What is the Rayleigh range?

The Rayleigh range z_R = πw₀²/(M²λ) is the distance from the beam waist where the beam radius increases by a factor of √2 (so beam area doubles). Within this range, the beam is approximately collimated; beyond it, divergence dominates.

What is M² (M-squared)?

M² is the beam quality factor. An ideal Gaussian beam has M² = 1. Real beams have M² > 1, meaning they diverge faster than an ideal beam at the same waist. It accounts for aberrations, multimode content, and intrinsic beam quality limitations. Typical values: HeNe ~1.05, single-mode laser diode ~1.1, multimode Nd:YAG ~2–10.

What is the depth of focus?

The confocal parameter b = 2z_R is the "depth of focus" — the range over which the beam stays nearly collimated. For tight focus (small w₀), this is very short; for loose focus (large w₀), much longer. Trade-off between tight focus and depth.

How is Gaussian beam different from a "ray"?

Rays are infinitely thin lines used in geometric optics; they ignore diffraction and beam intensity profile. Gaussian beams are the actual diffraction-limited beam shape of a laser, with finite minimum waist and finite divergence. Geometric optics is fine for slow systems (NA < 0.1); Gaussian beam math is needed for laser systems where intensity profile and diffraction matter.

How do I make a smaller focused spot?

Use a shorter wavelength (UV/blue laser instead of red/IR), higher NA optics (larger input beam to focusing lens), and improve beam quality (lower M² via single-mode operation). The fundamental limit is w₀_focused = λ × M² / (π × NA). At a given wavelength, single-mode beam, and high NA, you reach the diffraction limit.

How does divergence relate to waist size?

Inversely. θ = M² × λ / (π × w₀). Smaller waist → larger divergence. The product θ × w₀ = M²λ/π is constant for a given beam quality and wavelength. Two beams with the same M² × λ have the same beam-parameter-product; you can't cheat this with optics.

What's the focused spot size of a laser?

For a Gaussian beam focused by a lens: w₀_focused = M² × λ × f / (π × w_input), where f is focal length and w_input is input beam radius at the lens. Tighter focus needs larger input beam or smaller focal length. Maximum useful NA depends on the lens design; physical NA ≤ 1.0 (air), or up to ~1.4 (oil immersion).

Gaussian Beam Calculator

Q: How do I make a smaller focused spot?

Use a shorter wavelength (UV/blue laser instead of red/IR), higher NA optics (larger input beam to focusing lens), and improve beam quality (lower M² via single-mode operation). The fundamental limit is w₀_focused = λ × M² / (π × NA). At a given wavelength, single-mode beam, and high NA, you reach the diffraction limit.

Q: How does divergence relate to waist size?

Inversely. θ = M² × λ / (π × w₀). Smaller waist → larger divergence. The product θ × w₀ = M²λ/π is constant for a given beam quality and wavelength. Two beams with the same M² × λ have the same beam-parameter-product; you can't cheat this with optics.

Q: What's the focused spot size of a laser?

For a Gaussian beam focused by a lens: w₀_focused = M² × λ × f / (π × w_input), where f is focal length and w_input is input beam radius at the lens. Tighter focus needs larger input beam or smaller focal length. Maximum useful NA depends on the lens design; physical NA ≤ 1.0 (air), or up to ~1.4 (oil immersion).

Analyze Gaussian beam propagation parameters including beam waist, Rayleigh range, far-field divergence, beam diameter at any distance, and confocal parameter (depth of focus).

Gaussian beams are the natural output of most lasers. Unlike geometric-optics "rays," a real laser beam doesn't propagate as a perfect collimated cylinder — it has a finite minimum waist size and diverges away from it. The beam profile is a Gaussian function of distance from the optical axis, characterized by a "1/e² radius" w(z) at each propagation distance z. Two beams with the same total power but different waist sizes have dramatically different intensities and divergence angles.

This calculator handles the standard Gaussian beam propagation math. Given wavelength, beam waist (w₀, the minimum radius), propagation distance (z), and M² (beam quality factor — 1.0 for an ideal Gaussian, > 1 for real lasers with multiple modes), it returns the Rayleigh range (z_R), the beam radius at distance z, the far-field half-angle divergence, and the depth of focus (confocal parameter, b = 2z_R). Use this for laser cavity design, beam delivery system planning, fiber coupling, and any application where beam propagation matters.

The fundamental scaling: smaller w₀ → faster divergence; larger w₀ → less divergence. This is the diffraction limit applied to a laser. A 1mm diameter beam at 1064 nm has Rayleigh range ~2.4 meters and stays roughly collimated over that distance. A 10 µm beam at the same wavelength has Rayleigh range only 240 µm — it expands rapidly. This trade-off is the essence of laser optics: tight focus comes with shallow depth.

Inputs

Wavelength (nm)

Beam Waist w₀ (μm)

Radius at the beam waist (1/e² intensity)

Propagation Distance z (mm)

M² (Beam Quality)

1.0 for ideal Gaussian, >1 for real beams

Results

Rayleigh Range

49.63 mm

Divergence

2.01 mrad

Beam Radius at z

224.9 μm

Gaussian Beam Results

Parameter	Value
Beam Waist w₀	100 μm (200.0 μm diameter)
Rayleigh Range z_R	49.6302 mm
Divergence (half-angle)	2.0149 mrad (0.115445°)
Full Divergence Angle	4.0298 mrad
Beam Radius at z = 100 mm	224.94 μm
Beam Diameter at z = 100 mm	449.88 μm
Depth of Focus (confocal)	99.2604 mm
M² Factor	1
Wavelength	633 nm

Last updated: May 29, 2026

Formula

**Beam radius vs propagation distance:** w(z) = w₀ × √(1 + (z/z_R)²) Where w(z) is the beam radius at distance z from the waist. **Rayleigh range (where beam grows by √2 from waist):** z_R = π × w₀² / (M² × λ) Where: - **w₀**: beam waist radius (1/e² intensity) - **λ**: wavelength - **M²**: beam quality factor (1 for ideal Gaussian) **Far-field half-angle divergence:** θ = M² × λ / (π × w₀) (radians) This is the angle from the optical axis that the beam radius grows along far from the waist (z >> z_R). **Depth of focus (confocal parameter, b):** b = 2 × z_R = 2π × w₀² / (M² × λ) The depth over which the beam is roughly collimated. **Worked example: HeNe laser at 633 nm, w₀ = 0.5 mm** z_R = π × (500×10⁻⁶)² / (1 × 633×10⁻⁹) = π × 0.00000025 / 6.33×10⁻⁷ = π × 0.395 = 1.24 m Divergence θ = 633×10⁻⁹ / (π × 500×10⁻⁶) = 4.03 × 10⁻⁴ rad = 0.403 mrad = 0.023° (typical HeNe). Beam radius at z = 10 m: w(10) = 500 × √(1 + (10/1.24)²) = 500 × √(1 + 65) = 500 × 8.12 = 4.06 mm. Beam radius increased from 0.5 mm to 4 mm over 10 m — typical "barely-collimated" HeNe at moderate distances. **For a tightly focused beam (w₀ = 1 µm):** z_R = π × 1² × 10⁻¹² / 633×10⁻⁹ = 4.96 × 10⁻⁶ m = 5 µm A 1 µm focused spot has only 5 µm depth of focus. This is why micro-machining lasers use extreme depth control and Z-stages. **M² (M-squared) factor:** M² ≥ 1, with 1.0 being a perfect Gaussian (TEM₀₀ mode). - TEM₀₀ single-mode laser: M² ≈ 1.0 - Multimode HeNe: M² ~ 1.05–1.3 - Diode lasers (typical): M² ~ 1.1–1.5 (one axis), 5–30 (slow axis) — highly anisotropic - Industrial Nd:YAG (multimode): M² ~ 2–10 - Solid-state lasers (high power): M² ~ 5–50 A beam with M² = 4 has 4× the divergence of a perfect Gaussian, and 4× the focused spot size at the same f-number. **Intensity profile:** I(r, z) = (2P / πw(z)²) × exp(−2r² / w(z)²) Peak intensity at axis: I₀ = 2P / πw₀² (at the waist). For 1 W focused to w₀ = 10 µm: I₀ = 2 / (π × (10×10⁻⁶)²) = 6.4 × 10⁹ W/m² = 6.4 × 10⁵ W/cm². For 100 mW HeNe focused: ~10⁵ W/cm² — bright enough to damage retinas, hence safety glasses. **Coupling Gaussian beams into single-mode fibers:** A single-mode fiber has a mode field diameter (MFD) of ~10 µm at telecom wavelengths. To couple efficiently, the incoming Gaussian beam's waist must match the fiber's MFD — typically using a focusing lens with focal length sized to give w₀ ≈ MFD/2 at the fiber entrance. **Useful order-of-magnitude beams:** | Laser type | Typical w₀ | Divergence | z_R | |---|---|---|---| | HeNe (50 cm cavity) | 0.5 mm | 0.4 mrad | 1.2 m | | Diode laser collimated | 1 mm | 0.3 mrad | 5 m (slow axis longer) | | Pulsed Nd:YAG (1.06 µm) | 3 mm | 0.15 mrad | 27 m | | Fiber laser output | 0.1 mm | 4 mrad | 25 mm | | Focused laser machining spot | 10 µm | 30 mrad | 0.3 mm | | Telescope laser ranging | 50 mm | 4 µrad | 12 km |

How to use this calculator

Enter the laser wavelength (633 nm HeNe, 1064 nm Nd:YAG, 405 nm UV, etc.).
Enter the beam waist radius w₀. For a collimated laser output, look up the spec or measure the 1/e² radius.
Set M² to 1.0 for an ideal single-mode beam, or higher for multimode lasers (1.05-1.3 typical for HeNe, 1.1-1.5 for diode lasers).
Enter the propagation distance from waist. Use 0 for the waist itself; large z to see far-field behavior.
Read the Rayleigh range (collimation length), divergence angle, and beam radius at z.
For beam delivery design, ensure your target distance is within ~2-3 Rayleigh ranges for adequate collimation.

Worked examples

Laser pointer collimation

**Scenario:** A red laser pointer at 650 nm has w₀ = 0.5 mm. How collimated is it at 100 m? **Calculation:** z_R = π × (0.5×10⁻³)² / 650×10⁻⁹ = 1.21 m. At z=100 m: w = 0.5 × √(1 + (100/1.21)²) = 0.5 × 82.8 = 41.4 mm radius. Diameter at 100 m: ~83 mm. **Result:** Beam diameter is ~83 mm at 100 m distance — large enough to be visible on a wall but no longer "tightly focused." Most laser pointers at 1 mile (1.6 km) project a beam several meters across. Real divergence is usually worse than this Gaussian prediction (M² > 1, plus collimation imperfections).

Microscope objective for spot focusing

**Scenario:** Focus a 1064 nm laser beam (w₀_input = 2 mm) using a 0.5 NA objective. What's the focused waist radius? **Calculation:** Focused waist: w₀_out = λ × M² / (π × NA) ≈ 1064 × 10⁻⁹ / (π × 0.5) = 6.8 × 10⁻⁷ m = 0.68 µm. Rayleigh range at focus: z_R = π × (0.68 × 10⁻⁶)² / 1064 × 10⁻⁹ = 1.4 × 10⁻⁶ m = 1.4 µm. Depth of focus: ~2.8 µm. **Result:** Focused spot is sub-micron (radius 0.68 µm), with depth of focus only 2.8 µm. This is the fundamental laser micro-machining or microscopy resolution at this wavelength and NA. Going to higher NA (0.95 air, 1.4 oil) and shorter wavelength (532, 405 nm) makes spots smaller but depth shallower.

Single-mode fiber coupling

**Scenario:** Couple a collimated beam (w₀ = 2 mm) at 1550 nm into a single-mode fiber with MFD 10 µm. What lens focal length? **Calculation:** Target w₀_focused = 5 µm (radius). Need a lens that focuses 2 mm beam → 5 µm waist. Using paraxial: w₀_focused = λ × f / (π × w₀_input) → f = π × w₀_focused × w₀_input / λ = π × 5×10⁻⁶ × 2×10⁻³ / 1.55×10⁻⁶ = 20.3 mm focal length. **Result:** A 20 mm focal length lens couples the beam into the fiber's mode. Typical fiber-coupling lenses are 4–25 mm focal length depending on the input beam diameter. Coupling efficiency depends on alignment (offset reduces it sharply); typical bench setups achieve 60–80% coupling efficiency.

When to use this calculator

**Use Gaussian beam analysis for:**

- **Laser cavity design**: choosing mirror curvatures to support stable modes. - **Beam delivery systems**: getting laser power from source to target with adequate quality. - **Fiber coupling**: matching beam waist to fiber mode field diameter. - **Laser focusing optics**: minimum spot size vs depth of focus trade-off. - **Beam expansion**: spreading the beam to reduce intensity, often before re-focusing. - **Free-space optical communication**: beam pointing, divergence over long distances. - **Laser ranging and lidar**: divergence determines spot size at target distance. - **Holography and interferometry**: beam quality affects fringe contrast.

**Key trade-offs:**

- **Tight focus → shallow depth**: a 1 µm spot has 5 µm depth at 1064 nm. Can't have both. - **Long working distance → larger spot**: depth and spot diameter scale together via Rayleigh range. - **Higher M² → larger spot at same focal length**: hence "single-mode" emphasis in many applications.

**Practical guidance for beam manipulation:**

- **Beam expander**: increases w₀ → decreases divergence → longer Rayleigh range. Useful for long-distance pointing. - **Spatial filter**: pinhole at focused spot removes higher modes, reducing M² toward 1. - **Beam combining**: parallel beams add intensities at a target if coherently combined. - **Wavefront correction**: deformable mirrors compensate aberrations to reduce M².

**Common errors in real-world Gaussian beam math:**

- Using diameter where formulas use radius (off by 2×). - Using 1/e where 1/e² is the convention (1/e² is 2× the 1/e diameter). - Forgetting wavelength dependence: 1064 nm vs 532 nm beams diverge very differently. - Ignoring M² for real diode lasers — single-mode formulas don't apply to highly multimode beams. - Using paraxial approximation at high NA (where it breaks down).

**Conversions worth knowing:**

- 1/e² radius is the "standard" Gaussian beam radius. - 1/e radius = 1/e² radius × √(0.5) ≈ 0.707×. - FWHM (full-width half-max) = 1/e² diameter × √(ln2/2) ≈ 1.18×. - D₄σ (ISO 11146 standard): equal to 1/e² diameter for ideal Gaussian; equals 4× standard deviation.

**Far-field divergence intuition:**

- 1 mm beam at 1 µm wavelength: divergence ≈ 0.3 mrad → spreads by ~1 mm per 3 m. - 1 m beam (large): divergence ≈ 0.3 µrad → spreads by 1 m per 3 km. - 10 µm beam (focused spot): divergence ≈ 30 mrad → spreads by 10 mm per 1/3 m.

The product w₀ × θ = M² × λ / π is a "beam parameter product" — fundamental constant for a given M² and wavelength.

Common mistakes to avoid

Confusing waist radius w₀ with waist diameter 2w₀. Formulas use radius; specs often give diameter.
Using M² = 1 for real diode lasers. Most diode lasers have M² > 1, often very different on slow vs fast axes.
Confusing 1/e radius with 1/e² radius. The 1/e² is the standard for Gaussian beams; 1/e is smaller.
Treating Rayleigh range as a hard "collimation distance." Beam grows continuously; z_R is just where radius = √2 × w₀.
Forgetting M² when computing focused spot size. M² scales focused spot by the same factor as input divergence.
Using paraxial Gaussian beam formulas at high NA. For NA > 0.7, vector beam effects matter.
Assuming far-field formulas at near-field distances. Use the full w(z) formula for z near or below z_R.

Frequently Asked Questions

Sources & further reading

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