Frequency of $\exp(-iar^2)$ at $r_0$

Question

I have an oscillating function $$A(r)=A_0\exp\left(-i\frac{\lambda_0}{4\pi F}r^2\right)$$ This creates an oscillation with increasing frequency, depending on $r$. But how do I now get the oscillation frequency at a certain spot $r_0$? It would be easy if the equation depends on $r$ (then it would be $\frac{\lambda_0}{4\pi F}$), but not if it depends on $r^2$.

Answer 1

If you call the phase

$$ \phi(r) = \frac{\lambda_0}{4\pi F}r^2 $$

you could define the instantaneous frequency as

$$ k(r) = \frac{{\rm d}\phi}{{\rm d}r} = \frac{\lambda_0 r}{2\pi F} $$