , once separated by a pr i sm because o fthe i r (s l ight l y ) d i f fe rent ind ices o f re f rac t ion, re ta in the i r in tegr i ty and the i r index o fre f ract i on a f te rw a rd . H e a lso show ed tha t a beam o f w h i te l igh t w hose co lo rs havebeen separated by a pr i sm can be recombined by pass ing the separated beam througha second pr i sm to produ ce anoth er beam o f wh i te l i gh t , thus estab l ish ing that wh i tel ight i s comp osi te . Book I I ex tends H ooke 's w ork on the co lors o f th in p la tes and th infi lms (e.g., oi l s l icks and soap bubbles). By stud ying th e po si t ions of the co lors associatedwi th a convex lens p laced on f la t g lass (New ton 's r ings) , New ton fo und tha t maxim aoccur in re f lec t ion w hen a cer ta in condi t ion he ld. Had he wr i t te n the e quat ion fo r thecase o f no rmal re f lec t ion, i t wo uld have read

m Id -- (f i ts of reflection)2nHere d is the thickn ess o f the f i lm , n is the index of refraction , m is an od d i n teger ;the length / , w h ich is spec i f ic to a g iven co lor , is wh at in 1704 Ne wton ca lled the" in terva l o f the f i ts " but in ear l ie r work o f 1675 he ca l led "b igness." Book I I I p resentsh is exper imen ts and ideas on d i f f rac t ion, w hich Ne wton ca lled " in f lec t ion" because hetho ug ht o f i t as a d is turbance in the paths o f par t ic les o f l i gh t that p ass by an edge,caused by forces assoc ia ted w i th the edge.

Nev er thele ss , a t leas t in i tia l ly , h is phys ica l p ic tu re of l ight d id not involv e l ightpar t ic les a lone. In the 1670s he w rote: " the rays of l ight . . . . exci te v ib rat ions in thee the r ; . . , w i th va r ious co lou r s , acco rd ing to the i r b ignes s and mix tu re ; the b igges tw i th the s t ronges t co lo r s , r eds and ye l low s ; the l eas t w i th the w eakes t , b luesa n d v i o l e t s . . , m u c h a s n a t u r e m a k e s u s e o f s ev e ra l b ig n e ss e s to g e n e r a t e s o u n d sof d ivers tones ." T h u s , to N e w t o n , t h e v i b r a t i o n s o f t h e " e th e r" h a d c e r ta i n w a v e p r o -p e r ti e s, e v e n i f t h e l ig h t r a y s t h e m s e l v e s d i d n o t . N e w t o n i n d i c a t e d t h e " b i gn e ss " o fl igh t : " i t i s to be s uppo s ed tha t the e the r i s a v ib ra t ing m ed ium l ike a i r , on ly thev ib ra t ions a r e f a r m ore s w i f t and minu te ; tho s e o f a ir , ma de b y a ma n ' s o rd ina ryvo ice , s ucceed ing one ano the r a t more than ha l f a foo t , o r a foo t d i s t ance ; bu t o fe t h e r a t a l e s s d i s t a n c e t h a n t h e h u n d r e d t h o u s a n d t h p a r t o f a n i n c h . " B e c a u s eN e w t o n c o u l d n o t dr a w o n t h e w o r d w a v e l e n g t h , his verbal def in i t ion of " in tervalo f fi ts " t o t h e m o d e r n r e a d e r c o u l d c o r r e s p o n d t o e i t h e r a w a v e l e n g t h o r h a l f aw ave leng th . H ow ever , N e w t on s t a t e s tha t ye l low - orange has an in te rva l o f f it so f 1 /8 9 ,0 00 o f an inch , o r 285 nm, w h ic h i s ha l f o f ye l low - orang e ' s 570 nmw a v e l e n g t h .

Ne wton s tud ied sound and obta ined a va lue for its ve loc i ty in a ir that wa s abou t 25 %below the measured va lue. He a lso s tud ied wate r waves w i th wel l -def ine d w avelengths ,present ing an a rgum ent equ iva lent to )~ f = v [but more l ike (~)~)(2 f ) = v ] . Had he

1 of light, and hisons idered l igh t to be a wave, then f ro m h is es t imate o f the b igness ~)~ow n est imate o f i ts speed v (700,0 00 t imes the speed of sound, or abou t 2 .4 x 108 m/s) ,Ne wton cou ld have est imated the f requenc y f o f l i gh t to be on the order o f 1015 Hz. Thatwo uld have been cons is tent w i th the v iew of l i gh t as a rap id , sm al l -ampl i tude osc i ll a t ion.


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6 9 8 Chapter 16 E Optics

16 .6 Thoma s You ng - - In te r fe re nc eNewt on ' s r epu t a t i on cont i nued t o g row fo l l owi ng h i s dea t h , i n l a rge pa r t be -cause of the success of hi s Principia, re l a t i ng force t o mot i on , i n s t i mul a t i ngq uant i t a t i ve sc i en t i f i c deve l opment . B ut t o nont echni ca l r e ade rs l i k e B enjami nFrank l i n , Newt on ' s Opticks was t he more acce ss i b l e . Whoeve r was t o ca r ryo p t i c s f o r w a r d w a s g o i n g t o n e e d t h e c o n f i d e n c e t o b r e a k w i t h N e w t o n ' s o p -pos i t i on t o l i gh t a s a wave . Around 18 0 0 , Thomas Young (1773 -18 2 9) , edu-ca t ed broad l y bu t by profe s s i on a phys i c i an spec i a l i z i ng i n t he eye , en t e red t hepic ture .

Among o ther th ings , the po lymath Young (known to h is con temporar ies as "The Phe-n o m e n o n " ) e x p la i ne d v i s u a l accommodation (muscles at tache d to the lens chang e i tsshape, thus enabl ing the eye to focus at d i f ferent d istances); proposed the three-colortheory of color v is ion ( the f i rs t theory to propose that our percept ion of colors occurswithin the eye, rather than being an intr ins ic property of l ight) ; and made the f i rs t s tudyo f as t igmat ism (eyes tha t w hen v iewed head-on have an e l lip t ica l shape a lso have tw ofoca l po in ts , one fo r each ax is o f the e l l ipse) . He a lso in t roduced the te rm energy (inthe context of k inet ic energy), was responsib le for what is now known in e last ic i ty asYoung 's modu lus , and ana lyzed the e f fec t o f su r face tens ion on the wet t ing ang le o ff lu ids . He knew over ten languages and made ma jo r con t r ibu t ions to dec ipher ing thetw o unk now n Egyptian languages accom pany ing the Greek on the then recen t ly d iscov-ered R oset ta s tone . Neverthe less , in h is own judg me nt , Young 's mos t imp or tan t w orkwas in es tab l ish ing the w ave na tu re o f l igh t. Y oung 's works on l igh t as a wave inc ludedtw o endur ing demo nst ra t ions : (1 ) the p rev ious ly d iscussed r ipp le tank , fo r dem onst ra t -ing the in te r fe rence o f wate r waves ; and (2 ) the two-s l i t exper iment ( to be d iscussedshor tly ), fo r dem onst ra t ing the in te r fe rence o f l igh t waves. His wo rk w as no t imm ed i -a te ly recogn ized , u l t imate ly due to h is "undu ly conc ise and obscure" p resen ta t ion o fmathemat ica l de ta i l .

L i g h t as a W a v eIn 18 0 0 , t o a rgue i n suppo r t o f l i gh t a s a wave , Young q uot ed N ew t on ' s ana l ogyb e t w e e n t h e t o n e s o f s o u n d a n d t h e i r w a v e le n g th s , a n d b e t w e e n t h e c o l o rs o fl i gh t and t he i r wave l engt hs . To a rgue t ha t waves can be l oca l i z ed i n d i rec t i on( s o a s t o p r o d u c e s h a d o w s ) Y o u n g c o u n t e r e d N e w t o n ' s v i e w t h a t s o u n d w a v e s"d i ve rge eq ua l l y in a ll d irec t ions" by n o t i ng t ha t (1) t he so und of c annons i s mu chl a rge r i n t he forward t han t he back ward d i rec t i on ; and (2 ) " speak i ng t rumpe t s"( m e g a p h o n e s ) b y design produce sound t ha t i s l oca l i z ed i n d i rec t i on . Fur t he r ,t o a rgue aga i ns t " the N ew t oni an sys t em" o f l igh t a s an em i t t ed pa r t i c le , Youngnot ed t ha t (1) a s obse rved by Huygens , i t does no t ex pl a i n why t he ve l oc i t yof l i gh t is i nde pen den t o f t he i n t ens i t y o f i ts source ( t he ha rde r a ba l l ~ a t ypeof pa r t i c l e~ i s t h row n, t he fa s t e r it goes ); and (2 ) bo t h pa r t ia l r e f le c t ion an dre f rac t ion occur . B ot h of t he se ph eno m ena a re ea s il y ex pl a i ned by t he ana l ogousbehavi or o f sound waves .In Theo ry o f L igh t and Co lours (18 0 1) , Young s t a t ed t he principle o f in ter-ference, q uot ed a t t he chap t e r head i ng : W hen two Undu la t ions [waves] , f rom


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16.6 Thom as Young- - In ter ference 6 9 9

16~6~2

D if feren t Or ig ins , co incide e i the r per fec t ly or very ne ar ly in d irec t ion , the ir jo in t e f for ti s a C o m b i n a t i o n o f t h e M o t i o n s b e l o n g in g to ea ch .Th e imp l ica t ions o f th i s p r inc ip le w ere w orke d ou t p rev ious ly , in Sec t ion16 .2 . F igu re 16 .2 il lu s t ra te s the ad d i t ion o f tw o w aves o f the s ame am pl i tud ea n d w a v e l e n g th , w h e n c o m p l e t e l y i n p h a s e a n d w h e n c o m p l e t e l y o u t o f p h a se .Thes e cas es co r r es pond to ( 16 .1 ) and ( 16 .2 ) .C ohe re nceY o u n g a p p l i e d t h e p r i n c i p l e o f i n t e r f e re n c e t o t w o l i g h t w a v e s o n l y w h e n t h e yh a d t h e s a m e w a v e l e n g t h a n d o n l y w h e n t h e y h a d t h e s a m e s o u rc e , so t h e y w e r ei n p h a s e r e l a ti v e to o n e a n o t h e r o v e r a t i m e l o n g c o m p a r e d t o t h e r e s o l u t i o n t i m eof the eye . ( Th i s t ime i s abou t 1 /30 o f a s econd , a f ac t u s ed w hen mov ing p ic -tu res foo l the eye in to th ink ing tha t a r ap id s ucces s ion o f ind iv idua l pho tog rap hsi s a con t in uu m o f images . ) Such " in - phas e - nes s" i s kno w n as coherence . A typ ica ll igh t s ou rce has a coherence t ime T~ of abou t 10 - 9 s , co r r es pon d ing to a bou t 106osci l la t ions of the l ight , wi t h cha racter is t ic f re que ncy f ~ 0 .5 10 is Hz. S incel ight t ravels a t 3 108 m/s , th is r~ corre spon ds to w ha t is cal led a l o n g i t u d i n a lco h er ence l eng t h l ~ - cT ~ o f a b o u t 0 . 1 5 m . A " p h o t o g r a p h " o f s u c h a w a v e w o u l ds how tha t , s t a r t ing f rom a g iven w ave peak , w i th in the long i tud ina l coherencel e n g t h t h e o t h e r w a v e p e a k s w o u l d b e a p p r o x i m a t e l y a n i n t e g r a l n u m b e r o fw ave len g ths f rom the g iven w ave peak , b u t 10 ~ peaks aw ay the re w o u ld be s ig -n i f i can t dephas ing . A l igh t w ave a l so has a t r ans ver s e ex ten t g iven by it s t r a n s ver s ecoherence leng th . B o t h t y p e s o f c o h e r e n c e p r o v i d e l i m i t a ti o n s o n i n t e r f e r e n c e p h e -nom ena . Typ ica l com me rc ia l l a se r s have r~ 's on the o rde r o f 10 - 8 s . Las e r s w i thvery w e l l - de f ined f r eque nc ies can in p r inc ip le have r~ on the o rde r o f 102 s, bu tme chan ic a l s t ab i l ity o f the l a s er cav i ty r educes th i s to on the o rde r o f 10 - s s. Bycom par i s on , a be l l w i th f - 500 Hz m igh t have T~ on the o rde r o f 2 s, co r r e -s p o n d i n g t o a b o u t 1 0 0 0 os c il la ti o ns , o r a Q ~ s e e S e c t io n 1 4 . 3 ~ o f a b o u t 1 0 00 .F igu re 16 .20 g ives s chem at ics o f w ave f ron t s w i th good t r ans ver s e and long i tud i -na l coherence , w i th good long i tud ina l coh erence on ly , and w i th good t rans ver s ecoherence on ly .

m ~

I I

(a) (b) (c)Figure 16.20 Examples of wavefronts with v arying degrees of coherence: (a) goodlongitudinal and good transverse coherence, (b) good longitudinal coherence butweak transverse coherence, (c) good transverse coherence but weak longitudinalcoherence. D ashed line s in (b) and (c) indicate, for comparison, good longitudinaland good transverse coherence. An even better representation would vary thethickness of the dark lines at the w ave peak to indicate variations in amplitude.


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700 C h a p t e r 1 6 m O p t i c s

16.6.3 C o lo rs o f S t r iate d S ur face sY o u n g ' s f ir st d e t a i l e d a p p l i c a t i o n o f t h e p r i n c i p l e o f i n t e r f e r e n c e w a s t o t h ec o l o rs s o m e t i m e s s e e n w h e n l i g h t f r o m a s m a l l d i s t a n t l ig h t s o u r c e i s r e f l e c t e d o f f

8 2" I.. ~ S l ' b r-P 2 / ? " d -. /" - / . . 1 2

a = (n /4 ) -0 e l

Ray(zl/4) / $2

Figure16.21 N o r m a l i n t e r s e c t i o n w i t h t h e p a g eat P1 and P2 of a pa i r of l ines scra tched onY o u n g 's m i c r o m e t e r . T h e i n c i d e n t w a v e f r o n t$1-$2 sca t te rs to th e w ave fron t S ' I -S ~. Th epo in t P i s fo r purpose s o f compar i son . I n t heexperiment, 81-S2 and S '1-S~ were f ixed, and P2w a s r o t a t e d a b o u t P 1.

a s u r f a c e i n s c r i b e d w i t h t w on e a r b y n a r r o w , p a r a l l e l g r o o v e s .H e e x p e r i m e n t e d w i t h a m i -c r o m e t e r f o r w h i c h e a c h in -s c r i b e d l i n e w a s a c t u a l l y a p a i r o fl in e s s e p a r a t e d b y 0 . 0 0 0 1 i n c h ,s o d = 2 . 5 4 x 1 0 - 4 c m i n F i g u r e1 6 . 2 1 . ( T h e p a i r - t o - p a i r s e p a r a -t i o n o f 0 . 0 0 2 i n c h , w h i c h Y o u n gn o t e d , w a s n o t r e l e v a n t t o t h eo b s e r v e d e f f e c ts . ) F o r s u n l i g h ta t g l a n c i n g i n c i d e n c e ( ~ = 0 , o rP 1 - P 2 a l o n g t h e x - a x i s i n F i g u r e1 6 . 2 1 ), h e s a w o n l y b r i g h t r e d .W i t h ~ = J r / 4 - 0 , t h i s c o r r e -s p o n d e d t o 0 = ~ r /4 . O n r o t a -t io n o f t h e m i c r o m e t e r b y a ,

t h u s d e c r e a s i n g O , h e a l s o o b s e r v e d b r i g h t r e d f o r a n g l e s 0 o f 3 2 ~ 2 0 . 7 5 ~ a n d1 0 . 2 5 ~

~ Young'smicrometer( a) F o r Y o u n g ' s m i c r o m e t e r e x p e r i m e n t , r e l a te t h e d i f f e r e n c e i n p a t h l e n g t hr 2 - r l t o t h e l i n e s e p a r a t i o n d a n d t h e a n g l e 0 . ( b ) F r o m t h e d a t a , d e t e r m i n et h e w a v e l e n g t h ;v. ( T o Y o u n g , t h e " l e n g t h o f a n U n d u l a t i o n . " )Solution: ( a ) Cons ide r a gene ra l ro t a t i on ang l e a , wi th po in t s P1 and P2 sepa ra t edby d. By Figure 16.21, t he d i f ference in pa th length r2 - r l i s g iven by

)2 -- rl = P2 P - P1 P = d(c os a - sin a) = d sin ~ c os a - cos ~- sin cz ,w h e r e s in { = c os ~ - ~ h a s b e e n e m p l o y e d . W i t h s i n ( A - B ) - s in A c o s B -cos As in B and 0 = z r / 4 - ~ , t h is b e c o m e s

r2 r l = d ~/2 sin ( 4 )- a = , /2 d sin O.

(This gives r2 - rl = d for 0 - r r / 4 , as expec t ed . ) ( b ) For cons t ruc t i ve i n t e r fe r -ence , using this r2 - r l in (16.1) yie lds~/2 d sin 0 = m)v.

Taking d - 2 .54 x 10 - 4 cm, and m = 1, 2 , 3 , 4 to cor resp ond to 10.25 ~ 20.75 ~32 ~ and 4 5 ~ this gives )v = 63 9 n m, w hich inde ed l ies in the red. (Cha rac ter-i s t i c a l l y , Young mere ly s t a t ed wi thou t p roof t ha t t he ex t ra pa th d i f f e rence wasprop or t i ona l t o d si n 0 . ) No te t ha t shor t e r w ave l eng ths )~ ( e.g. , t owa rd t h e b lue )g ive i n t e r fe rence a t sma l l e r ang l e s 0 .


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16.6 T h o m a s Y o u n g - - I n te r fe r e n c e 701

C o lo r s o f T h i n F i lms a nd T h i n P la tesC o n s i d e r l i g h t o f w a v e l e n g t h s a t n e a r n o r m a l i n c i d e n c e o n a p la t e o f t h i c k n e s s dsu r rou nd e d b y a i r. I n F i gu re 16 .22 , t wo re f l e c t e d r a ys a re r e p re s e n t e d , one f romt h e f r o n t s u r f a c e a n d t h e o t h e r f r o m t h e b a c ksu r fa c e. O ur goa l is t o und e r s t a nd t he c o l o r ss e e n in r e fl e ct io n . A s r e c o g n i z e d b y N e w t o n , t ot h e e x t e n t t h a t l i g h t i s n o t a b s o r b e d w i t h i n t h ef i l m , m a x i m a i n r e f l e c t i o n c o r r e s p o n d t o m i n -i m a i n t r a nsm i s s i on , a nd v i c e ve r sa .In a na l yz i ng t h i s s i t ua t i on , Y o ung f i r st no t e dt h a t , f o r m o t i o n i n o n e d i m e n s i o n , a l i g h t o b -j e c t c o l l i d i ng wi t h a m a ss i ve ob j e c t wi l l c on -t i nue fo rwa rd i n t he s a m e d i re c t i on , a s i n F i gu re16 .23(a ) , w he re a s a m a ss i ve ob j e c t c o l l i d i ngwi t h a l i gh t ob j e c t wi l l bounc e ba c k ( i . e . , r e -Figure 16.22 P at hs o f t wo ra y s ve r se d i r e c t i on ) , a s i n F i gu re 1 6 .23(b ) . I f t here f lec ted to the eye a t nea r ve lo c i ty i s rep res en ted as a pos i t iv e am pl i tud enormal inc idence , t im es a s ine or cos ine , th i s reve rs a l in d i re c t ion

i n t he f i r s t c a se m a y be i n t e rp re t e d a s a pha sesh i f t o f 180 ~ Y oun g t he n a rgu e d ( c o r re c t l y ) t ha t a s i m i l a r pha s e sh i f t oc c u r s onr e f le c t io n w h e n l i g h t is i n c i d e n t o n a m o r e o p t i ca l ly d e n s e m a t e r i a l ( h i g h e r i n d e xof r e f ra c t i on ) . Thus , i n F i gu re 16 .22 , o f t he t w o re f l e c t e d r a ys, r a y 1 i nvo l ve s a180 + pha se sh i f t a nd r a y 2 doe s no t . Th i s ph a se sh i f t c o r re spo nds t o ha l f a wa v e -l e n g t h s i n t h e m e d i u m . W i t h n t h e i n d e x o f r e f r ac t io n , ( 1 5 . 5 9 ) g i ve s s - sso s sB e c a u s e o f t h i s e x t r a p h a s e s h if t, a t n e a r n o r m a l i n c i d e n c e t h e c o n d i t i o n t h a tt h e t w o r e f l e ct e d w a v e s m u s t y i e l d c o n s t r u c t iv e i n t e r f e r e n c e i s t h a t t h e e x t r a

p a t h l e n g t h r 2 - r ~ - 2 d m u s t c o r r e s p o n d t o a h a l f -i n t eg r a l n u m b e r m + 89 ofw a v e l e n g th s A ' = A / n . Thus , fo r c ons t ruc t i ve i n t e r fe re nc e on r e f l e c t i on ,

Figure 16.23 (a) Scattering of a light object off a heavy object. (b) Scattering of aheavy object off a light object.


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702 Chap ter 16 * Optics

S im i l a r ly , t h e co n d i t i o n f o r d e s t r u c t iv e i n t e r f e r en ce i s!iiill!!!iiiiiii~iiiii~ii!~i!!~i!!~~!!(!~ll ~ i!~!!!~i~i!~i~i!~i~i~i!~(~?(~i !~~ ~ ~ ,~ ~7% ii~i( ~iil84~!i!)~i~84 84 ~~~%~~~iiiiii~!ii~iii~i~i!~i~i~2 d : m = O , 1 , , i i ~ d e s t r u ~ v e ~ i i ~ !i ~ ~ i ii ! ~ !i i :i ! !i ~ ; ii ! i

9 i!ii!i!!T h e s e e q u a t i o n s a r e c o n s i s te n t w i t h N e w t o n ' s s t u d ie s , b u t g o b e y o n d t h e m i nth a t wav e len g th ap p ea r s , r a th e r t h an in t e r v a l o f f it s. E v en Y o u n g d id n o t u se t h ew o r d wavelength, b u t c l e ar l y h e h a d t h a t i d e a in m i n d .E q u a t io n s (1 6 .1 4 ) an d (1 6 .1 5 ) ap p ly t o t h e co lo r s o f t h in p l a t e s (F ig u r e16 .13) , soap f i lms, and o i l s l icks on water . I t exp la ins why , when such f i lmsa r e v e r y t h in , so d ( ( )~ , t h ey r e f l e c t n o l i g h t a t a l l [m = 0 in (1 6 .1 5 ) ] , b u t a ti n t e r m ed ia t e t h i ck n esse s t h e co lo r t h ey r e f l e c t d ep en d s u p o n f i lm th i ck n ess . In -t e r f e r en c e o f t h i s so r t ex p la in s t h e i r r i d e scen t co lo rs o f t h e w in g ca se s o f b ea t le s ,o f p e a c o c k a n d h u m m i n g b i r d f e a th e r s, a n d o f m o t h e r - o f -p e a r l . F o r a g i ve n o r d e rm , t h e l a r g e r t h e t h i ck n ess d , t h e l a r g e r t h e wav e len g th ;~ n ee d ed f o r t h e ch a r -a c t e ri s ti c i n t e rf e r e n c e m a x i m u m o r m i n i m u m . F i g ur e 1 6 . 2 4 i ll u s tr a te s a ch a r a c -t e r i s t ic t h in - f i lm in t e r f e r en c e p a t t e r n w h en th e t h i ck n ess o f t h e f i lm va r ie s , soth e wav e len g th l e ad in g to en h an ced r e f l e c t i o n v a r i e s . P l a t e 3 sh o ws th e sam ef igure in co lor . Pla te 4 shows soap bubbles . Pla te 5 shows peacock fea thers .Enjoy.

Fr o m h i s v a lu e s o f t h e w av e len g th s o f t h e d i f f e r en t co lo r s o f l i g h t , Y o u n g wasab le t o d e t e r m i n e th e co r r e sp o n d in g f r eq u en c ie s (o n th e o r d e r o f 1 0 i s H z ) , v ia)~ f = c (wh ich d id n o t ap p ea r i n h is p ap e r ) , f i n d in g th a t " th e ab so lu t e f r eq u e n cyex p r e ssed in n u m b er s i s t o o g r ea t t o b e d i s t i n c t ly co n ce iv ed . "

Figure 16.24 Oil slick, illustrating the different colors tha t can be seen on reflection asthe film thickness changes.


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16.6 T h o m a s Y o u n g h l n te r f er e n c e 703

Y o u n g a l so n o t e d t h a t i f a m a t e r i a l o f i n t e r m e d i a t e i n d e x o f r e f r a c t io n i s p l a c e db e t w e e n t h e a i r a n d t h e f ir st m e d i u m , t h e n b o t h r e f le c t i o ns i n v o l v e p h a s e s h i f tso f 1 8 0 ~ a n d t h e c o n d i t i o n s f o r c o n s t r u c t i v e i n t e r f e r e n c e a n d d e s t r u c t i v e i n t e r-f e r e n c e s w i t c h . ( P l a c i n g s a s sa f ra s o il , w i t h i n d e x o f r e f r a c t i o n 1 . 5 3 5 , b e t w e e nf l in t g la s s a n d c r o w n g l a ss , w i t h i n d i c e s o f r e f r a c t i o n 1 . 6 6 a n d 1 . 5 2, Y o u n g l a t e re s t a b l i s h e d t h a t s u c h a p h a s e s h i f t d o e s o c c u r . )~ Soap ilm interference

Co n s i d e r a s o ap f il m ( n = 1 .3 3 ) o f t h i ck n es s 5 4 0 n m , s u p p o r t ed o n a c i r cu l a rh o o p . ( a ) Fo r w h i t e l i g h t ( f ro m 4 0 0 n m t o 7 0 0 n m ) a t n o r m a l i n c i d en ce ,f i n d t h e r e f l e c t ed co l o r s t h a t a r e i n t en s i f i ed an d t h o s e t h a t a r e w eak en ed b yi n t e r f e r en ce . ( b ) Rep ea t f o r t r an s m i s s i o n , a s s u m i n g t h a t t h e l i g h t i s n e i t h e rs ca t t e r ed n o r ab s o r b ed i n t h e f il m . ( c ) Rep e a t f o r r e fl e c t i o n , i f t h e s o ap f i lmis f loa t ing on a su r f ace o f sassaf r as o i l (n = 1 .535) .Solu t ion: (a) For cons t ruc t ive in ter f er ence (16 .14) g ives m + 8 9where in our case 2 d n = 1436 nm . For )~ = 400 n m this gives m + 89= 3.59, sug-gest ing that m = 3 wil l g ive a solut ion for som e value o f X in the vis ible. Indeed,m = 3 cor r esponds to 410 n m (v io le t ) . Fur ther , m = 2 cor r esponds to 574 nm(yellow). (Larger m 's cor respond to ~ ' s in the u l t r av io le t , and sm al ler m 's to X's inthe inf rared.) For destruct ive inter ference, (16.15) gives m = 2 d n / ) ~ . For m = 3this gives ~. - 479 n m (blue -gree n). (Again, larger m 's cor respond to ~ ' s in theultraviolet , and smaller m 's lead to )~'s in the inf rared.) Thus, in ref lect ion th e f i lmwi l l appe ar to be a mix tu re o f v io le t and yel low, w i th b lue-green absen t . (b ) Inthe absence o f absorp t ion and sca t te r ing in the f ilm, the co lo rs tha t a r e r e f lec teds t rong ly w i l l be miss ing f rom the t r ansmi t ted l igh t ( thus v io le t and ye l low areabsen t f rom t r ansmiss ion) , and the co lo rs tha t a r e leas t re f lec ted wi l l be e nhan cedin t r ansmiss ion ( thus the f i lm wi l l appear to be b lue-green in t r ansmiss ion) . (c)W ith sassafras oi l in place, th ere is an extra 180 ~ pha se shif t, so 2 d = m ) ~ / n givescons t ruc t ive in ter fer ence , and 2d = (m + 89 g ives des t ruc t ive in ter f erence . Inref lec t ion , th i s w i ll g ive an enha nce me nt o f b lue-green and a weaken ing o f v i -olet and yel low. Thin-f i lm antiref lect ion coat ings, having an index of ref ract ionin term edia te be tw een tha t o f a ir and g lass , use th i s e f fec t .For th ick f i lms , the num ber o f waveleng ths g iv ing cons t ruc t ive in ter f er ence i slarge, an d they ar e so c lose togethe r in wav eleng th tha t , e f fec t ive ly , a ll waveleng thsare observed . As a consequence , in ter f er en ce ef f ec t s beco me less no t iceab le to th eeye.

~ New ton's rings: versushickness radiusY o u n g an a l y z ed N ew t o n ' s o r i g i n a l d a t a f o r t h i n f i l m s an d t h i n p l a t e s . F i g u r e1 6 .2 5 ( a ) d e p i c t s a co n v ex l en s o n a f la t p l a t e o f g la ss . T h e r e a r e t h r ee r e f le c -t i o n s: o f f t h e f l a t t o p s u r f ace o f t h e l en s ( n o t d r aw n ) , o f f t h e r o u n d b o t t o ms u r f ace o f t h e l ens , an d o f f t h e f l a t p l a t e o f g la ss . O n l y t h e l a t t e r t w o h av er e f l ec t ed b eam s c l o s e en o u g h t o ex h i b i t i n t e r f e r en ce . T h e e f f ec t i v e f i lm t h i ck -n es s d o f t h e a i r l ay e r v ar i e s w i t h d i s t an ce f r o m t h e cen t e r o f t h e l en s s o t h a tt h e co l o r s ch an g e a s t h i s d i s t an ce ch an g es .(a) For a g iven wa vel eng th )~, f ind d as a fun ct io n o f the ax ia l d i s tance rf r o m t h e cen t e r o f t h e l en s, o f r ad i u s o f cu r v a t u r e R . ( b ) F i n d t h e eq

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