Dynamical theory, stability, and experimental realization of nonlinear photonic-band-gap solitary waves in two and three dimensions

In 1993 John and Aközbek predicted, using a variational slowly-varying-envelope construction, that Kerr-nonlinear photonic band gap (PBG) crystals support finite-energy solitary waves inside their stop gaps: in a two-dimensional square lattice, solutions preserving the crystal symmetry occur for both signs of the Kerr coefficient, while symmetry-breaking solutions occur only for positive (focusing) nonlinearity. The prediction left the stationary structure only approximately known (the variational ansatz deviates near direct band edges), the stability and decay of metastable states lying in a stop gap but outside the complete PBG unanalyzed, the time-dependent dynamics in two and three dimensions without a theory (the higher-dimensional envelope matrices do not form a Dirac algebra, unlike the one-dimensional case), and experimental realization in low-loss, large- $\chi^{(3)}$ materials open. Later literature substantially advanced the stationary problem (effective nonlinear Dirac reduction with finite-element confirmation, direct numerical solution of the nonlinear Maxwell problem in finite crystals, and rigorous coupled-mode derivations) and stability analysis in prototypical nonlinear Dirac models, but a complete dynamical theory — mobility, scattering, stability under time-dependent perturbations, and the effective dynamical law — for the two- and three-dimensional PBG envelope equations, the decay of metastable out-of-gap states, and observation in complete-gap Kerr photonic crystals remain unresolved.

Background

A photonic band gap (PBG) material is a periodic dielectric whose linear band structure contains frequency intervals (stop gaps) in which electromagnetic waves cannot propagate; in two and three dimensions a complete PBG forbids propagation in all directions, and the relevant band edges occur at inequivalent symmetry points of the Brillouin zone (for a square lattice, the edge-center points $X$ and the corner points $M$ ; $\Gamma$ is the zone center). A crystal with a Kerr nonlinearity — a third-order susceptibility $\chi^{(3)}$ that makes the refractive index change in proportion to the local light intensity, with the sign distinguishing focusing (positive) from defocusing (negative) response — can trap light inside the gap: nonlinear shifting of the local bands creates a localized, finite-energy state, a gap soliton. One-dimensional gap solitons in fiber Bragg gratings were already understood in 1993 through two-component envelope equations whose matrix structure is built from Pauli matrices that anticommute, i.e. a Dirac algebra; this algebra underlies the exact 1D solutions and their relativistic (Dirac-like) time-dependent dynamics, with effective propagation velocity bounded by a scaled speed of light. John and Aközbek (Phys. Rev. Lett. 71, 1168 (1993)) extended this to higher dimensions. Expanding the Kerr-nonlinear wave equation in slowly varying envelopes about the $X$ points (a two-component spinor, selecting a single Bragg vector and hence breaking the square-lattice rotational symmetry) and about the four equivalent $X$ points or the fourfold-degenerate $M$ point (four-component spinors, preserving the crystal symmetry), they obtained coupled envelope partial differential equations containing first-order derivatives along the Bragg directions and second-order transverse dispersion. A variational (Ritz) method with analytic trial spinors — $\mathrm{sech}$ amplitude envelopes carrying arctangent phase kinks, which reproduce the known 1D exact solution — then gave the amplitude, size, phase structure, and total energy $U\sim[\lambda k_0^{d}\,\delta^{d/2-1}]^{-1}$ of the solitary waves, where $d$ is the spatial dimension, $\lambda$ the effective nonlinear coefficient, $k_0=\pi/a$ the Bragg wavevector for lattice period $a$ , and $\delta$ the squared frequency detuning from a band edge (so $U$ vanishes at the edge in $d=1$ , stays finite in $d=2$ , and diverges in $d=3$ ). They predicted threshold intensities of order $\mathrm{kW/cm^2}$ in candidate semiconductor systems (GaAs–AlAs structures, InGaAsP microdisk arrays) and proposed the states as elements of bistable all-optical switching in PBG materials. The construction, however, was static and approximate, and — crucially — the four-component envelope matrices $\gamma_i,\gamma_i'$ of the symmetric higher-dimensional equations do not satisfy the anticommutation relations of a Dirac algebra, so the relativistic dynamics of the 1D gap soliton was not expected to carry over, and no time-dependent theory was given. In the following decades the subject of nonlinear localized states in periodic media developed along several partly separate lines: coupled-mode and nonlinear-Schrödinger-type analyses of gap solitons in periodic potentials, spectral stability theory (stability decided by the eigenvalues of the operator obtained by linearizing the equations about the solitary wave) of nonlinear Dirac solitary waves, direct numerical solution of the nonlinear Maxwell problem in finite photonic crystals, and experiments on gap-type solitons in optically induced photonic lattices in photorefractive media (weak-contrast lattices with partial stop bands, rather than complete-PBG crystals).

Problem Statement

Establish a complete physical theory of the Kerr-nonlinear solitary waves that exist inside the stop gaps of two- and three-dimensional photonic band gap crystals, going beyond the static variational construction of John and Aközbek (1993). The linked questions, all serving the single objective of determining whether these gap solitons are real, robust, dynamical, and experimentally accessible objects (as required for their proposed use in bistable all-optical switching in PBG materials), are: (i) determine the solitary-wave families of the two-component ( $X$ -point, symmetry-breaking) and four-component (symmetric $X$ and $M$ point) envelope equations — including their precise behavior in the near-direct-band-edge regimes where the original variational ansatz deviates significantly from the exact solutions (notably near the lower direct band edge for positive Kerr coefficient $\lambda>0$ and the upper direct band edge for $\lambda<0$ ); (ii) determine the spectral stability (i.e., the growth rates of the operator obtained by linearizing the envelope equations about the solitary wave) and the dynamical stability of the in-gap states, and analyze the decay mechanisms and lifetimes of the metastable broken-symmetry solutions whose frequencies lie in a stop gap (e.g., within the $X$ stop gap but below the $M$ -point lower edge of an indirect-gap crystal, one whose complete gap opens between band extrema at inequivalent symmetry points) and are therefore predicted to couple to continuum radiation modes; (iii) construct the time-dependent dynamical theory of these higher-dimensional envelope equations — stability under time-dependent perturbations, mobility, collisions and scattering, and the effective (relativistic or nonrelativistic) dynamical law obeyed by the solitons — recognizing that the envelope matrices do not form a Dirac algebra, so the relativistic dynamics of the one-dimensional gap soliton does not carry over; and (iv) quantify the conditions for experimental realization: material loss, finite-size and boundary effects, and the refractive-index contrast and nonlinear response required for the predicted parameter regimes in low-loss materials with large $\chi^{(3)}$ at the operating frequency, in high-contrast photonic band gap crystals with a complete gap rather than weakly modulated lattices. These four linked questions jointly constitute the complete theory — (i)–(iii) as theoretical results and (iv) as a quantitative feasibility bound — and progress on any one of them stands as a self-contained partial result.

References

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