Stokes Vectors

Scott Prahl

April 2020

import numpy as np
import matplotlib.pyplot as plt

import pypolar.jones as jones
import pypolar.mueller as mueller
import pypolar.visualization as vis

np.set_printoptions(suppress=True)  # print 1e-16 as zero

Introduction

The basics of Stokes vectors and their implementation in the pypolar modules. The basic ways of creating Stokes vectors is show as well as simple visualization.

This notebook could use some more work.

Basic Polarization

light = mueller.stokes_horizontal()
print("Stokes vector for horizontally-polarized light")
print(light)

light = mueller.stokes_linear(0)
print("Stokes vector for 0° linearly polarized light")
print(light)

light = mueller.stokes_vertical()
print("Stokes vector for vertically-polarized light")
print(light)

light = mueller.stokes_linear(np.pi/2)
print("Stokes vector for 90° linearly polarized light")
print(light)

light = mueller.stokes_linear(np.radians(45))
print("Stokes vector for 45° linearly polarized light")
print(light)

light = mueller.stokes_linear(np.radians(-45))
print("Stokes vector for -45° linearly polarized light")
print(light)

light = mueller.stokes_linear(np.radians(30))
print("Stokes vector for 30° linearly polarized light")
print(light)

light = mueller.stokes_linear(np.radians(-60))
print("Stokes vector for -60° linearly polarized light")
print(light)


light = mueller.stokes_right_circular()
print("Stokes vector for right circularly polarized light")
print(light)

light = mueller.stokes_left_circular()
print("Stokes vector for left circularly polarized light")
print(light)

Stokes vector for horizontally-polarized light
[1 1 0 0]
Stokes vector for 0° linearly polarized light
[1. 1. 0. 0.]
Stokes vector for vertically-polarized light
[ 1 -1  0  0]
Stokes vector for 90° linearly polarized light
[ 1. -1.  0.  0.]
Stokes vector for 45° linearly polarized light
[1. 0. 1. 0.]
Stokes vector for -45° linearly polarized light
[ 1.  0. -1.  0.]
Stokes vector for 30° linearly polarized light
[1.        0.5       0.8660254 0.       ]
Stokes vector for -60° linearly polarized light
[ 1.        -0.5       -0.8660254  0.       ]
Stokes vector for right circularly polarized light
[1 0 0 1]
Stokes vector for left circularly polarized light
[ 1  0  0 -1]

Stokes vectors for elliptical polarization

Examples from Appendix A in Kliger.

azimuth = 0
ellipticity = 0.5
light = mueller.stokes_elliptical(1, azimuth, ellipticity)
print("Elliptically polarized (azimuth=%.2f°, ellipticity=%.3f)" % (azimuth, ellipticity))
print(light)

azimuth = 0
ellipticity = -0.5
light = mueller.stokes_elliptical(1, azimuth, ellipticity)
print("Elliptically polarized (azimuth=%.2f°, ellipticity=%.3f)" % (azimuth, ellipticity))
print(light)

azimuth = 90
ellipticity = 0.5
light = mueller.stokes_elliptical(1, np.radians(azimuth), ellipticity)
print("Elliptically polarized (azimuth=%.2f°, ellipticity=%.3f)" % (azimuth, ellipticity))
print(light)

azimuth = 90
ellipticity = -0.5
light = mueller.stokes_elliptical(1, np.radians(azimuth), ellipticity)
print("Elliptically polarized (azimuth=%.2f°, ellipticity=%.3f)" % (azimuth, ellipticity))
print(light)

azimuth = 45
ellipticity = 0.414
light = mueller.stokes_elliptical(1, np.radians(azimuth), ellipticity)
print("Elliptically polarized (azimuth=%.2f°, ellipticity=%.3f)" % (azimuth, ellipticity))
print(light)

azimuth = 45
ellipticity = -0.414
light = mueller.stokes_elliptical(1, np.radians(azimuth), ellipticity)
print("Elliptically polarized (azimuth=%.2f°, ellipticity=%.3f)" % (azimuth, ellipticity))
print(light)

azimuth = 25.38
ellipticity = 0.342
light = mueller.stokes_elliptical(1, np.radians(azimuth), ellipticity)
print("Elliptically polarized (azimuth=%.2f°, ellipticity=%.3f)" % (azimuth, ellipticity))
print(light)

azimuth = -64.62
ellipticity = -0.342
light = mueller.stokes_elliptical(1, np.radians(azimuth), ellipticity)
print("Elliptically polarized (azimuth=%.2f°, ellipticity=%.3f)" % (azimuth, ellipticity))
print(light)

Elliptically polarized (azimuth=0.00°, ellipticity=0.500)
[1.  0.6 0.  0.8]
Elliptically polarized (azimuth=0.00°, ellipticity=-0.500)
[ 1.   0.6  0.  -0.8]
Elliptically polarized (azimuth=90.00°, ellipticity=0.500)
[ 1.  -0.6  0.   0.8]
Elliptically polarized (azimuth=90.00°, ellipticity=-0.500)
[ 1.  -0.6  0.  -0.8]
Elliptically polarized (azimuth=45.00°, ellipticity=0.414)
[1.         0.         0.70736455 0.70684892]
Elliptically polarized (azimuth=45.00°, ellipticity=-0.414)
[ 1.          0.          0.70736455 -0.70684892]
Elliptically polarized (azimuth=25.38°, ellipticity=0.342)
[1.         0.50008973 0.61229734 0.61237426]
Elliptically polarized (azimuth=-64.62°, ellipticity=-0.342)
[ 1.         -0.50008973 -0.61229734 -0.61237426]

Stokes to Jones Conversion

S = mueller.stokes_horizontal()
print("Stokes vector for horizontally-polarized light")
print(S)
J = mueller.stokes_to_jones(S)
print("Jones conversion")
print(J)
print("Actual Jones vector")
print(jones.field_horizontal())

Stokes vector for horizontally-polarized light
[1 1 0 0]
Jones conversion
[1.+0.j 0.+0.j]
Actual Jones vector
[1 0]

S = mueller.stokes_vertical()
print("Stokes vector for vertically-polarized light")
print(S)
J = mueller.stokes_to_jones(S)
print("Jones conversion")
print(J)
print("Actual Jones vector")
print(jones.field_vertical())

Stokes vector for vertically-polarized light
[ 1 -1  0  0]
Jones conversion
[0. 1.]
Actual Jones vector
[0 1]

angle = 30*np.pi/180
S = mueller.stokes_linear(angle)
print("Stokes vector for linearly-polarized light at %.1f°"%(angle*180/np.pi))
print(S)
J = mueller.stokes_to_jones(S)
print("Jones conversion")
print(J)
print("Actual Jones vector")
print(jones.field_linear(angle))

Stokes vector for linearly-polarized light at 30.0°
[1.        0.5       0.8660254 0.       ]
Jones conversion
[0.8660254+0.j 0.5      +0.j]
Actual Jones vector
[0.8660254 0.5      ]

S = mueller.stokes_right_circular()
print("Stokes vector for right-circular-polarized light")
print(S)
J = mueller.stokes_to_jones(S)
print("Jones conversion")
print(J)
print("Actual Jones vector")
print(jones.field_right_circular())

Stokes vector for right-circular-polarized light
[1 0 0 1]
Jones conversion
[0.70710678+0.j         0.        +0.70710678j]
Actual Jones vector
[0.70710678+0.j         0.        +0.70710678j]

S = mueller.stokes_left_circular()
print("Stokes vector for left-circular-polarized light")
print(S)
J = mueller.stokes_to_jones(S)
print("Jones conversion")
print(J)
print("Actual Jones vector")
print(jones.field_left_circular())

Stokes vector for left-circular-polarized light
[ 1  0  0 -1]
Jones conversion
[0.70710678+0.j         0.        -0.70710678j]
Actual Jones vector
[0.70710678+0.j         0.        -0.70710678j]

Visualization

These only show the polarized component of the Stokes vector.

theta = np.radians(45)
v = mueller.stokes_linear(theta)
vis.draw_stokes_field(v)

theta = np.radians(-30)
v = mueller.stokes_linear(theta)
vis.draw_stokes_field(v)

v = mueller.stokes_left_circular()
vis.draw_stokes_animated(v)

Once Loop Reflect

v=mueller.stokes_right_circular()
vis.draw_stokes_animated(v)

Once Loop Reflect

azimuth = -64.62
ellipticity = -0.342
v = mueller.stokes_elliptical(1, np.radians(azimuth), ellipticity)
vis.draw_stokes_animated(v)

Once Loop Reflect

Field Interpretation

v=mueller.stokes_horizontal()
mueller.interpret(v)

I = 1.000
Q = 1.000
U = 0.000
V = 0.000

'not implemented yet'

v=mueller.stokes_vertical()
mueller.interpret(v)

I = 1.000
Q = -1.000
U = 0.000
V = 0.000

'not implemented yet'

theta = np.radians(45)
v=mueller.stokes_linear(theta)
mueller.interpret(v)

I = 1.000
Q = 0.000
U = 1.000
V = 0.000

'not implemented yet'

v=np.array([3*np.exp(-1j*np.pi), 3*np.exp(-1j*np.pi/3)])
mueller.interpret(v)

Stokes vector must have four real elements

0