Gaussian beam radius from the Lambert W function

Abstract The standard Gaussian intensity expression presents a long-standing analytical challenge: inverting the equation to determine the beam radius from off-axis intensity measurements typically requires numerical iteration. This paper closes this gap by introducing a Lambert W function (LW) framework that yields a direct, closed-form analytical solution for the beam radius. This novel approach offers new insights into beam propagation and provides significant computational efficiencies in radius inversion. To assess its practical viability for beam characterization, we benchmarked its performance for determining M 2 against the standard ISO 11146 second moment (SM) method. Monte Carlo simulations show that for an ideal Gaussian beam ( <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msup> <mml:mi>M</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo>=</mml:mo> <mml:mn>1.0</mml:mn> </mml:mrow> </mml:math> ), the LW method perfectly recovers all parameters in the noise-free case, highlighting its theoretical purity. In contrast, the SM method exhibits a small positive bias ( <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msup> <mml:mi>M</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mrow> <mml:mo>≈</mml:mo> </mml:mrow> <mml:mn>1.02</mml:mn> </mml:mrow> </mml:math> ) attributable to numerical artifacts such as discretization. Under noise, the simulations confirm an expected trade-off: the SM method offers superior precision (lower variance), while the LW method’s mean accuracy tracks the SM standard closely (within 2%–4%). The LW framework offers compelling advantages in its elegance, computational speed, and potential for implementation with simpler hardware. These attributes make it suitable for applications such as rapid diagnostics or analytical modeling, where such benefits are prioritized over ultimate precision in noisy environments.