Gouy phase

Remarks

The function $\psi$ is odd and increasing, with $\psi(z) \to \pm \pi/2$ as $z \to \pm\infty$. So the total Gouy shift through the focus is $\pi$, and half of it, $\pi/4$ on each side, occurs within one Rayleigh range of the waist. For other focused waves the Gouy phase is defined in the same way, as the lag of the phase on the axis behind $kz$. Its value depends on the wave.

Example: at $z = z_R$, the Gouy phase is $\arctan 1 = \pi/4$.

Non-example: a plane wave $e^{ikz}$ has on-axis phase exactly $kz$, so its Gouy phase is $0$ for all $z$; it has no focus.

With the opposite convention $e^{i\omega t - ikz}$, the Gouy phase appears with the opposite sign in the field.