Rayleigh Range Calculator

Inputs

Wavelength (λ)

nm um mm cm m

Typical lasers: 405 nm, 532 nm, 1064 nm.

Beam waist (w₀)

nm um mm cm m

Use 1/e² radius at focus for Gaussian beams.

Refractive index (n)

Air ≈ 1.000, fused silica ≈ 1.45.

Beam quality (M²)

M² = 1 is ideal; higher is less perfect.

Apply M² correction Yes, use M²-adjusted zR No, ideal Gaussian only

Controls which zR is used for derived values.

Output length unit nm um mm cm m

Affects zR, confocal parameter, and w(z).

Propagation distance (z)

nm um mm cm m

Used only for beam size w(z).

Calculate Reset

The Rayleigh range is the distance from the waist where the beam area doubles:

• Rayleigh range (ideal): zR = π · n · w₀² / λ
• M²-adjusted Rayleigh range (optional): zR,used = zR / M²
• Confocal parameter: b = 2 · zR
• Divergence half-angle (approx): θ = (M² · λ) / (π · n · w₀)
• Beam radius at distance z: w(z) = w₀ · √(1 + (z/zR)²)

Here λ is vacuum wavelength, n is refractive index, and w₀ is 1/e² radius.

1. Enter the laser wavelength and select its unit.
2. Enter the focused beam waist w₀ and its unit.
3. Set refractive index for the propagation medium.
4. Optionally set M² and choose whether to apply it.
5. Optionally enter a distance z to compute w(z).
6. Choose an output unit, then press Calculate.
7. Use Download buttons to export CSV or PDF results.

1) Rayleigh range in Gaussian optics

The Rayleigh range zR is a core length scale for a focused Gaussian beam. At z = zR, the beam radius increases by a factor of √2 and the cross‑sectional area doubles. For a waist radius w₀ and wavelength λ in a medium of refractive index n, the ideal relation is zR = π·n·w₀²/λ. This calculator keeps units consistent by converting all inputs to meters internally.

2) Why zR matters in optical design

Many alignment and tolerance questions reduce to “how quickly does the beam expand away from focus?” A larger zR means a longer region of near‑constant spot size, improving coupling into apertures, fibers, detectors, and nonlinear crystals. A smaller zR provides tighter focusing, but demands tighter positioning and surface quality.

3) Typical values for common lasers

With λ = 1064 nm and w₀ = 50 μm in air, an ideal beam yields zR ≈ 7.39 mm. Halving the waist to 25 μm reduces zR by a factor of four. At 532 nm with w₀ = 20 μm, the example table shows zR ≈ 2.36 mm.

4) Role of beam waist (w₀)

Because zR ∝ w₀², the waist dominates focal depth. Small changes in w₀ can strongly affect the confocal parameter b = 2zR. For microscopy, material processing, and tight‑aperture coupling, selecting a realistic w₀ is often more important than fine wavelength adjustments.

5) Influence of wavelength

For fixed w₀, zR decreases as wavelength increases (zR ∝ 1/λ). Shorter wavelengths therefore maintain a smaller spot over the same distance, which can benefit high‑resolution imaging and low‑divergence propagation. The calculator lets you explore this tradeoff rapidly across nanometer to meter scales.

6) Refractive index effects

In a medium, the effective wavelength becomes smaller, and zR scales with n. For the same physical waist, higher n increases zR and reduces the approximate divergence half‑angle reported here. This is useful for estimating focusing behavior inside glasses, polymers, and liquids.

7) Beam quality M² in real systems

Real beams are rarely perfect Gaussians. The beam quality factor M² is a practical way to account for extra divergence and reduced depth of focus. This calculator optionally applies the common engineering approximation zR,used = zR/M² and reports a divergence estimate θ = (M²·λ)/(π·n·w₀) for quick comparisons.

8) Reporting, verification, and exporting

Use the propagation distance input to compute w(z) and spot diameter at a specific location in your setup. After calculating, export the full result table as CSV for documentation or as a PDF summary for reports. Consistent, repeatable calculations reduce integration risk across optical subsystems.

1) Is w₀ the beam radius or diameter?

In Gaussian optics, w₀ is the 1/e² intensity radius at focus. The calculator also reports 2w(z) as the corresponding diameter at distance z.

2) Should I apply the M² option?

Use it when your laser is not close to a diffraction‑limited Gaussian. If you have a measured M² value, applying it gives a more conservative depth of focus and divergence estimate.

3) What refractive index should I enter?

Enter the refractive index of the medium where the beam propagates near the waist. For air use ~1.000; for fused silica use ~1.45; for water use ~1.33.

4) How accurate is the divergence value?

The divergence is a standard paraxial approximation for near‑Gaussian beams. It is best for small angles and well‑formed beams; strongly aberrated or clipped beams may deviate.

5) Why does zR change so much with waist?

Rayleigh range scales with the square of the waist: zR ∝ w₀². Doubling w₀ increases zR fourfold, extending the region of near‑constant spot size dramatically.

6) What is the confocal parameter used for?

The confocal parameter b = 2zR is a convenient measure of focal depth across the waist. It is often used for estimating interaction length in nonlinear optics and material processing.

7) Can I use this for fiber coupling?

Yes. Use zR and w(z) to gauge sensitivity to longitudinal misplacement and lens‑to‑fiber spacing. Combine with your fiber mode field radius to estimate coupling tolerance.