Graphical Constructions of Wavefronts and Waist Parameters in Gaussian Beam Optics

TL;DR

This work introduces graphical methods to determine key Gaussian-beam parameters from partial wavefront information, notably the beam waist plane and the Rayleigh range, using meridional diagrams and the circle method. It provides explicit algebraic relations for waist location from two wavefronts, and a complete set of geometric constructions to recover wavefronts, reduced radii, and Rayleigh spheres, as well as the imaging behavior of Gaussian beams through lenses via the double conjugation law. The main contributions include a practical circle-based procedure to locate the waist and quantify the Rayleigh range, a method to construct wavefronts from given waist and Rayleigh data, and a framework for analyzing beam imaging and resonator limits, including the non-uniqueness of waist position in symmetric confocal resonators. These graphical tools offer rapid, wavelength-agnostic insights for Gaussian-beam design and resonator analysis, with extensions to elliptical waists through orthogonal cross-sections.

Abstract

We provide several diagrams for the graphical determination of certain elements of a Gaussian beam based on prior knowledge of other elements. For example, these diagrams allow us to determine the plane of the beam waist and the Rayleigh range from knowledge of two wavefronts constituting the beam, or to determine the size of the light spot on a given wavefront. We also present a simple method for determining the waist position and the Rayleigh range of the image of a Gaussian beam formed by a lens.

Figures (10)

• Figure 1: A Gaussian beam. Spherical caps ${\mathcal{S}}_1$ and ${\mathcal{S}}_2$ are wavefronts. The wavefront ${\mathcal{W}}_0$, located at $W_0$, is a plane and contains the waist. Algebraic measures are positive if taken in the sense of light propagation. In the diagram, light is assumed to propagate from left to right, so that $D=\overline{V_1V_2}>0$, $d_1=\overline{W_0V_1}<0$ and $d_2=\overline{W_0V_2}>0$. Radii of curvature of wavefronts are $R_1=\overline{V_1C_1}>0$ and $R_2=\overline{V_2C_2}<0$.
• Figure 2: If ${\mathcal{S}}_1$ and ${\mathcal{S}}_2$ are the traces of two wavefronts of a Gaussian beam, the trace of the waist plane is the straight line passing through the points $I$ and $J$, defined as the intersections of the circles ${\mathcal{C}}_1$ and ${\mathcal{C}}_2$ having diameters $V_1C_1$ and $V_2C_2$. The waist is located at $W_0\equiv K$. The length of segment $KI$ is equal to the Rayleigh range $\zeta_0$. (The confocal parameter is $2\zeta_0=IJ$.) The diagram above is drawn for $R_2<0<R_1$. However, the method is also valid if $R_1$ and $R_2$ are of the same sign.
• Figure 3: Construction of a wavefront ${\mathcal{S}}$ knowing the waist plane (located at $W_0$) and the Rayleigh range $\zeta_0$. The scale is given by $W_0I=W_0J=\zeta_0$.
• Figure 4: Explanation for the construction of the reduced transverse radius $\zeta_d$ on the wavefront $S$ located at a distance $d$ from the waist plane ($d=\overline{W_0V}$). We obtain $\zeta_d=VJ'$.
• Figure 5: The reduced transverse radius on ${\mathcal{S}}$ is $\zeta_d=VI'$. The point $I'$ is the intersection of the straight lines supporting the segments $CI$ and $VP$.

Theorems & Definitions (5)

• Remark 3.1
• Remark 3.2
• Remark 3.3
• Remark 4.1
• Remark 5.1