Impact of phase modulation on the dynamics of temporal localized structures in injected Kerr microcavities

Abstract

We theoretically investigate how phase modulation alters the dynamics of temporal localized structures (TLSs) in vertically emitting Kerr micro-cavities under detuned optical injection operating in the normal dispersion regime. We show that the emergence of TLSs in general is governed by a synchronization between the imposed modulation and the intrinsic pulse dynamics. We perform a multi-parameter bifurcation analysis of the underlying delay-algebraic equation model in the uniform field limit and demonstrate that weakly nonlinear and dissipative Hermite-Gauss modes shape the dynamics of dark TLSs, leading to a complex hybrid bifurcation structure. Beyond the uniform field limit, both bright and dark modulated TLSs are shown to exist and to occupy distinct equilibrium positions within the cavity. An effective equation of motion for the TLS positions is derived, showing a good agreement with the full model.

Figures (9)

• Figure 1: (i) Schematic of a micro-cavity containing a Kerr medium. The micro-cavity has a round-trip time $\tau_\mathrm{c}$ and is coupled to an external cavity of the round-trip time $\tau$, closed by a mirror (orange) with reflectivity $\eta$ and phase shift $\phi$. The system is driven by a monomode injection field $Y_0$, and the feedback phase is modulated via periodic mirror motion. (ii) Intensity of a single temporal pulse circulating in the external cavity (blue) obtained by integrating Eqs. \ref{['subeq:KGTI_E']} and \ref{['subeq:KGTI_Y']} with $m=0.12$ and $\omega=0.0619$ plotted together with the feedback phase modulation (gray). Other parameters are $(\delta,\,h,\,\eta\,,\varphi_0\,,Y_0\,,\tau)=(1.5\,,2\,,0.75\,,0\,,0.55\,,100)$.
• Figure 2: (i) Difference of the the repetition frequency $\Omega$ of the pulses in the KGTI system \ref{['eq:whole_KGTI']} and the frequency of the phase modulation $\omega$. The data are obtained numerically (blue circles) and by path continuation (orange) for $m=0.12$. The horizontal axis is scaled by the natural frequency of the pulses $\omega_0$. (ii) The scaling of the beat period $T_\mathrm{beat}$ of the quasi-periodic dynamics near the critical frequency $\omega_\mathrm{c}$ in a double logarithmic scale. (iii) The Arnold tongue indicating the synchronization region as a function of the amplitude $m$ and frequency $\omega$ of the phase modulation. The dashed line corresponds to the panel (i). (iv), (v) present synchronized ($\omega=0.0619$) and quasi-periodic ($\omega=0.06198$) dynamics, respectively, using a two-time representation. The gray line indicates the phase modulation. (vi) The orange branch represents the path continuation of a synchronized solution (stable/unstable solutions in bold/thin), while the blue dots correspond to numerical time simulations, showing the integrated intensity in each round-trip. Other parameters are as in Fig. \ref{['fig:1']}.
• Figure 3: Bifurcation diagrams of Eqs. \ref{['subeq:KGTI_E']} and \ref{['subeq:KGTI_Y']} showing the integrated intensity $\langle|E|^2\rangle$ as a function of the injection $Y_0$ for different amplitudes of the modulation $m$. Panel (i) shows the CW (light gray) and the TLS (dark gray) branches in the background for $m=0$. Additionally, the modulated CW branches for $m=\{0.4,\,1,\,1.5,\,3.2,\,6.1\}\times10^{-3}$ are depicted in yellow, orange, blue, green and purple, respectively. The region marked with a dotted rectangle is shown in detail, together with the corresponding TLS branches in panels (ii) - (vi). Different black markers indicate points traced for increasing $m$. Panels (a)-(d) present exemplary profiles corresponding to the points marked with red dots in panels (i)-(vi), whereas the dotted gray line indicates the phase modulation. The system parameters are $(\delta,h,\eta,\varphi_0,\tau,\omega)=\left(1/\sqrt{3},2,0.99,0.3235\pi,300,0.0208\right)$.
• Figure 4: Two-parameter continuation of different fold points marked by black markers in Fig. \ref{['fig:3']} (ii)-(v) in the $(Y_0,\,m)$ plane showing the growth and reconnection of the TLSs isolas and modulated CW branches. Parameters are the same as in Fig. \ref{['fig:3']}.
• Figure 5: (i) First seven HG modes within a parabolic potential. (ii) Mode energy distribution associated with the TLS shown in blue in panel (iii). (iii) Comparison of TLS (blue) and superposition of HG modes weighted with the energy distribution from panel (ii) (green). The system parameters are as in Fig. \ref{['fig:3']} and $m=0.1$.