It is therefore important to understand the physics underlying the emergence of normal GVD solitons, and to develop reliable methods to create these curious states. These characteristics provide complementary functionalities to the current paradigm for laser synthesis 24 and optical clockwork 6 with anomalous dispersion Kerr combs. Beside interesting physics, normal GVD systems are advantageous for applications, including the relative ease of obtaining normal GVD, self-starting pulses 20, focused spectral power distribution, and high energy efficiency 23. Moreover, effectively bright-pulse states such as the platicon have been described through simulation 20 and can form through multi-frequency pumping 21 or the Raman effect 22. Details on normal GVD phase matching and the comb phases thus created are presented in Supplementary Section I and explored in 20.Įxperiments with dark-soliton states exhibit strikingly different behavior from its anomalous GVD counterpart, for example, the development of complex spectral modifications with pump-laser detuning 13. Still, particular nonlinear states have been observed through fortuitous mode-structure defects that create bands of anomalous GVD in an otherwise normal-GVD resonator to seed dark-soliton formation 10, 12. In particular, the LLE framework for optical states subject to normal GVD (and a positive nonlinear coefficient) is relatively sparse, since the nearly unconditional imbalance of GVD and nonlinearity at constant excitation suppresses phase matching 19. The LLE describes several states: the flat state of a sufficiently low-intensity pump laser oscillatory Turing patterns that extend over the entire resonator and localized bright and dark solitons, which are the canonical stationary states at anomalous and normal GVD, respectively. To understand the interplay of nonlinearity and either anomalous or normal GVD, we utilize the mean-field Lugiato-Lefever equation (LLE) 18 for the field in a Kerr resonator. More advanced device topologies promise to yield enhanced nonlinear laser sources, through coupled resonators 12, 13, dispersion engineering in nanophotoncis 14, 15, and inverse-design methods 16, 17. Dissipative Kerr solitons in microresonators enable ultraprecise optical-frequency metrology 6 and many other functionalities 7, 8, 9, 10, 11. Beyond illuminating complex systems, nonlinear-optical states are being applied and optimized. Simply the sign of GVD differentiates optical states, for example in the case of anomalous GVD that balances with the Kerr effect for soliton formation. However, the Kerr effect only represents half of the picture in the dynamics, since the frequency dependence of group-velocity dispersion (GVD or simply dispersion) typically controls what optical state forms at the balance against nonlinearity. We focus on a ubiquitous nonlinearity of materials, the Kerr effect 2, 3, which underlies fascinating behaviors in nonlinear optics, such as the formation of discrete states of patterns and pulses 4, 5. Similarly, in nonlinear optics, spatiotemporal laser patterns readily manifest in a medium, and their dynamics enable detailed, precise, and controllable tests of how light and matter interact. Fractals are a good example 1 in which mathematical relations produce intricate patterns, depending on a set of parameters that characterize the relations. The dark-to-bright continuum illuminates physical roles of Kerr nonlinearity, GVD, and laser propagation in a gapped nanophotonic medium.Ĭomplex systems generate patterns from a fundamental set of rules, which govern interactions between system components. These stationary temporal states are coherent frequency combs, featuring highly designable spectra and ultralow noise repetition-frequency and intensity characteristics. We access and explore this dark-to-bright pulse continuum by phase-matching with a photonic-crystal resonator, which mediates the competition of nonlinearity and normal GVD. Here, we introduce a resonator with unconditionally normal GVD and a single defect mode that supports both dark, reduced-intensity states and bright, enhanced-intensity states. Moreover, the dichotomy of states with locally enhanced or diminished field intensity depends critically on the relative sign of nonlinearity and either anomalous or normal GVD. In optics, diverse families of continuous and discrete states arise from balance of nonlinearity and group-velocity dispersion (GVD). Controlling nonlinearity leads to interesting states of matter and new applications. Nonlinearity is a powerful determinant of physical systems.

Dark pulse