Fit

We now have a first estimate for the optical thickness of the transparent layer on our substrate. Theoretically, this calculation method may also be employed to generate kinetic curves from sequentially recorded spectra, something we did in the past, but this evaluation proved to be less accurate than the values we received by fitting equation (3.29) to a chosen spectral range. This procedure is applied in theFit panel.

A.1.3. Fit

The Fit panel determines the closest fit between the parameters of equation (3.30) and the curve displayed in the axes element to give a more accurate value ofOT which may be used as a starting point for the kinetic curve. Before the fitting procedure can be set in motion, the refractive indices of the ambient medium (n₁), the transparent layer (n₂) and the underlying substrate (n₃) must be entered in the respective edit elements. As can be seen in figure A.2, the preset values pertain to an aluminum substrate covered

with an alumina film immersed in water, but they may be adapted to any other sample system.

With the pushbutton Simulate the m-file of the identical name is executed. The function simulates a spectrum for a sample with the given refractive indices and theOT determined in theStartvalue panel as follows:

1 function RIfS_param=RIfS_simulate(RIfS_param)

2 % Simulates the chosen data for the preset estimate of the Optical

3 % thickness and the refractive index of the sensing layer

4 % RIfS_param=RIfS_simulate(RIfS_param)

5 global RIfS_param;

6

7 % Specify the x−data range to be simulated

8 xdata=RIfS_param.xneu;

9

10 % First guess value for the optical thickness

11 OT1=RIfS_param.OT1(1);

12

13 % Estimate Fresnel−coefficients

14 RIfS_param.r1=(RIfS_param.n1−RIfS_param.n2)...

The Fresnel coefficients for both interfaces are calculated according to equation 3.23 for a sample irradiated perpendicular to the surface and entered into the simulation function

R = ^r

21+r²₂+2r₁r₂cos^4π_λ OT 1+r²₁r²₂+2r₁r₂cos^4π_λOT

!

Amp+Offset (A.1)

The function is derived from equation 3.30 which gives the reflectance of a non-absorbing, isotropic medium by inserting the term of equation 3.25 for normal incidence. The solution of the reflectance equation yields values between zero and one. The reflectivity

Appendix A: GUI Design

values calculated in SpectraSuite are given in percent. For this reason we multiply the equation with a factor to get as close to the displayed amplitude of the measured interference signal as possible and add a certain offset to the function since the measured values rarely reach zero. Amp andOffset can be adapted via the edit elements above the Simulate button shown in figure A.2. The simulated curve is displayed in blue, whereas the measured curve is shown in red. Once the closest match forAmp andOffset is found the actual fitting procedure begins.

The fitting procedure is specified in the RIfS_fitref m-file and operated via the Fit pushbutton element. It reads as follows:

1 function RIfS_param=RIfS_fitref(RIfS_param)

2 % Fits the measured data to a user defined equation to

3 % determine the optical thickness

4 % RIfS_param=RIfS_fitref(RIfS_param)

5 global RIfS_param;

6

7 % Specify the data to be fitted

8 xdata=RIfS_param.xneu;

9 ydata=RIfS_param.yneu;

10

11 % First guess value for the optical thickness

12 RIfS_param.OT1(2)=1;

13 RIfS_param.OT1(3)=1;

14 RIfS_param.OT1(4)=1;

15

16 % Estimate Fresnel−coefficients

17 RIfS_param.r1=(RIfS_param.n1−RIfS_param.n2)...

41

The fitting function differs from the simulation function in the series elements added

R = ^r

Theoretically, the modulation of an interference spectrum should vary around a fixed value, but for most samples this is not the case. As can be seen in figure A.2. Here a decline in the curve is visible. The phenomenon may arise from absorbing substances which are enclosed in the transparent layer or an inhomogeneous height of the film [103].

The three elements of the series expansion are fitted along with the optical thickness.

For each fit the results of the preceding fitting procedure are used as start values. This leads to more accurate values and a faster analysing process.

The employed fit process is lsqcurvefit a procedure taken from Matlab’s optimi-sation toolbox which is used to solve nonlinear curve-fitting problems in least-squares sense also known as the Levenberg-Marquardt algorithm. The function is utilised here in the syntax [214]:

OT = lsqcurvefit(myfun,Guess,xdata,ydata,[],[],options) lsqcurvefit starts at Guess (the previously determined starting parameters) and finds coefficients OTto best fit the nonlinear function myfun(OT,xdata) to the data ydata. The empty brackets would usually define a set of lower and upper bounds on the design variables in OT. We passed empty matrices for them since boundary values are not needed in our case. Using theoptionsstructure specifies the optimisation options for the minimisation. The predefined structureoptimset was used to define them the maximum number of iterations (options.MaxIter) the fitting process may run was increased, since the preset value did not result in an ideal fit. The values determined by the fit are passed on for further processing in the successive panels. The OT is saved as the variableRIfS_param.OT0and all values includingOT and the series expansion are deposited in the matrixRIfS_param.OT. The task that remains, is to calculate theOT of every recorded spectrum and plot them against time. The information still lacking is

Appendix A: GUI Design

the sampling rate in which the spectra are saved. Usually, a spectrum is recorded every five seconds so the Interval edit element is preset to this value. The conversion to a kinetic curve is carried out by the next two panels.

Figure A.2.:GUI to convert recorded spectra into kinetic curves. The axes element shows the measured spectrum in red, the simulated spectrum using the determined starting parameter of the opitcal thickness in blue and the actual fit according to equation (A.2) in green.