Tool · Beams and focusing

Laser Spot Size Calculator

How small will a lens focus a laser beam, and over what depth does the spot stay small? From the wavelength, the beam diameter at the lens, the focal length and M², the calculator gives the focused spot diameter, the Rayleigh range and depth of focus, the convergence angle, the peak intensity and the power lost to the lens aperture. Background: beam waist, Rayleigh range and beam quality.

Beam and lens

Wavelength (nm)

Beam diameter at the lens, 1/e² (mm)

Focal length (mm)

Beam quality M²

M² = 1 is a perfect TEM₀₀ beam; single-mode fiber lasers are typically 1.05 to 1.2. The beam waist is taken at the lens, which is close for a collimated beam.

Power and aperture

Laser power (W)

Lens clear aperture (mm)

The peak intensity is for a Gaussian focus. A clear aperture below about 1.5 times the beam diameter clips the beam and enlarges the spot.

Presets

Readouts

Beam radius through the focus

The hyperbola w(z) = w₀√(1 + (z/z_R)²). The dashed lines mark ±z_R, where the radius has grown by √2.

Spot diameter against input beam diameter

Doubling the beam at the lens halves the spot, until the clear aperture starts to clip it. The dot is the current beam.

Learn with it

Two short experiments. Each one sets the inputs, says where to look, and asks for a prediction before it shows the result.

Checked against

These checks run in your browser on every load. The focused waist, its position, the Rayleigh range and the clipping loss are compared with values worked out by hand.

Check	Expected	Computed	Tolerance

The expected values follow the Gaussian-beam thin-lens transformation (Self 1983) and the paraxial far-field divergence, evaluated by hand for the stated cases. The tolerance is the largest difference from Expected that still passes, relative to Expected or to 1, whichever is larger.

The model

A beam with 1/e² radius $w_{L} = D / 2$ at a thin lens of focal length $f$ , with its waist at the lens, has the input Rayleigh range $z_{R} = π w_{L}^{2} / (M^{2} λ)$ . Gaussian propagation through the lens gives a new waist

w_{0} = \frac{w_{L} (f / z_{R})}{\sqrt{1 + (f / z_{R})^{2}}} \approx \frac{M^{2} λ f}{π w_{L}}

a distance $f / (1 + (f / z_{R})^{2})$ behind the lens. For a collimated beam $z_{R}$ is far longer than $f$ , and the spot diameter takes the familiar form

2 w_{0} = \frac{4 M^{2} λ f}{π D}

Around the focus the radius grows as $w (z) = w_{0} \sqrt{1 + (z / z_{R}^{'})^{2}}$ with $z_{R}^{'} = π w_{0}^{2} / (M^{2} λ)$ , so the depth of focus over which the diameter stays within $\sqrt{2}$ of its minimum is $2 z_{R}^{'}$ . The peak intensity of a Gaussian spot is $2 P / (π w_{0}^{2})$ , and a clear aperture of diameter $A$ passes all but $e^{- 2 A^{2} / D^{2}}$ of the power. The results are paraxial and aberration-free: past a convergence half-angle of about 0.2 rad, or with a beam that overfills a simple lens, spherical aberration usually sets the spot size instead. M² treats a non-Gaussian beam as an embedded Gaussian in the second-moment sense, which predicts the second-moment diameter and not the shape of the spot.

Worked example

A 1064 nm fiber laser with $M^{2} = 1.1$ and a 10 mm beam, focused by a 100 mm lens, gives $2 w_{0} = 4 (1.1) (1.064 μ m) (100) / (π \cdot 10) = 14.90 μ m$ . The Rayleigh range is 149 µm, so the depth of focus is 298 µm, and the half-angle of the converging beam is 50 mrad. With 20 W the peak intensity is 22.93 MW/cm². Expanding the beam to 20 mm halves the spot to 7.451 µm and cuts the depth of focus fourfold to 74.5 µm; the 25 mm aperture then clips 4.4 % of the power.

Related: the Gaussian beam calculator propagates a beam to any distance, the Airy disk calculator covers uniformly filled apertures, and the focusing a laser beam article covers lens choice and alignment. See also depth of focus and beam expander.

References: S. A. Self, “Focusing of spherical Gaussian beams,” Applied Optics 22, 658–661 (1983). A. E. Siegman, Lasers, University Science Books (1986), chapter 17. B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, 2nd ed., Wiley (2007), chapter 3.