What is the 1/e² spot radius?

The 1/e² radius is the distance from the beam axis where the intensity drops to 1/e² (about 13.5%) of the peak value. It is the standard way to characterize Gaussian beam width.

M² (beam quality factor, also called "times-diffraction-limit") measures how closely a real beam resembles an ideal Gaussian. M² = 1 is a perfect Gaussian; typical diode lasers have M² between 1.1 and 2, CO₂ lasers often 1.0–1.2.

Does the formula assume the beam is collimated at the lens?

Yes. The formula w0 = M² × λ × f / (π × w_in) assumes a collimated (parallel) input beam. For a converging or diverging input beam, the effective object distance must be accounted for with the full Gaussian beam propagation (ABCD matrix) approach.

What is the Rayleigh range?

The Rayleigh range z_R is the distance from the waist at which the beam radius grows by a factor of √2. It defines the depth of focus: the region ±z_R around the waist where the beam stays near its minimum size.

Why does a larger input beam produce a smaller spot?

A wider input beam fills more of the lens aperture, which is equivalent to a higher numerical aperture, producing tighter focusing — just like a larger telescope mirror gives higher resolution.

Laser Spot Size Calculator

Name: Laser Spot Size Calculator
Availability: InStock
Author: Nham Vu

Calculate the focused 1/e² spot radius of a Gaussian laser beam from wavelength, beam quality M², input beam radius, and lens focal length.

Beam Parameters

Wavelength (λ)

Common: 532 nm (green), 1064 nm (Nd:YAG), 10600 nm (CO₂)

Beam Quality Factor (M²)

M² = 1 is ideal Gaussian; typical lasers: 1.0 – 3.0

Input Beam Radius (w_in)

1/e² radius of collimated beam at the focusing lens

Lens Focal Length (f)

Effective focal length of the focusing lens or objective

Enter beam parameters and click Calculate

Summary

Calculate the focused 1/e² spot radius of a Gaussian laser beam from wavelength, beam quality M², input beam radius, and lens focal length.

How it works

Enter the laser wavelength λ (e.g. 1064 nm for a fiber laser, 10.6 µm for CO₂).
Enter the beam quality factor M² (1 for an ideal Gaussian, 1.1–2 for typical diode lasers).
Enter the 1/e² input beam radius at the lens and the lens focal length f.
The calculator returns the focused spot radius w0 = M² × λ × f / (π × w_in), spot diameter, and Rayleigh range z_R.
Shorten the focal length or widen the input beam to get a smaller focused spot.

Use cases

Laser cutting and engraving — estimate the focused spot size to predict kerf width and resolution.
Laser material processing — choose a focal length to achieve a target spot size for a given power density.
Optical design — verify that a beam expander and focusing lens combination meets spot-size requirements.
Laser pointer and beam profiling experiments in physics laboratories.
Fiber coupling — estimate the focused spot needed to match a fiber core diameter.

Frequently Asked Questions

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Last updated: 2026-05-23 · Reviewed by Nham Vu