A Gaussian beam is the fundamental transverse electromagnetic mode (TEM₀₀) of a laser resonator. Its electric field amplitude follows a Gaussian profile, and the beam is fully characterised by its wavelength and a single spatial parameter — the beam waist radius $w_0$.
At any plane perpendicular to propagation, the intensity distribution is:
The quantity $w(z)$ is the 1/e² intensity radius — the distance at which intensity falls to $e^{-2} \approx 13.5\%$ of its on-axis peak. Setting $r = w$ in the exponential immediately gives $I(w)/I(0) = e^{-2}$.
The Gaussian profile is the unique transverse mode that propagates through a stable resonator without changing shape — only its size scales. It also represents the diffraction limit: for a given waist size, no beam can have smaller divergence than a Gaussian. In the paraxial approximation, any beam can be decomposed into Gaussian modes, making the Gaussian the natural basis for laser optics analysis.
Two fundamental parameters — the waist radius $w_0$ and the Rayleigh range $z_R$ — fully determine how a Gaussian beam evolves in free space.
As a Gaussian beam propagates away from its waist at $z = z_0$, the beam radius expands hyperbolically:
The Rayleigh range $z_R$ defines the boundary between the near field and far field:
At $|z - z_0| = z_R$, the beam radius is $w_0\sqrt{2}$ — the beam area has exactly doubled. Within one Rayleigh range the beam is approximately collimated; beyond it the beam expands like a cone. The closely related confocal parameter $b = 2z_R$ describes the total depth of focus — the axial extent over which the beam stays near its minimum size.
For $|z - z_0| \gg z_R$, the beam expands linearly with half-angle divergence:
This reveals the fundamental waist-divergence trade-off: the product $w_0 \cdot \theta = \lambda/\pi$ is invariant. Tighter focusing always produces faster divergence, and no optical system can improve both simultaneously.
Beyond beam size, Gaussian beams carry phase structure — curved wavefronts that determine how the beam focuses, and an additional phase shift through focus (the Gouy phase) with consequences in interferometry and resonator design.
The radius of curvature of the wavefront at position $z$ from the waist:
At the waist ($z = z_0$): $R \to \infty$ — the wavefront is flat, like a plane wave. The curvature reaches a minimum of $R = 2z_R$ at $z = z_0 \pm z_R$. In the far field: $R \approx z - z_0$, matching a spherical wave emanating from the waist.
A Gaussian beam acquires an extra phase shift relative to a plane wave as it propagates through focus:
The Gouy phase accumulates a total of $\pi$ radians from $-\infty$ to $+\infty$, with the steepest change near the waist (where $d\zeta/dz = 1/z_R$ at the waist itself). It shifts the on-axis phase of the full field by $e^{-i\zeta(z)}$, subtly altering interference patterns and resonance conditions.
In practice, the Gouy phase determines the mode spacing in laser resonators, produces measurable phase shifts in interferometers, and — for Hermite-Gaussian or Laguerre-Gaussian modes of order $(m+n)$ — is multiplied by $(m+n+1)$.
Real-world laser beams are rarely perfect TEM₀₀ modes. M² quantifies how a real beam deviates from the diffraction limit.
The beam propagation ratio $M^2 \geq 1$ is defined so that a real beam of waist $w_0$ and wavelength $\lambda$ behaves like an ideal Gaussian but with a modified divergence and Rayleigh range:
$M^2 = 1$ is the diffraction limit, achievable only by a perfect TEM₀₀ mode. Typical values: HeNe lasers $M^2 < 1.1$; diode-pumped solid-state $M^2 \approx 1.1$–$1.3$; multimode diodes $M^2 \gg 1$. A higher $M^2$ means the beam diverges faster and focuses to a larger spot than the diffraction limit predicts.
The beam propagation equations retain their Gaussian form exactly when $M^2$ is folded into $z_R$ and $\theta$ as above. This embedded-Gaussian model treats the real beam as if it were an ideal beam but with reduced Rayleigh range. It is the standard approach for propagating real beams through optical systems and is used throughout the Beam Dashboard.
All spatial properties of a Gaussian beam at a given plane — beam radius, wavefront curvature, Rayleigh range, and distance to waist — are encoded in a single complex number $q$. This makes $q$ the ideal state variable for propagation through optical systems.
The complex beam parameter is defined by:
Equivalently, at a distance $z$ measured from the beam waist:
Given $q$ at any plane, beam radius and wavefront curvature are recovered from:
The waist of the beam described by $q$ is located at $-\text{Re}(q)$ ahead of the current plane, and its Rayleigh range is $\text{Im}(q)$. The $q$ parameter is the ideal state variable for ABCD propagation: it carries complete beam information through the transform $q' = (Aq + B)/(Cq + D)$.
On the ABCD Designer's q(z) plot: $\text{Re}(q)$ tracks the signed distance to the nearest waist — it increases linearly in free space, jumps discontinuously at each lens (because the lens instantaneously changes wavefront curvature), and crosses zero at every waist location. $\text{Im}(q)$ is the Rayleigh range of the local beam — it stays constant during free-space propagation but changes at each lens, reflecting the new confocal geometry after focusing. Together the two traces give a complete picture of how the beam's focusing state evolves element by element.
In the paraxial approximation, any optical element can be described by a 2×2 matrix that maps an input ray $(y, \theta)$ — height and angle — to an output ray. The same matrices govern Gaussian beam propagation via the q-parameter.
For a cascade of elements, the system matrix is the ordered product of individual matrices — applied right to left, so the first element the beam encounters is the rightmost matrix:
The ABCD Designer computes this product, propagates $q$ through each element incrementally, and extracts $w(z)$ and $R(z)$ at every sample point. The live matrix display at the top of the results panel shows the full product $M_{\text{sys}} = \bigl[\begin{smallmatrix}A&B\\C&D\end{smallmatrix}\bigr]$ rendered in real time.
When a Gaussian beam passes through any optical system described by a composite matrix $M = \bigl[\begin{smallmatrix}A&B\\C&D\end{smallmatrix}\bigr]$, the output beam parameter is given by the ABCD law.
This single equation is the engine behind the ABCD Designer. At each element boundary, $q$ is updated via the element's $2\times 2$ matrix, and physical parameters are extracted from the updated $q$. In free space, $C = 0$, $D = 1$, and the law reduces to $q' = q + B = q + d$ — the beam parameter simply accumulates distance.
One of the most important practical problems in Gaussian optics is determining where a new beam waist forms after passing through a thin lens — and how large that waist is.
Consider a beam whose input waist $w_0$ is located a distance $s$ before a thin lens of focal length $f$. The beam arrives at the lens with $q_1 = s + iz_R$ (where $z_R = \pi w_0^2/\lambda$). After the lens and free-space propagation distance $s'$ to the new waist, working through the ABCD algebra gives:
This resembles the geometric thin-lens equation $1/s + 1/s' = 1/f$, but with $z_R$ appearing in the denominator. In the geometric limit ($z_R \to 0$), it reduces to the familiar formula exactly.
The "width" of a Gaussian beam can be defined in several ways. The Beam Dashboard supports all three major conventions and converts between them automatically.
Many real laser beams — particularly those from edge-emitting diode lasers — have different divergence in the two transverse planes. The Beam Dashboard handles these by propagating independent beam parameters for the x and y axes.
An astigmatic Gaussian beam is characterised by separate waist radii $w_{0x}$, $w_{0y}$ and corresponding Rayleigh ranges $z_{Rx} = \pi w_{0x}^2/\lambda$, $z_{Ry} = \pi w_{0y}^2/\lambda$. The beam radii along each axis evolve independently:
For an elliptical beam with total power $P$, the on-axis peak irradiance is:
The Gouy phase for an elliptical beam is the average: $\zeta(z) = \tfrac{1}{2}[\arctan((z-z_0)/z_{Rx}) + \arctan((z-z_0)/z_{Ry})]$, accumulating a total of $\pi$ radians as before.
For thin-lens ABCD transforms, $q_x$ and $q_y$ each transform through the same matrix independently, since the lens is treated as rotationally symmetric (no cross-coupling between axes).
All equations used in both tools, collected for quick reference.