Synchronously Pumped Dispersive Cavity

• Synchronously pumped dispersive cavity is an optical resonator that matches pump pulse repetition to its free spectral range while exploiting dispersion and nonlinearity.
• The intracavity dynamics exhibit symmetry breaking, multistability, and controlled frequency comb generation through engineered group-velocity dispersion.
• Experimental designs using fiber rings, microrings, and precise dispersion management enable ultrafast metrology and quantum mode generation.

A synchronously pumped dispersive cavity is an optical resonator, typically realized as a fiber ring or microresonator, in which a train of coherent pump pulses is injected at a repetition rate matched to the cavity’s round-trip frequency. The combination of optical dispersion and nonlinear effects within the cavity—most prominently group-velocity dispersion (GVD) and the optical Kerr or parametric nonlinearity—leads to complex multimode dynamics, frequency comb formation, spontaneous symmetry breaking, and dispersive tuning effects central to ultrafast photonics, frequency metrology, and quantum optics.

1. Fundamental Physical Principles

A synchronously pumped dispersive cavity is defined by two chief features: synchronization of the external pulsed pump to the cavity free-spectral-range (FSR), and the presence of dispersion in the intracavity medium. Synchronicity ensures that each circulating optical pulse constructively interferes with the incoming pump at every round-trip, maximizing intracavity field buildup. The dispersive medium (fiber, waveguide, or parametric crystal) introduces a frequency-dependent phase evolution characterized by a Taylor expansion of the propagation constant $k(\omega)$ about the carrier frequency: $$k(\omega) = k_0 + k'_0 (\omega-\omega_0) + \frac{1}{2}k''_0 (\omega-\omega_0)^2 + \cdots$$ Here, $k''_0$ quantifies the group-velocity dispersion (GVD), which leads to pulse broadening, spectral recoil, and detuning-dependent phase-matching. The system dynamics are further shaped by nonlinear effects: cubic (Kerr) in $\chi^{(3)}$ media, or quadratic (parametric) in $\chi^{(2)}$ systems. The mean intracavity field evolves according to a generalized, dispersive Lugiato–Lefever equation (LLE) or via discrete round-trip (Ikeda) maps (Xu et al., 2014, Schmidberger et al., 2013, Xu et al., 2020).

2. Mathematical Models and Analytical Framework

The standard modeling approach involves mean-field equations derived by averaging pulse propagation and boundary conditions over one round-trip. For a dissipative Kerr cavity under synchronous pulsed pumping, the normalized mean-field LLE is

$$\frac{\partial E}{\partial z} = [-1 + i(|E|^2 - \Delta) - i \eta\,\partial_\tau^2] E + S(\tau)$$

with $\tau$ the fast-time variable, $z$ the slow-time normalized to the photon lifetime, $\Delta$ the normalized detuning, $\eta = \mathrm{sign}(\beta_2)$ the GVD sign, and $S(\tau)$ the normalized pulsed pump envelope. Higher-order dispersion (e.g., third-order, $\beta_3$) is incorporated as $d_3\,\partial_\tau^3 E$ corrections.

For multimode quantum and frequency comb phenomena, mode expansions in Hermite–Gaussian bases are used, and the steady-state or dynamic equations are projected onto these temporal modes (Tikhonov et al., 3 Dec 2025, Averchenko et al., 26 Jul 2024). The spectral-domain steady-state solution for the field after many round trips is (Tikhonov et al., 3 Dec 2025): $$E_{\rm cav}(\omega) = \frac{\sqrt{1-\mathcal R}~E_{\rm in}(\omega)}{1 - \sqrt{\mathcal R}\,e^{i k(\omega)L}}$$ where $\mathcal R$ is the mirror power reflectivity, $L$ is the cavity length, and $E_{\rm in}(\omega)$ is the input pulse spectrum.

Key parameters are:

• Round-trip time $T_R = L/v_g$
• FSR ($1/T_R$)
• Cavity finesse $\mathcal{F} = \pi/\alpha$, with $\alpha$ total round-trip power loss
• Normalized desynchronization $\varepsilon = (\Delta\tau)/T_R$ (pump-cavity mismatch)
• GVD parameter $D_2$ in Heisenberg–Langevin or master equations for quantum fields.

3. Symmetry Breaking, Bifurcations, and Multistability

Synchronization and dispersion in the presence of nonlinearity give rise to multiple regimes of spontaneous symmetry breaking. For Kerr resonators with anomalous GVD ($\eta < 0$), increasing pump power beyond a first threshold induces a pitchfork bifurcation: the intracavity pulse shifts and becomes asymmetric in time, breaking $t\to -t$ symmetry despite a symmetric pump (Xu et al., 2014, Schmidberger et al., 2013, Copie et al., 2018). At a second, higher threshold, symmetry is restored as asymmetric solutions merge back to the symmetric branch.

In normalized LLE variables, the bifurcation scenario is:

• Below $X_{\mathrm{th},1}$: symmetric steady state
• $X_{\mathrm{th},1} < X < X_{\mathrm{th},2}$: two mirror-antisymmetric solutions coexist (pulse shifts forward or backward)
• Above $X_{\mathrm{th},2}$: only symmetric solution exists

Bistability and coexistence of symmetric/asymmetric states are observed over broad parameter windows, leading to generalized multistability in the cavity dynamics. Hysteresis and subcriticality are present depending on the detuning and pulse duration (Xu et al., 2014, Schmidberger et al., 2013). Higher-order dispersion weakly breaks the pitchfork degeneracy, favoring one handedness and smoothing the bifurcation onset.

4. Frequency Comb Generation and Dispersive Tuning

Synchronous pulsed pumping, combined with dispersion, enables frequency comb generation without the need for avoided mode crossings, even in normal-dispersion microresonators (Xu et al., 2020, Cheng et al., 2023). The key mechanism is the phase-matching of dispersive-wave (DW) sidebands, governed by

$$\Delta\phi(\Delta\omega) = \frac{\beta_2 L}{2}\Delta\omega^2 - \Delta\tau\,\Delta\omega + 2 \gamma L P_0 - \delta_0 = 0$$

for modulation-induced switching waves. The center frequency of the comb can be tuned by varying the pump-cavity desynchronization $\Delta\tau$: $\omega_c = \omega_p + \frac{\Delta\tau}{\beta_2 L}$ Enabling shifts of several THz and comb line-spacings from the FSR to higher harmonics via harmonic pumping.

Synchronous pumping in dispersion-engineered cavities—e.g., thin-film lithium niobate platforms—suppresses parasitic stimulated Raman scattering and achieves octave-spanning combs with high efficiency and programmable repetition rate (Cheng et al., 2023).

5. Quantum Temporal Modes and Dispersion-Induced Multimode Coupling

In synchronously pumped optical parametric oscillators (SPOPOs) and related dissipative-cavity systems, dispersion controls the resonance and coupling of orthogonal temporal modes ("supermodes") (Averchenko et al., 26 Jul 2024, Tikhonov et al., 3 Dec 2025). GVD both detunes the modal resonances ($C_{nn}$) and introduces coherent linear mixing between modes ($C_{nm}$ for $n\neq m$), as quantifiable from

$$C_{nm} = D\int dt\,s_n^*(t)\,\frac{d^2}{dt^2}s_m(t)$$

Mode entanglement, degradation of squeezing, and rotation of the squeezing ellipse arise when the dispersive coupling strengths approach the cavity decay rate. Analytical perturbative solutions are available up to second order in $C_{nm}/\gamma$, with a breakdown criterion $\left|N_\gamma/N_D~O_{nn}\right| < 1$ (where $O_{nn}$ is the mode-diagonal operator).

These features are critical for designing quantum-light sources with tailored multimode structure, and for homodyne detection and quantum information protocols exploiting temporal-mode selectivity.

6. Experimental Realizations and Design Implications

Experimental systems employ fiber-ring resonators, microring resonators, or bulk bow-tie cavities. Design criteria include:

• Precise matching of pump repetition rate to cavity FSR (typically $<$20 fs drift)
• Cavity finesse and loss optimized for both buildup and bandwidth control
• Dispersion management via tailored waveguide geometries, dielectric mirror coatings, and intracavity elements to constrain net GVD (e.g., to $|\beta_2| \lesssim 100$ fs$^2$ for sub-200 fs pulses) (Silfies et al., 2020)
• Use of on-chip pulse generators and electro-optic modulators for advanced platforms (Cheng et al., 2023)
• High peak pump powers ($\sim$10–30 W) for robust symmetry breaking and comb generation (Xu et al., 2014, Xu et al., 2020)

Multistable or symmetry-broken regimes are accessed and characterized via spectral asymmetry ($A$ factor), pulse time profiles (e.g., FROG retrieval), and dynamical bifurcation sequences. Quantum applications require careful engineering such that the modal-coupling parameter $N_\gamma/N_D$ remains in the perturbative regime for targeted modes (Tikhonov et al., 3 Dec 2025, Averchenko et al., 26 Jul 2024).

7. Impact, Technological Applications, and Outlook

Synchronous pumping in dispersive cavities underpins a broad range of applications:

• Frequency comb metrology with tunable center frequency and repetition rates, including in the visible regime via dispersion-managed enhancement cavities (Silfies et al., 2020)
• Ultrafast spectroscopy and ultrasensitive detection via path-length enhancement exceeding 190× (Silfies et al., 2020)
• On-chip frequency combs with high efficiency and Raman-suppression (Cheng et al., 2023)
• Quantum state generation in multimode SPOPOs and dispersive parametric devices, where the interplay of GVD and dissipative coupling structure the observable squeezing and entanglement properties (Averchenko et al., 26 Jul 2024, Tikhonov et al., 3 Dec 2025)
• Advanced cavity QED schemes exploiting parametric dispersive shifts for real-time control in quantum computing (Noh et al., 2021)

Ongoing research focuses on increasing accessible bandwidths, robust control of temporal mode structure, exploration of extreme nonlinear and multistable regimes, and the engineering of dispersive multimode networks for next-generation quantum and classical photonic systems.