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17. Jones Matrices & Mueller Matrices
Jones Matrices
Rotation of coordinates - the rotation matrix
Stokes Parameters and unpolarized light
Mueller MatricesR. Clark Jones
(1916 - 2004)
Sir George G. Stokes
(1819 - 1903) Hans Mueller
(1900 - 1965)
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2 2
1 1 = = →
+
x x
x
y y x yx y
E EE E
E E E EE E
Define the polarization state of a field as a 2D vector—
“Jones vector” —containing the two complex amplitudes:
Jones vectors describe the polarization
state of a wave
A few examples:
0° linear (x) polarization: Ey /Ex = 0
linear (arbitrary angle) polarization: Ey /Ex = tan α
right or left circular polarization: Ey /Ex= ±j
1
0
1
tanα
1
j
±
(normalized to length of unity)
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To model the effect of a medium on light's
polarization state, we use Jones matrices.
1 0E E= A
1 11 0 12 0
1 21 0 22 0
x x y
y x y
E a E a E
E a E a E
= +
= +
This yields:
1 0
0 0x
=
AFor example, an x-polarizer can be written:
0 0
1 0
0
1 0
0 0 0
x x
x
y
E EE E
E
= = =
ASo:
Since we can write a polarization state as a (Jones) vector, we use
matrices, A, to transform them from the input polarization, E0, to the
output polarization, E1.11 12
21 22
=
a a
a aA
This should be
thought of as a
transfer function.
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Other Jones matrices
A y-polarizer:0 0
0 1y
=
A
1 0
0 1HWP
= −
AA half-wave plate:
A quarter-wave plate:1 0
0
= ±
QWPj
A1 0 1 1
0 1
= ± ± j j
1 0 1 1
0 1 1 1
= − −
1 0 1 1
0 1 1 1
= − −
A half-wave plate rotates 45-degree-
polarization to -45-degree, and vice versa.
R. Clark Jones
(1916 - 2004)
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The orientation of a wave plate
matters.
Remember that a quarter-wave plate
only converts linear to circular if the
input polarization is ±45°.
If it sees, say, x polarization,
the input is unchanged.
Jones matrices are an
extremely useful way to
keep track of all this.
1 0 1 1
0 0 0j
= −
AQWP
Wave plate
w/ axes at
0° or 90°
0° or 90° Polarizer
Note: this little cube is a
cartoon representation of a
polarizer. Cube polarizers are
commonly used in optics.
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A wave plate example
What does a quarter-wave plate do if the input polarization is linear
but at an arbitrary angle?
( ) ( )1 11 0
tan tan0
=
jj α α
AQWP Ein Eout
For arbitrary α, this is an elliptical polarization.
α = 30°
α = 45°
α = 60°
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Jones Matrices for standard components
0 0
0 1
Vertical (y) linear
polarizer:
1 0
0 0
Horizontal (x)
linear polarizer:
1 11
1 12
Linear polarizer
at 45 degrees:
Linear polarizer
at −45 degrees:
1 11
1 12
− −
41 0
0
jej
−
πQuarter-wave plate,
fast axis vertical:
11
12
j
j
−
Right circular
polarizer:
11
12
j
j
−
Quarter-wave plate,
fast axis horizontal:41 0
0
jej
π
Left circular
polarizer:
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Rotated Jones matrices
( ) ( )
( )
0 0 1 1' and '
cos( ) sin( )
sin( ) cos( )
E R E E R E
R
θ θ
θ θθ
θ θ
= =
− =
Rotation of a vector by an angle θ means multiply by the rotation matrix:
where:
( ) ( ) ( ) ( ) ( )1
1 1 0 0'− = = = E R E R E R R R Eθ θ θ θ θA A
Rotating E1 by θ and inserting the identity matrix R(θ)-1 R(θ), we have:
( ) ( ) 1' R Rθ θ −=A AThus:
( ) ( ) ( ) ( ) ( )1 1
0 0 0 ' ' '− − = = = R R R E R R E Eθ θ θ θ θA A A
What about when the polarizer or wave plate responsible for
the transfer function A is rotated by some angle, θ ?
rotated Jones vector
of the input
rotated Jones vector
of the output
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Rotated Jones matrix for a polarizer
( )cos( ) sin( ) 1 0 cos( ) sin( )
sin( ) cos( ) 0 0 sin( ) cos( )xA
θ θ θ θθ
θ θ θ θ−
= −
( ) ( ) 1'
−= R Rθ θA A
cos( ) sin( ) cos( ) sin( )
sin( ) cos( ) 0 0
θ θ θ θθ θ
− =
2
2
cos ( ) cos( )sin( )
cos( )sin( ) sin ( )
θ θ θθ θ θ
=
( )1
0xA
ψψ
ψ
≈
for a small
angle ψ
Example: apply this to an x polarizer.
( ) 1/ 2 1/ 245
1/ 2 1/ 2xA
=
o
So, for example:
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To model the effect of many media on light's
polarization state, we use many Jones matrices.
1 3 2 1 0E E= A A A
The order may look counter-intuitive, but order matters!
The aggregate effect of multiple components or objects can be
described by the product of the Jones matrix for each one.
E1E0
A1 A2 A3
6444447444448}
}
input outputtransfer function
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Multiplying Jones Matrices
Crossed polarizers:
x
y z
1 0y xE E= A A0E
1E
x-pol
y-pol
0 0 1 0 0 0
0 1 0 0 0 0
= =
y xA A so no light leaks through.
( )0 0 1 0 0
0 1 0 0
ψψ
ψ ψ
= =
y xA A
Uncrossed polarizers
(by a slight angle ψ): 0E1E
rotatedx-pol
y-pol
( )00 0
0
x x
y y x
E E
E E Eψ
ψψ
= =
y xA A So Iout ≈ ψ2 Iin,x
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Multiplying Jones Matrices x
y z
0Ex-pol
45º-pol
1E
y-pol
Now, it is easy to compute how
inserting a third polarizer
between two crossed polarizers
leads to larger transmission.
1 45 0y xE E= A A A
45
1 1 0 00 0 1 02 21 00 1 1 1 0 022 2
= = y xA A A
Thus:,
1
, ,
00 0
11 02 2
x in
y in x in
EE
E E
= = The third polarizer, between the other two, makes the
transmitted wave non-zero.
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Natural light (e.g., sunlight, light bulbs, etc.)
is unpolarized
The direction of the E vector is
randomly changing. But, it is
always perpendicular to the
propagation direction.
polarized light natural light
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Light with very complex polarization
vs. position is "unpolarized."
If the polarization vs. position is unresolvable, we call this
“unpolarized.” Otherwise, we refer to this light as “locally
polarized” or “partially polarized.”
Light that has scattered multiple times, or that has scattered randomly, often becomes unpolarized as a result.
Here, light from the blue sky is
polarized, so when viewed
through a polarizer it looks
much darker. Light from clouds
is unpolarized, so its intensity is
reduced by only 50%.
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When the phases of the x- and y-polarizations
fluctuate, we say the light is "unpolarized."
As long as the time-varying relative phase, θx(t)–θy(t), fluctuates, the light
will not remain in a single polarization state and hence is unpolarized.
( ) ( )0
0
1
exp
−
y
y x
x
Ej t j t
Eθ θ
In practice, the
amplitudes are also
functions of time!
The polarization state (Jones vector) is:
where θx(t) and θy(t) are functions that vary on a time scale slower than
the period of the wave, but faster than you can measure.
( )( ){ }( )( ){ }
0
0
( , ) Re exp
( , ) Re exp
= − −
= − −
x x x
y y y
E z t E j kz t t
E z t E j kz t t
ω θ
ω θ
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Stokes Parameters
#0 detects total irradiance............................................I0
#1 detects horizontally polarized irradiance..........N...I1
#2 detects +45° polarized irradiance............................I2
#3 detects right circularly polarized irradiance.....NN.I3
We cannot use Jones vectors to describe something that is rapidly
fluctuating like this. So, to treat fully, partially, or unpolarized light, we
use a different scheme. We define "Stokes parameters."
Suppose we have four detectors, three with polarizers in front of them:
S0
≡≡≡≡ I0
S1
≡≡≡≡ 2I1– I
0S2
≡≡≡≡ 2I2– I
0S3
≡≡≡≡ 2I3– I
0
The Stokes parameters:
Note that these
quantities are time-
averaged, so even
randomly polarized
light will give a well-
defined answer.
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Interpretation of the Stokes Parameters
S0
≡≡≡≡ I0
S1
≡≡≡≡ 2I1– I
0S2
≡≡≡≡ 2I2– I
0S3
≡≡≡≡ 2I3– I
0
The Stokes parameters:
S0 = the total irradiance
S1 = the excess in intensity of light transmitted by a horizontal polarizer
over light transmitted by a vertical polarizer
S2 = the excess in intensity of light transmitted by a 45° polarizer over
light transmitted by a 135° polarizer
S3 = the excess in intensity of light transmitted by a RCP filter over light
transmitted by a LCP filter
What do we mean when we say ‘unpolarized light’?
All three of these excess quantities are zero
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Degree of polarization
( )1/22 2 2
1 2 3 0Degree of polarization= S +S +S / S= 1 for polarized light
= 0 for unpolarized light
If any of the excess quantities (S1, S2, or S3) are non-zero,
then the wave has some degree of polarization. We can
quantify this by defining the “degree of polarization”:
Note that this quantity can never be greater than unity,
since S0 is the total intensity.
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polarized part:
( )
2 2 2 + + ≡
1 2 3
2 1
2
3
S S S
SS
S
S
unpolarized part:
( )
2 2 2 − + + ≡
0 1 2 3
1
S S S S
0S
0
0
The Stokes vector
We can write the four Stokes parameters in vector form:
0
1
2
3
S
SS
S
S
≡
The Stokes vector S contain information about both the
polarized part and the unpolarized part of the wave.
S = S(1) + S(2)
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Stokes vectors
(and Jones
vectors for
comparison)
Sir George G. Stokes
(1819 - 1903)
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Mueller Matrices multiply Stokes vectors
To model the effects of more than one medium on the polarization
state, just multiply the input polarization Stokes vector by all of the
Mueller matrices:
Sout = M3 M2 M1 Sin
(just like Jones matrices multiplying Jones vectors, except that the
vectors have four elements instead of two)
SoutSin
M1 M2 M3
We can define matrices that multiply Stokes vectors,
just as Jones matrices multiply Jones vectors. These
are called Mueller matrices.
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Mueller Matrices
(and Jones
Matrices for
comparison)
With Stokes vectors and
Mueller matrices, we can
describe light with arbitrarily
complicated combination of
polarized and unpolarized light.
Hans Mueller
(1900 - 1965)