"Talking" Vector Solitons and their Polarization Conformity in Fabry-Pérot Cavities

. Optical frequency combs are applicable across many fields, especially in metrology. Through accounting for field polarization, the spontaneous symmetry breaking of vector temporal cavity solitons and combs in Kerr Fabry–Perot resonators is presented. This work can improve the generation of frequency combs through its control on the maximum number of soliton-pairs in a round trip and reveals the interesting phenomenon of dominant polarization conformity across all soliton-pairs in the cavity.


Introduction
Commonly described as "Rulers for Light", optical frequency combs and their generation are topics with widereaching research interest.This interest is especially owed to their diverse application in metrology; such as for high precision spectroscopy, optical atomic clocks, and navigation systems.One particular method for producing an optical frequency comb utilizes dissipative Kerr Temporal Cavity Solitons (TCS).
Our research concerns itself not only with the generation of TCS but also combining them with the phenomena known as Spontaneous Symmetry Breaking (SSB).Broadly defined, SSB describes the situation when two or more properties of a system suddenly change from having some equality between them (symmetric) to having an inequality (asymmetric) following an infinitely small change to some system parameter.
The SSB of light in Kerr resonators has seen a flurry of study and interest in the past half-decade, being predicted and observed for systems with counter-propagating light components [1][2][3], for orthogonally polarized light components [4,5], and also very recently for systems combining both counter-propogating light and orthogonal polarizations components together [6] and containing twinresonator photonic isomers [7].
Related to combs, there has further been great success in observing the SSB of TCS in specifically Kerr ring resonators [8,9].Addressing linear resonators however, while an anomalously dispersive Kerr Fabry-Pérot cavity, Fig. 1(a), has been shown to support temporal cavity solitons (TCS) [10], and further while they have also been shown to exhibit a spontaneous symmetry breaking (SSB) between the powers of orthogonally polarized light com- * e-mail: lewis.hill@mpl.mpg.deponents [4], dissimilar to the ring, the SSB of TCS in linear resonators has remained elusive -until now.

System, Model, and Results
To model anomalously dispersive Kerr Fabry-Pérot cavities, we derived the following coupled equations [11]: where E ± are the orthogonally polarized fields, E in accounts for the input laser with detuning θ to the closest cavity resonance, t and τ are both temporal variables but on the timescales of many, and a single, round-trip time, respectively, and A and B are constants that control the strengths of self-and cross-phase modulation effects [2].The angle-bracketed terms represent the averages of the encapsulated functions over a single cavity round-trip [12].Figure 1(b) shows an example of the sought SSB of TCS following the integration of Eq. (1).In the vector soliton pair, one notes that one polarization component is dominant over the other, thus realizing a linear cavity source of symmetry broken vectorial Kerr frequency combs, Fig. 1(c).We discuss here a number of further, fascinating and useful, results not possible in the previous Kerr ring studies.
In panel (d), we present a cavity detuning scan while tracking the maximum intracavity power of the two circulating field components.It indicates seven behavioural ).If we force a dominant polarization exchange (f.2) in a soliton pair, the system will evolve to conform back to a shared soliton peak dominant polarization (f.3).
regions: (1) flat profiles across the resonator, (2) symmetric turing patterns, (3) asymmetric patterns, (4) unstable patterns, (5) asymmetric breathing soliton pairs, (6) stable asymmetric soliton pairs, ( 7) symmetric soliton pairs.Panels (e) and (f) show two of the most intriguing results.These are both a result of the angle-bracketed terms, which cause a global coupling across the cavity.Related to the TCS pairs, these global coupling terms effectively mean that all the pairs are aware of, and influence, any others in the cavity.Shown in panel (e), one effect of the global coupling is that it limits, based upon detuning, the maximum viable number of TCS pairs coexisting within the resonator at any one time.This effect can thus guarantee a maximum of a single soliton pair for comb generation.The global coupling terms also cause a shared dominant polarization between all SSB TCS pair peaks to occur, panel (f), which is a conformity not present in the ring resonator systems.We note that this conformity is a robust phenomenon.From the panels within (f) one can see that even if we force a dominant polarization deviancy for one of the soliton pairs, the system will quickly evolve to reestablish conformity throughout.

FastFigure 1 .
Figure 1.(a) Studied FP system.(b) A single pair of symmetry broken TCS and (c) their corresponding frequency combs.(d) Tracking the maxima of the circulating fields' intensity profiles over a full cavity round trip reveals seven distinct regions, (1) flat profiles, (2) symmetric Turing patterns, (3) asymmetric patterns, (4) unstable patterns, (5) asymmetric breathing soliton pairs, (6) stable asymmetric soliton pairs, (7) symmetric soliton pairs.(e) The detuning range of a viable soliton base is limited by the number of soliton pairs present in the cavity.(f) All SSB TCS pairs always share the same dominant polarization (f.1).If we force a dominant polarization exchange (f.2) in a soliton pair, the system will evolve to conform back to a shared soliton peak dominant polarization (f.3).