Science calculator

Beam quality M².

Estimate M² from a beam's waist radius, wavelength and measured far-field half-angle, then compare its divergence with the diffraction limit.

Waist radius w₀ (1/e²)

Wavelength λ

Measured half-angle θ

Beam quality M²estimated

—

waist–divergence product

Ideal half-angle θ₀: —
Measured half-angle θ: —

Angle unit

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M^{2} = \frac{π w_{0} θ}{λ}

θ_{0} = \frac{λ}{π w_{0}}

M² beam-quality factor · w₀ waist radius · θ measured far-field half-angle · θ₀ ideal half-angle · λ wavelength

Assumptions

w₀ and θ are measured on the same beam axis using consistent 1/e² intensity-radius definitions.
θ is the asymptotic far-field half-angle, with aperture clipping and added optical power excluded.
The beam is approximately Gaussian-like and propagates paraxially in one homogeneous medium.
This waist–divergence estimate is not a substitute for a formal caustic measurement using second-moment widths; a physical beam has M² ≥ 1.

How it works

An ideal Gaussian has the smallest possible waist–divergence product at its wavelength. Its half-angle is λ divided by πw₀.

M² is the measured waist–divergence product divided by that ideal value. An M² of 1 is diffraction-limited; larger values indicate a beam that spreads more strongly for the same waist.

Measurement definitions matter. Mixing a diameter with a radius, a full angle with a half-angle, or 1/e² widths with another convention can change the answer by a fixed factor.

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