Science calculator
Gaussian beam divergence.
Enter waist radius, wavelength and beam quality M² to find the far-field divergence half-angle and full included angle.
Half-angle θM² = 1
—
from axis to 1/e² radius
- Full included angle 2θ
- —
- θ far-field 1/e² half-angle · w₀ waist radius · λ wavelength in the medium · M² beam-quality factor
Assumptions
- Paraxial Gaussian propagation and a waist identified by its 1/e² intensity radius.
- M² ≥ 1 scales the ideal Gaussian divergence using a single beam-quality value.
- The far-field angle is the asymptotic half-angle, not a near-field slope measured close to the waist.
- No aperture clipping, turbulence or additional optical power is included.
How it works
An ideal Gaussian beam cannot remain perfectly collimated. A smaller waist or longer wavelength produces a larger diffraction angle.
M² compares a real beam with the ideal limit. At fixed waist and wavelength, divergence increases directly with M².
Laser specifications may quote either half-angle or full-angle divergence. This page shows both and labels the convention explicitly; calculation retains full precision before display rounding.
- Free
- No ads
- No tracking
- No account
- Works offline once loaded
From the maker of Ans — a scientific calculator you can own. Same idea: a proper instrument, nothing in the way.