Download Polarized and unpolarised transverse waves

Polarized and unpolarised transverse waves
T. Johnson
13-02-15
Electromagnetic Processes In Dispersive Media, Lecture 6
1
Outline
•  The quarter wave plate
•  Set up coordinate system suitable for transverse waves
•  Jones calculus; matrix formulation of how wave polarization
changes when passing through polarizing component
–  Examples: linear polarizer, quarter wave plate, Faraday rotation
•  Statistical representation of incoherent/unpolarized waves
–  Stokes vector and polarization tensor
•  Poincare sphere
•  Muller calculus;
matrix formulation
for the transmission of
partially polarized waves
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Electromagnetic Processes In Dispersive Media, Lecture 6
2
Modifying wave polarization in a quarter wave plate (1)
•  Last lecture we noted that in birefringent crystals:
–  there are two modes: O-mode and X-mode
⎧ nO 2 = K⊥
⎪
⎨ 2
K⊥K||
⎪ n X =
2
2
K
sin
θ
+
K
cos
θ
⎩
⊥
||
⎧⎪eO (k) = (0 , 1 , 0)
⎨
⎪⎩e X (k) ∝ (K|| cosθ , 0 , K⊥ sinθ)
–  thus if K⊥ > K|| then nO ≥ n X
the O-mode has larger phase velocity
€
€ •  Next describe Quarter wave plates
–  uniaxial crystal; normal in z-direction
€
c
π /2
–  length L in the x-direction: L =
€
ω K|| − K⊥
–  Let a wave travel in the x-direction, then k is in the x-direction and θ=π/2
⎧⎪ n 2 = K
O
⊥
⎨ 2
⎪⎩ n X = K||
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⎧eO (k) = (0 , 1 , 0)
⎨€
⎩e X (k) = (0 , 0 , 1)
Electromagnetic Processes In Dispersive Media, Lecture 6
3
Modifying wave polarization in a quarter wave plate (2)
•  Plane wave ansats has to match dispersion relation
–  when the wave entre the crystal it will move slow, this corresponds to a
change in wave length, or k
ωnO ω
ωn X ω
kO =
=
K⊥ , k X =
=
K||
c
c
c
c
–  since the O- and X-mode travel at different speeds we write
€
E( t, x ) = ℜ{eO E O exp(ikO x − iωt) + e X E X exp(ikX x − iωt)}
•  Assume as initial condition a linearly polarized wave
€
E = [1 1 0] = (eO + e X ) ⇒ E O = E X = 1
⇒ E( t, x ) = ℜ{eO exp(ikO x − iωt) + e X exp(ikX x − iωt)}
= eO cos(kO x − ωt) + e X cos(kO x − ωt + Δkx) , Δk ≡ k X − kO
€
–  the difference in wave number causes the O- and X-mode to
drift in and out of phase with each other!
13-02-15
Electromagnetic Processes In Dispersive Media, Lecture 6
4
Modifying wave polarization in a quarter wave plate (3)
•  The polarization when the wave exits the crystal at x=L
E( t, x ) = [eO cos(kO L − ωt) + e X cos(kO L − ωt + ΔkL)] , Δkx = π /2
= [eO cos(kO L − ωt) − e X sin(kO L − ωt)]
€
L=
c
π /2
ω K|| − K⊥
– 
– 
– 
– 
– 
This is cicular polarization!
€
The crystal converts linear to circular polarization (and
vice versa)
Called a quarter wave plate; a common component in optical systems
But work only at one wave length – adapted for e.g. a specific laser!
In general, waves propagating in birefringent crystal change polarization
back and forth between linear to circular polarization
–  Switchable wave plates can be made from liquid crystal
•  angle of polarization can be switched by electric control system
•  Similar effect is Faraday effect in magnetoactive media
–  but the eigenmode are the circularly polarized components
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Electromagnetic Processes In Dispersive Media, Lecture 6
5
Optical systems
•  In optics, interferometry, polarometry, etc, there is an interest
in following how the wave polarization changes when passing
through e.g. an optical system.
•  For this purpose two types of calculus have been developed;
–  Jones calculus; only for coherent (polarized) wave
–  Muller calculus; for both coherent, unpolarised and partially polarised
•  In both cases the wave is given by vectors E and S (defined
later) and polarizing elements are given by matrixes J and M
E out = J • E in
Sout = M • Sin
13-02-15
€
Electromagnetic Processes In Dispersive Media, Lecture 6
6
The polarization of transverse waves
•  Lets first introduce a new coordinate system representing
vectors in the transverse plane, i.e. perpendicular to the k.
–  Construct an orthonormal basis for {e1,e2,κ}, where κ=k/|k|
–  The transverse plane is then given by {e1,e2}, where
e2
α
α
i i
e =e e
k
e1
–  where α=1,2 and ei , i=1,2,3 is any basis for R3
–  denote e1 the horizontal and e2 the vertical directions
€•  The electric field then has different component
representations: Ei (for i=1,2,3) and Eα (for α=1,2)
E i = eαi E α
–  similar for the polarization vector, eM
eM ,i = eαi eαM
€
€
The new coordinates provides 2D representations
13-02-15
Electromagnetic Processes In Dispersive Media, Lecture 6
7
Some simple Jones Matrixes
•  In the new coordinate system the Jones matrix is 2x2:
J
⎡ J11
= ⎢
⎣J 21
αβ
J12 ⎤
⎥
J 22 ⎦
α
E out
= J αβ E inβ
•  Example: Linear polarizer transmitting horizontal polarization
€
J
αβ
L,V
⎡1 0 ⎤
⎡1 0 ⎤ ⎡ E H ⎤ ⎡ E H ⎤
= ⎢
⎥ → ⎢
⎥ ⎢ ⎥ = ⎢ ⎥
⎣0 0 ⎦
⎣0 0 ⎦ ⎣ E V ⎦ ⎣ 0 ⎦
•  Example: Attenuator transmitting a fraction ρ of the energy
€
–  Note: energy ~ ε0|E|2
⎡1 0 ⎤
J Att (ρ) = ρ ⎢
⎥ →
⎣0 1 ⎦
αβ
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⎡1 0 ⎤⎡ E H ⎤
⎡ E H ⎤
ρ ⎢
⎥⎢ ⎥ = ρ ⎢ ⎥
⎣0 1 ⎦⎣ E V ⎦
⎣ E V ⎦
Electromagnetic Processes In Dispersive Media, Lecture 6
8
Jones matrix for a quarter wave plate
•  Quarter wave plates are birefringent
(have two different refractive index)
–  align the plate such that horizontal / veritical polarization
(corresponding to O/X-mode) has wave numbers k1 / k2
⎡ E H (x) ⎤ ⎡ E H (0)exp(ik1 x) ⎤
⎥
⎢
⎥ = ⎢
2
⎣ E V (x) ⎦ ⎣ E V (0)exp(ik x) ⎦
–  let the light entre the plate start at x=0 and exit at x=L
⎡e ik1 L
E(L) = ⎢
⎣ 0
€
0 ⎤ ⎡ E H (0) ⎤
⎥
⎥ ≡ J Ph E(L)
ik 2 L ⎢
e ⎦ ⎣ E V (0) ⎦
–  where Ph stands for phaser
•  Quarter wave plates chages the relative phase by π/2
⎡1 0 ⎤
€
1
2
ik 1 L
k L − k L = ±π /2 →JQ = e ⎢
⎥
⎣0 ±i ⎦
–  usually we considers only relative phase and skip factor exp(ik1L)
13-02-15
Electromagnetic Processes In Dispersive Media, Lecture 6
9
Jones matrix for a rotated birefringent media
•  If a birefringent media (e.g. quarter wave plates) is not aligned
with the axis of our coordinate system with axis of the media
–  then we may use a rotation matrix
⎡cos(θ) −sin(θ) ⎤
−1
R(θ) = ⎢
→
R
(θ) = R(−θ)
⎥
⎣ sin(θ) cos(θ) ⎦
EM2
e2
–  Eigenmode have directions as in the fig.:
€
⎡cos(θ) ⎤
⎡−sin(θ) ⎤
E(x) = E M 1 (x) ⎢
⎥ + E M 2 (x) ⎢
⎥
sin(θ)
cos(θ)
⎣
⎦
⎣
⎦
EM1
θ
e1
–  apply two rotation: first -θ and then +θ, i.e. no net rotation:
€
⎛
⎡ E M 1 (x) ⎤
⎡cos(θ) ⎤
⎡−sin(θ) ⎤⎞
E(x) = R(θ)R(−θ)⎜ E M 1 (x) ⎢
⎥ =
⎥ + E M 2 (x) ⎢
⎥⎟ = R(θ) ⎢
⎣ sin(θ) ⎦
⎣ cos(θ) ⎦⎠
⎣ E M 2 (x) ⎦
no net rotation ⎝
ik 1 x
⎡e ik 1 x
⎤
⎡
⎡
⎤
E M 1 (0)
0
e
0 ⎤
= R(θ) ⎢
⎥⎢
⎥
⎥ → J Ph = R(θ) ⎢
ik 2 x
ik 2 x
⎣ 0 e ⎦⎣ E M 2 (0) ⎦
⎣ 0 e ⎦
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Electromagnetic Processes In Dispersive Media, Lecture 6
10
Jones matrix for a Faraday rotation
•  Faraday rotation is similar to birefrigency, except that
eigenmodes have circularly polarized eigenvector
1 ⎡1⎤
1 ⎡ 1 ⎤
E(x) = E M 1 (x)
⎢ ⎥ + E M 2 (x)
⎢ ⎥
i
2 ⎣ ⎦
2 ⎣ −i ⎦
–  there is no useful rotation matrix
–  instead use a unitary matrix
1 ⎡1 −i ⎤
1 ⎡1 i ⎤
−1
U=
⎢
⎥ → U =
⎢
⎥
1
i
1
−i
2 ⎣
2 ⎣
⎦
⎦
€
⎛
⎡
⎤
⎡1⎤
⎡ 1 ⎤⎞
−1 E M 1 (x)
E(x) = U U⎜ E M 1 (x) ⎢ ⎥ + E M 2 (x) ⎢ ⎥⎟ = U ⎢
⎥ =
⎣i ⎦
⎣−i ⎦⎠
⎣ E M 2 (x) ⎦
⎝
ik 1 x
⎡e ik 1 x
⎤
⎡
⎤
⎡
⎤
E
(0)
0
e
0
M1
−1
= U −1 ⎢
→
J
=
U
⎢
⎥
⎥
2 ⎥ ⎢
FR
ik x
ik 2 x
⎣ 0 e ⎦⎣ E M 2 (0) ⎦
⎣ 0 e ⎦
−1
€
13-02-15
Electromagnetic Processes In Dispersive Media, Lecture 6
11
Outline
•  The quarter wave plate
•  Set up coordinate system suitable for transverse waves
•  Jones calculus; matrix formulation of how wave
polarization changes when passing through polarizing
component
–  Examples: linear polarizer, quarter wave plate, Faraday rotation
•  Statistical representation of incoherent/unpolarized waves
–  Stokes vector and polarization tensor
•  Poincare sphere
•  Muller calculus; matrix formulation for the transmission of
partially polarized waves
13-02-15
Electromagnetic Processes In Dispersive Media, Lecture 6
12
Incoherent/unpolarised
•  Many sources of electromagnetic radiation are not coherent
–  they do not radiate perfect harmonic oscillations (sinusoidal wave)
•  over short time scales the oscillations look harmonic
•  but over longer periods the wave look incoherent, or even stochastic
–  such waves are often referred to as unpolarised
•  To model such waves we will consider the electric field to be a stochastic
process, i.e. it has
–  an average: < Eα (t,x) >
–  a variance: < Eα (t,x) Eβ (t,x) >
–  a covariance: < Eα (t,x) Eβ (t+s,x+y) >
•  In this chapter we will focus on the variance, which we will refer to as the
intensity tensor
Iαβ = < Eα (t,x) Eβ (t,x) >
and the polarization tensor (where eM=E / |E| is the polarization vector)
pαβ = < eMα (t,x) eMβ (t,x) >
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Electromagnetic Processes In Dispersive Media, Lecture 6
13
The Stokes vector
•  It can be shown that the intensity tensor is hermitian
–  thus it can be described by four Stokes parameter {I,Q,U,V} :
1 ⎡ I + Q U − iV ⎤
αβ
I = ⎢
⎥
U
+
iV
I
−
Q
2 ⎣
⎦
•  A basis for hermitian matrixes is a set of unitary four matrixes:
€
αβ
1
τ
⎛1 0⎞
⎛1 0 ⎞
⎛0 1⎞
⎛0 −i⎞
αβ
αβ
αβ
= ⎜
⎟ , τ2 = ⎜
⎟ , τ3 = ⎜
⎟ , τ4 = ⎜
⎟
0
1
0
−1
1
0
i
0
⎝
⎠
⎝
⎠
⎝
⎠
⎝
⎠
–  where the last three matrixes are the Pauli matrixes
€
•  Define the Stokes vector: SA=[I,Q,U,V]
I
αβ
I αβ
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⎛ 1 0 ⎞
⎛ 0 1⎞
⎛0 −i⎞ ⎤
1 ⎡ ⎛1 0⎞
= ⎢I⎜
⎟ + Q⎜
⎟ + U⎜
⎟ + V ⎜
⎟ ⎥
2 ⎣ ⎝0 1⎠
⎝ 0 −1⎠
⎝ 1 0⎠
⎝ i 0 ⎠ ⎦
1
αβ αβ
= ταβ
A S A with inverse : S A = τ A I
2
Electromagnetic Processes In Dispersive Media, Lecture 6
14
Representations for the polarization tensor
•  The polarization tensor has similar representation
–  Note: trace(pαβ)=1, thus it is described by three parameter {q,u,v} :
αβ
p
1 ⎡⎛ 1 0⎞ ⎛1 0 ⎞ ⎛0 1⎞ ⎛ 0 −i⎞⎤
= ⎢⎜
⎟ + q⎜
⎟ + u⎜
⎟ + v⎜
⎟⎥
2 ⎣⎝ 0 1⎠ ⎝0 −1⎠ ⎝1 0⎠ ⎝ i 0 ⎠⎦
•  As we will show in the following slides the four terms above
€
represents different types of polarization
–  unpolarised (incoherent)
–  linear polarization
–  circular polarization
13-02-15
Electromagnetic Processes In Dispersive Media, Lecture 6
15
Examples
•  For example consider:
–  linearly polarised wave eMα=[1,0]
αβ
p
⎡1 ⎤
⎡1 0 ⎤ 1 ⎛ ⎡1 0 ⎤ ⎡1 0 ⎤⎞
= e e = ⎢ ⎥[1 0] = ⎢
⎥ = ⎜ ⎢
⎥ +1⎢
⎥⎟
⎣0 ⎦
⎣0 0 ⎦ 2 ⎝ ⎣0 1 ⎦ ⎣0 −1⎦⎠
α
M
β*
M
•  i.e. {q,u,v}={1,0,0}
–  rotate linearly polarization by 45o, eMα=[1,1] / 21/2
€
⎡1 1⎤ 1 ⎛ ⎡1 0 ⎤ ⎡0 1 ⎤⎞
1 ⎡1⎤
αβ
α β*
p = eM eM = ⎢ ⎥[1 1] = ⎢
⎥ = ⎜ ⎢
⎥ +1⎢
⎥⎟
2 ⎣1⎦
⎣1 1⎦ 2 ⎝ ⎣0 1 ⎦ ⎣1 0 ⎦⎠
•  i.e. {q,u,v}={0,1,0}
–  a circularly polarised wave, eMα=[1,-i] / 21/2
€
p
αβ
1 ⎡1⎤
1 ⎡1 −i ⎤ 1 ⎛ ⎡1 0 ⎤ ⎡0 −i ⎤⎞
= e e = ⎢ ⎥[1 −i] = ⎢
⎥ = ⎜ ⎢
⎥ +1⎢
⎥⎟
2 ⎣i ⎦
2 ⎣i 1 ⎦ 2 ⎝ ⎣0 1 ⎦ ⎣ i 0 ⎦⎠
α
M
β*
M
•  i.e. {q,u,v}={0,0,1}
13-02-15
€
Electromagnetic Processes In Dispersive Media, Lecture 6
16
The polarization tensor for unpolarized waves (1)
•  What are the Stokes parameters for unpolarised waves?
–  Let the eM1 and eM2 be independent stochastic variable
pαβ
⎛ e1M ⎞*
=< ⎜ 2 ⎟ (e1M
⎝ eM ⎠
⎡ < e1 *e1 > < e1 *e 2 >⎤
M
M
⎥
eM2 ) >= ⎢ M2 * 1M
*
2
2
⎣ < eM eM > < eM eM >⎦
–  Since eM1 and eM2 are uncorrelated the offdiagonal term vanish
pαβ
€
⎡<| e1M |2 >
0 ⎤
= ⎢
2 2 ⎥
<| eM | > ⎦
⎣ 0
1 2
M
2 2
M
=1
–  The vector eM is normalised: e + e
–  By symmetry (no physical difference between eM1 and eM2 )
€
€
1 2
M
2 2
M
e
=e
= 1/2
–  the polarization tensor
€ then reads
1 ⎡1 0 ⎤
αβ
p = ⎢
⎥
2 ⎣0 1 ⎦
–  i.e. unpolarised have {q,u,v}={0,0,0}!
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Electromagnetic Processes In Dispersive Media, Lecture 6
17
€
The polarization tensor for unpolarized waves (2)
•  Alternative derivation; polarization vector for unpolarized waves
–  Note first that the polarization vector is normalised
eM
2
2
= e1M + eM2
2
= 1 ~ cos2 (θ) + sin 2 (θ)
⎛e1M ⎞ ⎛e iφ 1 cos(θ)⎞
–  the polarization is complex and stochastic: ⎜ 2 ⎟ = ⎜ iφ
⎟
2
⎝eM ⎠ ⎝ e sin(θ) ⎠
•  where θ, φ and φ are
1
2
uniformly distibuted in [0,2π]
€
•  The corresponding polarization tensor
pαβ
⎛ e1M ⎞*
=< ⎜ 2 ⎟ (e1M
⎝ eM ⎠
−iφ 1 +iφ 2
€
⎛e −iφ 1 +iφ 1 cos(θ)cos(θ)
e
cos(θ)sin(θ)⎞
2
eM ) >=< ⎜ −iφ 2 +iφ 1
⎟ >
−iφ 2 +iφ 2
sin(θ)cos(θ) e
sin(θ)sin(θ) ⎠
⎝ e
–  here the average is over the three random variables θ, φ1 and φ2
€
1
pαβ =
(2π) 3
⎛
cos2 (θ)
e −iφ 1 +iφ 2 cos(θ)sin(θ)⎞ 1 ⎛1 0⎞
⎟ = ⎜
⎟
∫ dθ ∫ dφ1 ∫ dφ 2⎜e −iφ 2 +iφ 1 sin(θ)cos(θ)
2
0
1
2
sin (θ)
⎝
⎠
⎝
⎠
0
0
0
2π
2π
2π
–  i.e. unpolarised have {q,u,v}={0,0,0}!
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Electromagnetic Processes In Dispersive Media, Lecture 6
18
Poincare sphere
•  The polarised part of a wave field describes the normalised
vector {q/r,u/r,v/r} where r = q 2 + u 2 + v 2 is the degree of polarization
–  since this vector is real and normalised it will represent points on a
sphere, the so called Poincare sphere
€
u
r
•  Thus, any transverse wave field can
be described by
–  a point on the Poincare sphere
–  a degree of polarization, r
q
v
Poincare sphere
•  A polarizing element induces a motion on the sphere
–  e.g. when passing though a birefringent crystal we trace a circle on the
Poincare sphere
13-02-15
Electromagnetic Processes In Dispersive Media, Lecture 6
19
Outline
•  Set up coordinate system suitable for transverse waves
•  Jones calculus; matrix formulation of how wave
polarization changes when passing through polarizing
component
–  Examples: linear polarizer, quarter wave plate, Faraday rotation
•  Statistical representation of incoherent/unpolarized waves
–  Stokes vector and polarization tensor
•  Poincare sphere
•  Muller calculus; a matrix formulation for the transmission
of arbitrarily polarized waves
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Electromagnetic Processes In Dispersive Media, Lecture 6
20
Weakly anisotropic media
•  Next we will study Muller calculus for partially polorized waves
•  We will do so for weakly anisotropic media:
K αβ = n 0 2δαβ + ΔK αβ
€
–  where ΔKαβ is a small perturbation
–  although Muller calculus is not restricted to weak anisotropy
•  The wave equation
(n
2
)
− n 0 2 E α = ΔK αβ E α
–  when ΔKij is a small, the 1st order dispersion relation reads: n2≈n02
–  the left hand side can then be expanded to give
€
small, <<1
⎡ (n − n 0 ) ⎤
x
n − n = (n − n 0 )(n + n 0 ) = (n − n 0 )n 0 ⎢2 +
⎥ ≈ 2n 0 (n − n 0 )
n 0 ⎦
⎣
2
2
0
2n 0 (n − n 0 )E α ≈ ΔK αβ E α
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Electromagnetic Processes In Dispersive Media, Lecture 6
21
The wave equation as an ODE
•  Make an eikonal ansatz (assume k is in the x-direction):
⎛ ω
⎞ ⎛ ω
⎞
E = E (t)exp(ikx ) = E (t)exp⎜i n 0 x ⎟ exp⎜i ( n − n 0 ) x ⎟
⎝ c
⎠ ⎝ c
⎠
α
α
0
α
0
dE α
ω
ω
α
= i n 0 E + i (n − n 0 ) E α
dx
c
c
same expression as
on previous page!
•  The wave equation can then be written as
dE α
n 0ω α
ω
= −i
E +i
ΔK αβ E α
dx
c
2cn 0
€
–  describe how the wave changes when propagating through a media!
€
•  Wave equation for the intensity tensor:
dI αβ
d
iω
*
*
=
< E α E β >= ... =
ΔK αρδβσ − ΔK βσ δαρ I ρσ
dx
dx
2cn 0
(
€
13-02-15
Electromagnetic Processes In Dispersive Media, Lecture 6
)
22
The wave equation as an ODE
•  The wave equation the intensity tensor is not very convinient
αβ αβ
•  Instead, rewrite it in terms of the Stokes vector: SA = τA I
dSA
= (ρ AB − µ AB ) SB
dx
€
⎧
iω
H ,αρ βα ρβ
βρ
τA τB − ΔK H ,σβ τρσ
A τB )
⎪⎪ρ AB = 4cn ( ΔK
0
⎨
⎪µ = iω ΔK€H ,αρ τβα τρβ − ΔK H ,σβ τρσ τβρ
A B
A B )
⎪⎩ AB 4cn 0 (
–  we may call this the differential formulation of Muller calculus
–  symmetric matrix ρAB describes non-dissipative changes in polarization
€
–  and the antisymmetric
matrix µAB describes dissipation (absorption)
•  The ODE for SA has the analytic solution (cmp to the ODE y’=ky)
[
]
SA (x) = δ AB + (ρ AB − µ AB ) x +1/2(ρ AC − µ AC )(ρCB − µCB ) x 2 + ... SB (0) =
[
]
= exp (ρ AB − µ AB ) x SB (0) = M AB SB (0)
–  where MAB is called the Muller matrix
–  MAB represents entire optical components
€
•  we have a component based Muller calculus
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Electromagnetic Processes In Dispersive Media, Lecture 6
23
Examples of Muller matrixes
•  For illustration only – don’t memorise!
Linear polarizer
(45o transmission)
Linear polarizer
(Horizontal Transmission)
L, H
M AB
1
1
0
0
0
0
0
0
0 ⎤
⎥
0 ⎥
0 ⎥
⎥
0 ⎦
Quarter wave plate
(fast axis horizontal)
€
Q, H
M AB
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€
⎡1
⎢
1 ⎢1
=
2 ⎢0
⎢
⎣0
⎡1
⎢
0
⎢
=
⎢0
⎢
⎣0
0 ⎤ €
⎥
0 ⎥
0 0 −1⎥
⎥
0 1 0 ⎦
0 0
1 0
L,45
M AB
⎡ 1
⎢
1 ⎢ 0
=
2 ⎢ −1
⎢
⎣ 0
0 −1 0 ⎤
⎥
0 0 0 ⎥
0 1 0 ⎥
⎥
0 0 0 ⎦
Attenuating filter
(30% Transmission)
⎡1
⎢
0
Att
⎢
M AB (0.3) = 0.3
⎢0
⎢
⎣0
Electromagnetic Processes In Dispersive Media, Lecture 6
0
1
0
0
0
0
1
0
0 ⎤
⎥
0 ⎥
0 ⎥
⎥
1 ⎦
24
Examples of Muller matrixes
•  In optics it is common to connect a series of optical elements
•  consider a system with:
–  a linear polarizer and
–  a quarter wave plate
Q, H
L,45 in
SAout = M AB
M BC
SC
•  Insert unpolarised light, SAin=[1,0,0,0]
–  Step 1: Linear polariser transmit linear polarised light
€
T
L,45
S step1 = M BC
[1 0 0 0] = [1 0 −1 0]
T
–  Step 2: Quarter wave plate transmit circularly polarised light
€
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S
out
=M
Q, H
[1
T
0 −1 0] = [1 0 0 −1]
T
Electromagnetic Processes In Dispersive Media, Lecture 6
25