## Abstract

Ferroelectric materials tend to form macroscopic domains of electric polarization. These domains have different orientations and coexist in the medium being separated by domain walls. In general, symmetry and structure of ferroelectric domain walls differ from their parent materials and consequently lead to abundant physical properties. In this book chapter, we review the nonlinear optical effects which are bundled with ferroelectric domain walls or whose properties can be significantly enhanced by the presence of domain walls. In particular, we have reviewed Google Scholar articles from 2008 to 2018 using the keywords “nonlinear Čerenkov radiation from ferroelectrics”. We show that the spatially steep modulation of the second-order nonlinear optical coefficient across the domain wall leads to strong emission of the Čerenkov second harmonic in bulk materials. This feature also enables an effective nondestructive method for three-dimensional visualization and diagnostics of ferroelectric domain structures with very high resolution and high contrast.

### Keywords

- ferroelectric domain wall
- second harmonic generation
- nonlinear Čerenkov radiation
- nonlinear diffraction
- optical imaging

## 1. Introduction

The ferroelectric phenomenon was discovered in 1921 by J. Valasek during an investigation of the anomalous dielectric properties of Rochelle salt, NaKC_{4}H_{4}O_{6} · 4H_{2}O [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, 3, 4, 5, 6, 7, 8]. The ferroelectric materials generally consist of small uniform regions in which the spontaneous polarization points to the same 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, 5, 6, 7].

Lithium niobate (LiNbO_{3}) 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 LiNbO

_{3}crystals. Most importantly for nonlinear optical applications, the ferroelectric domains in LiNbO

_{3}crystal can be periodically aligned by using external stimuli such as external electric field [9] or intense light field [10, 11, 12, 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

k

_{2}and

k

_{1}represent wave vectors of the second harmonic and fundamental waves, respectively.

*is the magnitude of the reciprocal vector of the nonlinearity grating.*G

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, 15, 16]. In fact the steep change of the second-order (χ^{2}) 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

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.

## 2. Approach and methodology

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.

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

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

^{(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.

For a better understanding of the Čerenkov-type second harmonic generation at a single ferroelectric domain walls, the nonlinear optical interactions from a material system consisting of semi-infinite regions with different quadratic nonlinear responses

*) propagates along the boundary separating both media. To avoid any possible influence of the discontinuity in the linear polarization, the refractive index of the system is assumed to be homogenous.*w

The interaction of the fundamental and second harmonic waves in the nonlinear optical medium is described by the following system of coupled wave equations [23]:

In these equations

Here only the contributions from the diffraction and the quadratic nonlinearity are included, and no transient behavior or interface enhanced linear and/or nonlinear effects are considered.

We numerically solve the Eq. (1) by using the standard fast-Fourier-transform-based beam propagation method. We use the dispersion data of LiNbO_{3} crystal [24] in simulations. In Figure 3, we depict the far-field SH distributions versus the propagation distance, calculated with the fundamental beam propagating along two types of

_{3}[24]. It is clearly seen that the Čerenkov signal grows monotonically with the interaction distance, which is a typical feature of the longitudinally phase-matched nonlinear interactions. This simulation represents the experimental generation of nonlinear Čerenkov radiation on a single ferroelectric domain wall, across which the second-order nonlinear coefficient

The behavior becomes even clearer if we analytically deal with the frequency conversion process assuming the undepleted fundamental beam. In this case from Eq. (1), we can obtain the following formula to describe the strength of the nonlinear Čerenkov signal [15]:

in which * k*represents the transverse component of the Čerenkov second harmonic wave vector. According to Eq. (3) the amplitude of the Čerenkov signal is defined by the Fourier transform of the product of nonlinearity distribution function

_{c}

### 3.1. Nonlinear diffraction from multiple ferroelectric domain walls

When the fundamental beam is wide enough to cover multiple ferroelectric domain walls, the second harmonic shows more complicated far-field intensity distribution. In fact each domain wall can contribute toward its own Čerenkov second harmonic, and these harmonics will interfere with each other, leading to the so-called nonlinear diffraction [27]. As shown in Figure 4(a), the second harmonic pattern in this case generally consists of two types of spots [15]: (i) peripheral Čerenkov harmonic spots, situated relatively far from the fundamental beam at both sides of the diffraction pattern (top and bottom pairs in the figure), and (ii) central diffraction spots, grouped around the pump position, which is called nonlinear Raman-Nath diffraction, because of its close analogy to the linear Raman-Nath diffraction from a dielectric grating.

In fact Eq. (3) can still be used to calculate the Čerenkov second harmonic from multiple ferroelectric domain walls, except that the

Here

*being the duty cycle defined by the ratio of the length of the positive domains to the period of the*D

*being the beam width and*a

x

_{0}denoting the central position of the beam), the integral in Eq. (3) can be evaluated as

According to the definition of the Čerenkov second harmonic generation, the variation of the fundamental wavelength leads to the harmonic emission at different angles, i.e. the spatial frequency * k*in Eq. (5) changes. As a result, the intensity of the Čerenkov second harmonic signal varies as well considering the fact that different

_{c}

*corresponds to different Fourier coefficients*k

_{c}

*). When a wide fundamental beam is used, for instance,*a

*60 μm, the strength of the Čerenkov signal is very sensitive to the wavelength, showing a series of intensity peaks [see Figure 5(a)]. The emission is quite strong at these peak wavelengths (e.g., at*a =

λ

_{1}= 1.108 μm) but falls dramatically at the others (e.g., at λ

_{1}= 1.038 μm). Such a sensitive dependence of the Čerenkov second harmonic intensity on the fundamental wavelength is a typical characteristic of light interference in the case of multiple domain walls. It is very interesting to see that, depending on the value of beam width

*, the wavelength tuning shows weaker dependence on the wavelength, namely, the less contribution from the interference effect. Finally when the width of the fundamental beam becomes so narrow that it covers only a single-domain wall (e.g.,*a

*= 2 μm), all second harmonic peaks disappear, and the Čerenkov intensity exhibits monotonic dependence on wavelength.*a

In contrast to the Čerenkov emission defined by the fulfillment of the longitudinal phase-matching condition, the other group of the second harmonic diffraction spots (central spots located close to the pump in Figure 4(a)) only satisfies the transverse phase-matching (TPM) conditions, i.e.,

*th diffraction order,*m

*= 1,2,3… The external angles are then determined as follows:*m

where

The intensity of the nonlinear Raman-Nath second harmonic diffraction depends strongly on the duty cycle of the

*= 1) takes the maximum intensity at duty cycle*m

*= 0.5, while the second-order (*D

*= 1) exhibits two equal maxima at*m

*= 0.25 and 0.75, respectively.*D

We consider now the influence of the structure randomness of ferroelectric domain patterns on the Raman-Nath harmonic diffraction. It is well known that the fabrication process of periodic domain patterns in ferroelectric crystals often introduces some degree of randomness in otherwise fully periodic domain structure. For the collinear quasi-phase-matching frequency conversion processes, the randomness generally has a negative impact because it reduces frequency conversion efficiency. The situation becomes more complicated when it comes to the nonlinear Raman-Nath diffraction. As we show in Figure 7, the randomness of the domain pattern not only affects the efficiency of nonlinear diffraction but also leads to appearance of new emission peaks. We choose an average period

In Figure 8, we display the calculated dependence of the nonlinear Raman-Nath diffractions on the interaction distance. As the Raman-Nath interactions suffer from the phase mismatch in the longitudinal direction, their intensity oscillates with the interaction distance inside the crystal. Obviously the smaller the phase mismatch, the longer the oscillation period. With the parameters used in our calculation (fundamental wavelength

*= 60 μm, duty cycle D = 0.35), the largest oscillation period takes place at the fifth-order diffraction (*a

*= 5).*m

## 4. Application of Čerenkov harmonic generation to 3D imaging of ferroelectric domain patterns

High-quality visualization of ferroelectric domain structures plays a key role in understanding material property and better control of domain inversion process. However, due to the equality of antiparallel 180° ferroelectric domains in linear optical properties, the common imaging techniques do not apply to the detection of these domains. Currently the selective chemical etching [30] based on the different etching rates of antiparallel domains in hydrofluoric acid is still the most common method used for this purpose. Not only this is a destructive technique, but moreover, it is ineffective in revealing the internal domain structures hidden deep inside the crystals. To overcome some of these drawbacks, a large number of alternative approaches have been adapted to imaging ferroelectric domains, such as advanced electron microscopy [31] and piezo-forced microscopy [32]. However, most of these methods also fail in visualization of deep internal domain structures, which can be diverse and more complex than those on the surface [33]. For example, the inverted domains undergo sidewise expansion with depth [34], transform to preferred shapes depending on the crystallographic symmetry [35, 36], and sometimes merge to form a bigger structures [37]. The details about these domain formation processes such as when, where, and how they occur are very little known due to the lack of reliable techniques for three-dimensional visualization of domain patterns.

It has been shown in Section 2 that when a femtosecond laser beam is tightly focused to produce a focus that is narrow enough to cover a single ferroelectric domain wall, a pair of Čerenkov second harmonic beams will be generated. The Čerenkov signal disappears if the laser beam is moved away from the domain wall. In this way, by recording the second harmonic strengths at different positions inside the crystal, one can obtain a three-dimensional image reflecting the spatial distribution of ferroelectric domain walls (and subsequently domains) inside the crystal. This is a nondestructive imaging method and can offer sub-micrometer resolution because of its nonlinear optical mechanism [18]. This is a 3D optical method as it also enables one to reveal the details of inverted domains beneath the surface.

Figure 9 displays a schematic illustration of the nonlinear Čerenkov second harmonic imaging system. The fundamental femtosecond laser beam is provided here by a titanium-sapphire laser (Mai Tai, Spectra Physics, 80 MHz repetition rate and up to 12 nJ pulse energy). It is known that in the regime of a tightly focused fundamental beam, the Čerenkov process is insensitive to the wavelength of the fundamental wave. Therefore, this imaging system can operate at a wide range of wavelength limited only by the absorption edge of second harmonic and the total reflection condition. The latter condition means the Čerenkov harmonic emission angle has to be smaller than its total reflection angle so that the Čerenkov signals can get out from the sample for detection. For traditional nonlinear optical ferroelectric materials, such as LiNbO_{3} and LiTaO_{3} crystals, the Čerenkov angle becomes larger at shorter wavelength, so the fundamental wavelength used for the visualization cannot be shorter than the critical wavelength.

The main part of this imaging system is a commercial laser scanning confocal microscope (Zeiss, LSM 510 + Axiovert 200). The femtosecond laser beam is coupled into the confocal microscope and then illuminates the sample after being tightly focused by an

The advantages of this nonlinear Čerenkov imaging system include:

### 4.1. High contrast and high spatial resolution

A typical two-dimensional image of a ferroelectric domain structure obtained by the Čerenkov second harmonic microscope is shown in Figure 10(a). The quasi-periodic domain patterns, where the bright boundaries represent ferroelectric domain walls which facilitate stronger Čerenkov harmonic emissions, were clearly seen. Obviously the Čerenkov second harmonic microscope is capable of imaging ferroelectric domains with high contrast.

Figure 10(b) depicts the image of ferroelectric domain patterns obtained in an as-grown Sr_{0.28}Ba_{0.72}Nb_{2}O_{6} crystal, which process naturally random domain structures in two dimensions. It is seen that the Čerenkov method offers an exceptional spatial resolution and even domain boundaries separated by less than 250 nm can be easily resolved. This is below the diffraction limit for the excitation laser wavelength of 820 nm, owing to the mechanism of nonlinear optical interaction, i.e., the Čerenkov second harmonic signal can only be excited in the very central part of the laser beam’s focus.

### 4.2. Applicability to a wide range of materials

The imaging principle of the Čerenkov SHG microscope lies in the sensitivity of the Čerenkov emission on the existence of the spatial variation of the second-order nonlinearity

_{3}, LiTaO

_{3}, KTiOPO

_{4}, and Sr

_{0.28}Ba

_{0.72}Nb

_{2}O

_{6}crystals, as shown in Figure 11.

### 4.3. Capability for 3D imaging

As we described above, the scanning of laser focus in the X-Y plane enables us to obtain two-dimensional images of ferroelectric domains. Then if we stack a series X-Y plane images recorded at different depths inside the material, we can produce 3D images of domains. This is an advantage that cannot be met by the traditional domain imaging techniques. In Figure 12 we show a number of 3D images of ferroelectric domain patterns, which are formed in a congruent LiNbOb_{3} crystal [38]. From these images we can see how the initially circular-shaped domains transform to hexagons with depth [Figure 12(b)], how defects were formed during the domain inversion process [Figure 12(c)], and how the neighboring domains merge to form a bigger one [Figure 12(d)]. Revealing these details is essential for a full understanding of domain inversion and growth processes. This is also very useful for improving the quality of ferroelectric domain patterns, which is critical for a wide range of future applications.

## 5. Conclusion

We have investigated the nonlinear optical interactions that are strongly dependent on the existence of ferroelectric domain walls, i.e., the spatial variation of the second-order nonlinear coefficient

## Acknowledgments

The authors thank the Australian Research Council and Qatar National Research Fund (Grant No. NPRP 8-246-1-060) for financial supports. Mr. Xin Chen thanks China Scholarship Council (PhD Scholarship No. 201306750005). The authors thank Dr. Vito Roppo, Dr. Ksawery Kalinowski, Dr. Qian Kong, Dr. Wenjie Wang, Prof. Crina Cojocaru, and Prof. Joes Trull for their valued contributions to this work.

## Conflict of interest

There is no conflict of interest for this work.