Many of the green lasers that pump titanium-sapphire (Ti:sapphire) oscillators are intracavity-doubled neodymium lasers running on a few hundred longitudinal modes. Their output holds steady to a few hundredths of a percent. Look more closely at this output, though, and you’ll find the modes blinking on and off, switching in bursts tens of microseconds long. The model says every one of the 200 or so modes in the beam is doing this, but the beam is almost perfectly quiet. How can this be?

I have a personal stake in the question. At Spectra-Physics during the early 1980s, we built miniature diode-pumped neodymium lasers with a nonlinear doubling crystal inside the cavity, aiming to replace the inefficient argon-ion lasers then used for everything from printing to spectroscopy. The compact diode-pumped solid-state lasers (DPSSLs) were efficient and very small, but to my surprise they were extremely noisy, with intensity fluctuations of tens of percent, and sometimes chaotic. In 1986, I published the theory of why these compact intracavity-doubled lasers are noisy. My rate-equation model showed that the crystal couples the laser’s longitudinal modes through sum-frequency generation, and with two or three modes this coupling makes them fight—modes switch on and off in turn, irregularly when there are three. It became the accepted description of the “green problem.”

Years later, we found a cure that made no sense. Make the laser longer so it runs on many more modes—and the total noise almost disappears, while the individual modes become even noisier. My 1986 treatment, and the many-mode versions others built on it, didn’t explain what we saw. Although the physics was puzzling, the laser worked beautifully. My colleagues Bill Nighan and John Cole at Spectra-Physics designed the long-cavity Millennia laser system and the company sold tens of thousands of them. These laser systems have pumped Ti:sapphire oscillators and frequency combs worldwide ever since.

The answer to this puzzle, published this year in Physical Review A,¹ isn’t one anyone expected.

What the measurement reveals

The laser is familiar: A 42-cm cavity holds a neodymium vanadate gain element, which is pumped by a diode laser, and a lithium triborate (LBO) nonlinear crystal near an end mirror reflects both the infrared (IR) and the green so the green beam is double-passed through the nonlinear crystal on its way out. The gain bandwidth lets about 200 longitudinal modes oscillate at once. Every intracavity-doubled laser of this type has these parts.

A clue was hidden inside a measurement anyone with a Fabry-Pérot etalon can repeat. Take such a laser running on its usual 200 or so longitudinal modes and measure the intensity noise of the full green output—the specification is 0.05% root mean square (rms) from direct current to 10 MHz. Pass the same beam through an etalon that transmits only a few of these modes and look at what comes through on an identical detector. The transmitted modes are blinking on and off, each one modulated essentially 100%, in bursts tens of microseconds long, while their sum stays steady. Every mode is noisy and the beam is quiet.

Inserting the intracavity crystal makes the individual mode noise much worse, but the total intensity is far better than any simple addition of the individual mode intensities can produce. What is happening?

Why the 1986 model couldn’t see it

The 1986 equations, and the many papers built on them, track the intensity of each mode—but they discard the optical phase. It was the right simplification for two or three modes, where the phases turn out not to matter, and it kept the problem small enough for the computers of the time.

Cancellation depends entirely on the phases. Two hundred modes with random phases beat noisily against one another and produce an intensity that fluctuates on a nanosecond timescale within every round trip. But the same 200 modes with the right phase arrangement can produce an intensity constant in time. Which arrangement the laser adopts can’t be asked of a model that has thrown the phases away.

What the crystal does to the phases

My new model follows the amplitude and phase of every mode, several hundred of them, through the cavity one round trip at a time—through the gain medium with its spatial hole burning, through the nonlinear crystal twice, and back. The workstation that runs it is about 100,000x faster than the computers of 1986, and it made the 200-mode, phase-resolved model solvable numerically.

What the model shows can be stated without equations. The crystal isn’t a one-way converter. Green generated from one pair of IR modes is converted back into other IR modes, and the returned light carries the phase of the pair that produced it. This is the lever, and through it the crystal adjusts the phase of every mode on every pass—the adjustment isn’t random. A single quantity, the mean of the intensity squared divided by the square of the mean, which is one for a constant intensity and larger for anything bunched, can’t increase on a forward pass through the crystal; it’s a theorem, which my paper proves. The phase changes move the modes toward the arrangement that makes the total intensity constant, because this arrangement loses the least IR to the green. And the crystal reaches it within a fraction of a microsecond—far faster than the gain medium can respond.

The gain medium doesn’t care about the phases. Changing them costs the laser nothing in extracted power. So the phases are free to move and the nonlinear crystal moves them to create a nearly constant total intensity, while the division of that power among the modes is only weakly constrained and can drift. The modes trade power back and forth on the gain-recovery timescale, each one switching on and off. Two things keep the sum steady while this happens: The crystal’s coupling ties the modes’ gains and losses together so power leaving one mode is taken up across the rest of the set, not by any single partner, and the exchanges are collectively nearly zero-sum. And, on a much faster timescale, the phases are rearranged so the new set of mode powers still adds to a smooth waveform. A detector that sees the whole beam sees only the steady sum. The result? Loud, noisy modes, quiet total.

The same coupling that makes two modes fight makes 200 cooperate. With two modes the phases have no arrangement available that removes the beat, and the crystal’s coupling drives the competition. With hundreds, the arrangement exists, the crystal finds it, and the competition becomes invisible in the output.

What it means for the laser on the bench

Three things follow for anyone who builds or uses these lasers: First, a quiet total is not a sign of quiet modes. If an application filters the beam spectrally or relies on a subset of modes, it will see noise that the full beam doesn’t show.

Second, the number of modes is the design parameter and the nonlinear crystal is the stabilizer. The empirical cure of the 1990s, a longer cavity, moves the laser away from the few-mode regime where the coupling drives competition and into the many-mode regime where it drives cancellation. It was the right cure, for a reason that hadn’t been recognized.

Third, the crystal is doing more than doubling. It converts the harmonic back to the fundamental, couples every mode to every other, and moves their phases. A model that treats it only as a nonlinear loss to be minimized misses all of this, and it’s the part that determines whether the laser is quiet.

A new kind of phase-organized laser

Lasers whose modes hold a fixed phase relation are familiar, and it’s worth pointing out how this one differs from them. In a modelocked laser, the modes are locked in phase so that they add up to a short pulse once per round trip; the intensity is as far from constant as it can be.

The frequency-modulated laser of the 1960s is another way to approach a constant-intensity state—an intracavity phase modulator, driven at the cavity’s round-trip frequency, arranges the modes into a frequency-modulated field whose intensity would be constant if the arrangement were complete. It never quite is, because the limited bandwidth of the gain medium cuts off the outer modes the arrangement needs, and the residue appears as amplitude modulation. The arrangement is also imposed from outside, and the modulator fixes the mode amplitudes.

The closest modern relative is the quantum cascade laser, which can organize its own mode phases into a frequency-modulated comb with little intensity modulation and no modulator; the gain responds almost instantly and the mode amplitudes stay fixed. The laser described here is different from all three—no element in the cavity is driven. A nonlinear crystal, put there to make green, organizes the phases on its own, toward the arrangement that makes the intensity nearly constant rather than pulsed, and it does so while the mode amplitudes are free to wander. Modes switch on and off, the set of modes that carry the power changes on a time scale of tens of microseconds, and the phase arrangement is rebuilt each time within a fraction of a microsecond. It’s a passively phase-organized laser with a nearly constant intensity and a drifting spectrum, which is a combination the other phase-organized lasers don’t have, and it has been in commercial production for 30 years without being recognized as such.

A more general idea

Theoretical models of lasers often serve as the starting point for analyzing other systems of coupled oscillators. Hermann Haken built his theory of self-organization on the laser as the paradigm; the equations of coupled laser arrays and of arrays of Josephson junctions were shown to map onto each other, and both onto Yoshiki Kuramoto’s description of synchronization that’s now applied to everything from power grids to populations of neurons.

The result here depends only on the structure of the coupling, many oscillators sharing a quadratic nonlinearity through their summed field, and nothing in it is specific to neodymium or to lithium triborate. It will be interesting to see whether such related systems can be controlled in the same way by introducing a nonlinear element that couples the oscillators’ losses and phases so the parts stay noisy while the sum goes quiet. For 40 years the green problem was a problem that turns out to have been half of a mechanism, while the other half sat quietly within laboratories all along.

Forty years

I always find it rewarding when a mathematical model I’m working on produces results that fit experimental data that has puzzled me. Forty years ago, I was surprised by the noisy green emission from the compact DPSSLs I’d hoped would become a new product line for Spectra-Physics. I was very pleased when—after spending a few months coding my HP 9816 lab computer, with its 8-MHz processor, 128 KB of memory, and a fraction of a megaflop of computing speed—several hours of processor time generated traces that matched the green output of those few-mode lasers and explained why it was happening.

Forty years later, my workstation has 128 GB of memory, a processor running at 5 GHz on 48 threads, and several teraflops of computing speed. It powers through models with hundreds of modes, solving coupled differential equations with thousands of terms within a few minutes, and once again it has explained oscilloscope traces that at first seemed to me very bizarre.

I built my first lasers in 1974 and, even today, when I see an intense and quiet high-power green beam propagating across the lab, it seems like magic to me. The beauty of these human-made, collimated, coherent sources of light, and of the mathematics that predicts and explains how the magic occurs, still astounds me.

REFERENCE

1. T. M. Baer, Phys. Rev. A, 114, 033503 (2026), https://doi.org/10.1103/gf5b-vn5p; and T. Baer, J. Opt. Soc. Am. B, 3, 1175 (1986).