2. Approach and methodology

direction, called ferroelectric domains. The interfaces separating different domains in a crystal are called domain walls. For example, there are "180 walls" separating domains with oppositely orientated polarizations and "90 walls" separating regions with mutually perpendicular polarizations. The ferroelectric domain walls have symmetry and structure different from their parent materials and consequently possess many various physical properties including huge

Lithium niobate (LiNbO3) is a ferroelectric crystal with important photonics applications thanks to its excellent electro-optic, acousto-optic, and nonlinear optical properties. The crystal supports two distinct orientations of the spontaneous polarization along its optical (z) axis, i.e., only 180 domains exist in LiNbO3 crystals. Most importantly for nonlinear optical applications, the ferroelectric domains in LiNbO3 crystal can be periodically aligned by using external stimuli such as external electric field [9] or intense light field [10–13]. The alternative orientations of spontaneous polarization amounts to a spatial modulation of the second-order nonlinear coefficient of the crystal, an essential condition of the so-called quasi-phase-matching (QPM) technique, where the phase mismatch of a nonlinear optical process is compensated by one of the resulting reciprocal lattice vectors induced by the nonlinearity modulation. In the simplest case of second harmonic generation (SHG) in the medium, the quasi-phase-matching condition (which is equivalent to conservation of the momentum of interacting waves) can be expressed as , where k<sup>2</sup> and k<sup>1</sup> represent wave vectors of the second harmonic and fundamental waves, respectively. G is the magnitude of the reciprocal vector of the

It has been recently reported that efficient second-order nonlinear optical effects can also occur in an extreme case where only a single-domain wall was involved [14–16]. In fact the steep

ance of the so-called nonlinear Čerenkov radiation, whose emission angle is defined by the longitudinal phase-matching condition [17]. In case of frequency doubling via the Čerenkov second harmonic generation (ČSHG), the second harmonic signal is observed at the angle defined as [see Figure 1(a)]. The nonlinear Čerenkov interaction has been intensively investigated recently to fully understand all aspects of this fundamental

Figure 1. (a) The phase-matching diagram of Čerenkov-type second harmonic generation, where the harmonic emission angle is determined by the longitudinal phase-matching condition, i.e., . (b) Illustration of an experimental observation of the Čerenkov second harmonic generation at a single ferroelectric domain wall. FB, funda-

mental beam, and SH, second harmonic; represent the second-order nonlinear coefficient of the material [15].

) nonlinearity across the domain wall gives rise for the appear-

conductivity and anomalous dielectric responses [4–7].

nonlinearity grating.

22 Ferroelectrics and Their Applications

change of the second-order (χ<sup>2</sup>

The authors of this book chapter have been active in the field of nonlinear Čerenkov radiation from domain-engineered ferroelectric crystals for many years. Their latest research outcomes constitute the main body of this review. More details about these research works are available in the authors' publications, which have been correctly cited in the "References." Meanwhile, the authors have also reviewed other research groups' Google Scholar articles on this topic and have included some milestones in this chapter. These research progresses are organized into two categories according to the number of ferroelectric domain walls involved in the interaction, namely, the nonlinear Čerenkov radiations from a single-domain wall and those from multiple walls. In each category, not only experimental research but also theoretical treatment (using, e.g., the standard fast Fourier-transform-based beam propagation method) have been presented.
