Paraxial Gaussian Beam (TEM_00)

What you are seeing: the spatial intensity profile of a fundamental Gaussian laser beam as it propagates along the optical axis $z$ . The beam is narrowest at its waist $w_0$ (center of the plot), then expands hyperbolically. Each transverse slice has a Gaussian intensity profile $I(r, z) \propto \exp(-2 r^2 / w(z)^2)$ .

Three quantities define the beam: $w(z) = w_0\,\sqrt{1 + (z / z_R)^2},\quad R(z) = z\,[1 + (z_R / z)^2],\quad \eta(z) = \arctan(z / z_R),$ with Rayleigh range $z_R = \pi w_0^2 / \lambda$ . The far-field divergence half-angle is $\theta = \lambda / (\pi w_0)$ , the diffraction-limited bound.

The thin curves above the heatmap trace $\pm w(z)$ (1/e^2 spot radius). The yellow vertical bars mark the Rayleigh range; between them the beam stays close to its waist.

Figure 1. Paraxial Gaussian beam intensity

I(r, z)

through the beam waist with

\pm w(z)

overlay. Method: closed-form analytic profile.

w_00.120

lambda0.020

z range4.0

WHAT TO TRY

Shrink the waist w_0: a tighter waist has a shorter Rayleigh range, so the beam diverges into a wide cone right after the focus. A broad waist stays collimated far longer. Diffraction trades focus for reach.
Change the wavelength lambda: longer wavelengths diverge faster for the same waist, since the divergence angle scales as lambda over w_0. Bluer beams stay tight.
Widen the z range: zooming out shows the full hyperbolic envelope, the waist in the middle opening into the linear far-field cone whose half-angle is the diffraction limit.