Self-focusing

Self-focusing is the contraction of an intense light beam caused by its own effect on the refractive index. Through the optical Kerr effect the index rises with intensity, $n = n_0 + n_2 I$, so a beam that is brightest on its axis writes a graded-index lens into the medium. When the beam power exceeds a critical value, this lens overcomes diffraction and the beam shrinks until some other process stops it: ionization, multiphoton absorption, or material damage. For a Gaussian beam in fused silica the critical power is about 2.5 MW at 800 nm and 4.5 MW at 1064 nm; in air at 800 nm it is a few gigawatts. These are peak powers, reached easily by femtosecond and picosecond pulses: a 1 µJ, 100 fs pulse already carries about 10 MW.

Critical power

The critical power is set by the balance between nonlinear focusing and diffraction. For a Gaussian input beam, numerical solutions give

$$P_\text{cr} = \frac{3.77\,\lambda^2}{8\pi\,n_0\,n_2}$$

With $n_2 = 2.6 \times 10^{-20}$ m²/W for fused silica, $n_0$ = 1.4533 at 800 nm and 1.4496 at 1064 nm, the formula gives 2.54 MW and 4.51 MW. Older texts use $\lambda^2/(2\pi n_0 n_2)$, which gives 2.70 MW at 800 nm; the difference of about 6 percent reflects the beam profile assumed. Both forms carry a real uncertainty from $n_2$, which varies by tens of percent among measurements and depends on wavelength and pulse duration.

The critical power does not depend on the beam radius. A wider beam has lower intensity and a weaker nonlinear lens, but it also diffracts more slowly, and the two effects cancel. Focusing a beam more tightly therefore does not prevent self-focusing; only the power relative to $P_\text{cr}$ decides whether collapse occurs.

For a collimated beam above critical power, Marburger's empirical formula estimates the collapse distance:

$$z_\text{sf} = \frac{0.367\,k w_0^2}{\sqrt{\left(\sqrt{P/P_\text{cr}} - 0.852\right)^2 - 0.0219}}$$

For air, taking $n_2 \approx 3 \times 10^{-23}$ m²/W at 800 nm (published values range over roughly a factor of two), $P_\text{cr}$ is about 3.2 GW. A 1 mJ, 100 fs pulse with a 2 mm beam radius then has $P/P_\text{cr} \approx 3$ and collapses after about 13 m; at ten times critical power the distance falls to about 5 m.

Filamentation

In gases and transparent solids, collapse is arrested when the intensity becomes high enough to ionize the medium. The free-electron plasma lowers the refractive index and defocuses the beam, and the balance between Kerr focusing and plasma defocusing produces a filament: a narrow channel, about 100 µm across in air, that persists over distances much longer than the Rayleigh range. The intensity inside is clamped near $10^{13}$–$10^{14}$ W/cm². Filaments in air have been observed over hundreds of metres. In bulk solids, filamentation of femtosecond pulses is a standard route to supercontinuum generation, since the extreme intensity drives self-phase modulation and ionization together.

Damage and beam breakup

Self-focusing is a major source of optical damage in high-power lasers. Whole-beam self-focusing produces thin damage tracks inside glass components. In large beams well above critical power, small intensity ripples grow faster than the beam as a whole (small-scale self-focusing, described by Bespalov and Talanov in 1966), and the beam breaks into many filaments. High-energy Nd:glass laser chains limit this with spatial filters between amplifier stages, which remove the fastest-growing high-spatial-frequency ripples, and by keeping the accumulated nonlinear phase, the B-integral $B = (2\pi/\lambda)\int n_2 I\,dz$, to a few radians or less. Chirped pulse amplification was developed largely to keep the peak power inside the amplifier below these limits.

Uses

Self-focusing is also used deliberately. Kerr-lens mode locking relies on the self-focusing lens in the laser crystal to favour pulsed operation; intracavity peak powers approach $P_\text{cr}$, but the crystal is only a few millimetres long, too short for collapse. Femtosecond direct writing of waveguides in glass relies on the controlled deposition of energy near the focus, and filaments are studied for remote spectroscopy and lightning guiding.

Common questions

Does self-focusing depend on intensity or on power?

Whether a beam collapses depends on power; how quickly it collapses depends on intensity and beam size through the distance formula above. A beam below $P_\text{cr}$ will not collapse however tightly it is focused, although its focal spot can shift and shrink slightly.

Is thermal lensing a form of self-focusing?

Thermal lensing is also a self-induced lens, produced by absorbed power heating the medium, and it can be focusing or defocusing depending on the sign of $dn/dT$. It responds on millisecond timescales to average power, whereas Kerr self-focusing is effectively instantaneous and responds to peak power.

How does a material with negative n2 behave?

A medium with negative $n_2$, or one dominated by a plasma or thermal defocusing contribution, spreads the beam instead. This self-defocusing has no threshold and no collapse.

References: J. H. Marburger, "Self-focusing: theory," Prog. Quantum Electron. 4, 35 (1975); G. Fibich and A. L. Gaeta, "Critical power for self-focusing in bulk media and in hollow waveguides," Opt. Lett. 25, 335 (2000); A. Couairon and A. Mysyrowicz, "Femtosecond filamentation in transparent media," Phys. Rep. 441, 47 (2007); R. W. Boyd, Nonlinear Optics, 3rd ed. (Academic Press, 2008), Ch. 7.