Stokes Vectors

Scott Prahl

Feb 2026

%config InlineBackend.figure_format = 'retina'

import sys
import numpy as np
import matplotlib.pyplot as plt

plt.rcParams["animation.html"] = "jshtml"

if sys.platform == "emscripten":
    import micropip

    await micropip.install("pypolar")

from pypolar import jones
from pypolar import mueller
from pypolar import visualization as vis

Introduction

The basics of Stokes vectors and their implementation in the pypolar modules. This notebooks shows how to create Stokes vectors, convert to and from Jones vectors, visualize, and interpret them.

Creation

The Stokes vectors are numpy arrays of [I,Q,U,V]. There are a bunch of routines that support linear, circular, and elliptical polarization.

Linear Polarization

light = mueller.stokes_horizontal()
print("Stokes vector for horizontally-polarized light")
print(np.round(light, 4))

light = mueller.stokes_linear(0)
print("Stokes vector for 0° linearly polarized light")
print(np.round(light, 4))

light = mueller.stokes_vertical()
print("Stokes vector for vertically-polarized light")
print(np.round(light, 4))

light = mueller.stokes_linear(np.pi / 2)
print("Stokes vector for 90° linearly polarized light")
print(np.round(light, 4))

light = mueller.stokes_linear(np.radians(45))
print("Stokes vector for 45° linearly polarized light")
print(np.round(light, 4))

light = mueller.stokes_linear(np.radians(-45))
print("Stokes vector for -45° linearly polarized light")
print(np.round(light, 4))

light = mueller.stokes_linear(np.radians(30))
print("Stokes vector for 30° linearly polarized light")
print(np.round(light, 4))

light = mueller.stokes_linear(np.radians(-60))
print("Stokes vector for -60° linearly polarized light")
print(np.round(light, 4))

Circular Polarization

light = mueller.stokes_right_circular()
print("Stokes vector for right circularly polarized light")
print(np.round(light, 4))

light = mueller.stokes_left_circular()
print("Stokes vector for left circularly polarized light")
print(np.round(light, 4))

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   0.866 0.   ]
Stokes vector for -60° linearly polarized light
[ 1.    -0.5   -0.866  0.   ]
Stokes vector for right circularly polarized light
[1 0 0 1]
Stokes vector for left circularly polarized light
[ 1  0  0 -1]

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(np.round(light, 4))

azimuth = 0
ellipticity = -0.5
light = mueller.stokes_elliptical(1, azimuth, ellipticity)
print("Elliptically polarized (azimuth=%.2f°, ellipticity=%.3f)" % (azimuth, ellipticity))
print(np.round(light, 4))

azimuth = 90
ellipticity = 0.5
light = mueller.stokes_elliptical(1, np.radians(azimuth), ellipticity)
print("Elliptically polarized (azimuth=%.2f°, ellipticity=%.3f)" % (azimuth, ellipticity))
print(np.round(light, 4))

azimuth = 90
ellipticity = -0.5
light = mueller.stokes_elliptical(1, np.radians(azimuth), ellipticity)
print("Elliptically polarized (azimuth=%.2f°, ellipticity=%.3f)" % (azimuth, ellipticity))
print(np.round(light, 4))

azimuth = 45
ellipticity = 0.414
light = mueller.stokes_elliptical(1, np.radians(azimuth), ellipticity)
print("Elliptically polarized (azimuth=%.2f°, ellipticity=%.3f)" % (azimuth, ellipticity))
print(np.round(light, 4))

azimuth = 45
ellipticity = -0.414
light = mueller.stokes_elliptical(1, np.radians(azimuth), ellipticity)
print("Elliptically polarized (azimuth=%.2f°, ellipticity=%.3f)" % (azimuth, ellipticity))
print(np.round(light, 4))

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(np.round(light, 4))

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(np.round(light, 4))

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.     0.7074 0.7068]
Elliptically polarized (azimuth=45.00°, ellipticity=-0.414)
[ 1.      0.      0.7074 -0.7068]
Elliptically polarized (azimuth=25.38°, ellipticity=0.342)
[1.     0.5001 0.6123 0.6124]
Elliptically polarized (azimuth=-64.62°, ellipticity=-0.342)
[ 1.     -0.5001 -0.6123 -0.6124]

Stokes to Jones Conversion

S = mueller.stokes_horizontal()
print("Stokes vector for horizontally-polarized light")
print(np.round(S, 4))
J = mueller.stokes_to_jones(S)
print("Jones conversion")
print(np.round(J, 4))
print("Actual Jones vector")
print(np.round(jones.field_horizontal(), 4))

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(np.round(S, 4))
J = mueller.stokes_to_jones(S)
print("Jones conversion")
print(np.round(J, 4))
print("Actual Jones vector")
print(np.round(jones.field_vertical(), 4))

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(np.round(S, 4))
J = mueller.stokes_to_jones(S)
print("Jones conversion")
print(np.round(J, 4))
print("Actual Jones vector")
print(np.round(jones.field_linear(angle), 4))

Stokes vector for linearly-polarized light at 30.0°
[1.    0.5   0.866 0.   ]
Jones conversion
[0.866+0.j 0.5  +0.j]
Actual Jones vector
[0.866 0.5  ]

S = mueller.stokes_right_circular()
print("Stokes vector for right-circular-polarized light")
print(np.round(S, 4))
J = mueller.stokes_to_jones(S)
print("Jones conversion")
print(np.round(J, 4))
print("Actual Jones vector")
print(np.round(jones.field_right_circular(), 4))

Stokes vector for right-circular-polarized light
[1 0 0 1]
Jones conversion
[0.7071+0.j     0.    +0.7071j]
Actual Jones vector
[0.7071+0.j     0.    +0.7071j]

S = mueller.stokes_left_circular()
print("Stokes vector for left-circular-polarized light")
print(np.round(S, 4))
J = mueller.stokes_to_jones(S)
print("Jones conversion")
print(np.round(J, 4))
print("Actual Jones vector")
print(np.round(jones.field_left_circular(), 4))

Stokes vector for left-circular-polarized light
[ 1  0  0 -1]
Jones conversion
[0.7071+0.j     0.    -0.7071j]
Actual Jones vector
[0.7071+0.j     0.    -0.7071j]

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)
plt.show()

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

Once Loop Reflect

Field Interpretation

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

I = 1.000
Q = 1.000
U = 0.000
V = 0.000
Degree of polarization = 1.000
Fully polarized light
Linear polarization at 0.0 degrees CCW from x-axis

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

I = 1.000
Q = -1.000
U = 0.000
V = 0.000
Degree of polarization = 1.000
Fully polarized light
Linear polarization at 90.0 degrees CCW from x-axis

theta = np.radians(15)
v = mueller.stokes_linear(theta)
s = mueller.interpret(v)
print(s)

I = 1.000
Q = 0.866
U = 0.500
V = 0.000
Degree of polarization = 1.000
Fully polarized light
Linear polarization at 15.0 degrees CCW from x-axis

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

Malformed input: input has non-zero imaginary parts; Mueller/Stokes quantities must be real

v = np.array([1, 2])
s = mueller.interpret(v)
print(s)

Malformed input: expected a Stokes vector with shape (4,) or a Mueller matrix with shape (4, 4), got shape (2,)