A Gaussian beam is the fundamental transverse electromagnetic mode (TEM₀₀) of a laser resonator. Unlike a geometric ray of light, a Gaussian beam has a finite cross-section whose irradiance follows a bell-shaped profile. At any plane along the propagation axis $z$, the irradiance distribution is:
Here $w(z)$ is the 1/e² intensity radius — the distance from the beam axis at which the irradiance drops to $1/e^2 \approx 13.5\%$ of its peak value. The quantity $w_0$ is the beam waist, the minimum radius occurring at the focal plane $z = z_0$.
A Gaussian beam is fully characterized by two parameters: its waist radius $w_0$ and the wavelength $\lambda$. From these, all other properties follow. As the beam propagates away from the waist, it expands hyperbolically:
The Rayleigh range $z_R$ defines the boundary between the near field and the far field. Within one Rayleigh range of the waist ($|z - z_0| < z_R$), the beam is approximately collimated — its radius grows by no more than a factor of $\sqrt{2}$. The beam area doubles at $z = z_0 \pm z_R$, and the closely related confocal parameter $b = 2z_R$ describes the total depth of focus.
In the far field ($|z - z_0| \gg z_R$), the beam expands linearly and behaves like a cone of light with a well-defined half-angle divergence:
This reveals a fundamental trade-off: a tightly focused beam (small $w_0$) diverges rapidly, while a highly collimated beam (small $\theta$) requires a large waist. The product $w_0 \cdot \theta = \lambda / \pi$ is invariant — no optical system can reduce both simultaneously.
The wavefront of a Gaussian beam is not planar. Its radius of curvature varies along $z$:
At the waist, $R \to \infty$ (a flat wavefront). The curvature reaches a minimum of $R = 2z_R$ at $z = z_0 \pm z_R$, then asymptotically approaches $R \approx z$ in the far field, matching a spherical wave.
A Gaussian beam also acquires an extra phase shift relative to a plane wave as it passes through focus. This Gouy phase $\zeta(z) = \arctan\!\bigl((z-z_0)/z_R\bigr)$ accumulates a total of $\pi$ radians from $-\infty$ to $+\infty$, with the steepest change near the waist. It has measurable consequences in interferometric and resonator applications.
Real-world laser beams are rarely perfect Gaussian modes. The $M^2$ factor (also called the beam propagation ratio or beam quality factor) quantifies how a real beam deviates from the ideal. A real beam of quality $M^2 \geq 1$ behaves like an ideal Gaussian embedded within it, but with a modified divergence and Rayleigh range:
The beam propagation equations retain their form exactly when $M^2$ is folded into $z_R$ and $\theta$. This embedded Gaussian model is the standard approach used in this simulator — and throughout industry — for propagating real beams through optical systems.
When a Gaussian beam passes through a thin lens of focal length $f$, the ABCD (ray transfer matrix) formalism gives the new beam parameters. The complex beam parameter $q(z) = (z - z_0) + iz_R$ transforms as:
The lens creates a new waist $w_0'$ at a new location $z_0'$, both of which depend on where the lens sits relative to the original waist and on the ratio of the Rayleigh range to the focal length. The behavior differs significantly from geometric optics: the Gaussian lens equation includes a correction term $(z_R/f)^2$ that prevents the beam from focusing to a geometric point.