Gaussian Beam Calculator

Model laser beam spread with precise optical inputs. Compute waist, divergence, Rayleigh range, intensity quickly. Export results and plots for design and testing today.

Gaussian Beam Inputs

Use known waist mode, or compute a focused waist from a lens.

Lens mode uses w_f = (M² λ f)/(π w_in).
Examples: 532, 1064, 1550.
1.0 is near diffraction-limited.
This is 1/e² radius at the waist.
Quick check
Larger w₀ reduces divergence. Smaller w₀ increases intensity at focus.
1/e² radius at the lens plane.
Thin-lens approximation.
Focused waist output
The calculator derives w₀ at the focus, then evaluates at z.
Used for intensity estimates.
Use negative values for before the waist.
r=0 gives on-axis intensity.
Downloads appear after a successful calculation.

Example Data Table

Try these typical values to verify outputs.

Scenario λ (nm) w₀ (mm) z (mm) P (W) r (mm)
Near-ideal infrared beam 1064 1.0 0.50 500 10 0.25
Visible beam with modest M² 532 1.3 0.30 200 2 0.10
Lens focusing example 1550 1.1 0 1 0

For the lens row, switch to lens mode and use w_in=2.0 mm, f=100 mm.

Formula Used

  • θ = (M² · λ) / (π · w₀) is the half-angle divergence (radians).
  • zR = (π · w₀²) / (M² · λ) is the effective Rayleigh range.
  • w(z) = w₀ · √(1 + (z/zR)²) is the 1/e² beam radius at distance z.
  • R(z) = z · (1 + (zR/z)²) is the wavefront radius (Infinity at z=0).
  • ψ(z) = arctan(z/zR) is the Gouy phase.
  • I0(z) = 2P / (π · w(z)²), and I(r,z) = I0 · exp(-2r² / w(z)²).
  • Lens mode uses w_f = (M² · λ · f) / (π · w_in) to estimate the focused waist.

How to Use This Calculator

  1. Pick Known waist if you already know w₀ at the beam waist.
  2. Pick Lens focusing if you know the beam radius at a lens and its focal length.
  3. Enter wavelength, M², and optionally the optical power for intensity.
  4. Set z as the distance from the waist (or focus) to the point you care about.
  5. Set r to evaluate intensity off-axis; use r=0 for on-axis.
  6. Click calculate to see results above the form, then export CSV or PDF.

Wavelength drives geometric scaling

Beam divergence scales directly with wavelength, so longer infrared light spreads faster for a fixed waist. Here, λ sets Rayleigh range and far-field angle. Doubling λ doubles θ while shrinking zR by half when w₀ is constant. If you must deliver a small spot at long stand-off, shorter wavelengths reduce growth.

M² captures real beam quality

Laboratory beams rarely stay diffraction limited. The M² factor increases divergence and reduces effective Rayleigh range the same way as increasing wavelength. If M² rises from 1.0 to 1.5, divergence rises 50% and depth of focus drops to two thirds. Use measured M² from a profiler to keep designs realistic.

Rayleigh range explains depth of focus

The Rayleigh range zR is the distance where the beam radius grows by √2. Many alignment and processing tasks operate inside ±zR because spot size and intensity remain comparatively stable there. The confocal parameter 2zR is often quoted for machining and microscopy because it approximates the usable focus window for a chosen waist.

Lens mode supports quick focus estimates

When you know the beam radius at a lens, the thin-lens estimate w_f = (M² λ f)/(π w_in) predicts the focused waist. Increasing input radius at the lens tightens the focus, while longer focal length loosens it. This relation helps compare optics early, before full ABCD modeling or tolerance analysis.

Intensity planning connects power and radius

On-axis intensity varies as 1/w(z)², so modest spot changes create large energy-density shifts. At constant power, a 20% reduction in radius raises I₀ by about 56%. Off-axis intensity follows the exp(−2r²/w²) envelope, letting you estimate edge exposure, sensor saturation, or safety margins at specific radii. Use r to evaluate a detector pixel position or a burn threshold at a known offset.

Measurement and verification workflow

Validate inputs by measuring waist and M² using standard characterization practice and record wavelength from the laser specification. Then sweep z values to compare predicted w(z) against profiler data. If results disagree, check definitions: this tool uses the 1/e² radius, not FWHM. Also confirm that z is referenced to the waist or the focused spot, not the lens. Consistent units and definitions produce reliable decisions in alignment, cutting.


FAQs

What does w(z) represent?

w(z) is the 1/e² intensity radius of a TEM00 Gaussian beam at distance z. The diameter shown is 2w(z), which is common for beam size specifications.

Why is R(z) infinite at z=0?

At the waist, the wavefront is locally planar, so curvature is effectively zero and the radius of curvature tends to infinity. Away from the waist, the beam becomes curved.

Can I use negative z values?

Yes. Negative z represents positions before the waist location. The radius w(z) is symmetric in z, while the Gouy phase changes sign with z.

How accurate is the lens focusing estimate?

It is a thin-lens, paraxial approximation using the beam radius at the lens. It is useful for quick comparisons, but it ignores aberrations, truncation, and misalignment.

Does the intensity calculation include losses?

No. The intensity uses the entered optical power and assumes an ideal Gaussian profile without transmission losses. If your system has losses, reduce the power input accordingly.

How do I convert FWHM to 1/e² radius?

For a Gaussian intensity profile, FWHM ≈ w·√(2 ln 2). Therefore, w ≈ FWHM / √(2 ln 2). Use the converted w as the waist radius input.

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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.