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.
**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).
**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.
**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.
**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.
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Calculate reflectance of a single-layer thin-film optical coating at normal incidence with quarter-wave and half-wave analysis.
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Radius at the beam waist (1/e² intensity)
1.0 for ideal Gaussian, >1 for real beams
Rayleigh Range
49.63 mm
Divergence
2.01 mrad
Beam Radius at z
224.9 μm
| 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 |