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_0frac{w_0}{w(z)}H_mleft(sqrt{2}frac{x}{w(z)}right)H_nleft(sqrt{2}frac{y}{w(z)}right)expleft(-(x^2+y^2)left(frac{1}{w^2}-frac{ik}{2R(z)}right)right)expleft(ikz+iPhi (z)-omega tright)?
end{equation}

The Photon · Accepted Answer

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.

What is the coherence length of Hermite-Gaussian Modes?

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