17. jones matrices & mueller matrices...17. jones matrices & mueller matrices jones matrices...

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17. Jones Matrices & Mueller Matrices Jones Matrices Rotation of coordinates - the rotation matrix Stokes Parameters and unpolarized light Mueller Matrices R. Clark Jones (1916 - 2004) Sir George G. Stokes (1819 - 1903) Hans Mueller (1900 - 1965)

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Page 1: 17. Jones Matrices & Mueller Matrices...17. Jones Matrices & Mueller Matrices Jones Matrices Rotation of coordinates -the rotation matrix Stokes Parameters and unpolarized light Mueller

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)

Page 2: 17. Jones Matrices & Mueller Matrices...17. Jones Matrices & Mueller Matrices Jones Matrices Rotation of coordinates -the rotation matrix Stokes Parameters and unpolarized light Mueller

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)

Page 3: 17. Jones Matrices & Mueller Matrices...17. Jones Matrices & Mueller Matrices Jones Matrices Rotation of coordinates -the rotation matrix Stokes Parameters and unpolarized light Mueller

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.

Page 4: 17. Jones Matrices & Mueller Matrices...17. Jones Matrices & Mueller Matrices Jones Matrices Rotation of coordinates -the rotation matrix Stokes Parameters and unpolarized light Mueller

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)

Page 5: 17. Jones Matrices & Mueller Matrices...17. Jones Matrices & Mueller Matrices Jones Matrices Rotation of coordinates -the rotation matrix Stokes Parameters and unpolarized light Mueller

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.

Page 6: 17. Jones Matrices & Mueller Matrices...17. Jones Matrices & Mueller Matrices Jones Matrices Rotation of coordinates -the rotation matrix Stokes Parameters and unpolarized light Mueller

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°

Page 7: 17. Jones Matrices & Mueller Matrices...17. Jones Matrices & Mueller Matrices Jones Matrices Rotation of coordinates -the rotation matrix Stokes Parameters and unpolarized light Mueller

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:

Page 8: 17. Jones Matrices & Mueller Matrices...17. Jones Matrices & Mueller Matrices Jones Matrices Rotation of coordinates -the rotation matrix Stokes Parameters and unpolarized light Mueller

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

Page 9: 17. Jones Matrices & Mueller Matrices...17. Jones Matrices & Mueller Matrices Jones Matrices Rotation of coordinates -the rotation matrix Stokes Parameters and unpolarized light Mueller

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:

Page 10: 17. Jones Matrices & Mueller Matrices...17. Jones Matrices & Mueller Matrices Jones Matrices Rotation of coordinates -the rotation matrix Stokes Parameters and unpolarized light Mueller

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

Page 11: 17. Jones Matrices & Mueller Matrices...17. Jones Matrices & Mueller Matrices Jones Matrices Rotation of coordinates -the rotation matrix Stokes Parameters and unpolarized light Mueller

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

Page 12: 17. Jones Matrices & Mueller Matrices...17. Jones Matrices & Mueller Matrices Jones Matrices Rotation of coordinates -the rotation matrix Stokes Parameters and unpolarized light Mueller

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.

Page 13: 17. Jones Matrices & Mueller Matrices...17. Jones Matrices & Mueller Matrices Jones Matrices Rotation of coordinates -the rotation matrix Stokes Parameters and unpolarized light Mueller

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

Page 14: 17. Jones Matrices & Mueller Matrices...17. Jones Matrices & Mueller Matrices Jones Matrices Rotation of coordinates -the rotation matrix Stokes Parameters and unpolarized light Mueller

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%.

Page 15: 17. Jones Matrices & Mueller Matrices...17. Jones Matrices & Mueller Matrices Jones Matrices Rotation of coordinates -the rotation matrix Stokes Parameters and unpolarized light Mueller

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

Page 16: 17. Jones Matrices & Mueller Matrices...17. Jones Matrices & Mueller Matrices Jones Matrices Rotation of coordinates -the rotation matrix Stokes Parameters and unpolarized light Mueller

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.

Page 17: 17. Jones Matrices & Mueller Matrices...17. Jones Matrices & Mueller Matrices Jones Matrices Rotation of coordinates -the rotation matrix Stokes Parameters and unpolarized light Mueller

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

Page 18: 17. Jones Matrices & Mueller Matrices...17. Jones Matrices & Mueller Matrices Jones Matrices Rotation of coordinates -the rotation matrix Stokes Parameters and unpolarized light Mueller

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.

Page 19: 17. Jones Matrices & Mueller Matrices...17. Jones Matrices & Mueller Matrices Jones Matrices Rotation of coordinates -the rotation matrix Stokes Parameters and unpolarized light Mueller

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)

Page 20: 17. Jones Matrices & Mueller Matrices...17. Jones Matrices & Mueller Matrices Jones Matrices Rotation of coordinates -the rotation matrix Stokes Parameters and unpolarized light Mueller

Stokes vectors

(and Jones

vectors for

comparison)

Sir George G. Stokes

(1819 - 1903)

Page 21: 17. Jones Matrices & Mueller Matrices...17. Jones Matrices & Mueller Matrices Jones Matrices Rotation of coordinates -the rotation matrix Stokes Parameters and unpolarized light Mueller

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.

Page 22: 17. Jones Matrices & Mueller Matrices...17. Jones Matrices & Mueller Matrices Jones Matrices Rotation of coordinates -the rotation matrix Stokes Parameters and unpolarized light Mueller

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)