Gaussian beams and Rayleigh range

Use the intensity profile with radial dependence $\exp(-2r^2/w^2)$. The radius $w$ is therefore the distance where intensity falls to $1/e^2$ of its on-axis value, while the electric-field amplitude falls to $1/e$. The quoted beam diameter in this convention is $2w$. A radius based on half maximum or on $1/e$ intensity is a different number, so substituting it into the same Rayleigh-range formula creates a systematic error.

Integrate the intensity over circular rings to normalize it to total power. The result fixes peak irradiance as inversely proportional to squared radius. The Gaussian has no sharp edge: approximately 86.5 percent of its total power lies inside radius $w$, and some remains outside every finite circle. An aperture with radius equal to the quoted beam radius therefore clips appreciable power and generally perturbs the subsequent ideal Gaussian propagation.

For a waist of radius $w_0$ at $z=0$, the Rayleigh range is $\pi w_0^2/\lambda$, using wavelength in the uniform medium. At one Rayleigh range, the squared beam radius doubles, so the radius grows by a factor of square root two and the peak irradiance halves. The full confocal parameter is twice the Rayleigh range. These three related quantities should not be interchanged merely because all are sometimes described informally as a focal depth.

Far beyond the Rayleigh range, the radius grows approximately linearly with distance. The resulting divergence half-angle is $\lambda/(\pi w_0)$ in radians for an ideal fundamental Gaussian. Reducing waist radius therefore increases divergence; making a tighter focus does not make a more parallel outgoing beam. This result assumes paraxial angles and an ideal mode. A nonideal beam can have a larger beam-parameter product and requires a carefully defined beam-quality treatment.

The phase fronts are planar at the waist, even though the beam subsequently expands. Their curvature radius is $R(z)=z[1+(z_R/z)^2]$ away from the waist, with infinite-radius limit at zero. In the negative-time phasor convention, the field phase contains $kz+kr^2/(2R)-\psi$, where $\psi=\arctan(z/z_R)$ is the Gouy phase. The amplitude factor decreases as $w_0/w$, preserving integrated power as the cross-sectional area grows.

The Gouy contribution changes by pi between the distant incoming and outgoing sides of an ideal fundamental focus. It is a phase shift relative to an unconfined plane-wave reference, not an extra path length that all rays simply travel. Confusing a Gaussian envelope with a Gaussian intensity also causes errors: squaring the field doubles the exponent. Check that the radius convention, phase convention, and power normalization agree before combining formulas from different references.