In 1992, Grudinin, Richardson and Payne reported the first experimental observation of discrete, quantised jumps in the average output power of an all-fibre passively mode-locked figure-eight erbium fibre laser: as the pump power was reduced, the number of circulating soliton pulses dropped by integers, each pulse carrying nearly fixed energy (about 12 pJ), and pulse disappearance was identified as the mechanism of the jumps. The paper left three linked aspects unresolved: the exact process selecting which specific pulse (or group of pulses) is removed, the origin of an approximately 5% pulse-to-pulse soliton energy variation observed between jumps, and the challenge of controlling the pulse pattern in a predetermined manner to generate high-bit-rate soliton sequences. Later literature has largely settled the gross mechanism — the soliton energy quantum is fixed by the cavity energy balance and by gain competition among coexisting solitons, with pulse-number changes described as noise-triggered statistical events — and has substantially advanced deterministic soliton-number control. The open core is a quantitative, validated account of the few-percent pulse-to-pulse energy spread among coexisting solitons in a nonlinear-amplifying-loop-mirror cavity and of the selection rule governing which soliton (or soliton group) is annihilated at a pump-power step, together with a determination of whether, and by what rule, the pulse pattern can be commanded in a predetermined manner for high-repetition-rate soliton sequences.

Background

A mode-locked fibre laser emits a periodic train of short pulses; in passive mode locking no external modulation is applied, and pulse shaping is instead provided by a nonlinear element acting as an artificial saturable absorber — one that transmits intense pulses better than weak ones or continuous background light. In the figure-eight laser, two fibre loops are joined in a figure-eight shape: one loop contains an amplifier, and the whole nonlinear amplified loop mirror (NALM) — a balanced interferometer whose transmission depends on pulse intensity through the nonlinear phase shift accumulated in the amplified loop — provides the intensity-dependent loss that starts and sustains mode locking. Erbium-doped fibre amplifiers pumped at 980 nm provide the gain near 1550 nm.

In the anomalous-dispersion regime of standard fibre at 1550 nm, an optical pulse can propagate as a soliton: a pulse whose Kerr nonlinearity exactly balances chromatic dispersion, so that its shape is preserved. A fundamental soliton (soliton order N = 1) has a specific energy fixed by its duration and the fibre dispersion; higher-order solitons (N > 1) breathe periodically as they propagate. A soliton circulating in a laser cavity is not the ideal lossless soliton but a dissipative soliton: a pulse whose energy, continuously lost at cavity components and amplified in the gain fibre, reaches a stationary balance of gain, loss, dispersion and nonlinearity, so that its energy is pinned by the cavity operating point rather than freely chosen. This is the physical root of energy quantisation: when the cavity supports many coexisting solitons per round trip, each carries nearly the same energy quantum, and the average output power becomes a staircase in pump power, changing by one quantum whenever the number of solitons changes.

Grudinin, Richardson and Payne (Electronics Letters, 1992) measured this staircase in an all-fibre figure-eight erbium laser (450 fs solitons near 1558 nm, NALM loop of 35 m, 5 MHz round-trip frequency). From the size of an output-power step and the round-trip rate they inferred an energy of about 12 pJ per soliton, corresponding to a soliton order between 1.0 and 1.4, i.e. nearly fundamental solitons. Time-resolved measurements showed the pulse number dropping by integers (for example from 6 to 5) at each power discontinuity, sometimes with whole groups of pulses vanishing together, and spectral measurements showed the nonlinear amplifying loop mirror rejecting the non-solitonic content of the disappearing pulse. In the low-pump regime the pulses arranged themselves into ordered, motionless patterns, whereas at higher pump powers the pulse pattern was chaotic.

Several mechanisms for how coexisting solitons interact and how the laser decides its soliton number were established later. Gain competition means that all solitons in the cavity share one gain reservoir: the amplified spontaneous emission and the solitons deplete the same inversion, so a soliton with slightly more energy or a more favourable position can drive others below the lasing threshold. Komarov, Leblond and Sanchez (Physical Review A, 2005) showed theoretically that passively mode-locked lasers possess multiple stable pulse-number states with hysteresis between them. Tang, Zhao, Zhao and Liu (Physical Review A, 2005) showed numerically that soliton energy quantisation is a natural consequence of gain competition between multiple solitons, together with the peak-power-clamping effect of the cavity, and conjectured that the same mechanism applies to other soliton fibre lasers including the figure-eight type. Vodonos and co-workers (Physical Review Letters, 2004; Physical Review E, 2007) developed a statistical-mechanics description in which the abrupt creation or annihilation of pulses appears as cascaded first-order phase transitions of a pulse 'gas', each pulse carrying an almost constant energy quantum.

Two further threads matter for the third unresolved aspect. Harmonic mode locking is the operation of a mode-locked laser at a repetition rate that is an integer multiple of the fundamental round-trip frequency, so that many equally spaced pulses fill the cavity; it can be forced actively or arise passively through pulse-pulse interactions (for example soliton bunching driven by gain depletion and recovery), and it is the natural route to high-bit-rate soliton sequences. Related multi-soliton phenomena include bound states or soliton molecules — groups of solitons locked at fixed separations and relative phases, reviewed recently by Cui, Lu, Zhang, Mao and Malomed (Physics Reports, 2026) — whose quantised separations and binding energies (Komarov, Komarov and Sanchez, Physical Review A, 2009) are conceptually adjacent to, but distinct from, the quantisation of individual soliton energy.

Despite this mature framework, the 1992 discussion section's questions have only been answered in part. The field treats the identity of the soliton that dies at a pump-power step as intrinsically stochastic (Kbashi and Sergeyev, 2019, call soliton quantisation 'intrinsic and random'), no published account explains the magnitude of the few-percent pulse-to-pulse energy spread among coexisting solitons in the nonlinear-amplifying-loop-mirror cavity, and deterministic pattern control has been achieved mainly by adding hardware (a second independently pumped amplifier, intracavity spectral filtering) rather than by a predictive rule for the original single-pump figure-eight design.

Problem Statement

In an all-fibre passively mode-locked figure-eight erbium fibre laser whose solitons carry a nearly fixed energy quantum (about 12 pJ in the 1992 experiment) and whose average output power therefore changes in discrete steps as the pump power is varied — each step corresponding to the disappearance of one or a group of the circulating soliton pulses — the following linked aspects, raised by the original observation and not closed by subsequent literature, remain to be resolved: (1) identify, quantitatively and with validation against the observations reported for this cavity, the physical process that determines which specific pulse, or group of pulses, is removed when the pump power is reduced, including the role of the small (of order a few percent) pulse-to-pulse energy variations possibly associated with differences in polarisation states or central frequencies, and including why entire groups of pulses can vanish together; (2) explain the origin and magnitude of the approximately 5% variation in soliton pulse energy between jumps observed at the nonlinear-amplifying-loop-mirror transmission port; and (3) determine whether, and by what rule, the pulse pattern can be controlled in a predetermined manner — a predictive rule for the soliton number and arrangement as a function of cavity parameters — so as to generate high-bit-rate soliton sequences, or else establish why such predetermined control is not attainable. Together these aspects constitute one scientific objective: a predictive, validated account of how a passively mode-locked figure-eight laser with a given pump power distributes energy among, and selects among, its coexisting dissipative solitons, and of how that distribution can be commanded, with the inter-soliton energy spread demanded by part (2) developed as the quantitative prerequisite within the account demanded by part (1). Later literature has already established the gross mechanism (the soliton energy quantum is set by the cavity energy balance and by gain competition among coexisting solitons, with pulse-number changes occurring as noise-triggered events describable statistically), so the open target is the quantitative residual: the inter-soliton energy spread, the selection rule for annihilation events in the nonlinear-amplifying-loop-mirror cavity, and predetermined pattern control.

References

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