Beam Divergence Calculator
A Gaussian laser beam has a minimum radius (beam waist w0) and then diverges with half-angle theta = lambda / (pi w0) in the far field, where lambda is the wavelength. The beam remains approximately collimated within the Rayleigh range zR = pi w0^2 / lambda. At distance z from the waist, the beam radius is w(z) = w0 sqrt(1 + (z/zR)^2). This calculator computes the divergence half-angle, the Rayleigh range, and the beam radius at a user-specified distance. For real beams, multiply theta by the beam quality factor M^2 to get the actual divergence. Enter wavelength and beam waist radius in the same length unit (micrometers recommended for laser applications).
Gaussian beam divergence formulas
theta = M2 lambda / (pi w0)
zR = pi w02 / (M2 lambda)
w(z) = w0 sqrt(1 + (z / zR)2)
All quantities in consistent units. theta in radians (multiply by 1000 for mrad). w0 and lambda typically in micrometers; zR and z in meters (convert: 1 um = 1e-6 m).
Beam divergence in practice
- A typical HeNe laser (lambda = 633 nm) with w0 = 0.5 mm has theta approximately 0.40 mrad and zR approximately 1.24 m.
- Focusing a beam tighter (smaller w0) increases divergence: a 10 um waist at 633 nm diverges at approximately 20 mrad.
- High-power multimode lasers have M^2 of 10 to 100 or more, significantly increasing effective divergence.
- Single-mode fiber output has M^2 approximately 1, making it ideal for diffraction-limited applications.
Beam divergence: frequently asked questions
What is beam divergence?
Beam divergence is the angle at which a laser beam spreads as it propagates. For a Gaussian beam, the far-field half-angle divergence is theta = lambda / (pi w0), where lambda is the wavelength and w0 is the beam waist radius (the minimum beam radius at the focus). A smaller waist gives a larger divergence angle.
What is the Rayleigh range?
The Rayleigh range zR = pi w0^2 / lambda is the distance from the beam waist at which the beam radius expands by sqrt(2) times the waist. Within the Rayleigh range the beam is approximately collimated; beyond it the beam diverges at the far-field angle theta.
How does beam radius vary with distance?
The Gaussian beam radius at distance z from the waist is w(z) = w0 sqrt(1 + (z/zR)^2). At z = zR, w = w0 sqrt(2) approximately 1.414 w0. Far from the waist (z >> zR), w(z) approaches z x tan(theta) approximately z x theta (in radians).
Why does focusing a beam tighter increase divergence?
This is a consequence of the uncertainty principle for waves: confining a beam spatially (small w0) requires a wider spread of transverse wave vectors, which translates to a larger divergence angle. The product w0 x theta = lambda / pi is a constant (for an ideal Gaussian beam).
What is the M-squared (M^2) beam quality factor?
A real laser beam may have higher divergence than an ideal Gaussian. The M^2 (beam quality) factor accounts for this: actual divergence theta_real = M^2 x lambda / (pi w0). M^2 = 1 for an ideal Gaussian; most lasers have M^2 between 1 and 3.
Official sources
- Optica (formerly OSA), Gaussian Beam Optics. optica.org.
- NIST, Physical Measurement Laboratory. physics.nist.gov.
Reviewed by the CalculatorHub team, edited by James Graham, 15 June 2026. See our methodology.