1. Introduction

The ferroelectric phenomenon was discovered in 1921 by J. Valasek during an investigation of the anomalous dielectric properties of Rochelle salt, NaKC4H4O6 4H2O [1]. During the last few decades, the group of ferroelectric materials has been extended to over 250 pure materials and many more mixed crystal systems. They are intensively investigated because of a wide range of actual and potential applications of ferroelectric in critical fields such as electronics, nonvolatile memories, photonics, photovoltaics, etc. [2–8]. The ferroelectric materials generally consist of small uniform regions in which the spontaneous polarization points to the same

© 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited. © 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 conductivity and anomalous dielectric responses [4–7].

phenomenon and also because of a number of important actual and potential applications in nonlinear optical microscopy [18, 19], ultrashort pulse characterization [20], high harmonic

Nonlinear Optical Effects at Ferroelectric Domain Walls http://dx.doi.org/10.5772/intechopen.77238 23

In this chapter we review the latest research achievements in both experimental and theoretical studies of the nonlinear Čerenkov interactions that are closely associated with the existence of ferroelectric domain walls. In particular, we discuss two situations, namely, the nonlinear effects arising from a single ferroelectric domain wall and those coming from the multiple domain walls. We solve nonlinear coupled equations for the second harmonic generation and show how the efficiency of these nonlinear interactions depends on the structures of ferroelectric domain patterns and conditions of fundamental wave. These results are important for better understanding of second-order nonlinear optics and inspire optimizing the process for

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

3. Čerenkov-type second harmonic generation from a single ferroelectric

The experimental generation of the Čerenkov second harmonic is schematically illustrated in Figure 1(b). The fundamental beam (FB) generally propagates along a ferroelectric domain wall. A pair of beams at doubled frequency, i.e., the second harmonic (SH), is observed in the far field. Their emission angle agrees with that defined by the longitudinal phase-matching condition, i.e., as , where and are refractive indices of the fundamental and second harmonic waves, respectively. It is clear that the Čerenkov angle depends strongly on material properties. It is worth noting that the efficiency of Čerenkov harmonic generation in a single-domain (homogeneous χ(2)) crystals is low and its experimental observations have been scarce. As we show below, the emission of Čerenkov signal can be strongly enhanced by the presence of ferroelectric domain wall in the beam illuminated area.

generations [21], and functional materials analysis [22].

practical applications.

presented.

domain wall

2. Approach and methodology

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 nonlinearity grating.

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 change of the second-order (χ<sup>2</sup> ) nonlinearity across the domain wall gives rise for the appearance 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, fundamental beam, and SH, second harmonic; represent the second-order nonlinear coefficient of the material [15].

phenomenon and also because of a number of important actual and potential applications in nonlinear optical microscopy [18, 19], ultrashort pulse characterization [20], high harmonic generations [21], and functional materials analysis [22].

In this chapter we review the latest research achievements in both experimental and theoretical studies of the nonlinear Čerenkov interactions that are closely associated with the existence of ferroelectric domain walls. In particular, we discuss two situations, namely, the nonlinear effects arising from a single ferroelectric domain wall and those coming from the multiple domain walls. We solve nonlinear coupled equations for the second harmonic generation and show how the efficiency of these nonlinear interactions depends on the structures of ferroelectric domain patterns and conditions of fundamental wave. These results are important for better understanding of second-order nonlinear optics and inspire optimizing the process for practical applications.
