# Spot Size Calculator

Created by Davide Borchia
Last updated: Jul 01, 2022

Gaussian beams are particular modes of propagation of a light beam with a characteristic minimum spot size (waist): with our beam spot size calculator, you will learn how a lens intercepting a collimated Gaussian beam can focus it to a small beam spot size.

With our tool, you will discover:

• How to focus a beam to achieve the desired beam spot size;
• How to calculate a Gaussian beam spot size after focusing it with a lens;
• How to obtain the minimum spot size of a Gaussian beam;
• What is the depth of focus;
• The equation for the depth of focus: calculate the useful propagation length of a Gaussian beam.

Scientists often use lasers for materials characterization in their studies. The Sellmeier dispersion formula is one of the methods that need the light of a specific wavelength, and we have a tool dedicated to that topic!

## The optical set-up and Gaussian beams

We will assume we are lighting a lens with focal length $f$ with a collimated Gaussian beam with a diameter at the lens $d$.

The nature of the Gaussian beam causes it to focus not in a point but in a spot: the lens modifies (compresses) the beam, which, however, retains all its features.

As a general rule, the shorter the focal length of the lens, the smaller the beam spot size.

## How to calculate the spot size of a Gaussian beam

We calculate spot size of a Gaussian beam with the equation:

$S= \frac{4\cdot M^2\cdot \lambda \cdot f}{d\cdot \pi}$

Where:

• $M^2$ is the beam quality factor, a value greater or equal to $1$, which quantifies how much a Gaussian beam deviates from ideality ($M^2=1$);
• $\lambda$ is the wavelength of the light in the beam (we assume it to be monochromatic, or nearly so);
• $f$ is the focal length of the lens; and
• $d$ is the diameter of the beam at the lens.

Notice that $S$ is the diameter of the spot.

The spot size can't be reduced arbitrarily. The wavelength of the light and the focal length of the lens defines the beam spot at the diffraction limit. To calculate this value, substitute $M^2$ with the ideal value $1$ in the equation above.

Interestingly, the laser brightness doesn't affect the beam spot size, even though at higher powers it's likely that the beam will deviate from ideality.

🙋 Our calculator assumes by default that the beam is ideal: to change the value of the beam quality factor, click advanced mode: the variable will appear among the others.

The size of the beam spot is fundamental if you want to calculate the intensity of a pulsed laser, since it equals the ratio between the of the pulse and the beam area: find out more at our laser pulse calculator.

## How to calculate the depth of focus

After reaching its minimum spot size, a Gaussian beam retains its properties for a certain length, after which the divergence (a measure of how much the beam "opens") takes over. Coherence, both temporal and spatial, is lost, and the light resembles more and more a "normal" beam.

In optics, the length at which the beam still maintains a high degree of coherence and focusing is the depth of focus. We define it as two times the Rayleigh range:

$\text{dof}=2\cdot z_{\text{r}}$

The Rayleigh range is calculated in turn with:

$z_{\text{r}}= \frac{\pi}{M^2\cdot\lambda}\cdot\frac{S^2}{2}$

The value of the depth of focus is important in the planning of an experimental set-up since it defines the distance at which it's "safe" to place an optical element.

🙋 You can use our beam spot size calculator in reverse too! Find the focal length that allows you to achieve the desired beam spot size, or input the depth of focus to calculate the focal length.

Davide Borchia
Wavelength (λ)
nm
Diameter at the lens (d)
mm
Focal length (f)
mm
Depth of focus
mm
Beam spot size (S)
mm
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