I found the following text regarding laser cavities. But I do not understand why they use the second derivate instead of the first. Since the second can take any positive or negative value.
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The second derivative of a function is an indicator of "flatness", because it is the rate of change of the gradient (the rate of change of the rate of change). A flat peak has a very small changing gradient, and hence has a small second derivative. In contrast a very sharp peak has a very fast changing gradient, so has a large second derivative.
In the context of this $Q$-factor, a flat curve gives a small value of $|\omega_2|$ and hence a large value of $Q$, according to this definition. The fact that $\omega_2$ could be positive or negative seems irrelevant here, as we're taking the absolute value.
