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Mini Shell
Mini Shell
#!/usr/bin/env python
# -*- coding: utf-8 -*-
"""
The module implements routines to model the polarization of optical fields
and can be used to calculate the effects of polarization optical elements on
the fields.
- Jones vectors.
- Stokes vectors.
- Jones matrices.
- Mueller matrices.
Examples
========
We calculate a generic Jones vector:
>>> from sympy import symbols, pprint, zeros, simplify
>>> from sympy.physics.optics.polarization import (jones_vector, stokes_vector,
... half_wave_retarder, polarizing_beam_splitter, jones_2_stokes)
>>> psi, chi, p, I0 = symbols("psi, chi, p, I0", real=True)
>>> x0 = jones_vector(psi, chi)
>>> pprint(x0, use_unicode=True)
⎡-ⅈ⋅sin(χ)⋅sin(ψ) + cos(χ)⋅cos(ψ)⎤
⎢ ⎥
⎣ⅈ⋅sin(χ)⋅cos(ψ) + sin(ψ)⋅cos(χ) ⎦
And the more general Stokes vector:
>>> s0 = stokes_vector(psi, chi, p, I0)
>>> pprint(s0, use_unicode=True)
⎡ I₀ ⎤
⎢ ⎥
⎢I₀⋅p⋅cos(2⋅χ)⋅cos(2⋅ψ)⎥
⎢ ⎥
⎢I₀⋅p⋅sin(2⋅ψ)⋅cos(2⋅χ)⎥
⎢ ⎥
⎣ I₀⋅p⋅sin(2⋅χ) ⎦
We calculate how the Jones vector is modified by a half-wave plate:
>>> alpha = symbols("alpha", real=True)
>>> HWP = half_wave_retarder(alpha)
>>> x1 = simplify(HWP*x0)
We calculate the very common operation of passing a beam through a half-wave
plate and then through a polarizing beam-splitter. We do this by putting this
Jones vector as the first entry of a two-Jones-vector state that is transformed
by a 4x4 Jones matrix modelling the polarizing beam-splitter to get the
transmitted and reflected Jones vectors:
>>> PBS = polarizing_beam_splitter()
>>> X1 = zeros(4, 1)
>>> X1[:2, :] = x1
>>> X2 = PBS*X1
>>> transmitted_port = X2[:2, :]
>>> reflected_port = X2[2:, :]
This allows us to calculate how the power in both ports depends on the initial
polarization:
>>> transmitted_power = jones_2_stokes(transmitted_port)[0]
>>> reflected_power = jones_2_stokes(reflected_port)[0]
>>> print(transmitted_power)
cos(-2*alpha + chi + psi)**2/2 + cos(2*alpha + chi - psi)**2/2
>>> print(reflected_power)
sin(-2*alpha + chi + psi)**2/2 + sin(2*alpha + chi - psi)**2/2
Please see the description of the individual functions for further
details and examples.
References
==========
.. [1] https://en.wikipedia.org/wiki/Jones_calculus
.. [2] https://en.wikipedia.org/wiki/Mueller_calculus
.. [3] https://en.wikipedia.org/wiki/Stokes_parameters
"""
from sympy import sin, cos, exp, I, pi, sqrt, Matrix, Abs, re, im, simplify
from sympy.physics.quantum import TensorProduct
def jones_vector(psi, chi):
"""A Jones vector corresponding to a polarization ellipse with `psi` tilt,
and `chi` circularity.
Parameters
==========
``psi`` : numeric type or sympy Symbol
The tilt of the polarization relative to the `x` axis.
``chi`` : numeric type or sympy Symbol
The angle adjacent to the mayor axis of the polarization ellipse.
Returns
=======
Matrix :
A Jones vector.
Examples
========
The axes on the Poincaré sphere.
>>> from sympy import pprint, symbols, pi
>>> from sympy.physics.optics.polarization import jones_vector
>>> psi, chi = symbols("psi, chi", real=True)
A general Jones vector.
>>> pprint(jones_vector(psi, chi), use_unicode=True)
⎡-ⅈ⋅sin(χ)⋅sin(ψ) + cos(χ)⋅cos(ψ)⎤
⎢ ⎥
⎣ⅈ⋅sin(χ)⋅cos(ψ) + sin(ψ)⋅cos(χ) ⎦
Horizontal polarization.
>>> pprint(jones_vector(0, 0), use_unicode=True)
⎡1⎤
⎢ ⎥
⎣0⎦
Vertical polarization.
>>> pprint(jones_vector(pi/2, 0), use_unicode=True)
⎡0⎤
⎢ ⎥
⎣1⎦
Diagonal polarization.
>>> pprint(jones_vector(pi/4, 0), use_unicode=True)
⎡√2⎤
⎢──⎥
⎢2 ⎥
⎢ ⎥
⎢√2⎥
⎢──⎥
⎣2 ⎦
Anti-diagonal polarization.
>>> pprint(jones_vector(-pi/4, 0), use_unicode=True)
⎡ √2 ⎤
⎢ ── ⎥
⎢ 2 ⎥
⎢ ⎥
⎢-√2 ⎥
⎢────⎥
⎣ 2 ⎦
Right-hand circular polarization.
>>> pprint(jones_vector(0, pi/4), use_unicode=True)
⎡ √2 ⎤
⎢ ── ⎥
⎢ 2 ⎥
⎢ ⎥
⎢√2⋅ⅈ⎥
⎢────⎥
⎣ 2 ⎦
Left-hand circular polarization.
>>> pprint(jones_vector(0, -pi/4), use_unicode=True)
⎡ √2 ⎤
⎢ ── ⎥
⎢ 2 ⎥
⎢ ⎥
⎢-√2⋅ⅈ ⎥
⎢──────⎥
⎣ 2 ⎦
"""
return Matrix([-I*sin(chi)*sin(psi) + cos(chi)*cos(psi),
I*sin(chi)*cos(psi) + sin(psi)*cos(chi)])
def stokes_vector(psi, chi, p=1, I=1):
"""A Stokes vector corresponding to a polarization ellipse with ``psi``
tilt, and ``chi`` circularity.
Parameters
==========
``psi`` : numeric type or sympy Symbol
The tilt of the polarization relative to the ``x`` axis.
``chi`` : numeric type or sympy Symbol
The angle adjacent to the mayor axis of the polarization ellipse.
``p`` : numeric type or sympy Symbol
The degree of polarization.
``I`` : numeric type or sympy Symbol
The intensity of the field.
Returns
=======
Matrix :
A Stokes vector.
Examples
========
The axes on the Poincaré sphere.
>>> from sympy import pprint, symbols, pi
>>> from sympy.physics.optics.polarization import stokes_vector
>>> psi, chi, p, I = symbols("psi, chi, p, I", real=True)
>>> pprint(stokes_vector(psi, chi, p, I), use_unicode=True)
⎡ I ⎤
⎢ ⎥
⎢I⋅p⋅cos(2⋅χ)⋅cos(2⋅ψ)⎥
⎢ ⎥
⎢I⋅p⋅sin(2⋅ψ)⋅cos(2⋅χ)⎥
⎢ ⎥
⎣ I⋅p⋅sin(2⋅χ) ⎦
Horizontal polarization
>>> pprint(stokes_vector(0, 0), use_unicode=True)
⎡1⎤
⎢ ⎥
⎢1⎥
⎢ ⎥
⎢0⎥
⎢ ⎥
⎣0⎦
Vertical polarization
>>> pprint(stokes_vector(pi/2, 0), use_unicode=True)
⎡1 ⎤
⎢ ⎥
⎢-1⎥
⎢ ⎥
⎢0 ⎥
⎢ ⎥
⎣0 ⎦
Diagonal polarization
>>> pprint(stokes_vector(pi/4, 0), use_unicode=True)
⎡1⎤
⎢ ⎥
⎢0⎥
⎢ ⎥
⎢1⎥
⎢ ⎥
⎣0⎦
Anti-diagonal polarization
>>> pprint(stokes_vector(-pi/4, 0), use_unicode=True)
⎡1 ⎤
⎢ ⎥
⎢0 ⎥
⎢ ⎥
⎢-1⎥
⎢ ⎥
⎣0 ⎦
Right-hand circular polarization
>>> pprint(stokes_vector(0, pi/4), use_unicode=True)
⎡1⎤
⎢ ⎥
⎢0⎥
⎢ ⎥
⎢0⎥
⎢ ⎥
⎣1⎦
Left-hand circular polarization
>>> pprint(stokes_vector(0, -pi/4), use_unicode=True)
⎡1 ⎤
⎢ ⎥
⎢0 ⎥
⎢ ⎥
⎢0 ⎥
⎢ ⎥
⎣-1⎦
Unpolarized light
>>> pprint(stokes_vector(0, 0, 0), use_unicode=True)
⎡1⎤
⎢ ⎥
⎢0⎥
⎢ ⎥
⎢0⎥
⎢ ⎥
⎣0⎦
"""
S0 = I
S1 = I*p*cos(2*psi)*cos(2*chi)
S2 = I*p*sin(2*psi)*cos(2*chi)
S3 = I*p*sin(2*chi)
return Matrix([S0, S1, S2, S3])
def jones_2_stokes(e):
"""Return the Stokes vector for a Jones vector `e`.
Parameters
==========
``e`` : sympy Matrix
A Jones vector.
Returns
=======
sympy Matrix
A Jones vector.
Examples
========
The axes on the Poincaré sphere.
>>> from sympy import pprint, pi
>>> from sympy.physics.optics.polarization import jones_vector
>>> from sympy.physics.optics.polarization import jones_2_stokes
>>> H = jones_vector(0, 0)
>>> V = jones_vector(pi/2, 0)
>>> D = jones_vector(pi/4, 0)
>>> A = jones_vector(-pi/4, 0)
>>> R = jones_vector(0, pi/4)
>>> L = jones_vector(0, -pi/4)
>>> pprint([jones_2_stokes(e) for e in [H, V, D, A, R, L]],
... use_unicode=True)
⎡⎡1⎤ ⎡1 ⎤ ⎡1⎤ ⎡1 ⎤ ⎡1⎤ ⎡1 ⎤⎤
⎢⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎥
⎢⎢1⎥ ⎢-1⎥ ⎢0⎥ ⎢0 ⎥ ⎢0⎥ ⎢0 ⎥⎥
⎢⎢ ⎥, ⎢ ⎥, ⎢ ⎥, ⎢ ⎥, ⎢ ⎥, ⎢ ⎥⎥
⎢⎢0⎥ ⎢0 ⎥ ⎢1⎥ ⎢-1⎥ ⎢0⎥ ⎢0 ⎥⎥
⎢⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎥
⎣⎣0⎦ ⎣0 ⎦ ⎣0⎦ ⎣0 ⎦ ⎣1⎦ ⎣-1⎦⎦
"""
ex, ey = e
return Matrix([Abs(ex)**2 + Abs(ey)**2,
Abs(ex)**2 - Abs(ey)**2,
2*re(ex*ey.conjugate()),
-2*im(ex*ey.conjugate())])
def linear_polarizer(theta=0):
"""A linear polarizer Jones matrix with transmission axis at
an angle ``theta``.
Parameters
==========
``theta`` : numeric type or sympy Symbol
The angle of the transmission axis relative to the horizontal plane.
Returns
=======
sympy Matrix
A Jones matrix representing the polarizer.
Examples
========
A generic polarizer.
>>> from sympy import pprint, symbols
>>> from sympy.physics.optics.polarization import linear_polarizer
>>> theta = symbols("theta", real=True)
>>> J = linear_polarizer(theta)
>>> pprint(J, use_unicode=True)
⎡ 2 ⎤
⎢ cos (θ) sin(θ)⋅cos(θ)⎥
⎢ ⎥
⎢ 2 ⎥
⎣sin(θ)⋅cos(θ) sin (θ) ⎦
"""
M = Matrix([[cos(theta)**2, sin(theta)*cos(theta)],
[sin(theta)*cos(theta), sin(theta)**2]])
return M
def phase_retarder(theta=0, delta=0):
"""A phase retarder Jones matrix with retardance `delta` at angle `theta`.
Parameters
==========
``theta`` : numeric type or sympy Symbol
The angle of the fast axis relative to the horizontal plane.
``delta`` : numeric type or sympy Symbol
The phase difference between the fast and slow axes of the
transmitted light.
Returns
=======
sympy Matrix :
A Jones matrix representing the retarder.
Examples
========
A generic retarder.
>>> from sympy import pprint, symbols
>>> from sympy.physics.optics.polarization import phase_retarder
>>> theta, delta = symbols("theta, delta", real=True)
>>> R = phase_retarder(theta, delta)
>>> pprint(R, use_unicode=True)
⎡ -ⅈ⋅δ -ⅈ⋅δ ⎤
⎢ ───── ───── ⎥
⎢⎛ ⅈ⋅δ 2 2 ⎞ 2 ⎛ ⅈ⋅δ⎞ 2 ⎥
⎢⎝ℯ ⋅sin (θ) + cos (θ)⎠⋅ℯ ⎝1 - ℯ ⎠⋅ℯ ⋅sin(θ)⋅cos(θ)⎥
⎢ ⎥
⎢ -ⅈ⋅δ -ⅈ⋅δ ⎥
⎢ ───── ─────⎥
⎢⎛ ⅈ⋅δ⎞ 2 ⎛ ⅈ⋅δ 2 2 ⎞ 2 ⎥
⎣⎝1 - ℯ ⎠⋅ℯ ⋅sin(θ)⋅cos(θ) ⎝ℯ ⋅cos (θ) + sin (θ)⎠⋅ℯ ⎦
"""
R = Matrix([[cos(theta)**2 + exp(I*delta)*sin(theta)**2,
(1-exp(I*delta))*cos(theta)*sin(theta)],
[(1-exp(I*delta))*cos(theta)*sin(theta),
sin(theta)**2 + exp(I*delta)*cos(theta)**2]])
return R*exp(-I*delta/2)
def half_wave_retarder(theta):
"""A half-wave retarder Jones matrix at angle `theta`.
Parameters
==========
``theta`` : numeric type or sympy Symbol
The angle of the fast axis relative to the horizontal plane.
Returns
=======
sympy Matrix
A Jones matrix representing the retarder.
Examples
========
A generic half-wave plate.
>>> from sympy import pprint, symbols
>>> from sympy.physics.optics.polarization import half_wave_retarder
>>> theta= symbols("theta", real=True)
>>> HWP = half_wave_retarder(theta)
>>> pprint(HWP, use_unicode=True)
⎡ ⎛ 2 2 ⎞ ⎤
⎢-ⅈ⋅⎝- sin (θ) + cos (θ)⎠ -2⋅ⅈ⋅sin(θ)⋅cos(θ) ⎥
⎢ ⎥
⎢ ⎛ 2 2 ⎞⎥
⎣ -2⋅ⅈ⋅sin(θ)⋅cos(θ) -ⅈ⋅⎝sin (θ) - cos (θ)⎠⎦
"""
return phase_retarder(theta, pi)
def quarter_wave_retarder(theta):
"""A quarter-wave retarder Jones matrix at angle `theta`.
Parameters
==========
``theta`` : numeric type or sympy Symbol
The angle of the fast axis relative to the horizontal plane.
Returns
=======
sympy Matrix
A Jones matrix representing the retarder.
Examples
========
A generic quarter-wave plate.
>>> from sympy import pprint, symbols
>>> from sympy.physics.optics.polarization import quarter_wave_retarder
>>> theta= symbols("theta", real=True)
>>> QWP = quarter_wave_retarder(theta)
>>> pprint(QWP, use_unicode=True)
⎡ -ⅈ⋅π -ⅈ⋅π ⎤
⎢ ───── ───── ⎥
⎢⎛ 2 2 ⎞ 4 4 ⎥
⎢⎝ⅈ⋅sin (θ) + cos (θ)⎠⋅ℯ (1 - ⅈ)⋅ℯ ⋅sin(θ)⋅cos(θ)⎥
⎢ ⎥
⎢ -ⅈ⋅π -ⅈ⋅π ⎥
⎢ ───── ─────⎥
⎢ 4 ⎛ 2 2 ⎞ 4 ⎥
⎣(1 - ⅈ)⋅ℯ ⋅sin(θ)⋅cos(θ) ⎝sin (θ) + ⅈ⋅cos (θ)⎠⋅ℯ ⎦
"""
return phase_retarder(theta, pi/2)
def transmissive_filter(T):
"""An attenuator Jones matrix with transmittance `T`.
Parameters
==========
``T`` : numeric type or sympy Symbol
The transmittance of the attenuator.
Returns
=======
sympy Matrix
A Jones matrix representing the filter.
Examples
========
A generic filter.
>>> from sympy import pprint, symbols
>>> from sympy.physics.optics.polarization import transmissive_filter
>>> T = symbols("T", real=True)
>>> NDF = transmissive_filter(T)
>>> pprint(NDF, use_unicode=True)
⎡√T 0 ⎤
⎢ ⎥
⎣0 √T⎦
"""
return Matrix([[sqrt(T), 0], [0, sqrt(T)]])
def reflective_filter(R):
"""A reflective filter Jones matrix with reflectance `R`.
Parameters
==========
``R`` : numeric type or sympy Symbol
The reflectance of the filter.
Returns
=======
sympy Matrix
A Jones matrix representing the filter.
Examples
========
A generic filter.
>>> from sympy import pprint, symbols
>>> from sympy.physics.optics.polarization import reflective_filter
>>> R = symbols("R", real=True)
>>> pprint(reflective_filter(R), use_unicode=True)
⎡√R 0 ⎤
⎢ ⎥
⎣0 -√R⎦
"""
return Matrix([[sqrt(R), 0], [0, -sqrt(R)]])
def mueller_matrix(J):
"""The Mueller matrix corresponding to Jones matrix `J`.
Parameters
==========
``J`` : sympy Matrix
A Jones matrix.
Returns
=======
sympy Matrix
The corresponding Mueller matrix.
Examples
========
Generic optical components.
>>> from sympy import pprint, symbols
>>> from sympy.physics.optics.polarization import (mueller_matrix,
... linear_polarizer, half_wave_retarder, quarter_wave_retarder)
>>> theta = symbols("theta", real=True)
A linear_polarizer
>>> pprint(mueller_matrix(linear_polarizer(theta)), use_unicode=True)
⎡ cos(2⋅θ) sin(2⋅θ) ⎤
⎢ 1/2 ──────── ──────── 0⎥
⎢ 2 2 ⎥
⎢ ⎥
⎢cos(2⋅θ) cos(4⋅θ) 1 sin(4⋅θ) ⎥
⎢──────── ──────── + ─ ──────── 0⎥
⎢ 2 4 4 4 ⎥
⎢ ⎥
⎢sin(2⋅θ) sin(4⋅θ) 1 cos(4⋅θ) ⎥
⎢──────── ──────── ─ - ──────── 0⎥
⎢ 2 4 4 4 ⎥
⎢ ⎥
⎣ 0 0 0 0⎦
A half-wave plate
>>> pprint(mueller_matrix(half_wave_retarder(theta)), use_unicode=True)
⎡1 0 0 0 ⎤
⎢ ⎥
⎢ 4 2 ⎥
⎢0 8⋅sin (θ) - 8⋅sin (θ) + 1 sin(4⋅θ) 0 ⎥
⎢ ⎥
⎢ 4 2 ⎥
⎢0 sin(4⋅θ) - 8⋅sin (θ) + 8⋅sin (θ) - 1 0 ⎥
⎢ ⎥
⎣0 0 0 -1⎦
A quarter-wave plate
>>> pprint(mueller_matrix(quarter_wave_retarder(theta)), use_unicode=True)
⎡1 0 0 0 ⎤
⎢ ⎥
⎢ cos(4⋅θ) 1 sin(4⋅θ) ⎥
⎢0 ──────── + ─ ──────── -sin(2⋅θ)⎥
⎢ 2 2 2 ⎥
⎢ ⎥
⎢ sin(4⋅θ) 1 cos(4⋅θ) ⎥
⎢0 ──────── ─ - ──────── cos(2⋅θ) ⎥
⎢ 2 2 2 ⎥
⎢ ⎥
⎣0 sin(2⋅θ) -cos(2⋅θ) 0 ⎦
"""
A = Matrix([[1, 0, 0, 1],
[1, 0, 0, -1],
[0, 1, 1, 0],
[0, -I, I, 0]])
return simplify(A*TensorProduct(J, J.conjugate())*A.inv())
def polarizing_beam_splitter(Tp=1, Rs=1, Ts=0, Rp=0, phia=0, phib=0):
r"""A polarizing beam splitter Jones matrix at angle `theta`.
Parameters
==========
``J`` : sympy Matrix
A Jones matrix.
``Tp`` : numeric type or sympy Symbol
The transmissivity of the P-polarized component.
``Rs`` : numeric type or sympy Symbol
The reflectivity of the S-polarized component.
``Ts`` : numeric type or sympy Symbol
The transmissivity of the S-polarized component.
``Rp`` : numeric type or sympy Symbol
The reflectivity of the P-polarized component.
``phia`` : numeric type or sympy Symbol
The phase difference between transmitted and reflected component for
output mode a.
``phib`` : numeric type or sympy Symbol
The phase difference between transmitted and reflected component for
output mode b.
Returns
=======
sympy Matrix
A 4x4 matrix representing the PBS. This matrix acts on a 4x1 vector
whose first two entries are the Jones vector on one of the PBS ports,
and the last two entries the Jones vector on the other port.
Examples
========
Generic polarizing beam-splitter.
>>> from sympy import pprint, symbols
>>> from sympy.physics.optics.polarization import polarizing_beam_splitter
>>> Ts, Rs, Tp, Rp = symbols(r"Ts, Rs, Tp, Rp", positive=True)
>>> phia, phib = symbols("phi_a, phi_b", real=True)
>>> PBS = polarizing_beam_splitter(Tp, Rs, Ts, Rp, phia, phib)
>>> pprint(PBS, use_unicode=False)
[ ____ ____ ]
[ \/ Tp 0 I*\/ Rp 0 ]
[ ]
[ ____ ____ I*phi_a]
[ 0 \/ Ts 0 -I*\/ Rs *e ]
[ ]
[ ____ ____ ]
[I*\/ Rp 0 \/ Tp 0 ]
[ ]
[ ____ I*phi_b ____ ]
[ 0 -I*\/ Rs *e 0 \/ Ts ]
"""
PBS = Matrix([[sqrt(Tp), 0, I*sqrt(Rp), 0],
[0, sqrt(Ts), 0, -I*sqrt(Rs)*exp(I*phia)],
[I*sqrt(Rp), 0, sqrt(Tp), 0],
[0, -I*sqrt(Rs)*exp(I*phib), 0, sqrt(Ts)]])
return PBS
Zerion Mini Shell 1.0