Ho w Y o r k Univ e r s i ty
W
ashi ngt o n
S quar
e C ol l eg e
1 ath e m at i c s R e s e arc h
~
Gr oup
R e s ea r ch R ep o r t N o . EM- l }
un de r
C ontrac t N o .
DETERM I N AT I ON
FI EL DS
J o s eph
3 . K e ll e r
an d
H e rber fi E Kel l e r
.
W
ri t t en by :
T i t l e pa g e
M o r r i s Kl ine
P ro j
e c t Di re c t o r
l 6 Numb e r e d
A pr il
,
l9
n9
pag e s
C O NT EN T S
Ar
ticle
Pag e
Ab s t rac t
1
.
I nt r o duc t i o n
m Fo rmul a t i on o f the Pr o bl em
W
C al cul a t i o n o f th e
’
D i s cus s i o n
L
J ac ob i an
o f th e S olu t i o n
A pp e ndi x
AB STRAC T
By an ext e ns i o n o f o r di nary ge om e t ri cal op t i c s
( o r ac ou s t i c s ) t he i n t e ns i ty o f t he r e fl e c t e d an d t ran s mi t t e d
.
f i el d s
due t o a p o i n t s ou rc e i n th e p re s enc e o f
an arb i t rary i nt er fac e
b e t w e e n tw o
.
me di a i s found
.
A
p ar t i cul ar c o ns e que nc e o f th e s o lu t i on i s t he gene r al
l ens and
s u rfac e s
m i r r or
.
l aw
and th e e qua t i ons fo r t he cau s t i c
1
.
I nt r o duc t i o n
The cal cul a t i o n o f t he e l ec t romagne t i c o r ac ou s t i c f i e l d at any
W hen
p o i n t i n sp ac e
M axwe l l
t io n o f
'
tw o di ffe r en t m e di a ar e
t ai ne d
re s e n t ne c e s s i t at e s a s olu -
s e quat i o ns o r th e wav e e q uat i o n o f ac ou s t i c s w i t h app r o
b ounda ry c ondi t i ons i n ea c h c as e
p riat e
p
o nl y f o r t h e s imp l e s t
c o nf
.
Exac t s olu t i ons hav e b e en
i gur a t i o ns
l°2
¢
ob-
and appr ox im at e s o lu t i o ns
hav e b e e n f ound by sp e c i al m ean s i n s ome oth e r c as e s ( e
u
3
m e di a i s an arb i t r ar i l y curv e d thi n s h e l l
)
.
g
.
wh en o ne of t he
’
.
A
‘
g e ne r al p r oc e dur e f o r o b t aini ng an app r o x imat e s o lu t i o n t o an
el e c t r omagn e t i c o r ac ou s t ic p rob l e m i s th e m e th o d o f g e ome t r ic al op t i c s
or ac ous t i c s
.
By thi s i t is m eant t ha t t he f i e l d q uan t i t i e s p r op agat e
al o ng ray s wh i ch ar e de t e rm i n e d by t h e F e rma t p r i nc ip l e o f l e as t t im e
and t hat th e s e ray s ob ey
t he
,
law s o f r ef l e c t i o n an d r e frac t i o n at th e
int e r f ac e b e t we e n t wo di f fere n t me d i a
Fur t he rm o r e , i n t hi s me t h o d i t
6
t
hems
elv
e
s
s
i s a l s o p o s s ib l e t o de t e rmi ne t he f ie l d c omp o ne n t s
s ome .
!
t h in g wh i c h has no t o r dina r i ly b e en do ne in g e om e t r i c al op t i c s o r ac ous t i c s
Thu s
,
fo r exampl e , th e r e fl e c t e d
an
d
t rans m i t t e d fi e l d co mp o ne nt s at an
int e r fac e ar e r e l a t e d t o t he inc i de n t f i e l d c omp o ne n t s by t h e w e l l
Fre s ne l
f o rmul a s
o r t he c o rr e sp o ndi ng fo rmul ae i n ac ous t ic s
Als o
.
known
the
f i e l d c omp o nent s v ary al ong a ray i nv e r s ely a s th e s quar e r o o t o f the
v e l o c i t y an d o f t he ar e a of t he normal c r o s s s e c t i on o f an i nf i ni t e s ima l
t ub e o f ray s c o nt ain i ng t he ray in qu e s t i on
de t e rmi ne t he ge ome t r i c al
O
p t i c s or
a c oi s t i c s
b e exp e c t e d t ha t th i s s o lu t i o n wi l l b e
ful l s o lu t i on
c omp ar e d
to
.
an
fiel d
c o mp
l e t e ly
j
¢
-
I t is t o
ade q uat e app ro x imat i on t o th e
Ly at v e ry hi gh fr e q u enc i e s ( i
on
T he s e r e sul t s e nab l e o ne t o
.
s
.
a t wav el e n th s sma l l
g
th e d im e ns i ons o f th e
As a ma t t er o f fac t R K Luneb e rg has sh own qui t e ge ne r al ly that the
fi r s t t e rm s i n a symp t o t i c exp ans i o ns o f the s p ac e dep ende nc e o f the
e l e c t r i c and magnet i c f i e l d v e ct o r s i s tha t g iv e n by g e o m e t r i c al op t i c s
T h i s ma t e r ial wi l l appe ar i n a f o r t hc omi ng r ep o r t
.
.
.
"Fh a s e i s de t erm in e d by the p a t h l e ng t h al ong a ray
.
.
.
I n t h e p r e s e n t i nve s t iga t i on an i nc i den t f i e l d du e t o a po int
s ourc e i s a s sume d t o imp i n e up o n an arb i t rar i l y curv e d or f lat i nt erfac e
g
whi c h s ep arat e s t w o h om og ene ous and i s o t r op i c m e dia
t he r e f l e c t e d and t r ans m i t t e d f i e l d c o mp o ne nt s a t
mi ne d
oh
t he ba s i s o f g e om et r i c al
out l i ne ab ov e
O
i.
p t ics ,
s
.
.
any
g
p o i n t ar e det er -
by ap ply ing t h e th e ory
F o r th e c as e o f r e f l e c t i on th e p r obl em
.
The ma ni t u de s of
al r e ady b e e n
h as
’ 7
3
s o lv e d
but i t i s i nc lu de d h e r e b e caus e th i s can b e do ne wi th no addi ,
t i o nal
di f f i cul ty
T h e t rans mi s s i o n pr o bl em has b e en t r e at e d by the
8
3
0
an d the p r e s en t cal cul a t i on c ons t i t ut e s a ch e c k on
K i rchho ff me t ho d
.
Th e ch e c k f o r t h e
t ha t s o lut i o n
.
K i r chho ff
me t h o d appl i e d t o r e f l e c t i o n
i s c ont a i n e d he re as w e l l as i n r e f er e nc e 3
.
2 . F o r mul a t i o n
.
o
f
A p o i nt
.
th e P ro bl em
s ou r c e i s a s sum e d t o b e l o ca t e d at a p o in t
in a h om og ene ou s i s o t r op i c me dium w i th p r opagat i on sp e ed V
I
(
a z
2
m e d i um
,
y)
s epar at e s t hi s
m e di um
wi t h p r op agat i on sp e e d V
T
.
The
s
urfac e
fr o m a di ff e r en t h om oge ne ous i s o t rop i c
.
Th e magni t u de s o f th e r ef l e c t e d an d t rans m i t t e d f i el d c omp one nt s
at t he su r fac e c an b e fmund i n t e rm s o f the inc i dent fi el d c omp on ent s and
the F r e s ne l
.
f o rmul ae ( onl y f o r t ho s e sur fac e p o i nt s which c an b e c onnec t e d
t o th e s our c e by a s t r ai gh t l i ne s e gm ent ly ing in the fi rs t me di um )
Let
.
r ep re s en t t h e amp l i t ude o f any f i e l d c omp one nt at the p o int
due t o re fl e c t i on o r t r ans m i s s i o n
5
ge om e t r i cal p t i c s we h av e th e rel at i on
.
f r om
th e surfac e
.
Th en fr om
o
wh e r e
t hr ou gh
i s t he p o i nt in wh i ch a r e fl e c t e d o r t ran smi t t e d
i nt e r s e c t s t he
s
ur fac e
2
y
ra
is
'
the r e fl e c t e d or t rans mi t t e d f i e l d c omp one n t a t th e sur fac e , whe r e i
for r e fl e c t i on an d i
cul a t e d,
2
f or t ran s m i s s i on
.
l
Th e s e c ompo ne nt s c an b e cal -
as p r ev i ous ly s t at e d , fr om t he i nc i de nt f i el d c omp on ent at t he
sur fa c e
.
d cr
Th e quan t i t y
i s t h e ar e a i n wh ich an inf i ni t e s imal tub e o f
r e fl e c t e d o r t r ans mi t t e d rays c ont a i ni ng t he
cut s a pl an e p no rmal t o thi s
r ay
r ay
thr ough
and
a t th e p o int
( see
I
Fi g
'
p
d
.
O
is
‘
t he ar ea enc l o s e d by thi s s ame tub e o f r ay s o n a p lane
y
no rmal t o th e
ra
o f i nf i ni t e s imal
ar
i n qu e s t i o n a t t he p o i nt
dw
e a s th e rat i o
/ dG i s
I n t h e l imi t
Jus t t he Jac ob i an , J (
of
th e t ran s f o rmat i o n e s t ab l i sh e d by r e fl e t e d o r t r ans mi t t e d rays , whi ch
c
map s th e p lan e
p
'
i nt o t h e pl ane p
M am /
.
3
.
C al cul at i on
.2
)
J
1
H enc e
.
/2
we may w r i t e
a
o f th e J ac ob i an
un
t
be a
v e c t o r no rmal t o
i
Let
po i nt i ng i nt o t he f i r s t medium ;
fr om
r ay
at
t r ans m i t t e d
r ay
fl e c t i o n an d r e f rac t i o n
!s
that o f th e
(2)
T he dir e c t i on o f t h e
i s g iv e n by th e
.
4
m
ee
i 1
'
un i t
by
'
3
)
I
an d
i s th e uni t v e c t o r p o int ing
t o t he s our c e at
fl e c t e d
z '
vec tor
r e-
and
whe r e , fr om t h e law s o f
A pp end ix
1
1
'
r e-
]
(l
a
o
5
t
)
w
e“
v
)
2
t
If
.
(1
sin
He r e
n
mi s s i o n
.
e! 1
i
l f or r e fl e c t i on
,
i
2
f or r e frac t i on o r
The angl e s and uni t v e c t o r s ar e s h own i n Fig
.
l
.
t r ans -
-e
T he uni t
I and N hav e t he c omp on ent s
v e c t o rs
.
:
y
3.
y
e
z
2 ‘
( 3)
3
--L
E( x '
A
4
o
2 3
—
i :
0
whe r e :
2
t
i (
x t
,
y
I
u
2
.
(Kl
-X
?
V
‘
4
( 21
'
)
z .
2
and
2
The c o o r di na t e
m e d ium
sy s t e m
2
mus t b e s o ch o s e n tha t N p o i nt s int o the fi r s t
.
Fo r
o n t he sur fac e t he e quat i on o f
arb i t rary po i nt
t he r e fl e c t e d o r t rans m i t t e d
x'
x
1
thr ough th i s p o i nt i s
r ay
a
y
.
1
T
x
-
y
'
2 ‘
z
1
T
y
T
2
and thu s :
( h)
1
y
y
(z
'
z '
T
y
)
(
xi
’
y
t
g
z l
i
whe r e t he subs c r ip t s i ndi cat e c omp o ne nt s o f t he v e c t o r
F o r a f ix e d value o f
fac e
Fi g
8
g iv e n
by
e quat io ns ( h) ar e t h e
o nt o t he pl ane n
'
,
mapp i ng
g iv en by
z
=
~
o f th e sur -
c o ns t ( s e e
F rom th e s e e quat i on s w e c an c omput e th e J ac ob i an o f th i s
.
fo rmat i o n
(5)
.
z '
2
T
,
nam el y
,
"
-l !
)
3
8
e
8 y
x
x
'
57
?
ax
e?
"
-1
t rans -
F ig
( T ran s fo rmat i o ns f o r T ran smi s s i o n
2
.
Th e J ac o bi an wh ic h o c cur s in e qua t i o n ( 1 ) c an no w b e cal cul at e d s i nc e
a
a
n n
g
l
a
M
whe r e n “ i s a p l ane t hrou gh
Ja
)
g
g
and pa rall e l t o
t he n o rmal p lan e s p r ev i ou sly d ef i ne d and th e
.
n,
p and
p
no t a t i o n r ep r e s e nt s
.
'
ar e
t he
Jac o b i ans o f t he t rans f o rmat i o ns b e t we e n ind i c at e d su rfac e s de t e rmi ne d
I
b y me ans o f t he r e fl e c t e d or
t r an sm i t
t e d r ay s
J
.
P
s inc e th e p l ane s ar e par al l el by p a i rs an d t hu s c o r r e sp ondi n
g
i n de f in i ng th e t rans f o rmat i ons are e qual
J( L
p
N ow
t he
z - az
the
x-
is
and
T
I
)
I
Thu s w e ha v e :
8
.
..
p l ac e t he o r i g i n o f c o or di na t e s at t h e p o i nt
no rmal t o
S
and
p o s i t iv e
.
Th e p l ane
t ang e nt pl ane t o S a t t h e
p e nde d i nt o
a
.
Tay l o r
.
nd
‘
ne w
int o t he f i r s t
wi th
Furt he r le t
9
- e xe s b e p arall e l t o th e di re c t i o ns o f p ri nc ipal curv atur e
y
a t th i s p o i nt
(6)
u
.
ang l e s us e d
n ow b e c ome s the
o rig i n
.
xy p
me dium
.
l ane ( z =O ) and i s
The e quat i o n o f th e sur fac e
s e r i e s a round t he o r ig i n b e c om e s 9
th e
5
ex-
2a
whe r e
and 2b ar e t h e p r inc ip al cu rv a tur e s o f t he surfac e at th e o r igin
and th e p r im e s de no t e t h e c o o r dinat e s o f a sur fac e p o int r e l at iv e t o th e
new c o o r di na t e sy s t em
.
7
79
I n th e s e c o o rdi nat e s
r e fl e c t e d o r t ran smi t t e d r ays
4
i s th e Jac o b an o f th e
i
o f th e sur f ac e o n t h e
,
JQ’
mapp i n g
p l ane
,
by
T hi s
.
mapp i ng i s giv e n by
x = x'
2 '
t an
(7
)
y
an d
whe r e
y
P
(
an d
e quat i on s ( 6 ) and ( 7
) we
a
x'
c an
y
e
3
'
t an
k
)
y
b
h
q
'
t he ang l e s b e t w e e n a ray r e fl e c t e d ( o r
th e x- and y- ax e s r e sp e c t iv e ly
Fr om
ar e
t ran s mi t t e d ) at
J
=
a y
.
,
comput e
az
’
_
'
,
8 y
e r
“
’
'
fi nd th at , at th e o ri g i n ( L a 32 '
O ) , the ab ov e Jac ob ian
z
y
i s e qual t o o ne
U s i n t hi s r e s ul t i n the p r e c e di ng expr es s i on fo r
W
e
.
g
.
t
w e o bt ai n
P
wi t h t he
cho s en as ab ov e w e may c al cul at e
i n e quat i o n
D I
J ( a_
p
.
S
1
T}:
8
z
TI
(1
T
T2
0
2
2
-1
iT
a
(f
"
8 3
$
l
)
2
wh e r e t he p ar t ial de r iv at ive s
po ne nt e o f th e
o r i gi n ar e :
by us ing e quat i o ns
t h e ab ov e J ac o b i an :
1
J(
,
v e c t or
1
T
an d
a
"
a )
iT
8
X
“
ar e
8
2 *
iT
y
1 T
t o b e ev al ua t e d a t the or igi n
.
T he c om-
t he i r part i al de r iv a t iv e s evaluat e d at the
( 8 ) b e c om e s
p
J( _
p
%
2
‘
z
-
cos a
2< i
n
«
2
’
s
D
1
1
ax
2(
2
by
l
2
cos
1
n
2
2
2
1-
'
5-
by
f
am- 1
D
1
+ Lah
e
ul
cos
y
.
‘
D
1
cos
a
)
i l
L
2
n
)B
D
1
a1
m
-1
i
2
D
1
2“
:
0(
2( 1
Z
m ay
C(
;
2
Th i s e quat i on
2
2.
D
c os
l
f)
2
(x
i
4 W
'
A
2
D
E
l
1
b e exp r e s s e d in t e rm s o f g e om e t r i c p rop er t i e s o f th e sur -
G
c
ho i c e
= ltab
g
2)
Gll
2(
2
ax i
2G
n
G
1 1
by
t
?
2
2‘
th e di s tan c e s inv o lv e d ar e
2
s
1
!- a
y
l
2
l
,
al r e a
dy
z
Y
‘
1
s in
2
2
e
D
1
D
1
s in
D
1
13
and
)2
2
fac e an d di s t an c e s whi ch ar e indep e nde nt o f th e
9
Th e s e quant i t i e s ar e
(1
m
e( i - l
2
g iv e n
‘
f
,
and
of
c o or dinat e s
.
G an d G ar e r e sp e c t iv e ly t he m ean
m
u
g
o f t he
ature
Gl l i s th e cu r vatur e o f
u rfac e at th e p o int i nv o lv e d ;
s
in a p lan e c on t ai ni ng t he inc i de nt r ay
thi s p o i nt ;
D1
t he su rf ac e ,
and
t he Gau s s i an
an d
c rv
i s the di s t an c e al o n
an d
urvatur e
t he
sur fac e
t he no rma l t o th e su rf ac e at
t he i nc i de nt
g
c
fr om t he s ourc e t o
r ay
D i s t he di s tanc e al o ng a r e fl e c t e d o r t r an smi t t e d r am
fr om th e sur f ac e t o any p o int
on t hi s r ay
I n t e rm s o f t h e s e
r
.
i nvar i an t s e quat i o n ( l l ) b e c ome s
“ E
1
1
.
P
l
D
'
COS
9
2
v
( 29
'
c os
2
0
4
G31 tan
m
2
0
<
)
(
i
i l
u
cos
fi
‘
i-
c os
(X
i
( I n)
2
D
2
n
O
2
D
l
( 2Gm
c os
z
G11
4
'
r
c os
cos
g
wi t h t h e o r igi n at
OS
'
t an l )
G
N ow
2
any
!
t
4
c oe
(
n
un c ha ng e d
l
cos
r
'
t c o so<
f inal ly l e t t he
.
Equat i on
i n t he s e c o o r d inat e s p r ov i de d tha t
i
=
D
<
x -
1
x
0
( yl
-
y
o
( 21
-
w
e
(1 5 )
2
r
D
( 2c os
( If- 3' 0
2
v
2
(x
i
I f e qua t i on ( l l-L) i s us e d i n e quat i o n ( 1 ) the re fl e c t e d or
transm i t t e d f i e l d amp l i tu de i s giv en by :
(1 6)
D
1
p o int i n s pac e , l e t t h e s our c e be at
i n th e new c o or di na t e s
2
)
i
n
d
l e t a p o int o n t h e sur f ac e b e
ob s e rv at i o n p o int b e
1
is
1 0
s
i nc e w e c an exp r e s s
( h)
1
and y i n t e rm s o f
z '
,
y
‘
and
2
by
I n e quat i o n ( 1 6 ) J i s a func t i o na l symb o l f o r th e
.
m e ans o f e quat i ons
e xp r e s s
i on
.
on
t he
r i ght s i de o f e quat i o n ( 1 h) w i t h t he chang e s no t e d in
M
.
S o lut i o n
D i s cus s i on o f t he
"
l ev e l
I t i s o f int e r e s t t o inv e s t i gat e the
sur f a c e s
B
,
tha t i s
sur fac e s o n wh i ch a r e f l e c t e d o r t r an s mi t t e d f i e l d c ompone n t has a co ns t ant val ue
T o f ind such
.
su
r fac e s w e r e qui r e that
c o ast
Ea
W
in e qu at i o n
(1
e
th en s olv e th e r e sul t i ng e qua t i o n f or D an d o b tai n :
m
a n
7)
K
wh e r e
H
( i h)
Eq ua t i on s
.
an d
.
n am
e
ar e th e c o e ffi c i e nt s o f D and D
do ub l
x'
an d y '
.
e s ign i n e qua t i o n ( 1
and ar e o f spe c i al imp o r t anc e
E p- i ch
;
r e s p e c t iv ely i n e quat i on
F o r e a ch v alu e o f
.
7)
Th o s e l ev e l sur fac e s o n whi ch
l e t t i ng
e
) ar e e quat i on s f o r the l ev e l sur fac e s i n
( h) and ( 1 7
o f t he p arame t e r s
fac e s a s t he
.
.
E
B“
t
t her e ar e tw o such
in di c at e s
e rms
s ur -
.
i s i n f ini t e ar e c al l e d caus t i c s
Th e i r e quat i on s a r e e as ily o bt ai n ed by
i n th e e qu at i on s f o r th e l ev e l sur fac e s
.
T he di s t anc e
from a c au s t ic t o the sur fac e along a r e fl e c t e d o r t ran smi t t e d ray i s
us e ful
(1
.
T hi s di s t anc e i s giv e n by ( 1
H
8)
t hu s e ach r e fl e c t e d or t ransm i t t e d
7) wi th
E
1
co and i s :
2
r ay
i nt e r e e d s
t wo c aus t i c s
.
T h e two p o i n t s al ong a r e f l e c t e d o r t ransm i t t e d r ay at wh i ch
00
ar e c al l e d co n j ug at e p o i nt s
.
I f t he c aus t i c surf ac e s
int e r s e c t t h en t h e c o nj ugat e p o i nt s o n t he ray s thr ough th e int er s ec ti on
r
2) Y
-
0
c8
,
then
n;
2
2
G and t he p o int o f t ran sm i s s i on
m
g
i s an umb il i c al p o int
From e quat i on ( 1 8 ) t h e di s tanc e
.
i s g iv e n by
1
‘1
D
D
1
I n al l o f the ab ov e ca s e s th e d i s t anc e t o th e imag e i s exp re s s e d by the
ge ne ral fo rmul a
( 20
SI
)
1
’
(
l
-l
)
1
]
n
e
/
ogl
I f th e sour c e i s a t an inf i n—
whe r e t h e s ign t o b e ch o s en i s tha t o f G
i t e d i s t anc e fr om th e re fl ec t ing o r t rans m i t t i n g sur fac e , th e dis t anc e of
a r e sul t i ng p o int image ( whi ch i s c al l e d a fo c al p o i nt ) fr om the sur fac e
i s cal l e d th e f o c al l e n t h
f , and i s gi ven by e qua t i o n ( 20 ) wi th
g
1
( 21 )
,
i
1
_
g
]
n
D = <n
l
,
—
c
g
From e quat i on s ( 20 ) an d ( 21 ) w e ob t a in th e c omb in e d l ens and mi rr o r law :
( 22 )
1
ano t he r app l i c at i o n
As
to
a
sp he re
by van de r
.
Th e n
Po l
fr om t h e ear th
a c c oun t ,
t hei r
an d
.
.
W
,
we sp ec ial i z e the re fl e c t ing sur fac e
e q ua t i o n
( 1 H) y i e l ds the g e o me t r i cal fa c t o r ob tai ne d
B r e mm e r
or
,
1
%
t he r e fl e c t i o n o f an e l e c t ro magne t i c wav e
h e n t he Fre s ne l f o rmul a e an d t he pha s e ar e t ake n int o
c omp l e t e r e sul t ( f o r wav e l engths smal l c ompar e d t o t he
ear th ' s ra di u s ) i s o b t ai ned
.
APPE NDI X
To der iv e e qua t i o n ( 2 ) o f t h e t ex t p r op e r
v e c t o r s an d angl e s s h own in F ig
v e c t o r s and angl e s
we
1
.
we us e
'
the un it
From t h e de f i ni t i ons o f t h e s e
.
r e q ui r e :
2
3
4
N
2
1
:
2
3
l
(1 )
By th e l aw o f re fl e c t i on
an d fr om S ne l l ‘ s l aw n
( 3)
N
2
l
n
!l
4
wh e r e w e al way s t a ke t he p o s i t iv e s quar e r o o t i n
Eq
.
( 33
.
S i nc e th e
re-
fl e e t e d o r t rans m i t t e d ray mus t l i e i n t he p lane o f t he no rmal t o the
surfac e an d the i nc i de nt
r ay
( by t h e
l aw
o f r e f l e c t i o n and Sn e ll ' s law )
we hav e
e
A
l
B
l
-1
A
and
= 2 I
<
o
N)
l
13
1
= +1
=
=o
The s e c o nd s e t of c o e ff i c i en t s r ep r o duc e s t he i nc i dent ray an d s o mus t b e
di s c ar de d
(5)
.
Then t h e m os t gen er al
"
17
3
T v ec t o r i s g iv e n by :
M.
I f e qua t i o n ( M) i s u s e d in e quat i o n s ( 1 ) an d ( 3)
:
-
2
B
g
e
an d
n
n(
Y
1
.
i
r
we
=
n
a
_n I
(
B
The s e c o n d s e t o f c o e f f i c i en t s y i e l d a t r ans mi t t e d
s am e s i de o f t he n ormal a s th e inc i dent ray
di ct i o n t o th e
U s ing
l aw
o f r e frac t i on
and
ge t th e s o lut i ons
o
r ay
H ow ev e r ,
.
(6)
=
n1
-
(f
+
o
is
th i s i s in co nt r a-
s o t hi s s o lu t i on mus t b e ab andone d
t h e f ir s t s e t o f c o e ff i c i en t s g iv e s t h e g e ne ral
aI
whi c h l i e s on t he
2
.
’
T ve c t o r a s :
-
E q ua t i on s ( 5 ) and ( 6 ) c an now b e wr i t t e n i n t h e c omb ined fo rm :
a
T
-
(n
p
p
.
9
1 m
I
whi ch i s th e de s i r e d e q ua t i on
.
g“
«
f
(
)
Thi s c an al s o b e e xp re s s e d by th e s impl e
.
(8)
A
H
e xp r e s s i on :
-n
9
(n
co s Y
'
t-
c os
o<
Kell
er
D e t e rm i na t i o n
o
f
re fl e
c ted
8
’
s
H
u
A UTHO R
O
D e t e rm i n a t i o n
O
H
V
3
3
W
S
1
1
L
E
m
B
'
o
f
e c ted
re f l
T IT L E
e
u
p
a
x
n
o
a
p
u
p
d
e
l
N
"
n
m
a
a
m
a
m
g
m
5
9
m
2 3 19
Y U
.
M
Institute
.
athe m atical
4W
a shingto n
Plac e
N e w Y o rk 3, N
.
Y
.
of