Isotopic variations and Zeeman-like splitting in the spectra of nonlinear photonic meta-atoms

Abstract

We study photonic meta-atoms, a unique class of composite solitary wave, supported in nonlinear waveguides. We establish an analogy to one-dimensional soft-core atoms, allowing to describe the complex dynamics via concepts from atomic physics. Higher-order dispersive effects cause specific spectral resonances characteristic for the eigenspectrum of a meta-atom. We demonstrate that subtle changes in this level spectrum causes frequency shifts of the resonances. These shifts consist of isotopic and isomeric contributions that can be distinguished in terms of a simple model. We further demonstrate a generic mechanism that causes a Zeeman-like splitting of resonance lines.

Figures (9)

• Figure 1: Details of the model. (a) Dispersion, (b) inverse group-velocity, and (c) group-velocity dispersion. Domain of normal dispersion [$D_2(\Omega)>0$] is shaded gray. Dot and circle refer to the loci $\Omega_{\rm{S}}$ and $\Omega_{\rm{G}}$ of the soliton (S) and Gaussian pulse (G), respectively. (d) Trapping potential $V_{\rm{T}}$ with trapped states $\phi_n$ at eigenvalues $\kappa_n$ for $n=0,\ldots,4$. Dotted line (SR) denotes the short-range approximation of $V_{\rm{T}}$ (see text).
• Figure 2: Propagation dynamics demonstrating the excitation of resonant radiation. (a) Time-domain propagation dynamics of a soliton, and a superimposed Gaussian pulse (see text). (b) Corresponding spectrum. Dotted lines indicate zero-dispersion points. The arrow (labeled R) indicates the location of resonantly excited modes. The field $A_F$, enclosed by the box in (a), is defined by the spectrum enclosed by the box in (b).
• Figure 3: Spectral signature of meta-atom isotopes. (a) Graphical solution of the phase-matching condition (see text). (b-e) Resonance spectral lines for selected values of $t_0$. (b) $t_0=10$, (c) $t_0=11$, (d) $t_0=12$, and, (e) $t_0=13$. The spectral density function (SDF) is normalized as $\int_{0}^{\Omega_c}I(\Omega)~{\rm{d}}\Omega=1$. Black dotted line in (b-e) indicates cutoff frequency $\Omega_c$. Blue dotted lines in (b) indicate predicted resonances at $\Omega_{{\rm{R}},n}$ for $n=0,\ldots,4$.
• Figure 4: Splitting of spectral lines for a vibrating meta-atom. (a) Time-domain propagation dynamics, and, (b) spectrum, for soliton order $N_{\rm{S}}=1.5$ ($K_{\rm{S}}\approx0.026$). Dotted lines indicate zero-dispersion points. The field $A_F$, enclosed by the box in (a), is defined by the spectrum enclosed by the box in (b). (c) Shifting and splitting of the groundstate resonance, in excellent agreement with the phase matching results $\Omega_{{\rm{R}},0}^m$ for $m\in\{0,\pm1, 2,3,4\}$. $\Omega_{{\rm{R}},0}$ indicates the resonance for $N_{\rm{S}}=1$.
• Figure 5: Details on the phase-matching resonances in Figs. \ref{['fig:03']} and \ref{['fig:04']}. (a) Graphical solution of the phase-matching condition (\ref{['eq:PM_condition']}) for the $n\!=\!0$-resonances for the example in Fig. \ref{['fig:03']} (see main document). The horizontal dashed line indicates the sign-inverted wavenumber eigenvalue $\kappa_0\approx -0.0481$. Filled and open circles indicate the corresponding near and far resonance loci $\Omega_{{\rm{R}},0}^{({\rm{N}})}\approx 0.574$, and $\Omega_{{\rm{R}},0}^{({\rm{F}})}\approx -4.182$.