17. Jones Matrices & Mueller MatricesJones 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)

2 2

1 1

x xx

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 = 0linear (arbitrary angle) polarization: Ey /Ex = tan right or left circular polarization: Ey /Ex = ±j

10

1tan

1j

(normalized to length of unity)

To model the effect of a medium on light'spolarization 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 EE a E a E

This yields:

1 00 0x

AFor example, an x-polarizer can be written:

0 01 0

0

1 00 0 0

x xx

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 aa a

A

This should be thought of as a

transfer function.

Other Jones matrices

A y-polarizer: 0 00 1y

A

1 00 1HWP

A

A half-wave plate:

A quarter-wave plate: 1 00

QWP j

A1 0 1 10 1

j j

1 0 1 10 1 1 1

1 0 1 10 1 1 1

A half-wave plate rotates 45-degree-polarization to -45-degree, and vice versa.

R. Clark Jones(1916 - 2004)

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 10 0 0j

AQWP

Wave plate w/ axes at 0° or 90°

0° or 90° Polarizer

A wave plate exampleWhat does a quarter-wave plate do if the input polarization is linear but at an arbitrary angle?

1 11 0

tan tan0

jj

AQWP Ein Eout

For arbitrary , this is an elliptical polarization.

= 30° = 45° = 60°

Jones Matrices for standard components

Rotated Jones matrices

0 0 1 1' and '

cos( ) sin( )sin( ) cos( )

E R E E R E

R

Rotation of a vector by an angle means multiply by the rotation matrix:

where:

11 1 0 0' E R E R E R R R E A A

Rotating E1 by and inserting the identity matrix R()-1 R(), we have:

1' R R A AThus:

1 10 0 0 ' ' ' R R R E R R E E A A A

What about when the polarizer or wave plate responsible for the transfer function A is rotated by some angle, ?

rotated Jones vector of the input

rotated Jones vector of the output

Rotated Jones matrix for a polarizer

cos( ) sin( ) 1 0 cos( ) sin( )sin( ) cos( ) 0 0 sin( ) cos( )xA

1' R R A A

cos( ) sin( ) cos( ) sin( )sin( ) cos( ) 0 0

2

2

cos ( ) cos( )sin( )cos( )sin( ) sin ( )

1

0xA

for a small angle

Example: apply this to an x polarizer.

1/ 2 1/ 245

1/ 2 1/ 2xA

So, for example:

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.

E1E0A1 A2 A3

input outputtransfer function

Multiplying Jones Matrices

Crossed polarizers:

x

y z

1 0y xE E A A0E

1Ex-pol

y-pol

0 0 1 0 0 00 1 0 0 0 0

y xA A so no light leaks through.

0 0 1 0 00 1 0 0

y xA A

Uncrossed polarizers(by a slight angle ): 0E 1E

rotatedx-pol

y-pol

00 0

0x x

y y x

E EE E E

y xA A So Iout ≈ 2 Iin,x

Multiplying Jones Matrices x

y z

0Ex-pol

45º-pol

1Ey-pol

Now, it is easy to compute how inserting a third polarizer between two crossed polarizersleads 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 0 22 2

y xA A A

Thus:,

1, ,

00 011 02 2

x in

y in x in

EE

E E

The third polarizer, between the other two, makes the transmitted wave non-zero.

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

Light with very complex polarizationvs. 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%.

When the phases of the x- and y-polarizations fluctuate, we say the light is "unpolarized."

As long as the time-varying relative phase, x(t)–y(t), fluctuates, the light will not remain in a single polarization state and hence is unpolarized.

0

0

1

exp

yy x

x

Ej t j t

E

In practice, the amplitudes are also functions of time!

The polarization state (Jones vector) is:

where x(t) and y(t) are functions that vary on a time scale slower thanthe 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

Stokes Parameters

#0 detects total irradiance............................................I0

#1 detects horizontally polarized irradiance..........…...I1

#2 detects +45° polarized irradiance............................I2

#3 detects right circularly polarized irradiance.....…….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 – I0 S2 2I2 – I0 S3 2I3 – I0

The Stokes parameters:

Note that these quantities are time-averaged, so even randomly polarized light will give a well-defined answer.

Interpretation of the Stokes Parameters

S0 I0 S1 2I1 – I0 S2 2I2 – I0 S3 2I3 – I0

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 we mean when we say ‘unpolarized light’: All three of these excess quantities are zero

Degree of polarization

1/22 2 21 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.

This is not the same as the ‘degree of polarization’defined in the homework problem, which was only defined for fully polarized light.

polarized part:

2 2 2

1 2 3

2 1

2

3

S S SSSSS

unpolarized part:

2 2 2

0 1 2 3

1

S S S S0S00

The Stokes vector

We can write the four Stokes parameters in vector form:0

1

2

3

SS

SSS

The Stokes vector S contain information about both the polarized part and the unpolarized part of the wave.

S = S(1) + S(2)

Stokes vectors (and Jones vectors for comparison)

Sir George G. Stokes(1819 - 1903)

Mueller Matrices multiply Stokes vectors

To model the effects of more than one medium on the polarizationstate, 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)

SoutSinM1 M2 M3

We can define matrices that multiply Stokes vectors, just as Jones matrices multiply Jones vectors. These are called Mueller matrices.

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)