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Gaussian Beam Spot Size and Rayleigh Range Calculator

Every focusing, beam delivery and alignment decision on a laser line depends on how fast the beam grows away from its waist. This calculator gives the Rayleigh range, spot size at a chosen distance and divergence for real beams with an M squared above 1.

Your numbers

Beam waist radius (1/e2)

Wavelength

Beam quality M squared

1.0 is an ideal TEM00 beam. Single-mode fiber lasers are near 1.05 to 1.3; multimode lasers are far higher.

Distance from waist

1,000 mm

Your results

Rayleigh range

671.0511

Distance in mm over which the beam area doubles from the waist.

Beam radius at distance z

0.8973

1/e2 radius in mm.

Beam diameter at distance z

1.7946

mm.

Half-angle divergence

0.7451

mrad, far-field half angle.

Confocal parameter (2 x Rayleigh range)

1,342.1021

Usable depth of focus in mm using the sqrt(2) growth criterion.

Embedded Gaussian beam model for a stigmatic beam. Elliptical or astigmatic beams need separate X and Y calculations.

Save results:

Beam radius (mm) vs distance from waist (mm)

Sweeps Distance from waist from 0 to 3000, holding your other inputs fixed.

Tie beam quality checks to your maintenance plan

Optional expert review of how to schedule M squared and power checks inside your ERP maintenance module.

Optional. The tool and every export work without this. No spam.

How to use the calculator

Enter the 1/e2 waist radius of the beam, the wavelength in nanometers and the M squared value from the laser datasheet or your beam profiler. Move the distance slider to see the beam radius at that plane. The chart plots radius versus distance out to 3 meters, so you can see where the beam leaves your optics clear aperture or where the spot becomes too large for the process. Remember to use radius, not diameter, when you enter the waist.

Convert profiler D4sigma diameters to radii by halving them.
A smaller waist gives a shorter Rayleigh range and faster divergence.
Use the confocal parameter as a depth of focus guide, not a hard limit.

Formulas

The Rayleigh range is zR = pi w0^2 / (M2 lambda). The radius at distance z is w(z) = w0 sqrt(1 + (z / zR)^2). The far-field half-angle divergence is theta = M2 lambda / (pi w0). Real beams behave like an ideal Gaussian beam with a wavelength stretched by M2, which is why beam quality directly shortens the Rayleigh range. At z equal to zR the radius has grown by a factor of the square root of two, and the beam area has doubled.

Reading the result

A 1064 nm beam with a 0.5 mm waist and M squared of 1.1 has a Rayleigh range near 670 mm and a half-angle divergence of about 0.7 mrad. At 1 meter from the waist the radius is about 0.4 mm larger than at focus. Doubling the waist radius quadruples the Rayleigh range and halves the divergence, which is why beam expanders are used before long-distance delivery. If your measured spot grows faster than predicted, your effective M squared is worse than the datasheet.

Common mistakes

Mixing diameter and radius is the usual error and produces a factor of two in spot size and four in Rayleigh range. Another is using the datasheet M squared for a multi-kilowatt laser at full power, where thermal lensing can degrade beam quality. Also check that the waist location is where you think it is: the model measures z from the waist, not from the laser output aperture or the last lens.

Frequently Asked Questions

What is M squared?

M squared is the beam quality factor comparing a real beam to an ideal TEM00 Gaussian beam. It equals 1 for the ideal case and increases for multimode or distorted beams. It multiplies both divergence and the focused spot size at a given focal length.

Is Rayleigh range the same as depth of focus?

Related but not identical. The Rayleigh range is the distance where the beam area doubles. Depth of focus is often quoted as twice that, called the confocal parameter, but process tolerance may allow less.

Does the calculation work in air or glass?

It assumes the medium index is 1. In a medium of index n, the wavelength inside is shorter, so enter the in-medium wavelength, which is the vacuum wavelength divided by n.

Can I use it for diode lasers?

Only per axis. Diode beams are strongly astigmatic with different M squared in the fast and slow axes, so run the calculator once for each axis with its own waist and M squared.

If beam profile and calibration data live outside your ERP, Netray can link them to equipment, lot and maintenance records so drift in beam quality is traceable.

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