Multistep parametric processes in nonlinear optics

Energy transfer between different modes and phase-matching relations are fundamental concepts in nonlinear optics. Unlike nonparametric nonlinear processes such as self-action and self-focusing of light in a nonlinear Kerr-like medium, parametric processes involve several waves at different frequencies and they require special relations between the wave numbers and wave group velocities to be satisfied, the so-called phase-matching conditions. Parametric coupling between waves occurs naturally in nonlinear materials without inversion symmetry, when the lowest-order nonlinear effects are presented by quadratic nonlinearities, often called χ^{( 2 )} nonlinearities because they are associated with the second-order contribution ( ∼ χ^{( 2 )} E²) to the nonlinear polarization of a medium. Conventionally, the phase-matching conditions for most parametric processes in optics are implemented either by using anisotropic crystals (the so-called perfect phase matching), or in fabricated structures with a periodically reversed sign of the quadratic susceptibility (the so-called quasi-phase matching or QPM). The QPM technique is one of the leading technologies today, and it employs spatial scales (∼1-30 μm) which are compatible with the operational wavelengths of optical communication systems. Nonlinear effects produced by the quadratic intensity-dependent response of a transparent dielectric medium are usually associated with parametric frequency conversion, such as second harmonic generation (SHG). The SHG process is among the most intensively studied parametric interactions which may occur in a quadratic nonlinear medium. Moreover, recent theoretical and experimental results demonstrate that quadratic nonlinearities can also produce many of the effects attributed to nonresonant Kerr nonlinearities via cascading of several second-order parametric processes. Such second-order cascading effects can simulate third-order processes, in particular those associated with the intensity-dependent change of the medium refractive index (Stegeman, Hagan and Torner [1996]). Importantly, the effective (or induced) cubic nonlinearity resulting from a cascaded SHG process in a quadratic medium can be several orders of magnitude higher than that usually measured in centrosymmetric Kerr-like nonlinear media, and it is practically instantaneous. The simplest type of phase-matched parametric interaction is based on the simultaneous action of two second-order parametric sub-processes that belong to a single second-order interaction. For example, the so-called two-step cascading associated with type I SHG includes the generation of the second harmonic (SH), ω + ω = 2 ω, followed by the reconstruction of the fundamental wave through the down-conversion frequency-mixing process, 2 ω - ω = ω. These two sub-processes depend on only a single phase-matching parameter Δk. In particular, for nonlinear χ^{( 2 )} media with a periodic modulation of the quadratic nonlinearity, for QPM periodic structures, we have Δ k = k₂ - 2 k₁ + G_m, where k₁ = k ( ω ), k₂ = k ( 2 ω ) and G_m is the reciprocal vector of the periodic structure, G_m = 2 π m / Λ, where Λ is the lattice spacing and m is an integer. For a homogeneous bulk χ^{( 2 )} medium, we have G_m = 0. Multistep parametric interactions and multistep cascading represent a special type of second-order parametric processes that involve several different second-order nonlinear interactions; they are characterized by at least two different phase-matching parameters. For example, two parent processes of the so-called third-harmonic cascading are: (i) second-harmonic generation, ω + ω = 2 ω, and (ii) sum-frequency mixing, ω + 2 ω = 3 ω. Here, we may distinguish five harmonic sub-processes, and the multistep interaction results in their simultaneous action. Different types of multistep parametric processes include third-harmonic cascaded generation, two-color parametric interaction, fourth-harmonic cascading, difference-frequency generation, etc. Various applications of multistep parametric processes have been mentioned in the literature. In particular, multistep parametric interaction can support multi-color solitary waves, it usually leads to larger accumulated nonlinear phase shifts in comparison with simple cascading, it can be employed effectively for the simultaneous generation of higher-order harmonics in a single quadratic crystal, and it can be employed for the generation of a cross-polarized wave and frequency shifting in fiber-optics gratings. In general, simultaneous phase matching of several parametric processes cannot be achieved by traditional methods such as those based on the optical birefringence effect. However, the situation becomes different for media with a periodic sign change of the quadratic nonlinearity, as occurs in QPM structures or two-dimensional nonlinear photonic crystals. In this review, we describe the basic principles of simultaneous phase matching of two (or more) parametric processes in different types of one- and two-dimensional nonlinear quadratic optical lattices. We divide the different types of phase-matched parametric processes studied in nonlinear optics into two major classes, as shown in fig. 1, and discuss different types of parametric interactions associated with simultaneous phase matching of several optical processes in quadratic (or χ^{( 2 )}) nonlinear media, the so-called multistep parametric interactions. In particular, we provide an overview of the basic principles of double and multiple phase matching in engineered structures with sign-varying second-order nonlinear susceptibility, including different types of QPM optical superlattices, noncollinear geometry, and two-dimensional nonlinear quadratic photonic crystals (which can be considered two-dimensional QPM lattices). We also summarize the most important experimental results on the multi-frequency generation due to multistep parametric processes, and survey the physics and basic properties of multi-color optical solitons generated by these parametric interactions.