\hSSrSSKJrJr SSKJrJrJr SSKJ r SSK J r SSK J r Jr SSKJr SSKJr SS KJr S rSS jrS rSS jrSSjrSrSrSrSrSrSSjrg)u 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 )Ipi)Absimre)exp)sqrt)cossin)Matrix)simplify) TensorProductc[[*[U5-[U5-[U5[U5--[[U5-[U5-[U5[U5--/5$)uA Jones vector corresponding to a polarization ellipse with `psi` tilt, and `chi` circularity. Parameters ========== psi : numeric type or SymPy Symbol The tilt of the polarization relative to the `x` axis. chi : numeric type or SymPy Symbol The angle adjacent to the mayor axis of the polarization ellipse. Returns ======= Matrix : A Jones vector. Examples ======== The axes on the Poincaré sphere. >>> from sympy import pprint, symbols, pi >>> from sympy.physics.optics.polarization import jones_vector >>> psi, chi = symbols("psi, chi", real=True) A general Jones vector. >>> pprint(jones_vector(psi, chi), use_unicode=True) ⎡-ⅈ⋅sin(χ)⋅sin(ψ) + cos(χ)⋅cos(ψ)⎤ ⎢ ⎥ ⎣ⅈ⋅sin(χ)⋅cos(ψ) + sin(ψ)⋅cos(χ) ⎦ Horizontal polarization. >>> pprint(jones_vector(0, 0), use_unicode=True) ⎡1⎤ ⎢ ⎥ ⎣0⎦ Vertical polarization. >>> pprint(jones_vector(pi/2, 0), use_unicode=True) ⎡0⎤ ⎢ ⎥ ⎣1⎦ Diagonal polarization. >>> pprint(jones_vector(pi/4, 0), use_unicode=True) ⎡√2⎤ ⎢──⎥ ⎢2 ⎥ ⎢ ⎥ ⎢√2⎥ ⎢──⎥ ⎣2 ⎦ Anti-diagonal polarization. >>> pprint(jones_vector(-pi/4, 0), use_unicode=True) ⎡ √2 ⎤ ⎢ ── ⎥ ⎢ 2 ⎥ ⎢ ⎥ ⎢-√2 ⎥ ⎢────⎥ ⎣ 2 ⎦ Right-hand circular polarization. >>> pprint(jones_vector(0, pi/4), use_unicode=True) ⎡ √2 ⎤ ⎢ ── ⎥ ⎢ 2 ⎥ ⎢ ⎥ ⎢√2⋅ⅈ⎥ ⎢────⎥ ⎣ 2 ⎦ Left-hand circular polarization. >>> pprint(jones_vector(0, -pi/4), use_unicode=True) ⎡ √2 ⎤ ⎢ ── ⎥ ⎢ 2 ⎥ ⎢ ⎥ ⎢-√2⋅ⅈ ⎥ ⎢──────⎥ ⎣ 2 ⎦ )r rr r )psichis Y/var/www/auris/envauris/lib/python3.13/site-packages/sympy/physics/optics/polarization.py jones_vectorrasd~ A2c#h;s3x'#c(3s8*;;SX:c#h&S#c()::< ==cUnX2-[SU-5-[SU-5-nX2-[SU-5-[SU-5-nX2-[SU-5-n[XEXg/5$)uA 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⎦ )r r r )rrprS0S1S2S3s r stokes_vectorrsmn B S3ZAcE "B S3ZAcE "B S3ZB 22" ##rc Uup[[U5S-[U5S--[U5S-[U5S-- S[XR5-5-S[ XR5-5-/5$)u;Return the Stokes vector for a Jones vector ``e``. Parameters ========== e : SymPy Matrix A Jones vector. Returns ======= SymPy Matrix A Jones vector. Examples ======== The axes on the Poincaré sphere. >>> from sympy import pprint, pi >>> from sympy.physics.optics.polarization import jones_vector >>> from sympy.physics.optics.polarization import jones_2_stokes >>> H = jones_vector(0, 0) >>> V = jones_vector(pi/2, 0) >>> D = jones_vector(pi/4, 0) >>> A = jones_vector(-pi/4, 0) >>> R = jones_vector(0, pi/4) >>> L = jones_vector(0, -pi/4) >>> pprint([jones_2_stokes(e) for e in [H, V, D, A, R, L]], ... use_unicode=True) ⎡⎡1⎤ ⎡1 ⎤ ⎡1⎤ ⎡1 ⎤ ⎡1⎤ ⎡1 ⎤⎤ ⎢⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎥ ⎢⎢1⎥ ⎢-1⎥ ⎢0⎥ ⎢0 ⎥ ⎢0⎥ ⎢0 ⎥⎥ ⎢⎢ ⎥, ⎢ ⎥, ⎢ ⎥, ⎢ ⎥, ⎢ ⎥, ⎢ ⎥⎥ ⎢⎢0⎥ ⎢0 ⎥ ⎢1⎥ ⎢-1⎥ ⎢0⎥ ⎢0 ⎥⎥ ⎢⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎥ ⎣⎣0⎦ ⎣0 ⎦ ⎣0⎦ ⎣0 ⎦ ⎣1⎦ ⎣-1⎦⎦ r)r rr conjugater)eexeys rjones_2_stokesr#BsyPFB 3r7A:B *r7A:B *R<<>)**bLLN*++- ..rc[[U5S-[U5[U5-/[U5[U5-[U5S-//5nU$)uA 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) ⎦ r)r r r )thetaMs rlinear_polarizerr'qsRD UQE 3u: 56UCJ&E A 68 9A Hrc [[U5S-[[U-5[ U5S---S[[U-5- [U5-[ U5-/S[[U-5- [U5-[ U5-[ U5S-[[U-5[U5S---//5nU[[*U-S- 5-$)uA phase retarder Jones matrix with retardance ``delta`` at angle ``theta``. Parameters ========== theta : numeric type or SymPy Symbol The angle of the fast axis relative to the horizontal plane. delta : numeric type or SymPy Symbol The phase difference between the fast and slow axes of the transmitted light. Returns ======= SymPy Matrix : A Jones matrix representing the retarder. Examples ======== A generic retarder. >>> from sympy import pprint, symbols >>> from sympy.physics.optics.polarization import phase_retarder >>> theta, delta = symbols("theta, delta", real=True) >>> R = phase_retarder(theta, delta) >>> pprint(R, use_unicode=True) ⎡ -ⅈ⋅δ -ⅈ⋅δ ⎤ ⎢ ───── ───── ⎥ ⎢⎛ ⅈ⋅δ 2 2 ⎞ 2 ⎛ ⅈ⋅δ⎞ 2 ⎥ ⎢⎝ℯ ⋅sin (θ) + cos (θ)⎠⋅ℯ ⎝1 - ℯ ⎠⋅ℯ ⋅sin(θ)⋅cos(θ)⎥ ⎢ ⎥ ⎢ -ⅈ⋅δ -ⅈ⋅δ ⎥ ⎢ ───── ─────⎥ ⎢⎛ ⅈ⋅δ⎞ 2 ⎛ ⅈ⋅δ 2 2 ⎞ 2 ⎥ ⎣⎝1 - ℯ ⎠⋅ℯ ⋅sin(θ)⋅cos(θ) ⎝ℯ ⋅cos (θ) + sin (θ)⎠⋅ℯ ⎦ r)r r rrr )r%deltaRs rphase_retarderr,sN UQQuWc%j!m!;;3qw<U+CJ68C%L.#e*,SZ7E A AeG SZ] ::<= >A S!E!_ rc"[U[5$)uMA 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)⎠⎦ r,rr%s rhalf_wave_retarderr0s@ % $$rc([U[S- 5$)uA quarter-wave retarder Jones matrix at angle ``theta``. Parameters ========== theta : numeric type or SymPy Symbol The angle of the fast axis relative to the horizontal plane. Returns ======= SymPy Matrix A Jones matrix representing the retarder. Examples ======== A generic quarter-wave plate. >>> from sympy import pprint, symbols >>> from sympy.physics.optics.polarization import quarter_wave_retarder >>> theta= symbols("theta", real=True) >>> QWP = quarter_wave_retarder(theta) >>> pprint(QWP, use_unicode=True) ⎡ -ⅈ⋅π -ⅈ⋅π ⎤ ⎢ ───── ───── ⎥ ⎢⎛ 2 2 ⎞ 4 4 ⎥ ⎢⎝ⅈ⋅sin (θ) + cos (θ)⎠⋅ℯ (1 - ⅈ)⋅ℯ ⋅sin(θ)⋅cos(θ)⎥ ⎢ ⎥ ⎢ -ⅈ⋅π -ⅈ⋅π ⎥ ⎢ ───── ─────⎥ ⎢ 4 ⎛ 2 2 ⎞ 4 ⎥ ⎣(1 - ⅈ)⋅ℯ ⋅sin(θ)⋅cos(θ) ⎝sin (θ) + ⅈ⋅cos (θ)⎠⋅ℯ ⎦ rr.r/s rquarter_wave_retarderr2sH %A &&rcH[[U5S/S[U5//5$)uAn 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⎦ rr r )Ts rtransmissive_filterr6s$< DGQ>> 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⎦ rr4)r+s rreflective_filterr81s&: DGQ>> 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 ⎦ )r)rrr))r)rr)rr)r)rr)r rr rrinv)JAs rmueller_matrixr>QsQN  QB1  A AmA{{}55aeeg= >>rc \[[U5S[[U5-S/S[U5S[*[U5-[[U-5-/[[U5-S[U5S/S[*[U5-[[U-5-S[U5//5nU$)aA polarizing beam splitter Jones matrix at angle `theta`. Parameters ========== J : SymPy Matrix A Jones matrix. Tp : numeric type or SymPy Symbol The transmissivity of the P-polarized component. Rs : numeric type or SymPy Symbol The reflectivity of the S-polarized component. Ts : numeric type or SymPy Symbol The transmissivity of the S-polarized component. Rp : numeric type or SymPy Symbol The reflectivity of the P-polarized component. phia : numeric type or SymPy Symbol The phase difference between transmitted and reflected component for output mode a. phib : numeric type or SymPy Symbol The phase difference between transmitted and reflected component for output mode b. Returns ======= SymPy Matrix A 4x4 matrix representing the PBS. This matrix acts on a 4x1 vector whose first two entries are the Jones vector on one of the PBS ports, and the last two entries the Jones vector on the other port. Examples ======== Generic polarizing beam-splitter. >>> from sympy import pprint, symbols >>> from sympy.physics.optics.polarization import polarizing_beam_splitter >>> Ts, Rs, Tp, Rp = symbols(r"Ts, Rs, Tp, Rp", positive=True) >>> phia, phib = symbols("phi_a, phi_b", real=True) >>> PBS = polarizing_beam_splitter(Tp, Rs, Ts, Rp, phia, phib) >>> pprint(PBS, use_unicode=False) [ ____ ____ ] [ \/ Tp 0 I*\/ Rp 0 ] [ ] [ ____ ____ I*phi_a] [ 0 \/ Ts 0 -I*\/ Rs *e ] [ ] [ ____ ____ ] [I*\/ Rp 0 \/ Tp 0 ] [ ] [ ____ I*phi_b ____ ] [ 0 -I*\/ Rs *e 0 \/ Ts ] r)r r rr)TpRsTsRpphiaphibPBSs rpolarizing_beam_splitterrGsp 48Q$r( A.d2hA2d2h;s1T6{#:;T"X:q$r(A.qbbk#af+-q$r(;= >C JrN)r)r))r)rr)r)r)rrrr) __doc__sympy.core.numbersrr$sympy.functions.elementary.complexesrrr&sympy.functions.elementary.exponentialr(sympy.functions.elementary.miscellaneousr (sympy.functions.elementary.trigonometricr r sympy.matrices.denser sympy.simplify.simplifyr sympy.physics.quantumrrrr#r'r,r0r2r6r8r>rGrrrRsmRh'>>69?',/`=F{$|,.^$ N+\ %F$'N0B1@L?^<r