Laser Beam Divergence Calculator — Gaussian Beam Optics
Enter the laser wavelength, beam waist radius, and propagation distance to compute the far-field half-angle divergence, Rayleigh range, and beam radius at any distance along the beam path.
nm
μm
m
Far-field half-angle divergence of the Gaussian beam (1/e² definition)
- 1
Wavelength in metres
λ = 650 nm × 10⁻⁹ = 0.00000065 - 2
Beam waist in metres
w₀ = 500 μm × 10⁻⁶ = 0.0005 - 3
Half-angle (rad)
λ ÷ (π × w₀) = 0.000414Far-field divergence half-angle for a Gaussian beam: θ = λ / (π · w₀). - 4
Half-angle divergence θ
0.000414 × 1000 = 0.4138
How does this calculator work?
For an ideal Gaussian beam: half-angle divergence θ = λ/(π·w₀), Rayleigh range zR = π·w₀²/λ, and beam radius w(z) = w₀·√(1+(z/zR)²). Enter wavelength (nm), beam waist radius w₀ (μm), and distance z (m) to get divergence in mrad and beam radius in mm at your working distance.
Formula
How this is calculated
A Gaussian (TEM₀₀) laser beam has a minimum radius called the beam waist w₀ at the focus point. As the beam propagates away from the waist in either direction, it expands. The Rayleigh range zR = π·w₀²/λ is the distance at which the beam radius has grown to w₀·√2 (area doubled). Within the Rayleigh range the beam is effectively collimated; beyond it the beam diverges nearly linearly.
The far-field half-angle divergence θ = λ/(π·w₀) (in radians) describes the cone of propagation at large distances (z ≫ zR). The product θ·w₀ = λ/π is the beam parameter product (BPP) — a constant for a given beam quality. Real laser beams have M² ≥ 1 that multiplies the divergence: θ_real = M²·λ/(π·w₀); this calculator assumes an ideal Gaussian beam with M² = 1.
The beam radius at any distance z from the waist is w(z) = w₀·√(1 + (z/zR)²). For z ≪ zR the beam radius stays near w₀; for z ≫ zR it grows as θ·z. All formulas use the 1/e² intensity definition (standard in laser optics). Wavelength inputs are in nanometres, beam waist in micrometres, and distance in metres — with results displayed in milliradians and millimetres for practical engineering use.
Frequently asked questions
The Rayleigh range zR = π·w₀²/λ is the propagation distance over which the beam area doubles (radius grows to w₀√2). It separates the "near field" (z < zR, roughly collimated) from the "far field" (z > zR, diverging). Longer zR means you can focus the beam over a greater working distance.
M² (≥ 1) captures how much the beam diverges relative to a perfect Gaussian. For a real laser, multiply the ideal divergence θ by M². A single-mode fiber-coupled laser might have M² ≈ 1.05; a multimode diode bar could have M² > 20. This calculator assumes M² = 1.
This is a consequence of the Heisenberg uncertainty principle applied to photons: tightly confining the beam transversely (small w₀) introduces a large spread in the transverse momentum, which appears as a larger divergence angle. A 1 μm waist diverges much more than a 1 mm waist at the same wavelength.
Also known as
TG we-Calculate Editorial Team. (2026). Laser Beam Divergence Calculator — Gaussian Beam Optics [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/laser-beam-divergence-calculator
TG we-Calculate Editorial Team. "Laser Beam Divergence Calculator — Gaussian Beam Optics." TG we-Calculate. 2026. https://we-calculate.com/calculator/laser-beam-divergence-calculator.
TG we-Calculate Editorial Team, "Laser Beam Divergence Calculator — Gaussian Beam Optics," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/laser-beam-divergence-calculator
@misc{wecalculate_laser_beam_divergence_calculator, title = {Laser Beam Divergence Calculator — Gaussian Beam Optics}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/laser-beam-divergence-calculator}}, year = {2026}, note = {TG we-Calculate} }
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