Phase-locking, quenching, and dynamics of multiple FM oscillations in intracavity phase-modulated lasers: unresolved problems from the 1965 theory of FM laser oscillation

The 1965 theory of FM laser oscillation by Harris and McDuff explains how an intracavity phase modulator driven near the cavity axial-mode spacing converts a multimode laser into a single frequency-modulated (FM) oscillation whose sidebands have near-Bessel amplitudes, and predicts thresholds, output power, frequency pulling, distortion, and supermode conversion. The authors themselves identified what the theory leaves unresolved: whether and under what conditions several coexisting FM oscillations phase-lock instead of one quenching the rest — a regime in which the paper's reduced steady-state amplitude (a-) equations, including their factor-of-two cross-saturation result, demonstrably fail — together with the transient dynamics of such multi-oscillation states, the convergence domain of the iterative spectral method in the strongly distorted regime, and quantitative experimental validation of the predictions across parameter space. No existing treatment closes this question for the inhomogeneously broadened, intracavity-modulated lasers of the source. The surviving open core is a predictive theory of competition, phase-locking, and quenching among multiple FM oscillations, with the corresponding thresholds, cross-saturation, transient dynamics, method convergence limits, and experimental validation.

Background

In a multimode laser, many axial (longitudinal) cavity modes can oscillate simultaneously; in a medium with inhomogeneous broadening, such as the Doppler-broadened lines of a gas laser, different modes draw their gain from essentially independent velocity groups of inverted atoms, so the free-running multimode output is spectrally and temporally incoherent. A frequency-modulated (FM) laser is a laser containing an electro-optic phase modulator (a crystal whose refractive index is varied by an applied radio-frequency voltage) driven at a frequency $\nu_m$ close to the spacing $\Delta\Omega$ between adjacent axial modes; the time-varying refractive index parametrically couples each axial mode to its neighbors, so the cavity prefers to oscillate in a comb of modes whose amplitudes follow Bessel-function weights and whose phases are those of a sinusoidally frequency-modulated carrier. The optical output then has nearly constant intensity and a swept instantaneous frequency, in contrast to the short pulses of conventional mode locking. FM laser oscillation was demonstrated by Harris and Targ in 1964 and given a quantitative theory by Harris and McDuff (1964, 1965).

The 1965 theory starts from Lamb-type multimode self-consistency equations (the framework of Lamb's gas-laser theory, in which the field amplitudes are solved self-consistently together with the polarization each mode induces in the gain medium): each axial mode $n$ has amplitude $E_n$ and phase $\varphi_n$, cavity loss $\alpha_n$ and unsaturated gain $g_n$ from an atomic medium of arbitrary lineshape, and adjacent modes are coupled with coefficient $\delta$ (set by the strength of the phase perturbation), with detuning $\Delta\nu=\Delta\Omega-\nu_m$. In the linear (unsaturated) limit the normal modes are FM oscillations (also called supermodes): for each integer carrier index $q$ the amplitudes are $E_n=J_{n-q}(\Gamma)$, where $J$ is a Bessel function and $\Gamma=(c/L)\,\delta/\Delta\nu$ is the modulation depth ($c$ the speed of light, $L$ the cavity length), with zero relative phases and a uniform frequency shift. Because all supermodes overlap the same inverted population, they compete: the reduced steady-state amplitude equations for the supermode amplitudes $a_q$ (the a-equations) predict that the strongest FM oscillation quenches (suppresses) the others, and yield the notable result that one oscillation cross-saturates another twice as strongly as it saturates itself. Numerical integration of the full equations mapped the operating regimes versus detuning: a phase-locked pulsing state, an extinction region, and the steady-FM region; an iterative spectral procedure (based on an exact Green's-dyadic — the resolvent kernel of the coupled-mode operator — and a spectral inverse with Bessel eigenvectors) computes the distorted steady-state amplitudes and phases including saturation and mode pulling (the shift of the oscillation frequencies by the dispersive part of the atomic susceptibility), converging well for moderate distortion but slowly or not at all at large $\Gamma$ and small $\delta$. The theory also analyzed supermode conversion — recovering a single-frequency output from the FM comb with an external, counter-phased modulator, with near-unit predicted conversion efficiency — and showed that odd-harmonic intermode beats are sharply sensitive to displacement of the oscillation from the atomic line center, proposing this odd-beat signal as a frequency discriminator for stabilizing the FM laser.

The same numerical work, however, exhibited the theory's central unresolved point: for a nine-mode Doppler-broadened example, the a-equations predict a single surviving FM oscillation for $\Gamma\ge 2.3$, while the full equations show persistent multiple oscillations locked in definite phase relations. The 1965 paper itself flagged the consequences: phase-locking among FM oscillations invalidates the a-equation reduction and alters quenching thresholds and effective cross-saturation; the a-equations are only a steady-state stability test and do not describe transient dynamics; and the companion experiments of Ammann, McMurtry, and Oshman (1965) checked the theory only qualitatively. In the phase-locked (pulsed) region, Hong and Whinnery (1969) later showed that two distinct phase-locked states exist in the phase-modulated He-Ne laser and stated that it was not possible to predict which state would oscillate; Kuizenga and Siegman (1970) supplied a complete mode-locking theory only for homogeneously broadened lasers; and the modern theory of frequency-modulated combs in fast-gain semiconductor lasers (Opačak and Schwarz 2019 and successors) explains phase locking in a different gain-dynamics regime. What is still missing is a closed account of the multimode FM-laser dynamics in the regime the 1965 theory left open.

Problem Statement

Complete the theory of FM laser oscillation for intracavity phase-modulated, inhomogeneously broadened lasers in the regime where more than one FM oscillation (supermode) participates, as left open by Harris and McDuff (1965). The common objective is a predictive, experimentally validated account of multimode FM-laser operation, with the following jointly required results: (1) Determine the conditions — in the adjacent-mode coupling coefficient $\delta$, detuning $\Delta\nu$, modulation depth $\Gamma$, unsaturated gain, loss, and atomic lineshape — under which coexisting FM oscillations phase-lock into persistent multioscillation states with definite inter-oscillation phase relations, as observed in the full equations (e.g., for the nine-mode Doppler-broadened example at $\Gamma\ge 2.3$), versus the strongest oscillation quenching the others; and establish how phase-locking modifies the quenching thresholds and the effective cross-saturation between oscillations, thereby fixing the validity domain of the reduced steady-state a-equations and their factor-of-two cross-saturation prediction. (2) Construct the corresponding dynamical theory of transients — the approach to steady single-oscillation, phase-locked multioscillation, or persistent nonsteady states when multiple oscillations interact — which the a-equations, as a steady-state stability test, do not describe. (3) Characterize the domain of convergence (or failure) of the iterative spectral method for computing distorted steady-state amplitudes and phases, particularly in the strongly distorted regime of large $\Gamma$ and small $\delta$. (4) Provide quantitative experimental validation across parameter space of the predicted operating-regime maps, oscillation thresholds, frequency-pulling magnitudes, supermode conversion efficiencies, and the odd-beat frequency discriminator proposed for center-frequency stabilization. Results (2)–(4) all concern the same multi-oscillation regime whose steady-state boundary is fixed by (1).

References

1. S. E. Harris and O. P. McDuff, 'Theory of FM laser oscillation', IEEE Journal of Quantum Electronics 1(6), 245–262 (1965). DOI: 10.1109/jqe.1965.1072231, https://doi.org/10.1109/jqe.1965.1072231
2. S. E. Harris and O. P. McDuff, 'FM laser oscillation — theory', Applied Physics Letters 5(10), 205–206 (1964). DOI: 10.1063/1.1723589, https://doi.org/10.1063/1.1723589
3. S. E. Harris and R. Targ, 'FM oscillation of the He-Ne laser', Applied Physics Letters 5(10), 202–204 (1964). DOI: 10.1063/1.1723588, https://doi.org/10.1063/1.1723588
4. E. O. Ammann, B. J. McMurtry and M. K. Oshman, 'Detailed experiments on helium-neon FM lasers', IEEE Journal of Quantum Electronics 1(6), 263–272 (1965). DOI: 10.1109/JQE.1965.1072232, https://doi.org/10.1109/JQE.1965.1072232
5. S. E. Harris, M. K. Oshman, B. J. McMurtry and E. O. Ammann, 'Proposed frequency stabilization of the FM laser', Applied Physics Letters 7(7), 185–187 (1965). DOI: 10.1063/1.1754370, https://doi.org/10.1063/1.1754370
6. S. E. Harris, 'Stabilization and modulation of laser oscillators by internal time-varying perturbation', Applied Optics 5(10), 1639 (1966). DOI: 10.1364/AO.5.001639, https://doi.org/10.1364/AO.5.001639
7. L. M. Osterink, R. L. Byer and S. E. Harris, 'The internally phase-modulated laser with frequency selective output coupling', IEEE Journal of Quantum Electronics 2(4), 126 (1966). DOI: 10.1109/JQE.1966.1073845, https://doi.org/10.1109/JQE.1966.1073845
8. R. Targ and L. M. Osterink, 'A frequency-stabilized FM laser', IEEE Journal of Quantum Electronics 3(6), 270 (1967). DOI: 10.1109/JQE.1967.1074533, https://doi.org/10.1109/JQE.1967.1074533
9. G. Hong and J. R. Whinnery, 'Switching of phase-locked states in the intracavity phase-modulated He-Ne laser', IEEE Journal of Quantum Electronics 5(7), 367–376 (1969). DOI: 10.1109/JQE.1969.1076273, https://doi.org/10.1109/JQE.1969.1076273
10. D. L. Lyon and T. S. Kinsel, 'Transitions between mode-locked states in intracavity phase-modulated lasers', Applied Physics Letters 16(3), 89–91 (1970). DOI: 10.1063/1.1653132, https://doi.org/10.1063/1.1653132
11. D. J. Kuizenga and A. E. Siegman, 'FM and AM mode locking of the homogeneous laser — Part I: Theory', IEEE Journal of Quantum Electronics 6(11), 694–708 (1970). DOI: 10.1109/JQE.1970.1076343, https://doi.org/10.1109/JQE.1970.1076343
12. S. Longhi and P. Laporta, 'Time-domain analysis of frequency modulation laser oscillation', Applied Physics Letters 73(6), 720–722 (1998). DOI: 10.1063/1.121979, https://doi.org/10.1063/1.121979
13. Y.-F. Li, P. R. Herczfeld and L. M. Narducci, 'Modelocking and multimode instability in laser intracavity frequency modulation', IEEE Journal of Quantum Electronics 42(6), 525–537 (2006). DOI: 10.1109/JQE.2006.874062, https://doi.org/10.1109/JQE.2006.874062
14. N. Opačak and B. Schwarz, 'Theory of frequency-modulated combs in lasers with spatial hole burning, dispersion, and Kerr nonlinearity', Physical Review Letters 123, 243902 (2019). DOI: 10.1103/PhysRevLett.123.243902, https://doi.org/10.1103/PhysRevLett.123.243902 (preprint: https://arxiv.org/abs/1905.13635)
15. I. Heckelmann, M. Bertrand, A. Dikopoltsev, M. Beck, G. Scalari and J. Faist, 'Quantum walk comb in a fast gain laser', Science 382(6669), 434–438 (2023). DOI: 10.1126/science.adj3858, https://doi.org/10.1126/science.adj3858
16. M. Roy, T. Zeng and D. Burghoff, 'Self-frequency-modulated laser combs', Applied Physics Letters 125(7), 070503 (2024). DOI: 10.1063/5.0215583, https://doi.org/10.1063/5.0215583
17. M. Cuenca, H. Maestre and C. R. Fernández-Pousa, 'Modal theory of phase-modulated and frequency-shifting ring cavities', Advanced Photonics Research 6(12), 2500098 (2025). DOI: 10.1002/adpr.202500098, https://doi.org/10.1002/adpr.202500098
18. S. E. Harris and B. Buscaino, 'Technique for generating broadband FM light', Optics Letters 45(7), 2058 (2020). DOI: 10.1364/ol.383651, https://doi.org/10.1364/ol.383651
19. Correction to 'Theory of FM laser oscillation', IEEE Journal of Quantum Electronics 2(2), 49 (1966). DOI: 10.1109/jqe.1966.1073771, https://doi.org/10.1109/jqe.1966.1073771 (one-page erratum to the source paper)