newAD2 - optical characterization software

Tutorial 4 - modeling non-uniform samples

In this tutorial we will further show how to introduce sample non-uniformity, a source of depolarization.

Another feature of newAD2 that we will get to know in this tutorial is the ability to define the film thickness (and other parameters) separately for different types of measurement. This is useful for samples that do not have a constant thickness over the whole area, and because it is not always possible to target every measurement at exactly the same spot, we can expect differences in thickness or other parameters between different measurements. This is essentially the simplest model of non-uniformity along the surface, where we assume that the mean value of the parameter is different for each measurement. Even a minimal difference in the mean thickness can noticeably shift the interference pattern. We will start from the model presented in tutorial 3. This can be done in the modelfile in two ways. The first way involves defining two structural parameters, e.g. dfE and dfR, and then duplicating the boundary matrix and the measurement for each measurement type:
```
....
structural parameters:
dfR = 400 : d<sub>fR</sub> : [10,2000] nm
dfE = 400 : d<sub>fE</sub> : [10,2000] nm

.....

boundary systems:
ME = B(a,SiO2)*T(SiO2,dfE)*B(SiO2,cSi)
MR = B(a,SiO2)*T(SiO2,dfR)*B(SiO2,cSi)

sample measurements:
E = Surface(ME)
R = Surface(MR)

functions:
Is,Ic,In-UVV = IsIcIn(E)
R-UVV = R(R)
.....
```
A simpler way, however, is to use the function measurement = Delta(par). This defines a new variable delta_par_measurement for the chosen measurement measurement and parameter par. When calculating the Mueller matrix for the chosen measurement, the value of the parameter par is replaced with the value par+delta_par_measurement. The modelfile then looks like this: Download modelfile
We will use the same input data as in tutorial 3. Continue in the tutorial 3 directory, or create a new directory and download the input data files for reflectance and ellipsometry (sample X2551). First fit the parameters $d_{\rm f}$, $B_1$ and $\lambda_1$ with the value $\delta(d_{\rm f})=0$ fixed. We get the same result as in tutorial 3, which we display using the grapher:

Besides the fitted quantities $I_{\rm s}$, $I_{\rm c}$, $I_{\rm n}$ and $R$, we also displayed in the grapher the degree of polarization $P=\sqrt{I_{\rm s}^2+I_{\rm c}^2+I_{\rm n}^2}$, which is suitable for describing partially polarized light, and the quantity $\delta(R)=R_{\rm th}-R_{\rm exp}$, suitable for showing small differences between the theoretical and experimental values.
In the next step, perform the fit including the parameter $\delta(d_{\rm f})$:

As can be seen, introducing different thicknesses improved the reflectance fit by approximately 14% ($\chi^{\rm R}=3.8489\to3.3229$), while the ellipsometric data improved by 5% ($\chi^{\rm E}=21.223\to20.216$). The added parameter $\delta(d_{\rm f})$ improved the fit in terms of the positions of the interference maxima and minima in the measured quantities. However, it had no effect on the degree of polarization of light in the fit of the ellipsometric data.
One of the sources of depolarization of light is sample non-uniformity — that is, the fact that each part of the sample along the surface reflects light with a different polarization state, which results in partially polarized light. To achieve this within the model, we introduce the function measurement = Wedge(par) instead of measurement = Delta(par), which ensures that the resulting Mueller matrix of the measurement is calculated as $$ {\bf M} = \int {\bf M}(p) w(p) {\rm d} p \,, $$ where $p$ is a parameter that is non-uniform within the illuminated spot (in our case the film thickness $d_{\rm f}$) and $w(p)$ is the distribution function of this parameter. For the Wedge function, the parameter is assumed to be non-uniform such that its functional dependence has a constant gradient across the circular spot. The distribution function is then described by two parameters: the mean deviation $\delta(d_{\rm f})$ (the same parameter as in the Delta function) and the root-mean-square deviation $\sigma(d_{\rm f})$. We apply this function to both Mueller matrices, i.e. to matrix E as well as matrix R: Download modelfile
Since the mean film thicknesses corresponding to both measurements are now described by three parameters, we must fix either the parameter $\delta(d_{\rm f})$ or directly the parameter $d_{\rm f}$, so that we do not create a correlation between these parameters. We add only the parameters $\sigma(d_{\rm f})$ as free fit parameters:

Note that we set the parameters $\sigma(d_{\rm f})$ to a small non-zero value before fitting, because the derivatives of the calculated functions with respect to these parameters are zero at zero, and the fitting algorithm cannot move away from these values, even though the program calculates the function values correctly. It is clear from the results that introducing the parameters $\sigma(d_{\rm f})$ substantially improved the fit of the ellipsometric data, while the effect on the reflectance was minimal. In fact, $\chi$, expressing the agreement between experiment and theory for the reflectance, increased slightly.