Full cone angle (2θ)
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free engineering tools
Laser Beam Divergence Calculator
Compute the Gaussian beam divergence angle θ = M²·λ / (π·w₀) from wavelength, beam waist radius and beam quality. Solve for any of the four variables, with full-angle, Rayleigh range and beam parameter product readouts. 100% client-side — nothing leaves your browser.
Divergence solver
Radius, not diameter. Datasheet diameter? Halve it (or see FAQ).
M² = 1 is the diffraction limit. Diffraction-limited (TEM00 ideal)
θ half-angle divergence
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Rayleigh range zR
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Effective zR / M²
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Confocal parameter b
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Beam parameter product
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Beam quality
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Formula & how it works
θ = M² · λ / (π · w₀) (half-angle, radians)
zR = π · w₀² / λ (Rayleigh range)
zR = π · w₀² / λ (Rayleigh range)
For a perfect Gaussian (TEM00) beam M² = 1 and the first formula reduces to θ = λ/(πw₀). Real beams diverge M² times faster. The formula is the far-field (z ≫ zR) small-angle approximation, accurate for all practical laser divergences. This tool computes the angle only; to propagate the beam to a target distance, use our spot size calculator.
Related laser tools
Beam Expander Calculator
Laser Spot Size Calculator
Laser Brightness Calculator
Laser Linewidth Calculator
Scope: this tool = divergence angle only. Beam expander = telescope magnification (f₂/f₁). Spot size = beam radius w(z) at a propagation distance. Brightness/linewidth = other laser properties.
Frequently asked questions
What is laser beam divergence?
Laser beam divergence is the angle at which a laser beam spreads as it travels away from its source. Because of diffraction, no beam stays perfectly collimated: even the most ideal single-mode laser widens at a rate set by its wavelength and the minimum beam size at the waist. Divergence is usually expressed as a half-angle theta (from the beam axis to the 1/e-squared intensity edge); the full cone angle 2-theta is also commonly quoted. Milliradians (mrad) are the standard engineering unit: 1 mrad of divergence means the beam diameter grows by about 1 mm per metre of travel.
What is the laser beam divergence formula?
For a Gaussian beam, the far-field half-angle divergence is theta = M-squared times lambda divided by (pi times w-zero), where lambda is the wavelength, w-zero is the beam waist radius (1/e-squared radius at the focus), and M-squared is the beam quality factor. For a perfect diffraction-limited TEM00 beam, M-squared = 1 and the formula simplifies to theta = lambda / (pi times w-zero).
Example: what is the divergence of a 532 nm laser with a 1 mm waist?
With lambda = 532 nm, waist radius w-zero = 1 mm and M-squared = 1: theta = 532e-9 / (pi x 1e-3) = 1.6934e-4 rad = 0.1693 mrad (half-angle). The full cone angle is 2-theta = 0.3387 mrad, and the Rayleigh range is z-R = pi x w-zero-squared / lambda = 5.905 m.
Half-angle or full angle: which one should I use?
Both are used, so always check which convention a datasheet or paper quotes. The physics formula theta = M-squared x lambda / (pi x w-zero) gives the half-angle (axis to edge). The full cone angle (edge to edge through the axis) is simply 2-theta. This calculator shows both: it solves the half-angle and reports the full angle as a readout.
What is the M-squared (beam quality) factor?
M-squared measures how close a real beam is to a perfect Gaussian beam. M-squared = 1 is the diffraction limit (ideal TEM00), physically the best possible. Typical values: 1.0-1.3 for a good HeNe or DPSS green laser, 1.1-1.7 for collimated single-mode diode lasers, 3-10 for many industrial beams, and up to 25-30 for high-energy multimode lasers. Because divergence scales directly with M-squared, a beam with M-squared = 3 diverges three times faster than a perfect beam of the same waist size and wavelength.
Example: HeNe 632.8 nm, waist 0.5 mm, M-squared 1.1?
theta = 1.1 x 632.8e-9 / (pi x 0.5e-3) = 4.4314e-4 rad = 0.4431 mrad half-angle = 443.14 urad = 0.02539 degrees. Full cone angle = 0.8863 mrad. Rayleigh range z-R = pi x (0.5e-3)-squared / 632.8e-9 = 1.241 m.
How do you reduce the divergence of a laser beam?
Divergence is set by only three things: wavelength, waist radius and beam quality. In practice you reduce it by enlarging the beam waist, e.g. with a beam expander (a telescope that magnifies the beam diameter), because theta falls in inverse proportion to w-zero. A shorter wavelength also helps. Note that enlarging the waist lengthens the Rayleigh range too, so the beam stays collimated over a longer distance. Try our beam expander calculator to size the telescope.
What is the Rayleigh range?
The Rayleigh range z-R = pi x w-zero-squared / lambda is the distance from the waist over which the beam cross-sectional area doubles (the beam radius grows by the square root of 2). Inside roughly one Rayleigh range the beam is nearly collimated; beyond it, it expands linearly at the divergence angle. For real beams with M-squared > 1, the effective Rayleigh range is z-R / M-squared. The confocal parameter b = 2 x z-R spans the depth of focus.
Is the beam waist a radius or a diameter?
In Gaussian beam optics, w-zero is the 1/e-squared radius: the distance from the axis where the intensity falls to 1/e-squared (about 13.5%) of the peak. The full 1/e-squared diameter is 2 x w-zero. Datasheets often quote the diameter, so halve it before entering the waist radius. Entering the diameter by mistake makes your computed divergence exactly half of the true value.
How many degrees is 1 milliradian?
1 mrad = 0.0573 degrees. Laser divergences are usually quoted in mrad or urad because the numbers are tiny: a typical laser pointer divergence of 1 mrad is only about 0.06 degrees. This calculator converts between mrad, urad and degrees automatically.
Why is my laser pointer's divergence worse than the formula says?
Cheap laser pointers and multimode diodes have M-squared far above 1 (often 5-30) and an astigmatic, striped emitter, so the ideal formula theta = lambda / (pi x w-zero) underestimates their spread. Measure or look up the M-squared value for your device and enter it in the calculator; the result then accounts for the real beam quality. Typical green pointers measure around 1.2-1.5 mrad.
How is this different from beam expanders and spot size calculators?
This tool computes only the divergence angle theta from wavelength, waist radius and M-squared, plus derived readouts (full angle, Rayleigh range, beam parameter product). A beam expander calculator sizes the telescope that enlarges the waist to cut divergence; a spot size calculator propagates the beam to find its radius w(z) at a working distance. Use them together: compute divergence here, then propagate it there.
Free tool by NRTechStudio