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he he y f he b wWhen sunl ght is sca ered by ra ndrops, why is it ha colorf larcs appear n certain reg ons of he sky? Answer ng h s leques on has required all he resour es of ma hema i al ph

he r i bo is bri ge be ee het o cultures: oe s scie is s

like h ve lo g bee ch lle ge oescribe it. he scie i c escri io is

ofte su ose o be si le roblei geo e ric l o ics roble h

s solve lo g go h hol s i er es to y o ly s his oric l exercise

his is o so s isf c ory qu ti iveheory of the r i bo h s bee evel

o e o ly i the st fe ye rs. Moreover h heory i volves uch ore

h geo e ric l o ics i dr s o lle k o of the ure of gh . Allo

ce us be e for velike ro eries such s i erfere ce i r ctio dol riz io for r iclelike ro

erties such s he o e u c rrie by be of ligh .So e of the ost o erful ools of

he ic l hysics ere evise exlici ly o e l i h the roble of he

r i bo i h closely rel te roble s. I ee the r i bo h s serve s

ouchs o e for es i g theories of oics. Wi h he ore successful of thoseheories i is o ossible o escribe he

r i bo he ic lly h is o reic he is ribu io of light i he sky.he s e e ho s c lso be lieo rel e he o e such s he brigh

ri g of color c lle he glory eveo o her ki s of r i bo s such sto ic ucle r o es.

Scie i c i sigh h s o l ys beeelco e ithou reserv io s. Goe hero e h Ne o 's lysis of he r i

bo 's co ors oul "cri le N ture's he r . A si il r se i e s ex

resse by Ch rles L b Johe s i er r y i 1817 heyro ose to s : "Ne to 's he l h

co fusio o he ics. e he sci e is s ho h ve co ribu e o he heory of the r i bo re by o e s ise sitive o he r i bo 's be u y. I he

or s of Desc r es " he r i bo issuch re rk ble rve of N ure. . . h I coul h r ly choose oresui ble ex le for he lic io of y e ho .

he si gle brigh rc see f er r isho er or i the s r y of erf ll is

b H. zv g

the ri ry r i bo Cer i ly i s osco s icuous fe ure is i s s l sh of coors. hese v ry a goo e l i brigh ess

isti c ess but they l ys follohe s e seque ce viole is i er ost

ble i g gr u lly i h v rious sh esof blue gree yello d or ge i h re ou er os .

O her fe tures of he r i bo ref i er d i eed re no l ys res e . Higher in he sky h he ri rybo s the seco ry o e i hich thecolors e r i reverse or er i h redi er ost a d violet ou er ost. C refulobserv ion reve ls h t he region be

ee the o bo s is co si er blyrker h he surrou i g sky. Even

he he seco ry bo is ot iscerible he ri ry bo c be see o h ve "lighte si e d a "d rk si e.

he rk regio h s bee give he e Alex er's rk b f er heGreek hiloso her Alex er of A h ro isi s ho rs escribe i i bout

. . 200.A o her fe ure h is o ly so e

i es see is series of f i b s usully i k gree l er ely o he

i er si e of he ri ry bo (Eve ore r re y hey y e r o the ou er si e of he seco ry bo . hese"su er u er ry rcs re usu lly see os cle rly e r he o of the bo .

hey re y hi g bu co s icuous bu

hey h ve h jor i ue ce o theevelo e of heories of he r i bo .

he rs e o r io lly ex l ihe e r ce of he r i bo s

rob bly h of Aris otle. He ro oseh he r i bo is c u lly u usu l

ki of re ec io of su ligh froclou s. he light is re ec ed xe

gle givi g rise o circul r co eof "r i bo r ys. Aristo le hus ex

l i e correc ly he circul r sh e ofhe bo erceive h i is no

eri l objec i h e ite loc ioi he sky bu r her set of irec io s

lo g hich ligh is stro gly sc erei o he eyes of he observer.

he gle for e by he r i bo r ys he i ci e t su ligh s rst e

sure i 1266 by Roger Baco He mesure n gle of bout 42 egrees; these o ry bo is bou eight degrees higher i he sky. o y these a gles arecus o rily e sured fro the o posi e irec io so h t e easure the

o l ch ge i he irec io of the su 's r ys. he gle of he ri ry bo is

herefore 180 i us 42, or 13 8, egreeshis is c lle he r i bo a gle. he

gle of he seco ry bow s 130 egrees.After Aris o le's co jecture so e 17

ce uries ssed before fur her sig ic rogress s e i he theory of

he r i bo . I 1304 he Ger o kheo oric of Freiberg rejec ed Aris o

le's hy o hesis hat the ra bo resul sfro collec ive re ec ion by he r i

ro s i a c oud. He sugges e i s eth e ch ro is individu lly c ble of

ro uci g r i bo . Moreover he est e his co jec ure i ex eri e ts i h a g i e r i ro : a s herica sk

lle i h er. He as ab e o tr cethe h follo e by the ligh r ys h ke u he r i bo .

heo oric's di gs re i e l rgelyu k o for hree ce uries u til hey

ere i e e en ly re iscovere byDesc r es ho e oyed the s e

etho . Both heo oric nd Descartessho e h he r i bo is m de u of r ys h e ter dro let a re re ec e o ce fro the i er surf ce The sec

o ry bo co sis s of r ys th t h vu ergo e o i ernal re ectio s.Wi h e ch re ec ion so e light s los

hich is he in re son the second rybo is f i ter h n he ri ry o e.

heo oric Desc r es also o e thlo g e ch irectio i hi he a gul r

OUBLE A NBOW was photographed atohnstone Strait in Brit sh Columbia The

bright, inner band is the primary bow it isseparated from the fainter secondary bow bya region, called Alexander' da k band, that

is noticeab y darker than the surrounding sky.Below the primary bow are a few faint stripesof pink and green hey are supernumeraryarcs. The task of theory is give a quantitati e explanation for each of these features.

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� LIGHTED

� - - � SIDE

� - + c_ SECONDARYRAINBOW

ALEXANDER'SDARK BAND

z PRIMARY= = RAINBOWSUPERNUMERARYARCS

IGEOMETRY OF THE RA NBO is de erm ed by he sca ering ang e: he o a ang e hrough hich a ray of sun igh is ben by i s passage hrough a raindrop. Rays are s rong ysca ered a ang of 13 degrees and 30 degrees, giving rise respec ive y o he primary and he secondary rainbo s. Be een hose ang es very i e igh is de ec ed ha is he region ofA exander s dark band. The op imum ang es are s igh y di eren for each ave eng h of igh ,

i h he resu ha he co ors are dispersed no e ha he sequence of co ors in he secondary

bo is he reve e of ha in he primary bo . There is no sing e p ane in hich he rainbo es· he rainbo is mere y he se of direc ions a ong hich igh is sca ered o ard he observe .

ange c esp nd ng t the a nb wn y ne c at a t me c d be seen n

the ght catte ed by the g be henthe eye as m ved t a new p s t n sas t exp e the catte ng ang es the

the spe t a c s appea ed ne byne The d c and esca tes c nc d

ed that each the c s n the a nb wc mes t the eye m d e ent et wate d p ets

As The d c and e ca tes ea eda the ma n eat es the nb w canbe nde st d th gh a c ns de at n

the ght pass ng th gh a s ng e

REFLECT ON D REFRACT ON of igh a boundari be een and a er are hebasic even s in he crea ion of a rainbo . n re ec ion he ang e of incidence is equa o heang e of re ec i n. n refrac ion he ang e of he ansmi ed ray is de ermined by he proper iof he medium, as charac erized by i refrac ive index. Ligh en ering a medium i h a higherindex is ben o ard he norma . Ligh of di eren ave eng hs is refrac ed hrough s igh ydi eren ang his dependence of he refrac ive index on co or is ca ed dispe ion. Theoriof he rainbo of en dea separa e y i h each monochroma ic componen of inciden igh .

d p et The ndamenta p nc p esthat dete m ne the nat e the b w a eth se that g ve n the nte act n ghtw th t anspa ent med a name y e ect n and e act n

he aw e ect n s the am aand nt t ve y bv s p nc p e thatthe ang e e ect n m st eq a theang e nc dence The aw e act n

s s mewhat m e c mp cated he e

as the pa h a e ected ay s dete m ned ent e y by ge met y e act na s nv ves the p pe t es ght andthe p pe t es the med m

The speed ght n a vac m s nva an ; ndeed t s ne the nda

enta c nstants nat e The speed ght n a mate a med m n the

the hand s dete m ned by the p pet e the med m The at thespe ght n a vac m t the speed

n a bstance s ca ed the e act vendex that s bstance F a the n

dex s n y s ght y g eate than 1; wate t ab t 1 33

A ay ght pass ng m a ntwate s eta ded at the b nda y; tst es the s ace b q e y the change

n speed es ts n a change n d ect nThe s nes the ang es nc dence an e a t n a e a ways n c nstant at t each t e and the at s eq a t thatbetween the e act ve ndexes thetw mate a s Th s eq a ty s ca edSne 's aw a te eb d Sne wh

m ated t n 1621

p e m n y a a ys s the a nb wcan be bta ned by app y ng the

aws e ect n and e act n t the path a ay th gh a d p et Beca sethe d p et s ass med t be sphe ca ad ect ns a e eq va ent and the e s

n y ne s gn cant va ab e the d s p acement the nc dent ay m anax s pa s ng th gh the cente thed p et That d sp acement s ca ed the

mpact pa amete It anges m ewhen the ay c nc des w th the cent aax s t the ad s the d p et whenthe ay s tangent a

At th e su rf ace of the d rop let the in ci-d en t ra y is p ar tiall y refe ct ed. an d th is re ected ligh t w e s ha ll i de ntif y as th escat er ed r ay s of Class 1. Th e rem ai nin gligh t is tr an sm it ted in to t he dro plet (with a ch an ge in dir ec tio n cau sed b y ref ra c-tio n) a nd at t he ne xt su rfac e is a gain

par tial ly tr an sm itt ed (r ays of C lass 2 )a nd pa rtia lly r e ect ed. A t t he n extboun da ry th e r ef ec te d ra y is ag ain s pli tinto re ec ted a nd t ran sm itte d co m po- nen ts . a nd th e pr oces s con ti nu es inde - nit ely. T hus t he d ropl et give s r ise to a se ries of scatt e red ray s. usua lly w ith rap id ly d ec rea sing int en sit y Rays ofCl as s 1 r epr esen t dir ec t r e ec tion by th e d rop let a nd t ho se o f Cla ss 2 are dir ectl y

tra nsm itte d thr oug h it R ay s of C la ss 3a re t hose th a t e scap e t h e d rop l et a fte r o n e in ter nal r e e ctio n. a nd t h ey m ak e u p th e p rim a ry ra i nb o w Th e C las s 4 ra ys. hav ing u nd erg o ne tw o i nter nal re-

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ec io s give rise o he seco ry bo .R i bo s of higher or er re for e by r ys ki g ore co lic e ss ges bu hey re o or i rily visible.

For e ch cl ss of sc ere r ys hesc eri g gle v ries over i e r geof v lues s fu c io of he i

r e er. Si ce i su ligh he ro leis illu i e ll i c r e ers

si ul eously ligh is s ere i viru lly ll i re io s. I is o i ul o ligh hs hrough he ro le

co ribu e o he r i bo bu here rei i ely y o her hs ire

he ligh else here. W y he is esc ere i e si y e h e i e ici i y o he r i bo gle? I is ques

io heo ori i o o si er s er s rs rovi e by Des r es.

By lyi g e l s o re e io refr c io e h oi here rs rikes ir er bou r Dec r es i s ki gly o u e he hof y r ys i i e y i

r e ers. he r ys of l ss 3 re ofre o i i g i or e. W e he

i c r e er is zero hese r ys resc ere hrough gle of 180 egrees h is hey re b cks ere

o r he su h vi g sse hroughhe e er of he ro le bee reec e fro he f r ll. As he i

r e er i cre ses e i i e r ys re is l ce fro e ce er of e

ro le he sc eri g gle e re ses.Desc r es fou ho ever h his

re oes o co i ue s he ir e er is i cre se o i s xi u

v lue here he i ci e r y gr zes ero le a ge o i s sur ce. Is e he sc eri g gle sses hrough

i i u he he i r e eris bou seve eigh hs of he r ius of

he ro le here f er i i re sesgai he sc eri g gle e i i

u is 138 egrees.For r ys of l ss he sc eri g

gle is zero he he i c r e eris ero i o her or s he ce r l r y

s re ec e ice he co i ues i i sorigi l irec io . As he i c r e er i cre ses so oes he sc eri g -gle bu g i he re is eve u lly re

verse his i e 130 egrees. heCl ss 4 r ys h ve xi u sc eri ggle of 130 egrees s he ir e er is fur her i cre se hey be

b ck o r he for r sc eri g i rec o g i

B ec use ro le i su ligh is u ifor ly illu i e he i c

r e ers of he i i e r ys re u ifor ly is ribu e . he co ce r ioof sc ere ligh is herefore ex ec e

o be gre es here he sc eri g glev ries mos slo ly i h ch ges i hei ac p r e er I o her or s hesc ere ligh is brigh es here i g h

er oge her he i ci e r ys fro helarges r ge of i r e ers. he regio s of mi i u v ri io re hosesurro i g he xi u i i

u s eri g gles so he s eci l s us of he ri ry se o ry r i bo gles is ex l i e . Fur her ore si ce o r ys of l ss 3 or l ss

re sc ere i o he gul r regio beee 130 138 egrees Alex er srk b is lso ex l i e .Desc r es s heory be see ore

le rly by co si eri g i gi ry

o ul io of ro le s fro hi h lighis so eho s ere i h u ifor ie si y i ll ire io s. A sky lle i h

su h ro le s oul be u ifor ly brigh ll gles. I sky lle i h re l

er ro le s e s e o l illu iio is v il ble bu i is re is ribu e

Mos r s o he sky re i er hhey o l be i h u ifor s eri g

bu i e vi i i y of he r i bo glehere is brig r eri g o gr ully o he ligh e si e ore sh r ly

o he rk si e. he se o ry bo is si il r i e si y highligh ex e h iis rro er ll i s e ures re i er. I he r esi heory he regiobe ee he bo s is is i ly rker

h he sky else here if o ly r ys ofl ss 3 l ss exis e i oul be

qui e bl ck.he r esi r i bo i re rk

T -

IMPAPA AME E

bly si le he o e o . Brigh ess is fu c io of he r e hich he sceri g gle ch ges. h gle is i selfe er i e by jus o f c ors: he re

fr c ive i ex hich is ssu e o beco s he i c r e er

hi h is ssu e o be u i or ly is-ribu e . O e f or h h s o i u

e ce ll o he r i bo gle is size:

he geo e ry o s eri g is he s efor s ll lou ro le s for hel rge er lle globes e loye by

heo oric Des r es.

o f r e h ve ig ore o e of he osco s icuous fe ures of he r i

bo : i s colors. hey ere ex l i e ofcourse by Ne o i his ris ex eri e s of 1666. ose ex eri e s eo s r e o o ly h hi e ligh is ix ure of colors bu lso h he refr c ive i ex is i ere for e ch olor

he e ec c lle is ersio . I follo sh e ch color or ele g h of ligh

us h ve i s o r i bo gle he observe i ure is ollec io of

o ochro ic r i bo s e ch o esligh ly is l ce fro he ex .

Fro his e sure e s of he refr cive i ex Ne o l ul e h he

WA E OPLET

LASS 1

ATH OF L GHT through a droplet can be determined by applying the laws of geome ricaoptics. Each time the beam strikes he surface par of the light is re ected and pa is efracted.

ays re ecte directly from the surface are abeled rays of Class 1 those transmitted direct ythrough the droplet are designated Class The Class 3 rays emerge after one interna e ection it is these that give rise to the primary rainbow. The secondary bow is made up of C ass

rays, which have undergone two internal re ections. For rays of each c ass on y one factordetermines the value of the scattering angle. That factor is the impact parameter: the displacement of the incident ray from an axis that passes through the center of the d op e

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ainbow angle is 137 egr es 58 m nutesf ed light an 13 9 egrees 3 minutes

viole light. he i e ence betweenthes a gl s is o e eg ee 5 minutes

hich wo l be he i h of the ainbo if the ays of i i n sunl ght we e exa y p rall l llowing ha f a deg eefo h appar ia ete f the sunN ton obt in a to al width f tw

g s 15 i u s for the p ima y bois o n obser t ons we e n g d

agr n wi h his esultDes r es an New on between them

e able o a o nt fo al the m econspi ous fea u s o the a nb w

hey xplaine the exis ence p imaan s co a y bows and f the daban hat sepa ates the They ca c

e t e angula positions these eares an desc ibed the dispe s n the

scatte ed ght nt a spect m A th s was acc mpl shed w th n ge

met ca ptics The the neve theess had a ma a ng t c d n t

exp a n the s pe n me a a cs The unde stand ng f these seem ng m n

eat es eq es a m e s ph st catedv ew the natu e ight

The s pe nume a a cs appea nthe nne lighted s de f the p mab w In this angula egi n tw scatte ed ays of C ass 3 e e ge in the samedi ecti n; they a ise f om ncident a sthat have pa t pa a ete s n eachs de f the ai bo value Thus at ang ven angle sl gh ly g eate than the a nb w a g e he s a e ed light inc des ays h a e followed tw i

e ent pa hs hro gh he plet The a s eme ge at i r posi ons n the

s ace o the d plet but hey p oceedn the same i ec ionIn the time of Desca tes and ewto

these two cont butions to the sca e entensi y c ul be han le nly by si

p e ad ition. As a es lt the p e i eintensi y falls o s oo hly wi h eviat on f the ainbow a gle i h not ace supe n me a y ar s A allythe intensities o the two ays annot beadded because they a e not in epens u ces of a iation

The p ca e ect n e lying he su pe n me a a cs as isco e e i

803 b Th mas Y ng who showethat ght s capable f nte f en e a phen men n that was al ea y fa ilia

m the st d wate waves In a y med m the s pe p s ti n of waves can

ead eithe t e nf ce en (c est o

18 �

= - - - - - - - -- � -.

1

1

$1

<

1<Z

< 8

C

RAINBOW RA

2

ALEXAND R'S DARK BAND

IMPAC PA AM TER

B C

PRIMARY-RAINBOWRAY

SE ONDARYRAINBOW

RAY

DROPLET RADIUS -

A NBOW GL a s av a sp ia signi cance whethe sca ering a g s sid r d as a fu i f impac parameter. n h impa param s z r s a ring ang for aray f lass 3 s 1 0 d gr s ay pass s r ug nt r f thedr p e and is d y far su fa s aig a k a sun. As he impa param i r as s s a ering a g d r as s uev n ua ly a mi imum a g s a d T s ray f mi imum de ecti n is h rai w ay in diag am a f rays wi h impacparame ers ea sid f i ar s a r d r ug arg r ang es. Theminimum d s a u 13 d gr s a d gr a s centratio f s a r d ays s f und i vi ini y f is ang e Theresu ting nha m i in nsi y f sca er d igh s perceived as h p imary a n w T s c dary w s f rmed i asimi ar way x p ha h s a ring a g f C assrays fwhich i is mp s d in r as s a maximum ins ad f d creasingto a minimum Th maximum ies a ab u 130 d gr s. N rays fC ass 3 r C ass a r a ang s w 130 d gr s and 13 degrees exp ai i g ex s f xand r's dark a d. A he eftw C ass 3 ays wit impa param rs a sid f h rainb w valu em rg a sam s a ring a g is i rf r nc between ays su as h s w a giv s is sup r um rary ar s.

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cr or o canc lla ion (cr onro . Yo n d mon ra d in

rf r nc of li wav by pa in ain l b am of monoc roma ic li

ro wo pin ol and ob rvin al rna in bri an dark "frin

pro c d. I wa Yo n im lf w ooin o p r in nc of i di covry o p rn m rary arc of

rainbow T wo ray ca r in am ir c ion by a rain rop ar ric lyanalo o o e li pa in ro

wo pin o n Yo n ' xp rim n . an l v ry clo o rain ow an

l wo pa ro dropli r only li ly, and o wo ray

in rf r con r c iv ly. an l incr a , wo ray follow pa of

b an ially di r n l n . W n di r nc q al alf of wav n ,

in rf r nc i compl ly r civ a ll r a r an l b am r

inforc a ain T r l i a p rio icvaria ion in in n i y of ca r

li , s ri of al rna ly b i andark band .B ca e ca rin a l a w ic

n rf r nc app n o b con r cive are de rmin d by i r nc bw n wo pa l n , o an l ar

a c d by radi of ropl .T pa ern of p rn m rary arc(in con ra o rainbow an l i

r fore d p nd n on ro iz . Inlar r drop di r nc in pa l n

ncr a m c mor q ickly wi im pac param r an i do in malldrople s. H nce lar r roplar , e narrow r an lar para ionb w n p rn m rary arc i . Tarc can rar ly be di in i d if

ropl ar ar r an abo a millimr in diam r. T ov rlappin of col

or al o nd o wa o arc . Tiz d p nd nc of rn m rarixplain w y y ar a i r o n ar

op of bow rain rop n orow lar r a y fall

W i Yo n ' in rf r nc ory a l major f a r o rainbow

co ld b xplain , a l a in a q ali aiv and approxima way. W a wa

lackin wa a q an i a iv , ma ma ical ory capabl of pr ic in inn i y of ca r li a a f nc

ion of dropl iz an ca rin an l .Yo n ' xplana ion of p rn

m rary arc wa ba d on a wav o yof li . Paradoxically i pr dic ionfor o r id of rainbow, for r ion of l xan r' dark band, w rincon i n wi c a ory. T in

rf r nc ory, lik ori ofD car and N w on, pr ic d com

pl darkn in i r ion, a l aw n only ray of Cla 3 an Cla w r con d r d. S c an abr p ran i

on, ow v r, i no po ibl , b ca wave eory of li r q ir as arp bo ndari b w n li an

RAINBOW RAY

CONFLUENCE OF AYS scattered by a droplet g ves r se to caust cs, o "burn ng curvesA caust c s t e envelope �f a ay system Of spec al nterest s t e caust c of Class 3 rays, w c

as two branc es, a real branc and a "v al one t e latter s formed w en t e ays a e extended backward W en t e a nbow ay s produced n bot d rec ons, t app oac es t ebranc es of t s caust c A t eo y of t e ra nbow based on t e analys s of suc a caust c wasdev sed by George B ry av ng c osen an n t al wave front a surface pe pend cula atall po nts to t e rays of Class 3 A r was able to determ ne t e ampl tude d st but on n subse uent waves A weakness of t e t eory s t e need to guess t e ampl tudes of t e n t al waves

a ow b of n by di rac ion. T mo familiar manif a ion of di rac

ion i appar n b ndin of li or

o nd a of an opaq ob acl .In rainbow r i no r al ob acl ,b bo n ary b w n primarybow an dark band o ld x ibidi rac ion non l . T r a m nof i rac ion i a b l and di c l probl m in ma ma ical p y ic , and

b q n d v lopm n of o ry of rainbow wa im la d mainlyby or o olv i .

In 1835 Ric ar Po r of Univ ri y of Cambrid poin d o a

cro in of vario of li ray in adropl iv ri o ca ic c rv . ca ic or "b rnin c rv , r pr n

nv lop of a y m of ray and ialway a ocia d wi an n n i y i li . familiar ca ic i bri

c p ap c rv form d in a ac pw n nli i r c from i nn rwall . Ca ic , lik ra nbow, n r

ally av a li d id and a dark sidin n i y incr a con in o y p o ca ic, n drop abr p ly.

Po r ow a D car rainbow ray Cla 3 ray of minim m

ca rin an l can b r ard d a aca ic. ll o r ran mi d ray ofCla 3, w n x nd d o n ni y, ap proac D car ray from li

d id r ar no ray of cla on ark i . T ndin in n i y

of ca r d li in a rainbow i similar o probl m of d rm nin in n i y di rib ion in n i bor ood of a ca ic.

In 1838 an a mp o d rmin adi rib ion wa mad by Po er's Cambrid coll a G or B. iry. Hi

2

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i: woW (Z«

z

zW

DARK BAND PRIMARY BOW

SCAT ERING ANGLE

I RST SUPERNUMERARY I

PREDICTED INTENSI Y as a function of scattering angle is compared for three early theof the rainbow. In the geometric ana ysis of Descar es, intensity is in nite a the rainbow

; de ines smoot ly (withou supernumerary arcs) on the ighte side and falls oy zero on the dark side. The theory of Thomas Young, which is base on he nterfer

f g waves, pre icts supernumerary arcs bu retains the sharp ransition from in niteer in ensity. A ry's theory re ocates the peaks in the intensity curve and for he rst timevides (through di raction) an explana ion for gradual fa ing of the rainbow int shadow.

asoning as ba e o a prin iple of av propagation for u ate in the

century by Chri t aan Huygen ana abora e by ugustin Jean Fres-

This prin iple regar s every poin a ave front a being a sour e of

con ary pherical aves; the econ -a aves e ne a ne ave front an nce e ribe the propagation of the ave It follo s that if one ne theamplitu e of the aves over any one

p ete ave front the a plitu e is bu ion at any other point coul be re

nstructed The entire rainbo ou escribe rigorou y if e ne the

amplitu e istribution a ong a avent in a single rop et Unfortunate amp tu e is rib tion can sel o

deter ine ; a one can usually o isa e a reasonable guess or so e cho

ave front in the hope that it iad o a good approxi ationhe starting ave f ont chosen by

is a surface insi e the rople nora o al the ra s of C ass 3 an ith an

ction poin (a change n the sens ova u ) her n sec s he s

s a nbow ray The wave a p

a o g his wav ron w r st gh standa d assump ions in di rac ion Ai was th n

xp ss h intensit of the sca

tere light in the rainbo region inter s of a ne athe atica fun tionthen no n as the rainbo integral anto ay a le the iry function The a he ati a for of the iry function

ill not oncern us here; e shall con-entrate instea on its physica eaningThe intensity istribution pre i ted

by the iry function is ana ogous to thei raction pattern appearing in the

sha o of a straight e ge On the ight e i e of the pri ary bo there areoscil a ions in intensit tha corresponto the upernu erary arcs; the positionsan i ths of these pea s i er so e-

hat fro those pre icte by the Younginterference theory nother signi can

i tinc ion of the Ai heory is tha th axi u intensit of th rainbo fa sat an angle somewha gr a e than the

escartes ini um sca tering ang eThe escar es and Yo ng heo ies pre-

ic an in nite intensi at ha ang (be-ause of the caus c) Th Airy h ory

does no reach an in nit in ensi aany poin and a h sca t s ainbow ray h in nsi predic d s ess han ha th maxim na di rac ion

e ec s appea on h da s d o h rainbo inst ad o vanishi g ab ph n nsi y ap s awa smooth ith-

in A exan er s dar band

iry s ca cu ations ere for a monochromatic ainbow In o der o app his etho to a rainbow p oduce in sun

ight one mu superpos he ir pat-terns generate by h va ious ono-chro a ic co ponents To procee urthe an esc ibe the perceived i age ofthe ainbo equires a theory of co orvision

The purity of the rainbo colors iseter ine by the extent to hich theco ponent onoch o atic rainbo soverlap; that in turn is eter ined bythe rop et size Unifor ly arge rops( ith ia eters on the order of a fe illi eters) gene al y give bright rain-bo s ith pure co ors; ith ver s al

roplet ( ia eters of .01 illi eter oro) the over ap of co ors is so great that

the re ulting ight appears to be a ohite

A i por ant propert of ght that e have so far ignore is its state of

polarization Light i a transver e avethat is one in hi h the os illations are perpen i ular to he ire tion of propagation (Soun on the other han is a

ongitu ina vibration ) The orientationof he transver e os illation an be re-

o ve into o ponent alo g t o utua ly perpen i u ar axe ny li ht ray

an be e cribe in ter of these t oin epen ent s ate of inear po ariza-tion Sunlight i an in oheren ixtureof the t o in equal proportion it i often sai to be ran o ly polarize or

i p y unpolarize Re e tion an alter

i s a e of polarization an in that fa tlies he i portance of polarization tothe analysis of the rainbo

Let us con i er the re e tion of a ight ray trave ing insi e a ater roplet

hen i rea hes the boun ary of theroplet The plane of re e tion the

plane that contains both the inci entan he re e te rays prov es a conve- nient geo etric reference The po arization states of the in i ent ight can be

e ne as being para le o that p anean perpen icu ar to it For both polari-zations he re ectivity of the su ac isslight a ang es of inci ence near the per- pen icu ar an it rises very steep y neara cri i a ang e hose va ue is e e mined by the in ex o ef action Be-yon tha cr ti a angle he ay is o a re ected egard ess of po ariza io Ainterm dia ang es ho ever ec ivi-ty depends on po a a ion As h angleo incidence beco s sha owe a s adil a ger po ion o h pe p ndic ar po arized ompon n s e ect d or

h para compon n on he o her hand c iv a s fo i begins oinc as A on ang in par c a

ec vi o h para po a i d ave

va ish s n i ; tha av s o a yransm ed H nc o s n igh c d na ha a g h nte na c d a

s comp po a z d p rp ndicu a

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o e lane of e ec ion. e an le iscalle B ews e s an le, af e Da iB ews e , w o iscusse i s si ni cancein 1815

Li f o e ainbow is al osco le ely ola ize , as can be seenby lookin a a ainbow ou Polaoi sun lasses an o a in e lensesa oun e line of si . e s on o

la iza ion esul s f o a e a kable co-inci ence: e in e nal an le of inci-ence fo e ainbow ay is e y c oseo B ews e s an le. Mos of e a allel

co onen esca es in e ans i e ays of Class 2, lea in a e on e anceof e en icula ays in e ainbow.

W i e un e s an in a bo a e an a ia ion can be a e

as wa es, e eo y of e ain ow aseen enla e in sco e. us now en-

co ass new, in isible ainbows ouce in a o ic an nuclea sca e in .

n analo y be ween eo e ical o -

ics an classical a icle ec anics aal ea y been e cei e in 1831 by Wilia Rowan a il on, e is a e a ician. e analo ues of ays in eo e ical o ics a e a icle a ec o ies,an e ben in of a li ay on en e -in a e iu wi a i e en ef ac i ein ex co es on s o e e ec ion of a o in a icle un e e ac ion of afo ce. Pa icle sca e in analo ues ex-is fo any e ec s in o ics, inclu in

e ainbow.Consi e a collision be ween wo a -

o s in a as. s e a o s a oacf o a la e ini ial se a a ion, ey a ea s sub ec o a s ea ily inc easina ac ion. close an e, owe e , e elec on s ells of e a o s be in o in

e ene a e an e a ac i e fo ce i inis es. e y close an e i beco esan nc easin ly s on e ulsion.

s in e o ical ex e i en , ea o ic sca e in can be analyze by

acin e a s of e a o s as a func-ion of e i ac a a e e . Because

e fo ces a y a ually an con inu-ously, e a o s follow cu e a ec o- ies ns ea of c an in i ec ion su

enly, as a e boun a y be ween eia of i e in ef ac i e in ex. E en

ou so e of e a ec o ies a e a - e co lica e , eac i ac a a e eco es on s o a sin le e ec ion an le o eo e , e e is one a ec o y a e esen s a local axi u an ula e-

ec ion. a a ec o y u ns ou o bee one a akes e os e ec i e use

of e a ac i e in e ac ion be weena o s. s on c ncen a ion of sca -

e e a icles is ex ec e nea is an-le; is e ainbow an le fo e in e -

ac in a o s.A wa e ec anical ea en of e

a o ic an nuclea ainbows was fo -

ula e n 1959 by Kenne W. Fo ofB an eis Un e si y an Jo n . W ee-le of P ince on Uni e si y. n e fe ence

be wee a ec o ies e e in in esa e i ec ion i es ise o su e n ea y eaks in in ensi y. a icle sca ein analo ue of i y s eo y as alsobeen e i e

n a o ic ainbow was s obse ein 196 by E. un ausen an . Pau

y of e Uni e si y of Bonn in e sca

00

0

0

0

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(

0

0 I

0 0 0

e in of so iu a o s by e cu y ao s e ain ainbow eak an wosu e nu e a ies we e e ec e in o e

ecen ex e i en s oscilla ions on an e en ne scale a e been obse e . e ainbows easu e in ese ex e i en s ca y info a ion abou e in ea o ic fo ces. Jus as e o ical ain

O AL IN E NAL FLEC ON

PARALLEL

0

PLANE OF REFLEC ON - PERPENDICULAR

WA ER

AIR

EVANESCENWAVE

OLA IZA ION OF HE AINBOW results from di erential re ection An incident raycan be resolved into two components polarized parallel to and perpendicular to the plane ofre ection For a ray approaching an ai -water bou dary from inside a droplet the re ectivityof the surface depends o the angle of i cidence Beyond a critical angle both parallel and perpendicular components are totally re ected, although some light travels parallel to the surfaceas an "evanescent wave At lesser ang es the perpendicular component is re ected more e

ciently than the paral el one, and at one a gle in particular, Bre ster's angle, parallel-polarizedight is completely transmitted he angle of interna re ection for the rainbow ray falls near

Brewster's angle As a result light fro the rainbow has a strong perpendicular polarization

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AINBR JE T Y

SCATTE ING OF ATOMS BY ATOMS creates a part culate ra nb w T e r le play d pt cal scatter ng by the refrac ve ndex s played ere by n era rc s pr c p d

ference s t at the f rces vary s thly and c n nu us y, s trajec r es ne at appr ac es an t er e f rc be w n dgr w ng attract n d d g)but a cl se rang c es t ngl r pu s (

d g)A l cal ax mum n t e scatter ng angle c rresp nd pt ca b w s t e angle ade by the tra ect ry st e ect ve n us ng the at ract v pa th p tent a

�UU

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:

5

P IMA Y/ AINBOW

115 25

SCA ING ANGL (DEG EES)3 35

AT MIC AINBOW was detected by E Hundhausen and H auly f the Un vers ty f B nnn t cat er ng f s d u at ms by ercury at s T e sc lla ns n e nu ber f scatteredat s de ec ed c rresp nd a pr ary ra nb w and t tw supernu ary pea s A ra nb w

f t s nd e b d es nf rmat n ab u the strength and range f he nterat m c f rces

2

bo angle depends solel on the e act e ndex so the atom c a nbo angle

s dete m ned b the st ength o the att act e pa t o the nte act on S m lal the pos t ons o the supe nume pea s a e s ze dependent and the p o

de n o mat on about the ange o thente act on. Obse at ons o the same

k nd ha e now been made n the scatte

ng o atom c nucle . he A theo o the a nbo has had man sat s ng successes but t

conta ns one d stu b ng unce ta nt the need to guess the ampl tude d st but onalong the chosen n t al wa e ont Theassumpt ons emplo ed n ma ng thatguess a e plaus ble onl o athe a ge a nd ops In th s context s ze s best ex p essed n te ms o a s ze pa ametede ned as the at o o a d oplet s ccum e ence to the wa elength o thel ght. The s ze pa amete a es omabout 00 n og o m st to se e a thousand o a ge a nd ops. A s app ox mat on s plaus ble onl o d ops w th as ze pa amete g eate than about 5 000

It s on c that a p oblem as nt actable as the a nbow actuall has an exacsolut on and one that has been nown

o man ea s As soon as the elect o magnet c theo o l ght as p oposedb ames Cle axwell about a centu ago t became poss ble to g e a p ec se mathemat cal o mulat on o theopt al a nbo p oblem at s need ed s a computat on o the scatte ng oan elect omagnet c plane wa e b a ho

mogeneous sphe e. The solut on to as m la but sl ghtl eas e p oblem thescatte ng o sound wa es b a sphe e

as d scussed b se e al n est gato s notabl Lo d a le gh n the th centu . The solut on the obta ned cons st ed o an n n te se es o te ms called pa t al a es A solut on o the same

o m was ound o the elect omagnet c p oblem n 08 b Gusta e and ete W. Deb e

G en the ex stence o an exac solut on to the scatte ng p oblem t m ght

eem an eas matte o dete m ne all tseatu es nclud ng the p ec se cha acte

o the a nbow The p oblem o cou ses the need to sum the se es o pa t aa es each te m o wh ch s a athe

compl cated unct on The se es can bet uncated to g e an app ox mate solt on but th s p ocedu e s p act cal onl

n some cases The numbe o te ms that must be eta ned s o the same o de o magn t de as the s ze pa amete The pa t al a e se es s the e o e em nenl su ted to the t eatment o a e ghscatte ng wh ch s espon ble o theblue o the s n that case he scatte

ng pa t cles a e molecules and a e much smalle than the wa elength sothat one te m o the se es s enough othe a nbow p oblem s ze pa amete s upto se e al thousand must be cons de ed

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A g d app x mat n t t e s lut nby t e pa t al wave met d w uld equ e evaluat ng t e sum f seve alt usand c mpl cated te ms C mput

e s have been appl ed t t e tas but t e esults a e ap dly va y ng funct ns ft e s ze pa amete and t e scatte ng ang e s t at t e lab and c st qu c lybec me p b t ve Bes des a c mput

e can nly calculate nume cal s lut ns; t e s n ns g t nt t e p ys csf t e a nb w We a e t us n t e tanta

l z ng s tuat n f n w ng a f m f t e exact s lut n and yet be ng unable t ext act f m t an unde stand ng f t e p en mena t desc bes

e st steps t wa d t e es lut nf t s pa ad x we e ta en n t e

ea ly yea s f t e 20t centu y by t e mat emat c ans en P nca and GN Wats n T ey f un a met d ft ansf m ng t e pa t al wave se esw c c nve ges nly ve y sl wly nt astable value nt a ap dly c nve gent exp ess n T e tec n que as c me tbe n wn as t e Wats n t ansf mat n

as t e c mplex angula m mentum met d

It s n t pa t cula ly a d t see w yangula m mentum s nv lved n t e a nb w p blem alt ug t s less bv us w y "c mplex values f t e angula m mentum need t be c ns de edT e explanat n s s mplest n a c puscula t e y f l g t n w c a beam fl g t s ega ded as a st eam f t e pa tcles called p t ns Even t ug t e

p t n as n mass t d es t ansp t ene gy and m mentum n nve se p p t n t t e wavelengt f t e c esp nd ng l g t wave W en a p t nst es a wate d plet w t s me mpact pa amete g eate t an ze t e p t nca es an angula m mentum equal tt e p duct f ts l nea m mentum andt e mpact pa amete As t e p t nunde g es a se es f nte nal e ect ns t s e ect vely b t ng t e cente

f t e d plet Actually quantum mec an cs places add t nal c nst a nts nt s p cess On t e ne and t equ est at t e angula m mentum assume

nly ce ta n d sc ete values; n t e t et den es t at t e mpact pa amete canbe p ec sely dete m ned Eac d sc etevalue f angula m mentum c esp nds t ne te m n t e pa t al wavese es

In de t pe f m t e Wats n t ansf mat n values f t e angula m mentum t at a e c nvent nally ega d ed as be ng "unp ys cal must be ntduced F ne t ng t e angula m ment m must be all wed t va y c nt n

usly nstead f n quant zed un ts; m e mp tant t must be all wed t ange ve t e c mplex numbe s t set at nclude b t a eal c mp nent andan mag na y ne c nta n ng s me mult ple f t e squa e t f . T e

plane de ned by t ese tw c mp nentss efe ed t as t e c mplex angula

m mentum planeMuc s ga ned n etu n f t e mat

emat cal abst act ns f t e c mplexangula m mentum met d In pa t cula afte g ng ve t t e c mplexangula m mentum plane t ug t eWats n t ansf mat n t e c nt bu

t ns t t e pa t al wave se es can be ed st buted Instead f a g eat manyte ms ne can w w t just a few p nts called p les and saddle p nts nt e c mplex angula m mentum planeIn ecent yea s t e p les ave att actedg eat t e et cal nte est n t e p ys cs

f elementa y pa t cles In t at c ntextt ey a e usually called Regge p les afte t e Ital an p ys c st Tull Regge

B t p les and saddle p nts ave p ys cal nte p etat ns n t e a n

b w p blem C nt but ns f m ealsaddle p nts a e ass c ated w t t e d na y eal l g t ays we ave been c ns de ng t ug ut t s a t cle W atab ut c mplex saddle p nts? Imag na y c mplex numbe s a e d na ly ega ded as be ng unp ys cal s lut nst an equat n but t ey a e n t mean

ngless s lut ns In desc pt ns fwave p pagat n mag na y c mp nents a e usually ass c ated w t t edamp ng f t e wave ampl tude F example n t e t tal nte nal e ect n fa l g t ay at a wate a b unda y a

INCI ENT AY

l g t wave d es g "t ug t e l ngglass Its ampl tude s ap dly damped weve s t at t e ntens ty bec mes negl g ble w t n a dept n t e de fa s ngle wavelengt Suc a wave d es n t p pagate nt t e a ; nstead t bec mes attac ed t t e nte face betweent e wate and t e a t avel ng al ng t esu face; t s called an evanescent wave

T e mat emat cal desc pt n f t e evanescent wave nv lves t e mag na yc mp nents f a s lut n T e e ectcalled quantum mec an cal tunnel ng

n w c a pa t cle passes t ug a ptent al ba e w t ut cl mb ng ve t as a s m la mat emat cal bas s "C m plex ays als appea n t e s ad ws de f a caust c w e e t ey desc be t edamped ampl tude f t e d actedl g t waves

Regge p le c nt but ns t t e t ansf med pa t al wave se es a e ass c at ed w t su face waves f an t e ndT ese waves a e exc ted by nc dent ayst at st e t e sp e e tangent all Oncesuc a wave s launc ed t t avelsa und t e sp e e but t s c nt nuallydamped because t s eds ad at n tangent ally l e a ga den sp n le At eac p nt al ng t e wave s c cumfe ent al pat t als penet ates t e sp e eat t e c t cal angle f t tal nte nal e

ect n eeme g ng as a su face waveafte ta ng ne m e s c s tcutsIt s nte est ng t n te that hannesKeple c njectu ed n 584 t at "p n

SU FACEWAVE

IMP CTPA AMETEEQUA TO

OP ET

CRITICAAGLE

I _

RITNGLE

U FACEWAVE

COM LEX-ANGULA -MOMENTUM theory of the ra nbow beg ns w th the se vat othat a photon, o quant m of ght, nc dent on a drop et at some mpact parameter (wh ch cannot be e act y de ned ca es angu a moment m n the theo y, components of that angu amoment m are e ended to comp e va ues, that s, va ues conta n ng the square o t of 1

The consequences of th s p ocedure can be ustrated by the e amp e of a ray str k ng a opet angent a y The ray st mu ates su face waves, wh ch trave a ound the d op et and cont nuous y shed ad at on The ray can a so penet ate the drop et at the c t ca a g e fo tota

nterna e ect on, emerg ng e ther to fo m anothe surface wave or to epeat the sho tcut

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whee ays th s nd m ght be e-s ns b e the a nb w, but he aban-d ned the dea because t d d n ead t

he c ect a nb w ang eIn 937 the Dutch hys c s s Balthus

Van de P and H B emme a l edWats n s t ans mat n t the a nb w

blem, but they we e able t sh wn y tha y s a x mat n c uld bebta ned as a l m ng case In 965 I

de el ed an m ed e s n Wa -s n s me h d, and I a l ed the a nb w ble n 969 w h s e-what g eate success

n he s m le Ca es an ana ys s we sathat n the l ghted s de the a n-

b w the e a e tw ays eme g ng n thesame d ect n; at the a nb w angle

hese c alesce n the s ngle Desca tes ay m n mum de ec n and n theshad w s de they an sh In he c -

lex angula m en um lane, as I

ha e men ned, each ge me c ayc es nds t a ea sadd e nence n ma hema cal te ms a a nb ws e ely he c ll s n f tw saddle

«

>

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COMP EX·ANGU ARMOMENTUM _THEOR

nts n the c m lex angula m men-tum ane In the shad w eg n bey nd

he a nb w ang e the saddle nts d n t s m y d sa ea ; they bec mec m lex, that s, they de e mag - na y a ts The d acted ght n exande s da band a ses m a c m lexsaddle nt It s an exam le a "c m-

ex ay n the shad w s de a caust ccu e

I sh u d be n ed tha the ad t n he c ex angu a m men um me h-d d es n t m y tha ea l e s ut ns

t the a nb w b e we e w ngDesca es s ex lanat n the a yb w as the ay m n mum de ec n sby n means n al d, and he su e nu- me a y a cs can s l be ega ded as a

duct f n e fe ence, as Y ung -sed The c ex angula m en-

tum me h d s m ly g es a m e c m-ehens e acc unt ng the aths

a a lab e t a h t n n the a nb w e-

g n the s y, and t he eby ach e es e accu ate esul sIn 97 5 V jay Kha e the Un e s y R ches e made a de a led c a

AIRTHEOR

139SCATTERING ANG E (DEGREES)

s n th ee the es the a nb w they a x mat n, the "exact s u-

t n, bta ned by a c m ute summa-t n the a t a wa e se es, and the a nb w te ms n the c m ex angu a - m mentum meth d, ass c ated w th thec l s n tw sadd e nts thed m nant, e end cu a a zat nthe y the y equ es n y sma c ect ns w h n the ma y b w, and ts e s bec me a ec ab e n y n the eg n he su e nume a y a cs the sca e ed ays a zed a a e tthe scatte ng ane, h we e , y s a -

x ma n a s bad y the su e - nume a y a cs the exac s lut n sh ws m n ma whe e the y the y has max mum ntens ty, and ce e sa Th s se -

us a lu e s an nd ect esult the nea c nc dence between the angle

n e na e ect n the a nb w aysand B ews e s angle t B ewste s ang e the am tude the e ected ay

changes s gn, a change the y the yd es n t ta e nt acc unt s a esu t fthe change n s gn the nte e ence a ngd ec ns c es nd ng t the ea s n

4 4 4

Q N I IVE HEO IES of e ainbow p edic e in ensif e sca e ed ig as a func ion of e sca e i g ang e and a so

wi espec o d op e size and po a iza ion He e e p edic ions ofee eo ies a e p esen ed fo pa a e -po a ized ig sca e ed b

d op e s wi a ci cumfe ence equa o 1,500 wave eng hs of e igOne cu ve ep esen s e "exac so u ion o e ainbow p ob em,de ived f om James C e k Maxwe 's equa ions desc ibing e ec omagne ic adia ion he exac so u ion is e sum of an in ni e se iesof e ms, app oxima ed e e b adding up mo e an 1,500 comp i-

ca ed e ms fo eac poin emp o ed in p o ing e cu ve e Ai eo is c ea in disag eemen wi e exac so u ion, pa cu

a in e angu a egion of e supe nume a a cs e e he exacso u ion s ows oug s a e posi ions of Ai 's peaks The esu sob a ned b he comp ex-angu a -momen um me od, on he o e

and, co espond c ose o e exac so u ion, fai ing on o ep oduce sma , ig -f equenc osci a ions ese 8uc ua ions a e socia ed wi ano e op ica p enomenon in e a mosp e e, he g o ,w ic is a so exp ained b e comp ex-angu a -momen um eo

© 1977 SCIENTIFIC AMERICAN, INC

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the A s ut ns s dest uct ve nsteadf c nst uc veIn te ms f a ge sca e fea u es, such

as he p ma b w, the supe nume aa cs and the da s de d ac n pat

e n, the c mplex angu a m mentum esult ag ees qu e c se w th the exacs lut n Sma le scale uctuat ns nthe exact ntens t cu ve a e n t ep

duced as we l b the a nb w e ms nthe c mplex angu a m men um me hd On the the hand, the exac s u

t n, f a t p ca s ze pa amete f,5 0 equ es the summa n f m e

than 500 c mpl cated e ms; the c m plex angu a m mentum cu ve s b

a ned f m nl a few much s mp ete ms

he small es dual uctua ns n he exac ntens t cu ve a se m

h ghe de nte nal e ect ns a sbel ng ng t classes h ghe han C ass 3

Class 4 The a e f ttle mp ancef he p ma b w, but at a ge sca

e ng angles he c nt but n c eas es and nea he bac wa d d ect n bec mes d m nan The e hese a s a e esp ns b e an the fasc nat ng mete l g ca d sp a the g [see TheG , b H wa d C B ant and es n Ja m e; I IFI M I Ju ,

974].The g appea s as a ha spec

t a c l s s und ng he shad w anbse ve casts n c uds g; s

m st c mm n seen f m an a p aneng ab ve c uds It can a s be ex

p a ned th ugh he c mplex angu a m mentum he , but he exp ana n

s m e c mpl cated than hat he a nb w One se f c nt but ns heg c mes f m the su face waves desc bed b Regge p les that a e ass c a ed w th the angen a a s f Keple s p nwheel t pe M lt p e nte na e ect ns that happen t p duce c sed,sta shaped p l g ns pla an mp ant le, lead ng es nances, enhance ments n ntens t Such ge met c c nc dences a e ve much n the sp t Keple s the es

A sec nd mp ant set f c nt bu

t ns, dem nst ated b Kha e, s mhe shad w s de f h ghe de a nb ws that appea nea the bac wa d d ect n These c n but ns ep esenthe e ect f c mplex a s The th

de a nb w, med nl a few deg ees awa f m the bac wa d d ect n, s pa t cula e ect ve

F the h ghe de a nb ws A sthe w uld g ve nc ec esu ts b h p la za ns, and s the c mp exangula m mentum he mus be em

p ed One m ght thus sa he g sf med n pa f m he shad w a a nb w I s g at f ng d sc ve n

the elegan but seem ng abs ac he f c mp ex angu a m mentum an explana n f hese w natu a phe n mena, and nd he e an nexpec ed

n be ween hem.

Authors .LOOK NG

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2

Jo the f in

Com nd, Co rol ndCommunic ions

at .This system eng ineer ing corporation is d ffere t ! The size and scope of the projec s, the back

groun ds of he sta f and the techni cal envi ro ment a our headqua rters i Bed ord , Massachusetts, al lcombine to provide you with so i d, ong-range work that cha enges a l our techno og ca ski l

Projects l ike SATIN IV, he new SAC com ni cation s em usi g packe sw tch n g techn quesO JTIDS a sys em whic h com bine h pr nci pl of DMA, p read spec r odu lat o , rrorde ec ion and correction codes a d da a proce s for be r r -serv ce C3. A d here are cores ofother projec s that dai y cha enge ou r7 exper enc d s aff m bers, m ore t an ha f of who haveadvanced degrees

As for the environment, o ur comp ex is si uated on acres es outside of Boston, near

hi sto ic Lexi ngton and on cord H ere at he corporate headqua r ers, ou r s aff s backed up bspecia lized supp ort profession a s Bei g in he forefron of echno log is hat keeps sharp peoplecoming to M itre It a so keeps the m work n g here If yo know he C3 field, read on:

C M U R SYS MSProjects span the spectrum from systems prog amming w h 370/158 sof ware o developing cost mode s or

standardization in put ana yses sof wa e bu ild i g block development, compu er pe ormance va ua on, equ ements analysis for information process ng systems, in ell gence da a hand i ng a d su bsystem es documentat o .

C MMU C SMajor programs are unde ay in system des g and analysis, d ig tal adio comm un cat ons m c oprocesso

appli cations d ig ital c ircui t deve opment, satell e systems and ermi na s ard are and soft are for comm uni cations processors anti-jam analy s, mod ula on/codi ng ech ques, secure dig i al and ana og communicat onsperi phera s and sys em es and eva ua on

C MM D D C RNume rous projec s exist in a defense and ac cal a co rol sys ems eng neering f or dep oyab e and f xed

insta lat on militar systems. These pro ec s equ re op o c eng eers fo sys em a alys s and spec ica o,sof are developmen ntersystems eng nee ng , s mu la o and evalua on s u e l lance systems des gn and

ac ica contro sys ems enginee ngTo app y for hese career posi io , e o r r e, nc d n g sa a istor , o M r Dav d

Fin negan, The M ITRE Corporation , dl x r p k , edford, M ac se s7

THE =MITRE

O R P O R OA q l o loye

echnical excel ence t roug professiona c al e g