Calculator Inputs
Use one of three modes: derive waist from divergence, evaluate propagation from a known waist, or estimate focused waist from a lens.
Example Data Table
| Scenario | Mode | Wavelength | M² | Key Input | Waist Diameter | Rayleigh Range |
|---|---|---|---|---|---|---|
| Fiber laser alignment | Waist from Divergence | 1064 nm | 1.1 | 2.0 mrad full-angle | 745.100 µm | 372.550 mm |
| Green beam propagation | Propagation from Known Waist | 532 nm | 1.0 | 150 µm waist radius | 300.000 µm | 132.868 mm |
| Telecom beam focusing | Focused Waist from Lens | 1550 nm | 1.3 | 75 mm lens, 1.8 mm radius | 53.450 µm | 1.114 mm |
Formula Used
1) Effective Wavelength in a Medium
λeff = λ / n
Here, λ is vacuum wavelength and n is refractive index.
2) Waist from Far-Field Divergence
w₀ = (M² × λeff) / (π × θ)
θ is the half-angle divergence in radians. If you know full-angle divergence, divide it by two first.
3) Rayleigh Range
zR = πw₀² / (M² × λeff)
This is the axial distance where beam radius grows by √2 from the waist.
4) Beam Radius at Distance z
w(z) = w₀ × √(1 + (z / zR)²)
This describes Gaussian beam expansion away from the waist.
5) Wavefront Radius of Curvature
R(z) = z × [1 + (zR / z)²]
At the waist, the curvature is infinite because the wavefront is locally flat.
6) Lens-Focused Waist Approximation
w₀ ≈ (M² × λeff × f) / (π × win)
f is focal length and win is beam radius at the lens for an approximately collimated input beam.
7) Peak Intensity for a Gaussian Beam
I₀ = 2P / (πw²)
P is optical power and w is the beam radius at the location of interest.
How to Use This Calculator
- Select the calculation mode that matches your available beam data.
- Enter wavelength, beam quality, refractive index, power, and evaluation distance.
- Fill the mode-specific fields such as divergence, known waist, or lens data.
- Press Calculate Beam Waist to generate the result table.
- Review waist size, divergence, Rayleigh range, propagation size, curvature, and intensity.
- Use the CSV or PDF buttons to export the calculated engineering summary.
For best accuracy, keep units consistent and use realistic M² values from measured beam characterization.
Frequently Asked Questions
1) What is optical beam waist?
Beam waist is the location where a Gaussian beam has its minimum radius. It defines the smallest spot size and strongly affects divergence, intensity, and focusing behavior.
2) Why does the calculator ask for M²?
M² measures how far a real beam departs from an ideal Gaussian beam. Higher M² means poorer focusability, larger waist predictions, and shorter Rayleigh performance for the same optics.
3) Should I enter full-angle or half-angle divergence?
Enter full-angle divergence only in the divergence mode. The calculator automatically converts it to half-angle before applying the Gaussian beam waist equation.
4) What does Rayleigh range tell me?
Rayleigh range indicates how far the beam stays relatively tight around the waist. At this distance, the radius becomes √2 times the minimum waist radius.
5) Why use refractive index?
Propagation inside glass, liquids, or other media changes the effective wavelength. That shifts divergence and Rayleigh calculations, so refractive index improves practical optical modeling.
6) Is the focused waist formula exact?
No. It is a useful approximation for a near-collimated input beam and a simple lens. Strong aberrations, truncation, and non-Gaussian profiles require more detailed modeling.
7) What happens when z equals zero?
At z = 0, the beam is exactly at the waist. The radius is minimum, Gouy phase is zero, and wavefront curvature becomes infinite in the ideal model.
8) When should I export CSV or PDF?
Use CSV for spreadsheets and batch review. Use PDF for reports, documentation, procurement checks, design reviews, or sharing a locked calculation snapshot.