⇦Chinese Physics Letters, 2018, Vol. 35, No. 3, Article code 034203 Noncolinear Second-Harmonic Generation Pairs and Their Scatterings in Nd$^{3+}$:SBN Crystals with Needle-Like Ferroelectric Domains * Tian-Run Feng(冯天闰)1, Hui-Zhen Kang(康慧珍)2, Lei Feng(冯蕾)1, Jia Yang(杨嘉)2, Tian-Hao Zhang(张天浩)1**, Feng Song(宋峰)1, Jing-Jun Xu(许京军)1, Jian-Guo Tian(田建国)1, L. I. Ivleva3 Affiliations 1Photonics Research Center, School of Physics, The MOE Key Lab of Weak-Light Nonlinear Photonics, and Tianjin Key Lab of Photonics Materials and Technology for Information Science, Nankai University, Tianjin 300071 2Basic Department of Military Transportation College, Tianjin 300161 3Institute for General Physics of the Russian Academy of Science, Moscow 119991, Russia Received 18 December 2017, online 25 February 2018 *Supported by the National Natural Science Foundation of China under Grant No 61675101.
**Corresponding author. Email: zhangth@nankai.edu.cn Citation Text: Feng T R, Kang H Z, Feng L, Yang J and Zhang T H et al 2018 Chin. Phys. Lett. 35 034203 Abstract Second-harmonic generation in Nd$^{3+}$:SBN crystal with needle-like ferroelectric with aperiodic domain structures is investigated. Two pairs of second harmonic (SH) waves appearing in lines are observed in unpoled Nd$^{3+}$:SBN crystals with aperiodic needle-like domains. A pair of SH waves emit from the exit face, whose intensities are angle-dependent. The angular dependence is corresponding to the spatial frequency spectrum of the aperiodic domain structure. Another pair of SH waves emit from both the side surfaces, which are mainly the scattered SH waves by needle-like domain walls and obey the theory of Rayleigh scattering. DOI:10.1088/0256-307X/35/3/034203 PACS:42.65.Ky, 75.60.Ch, 42.70.Mp © 2018 Chinese Physics Society Article Text Efficient second-harmonic generation (SHG) depends critically on phase-matching (PM) conditions between the fundamental and the second-harmonic waves. In general, the wave vectors of fundamental and harmonic waves are parallel to each other and the momentum conservation law is fulfilled, $$ k_{2}=k_{1}+k'_{1},~~ \tag {1} $$ where $k_{1}$ and $k'_{1}$ are the wave vectors of the fundamental waves, and $k_{2}$ is the harmonic one. However, this rigorous collinear phase matching condition is difficult to realize. Then a 'phase corrective scheme', termed as the quasi-phase-matching (QPM) scheme, was proposed by Armstrong et al., and now it is applied in various practical nonlinear optical crystals.[1] Conventionally, this quasi-phase matching can be realized by using the vector $k_{\rm g}$ of the periodical inversed domain grating as an additional vector to compensate for the phase mismatching between the fundamental and SH waves, where $k_{2}=2k_{1}+k_{\rm g}$ is satisfied.[2-6] Recently, disordered nonlinear media attract researchers due to the relaxable QPM conditions and the tunable, efficient, broad bandwidth regime of the frequency conversion achieved in some media.[7-10] An example of a quadratic nonlinear medium with randomized domain structure is an unpoled strontium barium niobate (SBN) crystal. The domain structure provides a 2D continuous set of grating vectors $k_{\rm g}$ in SBN crystals, in the reciprocal or momentum space useful for frequency conversion, which has been demonstrated by different researchers using various configurations.[11-14] Considering the large length of the quasi-cylindrical ferroelectric domains in SBN crystals along the optical axis, the set of grating vectors for QPM-SHG will be located mainly in the plane perpendicular to the $c$-axis. The character of the generated SH emission strongly depends on the direction of propagation with respect to the $c$-axis. When the fundamental beam is propagated parallel to the $c$-axis of an SBN crystal, conical non-collinear phase-matched SH waves will be generated.[11] When the fundamental beam propagates perpendicular to the crystal $c$-axis, the SH will be generated in the $c$-plane, which is perpendicular to the $c$-axis.[12-14] However, for the latter configuration, the QPM mechanisms of the SHG present in the $c$-plane are still controversial up to now. Satoru et al. reported an observation of second harmonic diffuse green light generated from the domains by illuminating the domains of an unpoled SBN crystal with 1.06 μm radiation in 1998.[15] The intensity profile along the SHG emission curve was measured. However, these SHG lights which can clearly be seen with the naked eye would be impossible for power application because the second harmonic emission is from all the domains randomly located and illuminated, and the phase matching of the SHG light was very poor. As a result of all the above factors combined, two mechanisms have been proposed. One is the noncollinear QPM scheme corresponding to the whole $k_{\rm g}$ space,[12,13] and the other is the combination of a planar scattering of the fundamental radiation followed by efficient collinear QPM.[14] Both of the above point of views do not contain the clear quantitative studies on the QPM and physical investigation on the scattering mechanisms. However, from our theoretical and experimental studies, we found that when the fundamental beam is propagated perpendicular to the crystal $c$-axis, the SH waves should be two pairs and appear in lines. A pair of SH lights will emit from the exit face with angular depended intensities, which are excited by the incident fundamental waves through QPM, and the other pair of SH lights will emit from both the side surfaces, which are mainly composed of scattered light from the QPM SH waves generated in the main propagating path of the incident beam by walls of needle-like domains. In this Letter, we demonstrate clearly the character and mechanism of SHG in this configuration. The QPM is proved to be noncollinear and can be fulfilled in a wide spatial angle, where the corresponded additional momentum $k_{\rm g}$ is provided by the spatial frequency of aperiodic domain structure, and the angular depended SH intensity is corresponding to the spectrum of $k_{\rm g}$. Meanwhile, the mechanism and characteristics of one-dimensional scattering of the quasi-phase matched SHG by needle-like domains in an unpoled Nd$^{3+}$:SBN crystal are studied. The Nd$^{3+}$:SBN was grown by the modified Stepanov technique.[16] The point group at room temperature is 4 mm. Ferroelectric domains in the crystal are always parallel or anti-parallel, and their shapes are mostly needle-like. When the Nd$^{3+}$:SBN single crystal is cooled through its transition temperature in the absence of a poling field, numerous needle-like ferroelectric domains appear in it. In our experiments, an unpoled Nd$^{3+}$:SBN crystal with needle-like domains was cut along the principle axes with dimensions of 7.1 mm$\times$8.0 mm$\times$12.0 mm. The experimental setup is shown in Fig. 1. An extraordinarily polarized laser beam (semiconductor laser at a wavelength of 1064 nm, 75 mw) was focused into the crystal by a lens ($f=4$ mm). The images at the exit face are recorded by a digital camera. The second harmonic lights ($\lambda =532$ nm) can be observed by the naked eye in four spatial regions. A pair of SH waves emit from the exit face, and another pair of SH waves emit from both the side surfaces, which are the scattered lights of the SHG by the ferroelectric domain wall, as shown in Fig. 1.
cpl-35-3-034203-fig1.png
Fig. 1. (Color online) Scheme of the experimental setup.
The pair of green lights emitting from the exit face is $e$-polarized and in a wide spatial angular range, which display a symmetrical distribution about the direction of the fundamental light, as shown in Figs. 1 and 2. Figure 2(c) is the pair of SH lights concentrated by the lens behind the exit face, which cannot be seen simultaneously without the lens. The circular spot at the center in Fig. 2(c) is the exiting fundamental light. The intensities of the SH lights vary with their exit directions.
cpl-35-3-034203-fig2.png
Fig. 2. (Color online) [(a), (b)] The pair of SH lights emitting from the exit face imaged in different directions, respectively; and (c) the pair of SH lights concentrated by lens behind the exit face.
cpl-35-3-034203-fig3.png
Fig. 3. (Color online) (a) Probability density of spatial frequency spectrum as a function of $k_{\rm g}$ in needle-like domain structure arising from a one-dimensional array of 2000 randomly arranged needle-like domains. (b) The density of ferroelectric domains as a function of the domain size. The picture on the left shows the structure of chaotic domain arrangement with different sizes.
Nd$^{3+}$:SBN domains have quasi-cylindrical (needle-like) shape with the longest dimension parallel to the optical axis and with the diameter in size from 1 μm to 10 μm. The densities of ferroelectric domains are shown as a function of the domain size in the inset of Fig. 3. The random distribution of these domains is aperiodic. We arrange $n$ (=2000) needle-like domains according to the inset of Fig. 3 randomly, as shown in the left side of Fig. 3. The spatial location of needle-like domains is described as $$\begin{align} \sum\limits_{i=1}^n {\delta (x-x_i)}.~~ \tag {2} \end{align} $$ The corresponded spatial frequency spectrum and probability density can be calculated by the discrete Fourier transformation $$\begin{align} F(k_{\rm g})=\,&\sum\limits_{i=1}^n \int\limits_{-\infty }^{+\infty } \delta (x-x_i) e^{-ik_{\rm g} x}d_x =\sum\limits_{i=1}^n e^{-ik_{\rm g} x_i },~~ \tag {3a}\\ |F(k_{\rm g})|^2=\,&\sum\limits_{i=1}^n \sum\limits_{j=1}^n \cos k_{\rm g} (x_j -x_i).~~ \tag {3b} \end{align} $$ The probability density of spatial frequency $k_{\rm g}$ of the randomly distributed domains is depicted in Fig. 3. The most probable value of $k_{\rm g}$ is 3.15 μm$^{-1}$ and the full width at half maximum (FWHM) of the probability intensity profile is 0.85 μm$^{-1}$. For the arrangement of needle-like domains, $k_{\rm g}$ plays a role of additional momentum, the direction of $k_{\rm g}$ is perpendicular to the $c$-axis, i.e., in the direction of the needle-like domains. In this case, the phase matching is extended, and the momentum geometry for the quasi-phase matching of Eq. (4) is sketched in Fig. 4.
cpl-35-3-034203-fig4.png
Fig. 4. (Color online) Momentum diagram for Eq. (4). Here $k_{1}$, $k'_{1}$ and $k_{2}$ are perpendicular to the $c$-axis, and $k_{\rm g}$ is also perpendicular to the $c$-axis, the end of which is with a distribution indicated by the dashed line. The corresponding angle-dependent intensity distribution of the SH light wave is sketched by the curve on the right.
From the momentum diagram of Fig. 4, $k_{\rm g}$ is interpreted in a vectorial sense, whose direction varies in a wide spatial angle, whose end is located on the circle with a radius of $k_{2}$. Thus the QPM can be fulfilled in a wide spatial angle. This means that SH waves can emit in a wide spatial angle. The angle between the fundamental wave vector and the SH wave vector is denoted as $\theta$. The distribution of angular depended SH intensity is corresponding to the probability density of $k_{\rm g}$. Here the relationship between $k_{\rm g}$ and the angle $\theta$ can be expressed as $$\begin{align} \cos \theta =\frac{(k_1 +{k}'_1)^2+k_2^2 -k_{\rm g}^2 }{2(k_1 +{k}'_1)k_2}.~~ \tag {4} \end{align} $$ From Figs. 3 and 4, the most probable value of $k_{\rm g}$ (=3.15 μm$^{-1}$ yields an internal $\theta$ of 68$^{\circ}$, corresponding to an external angle of 15.4$^{\circ}$. In our experiments, the direction of fundamental beam was chosen to be strictly perpendicular to the $c$-axis. The intensity distribution of SH light as a function of the spatial angle at the exit face of the crystal is shown in Fig. 5. The angular distribution of the SH intensity is in accordance with the probability density of $k_{\rm g}$. The angle between the exit direction of maximum SH intensity and the incident direction of fundamental wave is 14.7$^{\circ}$ corresponding to a momentum of $k_{\rm g}=3.02$ μm$^{-1}$. The experimental results are in very good agreement with the theoretical analysis. Figure 6 shows the scattered fundamental waves and SH waves emitted from one of the side surfaces of the crystal. The scattered SH waves are $e$-polarized and have a planar intensity angular distribution in the $c$-plane as shown in Fig. 7(a). The sizes of Nd$^{3+}$:SBN domains are from 1 μm to 10 μm, and the needle-like ferroelectric domain walls are in the dimension of about 10 nm–100 nm, which are far less than the wavelength of fundamental waves and SH waves in the experiment. Thus it should be the Rayleigh scattering. Moreover, there are no fundamental and SH lights emitting from the two surfaces perpendicular to the $c$-axis. This is because the domain is a quasi-cylindrical (needle-like) shape with the longest dimension parallel to the optical axis, and the scattering on the domain walls should be one-dimensional.
cpl-35-3-034203-fig5.png
Fig. 5. Intensity distribution of SH light as a function of the spatial angle at the exit face of the crystal.
cpl-35-3-034203-fig6.png
Fig. 6. (Color online) The scattered SH light emitted at one of the side surfaces of the crystal in a narrow angle: (a) the scattered fundamental beam, and (b) the corresponding scattered SHG.
cpl-35-3-034203-fig7.png
Fig. 7. (Color online) (a) The intensity distribution of the scattered lights emitting at one of the side surfaces of the crystal varies with the external scattering angle. (b) Intensity distribution of SH light as a function of the distance between the incident path and the side surface of the crystal. The curve of circles is for the scattered fundamental beam, and the curve of triangles is for the scattered SH light.
From our theoretical and experimental studies, we conclude that the QPM mechanism should be a noncolinear one, whose momentum geometry is as described in Fig. 4. This can be confirmed by the angular distribution of SH intensity at the exit face, where no SH wave is emitting at the direction of the incident fundamental beam. This phenomenon and its characteristics have never been involved and described before, which cannot be explained by the proposition of the combination of a planar scattering of the fundamental radiation followed by efficient collinear QPM. Moreover, according to the theoretical analysis about the probability density of $k_{\rm g}$, the value of $k_{\rm g}$ for collinear QPM has a slight probability, which cannot support sufficient SHG. The pair of SH lights emitting from both the side surfaces are the scattered lights of the above described QPM SH waves generated in the main propagating path of the incident beam by the ferroelectric domain walls, few of which are excited by the scattered fundamental waves. This is in accordance with the theory of the Rayleigh scattering, in which the intensity of the scattered wave is inversely proportional to $\lambda^{4}$, where $\lambda$ is the wavelength. Thus fundamental waves are less scattered, most of which are propagating along the main path. The weak scattered fundamental lights cannot support bright enough SH light. The SHG is due to second order nonlinear polarization ${\boldsymbol P}$, which is determined by the nonlinear susceptibility tensor. For the noncentrosymmetric ferroelectric phase of Nd$^{3+}$:SBN with a fundamental field ${\boldsymbol E}=(E_{x}(\omega), 0, E_{z}(\omega))$, ${\boldsymbol P}$ can be written as $$\begin{align} \left[\begin{matrix} P_x(2\omega)\\ P_y(2\omega)\\ P_z(2\omega)\\ \end{matrix}\right]=\,&2\left[\begin{matrix} 0&0&0&0&d_{15}&0\\ 0&0&0&d_{24}&0&0\\ d_{31}&d_{32}&d_{33}&0&0&0\\ \end{matrix}\right]\\ &\cdot\left[\begin{matrix} E_x(\omega)^2\\ 0\\ E_z(\omega)^2\\ 0\\ 2E_x(\omega)E_z(\omega)\\ 0\\ \end{matrix}\right],~~ \tag {5} \end{align} $$ where $d_{31}=d_{32}$ and $d_{15}=d_{24}$. As mentioned above, the polarization of the two pairs of SH waves are detected, which were $e$-polarized. Few of $o$-polarized QPM SH waves can be detected because $d_{15}$ and $d_{24}$ are very small. Because the scattering is one-dimensional, the scattered light is only in the $c$-plane. For $e$-polarized light, the scattered light in the $c$-plane is still $e$-polarized, according to the theory of the Rayleigh scattering. Both the scattered fundamental and the SH lights have a relatively planar intensity angular distribution as shown in Fig. 7(a). It is because it will be further scattered to other directions subsequently once the light (fundamental or SH ones) is scattered by the domain wall. As a result, the scattered light will diffuse in the $c$-plane. When the path of the incident light is gradually close to the side surface (in the range of 0–0.76 mm), the intensity of the scattered fundamental and SH lights sharply increases and decreases, respectively, as shown in Fig. 7(b). In this situation, the incident fundamental beam will be scattered by the edge of the crystal, which results in its diffusing in the whole crystal and sharp intensity increasing out of the side surface. At the same time, following the decrease of the power density of the fundamental beam induced by the edge scattering, the conversion efficiency from the fundamental beam to the SH beam sharply decreases, where 0.76 mm is decided by the size of the focusing light spot at the input face. In conclusion, it is very important to correctly understand the mechanism and characteristics of the QPM in aperiodic structure materials. The results may offer a thought to study the SHG and other nonlinear optical phenomena even some other physical phenomena in aperiodic structures. Meanwhile, the angular distribution of scattered second harmonic light is compared with predictions following from certain quasi-phase matching conditions and scattering electromagnetic theory. It represents a successful attempt to study nonlinear scattering of needle-like domain structure in the crystal with micro-ferroelectric domain. The SHG is widely used to generate intense coherent light in special wavelength regions (far ultraviolet), which cannot be accessed by conventional lasers or in regions of frequency-doubled lasers where it has an efficiency advantage. 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