Revista Brasileira de Física, Vol. 11, NP 4, 1981
Polarization of Electron and Photon Beams
?. R. S. GOMES
Instituto de Física, Universidade Federal Fluminense, Niterdi, RJ.
Recebido eni 17 de Fevereiro de 1981
I n t h i s work t h e concepts o f p o l a r i z a t i o n o f photons and r e lativistic
electrons
a r e i n t r o d u c e d i n a s i m p l i f i e d form.
Although
t h e r e a r e i m p o r t a n t d i f f e r e n c e s i n t h e concepts o f photon and e l e c t r o n
polarization,
i t i s shown t h a t t h e saine f o r m a l ism may be used t o
c r i b e them, w h i c h i s v e r y u s e f u l i n t h e s t u d y oF i n t e r a c t i o n s
ned w i t h p o l a r i z a t i o n s o f b o t h
photons and e l e c t r o n s , as i n t h e
t o e l e c t r i c and Compton e f f e c t s .
Photons a r e c o n s i d e r e d , i n t h i s
as a s p e c i a l case o f r e l a t i v i s t i c s p i n 1 p a r t i c l e s ,
des-
conceryhowork,
rest
w i t h zero
mass.
Neste t r a b a l h o são apresentados, de uma forma
simplificada,
os c o n c e i t o s de p o l a r i z a ç ã o de f o t o n s e e l e t r o n s . Embora e x i s t a m
im-
p o r t a n t e s d i f e r e n ç a s nos c o n c e i t o s de p o l a r i z a ç a o de f o t 0 n s . e e l e t r o n s ,
mostra- se que o mesmo f o r m a l i s m o pode s e r usado p a r a d e s c r e v e - l o s ,
que é m u i t o ú t i l no e s t u d o de i n t e r a ç õ e s envolvendo
polarizações
o
de
ambos, f o t o n s e e l e t r o n s , como em e f e i t o s f o t o e l é t r i c o e Compton. Nest e t r a b a l h o os f o t o n s são c o n s i d e r a d o s como um caso e s p e c i a l de p a r t í c u l a s r e l a t i v í s t i c a s de s p i n 1 , com massa de repouso n u l a .
1. INTRODUCTION
Dur i n g a r e s e a r c h progi-am concerned x i t h c o r r e l a t i o n s
p o l a r i z ã t i o n s o f photons and e l e c t r o r ~ s i n r e l a t i v i s t i c
effect,
between
photoelectric
i t was n o t i c e d t h e l a c k o f a t e x t w h i c h b r i n g s t o g e t h e r t h e c o n -
c e p t s and s i m p l i f i e d d e s c r i p t i o n s o f p o l a r i z a t i o n o f e l e c t r o n and
t o n beams
.
That was t h e reason f o r t r y i n g t o w r i t e such a t e x t .
pho-
T h i s work s t a r t s w i t h t h e non r e l a t i v i s t i c d e s c r i p t i o n o f
l a r i z e d e l e c t r o n s and subsequently t h e r e l a t i v i s t i c d e s c r i p t i o n ,
powhen
t h e r e l a t i o n between t h e e l e c t r o n beam p o l a r i z a t i o n and t h e e x p e c t a t i o n
v a l u e o f t h e s p i n components o f t h e e l e c t r o n s i s n o t so o b v i o u s ,
v e 1oped
.
F o l l o w i n g t h e d e s c r i p t i o n o f s p i n 1/2 p a r t i c l e s
i s de-
polarization,
t h e f o r m a l ism i s extended f o r s p i n 1 p a r t i c l e s and t h e photon p o l a r i z a t i o n concept a r i s e s f r o m t h e f a c t t h a t photons a r e a s p e c i a l
case
of
r e l a t i v i s t i c s p i n 1 p a r t i c l e s w i t h z e r o r e s t mass.
AI though t h e concepts o f e l e c t r o n and photon p o l a r i z a t i o n s a r e
q u i t e d i f f e r e n t , i t i s shown t h a t t h e same f o r m a l i s m
to
can
be
applied
b o t h cases.
2. NON RELATIVISTIC DESCRIPTION OF POLARIZATION OF
OF ELECTRONS I n t h i s s e c t i o n some b a s i c r e s u l t s f r o m e l e m e n t a r y non r e l a t i v i s t i c quantum mechanics w i l l be p r e s e n t e d i n o r d e r
to
introduce
d e s c r i p t i o n o f t h e e l e c t r o n p o l a r i z a t i o n i n terms o f t h e s p i n
of
the
the
electrons.
The o b s e r v a b l e s p i n , r e p r e s e n t e d by t h e o p e r a t o r
s a t i s f i e s t h e commutation r e l a t i o n s c h a r a c t e r i s t i c o f a n g u l a r momentum
[si's 3.]= iEsk ( c y c l i c )
and can be wr i t t e n as
where ai a r e t h e P a u l i m a t r i c e s , w h i c h may be r e p r e s e n t e d as
968
The wave f u n c t i o n w h i c h d e s c r i b e s t h e s t a t e o f a p a r t i c l e
of
s p i n 1/2 i s a two component s p i n o r
and t h e most usual b a s i s t o r e p r e s e n t t h i s g e n e r a l s t a t e c o n s i s t s o f t h e
spinors
1
u, = ( o )
which a r e eigenfunctions o f a
z
(y)
(2.6)
w i t h e i g e n v a l u e s +Iand - 1 , r e s p e c t i v e l y .
Any g e n e r a l p u r e s t a t e
near c o m b i n a t i o n o f u, and u,
u, =
and
X
can,
t h e r e f o r e , be w r i t t e n as a
li-
(coherent states)
T h i s method o f r e p r e s e n t i n g a p u r e s t a t e i s c a l l e d Jones f o r m a l i s m , due
t o t h e s i m i l a r i t y w i t h t h e method i n t r o d u c e d by ~ o n e s ' t o d e s c r i b e
po-
larized light.
I f the state
c o n d i t i o n Ia1
1' +
x
i s n o r m a l i s e d such as < X I X > = 1, one g e t s
I a 2 I 2 = I, where
p a r t i c l e i s i n t h e s t a t e u,,
value+fi/;!
la,I2
Ia,
l2
i s the probabil i t y
that is, the probability o f
whenmeasuringthespincomponent
i s the p r o b a b i l i t y o f f i n d i n g
-
Z/2
in
z
that
finding
direction,
the
the
the
and
(stateu,).
Due t o t h e noncommutative r e l a t i o n s (2.2)
it i s not
possible
t o measure s i m u l t a n e o u s l y t h e t h r e e components o f t h e s p i n , w h i c h means
p h y s i c a l l y t h a t t h e measurement o f one component a f f e c t s t h e s p i n s t a t e .
However,
the operator s
2
= s2
x
+
s2
Y
+
s 2 c o m u t e s w i t h any canponent and
z
can be measured simul taneousl y w i t h any component. I t s
s(s
+
e i g e n v a lues i s
1)fi2 = 3fi2/4. T h e r e f o r e , when one says t h a t t h e s p i n o f t h e
t i c l e i s i n z direction,
i t means t h a t t h e s p i n component i n
z
pardirec-
:-
t i o n has t h e v a l u e l i / 2 b u t n o t h i n g can be s a i d about t h e o t h e ?
nents, o n l y t h a t s 2 + s 2 =
F2 1 E' =
x
Y
2
compo-
.
So f a r o n l y t h e s p i n o f a s i n g l e p a r t i c l e has
been
descri-
bed. Consider now a system o f e l e c t r o n s .
I f a11 t h e e l e c t r o n s a r e i n t h e same s p i n s t a t e ,
t h e system
(a1),
i s s a i d t o be i n a p u r e s p i n s t a t e X, =
and t h e s p i n d i r e c t i o n
is
2
s p e c i f i e d by
a1
and a,.
The s p i n o r t h a t d e s c r i b e s a s p i n i n an a r b i t r a r y d i r e c t i o n
g.2
i s such t h a t ( 2 . g ) ~ = hx, where
rator i n direction
ê
= sin
i s t h e p r o j e c t i o n o f t h e s p i n ope-
and A t h e c o r r e s p o d i n g e i g e n v a l u e .
the general expression f o r
6 cos
e as
4 êx +
sin
s i n $ í;.
Y
and s u b s t i t u t e s t h e Paul i m a t r i c e s as i n (2.4)
one g e t s t h e e i g e n v a l u e s X+ =
+
e
+
I f one
writes
cos 0 êz
i n t h e above e x p r e s s i o n ,
1 and X- = -1 w i t h
the
corresponding
wave f u n c t i o n s
are eigenfunctions o f
Thus, X+ and X-
2.g w i t h
+
fi/2 and
-
fi/2.
1 and - 1
direction
and r e p r e s e n t t h e s t a t e s where t h e s p i n i n t h e a r b i t r a r y
has t h e v a i u e s
+
eigenvalues
The d i r e c t i o n o f t h e s p i n ,
2,
2
i s deter-
mined by
The importante o f (2.9)
i s t h a t i t shows t h a t i t i s p o s s i b l e
t o c o n s t r u c t s t a t e s which a r e simultaneously eigenstates o f the
momen-
tum and s p i n component i n any a r b i t r a r y d i r e c t i o n s i n c e t h e s o l u t i o n o f
t h e Schrtidinger e q u a t i o n f o r a f r e e p a r t i c l e i s $ =
R e l a t i v i s t i c a l l y , as w i l l be shown i n s e c t i o n
(3),
x
exp
i
~2
(-.r -
i t i s only
~t).
possible
t o d e f i n e such s t a t e s i n t h e d i r e c t i o n o f t h e momentum.
The e i g e n v a l u e r e p r e s e n t s t h e r e s u l t o f a s i n g l e
measurement
and i f i n f o r m a t i o n about t h e average s p i n d i r e c t i o n o f a beam o f
elec-
t r o n s i s r e q u i r e d , one has t o c a l c u l a t e t h e e x p e c t a t i o n v a l u e
Pauli operator,
<g>.
of
the
I i s the i n t e n s i t y o f the
beam
The a x i a l v e c t o r
i s c a l l e d p o l a r i z a t i o n o f t h e beam, and
and can be w r i t t e n as
i
I = < U o > = X x
where
o.
i s t h e 2x2 u n i t y m a t r i x .
The f o u r q u a n t i t i e s
<o.>
i
Z
= 0 , 1,2,3
parameters, f i r s t i n t r o d u c e d f o r p o l a r i z e d l i g h t
are cal led
and gi v e
the
a
Stokes
complete
d e s c r i p t i o n o f t h e beam.
Using t h e r e p r e s e n t a t i o n (2.4)
t h e Stokes parameters
can be
written as:
Therefore,
=
<o.>
and
Z
from (2.10)
f r o m t h e d e f i n i t i o n o f t h e Stokes parameters
Pi
=
t h e r e l a t i o n between t h e p o l a r i z a t i o n v e c t o r and
t h e Stokes parameters i s
The degree o f p o l a r i z a t i o n i s d e f i n e d as
and has t h e v a l u e o f t h e u n i t y f o r a beam i n w h i c h a l l
a r e i n t h e same s p i n s t a t e ( t o t a l l y p o l a r i z e d beam).
the
electrons
A s i m p l e r e p r e s e n t a t i o n o f t h e beam i s ,
thus, g i v e n b y a f o u r
component column m a t r i x
Four q u a n t i t i e s a r e t h e r e f o r e r e q u i r e d t o d e s c r i b e t h e beam:
i t s inten-
s i t y and t h e t h r e e components o f t h e p o l a r i z a t i o n v e c t o r .
An a l t e r n a t i v e way o f d e s c r i b i n g t h e beam i s by
t h e d e n s i t y m a t r i x d e f i n e d as P =
Ix><XI
introducing
,
The d e n s i t y m a t r i x , as any 2x2 m a t r i x , can be expanded
as
l i n e a r c o m b i n a t i o n o f t h e P a u l i m a t r i x e s and t h e u n i t y m a t r i x . One
p e r t y o f t h e d e n s i t y m a t r i x i s t h a t f o r any o p e r a t o r A, t h e
a
pro-
expectation
v a l u e can be found by t h e r e l a t i o n
<A> =
t r a c e <pA>
trace p
As
P.
Z
=
<o .>, one can wr i t e
Z
w h i c h c a n b e e a s i l y v e r i f i e d b y u s i n g (2.4),
f r o m (2.12)
and (2.16)
t h e Stokes parameters
w h i c h can be w r i t t e n as
(2.12)and
(2.16).
Thus,
one can r e p r e s e n t t h e d e n s i t y m a t r i x i n terms
of
or,
i n terms o f t h e p o l a r i z a t i o n v e c t o r ,
So f a r o n l y t o t a l l y p o l a r i z e d e l e c t r o n s have been c o n s i d e r e d ,
t h a t i s , ensembles i n w h i c h a11 p a r t i c l e s a r e i n t h e same
spin
state.
E l e c t r o n s have two s p i n s t a t e s i n an a r b i t r a r y d i r e c t i o n , and beams composed o f e l e c t r o n s i n t h e s e two s t a t e s a r e r e p r e s e n t e d by a d e n s i t y
ma-
t r i x w h i c h i s t h e average o f t h e i n d i v i d u a l m a t r i c e s o f t h e two s t a t e s
The two o r t h o g o n a l s t a t e s rnay be w r i t t e n a s 9
and t h e average d e n s i t y m a t r i x i s
where w i s t h e r e l a t i v e p o p u l a t i o n o f t h e s t a t e
x,. The
r e s u l t a n t beam
i s s a i d t o be p a r t i a l l y p o l a r i z e d and the d e f i n i t i o n s o f t h e Stokes parameters (2.18),
(2.19)
s i t y m a t r i x i s used.
and (2.20) a r e s t i 11 v a l i d i f t h e average
I f t h e two s t a t e s a r e e q u a l l y populated,
d e s c r i bed by t h e Stokes v e c t o r
w
i.e.
= 1/2, t h e d e n s i t y m a t r i x becomes p = $ a o and t h e beam i s s a i d
u n p o l a r i z e d . A g e n e r a l p a r t i a l l y p o l a r i z e d beam i n t h e z
den=
to
be
direction
is
the z
d i r e c t i o n f i n d s an e i g e n v a l u e
+
A/2 and
Ia-
1
is
the
la+I2+la-l2
p r o b a b i l i t y o f f i n d i n g e i g e n v a l u e -fi/2.
where N+(N-)
(-h/2),
The p o l a r i z a t i o n o f t h e beam i s
i s t h e number o f measurements t h a t y i e l d t h a v a l u e
and N+
+
TI- i s t h e t o t a l nurnber o f measurements.
h/2
I f N+ o r N- i s
equal t o z e r o t h e beam i s t o t a l l y p o l a r i z e d ( / A I = l ) , i f + = N beam i s u n p o l a r i z e d (A = O), o t h e r w i s e t h e beam i s p a r t i a l l y
(o <
+
the
polarized
I A / < 1).
I f a beam i s a m i x t u r e o f i n d i v i d u a l subsystems w i t h d e f i n i -
t e spin states i n d i f f e r e n t directions, the polarization o f the
i s d e f i n e d as t h e average o f t h e p o l a r i z a t i o n v a l u e s o f t h e
(n) 1,
sybsystems w h i c h a r e i n p u r e s t a t e s
w h i c h can be r e w r i t t e n ,
x
f o l l o w i n g (2.10)
as
which can be s i m p l i f i e d t o
The d e n s i t y m a t r i x o f t h e ensemble i s g i v e n by
systern
individual
where 0'")
x (n
a r e the i n d i v i d u a l d e n s i t y m a t r i c e s o f the pure s t a t e s
as def ined i n (2.16).
The Stokes f o u r v e c t o r i s
'E:]
whet-e, now
P3
etc.
t h a t i s , the Stokes parameters o f the t o t a l enseinbles a r e t h e
sums
of
the i n d i v i d u a l Stokes parameters o f each p a r t i a 1 group o f electrons. The
r e l a t i o n (2.20)
i s s t i l l v a l i d , b u t w i t h the r e d e f i n i t i o n s (2.28)
and
(2.29).
An example may i l l u s t r a t e the m i x t u r e o f s t a t e s
Consider isn ensemble o f N
in
a beam.
electrons t o t a l l y polarized i n the z
t i o n mixetl w i t h another ensemble o f N e l e c t r o n s t o t a l l y
direc-
polarized
in
t h e . x d i i - e c t i o n . The resu t i n g p o l a r i z a t i o n i s , from (2.27)
-A
and t h e d t q r e e of p o l a r i z a t i o n i s A =
J1/4 + 1/4
=
JI/T
and
therefore
the r e s u l t a n t beam i s p a r t i a l l y p o l e r i z e d . The d e n s i t y m a t r i x i s ,
1
(2.28), (2.161, (2.4) and f r o n x ( ' ) =v%
x(') =
(A),
from
The Stokes parameters o f the t o t a l ensemble a r e
P, = 2N
P, = 2N/2
P2 = 2N/2
P, = O
which a r e the sums o f the i n d i v i d u a l Stokes parameters o f the two
systems.
sub-
To conclude t h e d i s c u s s i o n a b o u t p o l a r i z a t i o n o f e l e c t r o n s i t
can be s t a t e d t h a t an ensemble o f e l e c t r o n s i s s a i d t o be
polarized i f
t h e e l e c t r o n s p i n s have a p r e f e r e n t i a l o r i e n t a t i o n so t h a t t h e r e e x i s t s
a d i r e c t i o n f o r which t h e two p o s s i b l e s p i n s t a t e s a r e n o t e q u a l l y
po-
pulated.
3. POLARIZATION OF RELATIVISTIC ELECTRONS 1r285'7,10r11
II
The Schrodinge
quan-
equation i s not v a l i d i n r e l a t i v i s t i c
tum niechanics s i n c e i t i s n o t i n v a r i a n t under L o r e n t z
t r a n s f o r m a tion,
and t h e r e f o r e a new d e s c r p t i o n i s r e q u i r e d , w h i c h i s o b t a i n e d
by
the
D i r a c e q u a t i o n . The d e f i n t i o n o f p o l a r i z a t i o n i n terms o f t h e expectat i o n v a l u e o f t h e s p i n i s n o t L o r e n t z i n v a r i a n t and i s v a l i d
t h e r e s t frame o f t h e e l e c t r o n s .
only
in
I n t h i s section the r e l a t i v i s t i c trea-
tment o f p o l a r i z a t i o n o f e l e c t r o n s w i l l be d i s c u s s e d .
The Di r a c equat i o n says t h a t
a$
iz at = H$,
where t h e
Hami 1 t o -
n i a n i s g i v e n by
where a and
6, 4x4
herm i t i a n m a t r i c e s , a r e c a l l e d t h e D i r a c
and t h e wave f u n c t i o n $ i s a
matrices,
4 component column m a t r i x . The f o u r D i r a c
m a t r i c e s s a t i s f y t h e r el a t i o n s
The usual r e p r e s e n t a t i o n o f t h e D i r a c m a t r i c e s i s
where o, t h e Paul i m a t r i c e s , a r e r e p r e s e n t e d i n ( 2 . 4 ) .
A s o l u t i o n o f t h e D i r a c ' e q u a t i o n (3.1)
is
for a
free
particle
$(r,t)
= u exp
I-
v
T-
where
"p
E
((,p),
xU =
( c t , r)
- and
1
u i s a 4 component column m a t r i x u =
L4 3
From (3.11,
(3 . 3 ) , (2.4)
and (3.4)
one sees t h a t t h e s e t o f
f o u r e q u a t i o n s c o n t a i n e d i n t h e D i r a c e q u a t i o n s g i v e s non t r i v i a l sol u t i o n s i f the determinantl
v a n i s h e s . The c o n d i t i o n f o r non t r i v i a l s o l u t i o n s , t h e r e f o r e ,
leads t o
t h e energy r e l a t i o n
(3.5)
There a r e Four 1 i n e a r l y independent s o l u t ions, two belongimg t o t h e pos i t i v e energy, E
+
(which corresponds t o e l e c t r o n s ) and two b e l o n g i n g t o
t h e negat i v e energy, E-
(which corresponds t o pos i t r o n s )
are, a f t e r normalization:
.
The s o l u t i o n s
uA and uB correspond t o p o s i t i v e energy ( e l e c t r o n s ) and
U~
uD t o n e g a t i v e energy ( p o s i t r o n s ) . The g e n e r a l s o l u t i o n f o r p o s i t i -
Solutions
and
v e energy i s a 1 i n e a r c o m b i n a t i o n o f
where
u
=
a,uA + a2uB,t h a t i s ,
uA and uB
Each s o l u t i o n has two c o m p o n e n t s , ~ ~and u4 which,
i n t h e non
r e l a t i v i s t i c l i m i t a r e o f o r d e r o f V/C which i s s m a l l . These components
a r e c a l l e d t h e smal 1 components and t h e o t h e r s (ul and
up) a r e the l a r -
ge components, w h i c h i n non r e l a t i v i s t i c l i m i t correspond t o t h e
solu-
t i o n o f t h e Schrbdinger equation.
When one c a l c u l a t e s t h e t i m e v a r i a t i o n o f t h e a n g u l a r momentum
L of
a f r e e p a r t i c l e using the r e l a t i o n
=
1
[L,H]
one g e t s
which does n o t a g r e e w i t h t h e c l a s s i c a l and non
r e l a t i v i s t i c quantum
mechanics r e s u l t s where, f o r a f r e e p a r t i c l e ( o r p a r t i c l e under a
t r a l p o t e n t i a l ) t h e o r b i t a l a n g u l a r momentum i s a constan:
The o p e r a t o r which, added t o
L,
of
cen-
motion.
commutes w i t h H i s
where
and a a r e t h e non r e l a t i v i s t i c Paul i m a t r i c e s . The o p e r a t o r
s p i n o p e r a t o r o f t h e p a r t i c l e s and has eigenva l u e s 2 U 2 , w h i c h
is
the
means
t h a t a p a r t i c l e o b e y i n g t h e D i r a c e q u a t i o n has s p i n 1/2. T h e r e f o r e , one
can see t h a t t h e s p i n appears d i r e c t l y f rom t h e Di r a c equation. A1 though
t h i s d e r i v a t i o n was made f o r a s p e c i f i c r e p r e s e n t a t i o n o f t h e D i r a c matrices,
i t i s v a l i d f o r any r e p r e s e n t a t i o n .
I n t h e non r e l a t i v i s t i c case,
eigenfunctions o f
-ê
:.E,
i t i s possible
w i t h e i g e n v a l u e s '1,
to
f o r any a r b i t r a r y d i r e c t i o n
by t h e c o h e r e n t s u p e r - p o s i t i o n o f e i g e n s t a t e s o f oZ,
(2.9).
However, i n t h e r e l a t i v i s t i c case, t h a t i s no
Consider, f o r example,
the operator
04,
construct
as
shown
longer
in
possible.
a p p l i e d t o the general s o l u t i o n
o f the Dirac equation (3.8),
-a 2
-(alpz+a2
C
P
(;)-,
:+rn,c2
(a, (P,+~P~) - a , ~ ~ )
-C
:+m,c2
I
There i s no p o s s i b l e c o m b i n a t i o n o f v a l u e s o f a, and a 2 w h i c h makes t h e
s t a t e u an e i g e n s t a t e o f o;
I f the operator
03:
w i t h e i g e n v a l u e +1 o r - 1 , u n l e s s p x = p = 0.
Y
i s a p p l i e d t o t h e wave f u n c t i o n u,
such t h a t t h i s f u n c t i o n
can be e i g e n s t a t e o f
o&
is,
the
analogousl y,
py = pz = O. T h e r e f o r e , i t i s o n l y p o s s i b l e t o c o n s t r u c t
a r e s i m u l t a n e o u s l y e i g e n s t a t e s o f t h e momentum and t h e
condition
states
spin
which
component
a l o n g t h e momentum d i r e c t i o n . T h i s r e s u l t i s expected because
in
the
r e l a t i v i s t i c t r e a t m e n t i t i s n o t t h e s p i n w h i c h i s a c o n s t a n t o f motion,
b u t t h e t o t a l a n g u l a t momentum
E + 2,
Only i f
L=
O i s the
E direction,
spin
cons-
t h e cornponent
of
t h e o r b i t a l momenturn i n t h i s d i r e c t i o n , L e , v a n i s h e s and t h e r e f o r e
it
t a n t . For a p l a n e wave i n t h e a r b i t r a r y
i s posssible t o f i n d eigenstates o f Se.
i t can t h e r e f o r e be seen t h a t f o r r e l a t i v i s t i c e l e c t r o n s , o n l y
i n t h e i r r e s t frame, where ( P ~ ) ~(p,)
=
= (pZlB = O, i s i t p o s s i b l e t o
d R
f i n d e i g e n v a l u e s o f t h e s p i n component o p e r a t o r s , i n any a r b i t r a r y d i r e c t i o n . The s p i n p a r t o f t h e e i g e n f u n c t i o n (3.8),
u,=[21.
i n t h e r e s t frame i s
a1
w h i c h means t h a t o n l y t h e l a r g e components a r e n o t equal t o i e -
lol
r o . For l o w energy e l e c t r o n s t h e s m a l l components a r e so s m a l l t h a t
it
i s p o s s i b l e t o say t h a t one can f i n d e i g e n s t a t e s o f any s p i n component.
As t h e non r e l a t i v i s t i c P a u l i r n a t r i c e s used i n
t i o n o f t h e s p i n o p e r a t o r (3.9)
uz
and (3.10)
the
defini-
a r e i n a r e p r e s e n t a t i o n where
i s d i a g o n a l , i t i s c o n v e n i e n t t o choose t h e z a x i s as t h e d i r e c t i o n
o f t h e momentum, because some s i m p l i f i c a t i o n w i l l a r i s e ,
shown.
If
equation,
as
= p 2, t h e g e n e r a l s o l u t i o n , f o r e l e c t r o n s , o f
z
r e p r e s e n t e d i n (3.7) becomes
will
the
be
Dirac
The small components c a n be expressed i n terms o f t h e l a r g e cornponents,
u, = A u,
u, =
and
u,,= -ep
~+rn,c'
u, = -A u,
(3.13)
~+rn,c~
and t h e r e f o r e Jones f o r m a l i s m , used f o r non r e l a t i v i s t i c e l e c t r o n s , may
be a p p l i e d , b u t t h e d i f f e r e n c e i s t h a t now
ul and u2 a r e t h e l a r g e com-
ponents o f t h e D i r a c s p i n o r .
From (3.7)
one sees t h a t t h e r e a r e two independent s o l u t i o n s
u and uB f o r e l e c t r o n s o f momentum p , c o r r e s p o n d i n g t o o r t h o g o n a l s p i n
A
s t a t e s w i t h s p i n p a r a l l e l and a n t i p a r a l l e l t o t h e d i r e c t i o n o f t h e
mo-
men tum.
An a r b i t r a r y p u r e s t a t e may be w r i t t e n as IX> =aX+>
+ b!x->
and t h e d e n s i t y m a t r i x can be c o n s t r u c t e d .
I n t h e same f o r m as (2.16).
As any 2 x 2 m a t r i x , t h e d e n s i t y m a t r i x canbe
w r i t t e n as a l i n e a r c o m b i n a t i o n o f t h e Paul l r n a t r i c e s ,
which has t.he same f o r m o f (2.20),
and as b e f o r e (P,
,E)
are
d e f i n e d as
t h e Stokes parameters.
Ore has now t o i n t e r p r e t t h e
relation
t h e Stokes parameters and t h e p o l a r i z a t i o n v e c t o r w i t h t h e
value o f the spin f o r r e l a t i v i s t i c electrons.
between
expectation
In the electrons r e s t f r a -
me t h e former q u a n t i t i e s a r e s t i l l r e l a t e d t o t h e e x p e c t a t i o n v a l u e
of
t h e s p i n o p e r a t o r as
However,
i n t h e l a b o r a t o r y frame t h e t h r e e v e c t o r p o l a r i z a t i o n does n o t
have t h e same form,
since i t i s not a Lorentz covariant quantity.
r e l a t i o n between t h e Stokes parameters (and, consequently,
The
polari-
the
z a t i o n v e c t o r ) and t h e e x p e c t a t i o n v a l u e s o f t h e components o f t h e s p i n
o p e r a t o r a r e understood i n a s i m p l e s way i f one t a k e s
the
c h o i c e o f t a k i n g t h e z a x i s as t h e d i r e c t i o n o f momentum and
r e p r e s e n t a t i o n (2.4),
(3.3)
and ( 3 . 1 0 ) .
convenient
uses
I n t h i s case, as a l r e a d y
ted out, the s o l u t i o n s o f the Dirac equation a r e o f the form o f
the
poin-
(3.12)
and t h e q u a n t i t i e s a and b i n (3.14) may be i d e n t i f i e d as t h e largecomponents ai and a* o f t h e D i r ã c s p i n o r . The e x p e c t a t i o n v a l u e s
of
the
components o f t h e s p i n o p e r a t o r a r e w r i t t e n as
where lu> i s g i v e n by (3.12)
(3.101,
(2.4).
s ince
2E
E c m0c2
and O;
T h e r e f o r e , one has
= I + A ~ .
i s wri tten i n the
representations
<a!> =
2
<o;>
-I-(a:
(1
=
+
O
O '
0
0
0
A
-
a,
O
O
O
I
a IA
,O
O
1
O
A)
a:
A')
'
-(1
a i a:
(a, a:
+
a:
a,)
(1
-
'
1
1
O
,
'al
,
,-a,A
A')
i.A 2 )
Similarly
R e l a t i o n s (3.16),
z a t i o n , A I = PI/P,,
(3.17)
and (3.18)
show t h a t t h e l o n g i t u d i n a l
polari-
keeps t h e non r e l a t i v i s t i c meaning, because
However, t h e t r a n s v e r s e p o l a r i z a t i o n components
do
n o t have the
rneaning as e x p e c t a t i o n v a l u e o f t h e c o r r e s p o n d i n g s p i n components
same
be-
cause
T h e r e f o r e i n t h e r e s t frame t h e e x p e c t a t i o n
va l u e s
s p i n components have t h e same meaning as t h e p o l a r i z a t i o n
whereas f o r low energy e l e c t r o n s , w h i c h have E
%
2
m oc ,
of
the
components
t h e same
inter-
p r e t a t i o n i s s t i l l a reasonable approximation.
However, as t h e energy o f
t h e e l e c t r o n s i n c r e a s e s , t h e t r a n s v e r s e components o f t h e s p i n tend
z e r o whereas t h e p o l a r i z a t i o n components remain f i n i t e .
to
I n t h e extreme
r e l a t i v i s t i c l i m i t , the s p i n i s l o n g i t u d i n a l .
4. SPIN 1 PARTICLES 5'6''2''3
The f o r m a l i s m used i n s e c t i o n ( 2 ) f o r non r e l a t i v i s t i c
1/2 p a r t i c l e s w i l l now be extended f o r s p i n 1 p a r t i c l e s .
spin
The reason f o r
s t u d y i n g s p i n 1 p a r t i c l e s i s t h a t , as w i l l be shown, photons a r e a spec i a l case o f r e l a t i v i s t i c s p i n 1 p a r t i c l e s w i t h mass zero.
For non r e l a t i v i s t i c p a r t i c l e s w i t h s p i n 1, t h e s p i n
opera-
tor S
- s a t i s f i e d t h e same commutation r e l a t i o n s (2.2)
i' s2 a r e
where now t h e o p e r a t o r s S
where I
3x3 m a t r i c e s and
i s t h e u n i t 3x3 m a t r i x .
There a r e t h r e e p o s s i b l e s t a t e s o f t h e components o f the s p i n
a l o n g any d i r e c t i o n , c o r r e s p o n d i n g t o e i g e n v a l u e s m
I n a r e p r e s e n t a t i o n where S
Z
=
+A, O, - E .
i s d i a g o n a l , one can w r i t e
and t h e e i g e n s t a t e s o f SZ a r e t h e t h r e e cornponent s p i n o r s
+h, O, -h, r e s p e c t i v e l y .
corresponding t o eigenvalues
Any g e n e r a l p u r e s t a t e o f t h e p a r t i c l e can
terms o f t l i e o r t h o g o n a l bases formed by
x+,
X, and
be
written
X- a s
condi-
I f t h i s g e n e r a l s t a t e r e p r e s e n t s one p a r t i c l e t h e n o r m a l i z a t i o n
+
t i o n i s la,I2
in
l a o I 2 + l a - I 2 = 1 and i f i t r e p r e s e n t s a t o t a l l y
pola-
r i z e d beam, t h e same q u a n t i t y i s n o r m a l i z e d t o be t h e i n t e n s i t y o f
the
beam.
Through t h e commutation r e l a t i o n s (4.1)
ces c o r r e s p o n d i n g t o S
x
and S
Y
a r e found t o be,
The d e n s i t y m a t r i x p =
xXt
and (4.2)
in this
the matri-
representation
can be w r i t t e n as
I n analogy w i t h t h e P a u l i m a t r i c e s f o r s p i n
1/2
particles
one can d e f i n e
b u t i t i s no l o n g e r p o s s i b l e t o r e p r e s e n t any 3 x 3 m a t r i x i n
1 ,, az, a
Y
,O
2'
terms
I n o r d e r t o g i v e a complete r e p r e s e n t a t i o n o f a
of
general
3 x 3 m a t r i x f o r s p i n 1 p a r t i c l e s , n i n e q u a n t i t i e s must be s p e c i f i e d and
t h e c o n v e n t i o n a l r e p r e s e n t a t i o n s uses t h e f o l l o w i n g
t e n s o r opera t o r s 5 ' l2
set
of
spherical
The e x p e c t a t i o n v a l u e s o f t h e t e n s o r o p e r a t o r s T
S u b s t i t u t i n g (4.6)
and (4.9)- i n (4.10),
ii
are
the density
matrix
can be w r i t t e n as
i +iJ
1
t,,
+JT t 2 4t,-,+G
t2-,
The n i n e q u a n t i t i e s w h i c h d e s c r i b e t h e beam rnay
be
divided
i n t o t h r e e p a r t s : one component o f a t e n s o r o f r a n k z e r o , to,, w h i c h i s
t h e i n t e n s i t y o f t h e beam; t h r e e components o f a t e n s o r o f r a n k 1 , tli,
w h i c h f o r m t h e s o - c a l l e d v e c t o r p o l a r i z a t i o n ; f i v e components o f a t e n s o r o f r a n k 2, tZi, w h i c h f o r m t h e t e n s o r p o l a r i z a t i o n . U n l i k e t h e case
o f s p i n 1/2 p a r t i c l e s ,
t h e i n t e n s i t y and v e c t o r p o l a r i z a t i o n do r o t d e s -
c r i b e c o m p l e t e l y t h e beam, because t h e components o f t h e 2nd r a n k
ten-
s o r a r e a1 so o b s e r v a b l e s .
written,
The e x p e c t a t i o n v a l u e s o f t h e t e n s o r o p e r a t o r
Tij may
i n terms o f t h e m a t r i x elements o f t h e d e n s i t y r n a t r i x as:
be
As f o r s p i n 1/2 p a r t i c l e s , one can d e f i n e t h e Stokes parameters f o r spin 1 particles,
b u t now,
i n s t e a d o f f o u r , one has t o
define
n i n e components,
and t h e meaning o f each component i s n o t so c l e a r .
As i n t h e case o f s p i n 1/2 p a r t i c l e s ,
t h e ~ c h r g d i n ~ e equar
t i o n i s n o t v a l i d f o r r e l a t i v i s t i c s p i n 1 p a r t i c l e s , w h i c h must be desc r i b e d by a L o r e n t z c o v a r i a n t e q u a t i o n .
I n t h e case o f s p i n 1/2
c l e s t h i s e q u a t i o n i s t h e D i r a c e q u a t i o n (3.1).
parti-
P a r t i c l e s w i t h restmass
m o and s p i n z e r o a,re d e s c r i b e d by a s c a l a r f i e l d Q ( x ) w h i c h
satisfies
t h e Klein- Gordon e q u a t i o n 1 4
where
=
,
a a ,, LJ
= 0,1,2,3.
P a r t i c l e s w i t h s p i n 1 c o u l d p o s s i b l y be d e s c r i b e d by a
v e c t o r f i e l d which t r a n s f o r m s under L o r e n t z t r a n s f o r m a t i o n as )I
=
a
LJ
four
(x')
$ ( x ) ans whose components $ s a t i s f y t h e Klein- Gordon equat i o n
1iv v
Fi
=
However,
the four vector representation i s not a unique spin represent1 1
a t i o n s i n c e t h e f o u r dimensional r e p r e s e n t a t i o n D2D2 o f t h e p r o p e r Lo1
1
= D2 X D2 o f
r ( n t z group, w h i c h i s d e r i v e d f r o m t h e d i r e c t p r o d u c t Di
1
r e p r e s e n t a t i o n o f t h e t h r e e dimensional r o t a t i o n a l group,
i s reduced
t o t h e d i r e c t sum o f d i f f e r e n t s p a t i a l r o t a t i o n r e p r e s e n t a t i o n s ,
that
where Do i s t h e s c a l a r r e p r e s e n t a t i o n a s s o c i a t e d w i t h s p i n z e r o and
0'
i s t h e v e c t o r r e p r e s e n t a t i o n a s s o c i a t e d w i t h s p i n 1 . As t h e
represent-
a t i o n space o f D1 i s t h r e e d i m e n s i o n a l , t h e r e must
subsidiary
be
c o n d i t i o n w h i c h l i m i t s t h e number o f independent f i e l d
three,
one
components
i n order t o c u t out the scalar representation. This
which must a l s o be a L o r e n t z c o v a r i a n t ,
to
condition,
i s c a l l e d be L o r e n t z
condition
and can be w r i t t e n as14
a*
!J
The s e t o f equat i o n s
Fi
= o
(4.
(4. 15) and (4.17) d e s c r i b e s
partic
w i t h r e s t mass mo and a u n i q u e s p i n 1 and i s c a l l e d Proca e q u a t i o n s .
A s p e c i a l case o f p a r t i c l e s s a t i s f y i n g t h e
i s t h a t o f z e r o r e s t mass p a r t i c l e s .
Such p a r t i c l e s
w h i c h a r e t h e quanto o f e l e c t r o m a g n e t i c r a d i a t i o n .
t h e r e s t mass i n e q u a t i o n (4.15)
Proca
equations
a r e the
If
one
photons,
substitutes
w i t h zero, t h e equat i o n s becomes
which i s t h e w e l l known wave e q u a t i o n . The p o l a r i z a t i o n s t a t e s
of
the
photons w i l l be s t u d i e d i n t h e n e x t s e c t i o n .
As i n t h e case o f r e l a t i v i s t i c s p i n 1/2 p a r t i c l e s ,
i t i s not
possible t o construct eigenstates o f r e l a t i v i s t i c spin 1 p a r t i c l e s w i t h
a d e f i n i t e momentum i n an a r b i t r a r y d i r e c t i o n . There a r e , however,
si-
multaneous e i g e n s t a t e s o f t h e momentum and t h e s p i n component a l o n g t h e
momentum d i r e c t i o n s i n c e t h e component o f t h e o r b i t a l a n g u l a r
momentum
a l o n g t h e d i r e c t i o n o f momentum v a n i s h e s . Once more i t i s c o n v e n i e n t t o
choose t h e z a x i s as t h e d i r e c t i o n o f momentum, and t h e r e f o r e a
gene-
r a l pure s t a t e o f a r e l a t i v i s t i c s p i n 1 p a r t i c l e can be w r i t t e n
l i n e a r coisbination o f the e i g e n s t a t e s o f the s p i n component
d i r e c t i o n o f motion, corresponding t o eignevalues
(4.4),
(4.5),
+ z,
as
a
the
along
0, - E .
Equations
(4.7) a r e s t i l l v a l i d , w i t h t h e c o n d i t i o n t h a t
they
are
r e l a t e d w i t h t h e s p i n component a l o n g the d i r e c t i o n o f motion.
5. POLAAIIZATION OF PHOTONS 3,5*6r'3,'4
The electromagnetic f i e l d
A
= (;$,A),
as mentioned i n secFi
t i o n (4) i s a s p e c i a l case o f t h e f i e l d s which d e s c r i b e s p i n 1 p a r t i -
c l e s , w i t h t h e p a r t i c u l a r p r o p e r t y t h a t t h e quanta associated w i t h such
a f i e l d , the photons, have a zero r e s t mass. The Proca equations (4.15)
and (4.17) f o r the electromagnetic f i e l d become
w i t h the Lorentz c o n d i t i o n
I t i s important t o n o t i c e t h a t w h i l e t h e wave equations
(5.1)
are
a
p a r t i c u l a r case o f the more general Proca equations, t h e Lorentz condit i o n (5.2)
i s the same as f o r f i n i t e r e s t mass p a r t i c l e s .
The Lorentz c o n d i t i o n (5.2) does n o t exhaust
gauge trarisformation$ on the electrornagnetic f i e l d A
the
possible
and a v e r y u s e f u l
4
'
gauge i s the so c a l l e d Coulomb o r transverse gauge which says t h a t
I n the Coulomb gauge, from (5.2) and t h e second equation (5.3)
see t h a t the s c a l a r p o t e n t i a l
$ vanishes. This means
electromagnetic waves i t i s p o s s i b l e t o choose a gauge
one can
that
for
free
in
which
the
s c a l a r p o t e n t i a l vanishes and t h e e l e c t r o m a g n e t i c f i e l s i s t o t a l l y desc r i b e d by t h e p o t e r i t i a l v e c t o r
subjected t o the condition
and t h e c o n d i t i o n
that is,
dition
A
s a t i s f y i n g t h e wave e q u a t i o n
( 5 . 3 ) . The s o l u t i o n o f ( 5 . 4 ) i s a p l a n e wave
( 5 . 3 ) says t h a t
t h e e l e c t r o m a g n e t i c waves a r e t r a n s v e r s e . For t h i s reason con-
( 5 . 3 ) i s cal led the transversal i t y condition.
I t i s c o n v e n i e n t t o choose t h e z a x i s as t h e
p r o p a g a t i o n o f t h e f i e l d because i n t h i s case any v e c t o r
direction
of
operator
can
be w r i t t e n as
where
a r e e i g e n s t a t e s o f t h e s p i n c m p o n e n t a l o n g t h e z a x i s w i t h eigenva ues
+ E , 0, -Ti
r e s p e c t i v e l y , and
.9)
f o r m t h e s p h e r i c a l b a s i s w h i c h d i a g o n a l i s e s S Z , which i s t h e n represent e d by
( 4 . 3 ) and f o r m an i r r e d u c i b l e t e n s o r o f r a n k one which
forms under r o t a t i o n i n t h e same way as t h e s p h e r i c a l harmonics
Y,, and Y,,-,,r e s p e c t i v e l y .
In t h i s representation the
trans-
Y l l,
transversality
c o n d i t i o n (5.6)
f o r t h e e l e c t r o m a g n e t i c f i e l d a l lows a simpl i f i e d
form
o f t h e v e c t o r p o t e n t i a l t o be w r i t t e n as
4
where
x+
= A
+x+ +
,
A-X-
and X- a r e t h e e i g e n v e c t o r s o f S
Z
since Ao = O
w i t h eigenvalues
+
and
-E.
T h i s r e p r e s e n t a t i o n i s c a l l e d t h e h e l i c i t y r e p r e s e n t a t i o n and t h e s t a t e
X+
i s c a l l e d the s t a t e o f p o s i t i v e h e l i c i t y o r s t a t e o f r i g h t
p o l a r i z a t i o n (RCP),
t i o n o f momentum.
circular
c o r r e s p o n d i n g t o photons w i t h s p i n a l o n g t h e d i r e c -
X-
i s the.state o f negative h e l i c i t y , o r l e f t
l a r p o l a r i z a t i o n (LCP),
circu-
c o r r e s p o n d i n g t o photons w i t h s p i n o p p o s i t e
to
t h e d i r e c t i o n o f momentum. The h e l i c i t y , d e f i n e d as t h e c o m p c n e n t o f t h e
s p i n i n t h e d i r e c t i o n o f m o t i o n (S i n t h i s case) i s a c o n s t a n t o f
Z
t i o n s i n c e t h e component o f t h e o r b i t a l a n g u l a r momentum a l o n g t h e
r e c t i o n oF momentum i s zero,
modi-
t h a t i s JZ = Sz.
The t r a n s v e r s e c h a r a c t e r o f t h e e l e c t r o m a g n e t i c f i e l d
which
i s a consequence o f t h e v a n i s h i n g r e s t mass o f t h e photons can t h e r e f o r e be i n t e r p r e t e d t o correspond t o a z e r o p r o b a b i l i t y o f f i n d l n g a pho= O along the d i r e c t i o n o f motion,
which
S
means t h a t photons a r e l o n g i t u d i n a l l y p o l a r i z e d and have o n l y two pos-
t o n w i t h j p i n component rn
s i b l e s p i n s t a t e s , t h e r i g h t and l e f t c i r c u l a r p o l a r i z e d
states.
analogy w i t h t h e s t u d y o f s p i n 1/2 p a r t i c l e s one can say
that
In
as
the
v e l o c i t y o f t h e p a r t i c l e i n c r e a s e s t h e t r a n s v e r s e components of t h e s p i n
o f t h e p a r t i c l e decreases and i n t h e extreme r e l a t i v i s t i c
V = c , w h i c h i s t h e case o f t h e photons,
case
whete
t h e s t a t e c o r r e s p o n d i n g t o trans-
v e r s e s p i n vanishes and t h e p a r t i c l e has o n l y l o n g i t u d i n a l s p i n
compo-
n e n t . O f course, as t h e photons have a r e s t mass equal t o zero,
it
is
meaningless t o t r y t o d e f i n e t h e s p i n o f t h e photon i n i t s r e s t frame.
The massless c h a r a c t e r o f t h e photon, w h i c h
corresponds
to
a, = 0, leaves o n l y f o u r independent non v a n i s h i n g elements o f t h e dens i t y m a t r i x ,)+
'i'(.
and c o n s e q u e n t l y o n l y f o u r independent non v a n i s h i n g
expectation values o f t h e tensor operators
elements a r e t o ,=
la+I2 +
7
..
d e f i n e d i n (4.12).
These
la-12 w h i c h i s t h e i n t e n s i t y o f t h e beam,
which i s a measure o f t h e degree o f c i r c u l a r p o l a r i z a t i o n
6%:a-
-
t,,,
and
w h i c h a r e a s s o c i a t e d w i t h t h e degree o f l i n e a r p o l a r i z a t i o n .
S i n c e t h e r e a r e o n l y two p o s s i b l e s p i n s t a t e s f o r t h e
pho-
tons, one can a p p l y t h e same formalisrn used f o r non r e l a t i v i s t i c
elec-
t r o n s t o d e s c r i b e t h e p o l a r i z a t i o n o f photons, b u t
a
reinterpretation
o f t h e parameters i s necessary. A g e n e r a l p u r e s t a t e may be w r i t t e n
where
x+
and X- a r e two component s p i n o r s r e p r e s e n t i n g s t a t e s
of
and LCP, and la+I2 and la-I2 a r e t h e p r o b a b i l i t i e s o f f i n d i n g a
as
RCP
photon
i n such s t a t e s .
Alternatively,
the transverse c o n d i t i o n a l s o allows a repre-
sentation o f t h e vector potent ia1 as
and one can r e p r e s e n t a g e n e r a l p u r e s t a t e as
where x,and
xY
a r e states corresponding t o l i n e a r p o l a r i z a t i o n
two m u t u a l l y o r t h o g o n a l axes, x and
o f momentum, and a
,
y, p e r p e n d i c u l a r t o t h e
direction
/ag12a r e t h e p r o b a b i l i t i e s o f f i n d i n g a photon
i n such s t a t e s . T h i s f o r m a l i s m i s known as Jones f o r m a l i s m ,
f i r s t i n t r o d u c e d i n t h e a n a l y s i s o f g o l a r i z e d l i g h t 8 . The
(5.14)
along
and
was
x
state
in
i s c a l l e d t h e Jones v e c t o r and a l l t h e f o r m a l ism i s analogous t o
t h e c o h e r e n t s t a t e s used f o r non r e l a t i v i s t i c e l e c t r o n s ,
although
the
Jones v e c t o r i s a two component v e c t o r i n r e a l space w h i l e t h e f u n c t i o n s
d e s c r i b i n g t h e e l e c t r o n s a r e i n s p i n space ( s p i n o r s ) .
One can d e f i n e t h e d e n s i t y m a t r i x f o r a beam
of
t h e same p u r e s t a t e e x a c t l y as (2.161, and f o r a mixed beam
photons i n
as
A l s o , t h e d e n s i t y m a t r i x may be w r i t t e n i n terms o f t h e Stokes
t e r s as i n (2.19),
trices,
(2.20)
a l though t h e Stokes parameters and
i n t h e case o f photons, a r e n o t d e f i n e d i n r e a l space
(2.28).
parame-
Paul i mabut
in
~ o i n c a r éspace. The photon beam i s ,
thus, c o m p l e t e l y d e s c r i b e d
by
the
t h e d i f f e i - e n c e between t h e number o f photons p o l a r i z e d a l o n g t h e x
and
f o u r Stokes parameters:
t h e i n t e w i t y o f t h e beam i s t h e t o t a l number o f photons;
y axes,
ir; a measure o f t h e l i n e a r p o l a r i z a t i o n a l o n g t h e two t r a n s v e r -
se axes x and y ;
i s a rneasure o f t h e l i n e a r p o l a r i z a t i o n a l o n g axes r o t a t e d by I T /i ~
n rel a t i o n t o t h e axes o f p l ;
i s a measure o f t h e c i r c u l a r p o l a r i z a t i o n .
The p o l a r i z a t i o n i s d e f i n e d as
b u t t h e po,larization "vector"
i s n o t d e f i n e d i n r e a l space,
but i n the
P o i n c a r é space i n which t h e Stokes pararneters a r e d e f i n e d .
S t a t e s i n t h e h e l i c i t y and Jones r e p r e s e n t a t i o n d e f i n e d
(5.11)
and (5.13)
can be r e l a t e d by u s i n g t h e r e l a t i o n s (5.9)
the basis vectors.
r e p r e s e n t e d by
a,,
I n Jones r e p r e s e n t a t i o n s , t h e RCP and LCP s t a t e s a r e
1
and ( .) and by s u b s t i t u t i n g t h e r e l a t i o n s between
-2
a- w i t h a
a one g e t s
xt Y
1ax12
to,
t,,
1
A
+
i ( a {z* "Y
in
between
lay12 = Po ( i n t e n s i t y o f t h e beam)
a*a ) =
" Y
P,
(degree o f c I r c u l a r p o l a r i z a t i o n )
(Iaz/'
t,,,
- Iuy12 i i(a a* + a*u
X b
XY
) ) = P, ? i
li-
(combinations of
near p o l a r i z a t i o n ) w h i c h i s i n agreernent w i t h c h e i n t e r p r e t a t i o n o f
(5.11).
(h means equal except f o r c o n s t a n t s )
.
I am g r e a t l y i n d e b t e d t o Dr. Jarnes Byrne f o r
his
l i g h t e n i n g s u g g e s t i o n s and comments on t h e s u b j e c t m a t t e r and
t h i s work, which was done under h i s s u p e r v i s i o n a t t h e
very
en-
form
of
University
of
Sussex.
I would l i k e t o thank Dr. Amir C a l d e i r a f o r t h e v e r y h e l p f u l
d i s c u s s i o n s and s u g g e s t i o n s on t h e s e c t i o n s f o u r and f i v e o f t h i s t e x t .
Finally,
I would l i k e t o t h a n k t h e Conselho Nacional de
senvolv imento C i e n t í f i c o e T e c n o l ó g i c o (CNpq) f o r t h e i r f i n a n c i a l
p o r t d u r i n g t h e e l a b o r a t i o n o f t h i s work, and t h e P o n t i f í c i a
Desup-
Universi-
dade C a t ó l i c a do R i o de J a n e i r o (PUC-RJ) f o r p e r m í t t i n g me t o be absent
f r o m t h e i r P h y s i c s Department and f o r t h e i r f i n a n c i a 1 h e l p d u r i n g
part
o f t h i s work.
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