Calculator
Example Data Table
Sample scenarios using Gaussian mode (air, M² shown). Values are typical engineering ranges.
| Wavelength (nm) | Waist diameter (mm) | M² | Distance (m) | Beam diameter at distance (mm) |
|---|---|---|---|---|
| 532 | 1.000 | 1.100 | 2.000 | 1.79463 |
| 633 | 0.8000 | 1.000 | 5.000 | 5.10039 |
| 1064 | 2.000 | 1.300 | 10.00 | 9.02999 |
| 1550 | 1.500 | 1.000 | 20.00 | 26.3563 |
| 1064 | 0.5000 | 1.600 | 3.000 | 13.0150 |
Formula Used
Gaussian beam (waist-based)
Beam radius as a function of distance:
zR = π w0² / (M² λ)
w(z) = w0 √(1 + (z / zR)²)
D1/e²(z) = 2 w(z)
Divergence and intensity
Small-angle divergence from the waist:
θhalf ≈ (M² λ) / (π w0)
If power is given, irradiance is estimated over area πw²:
Iavg = P / (π w²), Ipeak = 2P / (π w²)
Linear divergence (far-field approximation)
Useful when a datasheet provides a full-angle divergence θ:
D(z) = D0 + 2 z · tan(θ/2)
Notes: w is the 1/e² radius. The refractive index adjusts wavelength inside a medium using λ = λ0 / n.
How to Use This Calculator
- Pick a method: Gaussian for most laser beams, or linear for datasheet divergence.
- Choose input and output units, then enter the propagation distance.
- For Gaussian mode, enter waist size, wavelength, refractive index, and M².
- Optionally add laser power to estimate average and peak irradiance.
- Press Calculate. Results appear above the form and download buttons become available.
Beam Expansion Guide
1) What “beam expansion” means in practice
A real laser spot grows as it propagates, so the same optical power spreads over a larger area. This calculator reports the 1/e² radius w(z) and diameter D(z)=2w(z), which are common in laser specifications and beam profilers. A larger w lowers irradiance, which affects cutting, marking, imaging, and safety limits.
2) Near-field vs far-field using Rayleigh range
The Rayleigh range zR marks the transition from a “mostly collimated” region to a strongly diverging one. Around z ≲ zR, the radius stays close to w0. For z ≫ zR, the radius increases almost linearly with distance. Many lab setups target zR of 0.5–5 m for stable spot sizes over benches.
3) Typical wavelengths and why they matter
Shorter wavelengths diverge less for the same waist. Common values include 532 nm (green), 633 nm (HeNe), 1064 nm (Nd:YAG / fiber), and 1550 nm (telecom). If you keep w0 fixed, doubling wavelength roughly doubles the divergence in Gaussian mode. This is one reason visible alignment beams can look “tighter” than IR beams.
4) Beam quality M² as a real-world correction
Ideal Gaussian beams have M²≈1. Many diode lasers are M² between 1.2–5, and multimode sources can exceed 10. Since zR ∝ 1/M² and divergence scales with M², a higher M² expands faster and reaches larger diameters at the same distance.
5) Converting between 1/e² and FWHM diameters
Some cameras or imaging specs use FWHM instead of 1/e². For a Gaussian intensity profile, the calculator estimates FWHM diameter as Dfwhm = w(z)·√(2 ln 2). As a rule of thumb, FWHM diameter is about 0.59× the 1/e² diameter.
6) Linear divergence method and datasheet angles
If you only know an initial diameter and a quoted full-angle divergence, the linear method is useful: D(z)=D0+2z·tan(θ/2). For small angles (like 0.5–5 mrad), tan(θ/2) is close to θ/2, so results are intuitive and match many “far-field” beam models.
7) Using power to estimate irradiance
When you enter power, the tool estimates average irradiance over area πw² and peak on-axis irradiance for a Gaussian beam. Example: a 5 W beam with w=1 mm has average irradiance near 1.59 MW/m². At w=2 mm, that drops by about 4×.
8) Practical tips for reliable inputs
Use the waist at the correct reference plane (often the focus). If you measured diameter, confirm whether it is 1/e² or FWHM. Keep units consistent, and treat M² and divergence as direction-dependent for asymmetric beams. For safety planning, use conservative power and worst-case spot sizes.
FAQs
1) What does the 1/e² diameter represent?
The 1/e² diameter is twice the radius where intensity drops to 13.5% of the peak. It is widely used in laser datasheets and beam profilers for consistent comparisons.
2) Should I enter waist as radius or diameter?
Enter whichever you know. If your instrument reports a beam diameter, choose “Diameter (2w0)”. If you already have the radius parameter w0 from theory, choose “Radius (w0)”.
3) Why does M² change my results?
M² models how non-ideal a beam is. Higher M² increases divergence and reduces Rayleigh range, so the beam grows faster with distance than an ideal Gaussian beam.
4) How is refractive index used here?
The calculator converts free-space wavelength to wavelength in the medium using λ = λ0 / n. This affects Rayleigh range and divergence when propagation occurs inside a material.
5) Is the linear method accurate for all distances?
It’s best when you are in far-field conditions and the divergence angle is known from a datasheet. Near the waist, Gaussian mode is more physically accurate.
6) What does “peak irradiance” assume?
Peak irradiance assumes a circular Gaussian intensity profile and uses the on-axis value derived from total power and beam radius. Real beams can deviate, especially with clipping or hotspots.
7) Can I use this for elliptical beams?
This tool assumes a circular beam. For elliptical beams, run the calculation separately for the fast and slow axes using different waist values, then treat the spot as an ellipse for area-based estimates.