The beams that add only if they differ

The brightness no lens can increase found that no arrangement of lenses and mirrors can make an image brighter than its source, and traced the bound to thermodynamics. The cone a fibre will accept and the cone light has to find to get out applied it to guides and to light trapped in a dense medium. The invariant that is a count divided the étendue by the square of the wavelength and found a count of modes, with a least value of one. The work a diluted beam will not do and the same cone, and a different arrival followed the consequences for energy and for timing.

A laser beam is the extreme case of all of that: a beam that fills a single mode, with the least étendue light can have. Which raises a practical question with an answer that surprises most people who first meet it. If one laser is not powerful enough, why not use several and add their beams? Industrial cutting, laser weapons, the lasers proposed to push light sails to other stars, all want more power in a single, narrow beam than any one laser can deliver. The étendue bound says the naive way cannot work, and it also says, precisely, which ways can.

Why copies cannot be merged #

Radiance — power per unit area per unit solid angle — is what a beam’s usefulness at a distance depends on, because a target far away receives power in proportion to the radiance of the source times the solid angle the target subtends. Étendue conservation says that passive optics cannot increase radiance: they can redistribute a beam’s area and angle but not shrink their product.

Each diffraction-limited laser beam occupies one mode, an étendue of about $\lambda^2$. Two such beams occupy two modes, and nothing lossless can squeeze them into one. Put them side by side and the combination has twice the power and twice the area, the same radiance. Overlap them at an angle and the combination has twice the power in twice the solid angle, the same radiance. Try to overlap them exactly, in the same place and the same direction, with a beam splitter used backwards, and half of the combined light leaves through the splitter’s other port — unless the two beams are in step, which is the case taken up below. The figure’s flat line is the bound: N copies of one mode are N modes, and a lossless device can rearrange modes but not merge them.

This is not a failure of engineering. It is the second law of thermodynamics applied to light, the same statement that forbids a lens from heating anything hotter than the Sun. Two lasers of the same wavelength and polarisation, run independently, are two independent sources, and merging their light into one mode would lower its entropy with nothing paying for it.

The thermodynamic reading can be made quantitative. The radiance of a beam in a single mode corresponds to a temperature: the temperature a black body would need to put the same power into that mode. A kilowatt laser at 1,060 nanometres with a linewidth of a megahertz carries about $5\times10^{15}$ photons per mode, and a black body would need a temperature of order $10^{20}$ kelvin to match it. Two such lasers are two bodies at that temperature, and merging their light into one mode would be making something hotter than either out of the two of them — which is exactly what a heat engine running between two equal temperatures cannot do.

What a beam splitter does with two inputs #

The beam splitter is where the argument is easiest to see in one piece of glass. A half-silvered mirror has two inputs and two outputs, and it is lossless: whatever power goes in comes out, divided between the outputs. Send one beam in and half leaves each way. Send two beams in, one through each input face, aligned so that they overlap on the way out, and each output carries a superposition of both.

What each output receives then depends on the relative phase of the two inputs. The splitter imposes a quarter-cycle difference between reflection and transmission, so at one relative phase the two beams interfere constructively at one output and destructively at the other, and all the power leaves through one port as a single beam with twice the radiance of either input. At the opposite phase it all leaves through the other port. At phases in between it divides. If the two lasers are independent, their relative phase wanders through every value, and averaged over time each output carries half the total — two beams in, two half-strength mixtures out, and no gain in radiance at all. The splitter did not fail. It obeyed energy conservation instantaneously and the étendue bound on average, and the average is all that independent sources give it.

Filled-aperture coherent combining is this, done in stages: a tree of splitters, each combining two beams whose phases a controller holds at the value that sends everything out of one port. With sixty-four lasers, six stages. Every stage must hold its phase, and a small error at any stage leaks a little power out of the unused port, where it is thrown away as heat.

A second label for the same mode #

A mode is not only a spatial pattern. A beam’s light is also labelled by its polarisation and by its frequency, and two beams that differ in one of those are not in the same mode even if they fill the same area in the same direction.

Polarisation gives the first, limited escape. A polarising beam splitter transmits one polarisation and reflects the other, so a beam polarised one way and a beam polarised at right angles can be sent into the splitter from its two input faces and leave as one beam, overlapping perfectly. That doubles the radiance once. It cannot be repeated: there are only two polarisations, and a third beam finds both taken.

Wavelength gives an escape with no such limit. A diffraction grating sends each wavelength in a different direction, and run backwards it can take beams of different wavelengths, each arriving from its own direction, and send them all out along one line.

The figure solves the grating equation for a grating of 1,480 lines per millimetre with every beam leaving at sixty degrees. Each wavelength must arrive at its own angle, a tenth of a degree apart per nanometre of wavelength near 1,050 nanometres, where the fibre lasers used for this operate. In practice the lasers are placed side by side at the focus of a lens, each at the position that sends its light onto the grating at the right angle, and each laser is forced to lase at exactly the wavelength its position requires by an external cavity that runs through the grating itself. A hundred lasers, each a nanometre from its neighbours in wavelength, come out as one beam of the quality of one laser, with a hundred times the power. The dispersion made of angles used the same angular spread of a grating to stretch and compress pulses; here it is used to fold a line of sources into a point.

Nothing in this breaks the étendue bound. The output beam is one spatial mode holding a hundred frequencies, and the étendue per frequency is exactly what each laser had. The radiance summed over all colours has risen a hundredfold because the beams were never copies of one another. The price is spectral: the combined beam spans a hundred nanometres, and anything downstream that cares about colour — a narrowband filter, a nonlinear crystal, an interferometer — sees a hundred separate lasers.

Adding fields instead of powers #

The last escape is to make the beams stop being separate. Two lasers whose phases are locked together — driven by a common seed and amplified in parallel, with each amplifier’s phase continuously corrected — are not two sources but one source with two parts, and their fields add rather than their powers.

The figure sets eight beams side by side and computes the pattern far away. With independent phases the intensities add, and the result is eight times the pattern of one beam, as broad as a single beam’s. With the phases locked, the fields add: on axis the field is eight times one beam’s and the intensity sixty-four times, in a peak whose width is set by the whole array rather than by one beam. The array has become a single aperture eight times wider, and a single mode of that aperture. Why two lamps never interfere found that two independent sources cannot produce steady fringes because their relative phase wanders; locking the phases is exactly what makes these fringes — the central peak and its neighbours — hold still.

The side peaks are where the étendue argument reappears. The eight beams, each covering half its share of the aperture, form a single mode of an aperture with gaps in it, and the fan of plane waves inside every beam found that the far field of any aperture is its content decomposed into directions. An aperture with periodic gaps sends power into periodic directions — a grating’s orders — and the power in the side peaks is lost to the central beam.

The figure plots the share of the power the central peak keeps against the fill factor. It rises almost in proportion, from under half for beams covering half the aperture to ninety per cent for a completely filled one, which is the share any uniformly lit aperture puts in its central lobe. For a two-dimensional array the loss goes roughly as the square of the fill factor, which is why tiled arrays of fibre lasers use lenslets to expand each beam until the tiles nearly touch. The alternative is to combine the beams before they leave, overlapping them with a cascade of beam splitters in which every stage sends both inputs out of one port, possible only because the phases are right: a filled aperture with no gaps, at the cost of a large and precisely aligned optical network.

Held to a twentieth of a wavelength #

Coherent combining asks something severe of the lasers: that their phases stay locked.

If each beam’s phase wanders randomly by $\sigma$ radians, the coherent part of the combined field is reduced by a factor $e^{-\sigma^2/2}$ in amplitude, and for many beams the share of the power left in the combined beam falls as $e^{-\sigma^2}$. The figure plots it. To keep ninety per cent, the phases must agree to about a third of a radian, a twentieth of a wavelength — fifty-five nanometres at the wavelength of a fibre laser. A fibre amplifier a few metres long drifts in phase by whole wavelengths as it warms by a fraction of a degree or vibrates, in milliseconds, so the phase of every beam must be measured and corrected continuously, typically by comparing each with the others and adjusting a phase modulator thousands of times a second. Arrays of dozens of fibre lasers have been locked this way; the difficulty grows with the number of beams and with the power, because a high-power amplifier’s phase noise is larger.

The two scalable methods therefore trade different things. Spectral combining needs no phase control but broadens the spectrum and needs many distinct wavelengths within the band an amplifier can serve. Coherent combining keeps a single wavelength but needs every phase held to a fraction of a wavelength and pays for any gaps. High-power systems increasingly use both — spectral combining of coherently combined groups.

Beams that are several modes already #

Most high-power lasers are not single-mode to begin with, and the same accounting applies to them in a form engineers use every day. A beam’s quality is quoted as a number $M^2$, the factor by which its divergence exceeds that of a perfect Gaussian beam of the same waist. A beam with $M^2 = 1$ fills one mode; one with $M^2 = 10$ in both directions has an étendue a hundred times larger and behaves, for this purpose, as a hundred modes.

That number, not the power, is what decides how small a spot the beam can be focused into at a given working distance, and so how narrow a cut it can make or how far it can carry its energy. The invariant that is a count found the count of modes to be the étendue divided by the wavelength squared, and $M^4$ is exactly that count for a beam. A kilowatt diode laser bar, whose many emitters are independent, may have an $M^2$ of hundreds in one direction; it is far more powerful than a single-mode fibre laser and far less bright. Much of the engineering of high-power lasers is therefore spent not on generating power but on keeping the mode count down while the power goes up — and when two or more lasers are to feed one beam, on making sure that the beams added are not copies.

Where it is used #

The largest proposed use is the rocket that leaves its fuel at home: a light sail pushed by a beam from the ground, where the distance over which the beam can deliver its energy is set by diffraction from the transmitting aperture. The only way to build a transmitting aperture kilometres across is as a phased array of many lasers, and the only way to make the array’s beam narrow is to lock the phases; the requirement to hold the phases of a hundred million lasers through the turbulent atmosphere is one of the reasons the proposal is regarded as far beyond present technology. Laser weapons use spectral and coherent combining of fibre lasers to reach a hundred kilowatts or more. Industrial cutting mostly does not bother: a cutting head a few centimetres from the metal does not need a narrow beam over kilometres, so several lasers are simply coupled into one fibre, accepting the loss of radiance the bound demands.

What the drawings cannot show #

The figures idealise the beams as identical, single-mode and perfectly collimated, and the arrays as one-dimensional rows. Real fibre lasers are close to single-mode but not exactly, and their small departures from a perfect Gaussian beam add a loss that no phase control recovers. Two-dimensional arrays, hexagonal or square, have patterns that the one-dimensional arithmetic only approximates. The phase-error curve assumes independent Gaussian errors with no correlation between beams; real errors are often correlated through shared mounts and temperatures, which helps, and have slow drifts that a control loop removes and fast ones it does not. And the grating figure ignores the finite width of each laser’s spectrum and the grating’s own efficiency, which is typically above ninety-five per cent and must be for the method to be worth using.

Still open: how many beams and how much power #

Spectral combining is limited by how many lasers of distinct, narrow, stable wavelengths can be packed into the band an amplifier can amplify, and by the heating of the grating, which distorts its surface and with it the combined beam. Coherent combining is limited by the phase control and by the nonlinear effects that grow in fibres at high power and broaden the spectrum, which makes phase-locking harder. Where the practical ceiling lies — a megawatt in a beam of near-perfect quality, from how many lasers, combined how — is being pursued in parallel by several approaches, and whether hybrid schemes can scale without the losses of each method multiplying is not settled.

The habit worth carrying away is to ask, before combining anything, in what respect the parts differ. Light’s brightness is conserved per mode, so identical beams can only be placed side by side; beams that differ in polarisation, in colour, or that have been locked in phase into one mode, can be added — and each escape is paid for, in the number of labels available, in spectral width, or in phases held to a twentieth of a wavelength with gaps that leak. A conservation law that forbids merging copies is an instruction to stop making copies.