The effect of cross-phase modulation on instability characteristics in negative refractive index-based oppositely directed tri-core couplers

We investigate modulation instability (MI) in an oppositely directed PIM-NIM-PIM three-core coupler with cross-phase modulation (XPM), using coupled-mode theory and linear stability analysis. XPM is considered between adjacent channels, and its influence is examined under normal, anomalous, and hybrid dispersion conditions. In the normal-dispersion regime, MI develops only above specific power-ratio thresholds, whereas anomalous dispersion produces threshold-free instability over broader frequency intervals. The XPM strength, sign, and spatial distribution substantially modify the MI gain, bandwidth, and sideband structure. XPM localized in the central NIM channel enhances the maximum gain, while positive XPM distributed across the three channels expands the unstable region and generates multibranch spectra. Mixed-sign XPM reduces the instability growth and produces fragmented sidebands. In the hybrid regime, the coexistence of normal- and anomalous-like channel dynamics results in separated instability domains and resonance-related sidebands. The stronger nonlinear response of the NIM channel further enriches the MI spectrum through its mediating role between the two outer PIM channels. Overall, the results show that suitable XPM configurations provide an effective way to control MI characteristics in oppositely directed tri-core couplers.