14. Matrix treatment of polarization
law s
Ampere' :
t
E E B
law s
Faraday' :
t
- B E
0 B
law electric s
Gauss' :
E
∂ + ∂
=
×
∇
∂
= ∂
×
∇
=
•
∇
=
•
∇
µε µσ
ε ρ
( ) ( )
( ) ( )
2 2
t B t
B
t B t
t B
t E E
t E E
B
∂
− ∂
∂
− ∂
=
∂
− ∂
∂ + ∂
∂
− ∂
=
×
∂ ∇ + ∂
×
∇
=
∂
× ∂
∇ +
×
∇
=
×
∇
×
∇
µε µσ
µε µσ
µε µσ
µε
µσ
( ) ( )
ε µε ρ
µσ
µε µσ
= ∇
∂
− ∂
∂
− ∂
∇
∴
∂ =
− ∂
∂
− ∂
∇
∴
−∇
=
∇
−
•
∇
∇
=
×
∇
×
∇
2 2 2
2 2 2
2 2
0
t E t
E E
t B t
B B
B B
B B
1 0 0
1 0 0
2 2 2 2
2 2 2
2 2 2 2
2 2 2
∂ =
− ∂
∇
⇒
∂ =
− ∂
∇
∂ =
− ∂
∇
⇒
∂ =
− ∂
∇
=
=
t E E v
t E E
t B B v
t B B
media, 0)
charge(
- free and
0) (
conducting -
non If
µε µε
ρ σ
velocity
group d :
dn n
n c dk
d
phase arbitrary
: velocity, phase
k :
e t)
(z,
: direction -
z in g propagatin tic waves
Monochroma
t kz i o
1
) (
1
−
+
−
+
=
=
ω ω
ω ω ϕ
ψ
ψ
ω ϕJones vectors
( )
x i y
oy ox i
oy i ox
t kz i oy i
ox
t kz i oy t
kz i ox
t kz i o
here, w
e E
E e
E e E
: vector
Jones
e E
j e
E i
e E j e
E i
e E t)
(z, E
y x x
x y
ϕ ϕ
ϕ
ϕ
∆
∆ ϕ
ϕ ϕ ω
ϕ ϕ ω
ω ϕ ω
−
=
→
+
=
+
=
=
−
+ + −
− +
−
) (
) ) (
(
) (
2 2
oy 2 ox
oy ox
2 2
o x
y
o y
x
x y
o o
x y
oy ox
oy
ox E
E E E 2
1 B
C , C B
E , A E
iC B
A C
B A
1 : 1/2) (m
or m
if POL.
ELLIPTIC
E i 2
2 ( by E
leads E
: wise -
cluck rotates
E when light
polarized circulary
- Right
E i 2
2 ( by E
leads E
: on - head viewed
wise, -
ck counterclu rotates
E when light
polarized circulary
- Left
2 i 1 :
if POL.
CIRCULAR
2 : 1 45 at
2
: 1 45 t
a :
orizontal h
:
vertical
2 1 : m
if POL.
LINEAR E E
Let,
= −
= +
=
=
⇒
+ +
+ +
≠
= −
→
−
=
⇒
=
→
=
⇒
± ±
=
−
=
− −
+
=
−
=
=
−
− ∆ϕ
α ϕ
∆
π π
ϕ
∆ ϕ π π ∆
ϕ π π ∆
ϕ π ϕ
ϕ
∆
α π α
ϕ ϕ
ϕ
∆
cos tan 2
, tan
1 2 ) 1
1 2 ) 1
1 2
1 1 1
1 1
0 1
0
sin cos
1 1
2
2
Jones matrices
−
+
→
−
= −
−
=
=
−
=
=
+
≡
=
+
−
+
−
β β
β β β
θ θ
π ε
ε ε
∆
π π
π π
ε ε ∆
ε ε
cos sin
sin cos
1 0
0 , 1
1 0
0 ), 1
2 (
0 0 , 1
0 0 ), 1
4 (
0 , 0 1 0
0
1 1
1 , 1
0 0
0 , 1
0 0
0 1 ,
2 / 2
/
4 / 4
/ '
'
: ) (
Rotator
e horizontal SA
HWP
e vertical
SA m
HWP
e i horizontal SA
QWP
e i vertical axis)
SA(slow
m QWP
/m 2 e
e e
e : plates) (wave
retarders Phase
2 1 45 at TA
vertical
TA
horizontal
axis) ssion
TA(transmi
polarizers
Linear
d c
b M a
: matrix
Jones E
E d c
b a E
E
i i
i i
x i y
i i
i
o y
x y
x
x y
x
o oy
ox
i i
i i
i
i i
o
o i
i
o o
i i
o
-45
E E
e i e e
i e e
e e M
45 /4
/8 2
-45
at Pol.
lliptical E
Pol.
circularly -
Left WP - Eighth
e e
45 at
Pol.
Linear
45 at
Pol.
Linear
horizon.)
HWP(SA
e i e i
Pol.
circularly -
Left 45 at
Pol.
Linear
horizon.)
QWP(SA
=
=
=
∴
+
= −
=
=
⇒
=
=
=
=
⇒ +
= −
−
−
⇒ +
+
= −
−
⇒ +
+
+ + + +
+
+ + +
+
+ +
α π
π ϕ
∆
α
π π π π
π
π π π
π
π π
, 1
2 1 2
1 1 2
1 1 2
1 1 2 1 0
0 1
0
0 1
1 1 2
1 1
1 2 1 1 0
0 1
1 2
1 1
1 2 1 0
0 1
8 / 4
/ 3 8
/ 4
/ 8
/
4 / 8
/ 2
/ 2
/
4 / 4
/
∑ − −
+
=
j j i j
fj m
Ne
) 2)
(( 2 0 1 2
)
0( ω
ε ε ω ω γ ωε
∑ − −
+
≅
=
j i j
j
f j m
Ne c
k c
o 2 0 (( 2 2) )
1 2
ω γ ω
ε ω ω
ω
ε ε
∑ − +
+ −
≅
=
j j j
j f j m
r Ne c k
n
(( 2 2)2 2 2) 2) ( 22 0 1 2
ω γ ω
ω
ω ω ω ε
∑ − +
≅
=
j j j
j f j c
m Ne ki
2) 2 )2 2 (( 2
0 2 2 2
ω γ ω
ω
γ εω
α Anomalous absorption: dn/dω< 0 between ω1 and ω!
At ω=ωj ; n-1=0 , above ωj ; n-1<0 (n<1): Phase velocity v=c/n>c!
∑ −
+
≅
j j
fj m
n
Ne2) ( 2
2 0 1 2
ω
ε ω Away from the resonance
ω εµ ) ( 0
= +
= since k iki
kr k
