software to fit optical spectra - Quantum Materials Group

2. Basics of optical data fitting
2.1. Non-linear modeling
The fitting or modeling of the experimental (in particular, optical) data is a process, when
someone looks for a meaningful model with a set of adjustable parameters and varies them in
order to get the best match of the model to experimental curves.
To find a proper model is usually a central problem, which a computer can hardly do on
its own (we humans are still proud to be smarter than the machines we make!). Once a model is
thought up and reasonable initial values of parameters are found, the second stage is to vary
them in order to get the best fit. To continue doing it by hand is usually not practical and even
doable, when there are more than 2 or 3 tangled parameters. It could be even dangerous,
because a ‘reasonably looking’ match might be rather far from the numerically-the-best one.
Fortunately, at this stage the fitting procedure can be very efficiently automated as described
below.
Suppose, we have a set of N experimental data points { x i , y i , σ i } (i = 1 ,…, N ), that
we want to fit. Here x i is the data coordinate 1 , y i is the data value and σ i is the data error bar.
Next, we take a model, which calculates the data value y as a function of x , and a set of
internal parameters { p 1 , 2 p … p M }: y = f ( x,
p1
K pM
) .
Let us construct a so-called ‘chi-square’ functional:
N
2 ⎛ yi
− f ( xi
, p1
K pM
) ⎞ 2
χ ≡ ∑ ⎜
⎟
⎜
= χ ( p1,
K p
i 1 σ ⎟
= ⎝
i ⎠
Equation 2-1
2
If we assume that all measured values y i are normally distributed with standard
deviations given by σ i , then ‘statistically-the-best’ match would correspond to the minimal
2
value of χ . Thus, the modeling is essentially the minimization of the chi-square with respect
to parameters. Therefore, the method itself is called the ‘least-square’ technique.
Of course, the error bars are determined not only by a statistical noise, but also by
systematic inaccuracies, which are very hard to estimate and are clearly not normally
distributed 2 . However, to move on, we suppose that they are somehow accounted for by the
values σ i .
When f ( x,
p1
K pM
) is a non-linear function of parameters (which is virtually always
the case in optical modeling), the so-called Levenberg-Marquardt algorithm of the chi-square
minimization is indispensable. We shall discuss it in the next section.
1 In the case of optical data i
x is usually the light frequency.
2 Honestly, I do not believe there is a ‘scientific way’ to deal with systematic error bars (tell me if I am wrong!)
Guide to RefFIT Page 8
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