CR L - 62 - 34

NEW YORK UNIVE RS ITY

Cou r a n t I n s t i t u te o f Ma th ema t i ca l S c i e n c e s

D i v i s i o n o f E l e c t r om a g n e t i c R e s e a r ch

R e s e a r c h R epo r t No . E M - l7l

OP T ICS

Mo r r i s Klin e

P r epa r e d f o r

E l e c t r on i c s R e s ea r ch Di r e c to r a t eAi r Of f i c e Ca mb r i dg e R e s e a r ch La bo r a to r i e s

O f f i c e o f Ae r o s pa c e R e s ea r c hUn i t e d S ta t e s Ai r F o r c e

Be d f o r d , Ma s s .

R ep r od u c t ion i n wh ole o r i n pa r t i s pe r m i t t e d f o r a n y u s e o f

t h e Un i te d S ta t e s G ove r n m en t .

AF l9 (6ofl ) 5238

P r o j e c t 5635

T a s k 56350

F eb r ua r y , 1 962

q u e s t s f o r a d d i t ion a l copie s by Ag en c i e s o f th e Depa r tm en t o f De f en s e

e i r c on t r a c to r s , a n d o th e r G ove r n m en t a g e n c ie s s hould b e d i r e c t e d t o

ARME D S ERVI CES TE CHN ICAL I NF ORMAT I ON AG ENCY

DOCUMENTS S ERVICE CENTERARL I NG TON HALL S TAT ION

ARLING TON 1 2 , VI RG IN I A

pa r t m e n t o f De f en s e c on t r a c to r s mu s t b e e s ta bli s h e d f o r AST I A s e r vi c e

ve th e i r ‘n e e d - to—know ‘

c e r t i f i e d by th e cog n i z a n t mi li ta r y a g e n cy o f

o j e c t o r con t r a c t .

1 o th e r pe r s on s a n d o r g a n i z a t i on s s hould a pply to th e

U . S . DEPARTMENT OF COMMERCE

OF F I CE OF TE CHN I CAL SERVI CES

WASHI NG TON 25 , D . C .

I . I NTRO DU C T IO N . It m ay s eem unnec e s s ary at t h i s l at e dat e to

di s cu s s the re l at ion s hi p of e lect romagnet ic t heory to g eomet ric a l O p

t ic s . The cont ent of both f i e l d s i s w e l l k nown and everyon e k now s

a l s o that geomet ric al Opt ic s i s the l im it f o r vani s hi ng wave l ength of

e le ct romag net ic t heo ry. Moreover, s inc e Maxwe l l' s theory s u pe rs e de s

t he o lde r geomet ri c a l opt i c s,pre s umab ly

,t hen

, g eomet ric a l opt ic s

cou l d be di s c arde d . The opt ica l indu s t ry cont i nue s to u s e it but pe r

hap s th at i s b ec au s e it i s behin d the t ime s .

The re are,however

,at le a s t t hre e m ajor re a s on s for purs u i ng an d

c l arify i ng t he re l at ion s hi p in que s t ion . The f i rs t i s t he pure ly t heo

r e t i c a l or ac a dem ic prob l em o f bu i l ding a mat hemat ic a l b ri dge between

the two doma i n s,e le ct romagnet ic t heory and geomet ric al opt i c s . The

o lde r b a s e s for a s s e r t i ng t hat geomet ric a l opt i c s i s a l im it i ng c a s e of

e l ect romagnet ic t heo ry are va gue and from a mat hem at ic a l s t andpoint

h igh ly un s at i s f acto ry .

The s econd m aj or re a s on for the inve s t ig at ion i s a pract ic a l one .

To s o lve prob l em s o f e l ect romagnet ic t heory,whet he r i n t he rang e of

radio fre que nc i e s or vi s ib l e l i ght fre que nc i e s,one s hou ld so lve Max

we l l 's e quat io n s with t he appropri at e i n it i a l and boundary condit ion s .

Howeve r,a s i s w e l l k nown

,Maxwe l l ' s e quat ion s c an b e s o lve d ex .

a c t ly i n on ly a few prob l em s . Henc e phy s ic i s t s and engi nee rs,e s

pe c i a lly t ho s e conc erne d with u lt ra - high fre que ncy prob l em s,have

re s o rt e d to the s impl er method s of geometric al opt ic s . Al though t he s e

methods have prove d rem ark ab ly eff ic ac iou s in the o pt ic al doma i n,

t hey are i nt ri n s ic a l ly l im it e d ; t hey do not furn i s h i nfo rmat ion about

s ome o f t he mos t im port ant phe nomena s uc h a s di ffract ion,pol ari z a

t ion , and inte rf e renc e , to s ay noth ing about the numeric a l accuracy

o f what geomet ric al opt ic s doe s y i e l d . Hence t he pract ic a l qu e s t io n

b ecome s whet he r t he e s t abl i s hment of a b ett e r l ink between Maxwe l l ' s

t heory and geomet ric a l opt ic s wi l l provide more accurat e approximat e

methods o f s o lving e l ect romagnet ic prob l em s . In s ofar a s ult ra—h igh

fre quency prob l em s are conc erned,t he an s wer

,ba s ed o n work o f t he

l a s t t e n years,c an alre ady be g iven aff i rm at ive ly . It i s a l s o a fact

that opt ic al peopl e are n o w look ing more and more i nto di f f ract ion

4 El ect romagnet ic Theo ry and Geomet ric a l Opt ic s

effect s and one m ig ht venture t hat the pract i c e of opt ic s i s on t he

verge of ent e ri ng into an e l ect romagnet ic t re atment o f opt ic al probl em s .

The inve s t i gat ion s e rve s a t hi rd pu rpo s e . In princ i pal it i s con

c erned W i t h t he re l at ion s hi p between a wave t heo ry and a n o n - periodic

phenomenon with the l atte r in some s en s e a l im it ing c a s e of t he wave

theo ry a s a par amete r,t he wave l ength i n the c a s e of e l ect romagnet ic

phenomena,goe s t o zero . Howeve r t here are m any b ranche s of phy s ic s

,

acou s t ic s,hydro dynam ic s

,magnetohydrodynamic s and qu antum me

c h a n i c s ,which a l s o t re at wave theori e s . Hence in e ac h c a s e there

s hou ld b e a corre s pondi ng " Opt ic al " t heo ry or if o n e exi s t s,a s i n t he

c a s e of qu antum mechan ic s,t he pre s ent t heory s hou l d s he d l ight on

the two compl ement ary domain s . We s hal l in fact s e e t hat t he e l ec

t r o m a g n e t i c i nve s t i g at ion s to be s u rv eye d here d o i n dee d l ead to new

cre at ion s or new in s ight s into ot he r b ranc he s of phy s ic s .

2 . SO M E RELEVANT H I STO RY . T o appreci ate j u s t what the prob lem of

reconc i l i ng geomet ric al opt ic s and e l ect romagnet ic t heory amount s to

we s hal l exam ine b ri ef ly the hi s toric al back ground .

The s c i enc e o f g eomet ric a l opt ic s wa s founde d in the s eve nt eent h

c entu ry . To the l aw of reflect ion,known s i nc e Euc l i d ' s day

,Ren é

D e s c a r t e s and Wi l l eb rord Sne l l added the l aw of ref ract ion ; Robe rt

B oyl e and Robert Hook e di s covere d int e r f e re nce ; O l af Rc'

i m e r e s t ab l i s hed

the fi n it ene s s o f t he ve loci ty of l ight ; F . M . Grima ldi an d Hooke di s

covere d di ff ract ion ; Era smu s Ba r t h o li n u s di s covered doubl e re fract ion

in Ice l and s par ; an d Newton di s covere d di s pe rs ion .

Two phy s ic al t heori e s o f l i ght were c re at e d in t he s eventee nth

c entury . Chri s t i a an Huygen s formu lat e d the " wave ” t heory of l ight 1

and New ton formu lat e d a theo ry o f propagat ion o f pa rt ic l e s2

Huygens

thought o f l ig ht a s a long itu dina l moti on of et he r and a s s prea di ng out

at a f in it e veloc ity from a po int s ourc e . The fa rthermo s t po s it ion

reac hed by t he l ight in s pac e fi l le d o ut a s urf ac e which he c a l le d the

f ront o f t he wave . In homogeneou s me di a th i s s urf ac e i s a s phere .

To expl ai n fu rt her h o w l ight propag ate s,Huygen s s u ppo s e d that when

the di s tu rbanc e re ache d any po int i n t he ether t hi s point im parte d

it s mot ion to a l l n e ig hboring po int s . Thu s if t he wave front at t ime

t , s hou l d b e t he s u rf ac e 8 1 and if P i s a typic a l po int on S , t he

point P commun ic at e d it s mot i on t o al l po int s in it s ne i ghborhood and

from P t he l ight s pre ad o ut i n a l l di rect i on s . It s ve loc ity i n t he s e

variou s direct ions depe nded upon t he nature of the me dium . Thu s in

s ome smal l i nt erval of time ( and in an i s ot ropic me dium ) the front of

the l i ght emanat ing f rom a point wou l d b e a s phere wit h P a s a ce nte r .

The s ame wou l d be t rue at any othe r point of t he s urf ac e S , exce pt

t hat t he radi i o f t he s phere s m ight di ffe r a s t he medium differ s along

S I The new po s it ion o f the f ront at s ome t ime t z gre at e r than t l

i s t he enve lope in t he mathemat ic a l s en s e o f the f am i ly o f s phere s

att ached o n e to e ach point of S I ( There i s accordi ng to thi s t heory

a l s o a b ackward wave . Thi s b ackward wave t roub l e d s c ient i s t s unt i l

Morri s Kl i ne 5

Kirc hhoff s howe d under h i s formu l at ion that it

doe s not exi s t . We s hal l not purs ue t hi s h i s

t o r i c a l point . ) T o expl a i n ref l ect ion and r e

f ract ion Huygen s s uppo s e d that t he s ame phe

n o m e n o n t ak e s pl ac e at each po int o n t he re

f le c t i n g o r ref ract i ng int e rf ac e when the f ront

re ache s it,exce pt

,o f cou rs e

,t hat no wave s

penet rat e t he re f l ect i ng s urfac e .

There are many more det a i l s to Huygen s '

theory which expla i n t he phe nomena of g e o

met ric a l opt ic s i nc ludi ng doub l e ref ract ion .

Howeve r,more re l evant for u s i s t h e f act t h at

Huygen s con s idere d l ight a s a s e ri e s of s uc

c e s s i v e im pu l s e s each t rave l l i ng a s a lre ady

de s c ribe d and he did not expl a i n t he re l at ion F i gu re

s hi p o f t he im pu l s e s t o e ach othe r . Thu s the

periodic ity o f l i ght i s not cont a ine d in Huygen s ' theory . Al s o,t hough

the phenomenon o f di ff ract ion had alre ady been ob s e rve d by Hook e and

Grima l di,Huygen s apparent ly di d not know it an d he di d n o t con s ide r

it t hough hi s t heory coul d have covere d at lea s t a c rude th eory of

diff ract ion .

The s econd maj or t heory of l ight wa s Newton' s . He s ugge s t e d in

oppos it ion t o Huyge n s ' " wave " th eory ,that a s ourc e o f l ig ht emit s a

s t re am o f part ic l e s i n a l l di rect io n s i n wh ic h the l ight propagat e s .

The s e par t ic l e s are di s t i nct from the et her i n whic h t he par t ic l e s move .

I n homogeneou s s pac e the s e pa rt ic l e s t rave l i n s t ra i ght l i n e s unl e s s

def l ect e d by fore ign bodi e s s uc h a s ref l ect i ng an d refract in g bodi e s .

New t on di d i nt ro duce a k ind of pe riodic ity,

” f it s ",whic h he u s e d to

expla i n b right and dark ri ng s appe ari ng in c e rt a i n phenomena o f r e f r a c

t ion . However,t he nature o f t he pe riodic ity wa s vagu e . Hi s t heo ry

was on the who le c ru de for t he variety of phenomena he t ri ed to embrac e

and he made many ad h o c a s s umpt ion s . Nevert he l e s s,Newton deve l

oped th i s m echan ic al t heory s o thoroughly that it s complet ene s s— i t

i nc l ude d diffract io n— and New ton ' s own gre at re put at ion c au s e d othe rs,

a s i de f rom Eu l e r,to acc e pt it fo r 100 years . Huyge n s ' work was

,on

the whole,ignored . Bot h men

,i nc i de nt al ly

,obt a ine d s ome ink l ing o f

pola ri z at ion through re a s on ing about doubl e re f ract ion in Ic e l and s par .

D e spite t he recogn it ion i n t he s event e enth c e ntury o f phenomena

s uc h a s diffract ion,a l im ite d t heo ry of l i ght c a l l e d geomet ric a l opt ic s

wa s e rect e d on the b a s i s of four pri nc i pl e s . I n homogeneou s me dia

l ight t rave l s in s t ra ight l ine s . The l ight ray s from a s ourc e t rave l out

inde pe ndent ly o f on e anot he r . That l ight ray s ob ey the l aw o f r e f le c

t i on wa s t he thi rd pri nc i p le,an d that t hey obey t he l aw of ref ract ion

f o r abru pt or cont inuou s c hange s i n t h e me dium wa s the fou rth . ( The

phenomenon o f doubl e refract i on i n c ry s t al s wa s emb rac ed by s uppo s ing

that the me dium ha s two i ndic e s of ref ract ion which depe nd upon po

s i t i o n and the direct ion o f the propagat ion .

6 E l ect romagnet ic Theo ry and Geomet rica l Opt ic s

All Of the s e l aws fo l low from F ermat 's Pri nc i pl e o f Le a s t T ime . Thi s

pri nc i pl e pre s uppo s e s t hat any me dium i s c haract eri z ed by a funct ion

n ( x ,y,2 ) cal l e d t he i ndex o f ref ract ion ( t he ab s o lut e i ndex o r i n dex

to a vacuum ) . The opt ic a l di s t anc e between t w o point s P I wit h co

ordi nat e s ( x 1 , y 1 z l and P2 with coordinate s ( xZ yz 2 2 ) over

any given pat h i s defi ne d t o be the l i ne int egra l

P2

I W e y , z ) as

P 1

t ak e n over t hat path . F ermat 's pri nc i pl e a s s t at e d by him an d others

fol lowi ng him,s ay s t hat t he o pt ic a l path

,the path which l ight actu al

ly t ake s , between P I and P2 i s that cu rve O f a l l tho s e j o i ning P I

and P2 which mak e s the value o f t he i ntegra l l e a s t . Thi s fo rmu l at ion

i s phy s ic a l ly incorrect,a s c an be s hown by example s

,an d t he correct

s t at ement i s t hat t he f i rs t vari at ion of t hi s i nt egra l,i n t he s en s e of the

c a lc u lu s o f vari at ion s,mu s t b e ze ro . Thi s princ i ple coul d b e and was

appl ie d t o t he de s ign o f numerou s opt ic a l i n s t rument s . It i s to b e

not e d t hat th i s pri nc i ple or any ot he r fo rm u lat ion O f g eomet ric a l O pt ic s

s ay s nothi ng about t he nature of l ight .

The mat hemat ic a l t heory o f geomet ric a l opt ic s rec e ived it s def ini

t ive formulat ion i n th e work of Wi l l i am R . Hami lton duri ng the ye ars

18 24 to Though Hami lton wa s aware of F re s ne l ' s work,whic h

we s hal l ment io n s hort ly,he wa s i ndiff e rent t o t he phy s ic a l i nte rpret a

t ion,that i s Huygens ' or Ne w t o n s '

s,and to a po s s ib l e ext e ns ion to in

c lude int e rfe renc e . He wa s concerned to bu i l d a de duct ive,m a t h e m a t i

c al s c i enc e Of opt ic s . Though h i s work i s de s cribe d a s geometric al O p

t i c s,he did i nc lude doubly ref ract i ng me dia ( which are somet ime s r e

garded a s out s i de the pa l e o f s t rict g eomet ric a l opt ic s ) and di s pe rs i on .

Hami lton ' s chie f i de a wa s a charact er i s t ic funct ion,o f which he

gave s evera l type s . The ba s ic o n e o f t he s e expre s s e d t he opt ic a l

l e ngt h of t he ray which j o i ned a po int in t he Obj ect s pac e to a point in

t he image s pac e a s a funct ion o f the po s it ion s o f t he s e two po int s . The

par t i a l derivat ive s of t hi s functi on g ive the di rect ion of t he l ight ray at

the po int i n que s t ion . Hami lton a l s o int roduc e d three othe r type s o f

c haract e ri s t ic funct ion s . He s hows t hat from a knowle dge of any o n e

of t he s e,a l l prob lem s in Opt ic s i nvolvi ng

,for exam ple

,l e n s e s

,mir

r o r s,cry s t a l s , and propagat ion in the atmo s phe re , c an be s o lve d .

F rom Hami lto n ' s work the e qu iva l enc e O f F ermat ' s princ i ple and

Huygen s ' pri nc i pl e i s c l e ar .

As we have a lre ady ob s e rve d,geomet ric a l Opt ic s c annot be regarde d

a s an adequ at e theory o f l ight bec au s e it doe s not t ake into account i n

t e r f e r e n c e,diff ract io n

,pol ari z at ion

,o r even a mea s ure o f the int en s ity

of l ight . I n t he ea rly part of t he n inet e ent h c entury new expe rimental

work by Thoma s Young,Augu s t i n F re s ne l

,E . L . M a ln u s

,D . F . I . Arago

,

B . B iot,D . B rews t er

,W . H . Wolla s ton and othe rs made it c l e ar t hat

a wave theory o f l i ght wa s ne eded t o account for al l the s e phe nomena .

F re s ne l exte nded Huygen s ' t heo ry by addi ng pe riodic ity in s pac e and

Morri s Kl i ne 7

t im e to Huygen s ' wave front s . Thereby int e rfe renc e wa s i nco rporat e d

and F re s ne l u s e d t he exte nded theo ry to expla in di ffract ion a s the

mutu a l i nt e rf e renc e o f t he s eco ndary wave s em itt e d by t ho s e port ion s

Of t he orig i na l wave f ront whic h have n o t been Ob s t ruct e d by the dif

fract i ng ob s tac l e .

U p t o th i s t ime ( 18 18 ) t hi nk i ng on the wave t heory of l ig ht (and for

that matt er even t he corpu s cu l ar theory ) had be en gu ide d by the anal

ogy wit h s ound . Young in 18 17 s ugge s t e d t ran s ve rs e rathe r t han long i

t u d i n a l wave mot ion . Young ' s s ugge s t i on c au s ed F re s ne l t o t h ink about

wave s i n s ol i ds and to s ugg e s t th at ri g i dity s hou l d g ive ri s e to t ran s

vers e wave s . Thi s i de a wa s import ant f o r th e yet to be deve lope d

theo ry o f wave s in e l a s t ic s o l i ds and al s o for the ether . He s ought

the n to b a s e the theo ry of l ight on the dynamic a l prope rt i e s O f ether .

However,F re s ne l 's t heoret ic a l foundati ons were i ncom plete and

even i ncon s i s t ent . He t rie d to expl a i n t he phys ica l nature O f l ig ht

propagat ing through i s ot rop ic and ani s ot ro pic me dia by reg ardi ng the

ether a s a qua s i—e l a s t ic me dium and the l ight a s a di s pl ac em ent of t heethe r pa rt ic le s . Wh en an et he r pa r t i c l e wa s di s p l ac e d , t he ot he r part i

c le s exe r te d a re s toring forc e pro port ional to di s pl ac eme nt . But t he

phenomena of i nte r f e re nc e,t he i nt en s ity in ref l ect ion a nd ref ract ion

,

and pa rt i cu l arly po l ari z at ion,l e d t o the co nc lu s ion that the vibrat ion s

Of the ethe r pa r t ic le s mu s t be t ran s ve rs e,where a s an e l a s t ic medium

c an s upport t ran s ve rs e and long itu di na l wave s . No r coul d the ethe r

be a rare ga s becau s e the re on ly long itu di na l wave s are t ran s mitt e d

a nd the re i s no e l a s t ic re s i s t anc e . Henc e F re s ne l a s s ume d h i s et he r

wa s inf i n it e ly compre s s ibl e . It wa s l ik e a g a s but with e l a s t ic ity i n

pl ac e o f vi s co s ity . The theory o f wave s in e l a s t ic medi a was n o t we l l

deve lope d in F re s ne l 's t ime s o that hi s approach wa s over - s im pl e,and

he cou l d n o t re adi ly e l im i nat e t he lo ng itu di na l wave s which an e l a s t ic

me dium c an s u pport .

A number o f g re a t ma t hemat ic a l phy s ic i s t s , C . L . Navi e r,S . D .

Po i s s on,A. L . Cauchy

,G . Gre en

,F . Neumann

,G . Lam é and I . W .

Strutt ( Lord Rayl e i g h ) work e d o n the theory of wave s i n e l a s t i c medi a

and the appl ic at io n o f t hi s t heory to l ight4

. I n al l th i s work t he ethe r

wa s an e l a s t ic me dium which exi s t e d i n i s ot ropic an d ani s ot ropic

me dia . Some o f t he theorie s s u ppo s ed that t he et her pa r t ic l e s int er

act e d wit h t he pa r t i c le s Of ponde rab l e matt e r t hrough which the l ight

p a s s e d . Thi s approac h to l ight wa s purs ue d eve n aft e r Maxwel l ' s t ime

but wa s never qu it e s at i s factory . One of t he pri nc i pa l di f f icu lt i e s wa s

to expl a i n away long itu dina l wave s . Anot he r w a s t he l ack o f a con

s i s t e nt exp lanat ion of t he phenome na of ref l ect ion and ref ract ion at the

bou ndarie s of i s ot ropic and ani s ot ropic me di a . A thi rd wa s that di s

pe r s i o n was not exp la i ne d .

O f a ddit io nal e ffort s prece di ng Maxwel l ' s work,we s ha l l ment ion

the work o f Jame s M a c C ulla g h . M a c C ulla g h i n 18 39 ( pub l i s hed 1848 )changed the nature of t he e l a s t ic s ol i d whic h re pre s e nt e d ethe r . I n

s t e ad O f a s o l i d which re s i s t s com pre s s ion an d di s to r t ion,he int roduce d

8 E lect romagnet i c Theo ry and Geomet ri ca l O pt ic s

one who s e pot ent i al e ne rgy de pe nd s on ly on the ro t at ion of the volume

e lement s . Wave s in M a c C ull‘a g h

's ethe r cou ld be on ly t ran s ve rs e an d

the vector e which re pre s ent e d a wave mot ion s at i s f i e d t he e qu at i onW e

at2

Moreover,div e O M a c C ulla g h did have to i nt roduce i ndepe ndent

boundary condit ion s . (Wh itt ak e r , fo l lowing Heavi s i de , po int s out t hat

t hi s e amount s t o t he magnet ic f i e l d i nt en s ity o f Maxwe l l .

Thi s s o l i d ethe r o f M a c C ulla g h place d di ff i cu lt ie s in t he way Of re p

re s ent i ng the re l at ion s hi p between ether and ordi nary matt e r ( when

l ig ht t rave l s t hrough matt e r ) and ob l ig ed him to po s tu l at e a part icu l ar

force ( l ate r c a l le d Kirc hhoff‘ s force ) i n order to expla i n the diffe ring

e la s t ic ity o f t he ethe r o n t he two s ide s o f a s urf ac e which s e pa rat e s

dive rs e ly re fract ing medi a . What i s s ig n i fi c ant about M a c C ulla g h's

work i s t hat h i s di ffe rent i al equ at ion s are c lo s e ly re l at e d to Maxwe l l 's

t hough phy s ic a l ly the forme r 's t heo ry bore no re l at ion t o e le ct romagnet i sm .

The mos t s at i s fac to ry theory o f l ight which we have today came

about not through the s tu dy of l ight pe r s e but t hrough t he development

of e lect ric ity and magnet i sm by C le rk Maxwel l . We shal l n o t purs ue

he re the h i s to ry Of t he re s e arc he s i n e l ect ric ity and magnet i s m of

Gau s s,Oe rs t e d

,Am pe re

,F araday

,Ri emann and others bec au s e t he ir

c ont ribut ion s are s t i l l t aught a s a ba s i s for Maxwe l l ' s e le ctromagnet ic

t heo ry and s o are l arge ly f ami l i ar . It i s we l l k nown that o n e of Max

we l l 's g re at di s coverie s wa s t he rea l i z at ion that l ight mus t be an e l ec

t r o m a g n e t i c phenomenon . Maxwe l l wrot e t o a fri e nd in January of 18 65" I have a pape r af loat

,with an e l ect romagnet ic t heo ry o f l ight

,which

't i l l I am convi nc e d to t he cont rary,I ho l d to be gre at gun s .

Though Maxwel l di d t ry un s ucc e s s fu l ly to Obtai n a m echanica l the

ory o f e l ect romag net ic phe nomena in te rm s o f pre s s ure s and t en s ion s

in an e l a s t ic me dium and afte r M axwel l,H . He rt z

,W . Thom son

,C .

A . B j erkne s and H . Po inc ar é t ri e d t o im prove mec hanic a l mode l s but

e qu al ly un s ucc e s s fu l ly,t he acc e ptanc e of Maxwe l l ' s theo ry marke d

the end O f e l a s t ic theori e s o f l i ght . The adopt ion of Maxwel l ' s t heo ry

me an s a l s o the adopt ion O f a pure ly mathemat ic a l vi ew,f o r the k now

le dge that l ight c on s i s t s o f a conj o ine d e l ectric and magnet ic f i e l d

trave l l i ng through s pace hardly expla i n s t he phys ic al natu re of l ight .

It mere ly re duce s the number o f mys t e ri e s in s c i enc e by compounding

one Of t hem .

We might ment ion that t he po s s ib i l ity o f l i nk ing l ight an d e lectro

magnet i s m wa s con s i dere d by s evera l pre dec e s s or s of Maxwe l l . Eu l er,

Young and F araday had s ugge s t e d thi s po s s ib i l ity on diffe rent grounds .

Ri emann had Ob s e rved the ident ity of t he ve loc ity of l ight wit h the rat io

o f the e lect ro s t at ic t o t he e l ect romagnet ic unit s of c harge and s o pro

d uc e d an ad hoc theo ry by exte nding the e l ect ro s tat ic po t e nt i a l equa

t ion

HA G p

A 4) 4 1Tp

Morr i s Kl ine 9

1 82

4)

c2 at 2

Thu s he had a wave mot ion which f o r the prope r va lu e o f 0 moved

wit h t he ve loc ity of l ight . However l ight wa s s t i l l a s c al ar i n t hi s

t heo ry n o r wa s the re any phy s ic al j u s t if ic at ion for addi ng 82

Maxwe l l 's a s s e r t ion tha t l ig ht i s an e l ect romagnet ic wave h a d

othe r arg ument s to recommend it t han the wave e qu at io n t o whic h hi s

e quat io ns re duc e and the f act t hat t he rat io of t he e l ect ro s t at i c to t he

e lect romagnet ic unit o f charge i s t he ve loc ity of l ight . It i s we l l

k nown that from the f i rs t t w o e quat ion s when expre s s ed in rect angu l ar

coordi nate s,fo r example

,and in a non - conduct i ng m edium one can

obta in for any com ponent o f E or H prec i s e ly t he s ame math emat ic a l

e quat ion w hich Navie r a nd Po i s s on had derive d for wave s i n an e l a s t ic

me dium 5 and the s e latt e r wave s di d expl a in m any of the phe nomena

O f l i ght . Moreove r,Maxwe l l ' s e quat io n s po s s e s s e d a s u peri or fe a

tu re . Navie r,Po i s s on and other work e rs i n the e l a s t ic t heory o f l ight

had to make the arb it ra ry a s s umpt ion that t he di l at at ion ( dive rge nc e )o f t he me dium i s 0 to e l im i nate long it u dina l wave s . I n Maxwe l l ' s

e qu at ion s th i s condit ion i s automat ic a l ly pre s ent,t hat i s

,div D 0

and div B 0 One cou ld a l s o derive from hi s e qu at ion s,a s He lm

holt z di d,t he proper bou nda ry condit ion s at an i nt e rf ac e b etwe en two

media wit hout addit iona l a s s um pt ion s . O f c ours e He r t z 's expe riment

a l conf i rm at ion s,pri nc i pa l ly t he exi s t e nc e o f t rave l l i ng e le c t r o m a g

net ic f i e l ds,at l e a s t s howed that radio wave s behave l i ke l ight w ave s .

One mu s t remember,howeve r

,t hat Maxwe l l ' s a s s e rt io n about l ight wa s

bol d and even que s t ionabl e i n hi s day . The s ourc e s of l ight ava i l ab l e

t he n and even up to the pre s e nt day are not monoc hromat ic an d s o no

fin e experim ent a l conf i rmat ion cou l d be expect e d . We are j u s t at t he

po int today,i n the deve lopme nt o f l a s e rs

,o f produc i ng cohe rent mon

o c h r o m a t i c l i ght .

Though the re are unre s o lve d di ff ic u lt i e s i n Maxwel l ' s t heory ,

chi e f ly i n connect ion with t he i nte ract ion o f e lect romagnet ic wave s

wit h matt e r ( t he s e prob l em s are , of cou rs e , be ing i nve s t i g at e d i n

quantum e l ect rodynamic s ) we mu s t acc ept a s our b e s t theory that

l ight i s an e l ect romagnet ic phenomenon s ubj e ct t o Maxwe l l 's e quat ion s .

Geomet ric a l opt i c s t he n c an be only an approximat e re pre s ent at ion i n

s evera l re s pect s . F ir s t,wave leng t h con s i derat ion s do not e nt e r , and

s o int e rfe renc e i s not t ak en into account . The vector c ha ract er Of t he

f ie l d , that i s , polari zat i on , and di ff ract ion , that i s , t he penet rat ion

o f the f i e l d beh in d ob s tac le s,are not incorporat e d . F i nal ly

,s i nc e

wave le ngt h con s i derat io ns do not e nt e r,ne it her doe s di s pe rs ion .

3 . EARLY EFFO RT S TO LI NK ELEC TRO M AG NETIC THEO RY AND GEO

M ETRIC AL O PT IC S . The f i rs t s i gn i fic ant e ffort t o de rive geomet ric al

Opt ic s from the e l ect romagnet ic t heo ry o f l ight i s due to Ki rchhoff .

Kirchhoff s ought a s t rong mathemat ic al foundat io n for l ight an d

10 E lect romagnet ic Theo ry and Geomet ric al Opt ic s

int roduc e d a modif ic at ion of Huygen ' s princ i pl e whic h incorporat ed

t he i nt e r f e renc e in s pac e an d t ime . ( The phy s ic al i nterpret at ion wa s

for him irre l evant . Si nc e l ight was re pre s e nt e d a s a s c al ar fu nct ion,

i n thi s re s pect Ki rchhoff ' s re pre s ent at ion of l ight i s not di rect ly re l e

vant . Moreover,a s i s we l l k nown , the re are di ffi cu lt i e s i n the u s e

of the Kirc hhoff - Huygen s princ i pl e which he t ri e d to ove rcome by t he

a s s um pt ion of rather arb it ra ry boundary condit ion s on the diff ract i ng

Ob s t ac l e and the s e l e ad t o mathemat ic a l i ncon s i s te nc ie s .

Neve rt he l e s s,i n 18 8 2 Kirc hhoff did s how

6 t hat when the wave

l ength o f t he s ourc e approac he s 0 t he wave fie l d g iven by the Kirc h

hoff i nt eg ral approache s t he fie l d g ive n by geomet ric a l opt ic s ; s pe c i f i

c al ly t he diffracte d f ie l d van i s he s and there i s s harp t ran s it ion between

the i l lum inat e d f ie l d and t he dark regi on . That i s,t he wave s be have

l ik e s t ra ight l ine s . Henc e the i dea wa s gene ra l ly acce pt e d by t he e nd

of the n inet e ent h c entu ry that geomet ric a l Opt ic s mu s t be s ome s o rt o f

l im it of e lect romagnet ic t heory a s t he wave l ength goe s to O

The mo s t wi de ly acce pte d argument f o r t he connect ion b etween

e lect romagnet ic theory and geomet ric a l Opt ic s i s t hat g ive n by Som

merf e ld and Runge who fo l lowed a s ugge s t ion of P . D ebye .

7 I n t h i s

argument a funct i on u whic h may re pre s e nt s om e com ponent o f E

o r a com ponent of a Hert z vecto r,i s a s s umed t o s at i s fy the s c a l ar

re duc e d wave equ at ion

A u kzu

where in k e /x Here 6 and p. may be funct ion s of po

s i t i o n and A i s t he vari ab l e wave le ngth i n t he i nhomogeneou s med

ium . The f ie l d i s g ene rat ed by a s ourc e,who s e fre quency i s co and

whos e wave l ength i n a con s t ant m edium 6 0 (I Q i s h o s o that

ko‘Ve o Ho w

Sommerf e ld and Runge now make t he a s s umpt ion that

u ( X ,y,2 ) A( X ,

y,Z ) e

lk° S ( X ’ y ’2 )

that i s,t hat u i s dete rm ine d by an ampl itud e func tion A a nd a pha s e

fu nct ion 8 The latte r,i nc i de nt a l ly

,i s c a l l e d the e i c o n a l funct ion

( becau s e , a s we s hal l s e e in a moment , it s at i s f i e s t he e i c o n a l di f

f e r e n t i a l e quat ion ) . While u wi l l va ry rap idly a s No approac he s

0 or ko approache s 00 it i s a s s umed that A and S do not varyrapidly i n x

,y and 2 ( re l at ive to the wave l ength ) and th at t hey r e

mai n bounded a s ko approac he s 00 The fo rm of ( 2 ) i s a gene ral

i z a t i o n o f t he fo rm of p lane wave s which exhib it s ome O f t he propert i e s

of geomet ric a l o pt ic s .

By di r ect dif fe re nt i at io n of ( 2 ) and s ub s t itut ion i n e quat ion ( 1)one Obta i n s

Morri s Kl ine 1 1

Zik o u [ % A 8 grad lo g A' grad S ] e AA 0

If we now divi de through by ké u and a s s ume t hat t he re s u lt i ng l a s t

t e rm o n t he l e ft s i de , name ly A A/kOZA rema i ns smal l a s ko be

come s inf in it e,t hen we may s at i s fy t he l a s t e qu at ion by re qu iri ng

that

( grad )2

n2

where n k/ko and

grad log A' grad S é—A S 0

Equat ion ( 3 ) i s c al l e d t he e i c o n a l di ffe re nt i al e quat ion and it s

s o lut ion s S con s t . are the wave s urf ac e s or wave front s of g e o m e t

r i c a l opt i c s . The s econd e qu at ion c an be writt en i n te rm s of t he di

r e c t i o n a l derivat ive O f log A in the direct io n Of g ra d S Since , by

|g r a d S | : n we may writ e

gra d log A + § A S O

and de not i ng the direct ional derivat ive in t he di rect ion o f gra d S by

d/ds we have

lo g A )+ §A S 0

The direct ion o f gra d S i s norma l t o t he s urf ac e S con s t . and so

e quat ion ( 5 ) give s us t he b ehavior o f log A along any norma l ( o r

t h o g o n a l t raj e ctory ) t o t he fam i ly of s u rf ac e s 8 con s t . or a long a

ray .

The fact t hat e quat ion ( 3 ) i s de rive d from t he s c al ar wave equat ion

by l ett i ng t o approach 0 and t he fact t hat th e e qu at ion s o Obt a ine d

i s t he e i c o n a l e quat ion a lre ady k nown in geomet ric a l Opt ic s and from

which al l o f geomet ri c a l opt ic s c a n be de rive d , provi de s the argument

for conc lu di ng t hat geomet ri c a l opt ic s c an be derive d from Maxwel l ' s

e quat io n s . Als o t he f act that t he am pl itu de A t ravel s a long t he ray s

i s i n accord with geom et ric a l opt i c s,though of cours e A may vary in

othe r di rect ion s n o t reve a l e d by the above derivat ion .

The Sommerf e ld—Runge derivat ion of geomet ric al Opt ic s i s O pe n t o

many obj ect ion s . The de rivat i on f rom the s c a l ar wave e quat ion i s n o t

s uff ic i e nt ly gene ra l i n th at not al l e l ect rom agnet ic probl em s c an be

reduc e d to t he s c a l ar wave equ at ion . However t hi s c rit ic i s m has been

met in that the s ame k ind of argument ha s be e n made fo r M axwel l ' s

e quat io n s . That i s,one a s s ume s

12 E lect romagnet ic Theory and Geometric a l O pt ic s

i ko S ( X a Y ,Z )

I

I

V ( X7 Y ,Z ) e

lkSO ( X ,

and o n e obt a in s t he e i c o n a l equ at ion fo r S and vector equat ion s for

u and v which are t he ana logue s of ( 5 ) above .

8

Though the Somme rfe l d - Runge proce dure c an be appl i ed t o Maxwe l l ' s

equat io n s a s we l l a s the s c a l ar wave equat i on,it i s n o t a s at i s factory

derivat ion o f geomet ric a l opt ic s from e lect romagnet ic t heory . The a s

s um pt ion ( 2 ) re pre s ent s a ve ry re s t rict e d c l a s s o f f i e l ds bec au s e it

a s s ume s that t he funct i on A i s inde pendent o f ko Thi s a s s umpt ion

i s fu lf i l l e d for pl ane wave s but i s n o t t ru e o f the f ie l ds encount ered

even in re lat ive ly s imple probl em s of propag at ion in unbou nde d medi a .

Hence t he argument s hows on ly that a ve ry re s t rict e d c l a s s of f i e l ds

give s ri s e t o a geomet ric a l optic s f i e l d . Secondly,the argument that

t he A and 8 det e rm ine d a s s o lut ion s o f ( 3 ) an d ( 4 ) are l im it s of t he

A and S in u Aelko 8 when k0 i s i nfi n it e i s incomplet e . The dif

f e r e n t i a l e quat ion s ( 3 ) and ( 4 ) are a l im it of the di ff e re nt i al equat ion

( 1) but th i s fact mu s t be brought to be ar o n t he s o lut ion s . Thirdly,

s ince in it i a l a nd boundary condit ion s play no ro le i n t he e nt i re de ri

vat io n the l im it i ng f ie l d det e rm ine d by A and 8 s e rve s n o purpos e

in re pre s ent i ng a geomet ric O pt ic s approxim at ion t o s ome de s ire d fi e l d .

F ina l ly,t he derivat ion s e em s to offe r no in s ight i nto t he re l at ion s hi p

between wave theory and geomet ric a l opt ic s which m ight b e u s e d t o

make s ome gradua l t ran s it i on from o n e to t he ot he r .

Anothe r proc e dure commonly u s e d to l ink geometri c a l opt ic s and

Maxwe l l ' s t heo ry i s t o t ak e t im e harmon ic pl ane wave s olut ion s o f

Maxwe l l ' s equat ion s a nd to apply the e l ect romagnet ic boundary c o n

d i t i o n s at a pl ane i nt e rf ace between two homoge neou s medi a . As a

con s e que nc e o n e de duce'

s t he l aw of re f l ect ion a nd Sne l l ' s l aw of r e

f ract io n . Thu s t he ba s ic laws o f geometric a l Opt ic s are derive d . The

s ame proc e dure i s u s ed in homogeneou s ani s ot ropic me di a . As a m at

t er o f f act,even the F re s ne l formu la s for t he am pl itu de s of t he r e f le c

t e d and refract e d wave s are a l s o derivabl e i n t hi s way .

There are s evera l ob j ect ion s to thi s procedure . P lane wave s and

plane boundarie s are e s pec i al ly s impl e . There i s no indic at ion from

suc h a derivat ion a s t o what may happen for curve d wave front s and

cu rved boundarie s . The argume nt i s commonly gi ven th at t he l aws of

pl ane wave s in homogeneou s media s uff i ce for t he ap proximat e e l ec

t r o m a g n e t i c t re atme nt of s uch phenomena in which the wave front s are

n o longe r pl ane but where t he curvature of t he wave front c an be neg

le c t e d over doma in s whos e l ine ar d imen s ion s are l arge compared t o the

wave le ngth of l ight . The analogou s remark i s ofte n made about curved

bou ndari e s . B ut i n geomet ric a l opt ic s t he l aws o f ref l ect i on and r e

f ract ion d o hol d for curve d front s and curve d boundarie s and even i n

i nhomogeneou s media . The s e fact s are not obta ined by the argument

Morri s Kl ine 13

ba s e d on pl ane w ave s .

Seco ndly,i n order t o u s e the re s u lt s obt a ine d from thi s argument

in geomet ric a l Opt i c a l prob l em s,t he pract ice i s to a s s ume t hat any

norma l to the wave front i s a ray and that e ach ray behave s at any one

point of an i nt e rf ac e a s t hough it were i nde pe ndent o f a l l t he ot her

ray s . B ut t he pl ane wave argument t re at s the i nf i n it e ly ext e nde d pl ane

wave and the i nf in ite pl ane bou ndary and the argument doe s not i s o l ate

what may happe n f o r any in div i du a l r a y at a s i ng l e bounda ry point . Yet

t he l aws are u s e d t hu s even at a po int on a cu rve d bounda ry s uch a s

the s u rf ac e of a len s .

Thirdly pl ane wave s have in f in ite ene rgy and are a h ighly ide a l

conc e pt . No phys ic a l s ourc e s ends out pl ane wave s . F ina l ly pl ane

wave s have a wave l e ngth . Si nc e thi s fact doe s n o t s how up in t he

l aws de rive d,it i s ignore d .

9

Al l one can re al ly s ay from t he s tu dy o f plane wave s i s t ha t they

obey s ome of t he l aws of geome t ric a l opt ic s but t hey d o not s uff i ce to

derive geomet ri c a l opt ic s f rom Maxw e l l 's e quat ion s .

4 . THE RE LATIO N SH I P OF G EO M ETRIC AL O PTIC S TO E L EC T RO M AG

NETIC THEO RY . I s hou l d now l ik e to pre s ent t w o new views of geo

met ric a l opt ic s from t he s t andpo int of e l ect romag net i c t heo ry . The

new viewpoint s are val i d in bot h i s ot ropic and an i s ot ropic me di a,but

I s ha l l t re at i s ot ropic me di a . We have Maxwe l l ' s e quat ion s,whic h

,

for s impl ic ity, I s ha l l t re at i n non - conduct i ng me di a

,name ly

curl

curl E g HC t

The t erm cont a i n i ng F to r s t rict ly the re a l part of 1/4 rr )F_ t

re p

re s e nt s a s ourc e curre nt de n s ity . I n the pre s ent di s cu s s ion it s rol e

i s irre levant an d one can su ppo s e in s t e ad th at i n it i a l valu e s o f E and

H which are func t ion s of x, y ,z and t are s pec ifi e d i n s t e ad .

There may a l s o be boundary condit ion s .

The fi rs t view of geomet ric a l opt ic s i s t hat t he geomet ric a l opti c s

f i e l d corre s pondi ng t o any e l ect rom agnet ic f ie l d at any po int ( x , y , z )of s pace con s i s t s o f the s i ngu la ri t i e s o f E and H a s funct ion s of

t ime t By the s ingu l arit i e s we me an,of cours e

,t he di s cont i nu it i e s

of E and H o r o f any of t he i r s ucc e s s ive t im e de rivat ive s a s fu nc

t ion s of t Thi s def i n it ion i s,i n a s en s e

,too ge ne ra l . If we wi s h

to obt a i n c l a s s ic a l g eometri ca l opt i c s we s hou l d re s t rict ours e lve s t o

s ingu l arit i e s wh ic h are f i n it e di s cont i nu it i e s wi t h re s pect to t ime i n

E H an d the ir s ucc e s s ive t ime de rivat ive s . There may very we l l b e

s ingu l arit i e s at which E and H are cont i nuou s,but s ome t ime der

i v a t i v e i s di s cont i nuou s o r where t he di s cont i nu it i e s of_

E_

and H are

f in it e but t ho s e of s ome t ime derivat ive are n o t .

14 El ect romagnet ic Theory and Geometric a l Opt ic s

B efore purs u ing t hi s conce pt ana lyti c al ly,l et u s examine it geo

metric a l ly . We s hal l con s i der two s pac e dimen s ion s . I f we s uppos e

that s ome s ou rc e loc at e d in t he

p l ane t = 0 beg in s to act at t ime

t 0 t he n we know that a fi e l d

s pre ads out into s pac e whic h at

a part icu l ar t ime t o covers only

a bounded reg ion of ( x , y , t )s pac e

,the s hade d reg ion i n

F ig . 2 . That i s,during t he

t ime 0 t t o the f i e l d wi l l

t rave rs e t he int e rior of a cone

which l i e s between t z 0 and

t : t o At a po int s uch a s P

o r ( x0 yo t o t he f ie l d wi l l

b e 0 f o r t < t o and at t z t o

t he re wi l l b e a j ump in the value

of E and H from 0 to a fi n it e

valu e . Thi s fi n it e valu e of E

and H i s the g eometric a l opt ic s

f i e l d at P Alt e rnat ive ly,the

geometric al opt ic s E and H

are t he l im it s appro ache d by

F igu re 2 and

a s t approache s t o t hrough

value s gre at er t han t o At t ime s t t o t he fi e l d may cont i nu e to

be n o n—zero a t the poin t s (x0 , y o , t ) but th i s fi e ld i s not a pa r t o f the

g eometrica l optic s f ie ld ; i t i s pa rt o f the wave fie ld E (x,y, t ) E (x, y , t )

whic h s a t i s fi e s Maxwe l l ' s equa t ion s . Thu s the g eometrica l opt ic s

fi e ld f o r a l l t va lu e s i s the s et o f_E_and H va lu e s whic h exi s t on ly

on th e s u rface o f the cone .

The cone it s e lf i s g ive n by s ome e quat ion d" ( x ,y,t ) 0 i n (x , y , t )

s pace . One c an int roduc e ray s i n t hi s s pac e - t im e p ictu re a s t he gen

e r a t o r s o f t hi s cone and fo l low the geometr i c al opt ic s f ie l d along s uch

a ray . ( Mathemat ic al ly t he s e ray s have a prec i s e def i n it ion as the bi

charact e ri s t ic s of Maxwell' s e quat ion s .

There i s a s econd geometric al picture whic h may be more u s e fu l i n

phy s ica l t hi nk i ng . At e ach t ime t t he locu s of ¢ ( x ,y,t ) 0 i s a

curve . We may plot the s e curve s a s a fam i ly o f curve s in (x , y )s pace ( F ig . 3 ) The s e cu rve s are t he wave front s of g eomet ric al O pt i c s . Analyt ic al ly

,we s uppo s e t hat ¢ ( x,

y,t ) 0 can b e writt en a s

t LIJ ( X , y )/c and for e ach va lue of t t here i s o n e cu rve of t hi s fam i ly

of wave front s . The u s u a l ray s of geometric a l opt ic s are ( in i s ot rop ic

me dia ) the ort hogonal traj ectori e s of t hi s fam i ly of wave f ront s . In

s ofar a s the geomet ric a l opt ic s f ie l d i s conc e r ne d,at e ach point on a

wave front and at t he t ime t o given by t he e quat ion llJ/C t o of t hi s

front t he valu e s o f E and H change from O for t t o to s ome

i

l

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16 El ect romagnet ic Theo ry and Geomet ric a l O pt ic s

Thu s fa r the approach t o geomet ric a l opt ic s i s no more t han a new

mathemat ic a l formu l at ion of c l a s s ic a l g eomet ric a l opti c s,but indee d

o n e whic h re l ate s geometric al opt ic s to Maxwel l ' s e quati on s . C la s

s i c a l geomet ric a l Opti c s become s the b ehavior o f s pec i a l valu e s o f t he

e l ec t romagnet ic fi e l d . Actua l ly thi s appro ach g ive s more t han c l a s s i

c al opt ic s,bec au s e it give s the vecto r ampl itu de s of t he geomet ric a l

opt ic f i e l ds and the F re s ne l l aws .

The above - de s c rib e d point o f view yie l d s a new in s ight at onc e .

Let u s return t o s pac e - t i m ell

Con s i der t he f i e l d ( F ig . 2 ) at ( x0 y o t o ) .

As t inc re a s e s beyond t o t he f i e l d E ( xo yo t ) H ( x0 ,y o

,t ) i s non

ze ro . Hence,i f E and H are analyt ic wi thi n t he cone

,both E and

H s hou ld be expre s s ib l e in powe r s erie s whos e vari ab l e i s t - t o which

re pre s ent s the t r ue f i e l d f o r t t o17“ The coeff ic i e nt s o f t he power

s eri e s f o r E f o r example,s hou ld be

_E_ t ( x0 yo t o ) ’ E t t

where we me an by the s e de rivat ive s t he valu e s a s s ume d by t he func

t ion s fo r t : tO

o r alte rnat ive ly the l im it s approac he d,fo r example

,

by a s t approache s t o t hrough va lu e s l arg er t han t o

TheValue s of E H an d thei r s ucc e s s ive t ime derivat ive s at t = t0 _

are 0 bec au s e f o r value s o f t t o the f i e l d ha s not re ac he d ( x0 yo)The quant it i e s E t ( xo yo to lr E t t are t hen di s conti nu it i e s of the

s ucc e s s ive t ime derivat ive s of E ( x , y ,t ) on the s urf ac e 4) O Since

t o M Xo y o )/c e ac h of the s e di s cont i nu it i e s may be expre s s e d a s a

funct ion o f x0 and y o only .

We may expre s s t he thought of the prec e ding paragraph in term s of

t he pure s pac e pictu re ( F ig . At any point ( x , y ) on the wave

front t o and at th e t ime t o , E , H,_E_ t , Ht ,

are di s

cont i nuou s a s fu nct ion s o i t However,f o r t t o and for po int s

( x, y ) on thi s wave f ront E and H are not ze ro and may be expre s s e d

a s Taylor ' s s erie s in powers of t - t oThu s under ei t he r i nt e rpret at ion we have the expan s ion s

E ( X 7 Y )t ) E ( X, Y ,

t o ) B t ( x 3Et t ( xy Y ,

t o )

for t t o

: O f o r t < t o

and the analogou s expan s ion s f o r H Sinc e t o llJ ( X ,y, )/c

f o r t

: O for t <

To obta in t he s e power s erie s we mu s t be abl e to c alcul ate t he c o

eff ic i e nt s . We have al re ady indic ate d how we can c alcul ate

E ( x , y , l/C ) Y ) The method which l ea ds t o i nformat ion about

Morri s Kl ine 17

t he di s cont inu it i e s of E and H t hem s e lve s,t hat i s

,which l eads to

the t ran s port e quat ion s:c an be ut i l i z e d 13 to obt a in l ine ar , f i r s t orde r,ordi nary diffe rent i al e qu at ion s f o r t he di s cont i nu it i e s i n E t , E t t

Ht , E t t ’ a s t he s e propagat e a long the generator s of Th e cone in

the s pac e—t ime picture or wit h the wave front or along t he ray s i n t he

s pace pictu re . The s e diffe rent i a l e quat ion s,which we c a l l the h igher

t ran s po rt e qu at ion s,can b e s o lved and s o we can obta in t he value s o f

t he s e di s cont i nu it i e s at any poi nt ( x , y ) at t he t ime t o LlJ/C

We can the n obta i n t he powe r serie s i n que s t ion and l e arn some

thi ng about the t ime - de pe nde nt f i e l d s E ( x, y , t ) H ( x ,y,t ) i n the

ne ig hborhood o f a wave front , t hat i s , f o r t ime s t

_

n e a r t he t ime t oat which E and H f i rs t become non - z ero at ( x, y ) St at e d ot her

wi s e,we Ea n o btafn t he s eri e s expan s ion s ( 9) for E a nd

_H_ i n whic hthe geomet ric a l o pt ic s f ie l d i s t he f i rs t t erm .

The s econd vi ew of geomet ric a l opt ic s to be pre s ent e d derive s from

con s ide ring t ime harmonic s o lut ion s o f Maxwel l ' s e qu at ion s . The

fie l ds we are de al ing with t he n have t he form ( we now u s e three s pace

vari abl e s )

E ( x , y, z ,t )

( 10 )

where i n u and v are compl ex vectors . The k ey re s u lt,phra s e d f o r

s im pl ic ity—o n t he

—a s s umpt ion t hat on ly o n e f am i ly o f wave front s exi s t s

,

i s th at

A x,z

E (X , y,Z )

Z )z )

B x z B x,z

N elkwx

’ y ’z ) —1 (

.

’ y ’ ‘ J

( lw )

( 12 )

w he re in the s eri e s are a s ymptot ic fo r l arge o.) and Ll} s at i s f i e s t he

e i c o n a l di ff e rent i a l e quat ion . The qu ant ity k i s Thu s t he

funct ion s 3 and x which are the ampl itu de s of t he t im e - ha rm onic

f i e l d vectors E and H may be repre s ente d a sym ptot i c a l ly by s eri e s

a sym ptot ic in_

l/w for—l a rg e co

Loo s e ly one c an now def ine the geometric al opt ic s f i e l d a s the l im it

fo r l arge 00 of t he f ie l d ampl itude s u and v Then the f irs t t e rm s o f

the s e two s eri e s are t he geomet ric a l Opt ic s f i e l d . The def i n it ion a s a

l im it for i nf in it e w i s n o t qu it e.

prope r bec au s e the f ir s t t e rm s o f t he

tw o s eri e s cont a in t he factor elkLl’ and the s e have n o l im it a s G ) be

come s i nfi n it e . One c an however s ay t hat t he geometric a l opt ic s f i e l d

con s i s t s of t he f i rs t t e rm s o f s e ri e s whic h are a s ym ptot ic for l arg e to

provi de d we n o w i nc lu de in geomet ric al opt ic s t he pha s e f actor elk“

Thi s f ie l d t hen i s not s t rict ly the c l a s s i c al g eometri c al opt i c s f i e l d but

cont ai n s an addit io na l and by no me an s unde s irabl e f e ature . We al s o

18 El ect romagnet ic Theory and Geomet ric a l Optic s

s ee c l early h o w thi s g eometrica l opt ic s f ie l d i s re l at e d t o t he fu l l

wave s o lut ion o f Maxwe l l ' s e quat ion s .

The i nt roduct io n o f thi s s econd def i nit ion of t he geomet ric a l Optic s

f i e l d ra i s e s t he que s t ion o f whet her it i s i de nt ic a l , exce pt f o r t he

pha s e factor,with the geomet ric a l opti c al f i e ld previou s ly int roduced

a s t he di s cont inu it ie s of E ( x , y , z , t ) an d H ( x,y,z,t ) The an swer

i s t hat t he ve ry derivat ion of t he s erie s ( 1 1) and ( 12 ) s hows t hat14

z E

Z H

A1 ( X ) Y )2 ) z § t ( n )

Z,ill/C )

B I ( X, Y ,Z ) z fl t ( X , Y )

Moreove r s i nc e we k now that t he above E,H

,Et ’

Ht

s at i s fy

l inea r,f i rs t order di ffe rent i a l e qu at ion s

,we know t h g t

l

t h e s ame i s

t rue for t he coeff ic i e nt s of t he a sym ptot ic s erie s and s o t he s e coef

f i c i e n t s c an be re adi ly det e rm i ne d . To obta in t he geomet ric a l opt ic s

f i e l d we have but to so lve t he e i c o n a l e quat i on

41,

2

; ia s mu st be done in any ca s e and then s o lve j u s t t he f i r s t t ran s port

e quati on s,o n e f o r AO o r E and t he othe r fo r B0 or H

”<

The l arge r m at hemat ica l poi nt o f i nt e re s t he re i s that if o n e i s s at

i s f i e d t o obta i n an a sym ptot ic s eri e s s o lut ion o f a t ime harmonic prob

l em in pl ac e of t he exact s o lut ion,he can re pl ac e the s o lut ion of

Maxwe l l ' s part i a l different i a l e quat ion s by the s o lut ion of a s eri e s of

f i rs t orde r ordi na ry diffe re nt i al e quat ion s . Thi s met hod mu st be di s

t i n g u i s h e d f rom obta i n i ng an exact s olut ion of Maxwe l l' s e qu at io ns in

t he fo rm of an int e gra l,s ay

,and then evaluat i ng the i nt eg ral a s ym p

t o t i c a lly by a met hod appropri at e to the a symptot ic eva luat ion of

int egral s .

Both views of geomet ric a l opt ic s not only re l ate t hi s theory di rect ly

to Maxwe l l ' s e quat ion s by prec i s e mathemati c a l connect ion s but a c

com pl i s h even more . Si nc e o n e c an ca lcu l at e t e rm s beyond t he f irs t

one s in the s e ri e s ( 1 1) and ( 12 ) t h i s vi ew o f t he re l at ion s h i p between

Opt ic s and el ect romagnet ic s perm it s u s to improve on g eomet ri ca l o p

t ic s a pproximat ions t o e lect rom agnet ic prob lem s . Lik ewi s e the T ay lor

s erie s expan s ion o f the t ime - de pen dent E and H in the ne ighborhood

of t o tux,y,z )/c improve s on the g eometric a l Opt ic s f i e l d in t he

direct ion o f t he fu l l t ime - depe nde nt s o lut ion . Thu s o u r new views Of

geometric a l opt ic s permit u s to m ak e bett e r approximat ion s to wave

s olut ion s than geomet ric a l opt ic s it s e lf . We s ee,i nc i dent a l ly

,that

we have s u ppl i ed the m athemat ic a l foundat ion f o r what Sommerf e l d

and Runge did .

Morri s Kl ine 19

The t heory di s cu s s e d t hu s f ar appl i e s to t he direct t ran s mi s s ion ,ref lect io n and refract ion in homoge neou s and inhomogeneou s i s ot ropic

me dia,and

,i n s ofar a s g eomet ric al Opt ic s a s a s tudy o f di s cont i nu it i e s

i s conc e rne d,it ha s a l s o b een ca rrie d out f o r homogeneou s an d i n h o m

o g e n e o u s ani s ot ropic me dia . Stat e d otherwi s e,whereve r t he ray s of

c la s s ic a l g eometric a l opt ic s had been def ine d , t he new theory appl i e s

al s o . F o r th i s c l a s s o f prob l em s one can obta i n a symptot ic s erie s

s o lut ion s corre s ponding t o g ive n s ourc e s,i n it i a l condi t io n s

,and

bou ndary condit ion s

5 . SO M E APPLIC ATIO N S OF THE THEO RY . The more c arefu l s tu dy of

the re l at ion s hi p o f geomet ric a l opt ic s to e l ect romagnet ic t heo ry ha s

s t imu l ate d a number o f inve s t i gat ion s and ha s thrown new l ig ht on

ol de r one s wit hi n t he domai n of e lect romagnet ic s and out s i de . We

s ee more c l e arly that t he propagat ion of di s cont i nu it i e s i s t he f i rs t

approximat ion t o aperi odic o r t ime de pe nde nt s o lut ion s of variou s

e quat io n s of mathemat ic a l phy s i c s and the a pproximat ion s obt ai ne d

by l ett i ng s om e paramete r appro ac h 00 are the f i rs t t e rm s i n a s ym p

t o t i c s e ri e s deve lopme nt s of t im e harmon ic f ie l d s or o f s o lut io ns of

t he t im e fre e e l l i pt ic part i a l dif fe rent i a l e qu at ion s . I s hou ld l ik e t o

g ive s ome indic at ion o f t he s cope of t he prob l em s encom pa s s e d by

the theo ry pre s ent e d in art ic l e 4 .

S inc e many e l ect rom agnet ic prob l em s ca n be t re at ed a s s c a l ar prob

lem s and s i nc e ot her b ranche s o f mathemat ic al phy s ic s involve e it her

s c al ar qu ant it i e s o r diffe re nt sy s t em s of part i a l di ff e re nt i a l e quat ion s,

I s hou l d l ik e to po int out f ir s t t hat t he t heo ry I have s k etc hed f o r Max

we l l ' s e quat ion s ha s been ext e nde d f i r s t .o f al l to t he g e ne ral l i ne ar

s econd orde r hype rbo l ic part i a l di ffe rent i a l e quat ion15

n nkb u

k+ c u z f

w

lhe r e i n u i s a funct ion of ( x l , x2 K

m ) and the coeff ic i ent s a

b and'

c are funct ion s o f x 1 , x2 , ’Xn - l

and Ui j

azu/Elx i c

'

i xJ Thu s t re at i ng xrla s t one may s tudy the behavior

o f t he di s cont i nuit i e s [u ] , [ut ] , [ut t ] , of u and obta in t ran s

po rt e qua t ion s f o r t he ir propag at ion a long what are c a l l e d the b i c h a r a c

t e r i s t i c s o f ( l ) or, i n t he ( x 1 , Xn - l)

- s pac e,along the ray s . One

may al s o di s cu s s the a symptot ic s erie s repre s e nt at ion of s o lut ion s

u ( x 1 , t hat i s,o f t ime—harmonic so lut i on s o f an d

al l o f the re lat ion s b etw een the t ime - depende nt s olut io n an d the t ime

inde pendent s o lut ion wh ic h ho ld for Maxwe l l ' s e quat ion s apply here t o o .

The theory ha s be e n f u rther exten de d 16 t o symmet ri c l i ne ar hype r

bo li c sy s t em s o f pa rt i a l diffe rent i a l e qu at ion s and thu s c an b e appl i e d

t o more compl ic at e d sy s t em s of f irs t o rde r pa r t i a l di f fe rent i a l equ at ion s

t han Maxwe l l ' s e quat ion s .

In s o far a s appl ic at ion s t o e l ect romagnet i c t heo ry are co nc e rne d,

20 E lect romagnet ic Theo ry and Geomet ric al O ptic s

t he appl icat ion s made in the l a s t t e n ye ars have b een numerou s . A

large numbe r o f s c a l ar prob lem s involvi ng s c att ering from the ext e rior

o f smooth bodie s ha s been t reat e d by Ke l l e r , Seck l er and Lewi s17

Sinc e the met hod of geomet rica l Opt ic s proper had b een avai l ab le the

prog re s s i n t hi s work i s t o obta i n improvement s ove r t he geometri ca l

o pt ic s f i e l d by c alcu l at i ng more te rm s o f t he a s ymptot ic s eri e s s o lu

t ion of s t e ady s t at e prob l em s . St i l l i n the doma in of e l ect romagnet ic

prob lem s I s hou l d al s o l ike t o cal l att e nt ion t o t he surpri s i ng re s u lt

obt ai ne d by Sc h e n s t e d18

Sc h e n s t e d c a lcu l at e d the a s ym ptot i c s e ri e s

for t he vector f i e l d diff ract e d by the ext erio r of a parabo lo i d o f revo

lut i o n when a plane wave i s i nc i dent along t he axi s ( the normal i s

di rect e d along t he axi s ) and found th at the a s ymptot ic s erie s con s i s t s

only of t he f ir s t t e rm . In th i s c a s e,the n

,the g eometric a l opt ic s f i e l d

i s al s o the exact e l ect romagnet ic s o lut ion .

Our t heory ha s an import ant b earing o n qu antum mechan ic s . In

e rect ing the sy s t em of wave mechanic s Sc h rOd i n g e r i n 1926 gave the

fo l lowi ng con s t ruct ion ” . He con s i de re d a part ic l e o f mas s m with

momentum p and tot a l ene rgy E in a f i e l d of forc e wit h pot e nt i al

V( x ,y,z ) Then Hami lton ' s pa rt ia l di ffe rent i a l e qu at ion f o r t he mot ion

i s

8W 8W 8WH

7 Y ’2?8X ay

78 2

where H i s t he Ham i lton ian funct ion f o r t he pa rt ic l e,name ly

,

1

2m

and W i s Hami lt on ' s pri nc i pa l fu nct ion . Thu s t he part i a l diffe rent i a l

e quat io n in t hi s c a s e i s

O

I n accordanc e wit h Ham i lton ' s t heo ry the princ i pa l funct ion c an be

writt e n a s

—E t S ( x,y,z )

where S i s Hami lton ' s charact eri s t ic func t ion . The e quat ion f o r S

now i s

2m (V—E ) 0

On the ba s i s of heuri s t ic con s i de rat ion s,Sc h rO d i n g e r n o w int ro

d uc e d a wave funct ion LP and wa s l e d t o t he t ime - i nde pen dent ( r ed u c e d ) Schro ding er e quat ion

( B - vw o

Morri s Kl i ne 21

where i n til i s a fu nct ion o f x,y and z Th i s de rivat ion o f the

Schro dinger e quat ion indic ate d that wave mechanic s i s i n s ome s en s e

a gene ra l i z at ion of c l a s s ic a l mechanic s in t he s ame vague way that

e l ect romagnet ic t heo ry appeare d in 1926 t o be a gene ra l i z at ion o f g e

o m e t r i c a l opt i c s . In fact Sc h rO d i n g e r was gu i de d by that ana logy an d

s poke o f ” work ing from t he Hami lton i an analogy o n the l ine s o f undu

la t o ry opt ic s .

I n 193 320B irk hoff s ugge s t ed that a s ym ptot ic s eri e s s o lut ion s f o r

the funct i on ll) i n ( 15 ) might b e obta i ne d by a s s umi ng a s e ri e s

kN e

S( v0 k

where S and t he V i are fu nct i on s of x , y and z By s ub s t itut ion

for Ll.) i n t he part i a l diffe rent i a l e quat ion ( 15 ) B i rk hoff obt a ine d a fi r s t

orde r non - l i ne ar pa r t i a l di ff e re nt i a l e qu at ion f o r S ( whic h corre s ponds

t o o ur e i c o n a l e qu at ion ) and s howed t hat t he Vns at i s fy a s y st em of

recurs ive ordi na ry diffe rent i a l e qu at ion s

dvn

d T

where An - l i s a k nown l ine ar dif fe re nt i a l expre s s ion i n v 0 v 1

Vn —lThe f i rs t orde r pa r t i a l diff erent i a l e quat ion ( e i c o n a l e qu at i on ) whic h

B irk hoff obt a in s i s t he Hami lton - jacob i e qu at ion from whic h Schrodinge r

s t a rt e d . Thu s on a pure ly formal b a s i s B irkhoff s howed that c l a s s ic a l

mechan ic s i s de rive d from w ave mechan ic s by t he i nt roduct ion o f an

a sym ptot ic s e ri e s in pract ic al ly t he s ame forma l way that Sommerfe l d

a n d Rung e de rive d t he e i c o n a l e qu at ion o f opti c s from t he s c al ar wave

equ at ion exce pt that B irkhoff a s s ume d a fu l l a s ymptot ic s e ri e s whe re

Sommerf e l d and Ru nge a s s ume d on ly the f ir s t t e rm . However,the pre

c i s e mathemat ic a l re l at ion s hi p of t he Sc h rO d i n g e r e qu at ion to the

orig in a l Hami lton - jacob i f i r s t o rde r part i a l diff e rent i a l e qu at ion re

maine d unc l ear .

It i s now apparent from our theo ry of a s ymptot ic s eri e s so lut ion o f

par ti a l di ffe re nt i al equat ion s that t he s e ri e s ( 16 ) adopt e d pure ly for

ma l ly by B irkhoff doe s in dee d provi de an a s ymptot ic s erie s s o lut ion o f

t he Sc h rO d i n g e r e qu at ion ( 15 ) and t hat t he corre s ponding e i c o n a l e qu a

t ion i s t he Hami lton - I a c o b i e quat ion

3

2m ( E - V)

Here S i s Ham i lton 's c haract eri s t ic fu nct ion . We now k now too t hat

thi s l a s t e quat ion hol ds prec i s e ly i n t he l im it for sm a l l h In othe r

words c l a s s ic a l mechanic s i s in de e d t he l im it i ng c a s e o f quantum

mechan ic s .

But o u r t heo ry goe s f art her i n th e doma in o f qu antum mech anic s .

2 2 E lect romagnet ic Theory and Geometric al O ptic s

To s olve the re duc e d Sc h r oding er e qu at ion f o r i t s e ig enfu nct ion s an d

e i ge nvalue s,phy s ic i s t s u s e d s e parat ion of vari ab l e s and obta ine d

the one—dimen s iona l ordi nary diff e re nt i a l e quat ion

2

llJ”

( X ) +8 ” m

( E 0 - oo < x < oo

whe re in m i s a ma s s, V( x ) i s the pot ent i al e ne rgy and E i s the

tot a l ene rgy ( and the e i genvalue paramet er ) and the n (1926 ) appl ie d

the approximat ion method now k nown a s the WKB method afte r it s i h

n o v a t o r s Went ze l , Krame rs and B ri l lou in t o approximate the e i g e n v a l

ue s . At thi s t ime ( 1926 ) t he prec i s e nature of t hi s approximat ion r e

mained unc l ear .

I n 1908 B irkhoff had given a theo ry o f a symptot ic s o lut ion of t he

n - t h orde r ordi nary di ffe rent i a l e quat ion

dmz d

n - lz n

+ p a ( x, p ) a o ( x , p ) zn n - ldx dx

and had s hown t hat e ach s o lut ion 2 1 c an be expre s s ed in t he a s ym p

t o t i c fo rm

wo

a —i

N e Z1

j = 0

whe re t he w i ( t ) are the s o lut ion s of the i ndic i a l equat io n

wn+ a ( x,

0 ) w : 0

and the Z i jc an be s ucc e s s ive ly det e rm ine d by s o lvi ng a recurs ive

sy s t em o f rat he r s im ple ordinary diffe re nt i a l e quat ion s . B irk hoff

s howed in t he s econd o f hi s 193 3 pape rs t hat hi s 1908 pape r re adi ly

covers t he o n e - dime n s iona l Sc hrodinger e quat ion and the firs t t e rm of

20 ) y ie lds the WKB s o lut ion of th i s re duce d or t ime fre e Sc hro di nge r

e qu at io n .

Our theo ry now pe rm it s u s t o s ay that t he f i rs t t e rm in th e s eri e s

name ly eks

vo whe re i n a l l thre e vari abl e s x,y and z are

pre s ent,i s the direct gene ra l i z at ion t o part i a l di ff erent i al equ at ion s

o f the WKB approximat ion u s e d in o n e —d i men s ional pr ob l em s .

Thi s l att e r po int may nee d and warrant e l aborat ion . Let u s con s i de r

the s econd orde r wave equ at ion

Our theo ry for the a s ym ptot ic s e ri e s s o lut ion o f t ime harmonic s o lu

t ion s of t hi s e quat i on t e l l s u s that t he t ime ha rm onic s o lut ion

u V ( x ,y,

can be re pre s ent ed in t he form

https://www.forgottenbooks.com/join

24 E l ect romagnet ic Theo ry and Geometric a l Opt ic s

d i ffe ren t ia l equa t ion s ca n be u s ed f o r the three—d imen s ional Sc hrod ing ere quat ion and othe r e qu at ion s when s e parat ion of vari abl e s i s not po s

s ibl e . Thu s Ke l le r?“4ha s de rive d th e half i nt eger quantum numbers for

t he thre e—dimen s iona l Sc h rO d i n g e r e quat ion by u s i ng the f ir s t t e rm of

the a s ymptot ic s erie s s olut ion for c); t hat i s,by a s s umi ng that ii ;

i s re pre s e nt e d a pproximate ly ( for l arge k or s ma l l h ) by

M

410 Z A

n z ln

and the cond i t ion that tho mu s t be s ing le - value d . The s ummat ion o f

t e rm s mere ly t ak e s care o f t he f act t hat t he re may be many s e rie s if

S i s mu lt i ple—value d o r i n opt ica l te rm s,i f many fam i l i e s of wave

front s pa s s a give n point .

The work de s c ribe d in the prec e di ng paragraph wa s appl ie d to the

Sc h r O d i n g e r e quat i o n i n unbounde d dom ai ns . However t he s ame method

ha s bee n u s e d to f ind a s ymptot ic value s o f t he l arge e igenvalu e s and

the corre s ponding e igenfunct io ns i n bounde d domain s and inde ed for

the re duc e d wave equat io n2 5

That i s,t he m et hod i s appl i ed t o

( A + k2

) u

where k i s t he e ig enva lue paramete r,the e quat ion i s val i d i n s ome

domain D and a boundary condi t ion,f o r example

,au/au 0 i s

impo s ed on the boundary B o f D The method wa s a l s o ap pl i e d to

the ( re duc ed Sc h rO d i n g e r e quat ion with a s phe ric a l ly s ymmet ric pote n

t i a l V( r ) name ly,

Azu - 1<

-2V( r ) ) u o

whe re - k2i s t he e i genvalue parameter . Here t he domain B i s al l

o f s pac e .

Wh ere a s t he appl ic at ion o f the theo ry o f art ic le 4 to qu ant um me

c h a n i c s ut i l i z e s t he t ime—harmon ic high fre que ncy point of view othera ppl ic at ion s rec e nt ly have made ut i l i z e d the s tudy o f di s c ont i nu it i e s .

Acou s t ic s had been deve loped from t he wave t heory point o f vi ew a l

mo s t from the s t art of t hi s s c i enc e . One c an however i nt roduc e a g e

o m e t r i c a l acou s t ic s , a s Ke l le r?‘ 6 and F r i e d la n d e r

Z7 have and f i nd that

t he po int of vi ew of di s cont inu it i e s pe rm it s o n e to s tudy weak s hock

wave s i n ga s e s . If o n e a s s ume s f o r a f lu i d mot ion that t he s hock

wave s i n t he me dium are we ak and s o can ignore t he i nt eract ion o f the

s hock and t he me dium behind t he s hock ( t he s i de into which t he s hock

i s proc e eding ) and i f o n e neg lect s vi s co s ity and he at conduct ion i n

the flu i d the n t he s hock wave s are t he di s cont i nu it i e s i n t he ( exce s s )pre s s ure a nd th e change o r di s cont i nu ity i n pre s s ure at the front i s

the s hock s t rength . One obta in s a s in geometric a l opt i c s an e i c o n a l

equat ion f o r t he wave or s hock front . The ray s a re o r thogon al to the

Morri s Kl i ne 25

front s and o n e derive s a t ran s port equ at ion f o r t he vari at ion o f t he

s hock s t re ngth along a ray . One can al s o t reat t h e ref l ect i on and

t ra ns mi s s ion o f t he s hock s acro s s boundarie s a s i n geomet ric a l Opt i c s .

It i s a l s o po s s ib le t o obta i n a s ymptot ic s eri e s s o lut ion s of t he l i n

e a r i z e d acou s t ic e quat ion s f o r pe riodic wave s of hig h fre que ncy by

u s i ng the t heory pre s ent e d e arl ie r for pe riodic s o lut ion s o f Maxwel l ' s

equat io n s or t he ge nera l s econd orde r s c a l ar equat ion28 Then the

theo ry fo r weak s hock s provide s an approximat ion to t he pe rio dic s o

l ut ion s in t he s ame way t hat geomet ri c a l Opt i c s i s an a pproxim at ion

to wave s o lut ion s o f Maxwe l l ' s equat ions .

The u s efu l ne s s o f a " geomet ri c a l Opt ic s " o f wate r wave theory a s

we l l a s of a symptot ic approxim at ion f o r hig h fre quency pe riodic wat er

wave s ha s a l s o b een f avorab ly con s i dere d?“9

F o r water wave s i n

s ha l low wate r t he wave ampl itu de u ( x ,y,t ) s at i s f i e s t he part i a l dif

f e r e n t i a l e quat ion

( g h ux )X

( g h uy)y

where i n g i s the acc e lerat io n due to gravity and h ( x ,y ) i s t he var

i able de pt h me a s ure d f rom the e qu i l ib ri um wate r s urf ac e . I n t hi s do

ma in o f appl i cat ion t he t re atment of b re ak e rs and s urf near a be ach

c an be handl ed eff ect ive ly by e it he r the s tudy of t he di s cont i nu it i e s

o f t he t ime dependent e quat io n or by exam ing t he high f re que ncy a p

proximat ion t o periodic wave s . We know of cour s e t hat t he f i rs t t e rm

of e it he r approximat ion i s the s ame except t hat t he pha s e factor ei k‘t’

i s pre s e nt i n t he l att e r c a s e ” .

An othe r c l a s s of appl ic at ion s de a l s wit h t he l i ne ari z ed e quat ion s

of mot ion in e l a s t i c i s ot ro pic me dia“ . Here for s ma l l ampl itu de s he ar

and com pre s s ion a l wave s o n e can obta i n t he propagat io n o f pu l s e s or

t he a s ymptot ic form of pe riodic wave s o f high fre qu ency in bot h ho

m o g e n e o u s and inhomogeneou s me dia . F o r homogeneou s i s ot ropi c

me di a t he l ine ari z ed e qu at ion of e l a s t ic wave mot ion i s

( At—p. ) W V a ) uvzu

Here u i s t he di s pl ac ement vector ( i n rect angul ar coordi nat e s ) , p

i s the den s ity of t he medium an d A and u are Lam é ' s con st ant s . A

more compl ic at e d equat i on hol d s for i nhomogeneou s me dia .

One s t art s wit h an a s ymptot ic s eri e s s olut ion o f t he form

00 A_ n

elcb ( S t )

where Arland S are fu ncti on s if x

, y and z and w i s t he an gu lar

f re que ncy o f the s o lut ion s s oug ht . I n th i s c a s e,o n e get s two diffe rent

26 El ect romagnet ic Theory and Geomet ric a l O pt ic s

e i c o n a l e quat ion s and two diff e re nt s et s of t ran s port equat ion s , one

f o r compre s s ional a nd one for t ran s ve rs e wave s ( becau s e the orig in al

dif fe re nt ia l e quat ion i s diffe re nt f rom Maxwel l ' s ) , but t he method of

obt a in ing the a symptot ic s erie s s olut ion s i s t hat s k etched above f o r

Maxwe l l 's e quat ion s . The " ray s " are the o rt hogonal t raj ectorie s to

the s olut ion s o f t he e i c o n a l e quation s .

Geomet ric a l o pt ic s a s t he s tudy o f di s cont inu it i e s ha s appl ic at ion

t o curre nt prob l em s of magnetohydrodynamic s . Here we are def i n it e ly

in the re alm of ani s ot ropic me dia . F or e l e ct rom agnet ic theo ry proper

the geomet ric a l o pt ic s o f an i s ot ropic me dia i s , a s in the c a s e of i s o

t ro pic me di a,t he t ran s port o f t he di s cont inu it ie s o f E and H F rom

thi s viewpoint we de rive f ir s t t he e i c o n a l e qu at ion o r t he Ham i ltoni an

a s it i s more commonly ca l l e d i n the c a s e of ani s ot ropic medi a . In

s uch me dia the ene rgy of the e l ect romagnet ic f ie l d doe s n o t propagate

a long t he norma l s to the wave front s but along d i s t i nct curve s c a l le d

ray s . The vari at ion of t he s e di s cont inu it ie s along the ray s a l s o s at i s

fy t ran s port e quat ions which prove to be f ir s t orde r ordi nary di ff e re nt i a l

e quat ion s .

The method o f e l ect romagnet ic s ha s been a ppl ie d to pl a smas . If one

approache s a pla s ma as a pe rfect ( n o n - vi s cou s ) com pre s s ibl e,inf in

i t e ly conduct ing f lu i d , o n e appl i e s t he e quat ion s o f f lu i d dynamic s and

e l ect rom agnet ic s . For weak s hock s t he e qu at ion s may be l ine ari ze d

and o n e obta in s four vector part i a l di ffe re nt i al e qu at ion s i n t he veloc ity

vector u the magnet ic f i e l d int en s ity H the den s ity p and t he

e nt ropy 8 per unit ma s s . A di s cont inu ity s urf ace i s o n e acro s s wh ic h

u H o r p i s di s cont i nuou s . F o r t he s e equ at ion s the s t udy o f t he

propagat ion o f the di s cont i nu it i e s le ads to t hre e fam i l i e s o f f ront s

( each wit h it s own s pee d ca l le d Alfve’

n,s low and fa s t ) in any one

normal di rect ion an d accord i ng ly three fami l i e s of rays . The s urf ac e

o f wave normal s i s according ly more compl icat e d than the F re s ne l s u r

face f o r c ry s t a l s .

\

I t i s the n po s s ib l e t o obt ain t ran s po r t e quat ion s for

e ach of t he di s cont inu it i e s along e ach fam i ly o f ray s . The res u lt s are

ext reme ly u s e fu l,f o r s uch s hock wave s can be ge ne rat e d” .

It i s a l s o po s s ibl e to apply the a s ymptot ic theo ry t o pe riodic wave s

of hi gh f re quency in pl a s ma s but t hi s ha s n o t bee n c arri e d out a s yet .

6 . SO M E O PEN PROB LEM S . The theory deve loped for Maxwe l l's e qua

t ion s pe rm it s u s to obt a i n u s efu l approximat e s o l ut ion s for t ime depen

de nt and for t ime harmonic probl em s provide d that t he corre s ponding

geomet ric al opt ic s approximat ion exi s t s,that i s

,i n prob lem s where

the wave front s and ray s o f c l a s s i ca l geomet ric opt ic s are de f ined .

Phy s ic a l ly th i s l im it at ion me an s a re s t rict ion t o pro pag at ion,ref l ect ion

and refract i on in homogeneou s and inhomogeneou s i s ot ropic and ani s o

t rop ic me dia . Even in t he s e phenomena,no c au s t ic s mus t be pre s ent .

I n view of the impo r tanc e o f di ffract ion phe nomena and in vi ew o f t he

diff icu lt i e s encount ere d i n s o lvi ng di ffract ion prob lems it would o f

cou r s e be highly de s i r abl e t o exte nd t he theory o f art ic l e 4 to cove r

Morri s Kl i ne 27

s uch prob l em s .

The fir s t di ff i cu lty o n e f ac e s i n att em pt i ng s uch an exte n s ion i s

t hat the t heory al re ady deve loped pre s u ppo s e s the exi s t enc e o f g e o

metric a l opti c s ; t hat i s , w e mu s t be abl e t o obt a in t he wave front s a s

s olut ion s o f t he e i c o n a l e quat ion and the i r o rt hogonal t raj ecto rie s,the

ray s . I n fact t he t ran s port e qu at ion s de s c ri be t he behavior of the c o

eff ic ie nt s o f t he a s ymptot ic s erie s a long the ray s . Ce rt ai n ly the n

when there are no wave front s and ray s,t he theo ry thu s far deve loped

ha s no meaning . Als o where the ray s form an enve lope or com e togeth e r

at a focu s,t he t ran s po rt e quat ion s bre ak down becau s e the pha s e func

t ion L)J ( X ,y,2 ) become s s ingu l ar . The fi rs t maj or s t e p in t he ext en s ion

of our t heory i s t o ext en d geomet ric a l opt ic s it s e lf . Th i s i de a ha s a l

re ady bee n t ack l e d by a number of men . It ha s be en deve lope d and

sy s t emat ica l ly handl e d by I . B . Ke l le r” w h o al s o s ugge s t s the un ify

ing princ i pl e t hat diffract e d ray s c an be obt a i ne d from an ext e n s ion of

F ermat ' s princ i pl e . Now F ermat 's pri nc i pl e for c l a s s ic a l g eomet ric a l

opt ic s i s de duc ibl e from Maxwel l ' s e quat ion s by the proce s s s k etche d

in art ic le 4 ( for t he very re a s on that geomet ric al Opt ic s i s de duc ibl e .

However,t he probl em rema in s a s to whethe r t he ext e nded F e rmat pri n

c i ple ,which e ncom pa s s e s ray s and wave front s n o t i n c l a s s ic a l geo

met ric a l opt ic s,c an be de duce d f rom Maxwe l l ' s e qu at ion s . The de

duct ion a l re a dy c arri e d o ut pre s uppo s e s E ( x , y , z , t ) H ( x ,y,z,t ) and

the i r s ucc e s s ive t ime derivat ive s have f init e di s cont inu it i e s on t he

wave front s . Thi s condit ion l im it s t he wave front s and ray s t o t hos e

o f geometric a l opt ic s .

Grant e d t he ext en s ion of geomet ric a l Opt ic s,the next s t ep i n dif

fract ion prob l em s i s to de rive t he form of t he a s ym ptot i c s e ri e s s olu

t ion which i s va l i d in di ffract ion reg ion s . The t heo ry al re a dy avai l ab l e

prove s that in t he c a s e o f pure propag at ion , re f l ect io n and re fract ion

t he form of t he a s ym ptot ic s eri e s i s th at o f a power s eri e s in l/o and

that t he s e rie s i s t ru ly a s ym ptot ic t o t he t ime harmon ic s o lut ion s o f

Maxwe l l ' s e qu at ion s . The corre s ponding s t e p i s m i s s ing for s eri e s

val i d in di ff ract ion reg ion s34

At the pre s ent writ ing a l l t hat we have

bee n ab l e to do i s t o a s s ume a form recommended by the a s ymptotic

expan s ion of s olut ion s obt a in e d i n an ent ire ly di ff e rent ma nne r . Thu s

the prob l em of di ffract ion by a c ircu l ar cyl i nde r c an be so lved and it s

s olut ion expanded a s ymptot ic a l ly . The form of t hi s a symptot ic s eri e s

or s ome gene ral i z at ion o f it ha s b ee n u s e d t o s o lve prob l em s involving

othe r s hape s .

The th ird s t e p wou l d be to l e arn how to dete rm ine the coeff ic i ent s

o f t he a symptot i c s e ri e s . If it s hou l d prove to be t he c a s e that t he s e

coeff ic i e nt s al s o s at i s fy t ran s port e quat ion s , t hen the in it i a l va lue s of

the s o lut ion s of the s e t ran s port e qu at ion s wou ld al s o have t o be

det e rmi ne d .

In view of the appl ic ab i l ity of t he t heo r y a lre ady deve lope d for e l ec

t r o m a g n e t i c s to many othe r b ranche s of phy s ic s,the prob l em s ju s t

s k e t c he d merit att ent io n . Though som e prog re s s ha s b een made beyond

28 E l ect romagnet ic Theo ry and Geomet ric a l Opt ic s

what wa s de s c ribe d in t he e arl i e r part s o f th i s pa per,the accompl i s h

ment s are not b road e nough to warrant att e nt ion i n t h i s s urvey .

F O O T NO TES

l . Huygen s,C . Tra it é de la L um i n e r e An Eng l i s h t ran s

lat ion i s avai l ab l e from the U n ive rs ity of“

Chic ago Pre s s,Chic ago

,

1 945 .

2 . Newton,1. O pt i ks ( 1704 ,

An Eng l i s h e dit ion i s avai l

ab l e f rom Dove r Publ ic at ion s , I nc . N . Y . 195 2 .

3 . Synge, I . L . and Conway

,W . The Mathemat ic a l Pape rs of

Wm . R . Hami lton,1,Cambri dge U n ivers ity Pre s s , London

,1931.

4 . A fu l l account o f the ve ry long s e ri e s o f e ffort s to deve lop an

e l a s t ic t heo ry o f l i ght i s g ive n by Wh itt aker,E . T . Hi s to ry o f t he

Theori e s of Aet her and E lect ric ity, V. 1, Re v . Ed . Thom as Ne l s on and

Son s,Lt d . London

,195 1.

5 . Kon ig,W . E le c t r o m a g n e t i s c h e L i c h t h e o r i e

,Handbuch der

Phy s ik,O l d E d .

,)O (

,p . 147

, I . Spri nge r,B e rl in

,1928 .

6 . Kirc hhoff ‘ s proof i s in An n . der Phy s ik 18,18 8 3

,p . 663

,

and in hi s Vorl e s unge n fiber Mathemat i s c he Phy s ik , T e ubn e r , Le i p z ig,

18 91, V. 2,p . 35 . An accou nt of it i s g ive n by KOn i g ,

W . E l ect ro

magnet i s che Li chttheorie,Handbuch de r Phy s ik

,O l d . E d . XX

,p . 167

ff . I . Springe r,B erl in

,1928 . An al te rnat ive proof whic h ut i l i ze s t he

t ran s format ion o f the Kirchhoff double inte gra l i nto t he i nc i dent f i e l d

plu s a l i ne int egra l i s g iven by B ak er,B . and C o p s o n ,

E . T . The

Mathem at i c al T heo ry of Huyg en s' Pri nc i ple

,Oxford U . Pre s s

,London

,

1939, p . 79, and Rub i n o w i c z ,A. D i e Be ug un g s w e lle i n de r Kirc hhoff

s che n Theori e de r B eugung,Wars aw

,1957, p . 166 ff .

7. Somme rf e l d,A . and Runge

, I . Anwendung de r Vektorrechnung

auf di e Grundlagen der geomet ri s chen O pt ik,Ann . der Phy s . 35

,191 1,

pp . 277- 298 . Al s o in Somme rf e ld,A. Opt ik

,2n d e d .

,Ak ademi s che

Verl ag s ge s e l l s chaft,Le i p z ig

,1959, p . 187 ff .

8 . F o r i s ot ropic me di a t he de rivat ion i s c arrie d out i n B orn,Max

,

and Wol f,E .

,Princ i p l e s of Opt ic s

,Perg amon Pre s s

,London

,1959,

p . 109. Re ferenc e s are g ive n t here t o orig in a l pape rs and to papers

i n whic h the analogou s proc e du re c an be employed in an i s ot ropic me di a .

9 . Actu al ly C lemen s Schaefe r i n hi s E i n f i i h r un g i n di e Theoret i s c he

Phy s ik,I I I

,1,p . 38 6 f f .

,W. D e Gruyt e r

,B erl in

,195 0 , avo id s t he

fre que ncy depe nde nc e at l e a s t in n o n - di s pe rs ive homogeneou s media

10 . Thi s re s u lt i s due t o R . K . Luneberg and can be found in Kl ine,

M .,An Asymptot ic So lut ion of Maxwe l l ' s E quat ion s

,Comm . on Pure

Morri s Kl i ne 29

and Appl . M a t h . 4, 1951, 225 - 262 . Als o i n Theo ry of E l ect romagnet ic

W ave s , A Sympo s ium ,Int e r s c i e nc e Pub .

, I n c . N . Y .,1951.

1 1 . W e co nt i nue to u s e tw o s pa ce vari ab l e s i n orde r t o i l lu s t rate

g eomet ric a l ly . However a l l s t at ement s apply w hen three s pac e var

i a ble s are pre s ent .

1 2 . The s e ri e s exi s t a n d converg e for va lu e s of t t o but do not

re pre s e nt t h e f ie l d .

1 3 . Kl ine,M .

, lo c . c i t .

14 . St ric t ly th e ident i fic at ion of t he coeff ic i e nt s of t he a symptot ic

s erie s w it h t he di s cont inu it i e s of t he t ime de pe ndent f i e l d pre s up po s e s

that t he s ame s ou rc e,s ay g ( x , y ,

'

z ) c re at e s both f i e l ds but t hat t he

t ime be hav ior of t he s ourc e i s e’ w t i n one ca s e an d i s n ( t ) the

Heavi s i de unit fu ncti on,i n t he ot her . Thi s fact c an be ig nore d in

s ome appl ic at ion s an d i s he l pfu l in othe rs .

The exi s t e nc e of t he a symptot i c s erie s ( 1 1) a n d ( 12 ) a n d t he re la

t ion s ( 13 ) are due to R . K . Luneb e rg . An expo s it ion c an b e found in

Kl i ne,M .

,lo c . c i t .

,1951.

1 5 . Kl i ne,M . Asymptot ic So lut ion of Line a r Hy perbo l ic Par ti a l

16. Lewis,Robe rt M . Asymptot ic Expa n s ion o f St e a dy - St at e So lu

t ion s of Symmetric Hype rbo l ic Li ne a r D i ff e rent i a l E quat ion s , I o u r . of

Ma t h . 8:Mech . ,7, 1958 , 593 - 628

17. Ke l l e r, I . B . Lewi s,R . M . a n d Se c k le r

,B . D . Asymptot ic

So lut ion of Some D if fract ion Prob l em s , Comm . on Pure and Appl . Math .

1 8 . Sc h e n s t e d , Cra ig E . E le c t romagnet ic and Acou s t ic Sc att ering

by a Sem i - Inf i n it e Body of Revo lution, Jour . Appl . Phy s . 26

,195 5 ,

306 - 8 .

19. Wh itt ak e r, E . T . His tory of the Theori e s o f Aether an d E l ec

t r i c i ty ,I I

,p . 270

,Thomas Ne l s on and Son s

,Lt d .

,London

,1953 .

20 . B irk hoff,Georg e D . Some remark s c onc e rn i ng Sc hrodinger 's

Wave Equa t ion,Proc . Nat ' l . Ac a d . of Sc i e nc e s

,19, 1933, 339- 344,

p . 475 ; Qu a ntum Mechan ic s and Asym ptoti c Se ri e s , Bu l l . A. M . S .

39, 1933 , 681- 700 ; The F oundat ion s of Quantum Mechan ic s , Compt e s

Rendu s d u Cong re s Int ernat . de s Math . O s lo,1936, 207- 225 . All of

Soc Provi de nc e,1950 .

21 .B i rk hoff

,Georg e D . On the Asymptot ic Charact er of the So lu

t ion s of Cert a in D if fe rent i a l Equat ion s Conta i ni ng a Parameter , Tran s .

Am er . Math . Soc . 9, 1908 , 219—231 .

30 E lect rom agnet ic Theo ry and Geometrica l Optic s

2 2 . The s e e quat ions wil l b e found in Kl i ne,M . lo c . c it . 1954 .

23 . Fo r a good de s cri pt ion of t he WKB method,s e e Kamke

,E .

Che l s e a Pub . C o .,N . Y .

,1948 , p . 138 . See al s o p . 276 .

24 . Ke l le r, I o s e ph B . Correct e d Bohr So m m e r f e ld

‘

Qu a n t um C o n

d i t i o n s for Non s e parab l e Sy s tem s,An nal s o f Phy s ic s

,4,1958 , 18 0

- 18 8 .

25 . Ke l le r, I . B . and Rubi now

,S . I . Asymptot ic So lut ion of E ig en

value Prob lem s,Annal s o f Phy s ic s

, 9, 1960 , pp . 24 - 75,and 10

,1960 ,

303 - 305 .

26. Ke l l e r, I . B . Geometric a l Acou s t ic s I . The Theory o f Weak

Shock Wave s, I . Appl . Phy s:

,25

,1954 , 938 - 947. Als o F rie dric h s

K .,and Kel le r

, I . B .,Geomet ric a l Acou s t ic s I I

, I . Appl . Phy s .,26

,

195 5 , 961- 966 . Wh it ham , G . B . On the Propag at ion of We ak Shock

Wave s, I o F lu id Mec h .

,1,1956, 290 - 318 .

27. F rie dl ande r,F . G .

,Sound Pu l s e s

,Cambri dge U niv . Pre s s

,

London,1958 .

28 . Se e F rie dric hs and Ke l le r,lo c . c it .

29. Lowe l l,Sherman C . The Pro pag at ion o f Wave s in Sha l low

Wate r,Comm . on Pure and Appl . Math .

,2,1949, pp . 275 - 291.

30 . To handle t he t ime periodic c a s e,Lowe l l re duc e d t o an ordi na ry

diffe re nt i a l e quat ion and appl ie d the WKB met hod . We now know that

t he approximat e s o lut i on obta ine d by thi s method i s the f irs t t e rm o f

our a symptot i c s e rie s .

3 1 . Kara l,F rank C . I r . and Ke l le r

, I o s e ph B . El a s t ic Wave

Propagat ion in Homogeneou s and Inhomogeneou s Medi a, I . Ac o u s .

3 2 . B a zer, I . and F l e i s c hman

,O . Pro pagat ion o f Weak Hydro

magnet ic D i s cont inu it i e s,Phy s ic s of F lu i ds

,2,1959, 366—378 .

33 . Ke l le r ' s work and the b e s t s i ng l e di s cu s s ion o f di ff racte d ray s

wi l l b e found in Ke l le r, I . B . A Geomet ric al Theory o f D iff ract ion . Thi s

pape r i s in Grave s,L . M . ed . Calcu lu s o f Vari at ion s and it s Appl ica

t ion s , Proc . o f Sym pos ia i n Appl . Math . VI I I , M c G r a w - Hi l l Book C o .

N . Y . 195 8 , pp . 27- 5 2 . Thi s pape r al s o cont a i n s refe renc e s to o ther

work on diffract ed ray s and g ive s appl icati on s .

34 . In a number of appl ic at i on s al re ady made o f t he a sym ptot ic

s eri e s val i d f o r propag at ion,re f l ect ion and refract ion

,t he aut hors have

00 na s s ume d the ex i s tenc e of a s e r ie s Z

n—O

vn ( x , y ,

z )/k o r s ome more

genera l form and have s ub s t itut ed in Maxwe l l ' s e qu at ion s or t he r e

d u c e d s ca lar wave equat ion . Thi s proc edu re i s j u s t i f i e d only a s a

convenienc e in pape rs which s eek to avoi d matt ers o f t heo ry and wi s h