733 lines
21 KiB
Python
733 lines
21 KiB
Python
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#!/usr/bin/env python
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# -*- coding: utf-8 -*-
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"""
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The module implements routines to model the polarization of optical fields
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and can be used to calculate the effects of polarization optical elements on
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the fields.
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- Jones vectors.
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- Stokes vectors.
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- Jones matrices.
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- Mueller matrices.
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Examples
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========
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We calculate a generic Jones vector:
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>>> from sympy import symbols, pprint, zeros, simplify
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>>> from sympy.physics.optics.polarization import (jones_vector, stokes_vector,
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... half_wave_retarder, polarizing_beam_splitter, jones_2_stokes)
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>>> psi, chi, p, I0 = symbols("psi, chi, p, I0", real=True)
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>>> x0 = jones_vector(psi, chi)
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>>> pprint(x0, use_unicode=True)
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⎡-ⅈ⋅sin(χ)⋅sin(ψ) + cos(χ)⋅cos(ψ)⎤
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⎢ ⎥
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⎣ⅈ⋅sin(χ)⋅cos(ψ) + sin(ψ)⋅cos(χ) ⎦
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And the more general Stokes vector:
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>>> s0 = stokes_vector(psi, chi, p, I0)
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>>> pprint(s0, use_unicode=True)
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⎡ I₀ ⎤
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⎢ ⎥
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⎢I₀⋅p⋅cos(2⋅χ)⋅cos(2⋅ψ)⎥
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⎢ ⎥
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⎢I₀⋅p⋅sin(2⋅ψ)⋅cos(2⋅χ)⎥
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⎢ ⎥
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⎣ I₀⋅p⋅sin(2⋅χ) ⎦
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We calculate how the Jones vector is modified by a half-wave plate:
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>>> alpha = symbols("alpha", real=True)
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>>> HWP = half_wave_retarder(alpha)
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>>> x1 = simplify(HWP*x0)
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We calculate the very common operation of passing a beam through a half-wave
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plate and then through a polarizing beam-splitter. We do this by putting this
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Jones vector as the first entry of a two-Jones-vector state that is transformed
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by a 4x4 Jones matrix modelling the polarizing beam-splitter to get the
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transmitted and reflected Jones vectors:
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>>> PBS = polarizing_beam_splitter()
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>>> X1 = zeros(4, 1)
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>>> X1[:2, :] = x1
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>>> X2 = PBS*X1
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>>> transmitted_port = X2[:2, :]
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>>> reflected_port = X2[2:, :]
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This allows us to calculate how the power in both ports depends on the initial
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polarization:
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>>> transmitted_power = jones_2_stokes(transmitted_port)[0]
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>>> reflected_power = jones_2_stokes(reflected_port)[0]
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>>> print(transmitted_power)
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cos(-2*alpha + chi + psi)**2/2 + cos(2*alpha + chi - psi)**2/2
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>>> print(reflected_power)
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sin(-2*alpha + chi + psi)**2/2 + sin(2*alpha + chi - psi)**2/2
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Please see the description of the individual functions for further
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details and examples.
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References
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==========
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.. [1] https://en.wikipedia.org/wiki/Jones_calculus
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.. [2] https://en.wikipedia.org/wiki/Mueller_calculus
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.. [3] https://en.wikipedia.org/wiki/Stokes_parameters
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"""
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from sympy.core.numbers import (I, pi)
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from sympy.functions.elementary.complexes import (Abs, im, re)
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from sympy.functions.elementary.exponential import exp
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from sympy.functions.elementary.miscellaneous import sqrt
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from sympy.functions.elementary.trigonometric import (cos, sin)
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from sympy.matrices.dense import Matrix
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from sympy.simplify.simplify import simplify
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from sympy.physics.quantum import TensorProduct
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def jones_vector(psi, chi):
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"""A Jones vector corresponding to a polarization ellipse with `psi` tilt,
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and `chi` circularity.
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Parameters
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==========
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psi : numeric type or SymPy Symbol
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The tilt of the polarization relative to the `x` axis.
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chi : numeric type or SymPy Symbol
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The angle adjacent to the mayor axis of the polarization ellipse.
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Returns
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=======
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Matrix :
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A Jones vector.
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Examples
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========
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The axes on the Poincaré sphere.
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>>> from sympy import pprint, symbols, pi
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>>> from sympy.physics.optics.polarization import jones_vector
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>>> psi, chi = symbols("psi, chi", real=True)
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A general Jones vector.
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>>> pprint(jones_vector(psi, chi), use_unicode=True)
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⎡-ⅈ⋅sin(χ)⋅sin(ψ) + cos(χ)⋅cos(ψ)⎤
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⎢ ⎥
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⎣ⅈ⋅sin(χ)⋅cos(ψ) + sin(ψ)⋅cos(χ) ⎦
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Horizontal polarization.
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>>> pprint(jones_vector(0, 0), use_unicode=True)
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⎡1⎤
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⎢ ⎥
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⎣0⎦
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Vertical polarization.
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>>> pprint(jones_vector(pi/2, 0), use_unicode=True)
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⎡0⎤
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⎢ ⎥
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⎣1⎦
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Diagonal polarization.
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>>> pprint(jones_vector(pi/4, 0), use_unicode=True)
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⎡√2⎤
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⎢──⎥
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⎢2 ⎥
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⎢ ⎥
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⎢√2⎥
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⎢──⎥
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⎣2 ⎦
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Anti-diagonal polarization.
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>>> pprint(jones_vector(-pi/4, 0), use_unicode=True)
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⎡ √2 ⎤
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⎢ ── ⎥
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⎢ 2 ⎥
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⎢ ⎥
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⎢-√2 ⎥
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⎢────⎥
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⎣ 2 ⎦
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Right-hand circular polarization.
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>>> pprint(jones_vector(0, pi/4), use_unicode=True)
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⎡ √2 ⎤
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⎢ ── ⎥
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⎢ 2 ⎥
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⎢ ⎥
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⎢√2⋅ⅈ⎥
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⎢────⎥
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⎣ 2 ⎦
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Left-hand circular polarization.
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>>> pprint(jones_vector(0, -pi/4), use_unicode=True)
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⎡ √2 ⎤
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⎢ ── ⎥
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⎢ 2 ⎥
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⎢ ⎥
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⎢-√2⋅ⅈ ⎥
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⎢──────⎥
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⎣ 2 ⎦
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"""
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return Matrix([-I*sin(chi)*sin(psi) + cos(chi)*cos(psi),
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I*sin(chi)*cos(psi) + sin(psi)*cos(chi)])
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def stokes_vector(psi, chi, p=1, I=1):
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"""A Stokes vector corresponding to a polarization ellipse with ``psi``
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tilt, and ``chi`` circularity.
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Parameters
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==========
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psi : numeric type or SymPy Symbol
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The tilt of the polarization relative to the ``x`` axis.
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chi : numeric type or SymPy Symbol
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The angle adjacent to the mayor axis of the polarization ellipse.
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p : numeric type or SymPy Symbol
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The degree of polarization.
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I : numeric type or SymPy Symbol
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The intensity of the field.
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Returns
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=======
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Matrix :
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A Stokes vector.
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Examples
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========
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The axes on the Poincaré sphere.
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>>> from sympy import pprint, symbols, pi
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>>> from sympy.physics.optics.polarization import stokes_vector
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>>> psi, chi, p, I = symbols("psi, chi, p, I", real=True)
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>>> pprint(stokes_vector(psi, chi, p, I), use_unicode=True)
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⎡ I ⎤
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⎢ ⎥
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⎢I⋅p⋅cos(2⋅χ)⋅cos(2⋅ψ)⎥
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⎢ ⎥
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⎢I⋅p⋅sin(2⋅ψ)⋅cos(2⋅χ)⎥
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⎢ ⎥
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⎣ I⋅p⋅sin(2⋅χ) ⎦
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Horizontal polarization
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>>> pprint(stokes_vector(0, 0), use_unicode=True)
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⎡1⎤
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⎢ ⎥
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⎢1⎥
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⎢ ⎥
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⎢0⎥
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⎢ ⎥
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⎣0⎦
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Vertical polarization
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>>> pprint(stokes_vector(pi/2, 0), use_unicode=True)
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⎡1 ⎤
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⎢ ⎥
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⎢-1⎥
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⎢ ⎥
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⎢0 ⎥
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⎢ ⎥
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⎣0 ⎦
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Diagonal polarization
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>>> pprint(stokes_vector(pi/4, 0), use_unicode=True)
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⎡1⎤
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⎢ ⎥
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⎢0⎥
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⎢ ⎥
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⎢1⎥
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⎢ ⎥
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⎣0⎦
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Anti-diagonal polarization
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>>> pprint(stokes_vector(-pi/4, 0), use_unicode=True)
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⎡1 ⎤
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⎢ ⎥
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⎢0 ⎥
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⎢ ⎥
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⎢-1⎥
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⎢ ⎥
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⎣0 ⎦
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Right-hand circular polarization
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>>> pprint(stokes_vector(0, pi/4), use_unicode=True)
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⎡1⎤
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⎢ ⎥
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⎢0⎥
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⎢ ⎥
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⎢0⎥
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⎢ ⎥
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⎣1⎦
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Left-hand circular polarization
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>>> pprint(stokes_vector(0, -pi/4), use_unicode=True)
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⎡1 ⎤
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⎢ ⎥
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⎢0 ⎥
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⎢ ⎥
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⎢0 ⎥
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⎢ ⎥
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⎣-1⎦
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Unpolarized light
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>>> pprint(stokes_vector(0, 0, 0), use_unicode=True)
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⎡1⎤
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⎢ ⎥
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⎢0⎥
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⎢ ⎥
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⎢0⎥
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⎢ ⎥
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⎣0⎦
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"""
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S0 = I
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S1 = I*p*cos(2*psi)*cos(2*chi)
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S2 = I*p*sin(2*psi)*cos(2*chi)
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S3 = I*p*sin(2*chi)
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return Matrix([S0, S1, S2, S3])
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def jones_2_stokes(e):
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"""Return the Stokes vector for a Jones vector ``e``.
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Parameters
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==========
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e : SymPy Matrix
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A Jones vector.
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Returns
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=======
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|||
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SymPy Matrix
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|||
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A Jones vector.
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|||
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Examples
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|||
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========
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|||
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|||
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The axes on the Poincaré sphere.
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|||
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>>> from sympy import pprint, pi
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>>> from sympy.physics.optics.polarization import jones_vector
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>>> from sympy.physics.optics.polarization import jones_2_stokes
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>>> H = jones_vector(0, 0)
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>>> V = jones_vector(pi/2, 0)
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>>> D = jones_vector(pi/4, 0)
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>>> A = jones_vector(-pi/4, 0)
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>>> R = jones_vector(0, pi/4)
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>>> L = jones_vector(0, -pi/4)
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>>> pprint([jones_2_stokes(e) for e in [H, V, D, A, R, L]],
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... use_unicode=True)
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⎡⎡1⎤ ⎡1 ⎤ ⎡1⎤ ⎡1 ⎤ ⎡1⎤ ⎡1 ⎤⎤
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⎢⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎥
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⎢⎢1⎥ ⎢-1⎥ ⎢0⎥ ⎢0 ⎥ ⎢0⎥ ⎢0 ⎥⎥
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⎢⎢ ⎥, ⎢ ⎥, ⎢ ⎥, ⎢ ⎥, ⎢ ⎥, ⎢ ⎥⎥
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⎢⎢0⎥ ⎢0 ⎥ ⎢1⎥ ⎢-1⎥ ⎢0⎥ ⎢0 ⎥⎥
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|
⎢⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎥
|
|||
|
|
⎣⎣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
|