Jones and Mueller polarization - Optics

## Python polarization

### Features

Py-pol is a Python library for Jones and Stokes-Mueller polarization optics. It has 4 main modules:

• jones_vector - Light polarization states in Jones formalism (2x1 vectors).

• jones_matrix - Optical elements polarization properties in Jones formalism (2x2 matrices).

• stokes - Light polarization states in Mueller-Stokes formalism (4x1 vectors).

• mueller - Optical elements polarization properties in Mueller-Stokes formalism (4x4 matrices).

Each one has its own class, with multiple methods for generation, operation and parameters extraction.

### Examples

#### Jones formalism

Generating Jones vectors and Matrices

from py_pol.jones_vector import Jones_vector
from py_pol.jones_matrix import Jones_matrix
from py_pol.utils import degrees

j0 = Jones_vector("j0")
j0.linear_light(angle=45*degrees)

m0 = Jones_matrix("m0")
m0.diattenuator_linear( p1=0.75, p2=0.25, angle=45*degrees)

m1 = Jones_matrix("m1")
m1.quarter_waveplate(angle=0 * degrees)

j1=m1*m0*j0

Extracting information form Jones Vector.

print(j0,'\n')
print(j0.parameters)
j0 = [+0.707; +0.707]

parameters of j0:
intensity        : 1.000 arb.u
alpha            : 45.000 deg
delay            : 0.000 deg
azimuth          : 45.000 deg
ellipticity angle: 0.000 deg
a, b             : 1.000  0.000
phase            : 0.000 deg
print(j1,'\n')
print(j1.parameters)
m1 * m0 @45.00 deg * j0 = [+0.530+0.000j; +0.000+0.530j]'

parameters of m1 * m0 @45.00 deg * j0:
intensity        : 0.562 arb.u
alpha            : 45.000 deg
delay            : 90.000 deg
azimuth          : 8.618 deg
ellipticity angle: -45.000 deg
a, b             : 0.530  0.530
phase            : 0.000 deg

Extracting information form Jones Matrices.

print(m0,'\n')
print(m0.parameters)
m0 @45.00 deg =
[+0.500, +0.250]
[+0.250, +0.500]

parameters of m0 @45.00 deg:
is_homogeneous: True
delay:          0.000 deg
diattenuation:  0.800
print(m1,'\n')
print(m1.parameters)
m1 =
[+1+0j, +0+0j]
[+0+0j, +0+1j]

parameters of m1:
is_homogeneous: True
delay:          90.000 deg
diattenuation:  0.000

#### Stokes-Mueller formalism

Generating Stokes vectors and Mueller matrices.

from py_pol.stokes import Stokes
from py_pol.mueller import Mueller
from py_pol.utils import degrees

j0 = Stokes("j0")
j0.linear_light(angle=45*degrees)

m1 = Mueller("m1")
m1.diattenuator_linear(p1=1, p2=0, angle=0*degrees)

j1=m1*j0

Extracting information from Stokes vectors.

Determining the intensity of a Stokes vector:

i1=j0.parameters.intensity()
print("intensity = {:4.3f} arb. u.".format(i1))
intensity = 1.000 arb. u.

Determining all the parameters of a Stokes vector:

print(j0,'\n')
print(j0.parameters)
j0 = [ +1;  +0;  +1;  +0]

parameters of j0:
intensity             : 1.000 arb. u.
amplitudes            : E0x 0.707, E0y 0.707, E0_unpol 0.000
degree polarization   : 1.000
degree linear pol.    : 1.000
degree   circular pol.: 0.000
alpha                 : 45.000 deg
delay                 : 0.000 deg
azimuth               : 45.000 deg
ellipticity  angle    : 0.000 deg
ellipticity  param    : 0.000
eccentricity          : 1.000
polarized vector      : [+1.000; +0.000; +1.000; +0.000]'
unpolarized vector    : [+0.000; +0.000; +0.000; +0.000]'
print(j1,'\n')
print(j1.parameters)
m1 * j0 = [+0.500; +0.500; +0.000; +0.000]

parameters of m1 * j0:
intensity             : 0.500 arb. u.
amplitudes            : E0x 0.707, E0y 0.000, E0_unpol 0.000
degree polarization   : 1.000
degree linear pol.    : 1.000
degree   circular pol.: 0.000
alpha                 : 0.000 deg
delay                 : 0.000 deg
azimuth               : 0.000 deg
ellipticity  angle    : 0.000 deg
ellipticity  param    : 0.000
eccentricity          : 1.000
polarized vector      : [+0.500; +0.500; +0.000; +0.000]'
unpolarized vector    : [+0.000; +0.000; +0.000; +0.000]'

Extracting information from Mueller matrices.

m2 = Mueller("m2")
m2.diattenuator_retarder_linear(D=90*degrees, p1=1, p2=0.5, angle=0)
delay = m2.parameters.retardance()
print("delay = {:2.1f}Âº".format(delay/degrees))
delay = 90.0Âº

There is a function in Parameters_Jones_Vector class, .get_all() that will compute all the parameters available and stores in a dictionary .dict_params(). Info about dict parameters can be revised using the print function.

print(m2,'\n')
m2.parameters.get_all()
print(m2.parameters)
m2 =
[+0.6250, +0.3750, +0.0000, +0.0000]
[+0.3750, +0.6250, +0.0000, +0.0000]
[+0.0000, +0.0000, +0.0000, +0.5000]
[+0.0000, +0.0000, -0.5000, +0.0000]

Parameters of m2:
Transmissions:
- Mean                  : 62.5 %.
- Maximum               : 100.0 %.
- Minimum               : 25.0 %.
Diattenuation:
- Total                 : 0.600.
- Linear                : 0.600.
- Circular              : 0.000.
Polarizance:
- Total                 : 0.600.
- Linear                : 0.600.
- Circular              : 0.000.
Spheric purity              : 0.872.
Retardance                  : 1.571.
Polarimetric purity         : 1.000.
Depolarization degree       : 0.000.
Depolarization factors:
- Euclidean distance    : 1.732.
- Depolarization factor : 0.000.
Polarimetric purity indices:
- P1                    : 1.000.
- P2                    : 1.000.
- P3                    : 1.000.

There are many types of Mueller matrices. The Check_Mueller calss implements all the checks that can be performed in order to clasify a Mueller matrix. They are stored in the checks field of Mueller class.

m1 = Mueller("m1")
m1.diattenuator_linear(p1=1, p2=0.2, angle=0*degrees)
print(m1,'\n')

c1 = m1.checks.is_physical()
c2 = m1.checks.is_homogeneous()
c3 = m1.checks.is_retarder()
print('The linear diattenuator is physical: {}; hogeneous: {}; and a retarder: {}.'.format(c1, c2, c3))
m1 =
[+0.520, +0.480, +0.000, +0.000]
[+0.480, +0.520, +0.000, +0.000]
[+0.000, +0.000, +0.200, +0.000]
[+0.000, +0.000, +0.000, +0.200]

The linear diattenuator is physical: True; hogeneous: True; and a retarder: False.

#### Drawings

The modules also allows to obtain graphical representation of polarization.

Drawing polarization ellipse for Jones vectors.

Drawing polarization ellipse for Stokes vectors with random distribution due to unpolarized part of light.

Drawing Stokes vectors in Poincare sphere.

### Authors

• Jesus del Hoyo <jhoyo@ucm.es>

• Luis Miguel Sanchez Brea <optbrea@ucm.es>

Universidad Complutense de Madrid, Faculty of Physical Sciences, Department of Optics Plaza de las ciencias 1, ES-28040 Madrid (Spain)

### References

1. Goldstein “Polarized light” 2nd edition, Marcel Dekker (1993).

1. Gil, R. Ossikovsky “Polarized light and the Mueller Matrix approach”, CRC Press (2016).

1. Brosseau “Fundamentals of Polarized Light” Wiley (1998).

1. Martinez-Herrero, P. M. Mejias, G. Piquero “Characterization of partially polarized light fields” Springer series in Optical sciences (2009).

1. Bennet “Handbook of Optics 1” Chapter 5 ‘Polarization’.

1. Chipman “Handbook of Optics 2” Chapter 2 ‘Polarimetry’.

1. Lu and RA Chipman, “Homogeneous and inhomogeneous Jones matrices”, J. Opt. Soc. Am. A 11(2) 766 (1994).

### Acknowlegments

This software was initially developed for the project Retos-ColaboraciÃ³n 2016 “Ecograb” (RTC-2016-5277-5) and “Teluro-AEI” (RTC2019-007113-3) of Ministerio de EconomÃ­a y Competitivdad (Spain) and the European funds for regional development (EU), led by Luis Miguel Sanchez-Brea.

## History

### 1.1.3

• bugfix zeroslike

• bugfix not taking shape = False as flatten the object.

### 1.1.2

• bugfix Jones_vector.parameters.global_phase

### 1.1.1

• from_list can also be used with lists of Py_pol objects, lists and tuples. Also, all elemeents may have more than one elemnt.

• minor bugfixes

### 1.1.0 (2022-3-31)

• Objects are now iterable

• 3D drawings changed to Plotly

• New density and existence methods

### 1.0.4 (2022-02-07)

• Bug fix related to variable limits

• Bug fixes

• Bug fixes

### 1.0.2 (2020-07-04)

• Implemented workaround of axis_equal issue.

### 1.0.0 (2020-06-04)

py_pol multidimensional. Alpha state

This is a big overhaul with many changes. All of them are based on the possibility of storing several vector/matrices in the same object. This reduces significantly the time required to perform the same operation to multiple vectors/matrices, using numpy methods instead of for loops. We have calculated that the reduction is around one order of magnitude.

New methods have been introduced. First, methods available for Mueller / Stoes modules have been created also for Jones (when possible). Also, some bugs and errors in the calculations have been solved.

Finally, some method and argument names have been changed to be consistent between different classes. Also, the default value of arguments with the same name have also been unified.

The biggest TO DO we have are tests. Right now, we only have tests for the Jones_vector class. However, we thought that it would be useful to release this version so the community can use it.

NOTE: Due to the change of argument and method names, this version is not compatible with the previous ones.

• Bug fixes.

### 0.2.1 (2019-09-04)

• Bug fixes.

• Solve incidents.

• Start to homogenize structures for both Jones and Stokes.

### 0.2.0 (2019-05-25)

pre-alpha state

• Stable version including tests

### 0.1.5 (2019-02-25)

• Jones_vector, Jones_matrix, Stokes works.

• Jones_vector: simplify function to represent better Jones vectors.

• tests drawing: Made tests for drawing

• Mueller is in progress.

• Functions = 9/10

• Documentation = 8/10

• Tutorial = 8/10.

• Examples = 8/10.

• Tests = 8/10

• Drawing = 10/10. Finished. Polarization ellipse for Jones and Stokes (partially random). Stokes on PoincarÃ© sphere.

• bug fixes

### 0.1.3 (2019-01-22)

• Fixed axis_equal issue.

• Jones_vector, Jones_matrix, Stokes works.

• Mueller is in progress.

• Functions = 9/10

• Documentation = 8/10

• Tutorial = 7/10.

• Examples = 6/10.

• Drawing = 0/10.

### 0.1.1 (2018-12-22)

• First release on PyPI in alpha state.

### 0.0.0 (2018-11-22)

First implementation of py_pol.