I try to recreate a fit in this paper on SHG on chiral surfaces.
I guess the question itself is too specific to get an answer, so I state these questions that are a bit more to the point (and leave my problem for how to recreate the data to the experts if there are any):
How to come up with these assupmtions: $\chi^{eee}_{zxx} = 1$ and $\chi^{eee}$ are real whereas $\chi^{eem}$ and $\chi^{mee}$ are imaginary ?
Can the fitresults be chosen arbitrarly as I stated down below or do I miss something? Is it the same problem as to try to find $a$ and $b$ in $f = ab \cdot x$ from fitting a line?
Here, upper indices $e$ and $m$ stand for electric or magnetic transitions and coordinates in the lower indices refer to the laboratory coordinate system. My geuss for $\chi^{eee}$ being real, is due to beeing far away from resonance.
Here is the detailed version of my struggle: The data fitting does not seem to be hard: It boils down to
$F(\phi) = A^2 (sin^2(\phi) + i cos^2(\phi)^2 $
$G(\phi) = A^2 sin^2(\phi) cos^2(\phi)(1-i)^2$
$H(\phi) = A^2 sin(\phi) cos(\phi) ( sin^2(\phi) + i cos^2(\phi) (1-i)$
And the fitfunction is $|E(\phi)|^2$ with $E(\phi) = f F(\phi)+ g G(\phi) + h H(\phi)$
And $f$,$g$ and $h$ are the fitparameters (complex valued) and I guess also $A$, but they dont report that in the results.
Now maybe this is also where the problem lies. The parameters are not independent right? At least if I change to different inital values, it converges but with different results. Since $E(\phi) = f F(\phi)+ g G(\phi) + h H(\phi) = A^2(f F'(\phi)+ g G'(\phi) + h H'(\phi))$ , where the function with apostrophe are the same but divided by $A^2$, then I can choose $A$ arbitrarily and $f$,$g$ and $h$ would just correct by the corresponding factor...
If I choose to fit without $A$, then I dont get the same results as they do. Is this correct as I see it? Meaning that the factor $A$ is to much to have an unique fit. Would be the same as try to find $a$ and $b$ in $f = ab \cdot x$ from fitting a line.
However, they say the fit results are under the following constraints: The angle of incidence is 30° and $\chi^{eee}_{zxx} = 1$ and $\chi^{eee}$ are real whereas $\chi^{eem}$ and $\chi^{mee}$ are imaginary. This may help since there exist the following relations:
$$ \begin{align} f_{TS}^{RS} &= \sin \theta \left( -2 \chi_{xyz}^{eee} \cos \theta - \chi_{xzx}^{eem} + \chi_{zzz}^{mee} \sin^2 \theta \right. \nonumber \\ & \quad \left. + \chi_{zxx}^{mee} \cos^2 \theta \pm 2 \chi_{xxz}^{mee} \cos^2 \theta \right), \\ g_{TS}^{RS} &= \sin \theta (\chi_{xxz}^{eem} + \chi_{zxx}^{mee}), \\ h_{TS}^{RS} &= \sin \theta \left[ 2 \chi_{zzz}^{eee} - (\chi_{xzy}^{eem} + \chi_{xyz}^{eem}) \cos \theta \mp 2 \chi_{xyz}^{mee} \cos \theta \right], \\ f_{TP}^{RP} &= \sin \theta \left( \chi_{zzz}^{eee} \sin^2 \theta + \chi_{zxx}^{eee} \cos^2 \theta + \mp 2 \chi_{xxz}^{eee} \cos^2 \theta \right. \nonumber \\ & \quad \left. - \chi_{zxy}^{eem} \cos \theta \pm \chi_{xzy}^{eem} \cos \theta + 2 \chi_{xyz}^{mee} \cos \theta \right), \\ g_{TP}^{RP} &= \sin \theta (\chi_{zxx}^{eee} - \chi_{zxy}^{eem} \cos \theta \mp \chi_{xyz}^{eem} \cos \theta), \\ h_{TP}^{RP} &= \sin \theta \left[ \mp 2 \chi_{xyz}^{eee} \cos \theta + (\chi_{zzz}^{eem} - \chi_{zxx}^{eem}) \sin^2 \theta \right. \nonumber \\ & \quad \left. \mp (\chi_{xzx}^{eem} + \chi_{xxz}^{eem}) \cos^2 \theta - 2 \chi_{xxz}^{mee} \right]. \end{align} $$
First, I have no clue how the assumtpions come about, second even if, I dont see how it helps much. Here are their fit results:
$f_{Rs} = 0.2 + i0.48 \quad g_{Rs} = i0.1 \quad h_{Rs} = 5.1$
$f_{Ts} = 0.2 + i0.59 \quad g_{Ts} = i0.1 \quad h_{Ts} = 5.1$
$f_{Rp} = -1.95 \quad g_{Rp} = 1-0.08 \quad h_{Rp} = 0.2+i0.36$
$f_{Tp} = 5.7 + i0.48 \quad g_{Tp} = 1 \quad h_{Tp} = -0.2-i0.8$
From the relation $g_{TS}^{RS} = \sin \theta (\chi_{xxz}^{eem} + \chi_{zxx}^{mee})$ and the assumptions I stated, it should be $g=sin(30)\cdot 1 = 0.5$ in contrast to $g=1$ as they show.
So, I am lost here. Any help is appreciated.
