Laser Rayleigh Range Spot Size Calculator
Laser Optical Engineering: Compute focused beam waist diameter ($2w_0$), Rayleigh range ($z_R$), Depth of Focus (DOF), and divergence per ISO 11146.
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Gaussian Beam Focusing & Rayleigh Range (ISO 11146)
Laser beam propagation through diffraction-limited optical elements is governed by Gaussian caustic geometry, where diffraction prevents light from focusing to an infinitesimal geometric point.
1. Focused Waist Diameter (2w₀)
When a collimated laser beam of diameter $D_0$ ($4\sigma$ standard) passes through a positive lens of focal length $f$, the focused waist diameter $2w_0$ is:
2w₀ = (4 · λ · f · M²) / (π · D₀)
2. Rayleigh Range & Hyperbolic Expansion
The Rayleigh range $z_R$ is the distance over which the beam area doubles:
z_R = (π · w₀²) / (λ · M²) Depth of Focus (DOF) = 2 · z_R
At any distance $z$ from the focal plane, the expanding beam radius $w(z)$ follows a hyperbola:
w(z) = w₀ · √[ 1 + (z / z_R)² ]
Frequently Asked Questions
What is the Rayleigh Range (z_R) and Depth of Focus (DOF)?
The Rayleigh range ($z_R$) is the axial distance along the optical propagation axis from the focused beam waist ($z=0$) to the point where the beam cross-sectional area doubles (or beam radius increases by a factor of $\sqrt{2} \approx 1.414$). The Depth of Focus (DOF) is the total symmetric range over which the beam remains tightly focused ($2 z_R$), defining the allowable axial working tolerance in laser cutting and drilling.
How does the beam quality factor M² affect focused spot size?
The dimensionless $M^2$ factor (per ISO 11146) quantifies how close a real laser beam is to a diffraction-limited fundamental Gaussian beam ($TEM_{00}$, where $M^2 = 1.0$). For a given focusing lens and raw beam diameter, both the focused spot diameter ($2w_0$) and the beam divergence half-angle ($\theta$) scale directly with $M^2$: $2w_0 = \frac{4 \lambda f M^2}{\pi D_0}$. Single-mode fiber lasers achieve $M^2 \approx 1.1$, whereas high-power multi-mode lasers typically have $M^2 \ge 4.0$.
What is the trade-off between spot size and depth of focus?
Spot size and depth of focus have an inverse quadratic relationship. Because $z_R = \frac{\pi w_0^2}{\lambda M^2}$, halving the spot diameter reduces the depth of focus by a factor of 4. High-precision micro-machining requires very short Rayleigh ranges (microns), requiring active optical autofocus systems to follow material surface undulations.