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 , 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.

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 and , as shown in Figure 2(a), are treated both numerically and analytically. We assume the fundamental Gaussian beam (wavelength and beam width w) 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.

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]:

$$\begin{split} \frac{\partial E\_1}{\partial z} &= \frac{i}{2k\_1} \nabla\_\perp^2 E\_1 - i \frac{\omega\_1^2 \chi^{(2)}(x)}{k\_1 c^2} E\_1^\* E\_2 e^{i(\mathbf{k}\_\flat - 2\mathbf{k})\_z}, \\ \frac{\partial E\_2}{\partial z} &= \frac{i}{2k\_2} \nabla\_\perp^2 E\_2 - i \frac{\omega\_2^2 \chi^{(2)}(x)}{2k\_2 c^2} E\_1^2 e^{i(2\mathbf{k}\_\flat - \mathbf{k})\_z}. \end{split} \tag{1}$$

In these equations and are the fundamental and SH frequencies, respectively. We assumed that the field can be decomposed into a superposition of these two frequencies, with stationary envelopes and fast oscillating term:

$$E = E\_1(x, z)e^{i\langle k, z - \omega, t\rangle} + E\_2(x, z)e^{i\langle k, z - \omega, t\rangle} + c.c.\tag{2}$$

fundamental wavelength in a good agreement with the calculated Čerenkov angle for the LiNbO3 [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 alters its sign. In Figure 3(b), we show the calculated SHG in a homogenous medium, i.e. . In this situation only the phase-mismatched, forward second harmonic signal is present. There is no trace of noncollinear Čerenkov harmonic signal. Compared with results shown in Figure 3(a), we confirm the presence of a sharp spatial variation of the

Figure 3. Far-field intensity of the second harmonic generation in composite media with fundamental beam propagating

Nonlinear Optical Effects at Ferroelectric Domain Walls http://dx.doi.org/10.5772/intechopen.77238 25

(a) along ferroelectric domain wall and (b) in a homogeneous medium.

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

in which kc 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 and the spatial distribution of the squared amplitude of the fundamental wave . Eq. (3) takes large value as long as its kernel undergoes a fast spatial variation. There are two ways to satisfy this condition. The first is to employ a spatial variation of the second-order nonlinearity in the transverse direction, e.g., propagating the fundamental wave along a ferroelectric domain wall [as shown in Figure 3(a)]. The other is to impose a strong spatial confinement to the fundamental beam, namely, to create

ð3Þ

nonlinearity forms a sufficient condition for efficient nonlinear Čerenkov radiation.

the following formula to describe the strength of the nonlinear Čerenkov signal [15]:

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 LiNbO3 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 boundary in nonlinear media. Figure 3(a) shows the SHG when the nonlinearity changes its sign across the boundary, i.e., . The strong emission of Čerenkov SHG is observed around 28.6� for the

Figure 2. Schematic of the simulation with second harmonic generation in optical media containing two layers of different nonlinear optical responses: .

Nonlinear Optical Effects at Ferroelectric Domain Walls http://dx.doi.org/10.5772/intechopen.77238 25

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 and , as shown in Figure 2(a), are treated both numerically and analytically. We assume the fundamental Gaussian beam (wavelength and beam width w) propagates along the boundary separating both media. To avoid any possible influence of the discontinuity in the linear polarization, the

The interaction of the fundamental and second harmonic waves in the nonlinear optical

In these equations and are the fundamental and SH frequencies, respectively. We assumed that the field can be decomposed into a superposition of these two frequencies, with

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 LiNbO3 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 boundary in nonlinear media. Figure 3(a) shows the SHG when the nonlinearity changes its sign across the boundary, i.e.,

Figure 2. Schematic of the simulation with second harmonic generation in optical media containing two layers of

. The strong emission of Čerenkov SHG is observed around 28.6� for the

ð1Þ

ð2Þ

medium is described by the following system of coupled wave equations [23]:

refractive index of the system is assumed to be homogenous.

stationary envelopes and fast oscillating term:

24 Ferroelectrics and Their Applications

different nonlinear optical responses: .

Figure 3. Far-field intensity of the second harmonic generation in composite media with fundamental beam propagating (a) along ferroelectric domain wall and (b) in a homogeneous medium.

fundamental wavelength in a good agreement with the calculated Čerenkov angle for the LiNbO3 [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 alters its sign. In Figure 3(b), we show the calculated SHG in a homogenous medium, i.e. . In this situation only the phase-mismatched, forward second harmonic signal is present. There is no trace of noncollinear Čerenkov harmonic signal. Compared with results shown in Figure 3(a), we confirm the presence of a sharp spatial variation of the nonlinearity forms a sufficient condition for efficient nonlinear Čerenkov radiation.

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]:

$$E\_{SH} \propto z \int\_{-\infty}^{+\infty} \chi^{(2)}(x) E\_1^2 \left(x, z\right) e^{ik\_+ x} dx,\tag{3}$$

in which kc 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 and the spatial distribution of the squared amplitude of the fundamental wave . Eq. (3) takes large value as long as its kernel undergoes a fast spatial variation. There are two ways to satisfy this condition. The first is to employ a spatial variation of the second-order nonlinearity in the transverse direction, e.g., propagating the fundamental wave along a ferroelectric domain wall [as shown in Figure 3(a)]. The other is to impose a strong spatial confinement to the fundamental beam, namely, to create

a spatially confined . This agrees with the presence of Čerenkov harmonic signal with an assumption of the well-defined rectangular profile of the fundamental beam in [25]. Similar effect was also reported in [26], which shows that a non-diffracting Bessel fundamental beam can lead to nonlinear Čerenkov radiation in a homogeneous crystal.

#### 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 is now a periodic function of spatial variable [28]. For 1D periodic domain pattern, the function can be expressed as the following Fourier series:

$$\chi^{\langle \mathfrak{I} \rangle}(x) = \sum\_{m=0,\pm 1,\pm 2} g\_m e^{imG\_b X}. \tag{4}$$

ð5Þ

27

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

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 kc in Eq. (5) changes. As a result, the intensity of the Čerenkov second harmonic signal varies as well considering the fact that different kc corresponds to different Fourier coefficients . In Figure 5, we show the wavelength response, i.e., the value of the Čerenkov SH generated by the fundamental Gaussian wave with different beam widths (a). 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 λ<sup>1</sup> = 1.108 μm) but falls dramatically at the others (e.g., at λ<sup>1</sup> = 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 a, 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

Figure 5. The spectral response of the Čerenkov SHG for different beam widths [28]. From (a) to (d), the beam widths of the fundamental wave are 60, 10, 5, and 2 μm, respectively. The plots are normalized to their individual maximum value.

dependence on wavelength.

Here is the primary reciprocal lattice vector (Λ is the modulation period of grating), the coefficients and with D being the duty cycle defined by the ratio of the length of the positive domains to the period of the structure. Considering a fundamental Gaussian beam, i.e., (with a being the beam width and x<sup>0</sup> denoting the central position of the beam), the integral in Eq. (3) can be evaluated as

Figure 4. (a) Scheme of Čerenkov SH emission in a 1D periodically poled LiNbO3 crystal. The right inset shows experimentally recorded far-field second harmonic image. The SH spots at small angles (θRN) represent the Raman-Nath emission, while the spot at bigger angles θ<sup>C</sup> is the Čerenkov second harmonics [15]. (b) The phase-matching diagram of the nonlinear Raman-Nath diffraction, which satisfies only the transverse phase-matching condition .

Nonlinear Optical Effects at Ferroelectric Domain Walls http://dx.doi.org/10.5772/intechopen.77238 27

$$\int \chi^{(2)}(x) E\_1^2(x) e^{ik\_+x} dx = a \{\pi/2\}^{12} \times \sum\_{m=0,\pm 1,\pm 2,\ldots} g\_m e^{-a^2(mG\_0+kJ)^2/8} e^{i(mG\_0+k\_0)x\_0}.\tag{5}$$

a spatially confined . This agrees with the presence of Čerenkov harmonic signal with an assumption of the well-defined rectangular profile of the fundamental beam in [25]. Similar effect was also reported in [26], which shows that a non-diffracting Bessel fundamental beam

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

In fact Eq. (3) can still be used to calculate the Čerenkov second harmonic from multiple ferroelectric domain walls, except that the is now a periodic function of spatial variable [28]. For 1D periodic domain pattern, the function can be expressed as the following Fourier series:

Here is the primary reciprocal lattice vector (Λ is the modulation period of grating), the coefficients and with D being the duty cycle defined by the ratio of the length of the positive domains to the period of the structure. Considering a fundamental Gaussian beam, i.e., (with a being the beam width and x<sup>0</sup> denoting the central position of the beam), the integral in Eq. (3) can

Figure 4. (a) Scheme of Čerenkov SH emission in a 1D periodically poled LiNbO3 crystal. The right inset shows experimentally recorded far-field second harmonic image. The SH spots at small angles (θRN) represent the Raman-Nath emission, while the spot at bigger angles θ<sup>C</sup> is the Čerenkov second harmonics [15]. (b) The phase-matching diagram of the nonlinear Raman-Nath diffraction, which satisfies only the transverse phase-matching condition .

ð4Þ

can lead to nonlinear Čerenkov radiation in a homogeneous crystal.

3.1. Nonlinear diffraction from multiple ferroelectric domain walls

from a dielectric grating.

26 Ferroelectrics and Their Applications

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 kc in Eq. (5) changes. As a result, the intensity of the Čerenkov second harmonic signal varies as well considering the fact that different kc corresponds to different Fourier coefficients . In Figure 5, we show the wavelength response, i.e., the value of the Čerenkov SH generated by the fundamental Gaussian wave with different beam widths (a). 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 λ<sup>1</sup> = 1.108 μm) but falls dramatically at the others (e.g., at λ<sup>1</sup> = 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 a, 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.

Figure 5. The spectral response of the Čerenkov SHG for different beam widths [28]. From (a) to (d), the beam widths of the fundamental wave are 60, 10, 5, and 2 μm, respectively. The plots are normalized to their individual maximum value.

In contrast to the Čerenkov emission defined by the fulfillment of the longitudinal phasematching 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., , for the mth diffraction order, m = 1,2,3… The external angles are then determined as follows:

$$m\sin\beta\_m = m\lambda\_2/\Lambda,\ m = 1,2,\ldots,\tag{6}$$

new emission peaks. We choose an average period m and consider that the domain width fluctuates randomly around its mean value. We consider four different degrees of randomness. From Figure 7(a)–(d), the randomness degree increases from 0 to 60%, which is defined by with representing the largest dispersion of the ferroelectric domain width. In the figure, we show the normalized harmonic strengths with respect to that of the first-order nonlinear Raman-Nath diffraction without any randomness, namely, . As shown in Figure 7(a), two intensity maxima, which correspond, respectively, to the first- and second-order Raman-Nath resonances, appear for the perfect periodic structure. They are marked as RN1 and RN2 in the figure, respectively. Increasing leads to the weakening of these two emission peaks and at the same time appearance of a few new ones, marked with indices N1, N2, and N3 in Figure 7(b)–(d). These new emitted signals become stronger and stronger

Figure 7. The influence of structure randomness of the ferroelectric domain patterns on the nonlinear Raman-Nath diffraction [29]. We take an average domain period m and consider four different degrees of structure randomness ranging from 0 to 60%. All emission strengths are normalized to that of the first-order Raman-Nath diffraction

Nonlinear Optical Effects at Ferroelectric Domain Walls http://dx.doi.org/10.5772/intechopen.77238 29

with , and finally their strengths can exceed those of the original emission resonances.

diffraction (m = 5).

without randomness, namely, RN1 in (a).

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 = 1.545 m, beam width a = 60 μm, duty cycle D = 0.35), the largest oscillation period takes place at the fifth-order

where is the SH wavelength. This is a generic condition that holds for any periodic structure and does not depend on its refractive index.

The intensity of the nonlinear Raman-Nath second harmonic diffraction depends strongly on the duty cycle of the grating [29]. In fact the impact of the duty cycle on the nonlinear Raman-Nath diffraction is very similar to that in linear diffraction on dielectric grating. The duty cycle directly determines the Fourier coefficient in Eq. (4), so it will cause the variation of the efficiency of nonlinear Raman-Nath diffraction. The detailed influence of duty cycle on the Raman-Nath diffraction from a periodic ferroelectric domain structure is shown in Figure 6. Agreeing quite well with the equation of Fourier coefficient , the first-order Raman-Nath harmonic diffraction (m = 1) takes the maximum intensity at duty cycle D = 0.5, while the second-order (m = 1) exhibits two equal maxima at D = 0.25 and 0.75, respectively.

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

Figure 6. The effect of duty cycle on the strengths of nonlinear Raman-Nath diffraction [29].

In contrast to the Čerenkov emission defined by the fulfillment of the longitudinal phasematching 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., , for the mth diffraction order, m = 1,2,3… The external angles

where is the SH wavelength. This is a generic condition that holds for any periodic

the second-order (m = 1) exhibits two equal maxima at D = 0.25 and 0.75, respectively.

Figure 6. The effect of duty cycle on the strengths of nonlinear Raman-Nath diffraction [29].

The intensity of the nonlinear Raman-Nath second harmonic diffraction depends strongly on the duty cycle of the grating [29]. In fact the impact of the duty cycle on the nonlinear Raman-Nath diffraction is very similar to that in linear diffraction on dielectric grating. The duty cycle directly determines the Fourier coefficient in Eq. (4), so it will cause the variation of the efficiency of nonlinear Raman-Nath diffraction. The detailed influence of duty cycle on the Raman-Nath diffraction from a periodic ferroelectric domain structure is shown in Figure 6. Agreeing quite well with the equation of Fourier coefficient , the first-order Raman-Nath harmonic diffraction (m = 1) takes the maximum intensity at duty cycle D = 0.5, while

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

ð6Þ

are then determined as follows:

28 Ferroelectrics and Their Applications

structure and does not depend on its refractive index.

Figure 7. The influence of structure randomness of the ferroelectric domain patterns on the nonlinear Raman-Nath diffraction [29]. We take an average domain period m and consider four different degrees of structure randomness ranging from 0 to 60%. All emission strengths are normalized to that of the first-order Raman-Nath diffraction without randomness, namely, RN1 in (a).

new emission peaks. We choose an average period m and consider that the domain width fluctuates randomly around its mean value. We consider four different degrees of randomness. From Figure 7(a)–(d), the randomness degree increases from 0 to 60%, which is defined by with representing the largest dispersion of the ferroelectric domain width. In the figure, we show the normalized harmonic strengths with respect to that of the first-order nonlinear Raman-Nath diffraction without any randomness, namely, . As shown in Figure 7(a), two intensity maxima, which correspond, respectively, to the first- and second-order Raman-Nath resonances, appear for the perfect periodic structure. They are marked as RN1 and RN2 in the figure, respectively. Increasing leads to the weakening of these two emission peaks and at the same time appearance of a few new ones, marked with indices N1, N2, and N3 in Figure 7(b)–(d). These new emitted signals become stronger and stronger with , and finally their strengths can exceed those of the original emission resonances.

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 = 1.545 m, beam width a = 60 μm, duty cycle D = 0.35), the largest oscillation period takes place at the fifth-order diffraction (m = 5).

image reflecting the spatial distribution of ferroelectric domain walls (and subsequently domains) inside the crystal. This is a nondestructive imaging method and can offer submicrometer 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.

Nonlinear Optical Effects at Ferroelectric Domain Walls http://dx.doi.org/10.5772/intechopen.77238 31

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 LiNbO3 and LiTaO3 crystals, the Čerenkov angle becomes larger at shorter wavelength, so the fundamental wave-

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 objective lens (Plan Apochromat, NA = 1.46). A pair of galvanometric mirrors is used to adjust the focus position in the X-Y plane, and a motorized stage is used to move the objective lens in the Z direction. To collect and detect the emitted second harmonic signal, a condenser lens and a

Figure 9. Schematic of the Čerenkov harmonic imaging system for visualization of ferroelectric domain patterns in a

nonlinear photonic crystal (NPC) [18].

length used for the visualization cannot be shorter than the critical wavelength.

Figure 8. Multi-order nonlinear Raman-Nath SHG as a function of crystal thickness (or interaction distance) [29].
