The beam that focuses itself

In the summer of 1964 a laser engineer named Michael Hercher fired a ruby laser into a block of glass and found that it had left behind a set of fine damage tracks — thin threads of shattered glass, a few micrometres wide, running along the beam’s path. The beam had been millimetres across. Nothing in the optics had focused it. Something in the glass had squeezed the light into filaments far narrower than it entered, and intense enough to break the glass along their length.

What had squeezed it was the beam itself. The year before, the Soviet physicist Gurgen Askar’yan had pointed out that a sufficiently intense beam might change the medium it crosses so as to make the medium a lens, and in 1964 Raymond Chiao, Elsa Garmire and Charles Townes worked out the condition: above a certain power — not a certain intensity, which is the surprising part — a beam of light focuses itself, and the narrower it gets, the harder it focuses.

An index that the light raises #

The refractive index of any transparent material rises slightly in a strong electric field. In glass the main cause is the electrons’ response becoming slightly non-linear: their displacement is no longer quite proportional to the field, and the leftover term, averaged over an optical cycle, raises the index in proportion to the light’s intensity,

$$n = n_0 + n_2\,I.$$

For fused silica $n_2$ is about $2.7\times10^{-20}\ \text{m}^2/\text{W}$. That is tiny: sunlight, at a kilowatt per square metre, changes silica’s index by three parts in $10^{17}$. A focused laser pulse at $10^{16}$ W/m² changes it by three parts in ten thousand — comparable to the difference between two kinds of glass. It is the third-harmonic term of the oscillator that answers at three times the question, which raises the index at the driving frequency as well as making light at three times it.

Now consider a beam brighter in the middle than at its edge, as every laser beam is. The index is raised most on the axis and least at the edge, so the glass the beam is crossing has become a graded-index lens, highest in the middle. The ray that bends without a surface found what a ray does in a medium whose index varies sideways: Fermat’s condition curves it towards the higher index. Every ray of the beam curves towards the axis.

The figure traces the rays through the index profile, holding the beam’s shape fixed, as a first look. The lens it writes is not a good lens: the index gradient is steepest about halfway out and weak at the edge, so rays near the axis cross it after 1.73 metres and rays from the edge only after 3.63. But a real beam does not hold its shape. As the rays converge the beam narrows, its intensity rises, its lens gets stronger, and the rays converge faster still. That feedback is the whole of what self-focusing is.

The size that does not matter #

Against the converging lens stands diffraction, which spreads any beam of finite width. Where rays stop being enough found that a beam of radius $w$ spreads at an angle of about $\lambda/\pi n_0 w$: a narrow beam spreads fast, a wide one slowly. The rays curving inward converge at an angle set by the index difference between axis and edge, roughly $\sqrt{2\,n_2 I/n_0}$, the same angle that decides whether a ray is trapped in a graded fibre.

Now compare them as the beam is made narrower at the same power. Halving the radius quarters the area, so the intensity rises fourfold and the self-focusing angle doubles. The diffraction angle also doubles. Both are proportional to $1/w$.

The lines on the figure are parallel. They never cross, so which effect wins cannot depend on how big the beam is — only on how much power it carries. Write the ratio of the two angles and the radius cancels:

$$\frac{\theta_{\text{sf}}}{\theta_{\text{d}}} \;\propto\; \frac{\sqrt{n_2 P/(n_0 w^2)}}{\lambda/(n_0 w)} \;=\; \sqrt{\frac{n_0 n_2 P}{\lambda^2}}.$$

The threshold is a power, $P_{\text{cr}} \sim \lambda^2/(n_0 n_2)$ with a numerical factor that the exact calculation fixes at $3.77/8\pi$ for a Gaussian beam. For fused silica at 1064 nm that is 4.3 megawatts. A beam a millimetre across and one a micrometre across face the same threshold; the narrow one reaches unimaginable intensities before it gets there, the wide one reaches it with intensities the glass shrugs off, and in both cases the beam does or does not collapse according to the same number.

This is why focusing a laser more tightly does not by itself make it self-focus, and why defocusing it does not prevent self-focusing. It is also why the effect is a property of beams in two transverse dimensions specifically. In one dimension — a pulse travelling along a fibre, where the competing spread is dispersion in time — the nonlinear term wins at short scales and loses at long ones, the two can balance stably, and the pulse two failures keep alive is the result. In two dimensions they scale identically and the balance is a knife-edge. In three, the nonlinearity wins at small scales and collapse is unavoidable.

A width that has to reach zero #

The ray picture is suggestive but not exact. What makes self-focusing more than a story is a result found in 1971 by Vlasov, Petrishchev and Talanov, which needs no approximation at all. The equation for a beam in a Kerr medium — the paraxial wave equation with the intensity-dependent index included, the two-dimensional cubic Schrödinger equation — has a conserved quantity whose consequence is that the beam’s mean square width, however its shape changes, is always a parabola in distance.

For a collimated Gaussian beam the parabola is

$$w^2(z) = w_0^2\left[1 + \left(1 - \frac{P}{P_G}\right)\frac{z^2}{z_R^2}\right], \qquad P_G = \frac{\lambda^2}{2\pi n_0 n_2},$$

where $z_R$ is the Rayleigh range, the distance over which the beam doubles its area at low power — the same distance the half cycle a focus adds found the Gouy phase being picked up over. At low power the width grows as diffraction requires. At exactly $P_G$ the bracket’s coefficient vanishes and the width never changes. Above $P_G$ the coefficient is negative and the parabola reaches zero at a finite distance, $z_R/\sqrt{P/P_G - 1}$.

A width cannot reach zero. The beam carries finite power, so a zero width means infinite intensity, and the theorem says that the equation predicts exactly that — at or before the distance where the parabola hits the axis. This is a proof by contradiction that the beam collapses: whatever its shape does in detail, its width is forced through zero, so the solution must have stopped existing first. The Gaussian threshold $P_G$ is about 7 per cent above the true critical power, because a Gaussian is not quite the shape that balances exactly; below $P_G$ a beam can still collapse if its shape is right, but above it, every beam must.

How far before it collapses #

The width law gives a bound. The actual collapse distance, computed by solving the equation numerically for a Gaussian input, was fitted by J. H. Marburger in 1975 to a formula that has been used ever since.

Just above the critical power the distance runs to infinity, because the beam is barely winning; well above it, the distance falls as the inverse square root of the power, so doubling the power brings the collapse only 30 per cent closer. The real collapse always comes before the bound, as it must, because the beam’s centre self-focuses faster than its mean width shrinks — the aberration the ray figure showed, now playing in the beam’s favour.

The numbers make sense of the damage tracks. In a laser rod of glass at a few times the critical power, with a beam a few millimetres across and a Rayleigh range of tens of metres, collapse comes within a few metres, which is longer than any rod. But a beam with a hot spot — a small region a few times brighter than its surroundings — carries that spot’s own small Rayleigh range, and if the power in the spot exceeds the critical power, the spot collapses on its own within centimetres. Hercher’s filaments were hot spots, each collapsing independently.

Every medium has its number #

The critical power goes as $\lambda^2/n_0 n_2$, so it varies enormously between materials.

Air has a nonlinear index about a thousand times smaller than glass, so its critical power is in the gigawatts. Carbon disulphide, a liquid whose rod-shaped molecules turn to line up with the light’s field, has one a hundred times larger than glass and a critical power of tens of kilowatts, and was the material of many of the first self-focusing experiments. Silicon in the telecommunications band is similar, which matters for photonic chips that carry high power in very small waveguides.

Continuous lasers almost never reach these powers. Pulsed lasers pass them routinely: a modest titanium–sapphire laser produces pulses of a hundred femtoseconds and a millijoule, a peak power of ten gigawatts, above the critical power of every solid and of air. So every high-power laser is designed around this number. The quantity engineers track is the B-integral, the extra phase the beam’s centre accumulates over its edge, $B = (2\pi/\lambda)\int n_2 I\,dz$; once it exceeds a few radians, hot spots begin to self-focus and the optics begin to break. The invention that made modern high-intensity lasers possible, chirped-pulse amplification, recognised with a Nobel Prize in 2018, is at bottom a way of keeping the B-integral small: the pulse is stretched in time a thousand-fold before amplification, so its peak power stays below the critical power while its energy grows, and compressed again only at the end, in vacuum, where there is nothing to focus it.

The plasma that stops the collapse #

The width law predicts infinite intensity, and in nature something always intervenes first. In glass it is damage. In air it is something more interesting. As a collapsing pulse’s intensity reaches about $10^{17}$ W/m², it begins to strip electrons from the air molecules by absorbing several photons at once, and the free electrons make a plasma. A plasma has a refractive index below one — the result the index that falls below one found for X-rays and for any light above the frequency below which nothing gets in — so the plasma on the axis lowers the index there and acts as a diverging lens.

The collapse is arrested. The beam’s core settles into a narrow channel about a tenth of a millimetre across, in which the self-focusing of the Kerr effect and the defocusing of the plasma balance dynamically, with the intensity clamped near the value at which ionisation begins. Energy is fed into the core from a broad surrounding reservoir of weaker light, and the result, a filament, can run for hundreds of metres — far beyond the Rayleigh range of its own tiny width — trailing a thin line of plasma. Pulses of many times the critical power break up into many filaments, each carrying a few critical powers. In 2023 a laser at the summit of the Säntis mountain in Switzerland used such filaments to guide lightning strikes along their ionised paths for tens of metres.

The same Kerr lens has been put to work more quietly. In a titanium–sapphire laser the crystal itself focuses the light more strongly when it is pulsed than when it is continuous, and a slit placed inside the cavity passes the focused, pulsed light preferentially. The laser chooses to pulse; the phases that turn a glow into pulses supplies the rest. Self-focusing, which destroyed the first high-power optics, is the mechanism by which the most widely used ultrafast lasers lock their modes.

The same equation in three other places #

The equation behind every figure here — a wave that spreads, plus a term that makes it travel faster or slower where it is stronger — is not special to light. It is the generic description of any nearly monochromatic wave in a weakly non-linear medium, and wherever it appears, the count of dimensions decides the outcome in the same way.

On deep water it appears in one dimension, for a train of swell travelling along the ocean. There, a crest that is slightly higher travels slightly faster, and the even wave train that cannot stay even found the result: a uniform train breaks up into groups, because the nonlinearity concentrates energy wherever it is already concentrated. In one dimension that concentration saturates — the groups become stable envelope solitons, and the extreme waves they produce are finite. In the transverse plane of a light beam the same instability has nowhere to stop.

In a Bose–Einstein condensate of atoms that attract one another weakly, the condensate’s wavefunction obeys the same cubic equation in three dimensions, with the trap’s confinement standing in for the beam’s finite width. Above a critical number of atoms the attraction wins over the spreading that quantum uncertainty enforces, and the condensate collapses. For lithium-7 the critical number is about a thousand atoms. When experimenters at JILA in 2001 used a magnetic field to make the interaction in a rubidium-85 condensate suddenly attractive, it shrank, and then ejected a burst of atoms in an explosion that was promptly named a Bosenova — the same arithmetic as a beam above its critical power, with atoms in place of watts.

And the one-dimensional case in time is the fibre soliton, where the pulse two failures keep alive is the stable balance that the two-dimensional beam cannot achieve. The difference between those two cases is the most useful thing the self-focusing calculation teaches: the same nonlinearity that gives telecommunications its most robust pulses destroys glass when it is allowed a second transverse direction to act in.

A finite-distance singularity of this kind has a cousin in a different equation. The front that steepens until it cannot found a sound wave whose crests travel faster than its troughs and catch them up after a finite distance, at which the smooth solution ceases to exist and a shock forms. There, the singularity is a slope becoming infinite; here, an intensity. In both, an equation that was perfectly well behaved at the start predicts its own breakdown, and in both, the physics it left out — viscosity in one case, plasma in the other — takes over exactly where it does.

A continuous beam in a medium that answers instantly #

The ray figure holds the beam’s profile fixed, which removes the feedback that defines the effect; it is there to show the shape of the lens the beam writes, not its evolution. The width law is exact for the paraxial cubic equation and for nothing else. Near collapse the beam’s angles are no longer small and the paraxial approximation fails; the index’s response is not instantaneous for molecular contributions like those in carbon disulphide and air; and the Kerr effect itself saturates. The width law proves the equation breaks down; it says nothing about what replaces it.

Every figure is for a continuous beam. A pulse is a sequence of slices of different power, each with its own collapse distance, so the brightest slice collapses first and the dimmer ones later or not at all, and the focus moves along the beam during the pulse — the moving focus that made Hercher’s damage tracks continuous lines rather than points. The nonlinear indices are typical published values, uncertain by tens of per cent and dependent on the pulse length, since slower mechanisms contribute more to longer pulses. And the critical power for a beam that is not Gaussian differs from the one drawn, by as much as the 7 per cent between $P_G$ and the true critical power, or more for beams with structure.

Still open: what a filament keeps and why #

Filaments in air have been studied for thirty years, and their basic balance — Kerr focusing against plasma defocusing, fed from a reservoir — is agreed. What is not agreed is the relative role of other mechanisms that could also arrest the collapse: higher-order Kerr terms that would make the index fall at the highest intensities, which some experiments around 2009 claimed to see and later work disputed; the role of the reservoir in sustaining the filament over long distances; and how far the clamped intensity can be controlled. Whether filaments can be steered, lengthened to kilometres reliably, or used to guide electrical discharges routinely depends on settling these questions, and practical applications — remote spectroscopy of the atmosphere, lightning protection, triggering of condensation — are being pursued before they are settled.

The general point is Fermat’s, in a medium that is written by the light passing through it. When the index a ray obeys depends on the beam the ray is part of, a beam brighter in its middle bends into itself; and when the bending and the spreading scale identically with size, only the power can decide which wins. Below 4.3 megawatts, a beam in glass behaves as linear optics says. Above it, linear optics has stopped describing what happens, and the glass finds out first.