Laser Beam Parameters Calculator
Compute key Gaussian laser beam properties — Rayleigh range, half-angle divergence, or beam radius at any distance from the waist. Essential for laser system design, focusing optics, and beam alignment tasks.
About this calculator
A Gaussian beam is characterised by its waist radius w₀, the point of tightest focus. The Rayleigh range z_R = π·w₀²/λ defines the distance over which the beam area doubles. Beyond z_R the beam diverges with half-angle θ = λ/(π·w₀). At any distance z from the waist the beam radius is w(z) = w₀·√(1 + (z/z_R)²). Wavelength λ must be in consistent units with w₀. These equations assume an ideal M² = 1 beam; real beams use M²·λ in place of λ. Understanding these parameters is critical for fibre coupling, laser machining spot size, and microscopy resolution.
How to use
Example: Laser wavelength λ = 532 nm = 5.32 × 10⁻⁷ m, beam waist w₀ = 1 mm = 0.001 m, distance z = 500 mm = 0.5 m. Step 1: z_R = π × (0.001)² / (5.32×10⁻⁷) ≈ 5.91 m. Step 2: w(z) = 0.001 × √(1 + (0.5/5.91)²) = 0.001 × √(1.00716) ≈ 1.00357 mm. The beam has expanded by only ~3.6 µm at 500 mm because z ≪ z_R. To find divergence: θ = 5.32×10⁻⁷/(π×0.001) ≈ 0.1693 mrad.
Frequently asked questions
What is the Rayleigh range of a laser beam and how is it used?
The Rayleigh range z_R = π·w₀²/λ is the distance from the beam waist at which the beam radius has grown by a factor of √2 and the beam area has doubled. It defines the depth of focus of the laser. Within one Rayleigh range on either side of the waist the beam is considered approximately collimated. Engineers use it to determine how far a focused laser can travel before it expands significantly, which is crucial for cutting, welding, and communications.
How does beam waist radius affect laser divergence angle?
Divergence θ = λ/(π·w₀) is inversely proportional to the waist radius. A tighter focus (smaller w₀) produces greater divergence, while a large collimated beam diverges very slowly. This is the diffraction limit in action: you cannot simultaneously have a very small spot and a very narrow divergence. In practice, this trade-off governs the design of beam expanders and focusing objectives for laser systems.
What does the M² beam quality factor mean for real laser beams?
M² (M-squared) quantifies how close a real beam is to the ideal Gaussian TEM₀₀ mode, which has M² = 1. A beam with M² = 2 diverges twice as fast as an ideal Gaussian and cannot be focused to as small a spot. Real diode lasers often have M² between 1.1 and 3, while multimode fibres can exceed 10. When using the beam radius formula, substitute M²·λ for λ to correctly predict beam size at any propagation distance.