Talk:DWBA

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 $|...|^{2}$ 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:

${\begin{aligned}I_{d}(q_{z})&=|T_{i}T_{f}F(+Q_{z1})+T_{i}R_{f}F(-Q_{z2})+R_{i}T_{f}F(+Q_{z2})+R_{i}R_{f}F(-Q_{z1})|^{2}\\&{\begin{aligned}=\,\,&T_{i}^{2}T_{f}^{2}|F(+Q_{z1})|^{2}&&+T_{i}^{2}T_{f}R_{f}F(+Q_{z1})F(-Q_{z2})\\&&&+T_{i}R_{i}T_{f}^{2}F(+Q_{z1})F(+Q_{z2})+T_{i}R_{i}T_{f}R_{f}F(+Q_{z1})F(-Q_{z1})\\&+T_{i}^{2}R_{f}^{2}|F(-Q_{z2})|^{2}&&+T_{i}^{2}T_{f}R_{f}F(+Q_{z1})F(-Q_{z2})\\&&&+T_{i}R_{i}T_{f}R_{f}F(+Q_{z2})F(-Q_{z2})+T_{i}R_{i}R_{f}^{2}F(-Q_{z1})F(-Q_{z2})\\&+R_{i}^{2}T_{f}^{2}|F(+Q_{z2})|^{2}&&+T_{i}R_{i}T_{f}^{2}F(+Q_{z1})F(+Q_{z2})\\&&&+T_{i}R_{i}T_{f}R_{f}F(+Q_{z2})F(-Q_{z2})+R_{i}^{2}T_{f}R_{f}F(-Q_{z1})F(+Q_{z2})\\&+R_{i}^{2}R_{f}^{2}|F(-Q_{z1})|^{2}&&+T_{i}R_{i}T_{f}R_{f}F(+Q_{z1})F(-Q_{z1})\\&&&+T_{i}R_{i}R_{f}^{2}F(-Q_{z1})F(-Q_{z2})+R_{i}^{2}T_{f}R_{f}F(-Q_{z1})F(+Q_{z2})\\\end{aligned}}\\\end{aligned}}$

Simplification

We can rearrange to:

${\begin{aligned}I_{d}(q_{z})=\,\,&T_{i}^{2}T_{f}^{2}|F(+Q_{z1})|^{2}+T_{i}^{2}R_{f}^{2}|F(-Q_{z2})|^{2}+R_{i}^{2}T_{f}^{2}|F(+Q_{z2})|^{2}+R_{i}^{2}R_{f}^{2}|F(-Q_{z1})|^{2}\\&+T_{i}^{2}T_{f}R_{f}F(+Q_{z1})F(-Q_{z2})\\&+T_{i}R_{i}T_{f}^{2}F(+Q_{z1})F(+Q_{z2})+T_{i}R_{i}T_{f}R_{f}F(+Q_{z1})F(-Q_{z1})\\&+T_{i}^{2}T_{f}R_{f}F(+Q_{z1})F(-Q_{z2})\\&+T_{i}R_{i}T_{f}R_{f}F(+Q_{z2})F(-Q_{z2})+T_{i}R_{i}R_{f}^{2}F(-Q_{z1})F(-Q_{z2})\\&+T_{i}R_{i}T_{f}^{2}F(+Q_{z1})F(+Q_{z2})\\&+T_{i}R_{i}T_{f}R_{f}F(+Q_{z2})F(-Q_{z2})+R_{i}^{2}T_{f}R_{f}F(-Q_{z1})F(+Q_{z2})\\&+T_{i}R_{i}T_{f}R_{f}F(+Q_{z1})F(-Q_{z1})\\&+T_{i}R_{i}R_{f}^{2}F(-Q_{z1})F(-Q_{z2})+R_{i}^{2}T_{f}R_{f}F(-Q_{z1})F(+Q_{z2})\\=\,\,&T_{i}^{2}T_{f}^{2}|F(+Q_{z1})|^{2}+T_{i}^{2}R_{f}^{2}|F(-Q_{z2})|^{2}+R_{i}^{2}T_{f}^{2}|F(+Q_{z2})|^{2}+R_{i}^{2}R_{f}^{2}|F(-Q_{z1})|^{2}\\&+2\times T_{i}^{2}T_{f}R_{f}F(+Q_{z1})F(-Q_{z2})\\&+2\times T_{i}R_{i}T_{f}^{2}F(+Q_{z1})F(+Q_{z2})\\&+2\times T_{i}R_{i}T_{f}R_{f}[F(+Q_{z1})F(-Q_{z1})+F(+Q_{z2})F(-Q_{z2})]\\&+2\times T_{i}R_{i}R_{f}^{2}F(-Q_{z1})F(-Q_{z2})\\&+2\times R_{i}^{2}T_{f}R_{f}xF(-Q_{z1})F(+Q_{z2})\\=\,\,&T_{i}^{2}T_{f}^{2}|F(+Q_{z1})|^{2}+T_{i}^{2}R_{f}^{2}|F(-Q_{z2})|^{2}+R_{i}^{2}T_{f}^{2}|F(+Q_{z2})|^{2}+R_{i}^{2}R_{f}^{2}|F(-Q_{z1})|^{2}\\&+2\times T_{i}^{2}T_{f}R_{f}F(+Q_{z1})F(-Q_{z2})+2\times T_{i}R_{i}T_{f}^{2}F(+Q_{z1})F(+Q_{z2})\\&+2\times T_{i}R_{i}T_{f}R_{f}[F(+Q_{z1})F(-Q_{z1})+F(+Q_{z2})F(-Q_{z2})]\\&+2\times T_{i}R_{i}R_{f}^{2}F(-Q_{z1})F(-Q_{z2})+2\times R_{i}^{2}T_{f}R_{f}xF(-Q_{z1})F(+Q_{z2})\\\end{aligned}}$

We can rewrite in a more compact form using the notation $T_{i}=T(\alpha _{i})$ and $F_{+1}=F(+Q_{z1})$:

${\begin{aligned}I_{d}(q_{z})=\,\,&T_{i}^{2}T_{f}^{2}|F_{+1}|^{2}+T_{i}^{2}R_{f}^{2}|F_{-2}|^{2}+R_{i}^{2}T_{f}^{2}|F_{+2}|^{2}+R_{i}^{2}R_{f}^{2}|F_{-1}|^{2}\\&+2\times T_{i}^{2}T_{f}R_{f}F_{+1}F_{-2}+2\times T_{i}R_{i}T_{f}^{2}F_{+1}F_{+2}\\&+2\times T_{i}R_{i}T_{f}R_{f}[F_{+1}F_{-1}+F_{+2}F_{-2}]\\&+2\times T_{i}R_{i}R_{f}^{2}F_{-1}F_{-2}+2\times R_{i}^{2}T_{f}R_{f}F_{-1}F_{+2}\end{aligned}}$

Expansion

Terms

If one expands the $|...|^{2}$ of the DWBA, one obtains 16 terms:

${\begin{matrix}&(T_{i}^{*}T_{f}^{*})&(T_{i}^{*}R_{f}^{*})&(R_{i}^{*}T_{f}^{*})&(R_{i}^{*}R_{f}^{*})\\(T_{i}T_{f})&T_{i}T_{i}^{*}T_{f}T_{f}^{*}&T_{i}T_{i}^{*}T_{f}R_{f}^{*}&T_{i}R_{i}^{*}T_{f}T_{f}^{*}&T_{i}R_{i}^{*}T_{f}R_{f}^{*}\\(T_{i}R_{f})&T_{i}T_{i}^{*}T_{f}^{*}R_{f}&T_{i}T_{i}^{*}R_{f}R_{f}^{*}&T_{i}R_{i}^{*}T_{f}^{*}R_{f}&T_{i}R_{i}^{*}R_{f}R_{f}^{*}\\(R_{i}T_{f})&T_{i}^{*}R_{i}T_{f}T_{f}^{*}&T_{i}^{*}R_{i}T_{f}R_{f}^{*}&R_{i}R_{i}^{*}T_{f}T_{f}^{*}&R_{i}R_{i}^{*}T_{f}R_{f}^{*}\\(R_{i}R_{f})&T_{i}^{*}R_{i}T_{f}^{*}R_{f}&T_{i}^{*}R_{i}R_{f}R_{f}^{*}&R_{i}R_{i}^{*}T_{f}^{*}R_{f}&R_{i}R_{i}^{*}R_{f}R_{f}^{*}\\\end{matrix}}$

${\begin{matrix}&(T_{i}^{*}T_{f}^{*})&(T_{i}^{*}R_{f}^{*})&(R_{i}^{*}T_{f}^{*})&(R_{i}^{*}R_{f}^{*})\\(T_{i}T_{f})&|T_{i}T_{f}|^{2}&|T_{i}|^{2}T_{f}R_{f}^{*}&T_{i}R_{i}^{*}|T_{f}|^{2}&T_{i}R_{i}^{*}T_{f}R_{f}^{*}\\(T_{i}R_{f})&|T_{i}|^{2}T_{f}^{*}R_{f}&|T_{i}R_{f}|^{2}&T_{i}R_{i}^{*}T_{f}^{*}R_{f}&T_{i}R_{i}^{*}|R_{f}|^{2}\\(R_{i}T_{f})&T_{i}^{*}R_{i}|T_{f}|^{2}&T_{i}^{*}R_{i}T_{f}R_{f}^{*}&|R_{i}T_{f}|^{2}&|R_{i}|^{2}T_{f}R_{f}^{*}\\(R_{i}R_{f})&T_{i}^{*}R_{i}T_{f}^{*}R_{f}&T_{i}^{*}R_{i}|R_{f}|^{2}&|R_{i}|^{2}T_{f}^{*}R_{f}&|R_{i}R_{f}|^{2}\\\end{matrix}}$

Equation

We take advantage of a more compact form using the notation $T_{i}=T(\alpha _{i})$ and $F_{+1}=F(+Q_{z1})$. The DWBA equation can thus be expanded as:

${\begin{aligned}I_{d}(q_{z})&=|T_{i}T_{f}F_{+1}+T_{i}R_{f}F_{-2}+R_{i}T_{f}F_{+2}+R_{i}R_{f}F_{-1}|^{2}\\&{\begin{aligned}=\,\,&|T_{i}T_{f}|^{2}|F_{+1}|^{2}&&+|T_{i}|^{2}T_{f}R_{f}^{*}F_{+1}F_{-2}^{*}\\&&&+T_{i}R_{i}^{*}|T_{f}|^{2}F_{+1}F_{+2}^{*}+T_{i}R_{i}^{*}T_{f}R_{f}^{*}F_{+1}F_{-1}^{*}\\&+|T_{i}R_{f}|^{2}|F_{-2}|^{2}&&+|T_{i}|^{2}T_{f}^{*}R_{f}F_{+1}^{*}F_{-2}\\&&&+T_{i}R_{i}^{*}T_{f}^{*}R_{f}F_{+2}^{*}F_{-2}+T_{i}R_{i}^{*}|R_{f}|^{2}F_{-1}^{*}F_{-2}\\&+|R_{i}T_{f}|^{2}|F_{+2}|^{2}&&+T_{i}^{*}R_{i}|T_{f}|^{2}F_{+1}^{*}F_{+2}\\&&&+T_{i}^{*}R_{i}T_{f}R_{f}^{*}F_{+2}^{*}F_{-2}+|R_{i}|^{2}T_{f}R_{f}^{*}F_{-1}^{*}F_{+2}\\&+|R_{i}R_{f}|^{2}|F_{-1}|^{2}&&+T_{i}^{*}R_{i}T_{f}^{*}R_{f}F_{+1}^{*}F_{-1}\\&&&+T_{i}^{*}R_{i}|R_{f}|^{2}F_{-1}F_{-2}^{*}+|R_{i}|^{2}T_{f}^{*}R_{f}F_{-1}F_{+2}^{*}\\\end{aligned}}\\\end{aligned}}$

Simplification

We can rearrange to:

${\begin{aligned}I_{d}(q_{z})=\,\,&|T_{i}T_{f}|^{2}|F_{+1}|^{2}+|T_{i}R_{f}|^{2}|F_{-2}|^{2}+|R_{i}T_{f}|^{2}|F_{+2}|^{2}+|R_{i}R_{f}|^{2}|F_{-1}|^{2}\\&+|T_{i}|^{2}T_{f}R_{f}^{*}F_{+1}F_{-2}^{*}+|T_{i}|^{2}T_{f}^{*}R_{f}F_{+1}^{*}F_{-2}\\&+T_{i}R_{i}^{*}|T_{f}|^{2}F_{+1}F_{+2}^{*}+T_{i}^{*}R_{i}|T_{f}|^{2}F_{+1}^{*}F_{+2}\\&+|R_{i}|^{2}T_{f}R_{f}^{*}F_{-1}^{*}F_{+2}+|R_{i}|^{2}T_{f}^{*}R_{f}F_{-1}F_{+2}^{*}\\&+T_{i}R_{i}^{*}|R_{f}|^{2}F_{-1}^{*}F_{-2}+T_{i}^{*}R_{i}|R_{f}|^{2}F_{-1}F_{-2}^{*}\\&+T_{i}R_{i}^{*}T_{f}R_{f}^{*}F_{+1}F_{-1}^{*}\\&+T_{i}R_{i}^{*}T_{f}^{*}R_{f}F_{+2}^{*}F_{-2}\\&+T_{i}^{*}R_{i}T_{f}R_{f}^{*}F_{+2}^{*}F_{-2}\\&+T_{i}^{*}R_{i}T_{f}^{*}R_{f}F_{+1}^{*}F_{-1}\\=\,\,&|T_{i}T_{f}|^{2}|F_{+1}|^{2}+|T_{i}R_{f}|^{2}|F_{-2}|^{2}+|R_{i}T_{f}|^{2}|F_{+2}|^{2}+|R_{i}R_{f}|^{2}|F_{-1}|^{2}\\&+|T_{i}|^{2}[T_{f}R_{f}^{*}F_{+1}F_{-2}^{*}+T_{f}^{*}R_{f}F_{+1}^{*}F_{-2}]\\&+|T_{f}|^{2}[T_{i}R_{i}^{*}F_{+1}F_{+2}^{*}+T_{i}^{*}R_{i}F_{+1}^{*}F_{+2}]\\&+|R_{i}|^{2}[T_{f}R_{f}^{*}F_{-1}^{*}F_{+2}+T_{f}^{*}R_{f}F_{-1}F_{+2}^{*}]\\&+|R_{f}|^{2}[T_{i}R_{i}^{*}F_{-1}^{*}F_{-2}+T_{i}^{*}R_{i}F_{-1}F_{-2}^{*}]\\&+[T_{i}R_{i}^{*}T_{f}R_{f}^{*}F_{+1}F_{-1}^{*}+T_{i}^{*}R_{i}T_{f}^{*}R_{f}F_{+1}^{*}F_{-1}]\\&+[T_{i}R_{i}^{*}T_{f}^{*}R_{f}F_{+2}^{*}F_{-2}+T_{i}^{*}R_{i}T_{f}R_{f}^{*}F_{+2}^{*}F_{-2}]\\\end{aligned}}$

We define $I_{+1}=|F_{+1}|^{2}$, and note that for any complex number $c$, it is true that $c+c^{*}=2\mathrm {Re} [c]$. Thus:

${\begin{aligned}I_{d}(q_{z})=\,\,&|T_{i}T_{f}|^{2}I_{+1}+|T_{i}R_{f}|^{2}I_{-2}+|R_{i}T_{f}|^{2}|I_{+2}+|R_{i}R_{f}|^{2}I_{-1}\\&+2|T_{i}|^{2}\mathrm {Re} [T_{f}R_{f}^{*}F_{+1}F_{-2}^{*}]\\&+2|T_{f}|^{2}\mathrm {Re} [T_{i}R_{i}^{*}F_{+1}F_{+2}^{*}]\\&+2|R_{i}|^{2}\mathrm {Re} [T_{f}R_{f}^{*}F_{-1}^{*}F_{+2}]\\&+2|R_{f}|^{2}\mathrm {Re} [T_{i}R_{i}^{*}F_{-1}^{*}F_{-2}]\\&+2\mathrm {Re} [T_{i}R_{i}^{*}T_{f}R_{f}^{*}F_{+1}F_{-1}^{*}]\\&+2\mathrm {Re} [T_{i}R_{i}^{*}T_{f}^{*}R_{f}F_{+2}^{*}F_{-2}]\\=\,\,&|T_{i}T_{f}|^{2}I_{+1}+|T_{i}R_{f}|^{2}I_{-2}+|R_{i}T_{f}|^{2}|I_{+2}+|R_{i}R_{f}|^{2}I_{-1}\\&+2|T_{i}|^{2}\mathrm {Re} [T_{f}R_{f}^{*}F_{+1}F_{-2}^{*}]+2|T_{f}|^{2}\mathrm {Re} [T_{i}R_{i}^{*}F_{+1}F_{+2}^{*}]\\&+2|R_{i}|^{2}\mathrm {Re} [T_{f}R_{f}^{*}F_{-1}^{*}F_{+2}]+2|R_{f}|^{2}\mathrm {Re} [T_{i}R_{i}^{*}F_{-1}^{*}F_{-2}]\\&+2\mathrm {Re} [T_{i}R_{i}^{*}T_{f}R_{f}^{*}F_{+1}F_{-1}^{*}]+2\mathrm {Re} [T_{i}R_{i}^{*}T_{f}^{*}R_{f}F_{+2}^{*}F_{-2}]\\\end{aligned}}$

Breaking into components

The experimental data $I_{d}(q_{z})$ can be broken into contributions from the transmitted channel $I_{d,\mathrm {Tc} }(qz)$ and reflected channel $I_{d,\mathrm {Rc} }(qz)$:

${\begin{aligned}I_{d}(q_{z})&=[|T_{i}T_{f}|^{2}+|R_{i}R_{f}|^{2}]I_{R}(+Q_{z1})+[|T_{i}R_{f}|^{2}+|R_{i}T_{f}|^{2}]I_{R}(+Q_{z2})\\&=[|T_{i}T_{f}|^{2}+|R_{i}R_{f}|^{2}]I_{R}(q_{z}-\Delta q_{z,\mathrm {Tc} })+[|T_{i}R_{f}|^{2}+|R_{i}T_{f}|^{2}]I_{R}(q_{z}-\Delta q_{z,\mathrm {Rc} })\\&=[|T_{i}T_{f}|^{2}+|R_{i}R_{f}|^{2}]I_{d,\mathrm {Tc} }(q_{z})+[|T_{i}R_{f}|^{2}+|R_{i}T_{f}|^{2}]I_{d,\mathrm {Rc} }(q_{z})\\&=|Tc|^{2}I_{d,\mathrm {Tc} }(q_{z})+|Rc|^{2}I_{d,\mathrm {Rc} }(q_{z})\\\end{aligned}}$

We define the ratio between the channels to be:

${\begin{aligned}w(q_{z})&={\frac {I_{d,\mathrm {Tc} }(q_{z})}{I_{d,\mathrm {Tc} }(q_{z})+I_{d,\mathrm {Rc} }(q_{z})}}\end{aligned}}$

Such that one can compute the two components from:

${\begin{aligned}I_{d}(q_{z})&=|Tc|^{2}(I_{d,\mathrm {Tc} }(q_{z}))+|Rc|^{2}(I_{d,\mathrm {Rc} }(q_{z}))\\I_{d}(q_{z})&=|Tc|^{2}(I_{d,\mathrm {Tc} }(q_{z}))+|Rc|^{2}\left({\frac {I_{d,\mathrm {Tc} }(q_{z})-wI_{d,\mathrm {Tc} }(q_{z})}{w}}\right)\\I_{d}(q_{z})&=I_{d,\mathrm {Tc} }(q_{z})\times \left(|Tc|^{2}+|Rc|^{2}{\frac {1}{w}}-|Rc|^{2}{\frac {w}{w}}\right)\\I_{d,\mathrm {Tc} }(q_{z})&={\frac {I_{d}(q_{z})}{|Tc|^{2}+{\frac {|Rc|^{2}}{w}}-|Rc|^{2}}}\\\end{aligned}}$

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:

${\begin{aligned}I_{d}(q_{z})&=|Tc|^{2}(I_{d,\mathrm {Tc} }(q_{z}))+|Rc|^{2}(I_{d,\mathrm {Rc} }(q_{z}))\\&=|Tc|^{2}\left({\frac {w}{1-w}}I_{d,\mathrm {Rc} }(q_{z})\right)+|Rc|^{2}(I_{d,\mathrm {Rc} }(q_{z}))\\\end{aligned}}$