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We analyze the conical second-harmonic generation via Cerenkov-type phase matching in two-dimensional decagonal nonlinear photonic quasi-crystals for linear, circular, and elliptical polarized fundamental beams. Depending on the polarization state, we observe either centrosymmetric or noncentrosymmetric intensity modulation of the second-harmonic rings. We explain this effect by employing the concept of effective quadratic nonlinearity combined with the hexagonal symmetry of the individual ferroelectric domains.

We observe experimentally a novel type of nonlinear diffraction in the process of two-wave mixing on a nonlinear quadratic grating.We demonstrate that when the nonlinear grating is illuminated simultaneously by two noncollinear beams, a second-harmonic diffraction pattern is generated by a virtual beam propagating along the bisector of the two pump beams. The observed iffraction phenomena is a purely nonlinear effect that has no analogue in linear diffraction

We analyze experimentally light scattering from 2 nonlinear gratings and observe two types of second-harmonic frequency-scattering processes. The first process is identified as Raman–Nath type nonlinear diffraction that is explained by applying only transverse phase-matching conditions. The angular position of this type of diffraction is defined by the ratio of the second-harmonic wavelength and the grating period. In contrast, the second type of nonlinear scattering process is explained by the longitudinal phase matching only, being insensitive to the nonlinear grating period.

Collinear third-harmonic generation at 526.7 nm was realized by the simultaneous phase matching of two second-order processes in a single quadratic crystal: second-harmonic generation (SHG) and sum-frequency mixing (SFM). The measured conversion efficiency was 12%. As a nonlinear medium a LiNbO₃ nonlinear photonic crystal with short-range order was used that allowed simultaneous phase matching by use of discrete reciprocal vector (for the SHG process) and continuous reciprocal vectors (for the SFM process). It was demonstrated that the third harmonic could be generated efficiently in such a crystal even if the intermediate process of SHG was not perfectly phase matched.

We describe the fabrication and characterization of partially ordered 2D photonic structures prepared by domain inversion in LiNbO3. Two types of structures have been prepared: (i) decagonal quasi-crystals designed by tiling the plane with different kinds of rhomb shaped tiles and (ii) a short-range ordered structure designed by placing randomly oriented basic units on a square lattice. Both types of structures exhibit rich sets of reciprocal vectors that may be used to satisfy the phase matching conditions for different nonlinear optical processes. As examples we describe collinear second harmonic generation of multiple wavelengths in a single decagonal nonlinear photonic quasi-crystal and cascaded third harmonic generation in a short range order nonlinear photonic crystal.

Short-range ordered nonlinear photonic crystals (SRO-NPC) are new, promising class of materials for optical frequency conversion. We describe a method for designing such SRO-NPC for efficient collinear third-harmonic generation (THG) at arbitrary wavelength. The method was used to prepare a LiNbO3 sample in which THG at 526.7 nm with 12% efficiency was achieved.

We study the second-harmonic generation via transversely matched interaction of two counterpropagating ultrashort pulses in X⁽²⁾ photonic structures. We show that the emitted second-harmonic wave attains the form of spatially expanding toroid with the initial thickness given by the cross correlation of the pulses. We demonstrate the formation of such toroidal waves in crystals with random ferroelectric domains as well as in annularly poled nonlinear photonic structures.

The authors study experimentally the transverse second-harmonic generation of counterpropagating pulses by a quasi-phase-matching in a medium with a random ferroelectric domain structure. The authors show that this parametric process results in a direct realization of the cross correlation of two optical signals and, therefore, it can be employed for direct characterizations of ultrashort pulses including their temporal structure and pulse front tilt.

We demonstrate that simultaneous phase- and group-velocity-matched cascaded parametric processes can be achieved in two-dimensional quadratic nonlinear photonic crystals by proper matching of noncollinear processes with front tilting of the fundamental pulse. We present examples of cascaded third- and fourth-harmonic generation in a poled LiNbO3 planar structure.

The generation of tetragonal crystals with third order nonlinearity in anisotropic optical media was discussed. Both linear and elliptically polarized vector solitons in tetragonal crystals were studied and their stability was discussed. It was shown that these vector solitons provide a nontrivial generalization to both conventional vector solitons of birefringent cubic media and parametric solitons supported by third order cascaded nonlinearities.

We describe a novel type of second-order cascading by which the signal that is cross polarized to the pump can be split with an efficiency of more than 100%. The process requires the simultaneous phase matching of two second-order parametric interactions, which can easily be achieved with recently fabricated two-dimensional nonlinear photonic crystals.

A concept of multistep cascading is applied to the problem of fourth-harmonic generation (FHG) in a single quadratic crystal, and a new model of parametric wave mixing is analyzed in detail. Important applications to the optical frequency division and efficient FHG as well as the realization of the double-phase-matching multistep cascading processes in engineered QPM structures with phase-reversal sequences are also suggested.

We investigate efficient fourth-harmonic generation in a single two-dimensional (2D) quadratically nonlinear photonic crystal. We propose a novel parametric process that starts with phase-matched generation of a pair of symmetric second-harmonic waves, which then interact to produce a fourth-harmonic wave that is collinear to the fundamental. We show that this process is more efficient than conventional fourth-harmonic-generation schemes by a factor that reaches 4 at low intensities and discuss how to design and optimize the nonlinear 2D photonic crystals that are implemented in LiNbO3 and LiTaO3.

We discuss several novel types of multi-component (temporal and spatial) envelope solitary waves that appear in fiber and waveguide nonlinear optics. In particular, we describe multi-channel solitary waves in bit-parallel-wavelength fiber transmission systems for high-performance computer networks, multi-color parametric spatial solitary waves due to cascaded nonlinearities of quadratic materials, and quasiperiodic envelope solitons due to quasi-phase-matching in Fibonacci optical superlattices.

We propose a novel type of cascading parametric interaction for generating a nonlinear phase shift in dielectric media with a quadratic nonlinear response based on two-frequency wave mixing of the fundamental and second-harmonic waves. Self-phase modulation of the fundamental wave results from a cascading process consisting of four second-order subprocesses, the direct and reverse subprocesses of Type I second-harmonic generation (SHG) and the direct and reverse subprocesses of Type II SHG. It is found analytically and numerically that the fundamental wave passing through a quadratic medium, tuned for simultaneous near phase matching for these two processes, collects 60% more nonlinear phase shift than does the corresponding two-step cascading. We also obtain the conditions for stationary waves (nonlinear modes) supported by such multistep cascading processes.