import sympy as sp
import numpy as np

Tp = sp.symbols('T^p_+, T^p_-', positive=True)
Ta = sp.symbols('T^a_+, T^a_-', positive=True)
I = sp.symbols('Ipp, Ipm, Imp, Imm', positive=True)  # noqa: E741

Polarization matrices for He3 cells and supermirror

From https://www.epj-conferences.org/articles/epjconf/abs/2023/12/epjconf_ecns2023_03004/epjconf_ecns2023_03004.html

I = sp.ImmutableDenseMatrix([[I[0]], [I[1]], [I[2]], [I[3]]])  # noqa: E741
I

$\displaystyle \left[\begin{matrix}Ipp\\Ipm\\Imp\\Imm\end{matrix}\right]$

TP = sp.ImmutableDenseMatrix(
    [
        [Tp[0], 0, Tp[1], 0],
        [0, Tp[0], 0, Tp[1]],
        [Tp[1], 0, Tp[0], 0],
        [0, Tp[1], 0, Tp[0]],
    ]
)
TP

$\displaystyle \left[\begin{matrix}T^{p}_{+} & 0 & T^{p}_{-} & 0\\0 & T^{p}_{+} & 0 & T^{p}_{-}\\T^{p}_{-} & 0 & T^{p}_{+} & 0\\0 & T^{p}_{-} & 0 & T^{p}_{+}\end{matrix}\right]$

TP.inv()

$\displaystyle \left[\begin{matrix}\frac{T^{p}_{+}}{\left(T^{p}_{+}\right)^{2} - \left(T^{p}_{-}\right)^{2}} & 0 & - \frac{T^{p}_{-}}{\left(T^{p}_{+}\right)^{2} - \left(T^{p}_{-}\right)^{2}} & 0\\0 & \frac{T^{p}_{+}}{\left(T^{p}_{+}\right)^{2} - \left(T^{p}_{-}\right)^{2}} & 0 & - \frac{T^{p}_{-}}{\left(T^{p}_{+}\right)^{2} - \left(T^{p}_{-}\right)^{2}}\\- \frac{T^{p}_{-}}{\left(T^{p}_{+}\right)^{2} - \left(T^{p}_{-}\right)^{2}} & 0 & \frac{T^{p}_{+}}{\left(T^{p}_{+}\right)^{2} - \left(T^{p}_{-}\right)^{2}} & 0\\0 & - \frac{T^{p}_{-}}{\left(T^{p}_{+}\right)^{2} - \left(T^{p}_{-}\right)^{2}} & 0 & \frac{T^{p}_{+}}{\left(T^{p}_{+}\right)^{2} - \left(T^{p}_{-}\right)^{2}}\end{matrix}\right]$

TA = sp.ImmutableDenseMatrix(
    [
        [Ta[0], Ta[1], 0, 0],
        [Ta[1], Ta[0], 0, 0],
        [0, 0, Ta[0], Ta[1]],
        [0, 0, Ta[1], Ta[0]],
    ]
)
TA

$\displaystyle \left[\begin{matrix}T^{a}_{+} & T^{a}_{-} & 0 & 0\\T^{a}_{-} & T^{a}_{+} & 0 & 0\\0 & 0 & T^{a}_{+} & T^{a}_{-}\\0 & 0 & T^{a}_{-} & T^{a}_{+}\end{matrix}\right]$

TA.inv()

$\displaystyle \left[\begin{matrix}\frac{T^{a}_{+}}{\left(T^{a}_{+}\right)^{2} - \left(T^{a}_{-}\right)^{2}} & - \frac{T^{a}_{-}}{\left(T^{a}_{+}\right)^{2} - \left(T^{a}_{-}\right)^{2}} & 0 & 0\\- \frac{T^{a}_{-}}{\left(T^{a}_{+}\right)^{2} - \left(T^{a}_{-}\right)^{2}} & \frac{T^{a}_{+}}{\left(T^{a}_{+}\right)^{2} - \left(T^{a}_{-}\right)^{2}} & 0 & 0\\0 & 0 & \frac{T^{a}_{+}}{\left(T^{a}_{+}\right)^{2} - \left(T^{a}_{-}\right)^{2}} & - \frac{T^{a}_{-}}{\left(T^{a}_{+}\right)^{2} - \left(T^{a}_{-}\right)^{2}}\\0 & 0 & - \frac{T^{a}_{-}}{\left(T^{a}_{+}\right)^{2} - \left(T^{a}_{-}\right)^{2}} & \frac{T^{a}_{+}}{\left(T^{a}_{+}\right)^{2} - \left(T^{a}_{-}\right)^{2}}\end{matrix}\right]$

Helium cell case

(TA @ TP).inv().factor()

$\displaystyle \left[\begin{matrix}\frac{T^{a}_{+} T^{p}_{+}}{\left(T^{a}_{+}\right)^{2} \left(T^{p}_{+}\right)^{2} - \left(T^{a}_{+}\right)^{2} \left(T^{p}_{-}\right)^{2} - \left(T^{a}_{-}\right)^{2} \left(T^{p}_{+}\right)^{2} + \left(T^{a}_{-}\right)^{2} \left(T^{p}_{-}\right)^{2}} & - \frac{T^{a}_{-} T^{p}_{+}}{\left(T^{a}_{+}\right)^{2} \left(T^{p}_{+}\right)^{2} - \left(T^{a}_{+}\right)^{2} \left(T^{p}_{-}\right)^{2} - \left(T^{a}_{-}\right)^{2} \left(T^{p}_{+}\right)^{2} + \left(T^{a}_{-}\right)^{2} \left(T^{p}_{-}\right)^{2}} & - \frac{T^{a}_{+} T^{p}_{-}}{\left(T^{a}_{+}\right)^{2} \left(T^{p}_{+}\right)^{2} - \left(T^{a}_{+}\right)^{2} \left(T^{p}_{-}\right)^{2} - \left(T^{a}_{-}\right)^{2} \left(T^{p}_{+}\right)^{2} + \left(T^{a}_{-}\right)^{2} \left(T^{p}_{-}\right)^{2}} & \frac{T^{a}_{-} T^{p}_{-}}{\left(T^{a}_{+}\right)^{2} \left(T^{p}_{+}\right)^{2} - \left(T^{a}_{+}\right)^{2} \left(T^{p}_{-}\right)^{2} - \left(T^{a}_{-}\right)^{2} \left(T^{p}_{+}\right)^{2} + \left(T^{a}_{-}\right)^{2} \left(T^{p}_{-}\right)^{2}}\\- \frac{T^{a}_{-} T^{p}_{+}}{\left(T^{a}_{+}\right)^{2} \left(T^{p}_{+}\right)^{2} - \left(T^{a}_{+}\right)^{2} \left(T^{p}_{-}\right)^{2} - \left(T^{a}_{-}\right)^{2} \left(T^{p}_{+}\right)^{2} + \left(T^{a}_{-}\right)^{2} \left(T^{p}_{-}\right)^{2}} & \frac{T^{a}_{+} T^{p}_{+}}{\left(T^{a}_{+}\right)^{2} \left(T^{p}_{+}\right)^{2} - \left(T^{a}_{+}\right)^{2} \left(T^{p}_{-}\right)^{2} - \left(T^{a}_{-}\right)^{2} \left(T^{p}_{+}\right)^{2} + \left(T^{a}_{-}\right)^{2} \left(T^{p}_{-}\right)^{2}} & \frac{T^{a}_{-} T^{p}_{-}}{\left(T^{a}_{+}\right)^{2} \left(T^{p}_{+}\right)^{2} - \left(T^{a}_{+}\right)^{2} \left(T^{p}_{-}\right)^{2} - \left(T^{a}_{-}\right)^{2} \left(T^{p}_{+}\right)^{2} + \left(T^{a}_{-}\right)^{2} \left(T^{p}_{-}\right)^{2}} & - \frac{T^{a}_{+} T^{p}_{-}}{\left(T^{a}_{+}\right)^{2} \left(T^{p}_{+}\right)^{2} - \left(T^{a}_{+}\right)^{2} \left(T^{p}_{-}\right)^{2} - \left(T^{a}_{-}\right)^{2} \left(T^{p}_{+}\right)^{2} + \left(T^{a}_{-}\right)^{2} \left(T^{p}_{-}\right)^{2}}\\- \frac{T^{a}_{+} T^{p}_{-}}{\left(T^{a}_{+}\right)^{2} \left(T^{p}_{+}\right)^{2} - \left(T^{a}_{+}\right)^{2} \left(T^{p}_{-}\right)^{2} - \left(T^{a}_{-}\right)^{2} \left(T^{p}_{+}\right)^{2} + \left(T^{a}_{-}\right)^{2} \left(T^{p}_{-}\right)^{2}} & \frac{T^{a}_{-} T^{p}_{-}}{\left(T^{a}_{+}\right)^{2} \left(T^{p}_{+}\right)^{2} - \left(T^{a}_{+}\right)^{2} \left(T^{p}_{-}\right)^{2} - \left(T^{a}_{-}\right)^{2} \left(T^{p}_{+}\right)^{2} + \left(T^{a}_{-}\right)^{2} \left(T^{p}_{-}\right)^{2}} & \frac{T^{a}_{+} T^{p}_{+}}{\left(T^{a}_{+}\right)^{2} \left(T^{p}_{+}\right)^{2} - \left(T^{a}_{+}\right)^{2} \left(T^{p}_{-}\right)^{2} - \left(T^{a}_{-}\right)^{2} \left(T^{p}_{+}\right)^{2} + \left(T^{a}_{-}\right)^{2} \left(T^{p}_{-}\right)^{2}} & - \frac{T^{a}_{-} T^{p}_{+}}{\left(T^{a}_{+}\right)^{2} \left(T^{p}_{+}\right)^{2} - \left(T^{a}_{+}\right)^{2} \left(T^{p}_{-}\right)^{2} - \left(T^{a}_{-}\right)^{2} \left(T^{p}_{+}\right)^{2} + \left(T^{a}_{-}\right)^{2} \left(T^{p}_{-}\right)^{2}}\\\frac{T^{a}_{-} T^{p}_{-}}{\left(T^{a}_{+}\right)^{2} \left(T^{p}_{+}\right)^{2} - \left(T^{a}_{+}\right)^{2} \left(T^{p}_{-}\right)^{2} - \left(T^{a}_{-}\right)^{2} \left(T^{p}_{+}\right)^{2} + \left(T^{a}_{-}\right)^{2} \left(T^{p}_{-}\right)^{2}} & - \frac{T^{a}_{+} T^{p}_{-}}{\left(T^{a}_{+}\right)^{2} \left(T^{p}_{+}\right)^{2} - \left(T^{a}_{+}\right)^{2} \left(T^{p}_{-}\right)^{2} - \left(T^{a}_{-}\right)^{2} \left(T^{p}_{+}\right)^{2} + \left(T^{a}_{-}\right)^{2} \left(T^{p}_{-}\right)^{2}} & - \frac{T^{a}_{-} T^{p}_{+}}{\left(T^{a}_{+}\right)^{2} \left(T^{p}_{+}\right)^{2} - \left(T^{a}_{+}\right)^{2} \left(T^{p}_{-}\right)^{2} - \left(T^{a}_{-}\right)^{2} \left(T^{p}_{+}\right)^{2} + \left(T^{a}_{-}\right)^{2} \left(T^{p}_{-}\right)^{2}} & \frac{T^{a}_{+} T^{p}_{+}}{\left(T^{a}_{+}\right)^{2} \left(T^{p}_{+}\right)^{2} - \left(T^{a}_{+}\right)^{2} \left(T^{p}_{-}\right)^{2} - \left(T^{a}_{-}\right)^{2} \left(T^{p}_{+}\right)^{2} + \left(T^{a}_{-}\right)^{2} \left(T^{p}_{-}\right)^{2}}\end{matrix}\right]$

Same denominator in all entries, maybe easier to evaluate this way:

d = (Ta[0] ** 2 - Ta[1] ** 2) * (Tp[0] ** 2 - Tp[1] ** 2)
d

$\displaystyle \left(\left(T^{a}_{+}\right)^{2} - \left(T^{a}_{-}\right)^{2}\right) \left(\left(T^{p}_{+}\right)^{2} - \left(T^{p}_{-}\right)^{2}\right)$

(d * (TA @ TP).inv()).simplify()

$\displaystyle \left[\begin{matrix}T^{a}_{+} T^{p}_{+} & - T^{a}_{-} T^{p}_{+} & - T^{a}_{+} T^{p}_{-} & T^{a}_{-} T^{p}_{-}\\- T^{a}_{-} T^{p}_{+} & T^{a}_{+} T^{p}_{+} & T^{a}_{-} T^{p}_{-} & - T^{a}_{+} T^{p}_{-}\\- T^{a}_{+} T^{p}_{-} & T^{a}_{-} T^{p}_{-} & T^{a}_{+} T^{p}_{+} & - T^{a}_{-} T^{p}_{+}\\T^{a}_{-} T^{p}_{-} & - T^{a}_{+} T^{p}_{-} & - T^{a}_{-} T^{p}_{+} & T^{a}_{+} T^{p}_{+}\end{matrix}\right]$

Supermirror case

f1, f2 = sp.symbols('f_1:3')
F1_parallel = sp.ImmutableDenseMatrix(
    [
        [1, 0, 0, 0],
        [0, 1, 0, 0],
        [1 - f1, 0, f1, 0],
        [0, 1 - f1, 0, f1],
    ]
)
F1_parallel

$\displaystyle \left[\begin{matrix}1 & 0 & 0 & 0\\0 & 1 & 0 & 0\\1 - f_{1} & 0 & f_{1} & 0\\0 & 1 - f_{1} & 0 & f_{1}\end{matrix}\right]$

F1_parallel.inv()

$\displaystyle \left[\begin{matrix}1 & 0 & 0 & 0\\0 & 1 & 0 & 0\\\frac{f_{1} - 1}{f_{1}} & 0 & \frac{1}{f_{1}} & 0\\0 & \frac{f_{1} - 1}{f_{1}} & 0 & \frac{1}{f_{1}}\end{matrix}\right]$

(F1_parallel.inv() @ F1_parallel).simplify()

$\displaystyle \left[\begin{matrix}1 & 0 & 0 & 0\\0 & 1 & 0 & 0\\0 & 0 & 1 & 0\\0 & 0 & 0 & 1\end{matrix}\right]$

F2_parallel = sp.ImmutableDenseMatrix(
    [
        [1, 0, 0, 0],
        [1 - f2, f2, 0, 0],
        [0, 0, 1, 0],
        [0, 0, 1 - f2, f2],
    ]
)
F2_parallel

$\displaystyle \left[\begin{matrix}1 & 0 & 0 & 0\\1 - f_{2} & f_{2} & 0 & 0\\0 & 0 & 1 & 0\\0 & 0 & 1 - f_{2} & f_{2}\end{matrix}\right]$

F1_antiparallel = sp.ImmutableDenseMatrix(
    [
        [f1, 0, 1 - f1, 0],
        [0, f1, 0, 1 - f1],
        [0, 0, 1, 0],
        [0, 0, 0, 1],
    ]
)
F1_antiparallel

$\displaystyle \left[\begin{matrix}f_{1} & 0 & 1 - f_{1} & 0\\0 & f_{1} & 0 & 1 - f_{1}\\0 & 0 & 1 & 0\\0 & 0 & 0 & 1\end{matrix}\right]$

F2_antiparallel = sp.ImmutableDenseMatrix(
    [
        [1, 1 - f2, 0, 0],
        [0, f2, 0, 0],
        [0, 0, 1, 1 - f2],
        [0, 0, 0, f2],
    ]
)
F2_antiparallel

$\displaystyle \left[\begin{matrix}1 & 1 - f_{2} & 0 & 0\\0 & f_{2} & 0 & 0\\0 & 0 & 1 & 1 - f_{2}\\0 & 0 & 0 & f_{2}\end{matrix}\right]$

(F1_antiparallel @ F1_antiparallel.inv()).simplify()

$\displaystyle \left[\begin{matrix}1 & 0 & 0 & 0\\0 & 1 & 0 & 0\\0 & 0 & 1 & 0\\0 & 0 & 0 & 1\end{matrix}\right]$

(F2_antiparallel @ F2_antiparallel.inv()).simplify()

$\displaystyle \left[\begin{matrix}1 & 0 & 0 & 0\\0 & 1 & 0 & 0\\0 & 0 & 1 & 0\\0 & 0 & 0 & 1\end{matrix}\right]$

(TP.inv() @ F1_parallel.inv()).simplify()

$\displaystyle \left[\begin{matrix}\frac{T^{p}_{+} f_{1} - T^{p}_{-} \left(f_{1} - 1\right)}{f_{1} \left(\left(T^{p}_{+}\right)^{2} - \left(T^{p}_{-}\right)^{2}\right)} & 0 & - \frac{T^{p}_{-}}{f_{1} \left(\left(T^{p}_{+}\right)^{2} - \left(T^{p}_{-}\right)^{2}\right)} & 0\\0 & \frac{T^{p}_{+} f_{1} - T^{p}_{-} \left(f_{1} - 1\right)}{f_{1} \left(\left(T^{p}_{+}\right)^{2} - \left(T^{p}_{-}\right)^{2}\right)} & 0 & - \frac{T^{p}_{-}}{f_{1} \left(\left(T^{p}_{+}\right)^{2} - \left(T^{p}_{-}\right)^{2}\right)}\\\frac{T^{p}_{+} \left(f_{1} - 1\right) - T^{p}_{-} f_{1}}{f_{1} \left(\left(T^{p}_{+}\right)^{2} - \left(T^{p}_{-}\right)^{2}\right)} & 0 & \frac{T^{p}_{+}}{f_{1} \left(\left(T^{p}_{+}\right)^{2} - \left(T^{p}_{-}\right)^{2}\right)} & 0\\0 & \frac{T^{p}_{+} \left(f_{1} - 1\right) - T^{p}_{-} f_{1}}{f_{1} \left(\left(T^{p}_{+}\right)^{2} - \left(T^{p}_{-}\right)^{2}\right)} & 0 & \frac{T^{p}_{+}}{f_{1} \left(\left(T^{p}_{+}\right)^{2} - \left(T^{p}_{-}\right)^{2}\right)}\end{matrix}\right]$

(TP.inv() @ F1_antiparallel.inv()).simplify()

$\displaystyle \left[\begin{matrix}\frac{T^{p}_{+}}{f_{1} \left(\left(T^{p}_{+}\right)^{2} - \left(T^{p}_{-}\right)^{2}\right)} & 0 & \frac{T^{p}_{+} \left(f_{1} - 1\right) - T^{p}_{-} f_{1}}{f_{1} \left(\left(T^{p}_{+}\right)^{2} - \left(T^{p}_{-}\right)^{2}\right)} & 0\\0 & \frac{T^{p}_{+}}{f_{1} \left(\left(T^{p}_{+}\right)^{2} - \left(T^{p}_{-}\right)^{2}\right)} & 0 & \frac{T^{p}_{+} \left(f_{1} - 1\right) - T^{p}_{-} f_{1}}{f_{1} \left(\left(T^{p}_{+}\right)^{2} - \left(T^{p}_{-}\right)^{2}\right)}\\- \frac{T^{p}_{-}}{f_{1} \left(\left(T^{p}_{+}\right)^{2} - \left(T^{p}_{-}\right)^{2}\right)} & 0 & \frac{T^{p}_{+} f_{1} - T^{p}_{-} \left(f_{1} - 1\right)}{f_{1} \left(\left(T^{p}_{+}\right)^{2} - \left(T^{p}_{-}\right)^{2}\right)} & 0\\0 & - \frac{T^{p}_{-}}{f_{1} \left(\left(T^{p}_{+}\right)^{2} - \left(T^{p}_{-}\right)^{2}\right)} & 0 & \frac{T^{p}_{+} f_{1} - T^{p}_{-} \left(f_{1} - 1\right)}{f_{1} \left(\left(T^{p}_{+}\right)^{2} - \left(T^{p}_{-}\right)^{2}\right)}\end{matrix}\right]$

(F2_parallel.inv() @ TA.inv()).simplify()

$\displaystyle \left[\begin{matrix}\frac{T^{a}_{+}}{\left(T^{a}_{+}\right)^{2} - \left(T^{a}_{-}\right)^{2}} & - \frac{T^{a}_{-}}{\left(T^{a}_{+}\right)^{2} - \left(T^{a}_{-}\right)^{2}} & 0 & 0\\\frac{T^{a}_{+} \left(f_{2} - 1\right) - T^{a}_{-}}{f_{2} \left(\left(T^{a}_{+}\right)^{2} - \left(T^{a}_{-}\right)^{2}\right)} & \frac{T^{a}_{+} - T^{a}_{-} \left(f_{2} - 1\right)}{f_{2} \left(\left(T^{a}_{+}\right)^{2} - \left(T^{a}_{-}\right)^{2}\right)} & 0 & 0\\0 & 0 & \frac{T^{a}_{+}}{\left(T^{a}_{+}\right)^{2} - \left(T^{a}_{-}\right)^{2}} & - \frac{T^{a}_{-}}{\left(T^{a}_{+}\right)^{2} - \left(T^{a}_{-}\right)^{2}}\\0 & 0 & \frac{T^{a}_{+} \left(f_{2} - 1\right) - T^{a}_{-}}{f_{2} \left(\left(T^{a}_{+}\right)^{2} - \left(T^{a}_{-}\right)^{2}\right)} & \frac{T^{a}_{+} - T^{a}_{-} \left(f_{2} - 1\right)}{f_{2} \left(\left(T^{a}_{+}\right)^{2} - \left(T^{a}_{-}\right)^{2}\right)}\end{matrix}\right]$

(F2_antiparallel.inv() @ TA.inv()).simplify()

$\displaystyle \left[\begin{matrix}\frac{T^{a}_{+} f_{2} - T^{a}_{-} \left(f_{2} - 1\right)}{f_{2} \left(\left(T^{a}_{+}\right)^{2} - \left(T^{a}_{-}\right)^{2}\right)} & \frac{T^{a}_{+} \left(f_{2} - 1\right) - T^{a}_{-} f_{2}}{f_{2} \left(\left(T^{a}_{+}\right)^{2} - \left(T^{a}_{-}\right)^{2}\right)} & 0 & 0\\- \frac{T^{a}_{-}}{f_{2} \left(\left(T^{a}_{+}\right)^{2} - \left(T^{a}_{-}\right)^{2}\right)} & \frac{T^{a}_{+}}{f_{2} \left(\left(T^{a}_{+}\right)^{2} - \left(T^{a}_{-}\right)^{2}\right)} & 0 & 0\\0 & 0 & \frac{T^{a}_{+} f_{2} - T^{a}_{-} \left(f_{2} - 1\right)}{f_{2} \left(\left(T^{a}_{+}\right)^{2} - \left(T^{a}_{-}\right)^{2}\right)} & \frac{T^{a}_{+} \left(f_{2} - 1\right) - T^{a}_{-} f_{2}}{f_{2} \left(\left(T^{a}_{+}\right)^{2} - \left(T^{a}_{-}\right)^{2}\right)}\\0 & 0 & - \frac{T^{a}_{-}}{f_{2} \left(\left(T^{a}_{+}\right)^{2} - \left(T^{a}_{-}\right)^{2}\right)} & \frac{T^{a}_{+}}{f_{2} \left(\left(T^{a}_{+}\right)^{2} - \left(T^{a}_{-}\right)^{2}\right)}\end{matrix}\right]$

(((F2_parallel.inv() @ TA.inv()) @ I) * (Ta[0] ** 2 - Ta[1] ** 2)).simplify()

$\displaystyle \left[\begin{matrix}- Ipm T^{a}_{-} + Ipp T^{a}_{+}\\\frac{Ipm \left(T^{a}_{+} - T^{a}_{-} \left(f_{2} - 1\right)\right) + Ipp \left(T^{a}_{+} \left(f_{2} - 1\right) - T^{a}_{-}\right)}{f_{2}}\\- Imm T^{a}_{-} + Imp T^{a}_{+}\\\frac{Imm \left(T^{a}_{+} - T^{a}_{-} \left(f_{2} - 1\right)\right) + Imp \left(T^{a}_{+} \left(f_{2} - 1\right) - T^{a}_{-}\right)}{f_{2}}\end{matrix}\right]$

(((TP.inv() @ F1_parallel.inv()) @ I) * (Tp[0] ** 2 - Tp[1] ** 2)).simplify()

$\displaystyle \left[\begin{matrix}\frac{- Imp T^{p}_{-} + Ipp \left(T^{p}_{+} f_{1} - T^{p}_{-} \left(f_{1} - 1\right)\right)}{f_{1}}\\\frac{- Imm T^{p}_{-} + Ipm \left(T^{p}_{+} f_{1} - T^{p}_{-} \left(f_{1} - 1\right)\right)}{f_{1}}\\\frac{Imp T^{p}_{+} + Ipp \left(T^{p}_{+} \left(f_{1} - 1\right) - T^{p}_{-} f_{1}\right)}{f_{1}}\\\frac{Imm T^{p}_{+} + Ipm \left(T^{p}_{+} \left(f_{1} - 1\right) - T^{p}_{-} f_{1}\right)}{f_{1}}\end{matrix}\right]$

(TP.inv() @ F1_parallel.inv() @ F2_parallel.inv() @ TA.inv()).simplify()

$\displaystyle \left[\begin{matrix}\frac{T^{a}_{+} \left(T^{p}_{+} f_{1} - T^{p}_{-} \left(f_{1} - 1\right)\right)}{f_{1} \left(\left(T^{a}_{+}\right)^{2} - \left(T^{a}_{-}\right)^{2}\right) \left(\left(T^{p}_{+}\right)^{2} - \left(T^{p}_{-}\right)^{2}\right)} & - \frac{T^{a}_{-} \left(T^{p}_{+} f_{1} - T^{p}_{-} \left(f_{1} - 1\right)\right)}{f_{1} \left(\left(T^{a}_{+}\right)^{2} - \left(T^{a}_{-}\right)^{2}\right) \left(\left(T^{p}_{+}\right)^{2} - \left(T^{p}_{-}\right)^{2}\right)} & - \frac{T^{a}_{+} T^{p}_{-}}{f_{1} \left(\left(T^{a}_{+}\right)^{2} - \left(T^{a}_{-}\right)^{2}\right) \left(\left(T^{p}_{+}\right)^{2} - \left(T^{p}_{-}\right)^{2}\right)} & \frac{T^{a}_{-} T^{p}_{-}}{f_{1} \left(\left(T^{a}_{+}\right)^{2} - \left(T^{a}_{-}\right)^{2}\right) \left(\left(T^{p}_{+}\right)^{2} - \left(T^{p}_{-}\right)^{2}\right)}\\\frac{\left(T^{a}_{+} \left(f_{2} - 1\right) - T^{a}_{-}\right) \left(T^{p}_{+} f_{1} - T^{p}_{-} \left(f_{1} - 1\right)\right)}{f_{1} f_{2} \left(\left(T^{a}_{+}\right)^{2} - \left(T^{a}_{-}\right)^{2}\right) \left(\left(T^{p}_{+}\right)^{2} - \left(T^{p}_{-}\right)^{2}\right)} & \frac{\left(T^{a}_{+} - T^{a}_{-} \left(f_{2} - 1\right)\right) \left(T^{p}_{+} f_{1} - T^{p}_{-} \left(f_{1} - 1\right)\right)}{f_{1} f_{2} \left(\left(T^{a}_{+}\right)^{2} - \left(T^{a}_{-}\right)^{2}\right) \left(\left(T^{p}_{+}\right)^{2} - \left(T^{p}_{-}\right)^{2}\right)} & \frac{T^{p}_{-} \left(- T^{a}_{+} \left(f_{2} - 1\right) + T^{a}_{-}\right)}{f_{1} f_{2} \left(\left(T^{a}_{+}\right)^{2} - \left(T^{a}_{-}\right)^{2}\right) \left(\left(T^{p}_{+}\right)^{2} - \left(T^{p}_{-}\right)^{2}\right)} & \frac{T^{p}_{-} \left(- T^{a}_{+} + T^{a}_{-} \left(f_{2} - 1\right)\right)}{f_{1} f_{2} \left(\left(T^{a}_{+}\right)^{2} - \left(T^{a}_{-}\right)^{2}\right) \left(\left(T^{p}_{+}\right)^{2} - \left(T^{p}_{-}\right)^{2}\right)}\\\frac{T^{a}_{+} \left(T^{p}_{+} \left(f_{1} - 1\right) - T^{p}_{-} f_{1}\right)}{f_{1} \left(\left(T^{a}_{+}\right)^{2} - \left(T^{a}_{-}\right)^{2}\right) \left(\left(T^{p}_{+}\right)^{2} - \left(T^{p}_{-}\right)^{2}\right)} & - \frac{T^{a}_{-} \left(T^{p}_{+} \left(f_{1} - 1\right) - T^{p}_{-} f_{1}\right)}{f_{1} \left(\left(T^{a}_{+}\right)^{2} - \left(T^{a}_{-}\right)^{2}\right) \left(\left(T^{p}_{+}\right)^{2} - \left(T^{p}_{-}\right)^{2}\right)} & \frac{T^{a}_{+} T^{p}_{+}}{f_{1} \left(\left(T^{a}_{+}\right)^{2} - \left(T^{a}_{-}\right)^{2}\right) \left(\left(T^{p}_{+}\right)^{2} - \left(T^{p}_{-}\right)^{2}\right)} & - \frac{T^{a}_{-} T^{p}_{+}}{f_{1} \left(\left(T^{a}_{+}\right)^{2} - \left(T^{a}_{-}\right)^{2}\right) \left(\left(T^{p}_{+}\right)^{2} - \left(T^{p}_{-}\right)^{2}\right)}\\\frac{\left(T^{a}_{+} \left(f_{2} - 1\right) - T^{a}_{-}\right) \left(T^{p}_{+} \left(f_{1} - 1\right) - T^{p}_{-} f_{1}\right)}{f_{1} f_{2} \left(\left(T^{a}_{+}\right)^{2} - \left(T^{a}_{-}\right)^{2}\right) \left(\left(T^{p}_{+}\right)^{2} - \left(T^{p}_{-}\right)^{2}\right)} & \frac{\left(T^{a}_{+} - T^{a}_{-} \left(f_{2} - 1\right)\right) \left(T^{p}_{+} \left(f_{1} - 1\right) - T^{p}_{-} f_{1}\right)}{f_{1} f_{2} \left(\left(T^{a}_{+}\right)^{2} - \left(T^{a}_{-}\right)^{2}\right) \left(\left(T^{p}_{+}\right)^{2} - \left(T^{p}_{-}\right)^{2}\right)} & \frac{T^{p}_{+} \left(T^{a}_{+} \left(f_{2} - 1\right) - T^{a}_{-}\right)}{f_{1} f_{2} \left(\left(T^{a}_{+}\right)^{2} - \left(T^{a}_{-}\right)^{2}\right) \left(\left(T^{p}_{+}\right)^{2} - \left(T^{p}_{-}\right)^{2}\right)} & \frac{T^{p}_{+} \left(T^{a}_{+} - T^{a}_{-} \left(f_{2} - 1\right)\right)}{f_{1} f_{2} \left(\left(T^{a}_{+}\right)^{2} - \left(T^{a}_{-}\right)^{2}\right) \left(\left(T^{p}_{+}\right)^{2} - \left(T^{p}_{-}\right)^{2}\right)}\end{matrix}\right]$

(TP.inv() @ F1_antiparallel.inv() @ F2_antiparallel.inv() @ TA.inv()).simplify()

$\displaystyle \left[\begin{matrix}\frac{T^{p}_{+} \left(T^{a}_{+} f_{2} - T^{a}_{-} \left(f_{2} - 1\right)\right)}{f_{1} f_{2} \left(\left(T^{a}_{+}\right)^{2} - \left(T^{a}_{-}\right)^{2}\right) \left(\left(T^{p}_{+}\right)^{2} - \left(T^{p}_{-}\right)^{2}\right)} & \frac{T^{p}_{+} \left(T^{a}_{+} \left(f_{2} - 1\right) - T^{a}_{-} f_{2}\right)}{f_{1} f_{2} \left(\left(T^{a}_{+}\right)^{2} - \left(T^{a}_{-}\right)^{2}\right) \left(\left(T^{p}_{+}\right)^{2} - \left(T^{p}_{-}\right)^{2}\right)} & \frac{\left(T^{a}_{+} f_{2} - T^{a}_{-} \left(f_{2} - 1\right)\right) \left(T^{p}_{+} \left(f_{1} - 1\right) - T^{p}_{-} f_{1}\right)}{f_{1} f_{2} \left(\left(T^{a}_{+}\right)^{2} - \left(T^{a}_{-}\right)^{2}\right) \left(\left(T^{p}_{+}\right)^{2} - \left(T^{p}_{-}\right)^{2}\right)} & \frac{\left(T^{a}_{+} \left(f_{2} - 1\right) - T^{a}_{-} f_{2}\right) \left(T^{p}_{+} \left(f_{1} - 1\right) - T^{p}_{-} f_{1}\right)}{f_{1} f_{2} \left(\left(T^{a}_{+}\right)^{2} - \left(T^{a}_{-}\right)^{2}\right) \left(\left(T^{p}_{+}\right)^{2} - \left(T^{p}_{-}\right)^{2}\right)}\\- \frac{T^{a}_{-} T^{p}_{+}}{f_{1} f_{2} \left(\left(T^{a}_{+}\right)^{2} - \left(T^{a}_{-}\right)^{2}\right) \left(\left(T^{p}_{+}\right)^{2} - \left(T^{p}_{-}\right)^{2}\right)} & \frac{T^{a}_{+} T^{p}_{+}}{f_{1} f_{2} \left(\left(T^{a}_{+}\right)^{2} - \left(T^{a}_{-}\right)^{2}\right) \left(\left(T^{p}_{+}\right)^{2} - \left(T^{p}_{-}\right)^{2}\right)} & - \frac{T^{a}_{-} \left(T^{p}_{+} \left(f_{1} - 1\right) - T^{p}_{-} f_{1}\right)}{f_{1} f_{2} \left(\left(T^{a}_{+}\right)^{2} - \left(T^{a}_{-}\right)^{2}\right) \left(\left(T^{p}_{+}\right)^{2} - \left(T^{p}_{-}\right)^{2}\right)} & \frac{T^{a}_{+} \left(T^{p}_{+} \left(f_{1} - 1\right) - T^{p}_{-} f_{1}\right)}{f_{1} f_{2} \left(\left(T^{a}_{+}\right)^{2} - \left(T^{a}_{-}\right)^{2}\right) \left(\left(T^{p}_{+}\right)^{2} - \left(T^{p}_{-}\right)^{2}\right)}\\\frac{T^{p}_{-} \left(- T^{a}_{+} f_{2} + T^{a}_{-} \left(f_{2} - 1\right)\right)}{f_{1} f_{2} \left(\left(T^{a}_{+}\right)^{2} - \left(T^{a}_{-}\right)^{2}\right) \left(\left(T^{p}_{+}\right)^{2} - \left(T^{p}_{-}\right)^{2}\right)} & \frac{T^{p}_{-} \left(- T^{a}_{+} \left(f_{2} - 1\right) + T^{a}_{-} f_{2}\right)}{f_{1} f_{2} \left(\left(T^{a}_{+}\right)^{2} - \left(T^{a}_{-}\right)^{2}\right) \left(\left(T^{p}_{+}\right)^{2} - \left(T^{p}_{-}\right)^{2}\right)} & \frac{\left(T^{a}_{+} f_{2} - T^{a}_{-} \left(f_{2} - 1\right)\right) \left(T^{p}_{+} f_{1} - T^{p}_{-} \left(f_{1} - 1\right)\right)}{f_{1} f_{2} \left(\left(T^{a}_{+}\right)^{2} - \left(T^{a}_{-}\right)^{2}\right) \left(\left(T^{p}_{+}\right)^{2} - \left(T^{p}_{-}\right)^{2}\right)} & \frac{\left(T^{a}_{+} \left(f_{2} - 1\right) - T^{a}_{-} f_{2}\right) \left(T^{p}_{+} f_{1} - T^{p}_{-} \left(f_{1} - 1\right)\right)}{f_{1} f_{2} \left(\left(T^{a}_{+}\right)^{2} - \left(T^{a}_{-}\right)^{2}\right) \left(\left(T^{p}_{+}\right)^{2} - \left(T^{p}_{-}\right)^{2}\right)}\\\frac{T^{a}_{-} T^{p}_{-}}{f_{1} f_{2} \left(\left(T^{a}_{+}\right)^{2} - \left(T^{a}_{-}\right)^{2}\right) \left(\left(T^{p}_{+}\right)^{2} - \left(T^{p}_{-}\right)^{2}\right)} & - \frac{T^{a}_{+} T^{p}_{-}}{f_{1} f_{2} \left(\left(T^{a}_{+}\right)^{2} - \left(T^{a}_{-}\right)^{2}\right) \left(\left(T^{p}_{+}\right)^{2} - \left(T^{p}_{-}\right)^{2}\right)} & - \frac{T^{a}_{-} \left(T^{p}_{+} f_{1} - T^{p}_{-} \left(f_{1} - 1\right)\right)}{f_{1} f_{2} \left(\left(T^{a}_{+}\right)^{2} - \left(T^{a}_{-}\right)^{2}\right) \left(\left(T^{p}_{+}\right)^{2} - \left(T^{p}_{-}\right)^{2}\right)} & \frac{T^{a}_{+} \left(T^{p}_{+} f_{1} - T^{p}_{-} \left(f_{1} - 1\right)\right)}{f_{1} f_{2} \left(\left(T^{a}_{+}\right)^{2} - \left(T^{a}_{-}\right)^{2}\right) \left(\left(T^{p}_{+}\right)^{2} - \left(T^{p}_{-}\right)^{2}\right)}\end{matrix}\right]$

Now we will test if the flipper matrices together with the polarizer/analyzer matrices yield physically useful results for the examples:

1. collinear magnetic moments & no chiral terms, i.e., P=F=1 & S+-=S-+

2. chiral magnetic moments with S++=S–=0 & S+- :nbsphinx-math:`neq `S-+

S_test_i = sp.symbols('Spp, Spm, Smp, Smm', positive=True)
S_test_i = sp.ImmutableDenseMatrix([[0], [S_test_i[1]], [S_test_i[2]], [0]])
S_test_i

$\displaystyle \left[\begin{matrix}0\\Spm\\Smp\\0\end{matrix}\right]$

((TA @ F2_parallel @ F1_parallel @ TP) @ S_test_i).simplify()

$\displaystyle \left[\begin{matrix}Smp T^{p}_{-} \left(T^{a}_{+} - T^{a}_{-} \left(f_{2} - 1\right)\right) + Spm T^{a}_{-} T^{p}_{+} f_{2}\\- Smp T^{p}_{-} \left(T^{a}_{+} \left(f_{2} - 1\right) - T^{a}_{-}\right) + Spm T^{a}_{+} T^{p}_{+} f_{2}\\Smp \left(T^{a}_{+} - T^{a}_{-} \left(f_{2} - 1\right)\right) \left(T^{p}_{+} f_{1} - T^{p}_{-} \left(f_{1} - 1\right)\right) - Spm T^{a}_{-} f_{2} \left(T^{p}_{+} \left(f_{1} - 1\right) - T^{p}_{-} f_{1}\right)\\- Smp \left(T^{a}_{+} \left(f_{2} - 1\right) - T^{a}_{-}\right) \left(T^{p}_{+} f_{1} - T^{p}_{-} \left(f_{1} - 1\right)\right) - Spm T^{a}_{+} f_{2} \left(T^{p}_{+} \left(f_{1} - 1\right) - T^{p}_{-} f_{1}\right)\end{matrix}\right]$

S_test_ii = sp.symbols('Spp, Spm, Smp, Smm', positive=True)
S_test_ii = sp.ImmutableDenseMatrix(
    [[S_test_ii[0]], [S_test_ii[1]], [S_test_ii[1]], [S_test_ii[3]]]
)
S_test_ii

$\displaystyle \left[\begin{matrix}Spp\\Spm\\Spm\\Smm\end{matrix}\right]$

((TA @ F2_antiparallel) @ S_test_ii).simplify()

$\displaystyle \left[\begin{matrix}- Spm \left(T^{a}_{+} \left(f_{2} - 1\right) - T^{a}_{-} f_{2}\right) + Spp T^{a}_{+}\\Spm \left(T^{a}_{+} f_{2} - T^{a}_{-} \left(f_{2} - 1\right)\right) + Spp T^{a}_{-}\\- Smm \left(T^{a}_{+} \left(f_{2} - 1\right) - T^{a}_{-} f_{2}\right) + Spm T^{a}_{+}\\Smm \left(T^{a}_{+} f_{2} - T^{a}_{-} \left(f_{2} - 1\right)\right) + Spm T^{a}_{-}\end{matrix}\right]$

Now trying the same with the to 2-dim converted matrices

S_test_iii = [[0], [4], [5], [0]]
f1_test = 0.9
f2_test = 0.9
F1_parallel_test = np.array(
    [
        [1, 0, 0, 0],
        [0, 1, 0, 0],
        [1 - 0.9, 0, 0.9, 0],
        [0, 1 - 0.9, 0, 0.9],
    ]
)
F2_parallel_test = np.array(
    [
        [1, 0, 0, 0],
        [1 - 0.9, 0.9, 0, 0],
        [0, 0, 1, 0],
        [0, 0, 1 - 0.9, 0.9],
    ]
)
Tp_test = np.array([0.8, 0.2])
TP_test = np.array(
    [
        [Tp_test[0], 0, Tp_test[1], 0],
        [0, Tp_test[0], 0, Tp_test[1]],
        [Tp_test[1], 0, Tp_test[0], 0],
        [0, Tp_test[1], 0, Tp_test[0]],
    ]
)
Ta_test = np.array([0.8, 0.2])
TA_test = np.array(
    [
        [Ta_test[0], Ta_test[1], 0, 0],
        [Ta_test[1], Ta_test[0], 0, 0],
        [0, 0, Ta_test[0], Ta_test[1]],
        [0, 0, Ta_test[1], Ta_test[0]],
    ]
)
TA_test

array([[0.8, 0.2, 0. , 0. ],
       [0.2, 0.8, 0. , 0. ],
       [0. , 0. , 0.8, 0.2],
       [0. , 0. , 0.2, 0.8]])

(TP_test @ TA_test) @ S_test_iii

array([[1.44],
       [2.76],
       [3.36],
       [1.44]])

ground_truth = np.array([0, 4, 5, 0])
analyzer = np.array([[0.8, 0.2], [0.2, 0.8]])
polarizer = np.array([[0.8, 0.2], [0.2, 0.8]])
identity = np.array([[1.0, 0.0], [0.0, 1.0]])
intensity = np.kron(identity, analyzer) @ np.kron(polarizer, identity) @ ground_truth
intensity

array([1.44, 2.76, 3.36, 1.44])

Latex formatting

(For convenience)

# Latex formatted for copying to other docs
for s in (
    r'\hat{T}_P^{-1} = ' + sp.latex(TP.inv().simplify()),
    r'\hat{T}_A^{-1} = ' + sp.latex(TA.inv().simplify()),
    r'\hat{F}_1^{-1} = ' + sp.latex(F1_parallel.inv().simplify()),
    r'\hat{F}_2^{-1} = ' + sp.latex(F2_parallel.inv().simplify()),
):
    print(s, end='\n\n')

\hat{T}_P^{-1} = \left[\begin{matrix}\frac{T^{p}_{+}}{\left(T^{p}_{+}\right)^{2} - \left(T^{p}_{-}\right)^{2}} & 0 & - \frac{T^{p}_{-}}{\left(T^{p}_{+}\right)^{2} - \left(T^{p}_{-}\right)^{2}} & 0\\0 & \frac{T^{p}_{+}}{\left(T^{p}_{+}\right)^{2} - \left(T^{p}_{-}\right)^{2}} & 0 & - \frac{T^{p}_{-}}{\left(T^{p}_{+}\right)^{2} - \left(T^{p}_{-}\right)^{2}}\\- \frac{T^{p}_{-}}{\left(T^{p}_{+}\right)^{2} - \left(T^{p}_{-}\right)^{2}} & 0 & \frac{T^{p}_{+}}{\left(T^{p}_{+}\right)^{2} - \left(T^{p}_{-}\right)^{2}} & 0\\0 & - \frac{T^{p}_{-}}{\left(T^{p}_{+}\right)^{2} - \left(T^{p}_{-}\right)^{2}} & 0 & \frac{T^{p}_{+}}{\left(T^{p}_{+}\right)^{2} - \left(T^{p}_{-}\right)^{2}}\end{matrix}\right]

\hat{T}_A^{-1} = \left[\begin{matrix}\frac{T^{a}_{+}}{\left(T^{a}_{+}\right)^{2} - \left(T^{a}_{-}\right)^{2}} & - \frac{T^{a}_{-}}{\left(T^{a}_{+}\right)^{2} - \left(T^{a}_{-}\right)^{2}} & 0 & 0\\- \frac{T^{a}_{-}}{\left(T^{a}_{+}\right)^{2} - \left(T^{a}_{-}\right)^{2}} & \frac{T^{a}_{+}}{\left(T^{a}_{+}\right)^{2} - \left(T^{a}_{-}\right)^{2}} & 0 & 0\\0 & 0 & \frac{T^{a}_{+}}{\left(T^{a}_{+}\right)^{2} - \left(T^{a}_{-}\right)^{2}} & - \frac{T^{a}_{-}}{\left(T^{a}_{+}\right)^{2} - \left(T^{a}_{-}\right)^{2}}\\0 & 0 & - \frac{T^{a}_{-}}{\left(T^{a}_{+}\right)^{2} - \left(T^{a}_{-}\right)^{2}} & \frac{T^{a}_{+}}{\left(T^{a}_{+}\right)^{2} - \left(T^{a}_{-}\right)^{2}}\end{matrix}\right]

\hat{F}_1^{-1} = \left[\begin{matrix}1 & 0 & 0 & 0\\0 & 1 & 0 & 0\\\frac{f_{1} - 1}{f_{1}} & 0 & \frac{1}{f_{1}} & 0\\0 & \frac{f_{1} - 1}{f_{1}} & 0 & \frac{1}{f_{1}}\end{matrix}\right]

\hat{F}_2^{-1} = \left[\begin{matrix}1 & 0 & 0 & 0\\\frac{f_{2} - 1}{f_{2}} & \frac{1}{f_{2}} & 0 & 0\\0 & 0 & 1 & 0\\0 & 0 & \frac{f_{2} - 1}{f_{2}} & \frac{1}{f_{2}}\end{matrix}\right]