Estia

docs/user-guide/estia-data-reduction.md

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+# Polarization data reduction for ESTIA
+
+Based on https://confluence.ess.eu/display/ESTIA/Polarised+Neutron+Reflectometry+%28PNR%29+-+Reduction+Notes
+
+## Model
+
+Intensity in the detector is related to the reflectivity of the sample by the model
+```math
+\begin{bmatrix}
+I^{+} \\
+I^{-}
+\end{bmatrix}
+\big(\lambda, j\big)
+=
+D(\lambda, j)
+\begin{bmatrix}
+1 - a^{\uparrow} & 1 - a^{\downarrow} \\
+a^{\uparrow} & a^{\downarrow}
+\end{bmatrix}
+\begin{bmatrix}
+(1 - f_2) & f_2 \\ f_2 & (1 - f_2)
+\end{bmatrix}
+\begin{bmatrix}
+R^{\uparrow\uparrow} & R^{\downarrow\uparrow} \\
+R^{\uparrow\downarrow} & R^{\downarrow\downarrow}
+\end{bmatrix}
+\begin{bmatrix}
+(1 - f_1) & f_1 \\ f_1 & (1 - f_1)
+\end{bmatrix}
+\begin{bmatrix}
+1 - p^{\uparrow} \\
+1 - p^{\downarrow}
+\end{bmatrix}
+```
+where
+
+* $I^+$ is the intensity of the neutron beam transmitted by the analyzer
+and $I^-$ is the intensity of the neutron beam reflected by the analyzer,
+* $R^\cdot$ are the reflectivities of the sample,
+  - $R^{\uparrow\uparrow}$ is the fraction of incoming neutrons with spin up that are reflected with spin up,
+  - $R^{\uparrow\downarrow}$ is the fraction of incoming neutrons with spin up that are reflected with spin down,
+  - etc..
+* $a^\uparrow$ is the analyzer reflectivity for spin up neutrons and $a^\downarrow$ is the analyzer reflectivity for spin down neutrons,
+* $p^\uparrow$ is the polarizer reflectivity for spin up neutrons and $p^\downarrow$ is the polarizer reflectivity for spin down neutrons,
+* $f_1$ is the probability of spin flip by the polarizer spin flipper, $f_2$ is the probability of spin flip by the analyzer spin flipper
+* $D$ represents the inhomogeneity from the beam- and detector efficiency (and all other polarization unrelated terms).
+
+## Reducing a measurement
+
+If the sample is measured at two different flipper settings $f_1=0, f_2=0$ and $f_1=1, f_2=0$, then we have four measurement in total:
+```math
+\begin{bmatrix}
+I^{0+} \\
+I^{0-}
+\end{bmatrix}
+\big(\lambda, j\big)
+=
+D(\lambda, j)
+\begin{bmatrix}
+1 - a^{\uparrow} & 1 - a^{\downarrow} \\
+a^{\uparrow} & a^{\downarrow}
+\end{bmatrix}
+\begin{bmatrix}
+R^{\uparrow\uparrow} & R^{\downarrow\uparrow} \\
+R^{\uparrow\downarrow} & R^{\downarrow\downarrow}
+\end{bmatrix}
+\begin{bmatrix}
+1 - p^{\uparrow} \\
+1 - p^{\downarrow}
+\end{bmatrix}
+```
+```math
+\begin{bmatrix}
+I^{1+} \\
+I^{1-}
+\end{bmatrix}
+\big(\lambda, j\big)
+=
+D(\lambda, j)
+\begin{bmatrix}
+1 - a^{\uparrow} & 1 - a^{\downarrow} \\
+a^{\uparrow} & a^{\downarrow}
+\end{bmatrix}
+\begin{bmatrix}
+R^{\uparrow\uparrow} & R^{\downarrow\uparrow} \\
+R^{\uparrow\downarrow} & R^{\downarrow\downarrow}
+\end{bmatrix}
+\begin{bmatrix}
+1 - p^{\downarrow} \\
+1 - p^{\uparrow}
+\end{bmatrix}.
+```
+
+To simplify the above, collect the terms in the matrix $\mathbf{a}$
+```math
+\begin{bmatrix}
+I^{0+} \\
+I^{0-} \\
+I^{1+} \\
+I^{1-}
+\end{bmatrix}
+\big(\lambda, j\big)
+=
+D(\lambda, j)
+\mathbf{a}(\lambda)
+\begin{bmatrix}
+R^{\uparrow\uparrow} \\
+R^{\uparrow\downarrow} \\
+R^{\downarrow\uparrow} \\
+R^{\downarrow\downarrow}
+\end{bmatrix}
+(Q(\lambda, j)).
+```
+
+To compute the reflectivities, integrate over a region of (almost) constant $Q$
+```math
+\int_{Q\in[q_{n}, q_{n+1}]}
+\mathbf{a}^{-1}(\lambda)
+\begin{bmatrix}
+I^{0+} \\
+I^{0-} \\
+I^{1+} \\
+I^{1-}
+\end{bmatrix}
+\big(\lambda, j\big)
+d\lambda dj
+\approx
+\int_{Q\in[q_{n}, q_{n+1}]}
+D(\lambda, j)
+d\lambda dj
+\begin{bmatrix}
+R^{\uparrow\uparrow} \\
+R^{\uparrow\downarrow} \\
+R^{\downarrow\uparrow} \\
+R^{\downarrow\downarrow}
+\end{bmatrix}
+(q_{n+\frac{1}{2}}).
+```
+The integral on the righ-hand-side can be evaluated using the reference measurement, call evaluated integral $\bar{D}(q_{n+{\frac{1}{2}}})$.
+$R$ was moved outside of the integral because if $Q$ is almost constant so is $R(Q)$.
+
+Finally we have
+```math
+\int_{Q\in[q_{n}, q_{n+1}]}
+\mathbf{a}^{-1}(\lambda)
+\bar{D}^{-1}(q_{n+{\frac{1}{2}}})
+\begin{bmatrix}
+I^{0+} \\
+I^{0-} \\
+I^{1+} \\
+I^{1-}
+\end{bmatrix}
+\big(\lambda, j\big)
+d\lambda dj
+\approx
+\begin{bmatrix}
+R^{\uparrow\uparrow} \\
+R^{\uparrow\downarrow} \\
+R^{\downarrow\uparrow} \\
+R^{\downarrow\downarrow}
+\end{bmatrix}
+(q_{n+\frac{1}{2}}).
+```
+
+### How to use the reference measurement to compute the integral over $D(\lambda, j)$?
+
+For a reference measurement using flipper setting $f_1=0, f_2=0$ we have
+```math
+\begin{bmatrix}
+I_{ref}^{+} \\
+I_{ref}^{-}
+\end{bmatrix}
+\big(\lambda, j\big)
+=
+D(\lambda, j)
+\begin{bmatrix}
+1 - a^{\uparrow} & 1 - a^{\downarrow} \\
+a^{\uparrow} & a^{\downarrow}
+\end{bmatrix}
+\begin{bmatrix}
+R_{ref}^{\uparrow\uparrow} & R_{ref}^{\downarrow\uparrow} \\
+R_{ref}^{\uparrow\downarrow} & R_{ref}^{\downarrow\downarrow}
+\end{bmatrix}
+\begin{bmatrix}
+1 - p^{\uparrow} \\
+1 - p^{\downarrow}
+\end{bmatrix}.
+```
+But in practice, the analyzer/polarizer will be efficient enough to make only one of $I_{ref}^\pm$ have enough intensity to be useful. For example:
+```math
+\frac{I_{ref}^{+}(\lambda, j)}{r^+(\lambda, j)}
+=
+D(\lambda, j)
+```
+where $r^+$ is a known term involving the reflectivity of the supermirror and the pol-/analyzer efficiencies.
+The expression for $D$ above can be used to evaluate integrals of $D$,
+but in this case only in the region of the detector where the transmitted beam hits, because we only got data in that region from our reference measurement.
+
+To measure $D$ for the entire detector we need to make several reference measurements with different flipper settings so that every part of the detector is illuminated in at least one measurement.
+It might be unecessary to use all 4 flipper settings, but to illustrate the idea imagine we make reference measurements using all 4 flipper settings:
+```math
+\begin{bmatrix}
+I_{ref}^{00+} \\
+I_{ref}^{00-}
+\end{bmatrix}
+\big(\lambda, j\big)
+=
+D(\lambda, j)
+\begin{bmatrix}
+1 - a^{\uparrow} & 1 - a^{\downarrow} \\
+a^{\uparrow} & a^{\downarrow}
+\end{bmatrix}
+\begin{bmatrix}
+R_{ref}^{\uparrow\uparrow} & R_{ref}^{\downarrow\uparrow} \\
+R_{ref}^{\uparrow\downarrow} & R_{ref}^{\downarrow\downarrow}
+\end{bmatrix}
+\begin{bmatrix}
+1 - p^{\uparrow} \\
+1 - p^{\downarrow}
+\end{bmatrix}
+```
+```math
+\begin{bmatrix}
+I_{ref}^{01+} \\
+I_{ref}^{01-}
+\end{bmatrix}
+\big(\lambda, j\big)
+=
+D(\lambda, j)
+\begin{bmatrix}
+1 - a^{\uparrow} & 1 - a^{\downarrow} \\
+a^{\uparrow} & a^{\downarrow}
+\end{bmatrix}
+\begin{bmatrix}
+0 & 1 \\ 1 & 0
+\end{bmatrix}
+\begin{bmatrix}
+R_{ref}^{\uparrow\uparrow} & R_{ref}^{\downarrow\uparrow} \\
+R_{ref}^{\uparrow\downarrow} & R_{ref}^{\downarrow\downarrow}
+\end{bmatrix}
+\begin{bmatrix}
+1 - p^{\uparrow} \\
+1 - p^{\downarrow}
+\end{bmatrix}
+```
+```math
+\begin{bmatrix}
+I_{ref}^{10+} \\
+I_{ref}^{10-}
+\end{bmatrix}
+\big(\lambda, j\big)
+=
+D(\lambda, j)
+\begin{bmatrix}
+1 - a^{\uparrow} & 1 - a^{\downarrow} \\
+a^{\uparrow} & a^{\downarrow}
+\end{bmatrix}
+\begin{bmatrix}
+R_{ref}^{\uparrow\uparrow} & R_{ref}^{\downarrow\uparrow} \\
+R_{ref}^{\uparrow\downarrow} & R_{ref}^{\downarrow\downarrow}
+\end{bmatrix}
+\begin{bmatrix}
+0 & 1 \\ 1 & 0
+\end{bmatrix}
+\begin{bmatrix}
+1 - p^{\uparrow} \\
+1 - p^{\downarrow}
+\end{bmatrix}
+```
+```math
+\begin{bmatrix}
+I_{ref}^{11+} \\
+I_{ref}^{11-}
+\end{bmatrix}
+\big(\lambda, j\big)
+=
+D(\lambda, j)
+\begin{bmatrix}
+1 - a^{\uparrow} & 1 - a^{\downarrow} \\
+a^{\uparrow} & a^{\downarrow}
+\end{bmatrix}
+\begin{bmatrix}
+0 & 1 \\ 1 & 0
+\end{bmatrix}
+\begin{bmatrix}
+R_{ref}^{\uparrow\uparrow} & R_{ref}^{\downarrow\uparrow} \\
+R_{ref}^{\uparrow\downarrow} & R_{ref}^{\downarrow\downarrow}
+\end{bmatrix}
+\begin{bmatrix}
+0 & 1 \\ 1 & 0
+\end{bmatrix}
+\begin{bmatrix}
+1 - p^{\uparrow} \\
+1 - p^{\downarrow}
+\end{bmatrix}.
+```
+
+Summing all 8 measurements gives us an expression for $D$ that ought to be valid for the entire detector:
+```math
+\frac{
+I_{ref}^{00+}(\lambda, j) +
+I_{ref}^{00-}(\lambda, j) +
+I_{ref}^{01+}(\lambda, j) +
+I_{ref}^{01-}(\lambda, j) +
+I_{ref}^{10+}(\lambda, j) +
+I_{ref}^{10-}(\lambda, j) +
+I_{ref}^{11+}(\lambda, j) +
+I_{ref}^{11-}(\lambda, j)
+}{
+r^{00+}(\lambda, j) +
+r^{00-}(\lambda, j) +
+r^{01+}(\lambda, j) +
+r^{01-}(\lambda, j) +
+r^{10+}(\lambda, j) +
+r^{10-}(\lambda, j) +
+r^{11+}(\lambda, j) +
+r^{11-}(\lambda, j)
+}
+=
+D(\lambda, j).
+```