What is the coherence length of Hermite-Gaussian Modes?

Question

Can one derive the coherence length of a Hermite-Gaussian Mode just from the field \begin{equation} \small E_{m,n}(\vec{r})=E_0\frac{w_0}{w(z)}H_m\left(\sqrt{2}\frac{x}{w(z)}\right)H_n\left(\sqrt{2}\frac{y}{w(z)}\right)\exp\left(-(x^2+y^2)\left(\frac{1}{w^2}-\frac{ik}{2R(z)}\right)\right)\exp\left(ikz+i\Phi (z)-\omega t\right)? \end{equation}

Answer 1

No, the coherence length cannot be derived from the field equation.

Or rather, the equation is an approximation, made with the assumption of infinite coherence length.

A real source will have $\omega$ (and thus $k$) varying at least slightly over time. It's the characteristic period of this variation of $\omega$ that produces a finite coherence length.