Focusing a Laser Beam: Spot Size, Depth of Focus and Damage

Scope

This article brings together what sets the size and shape of a focused laser spot and what the focus can do: the spot size formula, its trade against depth of focus, lens choice and aberrations, the irradiance and fluence at the focus, and the damage thresholds that limit them. It is written for collimated input beams focused by a single lens, the usual case on the bench and in materials processing. The Gaussian Beam Calculator computes the same quantities for any input, and the reverse task, making a diverging beam parallel, is covered in How to Collimate a Laser Beam.

Spot size

A collimated Gaussian beam of $1/e^2$ diameter $D$ at a lens of focal length $f$ focuses to a waist of $1/e^2$ diameter

$$2w_0 \;=\; \frac{4\lambda f M^2}{\pi D},$$

where $M^2$ is the beam quality factor, 1 for an ideal Gaussian beam and larger for real multimode beams (see M² Beam Quality Measurement). The spot shrinks with a shorter focal length or a larger input beam, which is why a beam expander is placed before a focusing lens when a small spot is needed. The ratio $f/D$ is the working f-number of the focus.

The trade against depth of focus

The distance over which the focused beam stays within $\sqrt{2}$ of its waist radius is twice its Rayleigh range,

$$2z_R \;=\; \frac{2\pi w_0^2}{\lambda M^2},$$

which falls as the square of the spot size. For a 1064 nm beam of 6 mm diameter with $M^2$ = 1:

Halving the spot size cuts the depth of focus by four. A process that needs a tolerant focus, such as cutting thick material or working on an uneven surface, therefore uses a larger spot than the smallest the optics could give.

Lens choice and orientation

The formula above assumes the lens adds no aberration. For a singlet this holds when the working f-number is large; as it falls toward a few, spherical aberration enlarges the spot beyond the Gaussian value and spreads the focus along the axis.

• Plano-convex singlets give their smallest aberration for a collimated input when the curved side faces the collimated beam.
• Best-form and aspheric lenses reduce spherical aberration further; an asphere can reach the diffraction limit at small f-numbers where a singlet cannot.
• Achromatic doublets correct chromatic aberration as well, which matters for broadband or multi-wavelength beams.
• Windows and cover glasses between the lens and the focus shift the focus away from the lens by $t(1 - 1/n)$ for a plate of thickness $t$ and index $n$, and add spherical aberration at small f-numbers.

A lens whose clear aperture is less than about 1.5 times the beam diameter clips the beam, adding diffraction rings around the spot; if the beam is truncated hard, the focus approaches the Airy pattern of a uniformly filled aperture rather than a Gaussian spot.

Irradiance and fluence at the focus

The peak irradiance of a Gaussian spot of power $P$ is twice its average over the $1/e^2$ area:

$$I_0 \;=\; \frac{2P}{\pi w_0^2}.$$

For pulses of energy $E$ the peak fluence is, likewise, $F_0 = 2E/(\pi w_0^2)$. For the 100 mm case above ($w_0$ = 11.3 μm), 1 W of continuous power gives a peak irradiance of 5.0 × 10⁵ W/cm², and a 100 μJ pulse gives a peak fluence of 50 J/cm². These are far above what most optical coatings survive, which is why the focus must never fall on an optic.

Damage thresholds

Laser-induced damage thresholds (LIDT) are quoted as a fluence (J/cm²) for pulsed beams, at a stated wavelength, pulse duration and repetition rate, and often as a linear power density (W/cm, power divided by beam diameter) for continuous beams, where the failure is thermal. Two conventions make comparisons treacherous:

• Peak or average fluence. A threshold quoted as average fluence over the $1/e^2$ area is half the peak fluence of the same beam. Compare the beam's peak fluence with a threshold only when the threshold is also defined at the peak.
• Pulse duration. For pulses in the nanosecond range, vendors commonly scale thresholds with the square root of the pulse duration. The rule is approximate, and it does not hold for picosecond and femtosecond pulses, where the damage mechanism changes.

A working margin of a factor of two below the specified threshold is common practice; contamination and defects lower the real threshold of an optic in use.

Measuring the spot

A focused spot of tens of micrometres is smaller than most camera pixels can resolve directly. Three approaches work:

1. Knife edge through the focus, as in the knife-edge method, which resolves spots of a few micrometres.
2. A camera with magnification, imaging the focal plane onto the sensor with a microscope objective of known magnification, while keeping the irradiance on the sensor below saturation.
3. The caustic scan of the M² measurement, which fits widths through the focus and gives the waist, its position and $M^2$ together.

References: A. E. Siegman, Lasers (University Science Books, 1986), chapters on Gaussian beams; S. A. Self, "Focusing of spherical Gaussian beams," Applied Optics 22, 658 (1983); ISO 21254-1, Lasers and laser-related equipment: Test methods for laser-induced damage threshold.