Compute Gaussian beam waist from key optical measurements. Include wavelength, M², divergence, and propagation distance. Get Rayleigh range estimates for precise alignment decisions today.
This calculator uses standard Gaussian-beam relations with a configurable beam-quality factor M².
All radii use the 1/e² convention. Divergence uses the far-field half-angle.
| Mode | λ | M² | Input | Computed w0 | Computed zR |
|---|---|---|---|---|---|
| Divergence | 1064 nm | 1.0 | θ = 1.0 mrad (half-angle) | 338.6 µm | 33.9 cm |
| Rayleigh | 532 nm | 1.2 | zR = 15 cm | 174.4 µm | 15 cm |
| Spot size | 633 nm | 1.0 | w(50 cm)= 1.2 mm | ~0.23 mm | ~0.26 m |
These examples are illustrative. Use your instrument conventions for radius and angle.
The waist w0 is the minimum 1/e² radius of a Gaussian beam and the anchor for propagation calculations. Once w0 is known, you can predict spot size, divergence, depth of focus, and coupling efficiency. In lab optics, waist values often range from tens of micrometers (tight focusing) to several millimeters (collimated systems).
This calculator expects radii, not diameters. If your beam profiler reports a diameter (for example, 1/e² diameter or FWHM), convert it to radius before entering. A common mistake is entering a 2 mm diameter as 2 mm radius, which doubles w(z) and can inflate the computed waist and Rayleigh range by large factors.
For a fixed divergence half-angle, the waist scales linearly with wavelength: w0 ∝ λ. For example, keeping θ constant, a 1064 nm beam will have roughly double the waist of a 532 nm beam. Typical laser wavelengths used in alignment and metrology include 405 nm, 532 nm, 633 nm, 780 nm, and 1064 nm.
M² captures how far a beam deviates from an ideal Gaussian. Values near 1.0 indicate high beam quality, while 1.2–2.0 is common for many diode and fiber sources. Higher M² increases divergence for a given waist, or increases the required waist for a measured divergence, affecting focusing performance and system brightness.
Divergence is most reliable when measured in the far field where the beam expands nearly linearly. Use the half-angle if your instrument reports full-angle; the calculator can convert that choice automatically. Typical small-divergence beams might be 0.2–2 mrad half-angle, while strongly focused or multimode beams can be larger, depending on optics and M².
The Rayleigh range zR is the distance from the waist where the beam area doubles, meaning the radius grows by √2. The confocal parameter 2 zR is often used as a practical “depth of focus” metric in scanning, cutting, microscopy, and free-space coupling. Larger waists produce longer zR for a fixed wavelength.
If you know a spot size w(z) at a distance z, the calculator solves the Gaussian propagation equation to recover w0. If it reports “inconsistent inputs,” the measured spot may be too small for the given wavelength and distance, or the measurement convention (radius/diameter) may be mismatched.
For documentation, report the waist unit, the divergence unit, and confirm the half-angle convention. The beam parameter product BPP = w0·θ is exported in m·rad, making it convenient for comparing systems across wavelengths. Use CSV for lab notebooks and PDF for quick sharing in alignment or design reviews.
It is the minimum Gaussian beam radius at the 1/e² intensity level. This waist is the reference point used to compute Rayleigh range, divergence half-angle, and spot size behavior versus distance.
Many laser and optics formulas are defined using the 1/e² radius because it links cleanly to Gaussian propagation relations. If your device reports diameter or FWHM, convert to 1/e² radius first.
Either is fine—select the option that matches your data. The waist equation uses half-angle divergence, so the calculator divides full-angle by two before computing w0.
No. By definition, M² ≥ 1. If a measurement suggests less than 1, it usually indicates uncertainty, a calibration issue, or a mismatch between the reported beam width definition and the Gaussian model.
Your entered spot size is too small for the provided distance, wavelength, and M² under Gaussian propagation. Double-check radius vs diameter, the distance unit, and whether the beam is measured near a true waist.
The beam parameter product is BPP = w0·θ using waist radius and divergence half-angle. It is reported in m·rad and is useful for comparing beam quality and focusing capability across different systems.
Accuracy depends on your measurements and conventions. Small errors in divergence or spot size can noticeably change w0 and zR. Use consistent 1/e² radii, verify units, and prefer far-field divergence or calibrated profiler data.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.