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Starting from the infinite-dimensional Ikeda map, we derive an extendedtemporal Lugiato-Lefever equation that may account for the effects of theconjugate electromagnetic fields (also called `negative frequency fields'). Inthe presence of nonlinearity in a ring cavity, these fields lead to new formsof modulational instability and resonant radiations. Numerical simulationsbased on the new extended Lugiato-Lefever model show that thenegative-frequency resonant radiations emitted by ultrashort cavity solitonscan impact Kerr frequency comb formation in externally pumped temporal opticalcavities of small size. Our theory is very general, is not based on theslowly-varying envelope approximation, and the predictions are relevant to allkinds of resonators, such as fiber loops, microrings and microtoroids.

Extended temporal Lugiato-Lefever equation and the effect of conjugate fields in optical resonator frequency combs