What is the formula for GaussianBeam?

Date	Category
2025-09-05 17:29:25	Sources

The GaussianBeam source has the following scalar field amplitude, in cylindrical coordinates:

$u(r,z)=\frac{w_0}{w(z)}e^{-\frac{r^2}{w(z)^2}}e^{i(zk_0 + \frac{r^2k_0}{2R(z)} - \psi_g)}$
where:

• $z$ is the propagation direction
• $k_0=\frac{2\pi nf}{c}$ are the wavenumbers of the frequencies $f$ where the beam is sampled
• $w_0$ is the beam waist
• $w(z)=w_0\sqrt{1 + \frac{(z + z_0)^2}{z_r}}$, where $z_0$ is the waist distance and $z_r=\frac{1}{2}w_0^2k_0$ is the Rayleigh range
• $R(z)=z(1 + (\frac{z_r}{z})^2)$ is the radius of curvature of the wavefront at $z$
• $\psi_g=\arctan(\frac{z+z_0}{z_r})-\arctan(\frac{z_0}{z_r})$ is the Gouy phase

See the code here.