The_Cambridge_Handbook_of_Physics_Formulas

170 Optics
8.6 Polarisation
Elliptical polarisation a
Elliptical
polarisation
Polarisation
angle b
Ellipticity c
E =(E 0x ,E 0y e iδ )e i(kz−ωt)
(8.80)
tan2α = 2E 0xE 0y
E0x 2 cosδ
−E2 0y
(8.81)
e = a−b
a
(8.82)
E electric field
k wavevector
z propagation axis
ωt angular frequency ×
time
E 0x x amplitude of E
E 0y y amplitude of E
δ relative phase of E y
with respect to E x
α polarisation angle
e
a
b
ellipticity
semi-major axis
semi-minor axis
I(θ) transmitted intensity
Malus’s law d I(θ)=I 0 cos 2 θ (8.83) I 0 incident intensity
θ polariser–analyser
angle
a See the introduction (page 161) for a discussion of sign and handedness conventions.
b Angle between ellipse major axis and x axis. Sometimes the polarisation angle is
defined as π/2−α.
c This is one of several definitions for ellipticity.
d Transmission through skewed polarisers for unpolarised incident light.
E 0y
y
b
α
θ
a
x
E 0x
Jones vectors and matrices
Normalised
electric field a E =
Example
vectors:
E x =
(
Ex
)
; |E| = 1 (8.84)
E y
( 1
E
0)
45 = √ 1 ( ) 1
2 1
E r = 1 √
2
(
1
−i
)
E l = 1 √
2
(
1
i
Jones matrix E t = AE i (8.85)
)
E
E x
E y
E 45
E r
E l
E t
E i
A
electric field
x component of E
y component of E
45 ◦ to x axis
right-hand circular
left-hand circular
transmitted vector
incident vector
Jones matrix
Example matrices:
( )
1 0
Linear polariser ‖ x
0 0
( )
Linear polariser at 45 ◦ 1 1 1
2 1 1
( )
1 1 i
Right circular polariser
2 −i 1
( )
λ/4 plate (fast ‖ x) e iπ/4 1 0
0 i
( )
0 0
Linear polariser ‖ y
0 1
( )
Linear polariser at −45 ◦ 1 1 −1
2 −1 1
( )
1 1 −i
Left circular polariser
2 i 1
( )
λ/4 plate (fast ⊥ x) e iπ/4 1 0
0 −i
a Known as the “normalised Jones vector.”