Pulsed Laser Focus and Damage Threshold

What is modelled

The beam leaves the laser as a Gaussian beam of a given radius and quality factor M2, and a thin lens focuses it. Because light is a wave, the spot cannot shrink to a point: it reaches a smallest radius, the waist, and then spreads out again. The distance over which the beam stays almost as narrow is the Rayleigh range. The energy of one pulse is spread over the area of the beam, and that energy per area is the fluence; a surface is damaged when the fluence is above its threshold. A short pulse packs the same energy into a very short time, which gives a very high peak power.

Equations

• Waist w0 = M2 λ f / (π win), Rayleigh range zR = π w02 / (M2 λ), radius w(z) = w0 √(1 + (z / zR)2).
• On-axis fluence F = 2E / (π w2); the damage zone is where F > Fth.
• Gaussian pulse of duration τ (FWHM): peak power P = 0.9394 E / τ.

Checks run in the page

• The waist and the Rayleigh range against a numerical beam-parameter propagation through lens and free space.
• The fluence summed over rings against the pulse energy.
• The pulse power integrated over time against the energy.
• The damage length found by a search against the closed form.

Honest simplifications

An ideal thin lens with no aberration, a Gaussian beam in space and in time, and a fixed damage threshold. Real thresholds depend on pulse duration, wavelength, surface quality and many shots, and at very high intensity air breaks down and the focus changes, so these numbers show the scaling and are not safety or design values.

• Press Tight focus, short lens and watch the waist shrink and the Rayleigh range collapse.
• Press Poor beam quality and see how a larger M2 makes the spot bigger and the focus longer.
• Press Fragile optic and move the target plane to see how far from the focus the fluence is safe.