Existence, stability, and dynamics of SIT-NLS solitons in the coupled Kerr-Maxwell-Bloch fiber system under detuning, inhomogeneous broadening, and power mismatch

In an erbium-doped nonlinear optical fiber, the resonant dopants and the host Kerr nonlinearity act on the same pulse: the two-level medium supports self-induced transparency (SIT) solitons (lossless $2\pi$ pulses), while group-velocity dispersion balanced by Kerr self-phase modulation supports nonlinear Schrödinger (NLS) solitons. Nakazawa, Yamada and Kubota (Phys. Rev. Lett. 66, 2625 (1991)) showed numerically that the two can coexist as a single $2\pi$-(N=1) mixed pulse, an SIT-NLS soliton, whose phase rotation is governed solely by the NLS component and whose pulse delay is governed solely by the SIT component, with higher-area pulses splitting into fundamental $2\pi$ pieces — but only in an idealized regime: zero detuning, homogeneous broadening, a matched power scale ($W=1$), and several ad hoc modeling simplifications. What remains open is a systematic theory of the mixed SIT-NLS soliton outside that regime: its existence and SIT/NLS role partition under nonzero detuning; the effects of inhomogeneous broadening and of the full spatially averaged coherence; the dynamics of mismatched regimes ($W\neq 1$); and an analytic linear/spectral stability characterization of the $2\pi$-(N=1) soliton and of the higher-order splitting.

Background

Two mechanisms produce robust solitary pulses in fibers. Self-induced transparency (SIT), discovered by McCall and Hahn, is the coherent interaction of a resonant pulse with an absorbing ensemble of two-level atoms: when the pulse area, the time integral of the Rabi frequency (proportional to the field envelope), equals $2\pi$, the medium is driven from the ground state to full inversion and back, so the pulse propagates without net absorption with a well-defined delay; the pulse dynamics is described by the Maxwell-Bloch equations, which couple the field envelope to the population difference $F$ and the slowly varying coherence (dipole polarization) $M$ of the transition. Separately, in a fiber whose host glass has a Kerr (intensity-dependent refractive index) nonlinearity, group-velocity dispersion balanced by Kerr self-phase modulation is governed by the nonlinear Schrödinger (NLS) equation; its fundamental soliton is the order-$N=1$ sech-shaped envelope pulse, and higher-order ($N>1$) solitons undergo periodic internal breathing and can split. Hasegawa and Tappert introduced this fiber soliton in 1973. An erbium-doped nonlinear fiber combines both: the dopants provide a resonant two-level (or quasi-two-level) transition while the host provides Kerr nonlinearity and dispersion, so the two soliton mechanisms act simultaneously on one pulse.

Nakazawa, Yamada and Kubota analyzed this situation in a 1991 Physical Review Letter, with an extended account in Phys. Rev. A 44, 5973 (1991), using a coupled system for the slowly varying envelope, the population difference, and the coherence, with characteristic power scales $P_{0(\mathrm{NLS})}$ (set by dispersion, Kerr coefficient, and modal area) and $P_{0(\mathrm{SIT})}$ (set by the dipole moment, pulse width, and effective area). The ratio $W^{2}=P_{0(\mathrm{NLS})}/P_{0(\mathrm{SIT})}$ emerges as the key matching parameter: only when $W=1$ does a fundamental $N=1$ NLS soliton amplitude carry exactly the $2\pi$ SIT pulse area, which forces the closed-form material condition $n_{2}=c|p_{21}|^{2}|k''|/(\omega\hbar^{2})$ linking the Kerr coefficient $n_{2}$, the group-velocity dispersion $k''$, and the dipole matrix element $p_{21}$. In this matched regime they demonstrated, by direct numerical integration, a stable mixed soliton with total SIT area $2\pi$ and NLS order $N=1$ (the $2\pi$-(N=1) SIT-NLS soliton), showed that its dipole phase evolves as $\phi(q)=-\pi/2+\alpha q$ with $\alpha=\tfrac{1}{2}(2\eta)^{2}$ set entirely by the NLS nonlinear phase ($\eta$ the soliton amplitude parameter, $q=z/Z_{0}$ the normalized propagation distance), and that the pulse delay equals that of a pure SIT soliton, so the phase is controlled by the NLS part and the delay by the SIT part — but this role separation is stated for zero detuning, where detuning $\xi$ measures the offset of the carrier from the transition frequency. They also found that $4\pi$ and $6\pi$ mixed pulses always split into multiple $2\pi$-(N=1) solitons.

All of these conclusions rest on idealizations: exact resonance ($\xi=0$); homogeneous broadening, i.e., all dopant atoms are identical so the spatially averaged coherence $\langle M\rangle$ entering the envelope equation is replaced by the local $M$ (as opposed to inhomogeneous broadening, where atoms have a distribution of transition frequencies); the identification of the SIT core cross section $S$ with the modal effective area $A_{\mathrm{eff}}$; replacement of the averaged-coherence coupling coefficient in the numerical runs by a scalar; and stability claims supported only by finite-distance numerical propagation rather than by linear stability analysis, the standard method in which the PDE is linearized about the soliton and the spectrum of the resulting operator determines whether small perturbations grow. What happens to the mixed SIT-NLS soliton when these idealizations are removed was left open.

Problem Statement

For the coupled Kerr-Maxwell-Bloch system describing pulse propagation in a resonantly doped nonlinear fiber (the field envelope governed by group-velocity dispersion, Kerr self-phase modulation, and the resonant polarization; the dopants described by the population difference and coherence of a two-level transition), establish a systematic theory of the mixed SIT-NLS soliton beyond the idealized regime in which Nakazawa, Yamada and Kubota demonstrated it. Specifically: (i) determine whether, and in what form, mixed $2\pi$-(N=1) SIT-NLS solitons exist and propagate for nonzero detuning $\xi\neq 0$ of the carrier from the transition frequency, and how detuning modifies the observed partition of roles in which the nonlinear phase is set by the NLS (Kerr) component and the pulse delay by the SIT component; (ii) determine the effect of inhomogeneous broadening of the transition (equivalently, of retaining the full averaged coherence $\langle M\rangle$ rather than the local $M$) on the existence, stability, and higher-order splitting dynamics of the mixed soliton; (iii) characterize the full dynamics of the mismatched regime $W\neq 1$, in which the NLS and SIT power scales differ and a $2\pi$ pulse no longer corresponds to the fundamental NLS soliton amplitude, including any alternative mixed, transformed, or decaying states; (iv) provide both an analytic stability characterization (linear stability analysis or spectral analysis of the linearization about the soliton) of the $2\pi$-(N=1) SIT-NLS soliton, and an analytic account of the splitting of higher-area ($4\pi$, $6\pi$, etc.) mixed pulses into fundamental $2\pi$ pulses; and (v) map the sensitivity of the coexistence and matching conditions (including the required relation among Kerr coefficient, dispersion, and dipole moment) to the modeling simplifications ($S=A_{\mathrm{eff}}$, the scalar averaged-coherence coefficient) and to realistic doped-fiber parameters. These five aspects (i)-(v) are coupled parts of one systematic theory. The common objective is to replace the numerically demonstrated, ideal-regime picture of the mixed SIT-NLS soliton by a verifiable account of when such coexistence survives the non-ideal conditions of a real resonant fiber.

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