Quantifying convolution with Gaussian/Lorenzian

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Diagonal I
Diagonal I 2016-7-8
Briefly, I am exciting a thin film of upconverter (fluorescent) particles with an IR laser. The laser has a sharpness profile that I am comparing to the emission profile from two different types of particles. In both cases, there is broadening (i.e. it goes from an almost square linescan to a more Gaussian-looking curve). I would like to quantify this broadening to compare it between the particles. I know I can just compare the width at 1/2 max, but I think I would get more information from comparing the degree of convolution. In other words, I want to see how many Guaussians (and the width of each one) give me the emission profile when convoluted with the laser profile. Here are what the profiles look like:

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