How to Measure Laser Beam Divergence: Two-Point and Focal-Plane Methods

Scope

This article gives the procedures for measuring the far-field divergence angle of a laser beam: the two-point method, which measures the beam's width at two distances, and the focal-plane method, which uses a lens to bring the far field onto the bench. It covers the choice of width definition, the half-angle and full-angle conventions, and the conditions under which each method is valid. The full propagation measurement of ISO 11146, which fits the beam's width along its caustic and yields the waist, the divergence and the beam quality factor together, is covered in M² Beam Quality Measurement; the divergence of edge-emitting laser diodes, which differs strongly between their two axes, is described in the far-field divergence entry.

What is being measured

A beam's width grows with distance from its waist. Far from the waist, the growth becomes linear, and the divergence is the angle of that asymptote. For a Gaussian beam of waist radius $w_0$ at wavelength $\lambda$, the $1/e^2$ radius at distance $z$ from the waist is

$$w(z) \;=\; w_0\sqrt{1 + (z/z_R)^2},\qquad z_R = \frac{\pi w_0^2}{\lambda},$$

and the far-field half-angle divergence is

$$\theta \;=\; \frac{\lambda}{\pi w_0}.$$

A beam with beam quality factor $M^2$ diverges $M^2$ times faster than a Gaussian with the same waist. The Gaussian Beam Calculator evaluates these relations for other waists and wavelengths. The Rayleigh range $z_R$ sets the scale: the beam's width grows linearly only at distances several times $z_R$ from the waist.

Half angle or full angle. Divergence is quoted both as the half angle $\theta$ and as the full angle $2\theta$, and datasheets do not always say which. A reported value should always state which convention and which width definition were used.

Width definition. The width can be defined at $1/e^2$ of the peak intensity, at half maximum (FWHM), or as the second-moment (D4σ) width. For a Gaussian beam the $1/e^2$ and D4σ widths are equal and the FWHM is 0.589 of them; for other beams the definitions differ by amounts that depend on the profile, and ISO 11146 specifies the second-moment width. Knife-edge scans, slit scans and camera profilers each measure one of these definitions more naturally than the others (see knife-edge measurement).

Two-point method

Measure the beam's width $D_1$ and $D_2$ at two distances $z_1$ and $z_2$ along the beam, and compute the full-angle divergence as

$$2\theta \;\approx\; \frac{D_2 - D_1}{z_2 - z_1}.$$

The estimate is accurate only if both planes are in the far field. Inside a few Rayleigh ranges the width still grows more slowly than its asymptote, and the method underestimates the divergence. For a 633 nm beam with a 0.4 mm waist radius, $\theta$ = 0.504 mrad and $z_R$ = 0.79 m; measuring from the waist:

A beam with a large waist has a long Rayleigh range and may not reach its far field anywhere on the bench. The waist position is also often unknown, so the two-point result is best treated as a lower bound unless a third measurement confirms that the width is growing linearly.

Procedure

1. Align the profiler or knife-edge stage on a rail parallel to the beam, so it can be moved along the beam without re-centering.
2. Measure the width at several distances, at least three, spanning as long a path as the bench allows. Fold the beam with mirrors if needed; each mirror must be flat and large enough not to clip the beam.
3. Plot width against distance. If the points lie on a straight line, the slope is the full-angle divergence. If they curve, the planes are not in the far field, and the focal-plane method or a full caustic fit is needed.

Focal-plane method

A lens maps angles in the incoming beam to positions in its back focal plane: a ray arriving at angle $\alpha$ crosses the focal plane at a distance $f\alpha$ from the axis, whatever its position on the lens. The beam's profile in the back focal plane is therefore its far-field angular distribution scaled by $f$, and the divergence follows from one width measurement:

$$\theta \;=\; \frac{w_f}{f},\qquad 2\theta \;=\; \frac{D_f}{f},$$

where $w_f$ and $D_f$ are the radius and diameter at the focal plane. The relation holds for any beam, not only a Gaussian one, and wherever the lens is placed along the beam, because it measures angles rather than sizes.

For the beam above, a lens of $f$ = 0.5 m produces a focal-plane spot of radius 252 μm, easily measured with a camera or knife edge.

Procedure

1. Choose a lens whose focal length makes the focal spot large enough to measure accurately: several tens of camera pixels, or many knife-edge steps, across. The lens aperture must pass the whole beam without clipping, at least 1.5 times the beam's $1/e^2$ diameter at the lens.
2. Place the lens in the beam, perpendicular to it and centered.
3. Locate the back focal plane by measuring the focal length's distance from the lens's principal plane, or, more reliably, by finding the plane of the smallest spot. For a collimated or nearly collimated input the smallest spot lies at the focal plane; for a strongly diverging input it lies beyond it, and the measurement must be taken at the true focal distance $f$, not at the smallest spot.
4. Measure the width in the focal plane with the chosen definition, and compute the divergence.
5. For an elliptical or astigmatic beam, measure the widths along both principal axes separately.

Error sources

The focal length has to be known to the accuracy wanted in the result; a stock lens's nominal focal length is typically specified to about a percent, and it changes with wavelength. Aberrations enlarge the focal spot and make the divergence read high, so use a lens with a long focal length relative to the beam size, or a lens corrected for the wavelength. The camera's pixel size and any saturation limit the width measurement as in any beam profile; attenuate with neutral-density filters placed before the lens rather than after it, where their surfaces could distort the focus.

Verification

Measure a beam whose divergence is known, such as the output of a single-mode fiber of known mode field diameter at a known wavelength, whose far field is close to Gaussian with $\theta = \lambda/(\pi w_0)$. For a 10.4 μm MFD at 1550 nm that is 94.9 mrad half angle. Agreement within a few percent confirms the width definition, the focal length and the scaling.

Common failure modes

Near-field measurement. A two-point measurement inside a few Rayleigh ranges reads low.

Half and full angle confused. A factor of two, and the most common disagreement between a datasheet and a bench result.

Mixed width definitions. A FWHM divergence compared with a $1/e^2$ specification differs by a factor of 0.589 for a Gaussian beam, and by an unknown factor for other beams.

Clipping. An aperture that cuts the beam's wings narrows the measured width and makes the divergence read low, while diffraction from the edge adds structure to the far field.

Unstable pointing. Beam wander during a slow knife-edge or slit scan broadens the measured width; a camera captures the profile in one frame and averages several.

References: ISO 11146-1, Lasers and laser-related equipment: Test methods for laser beam widths, divergence angles and beam propagation ratios, Part 1: Stigmatic and simple astigmatic beams; A. E. Siegman, Lasers (University Science Books, 1986), chapters on Gaussian beams and beam quality; H. Kogelnik and T. Li, "Laser beams and resonators," Applied Optics 5, 1550 (1966).