Difference between revisions of "Talk:DWBA"

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(Created page with "<math> \begin{align} I_d(q_{z}) & = | T_i T_f F(+Q_{z1}) + T_i R_f F(-Q_{z2}) + R_i T_f F(+Q_{z2}) + R_i R_f F(-Q_{z1}) | ^{2} \\ & \begin{align} ...")
 
(→‎Breaking into components)
 
(32 intermediate revisions by the same user not shown)
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 +
==DWBA Equation in thin film==
 +
Using the notation <math>T_i = T(\alpha_i)</math> for compactness, the DWBA equation inside a thin film can be written:
 +
 +
<math>
 +
\begin{align}
 +
I_d(q_{z}) & =  |
 +
      T_i T_f F(+Q_{z1})
 +
    + T_i R_f F(-Q_{z2})
 +
    + R_i T_f F(+Q_{z2})
 +
    + R_i R_f F(-Q_{z1})  | ^{2} \\
 +
\end{align}
 +
</math>
 +
 +
==Expansion (incorrect)==
 +
'''WARNING: This incorrectly ignores the complex components.'''
 +
===Terms===
 +
If one expands the <math>|...|^2</math> of the DWBA, one obtains 16 terms:
 +
 +
<math>
 +
 +
\begin{matrix}
 +
          &  (T_i T_f)    &  (T_i R_f)      &  (R_i T_f)    &  (R_i R_f)    \\
 +
(T_i T_f)  &  T_i^2T_f^2    &  T_i^2 T_f R_f  & T_iR_iT_f^2    &  T_iR_iT_fR_f  \\
 +
(T_i R_f)  &  T_i^2T_fR_f  &  T_i^2R_f^2    &  T_iR_iT_fR_f  &  T_iR_iR_f^2  \\
 +
(R_i T_f)  &  T_iR_iT_f^2  &  T_iR_iT_fR_f  &  R_i^2T_f^2    &  R_i^2T_fR_f  \\
 +
(R_i R_f)  &  T_iR_iT_fR_f  &  T_iR_iR_f^2    &  R_i^2T_fR_f  &  R_i^2R_f^2    \\
 +
\end{matrix}
 +
 +
</math>
 +
 +
===Equation===
 +
The equation can thus be expanded as:
 +
 
<math>
 
<math>
 
\begin{align}
 
\begin{align}
Line 9: Line 42:
 
     & \begin{align}
 
     & \begin{align}
 
         = \,\, &  T_i^2 T_f^2 | F(+Q_{z1}) |^2 && + T_i^2 T_f R_f F(+Q_{z1})F(-Q_{z2}) \\
 
         = \,\, &  T_i^2 T_f^2 | F(+Q_{z1}) |^2 && + T_i^2 T_f R_f F(+Q_{z1})F(-Q_{z2}) \\
−
           & && + T_i R_i T_f R_f F(+Q_{z1})F(+Q_{z2}) + T_i R_i T_f R_f F(+Q_{z1}) F(-Q_{z1})  \\
+
           & && + T_i R_i T_f^2 F(+Q_{z1})F(+Q_{z2}) + T_i R_i T_f R_f F(+Q_{z1}) F(-Q_{z1})  \\
  
 
           & + T_i^2 R_f^2 | F(-Q_{z2}) |^2 && + T_i^2T_fR_f F(+Q_{z1}) F(-Q_{z2}) \\
 
           & + T_i^2 R_f^2 | F(-Q_{z2}) |^2 && + T_i^2T_fR_f F(+Q_{z1}) F(-Q_{z2}) \\
Line 22: Line 55:
 
         \end{align} \\
 
         \end{align} \\
  
 +
\end{align}
 +
</math>
 +
 +
===Simplification===
 +
We can rearrange to:
 +
 +
 +
<math>
 +
\begin{align}
 +
I_d(q_{z}) = \,\, & T_i^2 T_f^2 | F(+Q_{z1}) |^2 + T_i^2 R_f^2 | F(-Q_{z2}) |^2 + R_i^2 T_f^2 | F(+Q_{z2}) |^2 + R_i^2 R_f^2 | F(-Q_{z1}) |^2 \\
 +
 +
        & + T_i^2 T_f R_f F(+Q_{z1})F(-Q_{z2}) \\
 +
        & + T_i R_i T_f^2 F(+Q_{z1})F(+Q_{z2}) + T_i R_i T_f R_f F(+Q_{z1}) F(-Q_{z1})  \\
 +
 +
        & + T_i^2T_fR_f F(+Q_{z1}) F(-Q_{z2}) \\
 +
        & + T_i R_i T_f R_f F(+Q_{z2})F(-Q_{z2}) + T_i R_i R_f^2 F(-Q_{z1}) F(-Q_{z2})  \\
 +
 +
        & + T_i R_i T_f^2 F(+Q_{z1}) F(+Q_{z2}) \\
 +
        & + T_i R_i T_f R_f F(+Q_{z2})F(-Q_{z2}) + R_i^2 T_f R_f F(-Q_{z1}) F(+Q_{z2})  \\
 +
 +
        & + T_i R_i T_f R_f F(+Q_{z1}) F(-Q_{z1}) \\
 +
        & + T_i R_i R_f^2 F(-Q_{z1})F(-Q_{z2}) + R_i^2 T_f R_f F(-Q_{z1}) F(+Q_{z2})  \\
 +
 +
   
 +
 +
= \,\, & T_i^2 T_f^2 | F(+Q_{z1}) |^2 + T_i^2 R_f^2 | F(-Q_{z2}) |^2 + R_i^2 T_f^2 | F(+Q_{z2}) |^2 + R_i^2 R_f^2 | F(-Q_{z1}) |^2 \\
 +
        & + 2 \times T_i^2T_fR_f F(+Q_{z1})F(-Q_{z2}) \\
 +
        & + 2 \times T_iR_iT_f^2 F(+Q_{z1})F(+Q_{z2}) \\
 +
        & + 2 \times T_i R_i T_f R_f [ F(+Q_{z1})F(-Q_{z1}) + F(+Q_{z2})F(-Q_{z2}) ]  \\
 +
        & + 2 \times T_iR_iR_f^2 F(-Q_{z1})F(-Q_{z2}) \\
 +
        & + 2 \times R_i^2T_fR_fx F(-Q_{z1})F(+Q_{z2}) \\
 +
 +
 +
= \,\, & T_i^2 T_f^2 | F(+Q_{z1}) |^2 + T_i^2 R_f^2 | F(-Q_{z2}) |^2 + R_i^2 T_f^2 | F(+Q_{z2}) |^2 + R_i^2 R_f^2 | F(-Q_{z1}) |^2 \\
 +
        & + 2 \times T_i^2T_fR_f F(+Q_{z1})F(-Q_{z2})
 +
          + 2 \times T_iR_iT_f^2 F(+Q_{z1})F(+Q_{z2}) \\
 +
        & + 2 \times T_i R_i T_f R_f [ F(+Q_{z1})F(-Q_{z1}) + F(+Q_{z2})F(-Q_{z2}) ]  \\
 +
        & + 2 \times T_iR_iR_f^2 F(-Q_{z1})F(-Q_{z2})
 +
          + 2 \times R_i^2T_fR_fx F(-Q_{z1})F(+Q_{z2}) \\
 +
 +
 +
\end{align}
 +
</math>
 +
 +
We can rewrite in a more compact form using the notation <math>T_i = T(\alpha_i)</math> and <math>F_{+1} = F(+Q_{z1})</math>:
 +
 +
<math>
 +
\begin{align}
 +
I_d(q_{z})
 +
= \,\, & T_i^2 T_f^2 | F_{+1} |^2 + T_i^2 R_f^2 | F_{-2} |^2 + R_i^2 T_f^2 | F_{+2} |^2 + R_i^2 R_f^2 | F_{-1} |^2 \\
 +
        & + 2 \times T_i^2T_fR_f F_{+1}F_{-2}
 +
          + 2 \times T_iR_iT_f^2 F_{+1}F_{+2} \\
 +
        & + 2 \times T_i R_i T_f R_f [ F_{+1}F_{-1} + F_{+2}F_{-2} ]  \\
 +
        & + 2 \times T_iR_iR_f^2 F_{-1}F_{-2}
 +
          + 2 \times R_i^2T_fR_f F_{-1}F_{+2}
 +
\end{align}
 +
</math>
 +
 +
==Expansion==
 +
 +
===Terms===
 +
If one expands the <math>|...|^2</math> of the DWBA, one obtains 16 terms:
 +
 +
<math>
 +
 +
\begin{matrix}
 +
          &  (T_i^* T_f^*)        &  (T_i^* R_f^*)      &  (R_i^* T_f^*)    &  (R_i^* R_f^*)    \\
 +
(T_i T_f)  &  T_i T_i^* T_f T_f^*  &  T_i T_i^* T_f R_f^*  & T_i R_i^* T_f T_f^*    &  T_i R_i^* T_f R_f^*  \\
 +
(T_i R_f)  &  T_i T_i^* T_f^* R_f  &  T_i T_i^* R_f R_f^*    &  T_i R_i^* T_f^* R_f  &  T_i R_i^* R_f R_f^*  \\
 +
(R_i T_f)  &  T_i^* R_i T_f T_f^*  &  T_i^* R_i T_f R_f^*  &  R_i R_i^* T_f T_f^*    &  R_i R_i^* T_f R_f^*  \\
 +
(R_i R_f)  &  T_i^* R_i T_f^* R_f  &  T_i^* R_i R_f R_f^*    &  R_i R_i^* T_f^* R_f  &  R_i R_i^* R_f R_f^*    \\
 +
\end{matrix}
 +
 +
</math>
 +
 +
 +
<math>
 +
 +
\begin{matrix}
 +
          &  (T_i^* T_f^*)        &  (T_i^* R_f^*)      &  (R_i^* T_f^*)    &  (R_i^* R_f^*)    \\
 +
(T_i T_f)  &  |T_i T_f|^2  &  |T_i|^2 T_f R_f^*  & T_i R_i^* |T_f|^2    &  T_i R_i^* T_f R_f^*  \\
 +
(T_i R_f)  &  |T_i|^2 T_f^* R_f  &  |T_i R_f|^2    &  T_i R_i^* T_f^* R_f  &  T_i R_i^* |R_f|^2  \\
 +
(R_i T_f)  &  T_i^* R_i |T_f|^2  &  T_i^* R_i T_f R_f^*  &  |R_i T_f|^2    &  |R_i|^2 T_f R_f^*  \\
 +
(R_i R_f)  &  T_i^* R_i T_f^* R_f  &  T_i^* R_i |R_f|^2    &  |R_i|^2 T_f^* R_f  & | R_i R_f |^2    \\
 +
\end{matrix}
 +
 +
</math>
 +
 +
 +
 +
 +
===Equation===
 +
We take advantage of a more compact form using the notation <math>T_i = T(\alpha_i)</math> and <math>F_{+1} = F(+Q_{z1})</math>. The DWBA equation can thus be expanded as:
 +
 +
<math>
 +
\begin{align}
 +
I_d(q_{z}) & =  |
 +
      T_i T_f F_{+1}
 +
    + T_i R_f F_{-2}
 +
    + R_i T_f F_{+2}
 +
    + R_i R_f F_{-1}  | ^{2} \\
 +
 +
    & \begin{align}
 +
        = \,\, &  |T_i T_f|^2 | F_{+1} |^2 && + |T_i|^2 T_f R_f^* F_{+1}F_{-2}^* \\
 +
          & && + T_i R_i^* |T_f|^2 F_{+1}F_{+2}^* + T_i R_i^* T_f R_f^* F_{+1} F_{-1}^*  \\
 +
 +
          & + |T_i R_f|^2 | F_{-2} |^2 && + |T_i|^2T_f^*R_f F_{+1}^* F_{-2} \\
 +
          & && + T_i R_i^* T_f^* R_f F_{+2}^*F_{-2} + T_i R_i^* |R_f|^2 F_{-1}^* F_{-2}  \\
 +
 +
          & + |R_i T_f|^2 | F_{+2} |^2 && + T_i^* R_i |T_f|^2 F_{+1}^* F_{+2} \\
 +
          & && + T_i^* R_i T_f R_f^* F_{+2}^*F_{-2} + |R_i|^2 T_f R_f^* F_{-1}^* F_{+2}  \\
 +
 +
          & + |R_i R_f|^2 | F_{-1} |^2 && + T_i^* R_i T_f^* R_f F_{+1}^* F_{-1} \\
 +
          & && + T_i^* R_i |R_f|^2 F_{-1}F_{-2}^* + |R_i|^2 T_f^* R_f F_{-1} F_{+2}^*  \\
 +
 +
        \end{align} \\
 +
 +
\end{align}
 +
</math>
 +
 +
===Simplification===
 +
We can rearrange to:
 +
 +
<math>
 +
\begin{align}
 +
I_d(q_{z}) = \, \, &  |T_i T_f|^2 | F_{+1} |^2 + |T_i R_f|^2 | F_{-2} |^2 + |R_i T_f|^2 | F_{+2} |^2 + |R_i R_f|^2 | F_{-1} |^2 \\
 +
 +
    & + |T_i|^2 T_f R_f^* F_{+1}F_{-2}^* + |T_i|^2T_f^*R_f F_{+1}^* F_{-2} \\
 +
 +
    & + T_i R_i^* |T_f|^2 F_{+1}F_{+2}^* + T_i^* R_i |T_f|^2 F_{+1}^* F_{+2} \\
 +
 +
    & + |R_i|^2 T_f R_f^* F_{-1}^* F_{+2} + |R_i|^2 T_f^* R_f F_{-1} F_{+2}^* \\
 +
 +
    & + T_i R_i^* |R_f|^2 F_{-1}^* F_{-2} + T_i^* R_i |R_f|^2 F_{-1}F_{-2}^*\\
 +
 +
    & + T_i R_i^* T_f R_f^* F_{+1} F_{-1}^* \\
 +
    & + T_i R_i^* T_f^* R_f F_{+2}^*F_{-2} \\
 +
    & +  T_i^* R_i T_f R_f^* F_{+2}^*F_{-2} \\
 +
    & +  T_i^* R_i T_f^* R_f F_{+1}^* F_{-1} \\
 +
 +
= \, \, &  |T_i T_f|^2 | F_{+1} |^2 + |T_i R_f|^2 | F_{-2} |^2 + |R_i T_f|^2 | F_{+2} |^2 + |R_i R_f|^2 | F_{-1} |^2 \\
 +
 +
    & + |T_i|^2 [ T_f R_f^* F_{+1}F_{-2}^* + T_f^*R_f F_{+1}^* F_{-2} ] \\
 +
 +
    & + |T_f|^2 [ T_i R_i^*  F_{+1}F_{+2}^* + T_i^* R_i F_{+1}^* F_{+2} ] \\
 +
 +
    & + |R_i|^2 [ T_f R_f^* F_{-1}^* F_{+2} + T_f^* R_f F_{-1} F_{+2}^* ] \\
 +
 +
    & + |R_f|^2 [ T_i R_i^*  F_{-1}^* F_{-2} + T_i^* R_i F_{-1}F_{-2}^* ]\\
 +
 +
    & + [ T_i R_i^* T_f R_f^* F_{+1} F_{-1}^* + T_i^* R_i T_f^* R_f F_{+1}^* F_{-1} ] \\
 +
    & + [ T_i R_i^* T_f^* R_f F_{+2}^*F_{-2} + T_i^* R_i T_f R_f^* F_{+2}^*F_{-2} ] \\
 +
 +
\end{align}
 +
</math>
 +
 +
We define <math>I_{+1}=|F_{+1}|^2</math>, and note that for any complex number <math>c</math>, it is true that <math>c+c^*=2 \mathrm{Re}[c]</math>. Thus:
 +
 +
<math>
 +
\begin{align}
 +
I_d(q_{z})
 +
= \, \, &  |T_i T_f|^2 I_{+1} + |T_i R_f|^2 I_{-2} + |R_i T_f|^2 | I_{+2} + |R_i R_f|^2 I_{-1} \\
 +
 +
    & + 2 |T_i|^2 \mathrm{Re}[ T_f R_f^* F_{+1}F_{-2}^* ] \\
 +
 +
    & + 2 |T_f|^2 \mathrm{Re}[ T_i R_i^*  F_{+1}F_{+2}^* ] \\
 +
 +
    & + 2 |R_i|^2 \mathrm{Re}[ T_f R_f^* F_{-1}^* F_{+2} ] \\
 +
 +
    & + 2 |R_f|^2 \mathrm{Re}[ T_i R_i^*  F_{-1}^* F_{-2} ]\\
 +
 +
    & + 2 \mathrm{Re}[ T_i R_i^* T_f R_f^* F_{+1} F_{-1}^* ] \\
 +
    & + 2 \mathrm{Re}[ T_i R_i^* T_f^* R_f F_{+2}^*F_{-2} ] \\
 +
 +
= \, \, &  |T_i T_f|^2 I_{+1} + |T_i R_f|^2 I_{-2} + |R_i T_f|^2 | I_{+2} + |R_i R_f|^2 I_{-1} \\
 +
 +
    & + 2 |T_i|^2 \mathrm{Re}[ T_f R_f^* F_{+1}F_{-2}^* ] + 2 |T_f|^2 \mathrm{Re}[ T_i R_i^*  F_{+1}F_{+2}^* ] \\
 +
 +
    & + 2 |R_i|^2 \mathrm{Re}[ T_f R_f^* F_{-1}^* F_{+2} ] + 2 |R_f|^2 \mathrm{Re}[ T_i R_i^*  F_{-1}^* F_{-2} ]\\
 +
 +
    & + 2 \mathrm{Re}[ T_i R_i^* T_f R_f^* F_{+1} F_{-1}^* ] + 2 \mathrm{Re}[ T_i R_i^* T_f^* R_f F_{+2}^*F_{-2} ] \\
 +
\end{align}
 +
</math>
 +
 +
==Breaking into components==
 +
The experimental data <math>I_d(q_z)</math> can be broken into contributions from the transmitted channel <math>I_{d,\mathrm{Tc}}(qz)</math> and reflected channel <math>I_{d,\mathrm{Rc}}(qz)</math>:
 +
 +
<math>
 +
\begin{align}
 +
I_d(q_{z})
 +
  & = [ | T_i T_f|^2 + |R_i R_f|^2 ] I_{R}(+Q_{z1}) + [ |T_i R_f|^2 + |R_i T_f|^2 ] I_{R}(+Q_{z2}) \\
 +
  & = [ | T_i T_f|^2 + |R_i R_f|^2 ] I_{R}(q_z-\Delta q_{z,\mathrm{Tc}}) + [ |T_i R_f|^2 + |R_i T_f|^2 ] I_{R}(q_z-\Delta q_{z,\mathrm{Rc}}) \\
 +
  & = [ | T_i T_f|^2 + |R_i R_f|^2 ] I_{d,\mathrm{Tc}}(q_z) + [ |T_i R_f|^2 + |R_i T_f|^2 ] I_{d,\mathrm{Rc}}(q_z) \\
 +
  & = |Tc|^2 I_{d,\mathrm{Tc}}(q_z) + |Rc|^2 I_{d,\mathrm{Rc}}(q_z) \\
 +
\end{align}
 +
</math>
 +
 +
We define the ratio between the channels to be:
 +
 +
<math>
 +
\begin{align}
 +
w (q_z)
 +
  & = \frac{ I_{d,\mathrm{Tc}}(q_z) }{ I_{d,\mathrm{Tc}}(q_z) + I_{d,\mathrm{Rc}}(q_z) }
 +
\end{align}
 +
</math>
 +
 +
Such that one can compute the two components from:
 +
 +
<math>
 +
\begin{align}
 +
I_d(q_{z}) & = |Tc|^2 ( I_{d,\mathrm{Tc}}(q_z) ) + |Rc|^2 ( I_{d,\mathrm{Rc}}(q_z) ) \\
 +
I_d(q_{z}) & = |Tc|^2 ( I_{d,\mathrm{Tc}}(q_z) ) + |Rc|^2 \left ( \frac{ I_{d,\mathrm{Tc}}(q_z) - w I_{d,\mathrm{Tc}}(q_z) }{w}  \right ) \\
 +
I_d(q_{z}) & = I_{d,\mathrm{Tc}}(q_z) \times \left ( |Tc|^2  +  |Rc|^2 \frac{ 1}{w}  - |Rc|^2 \frac{w }{w}  \right ) \\
 +
I_{d,\mathrm{Tc}}(q_z)  & = \frac{ I_d(q_{z}) }{  |Tc|^2  +  \frac{ |Rc|^2 }{w}  - |Rc|^2  } \\
 +
\end{align}
 +
</math>
 +
 +
and:
 +
 +
 +
<math>
 +
\begin{align}
 +
I_{d,\mathrm{Rc}}(q_z) & = \frac{ I_d(q_{z}) - |Tc|^2 I_{d,\mathrm{Tc}}(q_z) }{|Rc|^2} \\
 +
    & = \frac{ I_d(q_{z}) }{|Rc|^2} - \frac{|Tc|^2}{|Rc|^2} I_{d,\mathrm{Tc}}(q_z)
 +
\end{align}
 +
</math>
 +
 +
or:
 +
 +
<math>
 +
\begin{align}
 +
I_d(q_{z}) & = |Tc|^2 ( I_{d,\mathrm{Tc}}(q_z) ) + |Rc|^2 ( I_{d,\mathrm{Rc}}(q_z) ) \\
 +
    & = |Tc|^2 \left( \frac{w}{1-w}  I_{d,\mathrm{Rc}}(q_z) \right) + |Rc|^2 ( I_{d,\mathrm{Rc}}(q_z) ) \\
 +
I_{d,\mathrm{Rc}}(q_z) & = \frac{ I_d(q_{z}) }{|Tc|^2 \frac{w}{1-w} + |Rc|^2}
 
\end{align}
 
\end{align}
 
</math>
 
</math>

Latest revision as of 10:21, 5 April 2018

DWBA Equation in thin film

Using the notation Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T_i = T(\alpha_i)} for compactness, the DWBA equation inside a thin film can be written:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} I_d(q_{z}) & = | T_i T_f F(+Q_{z1}) + T_i R_f F(-Q_{z2}) + R_i T_f F(+Q_{z2}) + R_i R_f F(-Q_{z1}) | ^{2} \\ \end{align} }

Expansion (incorrect)

WARNING: This incorrectly ignores the complex components.

Terms

If one expands the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |...|^2} of the DWBA, one obtains 16 terms:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{matrix} & (T_i T_f) & (T_i R_f) & (R_i T_f) & (R_i R_f) \\ (T_i T_f) & T_i^2T_f^2 & T_i^2 T_f R_f & T_iR_iT_f^2 & T_iR_iT_fR_f \\ (T_i R_f) & T_i^2T_fR_f & T_i^2R_f^2 & T_iR_iT_fR_f & T_iR_iR_f^2 \\ (R_i T_f) & T_iR_iT_f^2 & T_iR_iT_fR_f & R_i^2T_f^2 & R_i^2T_fR_f \\ (R_i R_f) & T_iR_iT_fR_f & T_iR_iR_f^2 & R_i^2T_fR_f & R_i^2R_f^2 \\ \end{matrix} }

Equation

The equation can thus be expanded as:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} I_d(q_{z}) & = | T_i T_f F(+Q_{z1}) + T_i R_f F(-Q_{z2}) + R_i T_f F(+Q_{z2}) + R_i R_f F(-Q_{z1}) | ^{2} \\ & \begin{align} = \,\, & T_i^2 T_f^2 | F(+Q_{z1}) |^2 && + T_i^2 T_f R_f F(+Q_{z1})F(-Q_{z2}) \\ & && + T_i R_i T_f^2 F(+Q_{z1})F(+Q_{z2}) + T_i R_i T_f R_f F(+Q_{z1}) F(-Q_{z1}) \\ & + T_i^2 R_f^2 | F(-Q_{z2}) |^2 && + T_i^2T_fR_f F(+Q_{z1}) F(-Q_{z2}) \\ & && + T_i R_i T_f R_f F(+Q_{z2})F(-Q_{z2}) + T_i R_i R_f^2 F(-Q_{z1}) F(-Q_{z2}) \\ & + R_i^2 T_f^2 | F(+Q_{z2}) |^2 && + T_i R_i T_f^2 F(+Q_{z1}) F(+Q_{z2}) \\ & && + T_i R_i T_f R_f F(+Q_{z2})F(-Q_{z2}) + R_i^2 T_f R_f F(-Q_{z1}) F(+Q_{z2}) \\ & + R_i^2 R_f^2 | F(-Q_{z1}) |^2 && + T_i R_i T_f R_f F(+Q_{z1}) F(-Q_{z1}) \\ & && + T_i R_i R_f^2 F(-Q_{z1})F(-Q_{z2}) + R_i^2 T_f R_f F(-Q_{z1}) F(+Q_{z2}) \\ \end{align} \\ \end{align} }

Simplification

We can rearrange to:


Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} I_d(q_{z}) = \,\, & T_i^2 T_f^2 | F(+Q_{z1}) |^2 + T_i^2 R_f^2 | F(-Q_{z2}) |^2 + R_i^2 T_f^2 | F(+Q_{z2}) |^2 + R_i^2 R_f^2 | F(-Q_{z1}) |^2 \\ & + T_i^2 T_f R_f F(+Q_{z1})F(-Q_{z2}) \\ & + T_i R_i T_f^2 F(+Q_{z1})F(+Q_{z2}) + T_i R_i T_f R_f F(+Q_{z1}) F(-Q_{z1}) \\ & + T_i^2T_fR_f F(+Q_{z1}) F(-Q_{z2}) \\ & + T_i R_i T_f R_f F(+Q_{z2})F(-Q_{z2}) + T_i R_i R_f^2 F(-Q_{z1}) F(-Q_{z2}) \\ & + T_i R_i T_f^2 F(+Q_{z1}) F(+Q_{z2}) \\ & + T_i R_i T_f R_f F(+Q_{z2})F(-Q_{z2}) + R_i^2 T_f R_f F(-Q_{z1}) F(+Q_{z2}) \\ & + T_i R_i T_f R_f F(+Q_{z1}) F(-Q_{z1}) \\ & + T_i R_i R_f^2 F(-Q_{z1})F(-Q_{z2}) + R_i^2 T_f R_f F(-Q_{z1}) F(+Q_{z2}) \\ = \,\, & T_i^2 T_f^2 | F(+Q_{z1}) |^2 + T_i^2 R_f^2 | F(-Q_{z2}) |^2 + R_i^2 T_f^2 | F(+Q_{z2}) |^2 + R_i^2 R_f^2 | F(-Q_{z1}) |^2 \\ & + 2 \times T_i^2T_fR_f F(+Q_{z1})F(-Q_{z2}) \\ & + 2 \times T_iR_iT_f^2 F(+Q_{z1})F(+Q_{z2}) \\ & + 2 \times T_i R_i T_f R_f [ F(+Q_{z1})F(-Q_{z1}) + F(+Q_{z2})F(-Q_{z2}) ] \\ & + 2 \times T_iR_iR_f^2 F(-Q_{z1})F(-Q_{z2}) \\ & + 2 \times R_i^2T_fR_fx F(-Q_{z1})F(+Q_{z2}) \\ = \,\, & T_i^2 T_f^2 | F(+Q_{z1}) |^2 + T_i^2 R_f^2 | F(-Q_{z2}) |^2 + R_i^2 T_f^2 | F(+Q_{z2}) |^2 + R_i^2 R_f^2 | F(-Q_{z1}) |^2 \\ & + 2 \times T_i^2T_fR_f F(+Q_{z1})F(-Q_{z2}) + 2 \times T_iR_iT_f^2 F(+Q_{z1})F(+Q_{z2}) \\ & + 2 \times T_i R_i T_f R_f [ F(+Q_{z1})F(-Q_{z1}) + F(+Q_{z2})F(-Q_{z2}) ] \\ & + 2 \times T_iR_iR_f^2 F(-Q_{z1})F(-Q_{z2}) + 2 \times R_i^2T_fR_fx F(-Q_{z1})F(+Q_{z2}) \\ \end{align} }

We can rewrite in a more compact form using the notation Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T_i = T(\alpha_i)} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F_{+1} = F(+Q_{z1})} :

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} I_d(q_{z}) = \,\, & T_i^2 T_f^2 | F_{+1} |^2 + T_i^2 R_f^2 | F_{-2} |^2 + R_i^2 T_f^2 | F_{+2} |^2 + R_i^2 R_f^2 | F_{-1} |^2 \\ & + 2 \times T_i^2T_fR_f F_{+1}F_{-2} + 2 \times T_iR_iT_f^2 F_{+1}F_{+2} \\ & + 2 \times T_i R_i T_f R_f [ F_{+1}F_{-1} + F_{+2}F_{-2} ] \\ & + 2 \times T_iR_iR_f^2 F_{-1}F_{-2} + 2 \times R_i^2T_fR_f F_{-1}F_{+2} \end{align} }

Expansion

Terms

If one expands the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |...|^2} of the DWBA, one obtains 16 terms:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{matrix} & (T_i^* T_f^*) & (T_i^* R_f^*) & (R_i^* T_f^*) & (R_i^* R_f^*) \\ (T_i T_f) & T_i T_i^* T_f T_f^* & T_i T_i^* T_f R_f^* & T_i R_i^* T_f T_f^* & T_i R_i^* T_f R_f^* \\ (T_i R_f) & T_i T_i^* T_f^* R_f & T_i T_i^* R_f R_f^* & T_i R_i^* T_f^* R_f & T_i R_i^* R_f R_f^* \\ (R_i T_f) & T_i^* R_i T_f T_f^* & T_i^* R_i T_f R_f^* & R_i R_i^* T_f T_f^* & R_i R_i^* T_f R_f^* \\ (R_i R_f) & T_i^* R_i T_f^* R_f & T_i^* R_i R_f R_f^* & R_i R_i^* T_f^* R_f & R_i R_i^* R_f R_f^* \\ \end{matrix} }


Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{matrix} & (T_i^* T_f^*) & (T_i^* R_f^*) & (R_i^* T_f^*) & (R_i^* R_f^*) \\ (T_i T_f) & |T_i T_f|^2 & |T_i|^2 T_f R_f^* & T_i R_i^* |T_f|^2 & T_i R_i^* T_f R_f^* \\ (T_i R_f) & |T_i|^2 T_f^* R_f & |T_i R_f|^2 & T_i R_i^* T_f^* R_f & T_i R_i^* |R_f|^2 \\ (R_i T_f) & T_i^* R_i |T_f|^2 & T_i^* R_i T_f R_f^* & |R_i T_f|^2 & |R_i|^2 T_f R_f^* \\ (R_i R_f) & T_i^* R_i T_f^* R_f & T_i^* R_i |R_f|^2 & |R_i|^2 T_f^* R_f & | R_i R_f |^2 \\ \end{matrix} }



Equation

We take advantage of a more compact form using the notation Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T_i = T(\alpha_i)} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F_{+1} = F(+Q_{z1})} . The DWBA equation can thus be expanded as:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} I_d(q_{z}) & = | T_i T_f F_{+1} + T_i R_f F_{-2} + R_i T_f F_{+2} + R_i R_f F_{-1} | ^{2} \\ & \begin{align} = \,\, & |T_i T_f|^2 | F_{+1} |^2 && + |T_i|^2 T_f R_f^* F_{+1}F_{-2}^* \\ & && + T_i R_i^* |T_f|^2 F_{+1}F_{+2}^* + T_i R_i^* T_f R_f^* F_{+1} F_{-1}^* \\ & + |T_i R_f|^2 | F_{-2} |^2 && + |T_i|^2T_f^*R_f F_{+1}^* F_{-2} \\ & && + T_i R_i^* T_f^* R_f F_{+2}^*F_{-2} + T_i R_i^* |R_f|^2 F_{-1}^* F_{-2} \\ & + |R_i T_f|^2 | F_{+2} |^2 && + T_i^* R_i |T_f|^2 F_{+1}^* F_{+2} \\ & && + T_i^* R_i T_f R_f^* F_{+2}^*F_{-2} + |R_i|^2 T_f R_f^* F_{-1}^* F_{+2} \\ & + |R_i R_f|^2 | F_{-1} |^2 && + T_i^* R_i T_f^* R_f F_{+1}^* F_{-1} \\ & && + T_i^* R_i |R_f|^2 F_{-1}F_{-2}^* + |R_i|^2 T_f^* R_f F_{-1} F_{+2}^* \\ \end{align} \\ \end{align} }

Simplification

We can rearrange to:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} I_d(q_{z}) = \, \, & |T_i T_f|^2 | F_{+1} |^2 + |T_i R_f|^2 | F_{-2} |^2 + |R_i T_f|^2 | F_{+2} |^2 + |R_i R_f|^2 | F_{-1} |^2 \\ & + |T_i|^2 T_f R_f^* F_{+1}F_{-2}^* + |T_i|^2T_f^*R_f F_{+1}^* F_{-2} \\ & + T_i R_i^* |T_f|^2 F_{+1}F_{+2}^* + T_i^* R_i |T_f|^2 F_{+1}^* F_{+2} \\ & + |R_i|^2 T_f R_f^* F_{-1}^* F_{+2} + |R_i|^2 T_f^* R_f F_{-1} F_{+2}^* \\ & + T_i R_i^* |R_f|^2 F_{-1}^* F_{-2} + T_i^* R_i |R_f|^2 F_{-1}F_{-2}^*\\ & + T_i R_i^* T_f R_f^* F_{+1} F_{-1}^* \\ & + T_i R_i^* T_f^* R_f F_{+2}^*F_{-2} \\ & + T_i^* R_i T_f R_f^* F_{+2}^*F_{-2} \\ & + T_i^* R_i T_f^* R_f F_{+1}^* F_{-1} \\ = \, \, & |T_i T_f|^2 | F_{+1} |^2 + |T_i R_f|^2 | F_{-2} |^2 + |R_i T_f|^2 | F_{+2} |^2 + |R_i R_f|^2 | F_{-1} |^2 \\ & + |T_i|^2 [ T_f R_f^* F_{+1}F_{-2}^* + T_f^*R_f F_{+1}^* F_{-2} ] \\ & + |T_f|^2 [ T_i R_i^* F_{+1}F_{+2}^* + T_i^* R_i F_{+1}^* F_{+2} ] \\ & + |R_i|^2 [ T_f R_f^* F_{-1}^* F_{+2} + T_f^* R_f F_{-1} F_{+2}^* ] \\ & + |R_f|^2 [ T_i R_i^* F_{-1}^* F_{-2} + T_i^* R_i F_{-1}F_{-2}^* ]\\ & + [ T_i R_i^* T_f R_f^* F_{+1} F_{-1}^* + T_i^* R_i T_f^* R_f F_{+1}^* F_{-1} ] \\ & + [ T_i R_i^* T_f^* R_f F_{+2}^*F_{-2} + T_i^* R_i T_f R_f^* F_{+2}^*F_{-2} ] \\ \end{align} }

We define Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle I_{+1}=|F_{+1}|^2} , and note that for any complex number Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c} , it is true that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c+c^*=2 \mathrm{Re}[c]} . Thus:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} I_d(q_{z}) = \, \, & |T_i T_f|^2 I_{+1} + |T_i R_f|^2 I_{-2} + |R_i T_f|^2 | I_{+2} + |R_i R_f|^2 I_{-1} \\ & + 2 |T_i|^2 \mathrm{Re}[ T_f R_f^* F_{+1}F_{-2}^* ] \\ & + 2 |T_f|^2 \mathrm{Re}[ T_i R_i^* F_{+1}F_{+2}^* ] \\ & + 2 |R_i|^2 \mathrm{Re}[ T_f R_f^* F_{-1}^* F_{+2} ] \\ & + 2 |R_f|^2 \mathrm{Re}[ T_i R_i^* F_{-1}^* F_{-2} ]\\ & + 2 \mathrm{Re}[ T_i R_i^* T_f R_f^* F_{+1} F_{-1}^* ] \\ & + 2 \mathrm{Re}[ T_i R_i^* T_f^* R_f F_{+2}^*F_{-2} ] \\ = \, \, & |T_i T_f|^2 I_{+1} + |T_i R_f|^2 I_{-2} + |R_i T_f|^2 | I_{+2} + |R_i R_f|^2 I_{-1} \\ & + 2 |T_i|^2 \mathrm{Re}[ T_f R_f^* F_{+1}F_{-2}^* ] + 2 |T_f|^2 \mathrm{Re}[ T_i R_i^* F_{+1}F_{+2}^* ] \\ & + 2 |R_i|^2 \mathrm{Re}[ T_f R_f^* F_{-1}^* F_{+2} ] + 2 |R_f|^2 \mathrm{Re}[ T_i R_i^* F_{-1}^* F_{-2} ]\\ & + 2 \mathrm{Re}[ T_i R_i^* T_f R_f^* F_{+1} F_{-1}^* ] + 2 \mathrm{Re}[ T_i R_i^* T_f^* R_f F_{+2}^*F_{-2} ] \\ \end{align} }

Breaking into components

The experimental data Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle I_d(q_z)} can be broken into contributions from the transmitted channel Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle I_{d,\mathrm{Tc}}(qz)} and reflected channel Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle I_{d,\mathrm{Rc}}(qz)} :

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} I_d(q_{z}) & = [ | T_i T_f|^2 + |R_i R_f|^2 ] I_{R}(+Q_{z1}) + [ |T_i R_f|^2 + |R_i T_f|^2 ] I_{R}(+Q_{z2}) \\ & = [ | T_i T_f|^2 + |R_i R_f|^2 ] I_{R}(q_z-\Delta q_{z,\mathrm{Tc}}) + [ |T_i R_f|^2 + |R_i T_f|^2 ] I_{R}(q_z-\Delta q_{z,\mathrm{Rc}}) \\ & = [ | T_i T_f|^2 + |R_i R_f|^2 ] I_{d,\mathrm{Tc}}(q_z) + [ |T_i R_f|^2 + |R_i T_f|^2 ] I_{d,\mathrm{Rc}}(q_z) \\ & = |Tc|^2 I_{d,\mathrm{Tc}}(q_z) + |Rc|^2 I_{d,\mathrm{Rc}}(q_z) \\ \end{align} }

We define the ratio between the channels to be:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} w (q_z) & = \frac{ I_{d,\mathrm{Tc}}(q_z) }{ I_{d,\mathrm{Tc}}(q_z) + I_{d,\mathrm{Rc}}(q_z) } \end{align} }

Such that one can compute the two components from:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} I_d(q_{z}) & = |Tc|^2 ( I_{d,\mathrm{Tc}}(q_z) ) + |Rc|^2 ( I_{d,\mathrm{Rc}}(q_z) ) \\ I_d(q_{z}) & = |Tc|^2 ( I_{d,\mathrm{Tc}}(q_z) ) + |Rc|^2 \left ( \frac{ I_{d,\mathrm{Tc}}(q_z) - w I_{d,\mathrm{Tc}}(q_z) }{w} \right ) \\ I_d(q_{z}) & = I_{d,\mathrm{Tc}}(q_z) \times \left ( |Tc|^2 + |Rc|^2 \frac{ 1}{w} - |Rc|^2 \frac{w }{w} \right ) \\ I_{d,\mathrm{Tc}}(q_z) & = \frac{ I_d(q_{z}) }{ |Tc|^2 + \frac{ |Rc|^2 }{w} - |Rc|^2 } \\ \end{align} }

and:


Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} I_{d,\mathrm{Rc}}(q_z) & = \frac{ I_d(q_{z}) - |Tc|^2 I_{d,\mathrm{Tc}}(q_z) }{|Rc|^2} \\ & = \frac{ I_d(q_{z}) }{|Rc|^2} - \frac{|Tc|^2}{|Rc|^2} I_{d,\mathrm{Tc}}(q_z) \end{align} }

or:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} I_d(q_{z}) & = |Tc|^2 ( I_{d,\mathrm{Tc}}(q_z) ) + |Rc|^2 ( I_{d,\mathrm{Rc}}(q_z) ) \\ & = |Tc|^2 \left( \frac{w}{1-w} I_{d,\mathrm{Rc}}(q_z) \right) + |Rc|^2 ( I_{d,\mathrm{Rc}}(q_z) ) \\ I_{d,\mathrm{Rc}}(q_z) & = \frac{ I_d(q_{z}) }{|Tc|^2 \frac{w}{1-w} + |Rc|^2} \end{align} }

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