Three-Wave Mixing of Dipole Solitons in One-Dimensional\\ Quasi-Phase-Matched Nonlinear Crystals

Yuxin Guo(郭宇欣)$^{1}$, Xiaoxi Xu(徐晓喜)$^{1}$, Zhaopin Chen(陈招拼)$^{2}$, Yangui Zhou(周延桂)$^{1,3}$,\\ Bin Liu(刘彬)$^{1,3}$, Hexiang He(和河向)$^{1,3\hyperlink{s}{*}}$, Yongyao Li(黎永耀)$^{1,3}$, and Jianing Xie(谢嘉宁)$^{1,3\hyperlink{s}{*}}$

doi:10.1088/0256-307X/41/1/014204

⇦Chinese Physics Letters, 2024, Vol. 41, No. 1, Article code 014204⇨ Three-Wave Mixing of Dipole Solitons in One-Dimensional Quasi-Phase-Matched Nonlinear Crystals Yuxin Guo (郭宇欣)1, Xiaoxi Xu (徐晓喜)1, Zhaopin Chen (陈招拼)2, Yangui Zhou (周延桂)1,3, Bin Liu (刘彬)1,3, Hexiang He (和河向)1,3*, Yongyao Li (黎永耀)1,3, and Jianing Xie (谢嘉宁)1,3* Affiliations 1School of Physics and Optoelectronic Engineering, Foshan University, Foshan 528000, China 2Physics Department and Solid-State Institute, Technion, Haifa 32000, Israel 3Guangdong-Hong Kong-Macao Joint Laboratory for Intelligent Micro-Nano Optoelectronic Technology, School of Physics and Optoelectronic Engineering, Foshan University, Foshan 528000, China Received 14 November 2023; accepted manuscript online 14 December 2023; published online 24 January 2024 ^*Corresponding authors. Email: sysuhhx@163.com; xiejianingfs@126.com Citation Text: Guo Y X, Xu X X, Chen Z P et al. 2024 Chin. Phys. Lett. 41 014204 Abstract A quasi-phase-matched technique is introduced for soliton transmission in a quadratic $[\chi^{(2)}]$ nonlinear crystal to realize the stable transmission of dipole solitons in a one-dimensional space under three-wave mixing. We report four types of solitons as dipole solitons with distances between their bimodal peaks that can be laid out in different stripes. We study three cases of these solitons: spaced three stripes apart, one stripe apart, and confined to the same stripe. For the case of three stripes apart, all four types have stable results, but for the case of one stripe apart, stable solutions can only be found at $\omega_{1}=\omega_{2}$, and for the condition of dipole solitons confined to one stripe, stable solutions exist only for Type1 and Type3 at $\omega_{1}=\omega_{2}$. The stability of the soliton solution is solved and verified using the imaginary time propagation method and real-time transfer propagation, and soliton solutions are shown to exist in the multistability case. In addition, the relations of the transportation characteristics of the dipole soliton and the modulation parameters are numerically investigated. Finally, possible approaches for the experimental realization of the solitons are outlined.

DOI:10.1088/0256-307X/41/1/014204 © 2024 Chinese Physics Society Article Text Quadratic $[\chi^{(2)}]$ nonlinear effects are widely leveraged in a variety of fields, including physics,[1,2] chemistry,[3] materials science,[4,5] optics,[6-8] communications,[9] and so on. The $\chi^{(2)}$ nonlinear effect refers to the fact that in the field of optics, when the light intensity is sufficiently large, the electric field of light has a $\chi^{(2)}$ effect on the polarizability of the medium, resulting in optical frequency conversion processes such as second harmonic generation,[10-12] sum frequency generation,[13,14] difference frequency generation,[15-17] and optical parametric amplification.[18-20] This effect is widely used in the frequency conversion processes in optical fields,[21-23] such as laser frequency doubling, electro-optical modulation, optical storage, and optical switching. It is well known that $\chi^{(2)}$ nonlinear photonic crystals are crystals that show more than two nonlinear optical effects for strong laser electric fields. Commonly used $\chi^{(2)}$ nonlinear optical crystals include potassium dihydrogen phosphate (KDP), ammonium dihydrogen phosphate, potassium dideuterium phosphate (DKDP), and lithium niobate (LiNbO$_3$).[24-29] Notably, $\chi^{(2)}$ nonlinear optical crystals are a functional material, in which the frequency doubling (or “frequency”) crystals can be used for frequency conversion of the laser wavelength, thus expanding the tunable range of lasers, which offers an important application value in the field of laser technology.[30-32] A soliton is a typical phenomenon in a nonlinear system, and it has attracted the interest of researchers from areas of quantum physics,[33,34] cold atoms,[35-37] condense matter,[38-41] optics,[42,43] etc. Here, we focus on optical soliton transport in nonlinear optical field modulation; $\chi^{(2)}$ nonlinear optical crystals show strong, and fast nonlinear optical responses are proved to be favorable platforms for studying soliton transport.[42,43] They can create ideal conditions for optical soliton transport due to their unique nonlinear effect and have been realized for a variety of solitons in some studies.[44,45] However, although $\chi^{(2)}$ nonlinear optical crystals can support transmission of solitons, they only support optical solitons in the fundamental state in one and two dimensions (1D and 2D),[46,47] and dipole solitons are not easily stabilized,[48,49] which features a spontaneous separation between two peaks.[50] Due to the different refractive indices of input and output waves with different wavelengths in a $\chi^{(2)}$ nonlinear photonic crystal, which results in different wave vectors, phase matching is impossible under natural conditions.[51,52] To overcome this problem, it becomes particularly important to find suitable conditions for phase matching. Recently, a checkerboard structure was introduced in the medium of $\chi^{(2)}$ nonlinear photonic crystals to realize the stable transmission of vortex solitons in three-dimensional (3D) space.[53] The quasi-phase-matched (QPM) technique was first proposed by Bloembergen in 1962 and, currently, it has been developed into a more mature technique.[51-57] In the QPM technique, the phase difference between different wavelengths was compensated for by periodically modulating the nonlinear polarizabilities of nonlinear photonic crystals,[58-63] thus improving the conversion efficiency of nonlinear optical frequencies.[64-67] Inspired by this report,[53] we introduce a QPM technique for 1D dipole soliton transmission in $\chi^{(2)}$ nonlinear photonic crystals, which realizes the stable transmission of a 1D dipole soliton in the second harmonic case.[50] While the second harmonic generation is typically a special case of three-wave mixing, we generalize it to the more universal and compatible case of three-wave mixing.[68-73] By QPM between the three waves, there are four types of dipole solitons that are obtained via the phase-matching condition.

Fig. 1. (a) Input plane ($Z=0$), with purple representing a position with phase $\pi$ and white for phase 0. A position with phase 0 and a position with phase $\pi$ constitute a cycle. $L$ is the length of a cycle, and $D$ is the length of phase 0 in a period ($D=L/2$). (b) The one-dimensional model of the crystal; at a position with phase 0, the beam is modulated in the $Z$-direction, as shown in (c1); at a position with phase $\pi$, the beam is modulated in the $Z$-direction, as shown in (c2). [(c1), (c2)] The duty cycle in the $Z$-axis positive polarization direction is fixed at 1/2.

We explore whether the 1D dipole soliton can be stabilized by the QPM technique for transmission in the presence of three-wave mixing. The solution of the dipole soliton obtained by the imaginary time propagation (ITP) of convergence is used,[74,75] and the stability of the soliton solution is verified using the real-time propagation (RTP) method.[76,77] Through exploration of the solution space, we find that the dipole soliton can be stably transmitted in a $\chi^{(2)}$ nonlinear photonic crystal under three-wave mixing with the introduction of QPM, and soliton solutions exist in the multistability case. In this Letter, first we introduce the simulation model. Then, we present the numerical results and discussions. The experimental parameters are estimated. Finally, we provide conclusions. Model. We use the QPM technique to modulate the solitons in the transmission direction $Z$. The equations for the evolution of the amplitudes of the idler wave, pump wave, and signal wave along propagation distance $Z$ are \begin{align} &i\partial_{Z}A_{1}=-\frac{1}{2k_{1}}\partial_{X}^2A_{1} -\frac{2d(Z,\,X)\omega_{1}}{cn_{1}}A_{2}^{*}A_{3}e^{-i\Delta k_{0}Z} , \tag {1} \\ &i\partial_{Z}A_{2}=-\frac{1}{2k_{2}}\partial_{X}^2A_{2} -\frac{2d(Z,\,X)\omega_{2}}{cn_{2}}A_{1}^{*}A_{3}e^{-i\Delta k_{0}Z} , \tag {2} \\ &i\partial_{Z}A_{3}=-\frac{1}{2k_{3}}\partial_{X}^2A_{3} -\frac{2d(Z,\,X)\omega_{3}}{cn_{3}}A_{1}A_{2}e^{i\Delta k_{0}Z} , \tag {3} \end{align} where $A_{1,\, 2,\, 3}$ are the slow-varying amplitudes of the idler, pump, and signal waves, respectively; $k_{1,\, 2,\, 3}$, $\omega_{1,\, 2,\, 3}$ ($\omega_{1}+\omega_{2}=\omega_{3}$), and $n_{1,\, 2,\, 3}$ are the wave vector, angular frequency, and refractive index in a nonlinear crystal for the idler, pump, and signal waves, respectively; $c$ is the speed of light in vacuum; and $\Delta k_{0}=k_{1}+k_{2}-k_{3}$ is the phase mismatch factor. Furthermore, we define $d(X,\,Z)=d(Z)\sigma(X)$, which is used to describe the modulation coefficient of the photonic crystal at QPM. Here $\sigma(X)$ is a modulation function in the $X$-direction and can be written as a symbolic function, we can see from Fig. 1(a) that $L$ is the length of one modulation cycle in the input plane and $D$ is the length of phase 0 in a period: \begin{align} \sigma(X)=-\mathrm{sgn}[\cos(\pi X/D)],~~(D=L/2), \tag {4} \end{align} $d(Z)$ is a modulation function in the $Z$-direction. It can be written as a Fourier series expansion in the form:[23,78,79] \begin{align} d(Z)=d_{0}\sum^{}_{m\neq0}\frac{2}{\pi m}\sin\Big(\frac{\pi m}{2}\Big)\exp\Big(i\frac{2\pi mZ}{\varLambda}\Big), \tag {5} \end{align} where $d_{0}$ is the $\chi^{(2)}$ polarizability tensor. As shown in Figs. 1(c1) and (c2), in Eq. (5), the adjacent positive polarized part and negative polarized part construct a modulation cycle whose period is $\varLambda$. The polarized modulation duty cycle is 1/2 [shown in Fig. 1(c1)]. To fit the phase matching, we take the value of $m = \pm1$ into Eq. (5). In Eqs. (1) and (2), $m=1$, and in Eq. (3), $m=-1$. By applying rescalings[80,81] \begin{align} &u={A}_{1}\sqrt{\frac{n_{1}}{\omega_{1}I_{0}}} \exp\Big[i(\Delta k-\frac{2\pi}{\varLambda})Z\Big], \notag\\ &v={A}_{2}\sqrt{\frac{n_{2}}{\omega_{2}I_{0}}} \exp\Big[i(\Delta k-\frac{2\pi}{\varLambda})Z\Big], \notag\\ &w={A}_{3}\sqrt{\frac{n_{3}}{\omega_{3}I_{0}}} \exp\Big[i(\Delta k-\frac{2\pi}{\varLambda})Z\Big], \notag\\ &I_{0}=\Big(\frac{n_{1}}{\omega_{1}}+\frac{n_{2}}{\omega_{2}} +\frac{n_{3}}{\omega_{3}}\Big)|A_{0}|^{2},~~z^{-1}_{\rm d}=\frac{4d_{0}}{\pi c}\sqrt{\frac{\omega_{1}\omega_{2}\omega_{3}}{n_{1}n_{2}n_{3}}I_{0}}, \notag\\ &\varDelta=z_{\rm d}\Big(\Delta k-\frac{2\pi}{\varLambda}\Big),~~z=\frac{Z}{z_{\rm d}},~~x=X\sqrt{\frac{k_{1}}{z_{\rm d}}}, \notag\\ &\gamma_{11}=k_{1}/k_{1}=1,~~\gamma_{21}=k_{2}/k_{1},~~\gamma_{31}=k_{3}/k_{1}, \tag {6} \end{align} we obtain the three dimensionless equations: \begin{align} &i\partial_{z}u=-\frac{1}{2\gamma_{11}} \frac{\partial^{2}u}{\partial_{{x}^{2}}} - u\varDelta - \sigma(x)v^{*}w, \tag {7} \\ &i\partial_{z}v=-\frac{1}{2\gamma_{21}} \frac{\partial^{2}v}{\partial_{{x}^{2}}} - v\varDelta- \sigma(x)u^{*}w, \tag {8} \\ &i\partial_{z}w=-\frac{1}{2\gamma_{31}} \frac{\partial^{2}w}{\partial_{{x}^{2}}} - w\varDelta- \sigma(x)uv, \tag {9} \end{align} where $\varDelta$ is the wave vector mismatch parameter (detuning).[82,83] If we assume \begin{align} &n_{1}\approx n_{2}\approx n_{3}=n,~~\omega_{1}=\omega, \notag\\ &\omega_{2}=\zeta\omega,~~\omega_{3}=(1+\zeta)\omega, \tag {10} \end{align} where $\zeta$ is a proportionality coefficient, and when $\zeta=1$, it is the case of second harmonic, a special case of the three-wave mixing, then we have \begin{align} \gamma_{11}=1,~~\gamma_{21}=\zeta,~~\gamma_{31}=\zeta+1. \tag {11} \end{align} The total Hamiltonian and power are:[84-86] \begin{align} H=\,&\int\frac{1}{2}\Big(|\partial_{x}u|^{2}+\frac{1}{\zeta}|\partial_{x}v|^{2}+\frac{1}{\zeta+1}|\partial_{x}w|^{2}\Big)\notag\\ &-\varDelta(|u|^{2}+|v|^{2}+|w|^{2})-[\sigma(x)u^{*}v^{*}w+{\rm c.c.}]dx,\notag\\ P=\,&\int{(|u|^{2}+|v|^{2}+2|w|^{2})}dx. \tag {12} \end{align} The steady-state soliton solution of the dipole soliton is expressed in terms of the propagation constant $\beta$ of the three waves as \begin{align} &u(z,\,x)=u(x)e^{i\beta_{1} z}, \tag {13} \\ &v(z,\,x)=v(x)e^{i\beta_{2} z}, \tag {14} \\ &w(z,\,x)=w(x)e^{i\beta_{3} z}, \tag {15} \end{align} where $u(x)$, $v(x)$, and $w(x)$ represent the stationary shapes of the idler wave, pump wave, and signal wave, with propagation constants $\beta_{1}$, $\beta_{2}$, and $\beta_{3}$,[87,88] respectively. The following relationship exists among the three propagation constants: \begin{align} \beta_{3}=\beta_{1}+\beta_{2}. \tag {16} \end{align} Numerical Results. To explore the condition of the stable transport of dipole solitons in a 1D photonic crystal with the QPM technique under three-wave mixing process, a series of probes are made. The dipole soliton define here means a bimodal structure of the wavefront spatial distribution. Numerical studies have found that in the case of the idler wave as the opposite direction dipole, the double peaks of the dipole solitons of the pump light are then positive–negative, negative–positive, positive–positive, and negative–negative. According to the phase matching relationship of the three waves, we can obtain the corresponding four dipole solitons of the signal light determined jointly by the structure of the idler and pump light. Therefore, the system supports four different types of dipole solitons, as shown in Table 1.

Fig. 2. Soliton transmission results for the five-strip dipole, where $u$ is the idler wave, $v$ is the pump wave, and $w$ is the signal wave. [(a1)–(a3)] The dipole soliton virtual time transmission results for Type1. [(b)–(d)] The results for Type2–Type4, where the parameters are ($P$, $L$, $\varDelta$, $\zeta$) = (15, 8, 0, 1). $P$ [the total power, see Eq. (12)], $L$ [the period length, see Fig. 1(a) and Eq. (4)], $\varDelta$ [the detuning quantities, defined in Eqs. (7)-(9)] and $\zeta$ [the proportionality coefficient, defined in Eq. (10)].

Table 1. Four types of dipole solitons.
Type	$u$($\omega_{1}$)	$v$($\omega_{2}$)	$w$($\omega_{3}$)
Type1	$+ -$	$+ -$	$+ +$
Type2	$+ -$	$- +$	$- -$
Type3	$+ -$	$+ +$	$+ -$
Type4	$+ -$	$- -$	$- +$
[$+$] Wave crest upwards. [$-$] Wave crest downwards.

These solitons can be bimodal and can have different spacings. These spacings can be characterized by the stripes of the QPM. We focus on the object of the dipole bimodal spacing for three stripes, one stripe, and forms confined to the same stripe. The solution of Eqs. (7)-(9) under three stripes with two peaks spaced apart is obtained by convergence using the ITP method, as shown in Fig. 2, and we observe that the characterization of its soliton is consistent with Table 1. The four types of transmission maps obtained with the RTP method are shown in Fig. 3. Together with the stripes where the two wave crests of the soliton are located, we refer to this case as a five-strip dipole. The results in Figs. 2 and 3 show that the 1D dipole soliton can be stably transported in the $\chi^{(2)}$ nonlinear photonic crystal in the three-wave mixing case. Notably, in the example given above, the two wave peaks of the dipole soliton are separated by three phase stripes, while in the case of one phase stripe, the range of the soliton's stable transmission in the three-wave mixing case is very limited. This case can only be stabilized in a very small numerical range around the proportionality coefficient $\zeta=1$ for all four types. All numerical results are obtained for the case of dipole solitons with three phase stripes separating the two wave peaks, and the dipole solitons are solved and stabilized in the parameter space as described.

Fig. 3. RTP of the five-strip dipole soliton at ($P$, $L$, $\varDelta$, $\zeta$) = (15, 8, 0, 1). [(a)–(d)] Corresponding to Type1–Type4, respectively.

Fig. 4. The variation in the propagation constant ($\beta_{1}$ [see Eq. (13)]) with different parameters for the four types. The red circle represents Type1, the black triangle represents Type2, the purple diamond represents Type3, and the blue star is Type4. (a) $\beta_{1}$ with $P$ for different $\zeta$ ($\zeta=0.5$, 1, 1.5) and other parameters ($L$, $\varDelta$) = (8, 0); (b) $\beta_{1}$ with $\varDelta$ for different $\zeta$ ($\zeta=0.5$, 1, 1.5) and other parameters ($P,\, L$) = (15, 8); (c) $\beta_{1}$ with $L$ for different $\zeta$ ($\zeta=0.5$, 1, 1.5) and other parameters ($P$, $\varDelta$) = (15, 0); (d) $\beta_{1}$ with $\zeta$ for different $P$ ($P=10$, 15) and other parameters ($L$, $\varDelta$) = (8, 0).

To further investigate the transport properties of dipole solitons in crystals, we perform a numerical analysis of the variation in the propagation constant with each control parameter as an example. The propagation constants of the four types show a simple merger. Figure 4(a) shows that the propagation constant $\beta_{1}$ increases with the total power $P$ for the proportionality coefficient ($\zeta=0.5$, 1, 1.5), i.e., the higher the $P$ is, the more stable the soliton transmission is, which satisfies the Vakhitov–Kolokolov criterion, i.e., the slope $d\beta/dP>0$, which is a necessary condition for the formation of stable solitons in a self-focusing medium. Moreover, at the same $P$, $\beta_{1(\zeta=0.5)}>\beta_{1(\zeta=1)}>\beta_{1(\zeta=1.5)}$ exists. In Fig. 4(b), the propagation constant shows a positive correlation with the amount of following detuning, i.e., the larger the $\varDelta$ is, the more stable the soliton transmission is. When $\varDelta=-0.5$, $\beta_{1(\zeta=1.5)}>\beta_{1(\zeta=1)}>\beta_{1(\zeta=0.5)}$, and as $\varDelta$ increases, it gradually becomes $\beta_{1(\zeta=0.5)}>\beta_{1(\zeta=1)}>\beta_{1(\zeta=1.5)}$. However, there is still $\beta_{1(\zeta=0.5)}>\beta_{1(\zeta=1)}>\beta_{1(\zeta=1.5)}$ in the slope. In Fig. 4(c), the propagation constant $\beta_{1}$ shows a positive correlation with the period length $L$. With other control parameters fixed, as $L$ continues to increase, the growth of the propagation constant with the increase in the period length becomes very slow when $L>9$. Upon continued increases of $L$, $\beta_{1}$ no longer increases, and the dipole soliton reaches its most stable state for this parameter space. In Fig. 4(d), the propagation constant $\beta_{1}$ shows a negative correlation with the proportionality coefficients $\zeta$, which mirrors the results of Figs. 4(a)–4(c). The numerical results show that $P_{1}/P_{2}=\omega_{2}/\omega_{1}=\zeta$ ($P_{1}$ is the power of the idler wave and $P_{2}$ is the power of the pump light), the smaller the $\zeta$ is, the greater the power of the pump light is, and increasing the power of the pump light improves the intensity of the idler light and the signal light in the mixing process. Therefore, more energy can be used for mixing, thus increasing the likelihood of realizing the phase matching condition. This effect helps to improve the mixing efficiency and transmission stability.

Fig. 5. The variation in power with the scaling factor. The red circle represents Type1, the black triangle represents Type2, the purple diamond represents Type3, and the blue star is Type4. (a) Variation of $P_{1}/P$ with $\zeta$ ($\zeta=0.5$, 1, 1.5, 2) and other parameters ($P$, $L$) = (15, 8); (b) variation of $P_{2}/P$ with $\zeta$ ($\zeta=0.5$, 1, 1.5, 2) and other parameters ($P$, $L$) = (15, 8); (c) variation of $P_{3}/P$ with $\zeta$ ($\zeta=0.5$, 1, 1.5, 2) and other parameters ($P$, $L$) = (15, 8).

Fig. 6. Soliton transmission results for the three-strip dipole, where $u$ for the idler wave, $v$ for the pump wave, and $w$ for the signal wave. [(a1)–(a3)] The dipole soliton virtual time transmission results for Type1, and [(b)–(d)] the results for Type2–Type4, where the parameters are ($P$, $L$, $\varDelta$, $\zeta$) = (25, 8, 0, 1).

Fig. 7. RTP of the three-strip dipole soliton at ($P$, $L$, $\varDelta$, $\zeta$) = (25, 8, 0, 1) for the idler component, a stable transport example. [(a)–(d)] Corresponding to the four types, i.e., Type1–Type4, respectively.

Figures 5(a) and 5(b) show that the ratio of the power of the idler light to the total power, $P_{1}/P$, is positively correlated with the proportionality coefficient $\zeta$, and the ratio of the power of the pump light to the total power, $P_{2}/P$, is negatively correlated with the proportionality coefficient $\zeta$. Figure 5(c) shows that the power of the signal light is positively correlated with the proportionality coefficient $\zeta$ when $\zeta < 1$, and negatively correlated with the proportionality coefficient $\zeta$ when $\zeta>1$. Notably, the power of signal light reaches the highest when $\zeta=1$, which is the case under the second harmonic, i.e., the highest conversion efficiency in the three-wave mixing process. For any value of $\zeta$, $P_{1}/P+P_{2}/P+P_{3}/P=1$ exists. The solution of the equation under one stripe at two wave peak intervals obtained by convergence using the ITP method is shown in Fig. 6, and the four types of transmission maps obtained with the RTP method are shown in Figs. 7 and 8. With the stripes where the two wave crests of the soliton are located, we refer to this case as a three-strip dipole. In this case, there are stable solutions for all four types, but there are also unstable solutions.

Fig. 8. RTP of the three-strip dipole soliton at ($P$, $L$, $\varDelta$, $\zeta$) = (10, 8, 0, 1) for the idler component, an unstable transport example. [(a)–(d)] Corresponding to Type1–Type4, respectively.

Fig. 9. Soliton transmission results for the single-strip dipole, where $u$ for the idler wave, $v$ for the pump wave, and $w$ for the signal wave. [(a1)–(a3)] The dipole soliton virtual time transmission results for Type1, and [(b1)–(b3)] the results for Type3, where the parameters are ($P$, $L$, $\varDelta$, $\zeta$) = (40, 8, 0, 1).

The solution of the equation under zero stripe at two wave peak intervals obtained by convergence using the ITP method is shown in Fig. 9, and the two types of transmission maps obtained with the RTP method are shown in Fig. 10. Because the two wave crests of the soliton are in one stripe, we refer to this case as a single-strip dipole. In this case, only Type1 and Type3 have solutions, but only Type1 has stable intervals, and Type3 has no stable solutions in the parameter space considered here. Figure 11 shows that the propagation constant $\beta_{1}$ is positively correlated with the total power $P$ in all four types, and a single-strip dipole exists only for Type1 and Type3. We have explored the stabilization intervals of solitons using the RTP method and numerical results show that: In Type1, the three-strip dipole is stable when $1.2\leqslant P\leqslant63.9$, the single-strip dipole is stable when $P\geqslant37.7$, and there is a multistable situation. In Type2, the three-strip dipole is stable when $P\geqslant23.4$, and there is no single-strip dipole. In Type3, the three-strip dipole is stable for intervals $13.5\leqslant P\leqslant36.4$, and there is no stable interval for the single-strip dipole. In Type4, the three-strip dipole is stable when $P\geqslant13.4$, and there is no single-strip dipole. All the solutions are verified by direct numerical simulations everywhere along curves shown on Fig. 11. Figure 12 shows that when $L\leqslant7$, the propagation constant $\beta_{1}$ is positively correlated with the period length $L$ in all four types, followed by a slight downward trend, and when the period grows to a certain length, $\beta_{1}$ no longer changes. We have explored the stabilization intervals of solitons using the RTP method and numerical results show that: in Type1, the three-strip dipole is stable when $2\leqslant L\leqslant10$, and the single-strip dipole is stable when $11\leqslant P\leqslant38$; in Type2, the three-strip dipole is stable when $8\leqslant L\leqslant38$, and there is no stable interval for the single-strip dipole; in Type3, the three-strip dipole is stable for intervals $8\leqslant L\leqslant9$, and the single-strip dipole is stable when $30\leqslant L\leqslant50$; and in Type4, the three-strip dipole is stable when $L\geqslant8$, and there is no single-strip dipole.

Fig. 10. RTP of the single-strip dipole soliton at ($P$, $L$, $\varDelta$, $\zeta$) = (40, 8, 0, 1) for the idler component. [(a), (b)] Corresponding to Type1 and Type3, respectively.

Fig. 11. The variation in $\beta_{1}$ with different $P$ for the four types. The black solid line represents the stable case with three-strip dipole, and the black dashed line represents the unstable case. The solid red line represents the stable case with the single-stripe dipole, and the dashed red line represents the unstable case. [(a)–(d)] Variation in $\beta_{1}$ with $P$ ($P=[0,\,100]$) for the four types. Other parameters are ($L$, $\varDelta$, $\zeta$) = (8, 0, 1).

Fig. 12. The variation in $\beta_{1}$ with different $L$ for the four types. The black solid line represents the stable case with the three-strip dipole, and the black dashed line represents the unstable case. The solid red line represents the stable case with the single-stripe dipole, and the dashed red line represents the unstable case. [(a)–(d)] Variation in $\beta_{1}$ with $L$ ($L\leqslant$50) for the four types. Other parameters are ($P$, $\varDelta$, $\zeta$) = (25, 0, 1).

Estimation of Experimental Parameters. The proposed 1D nonlinear photonic crystal is fabricated within a periodically poled lithium niobate crystal possessing a $\chi^{(2)}$ polarizability tensor factor $\chi^{(2)}$ of $d_0=27$ pm/V. When the proportionality coefficient $\zeta=1$, we selected idler waves and pump waves with a wavelength of 1064 nm and corresponding signal light at 532 nm, and both exhibited similar refractive indices, $n_{1}\approx n_{2}\approx2.2$. An optical field amplitude $A_0=100$ kV/cm is uniformly applied within a $\chi^{2}$ nonlinear crystal. Utilizing Eq. (6), the resulting characteristic distance $z_{\rm d}$ is found to be 0.125 cm. Table 2 provides the dimensionless covariates for the units involved in the analysis. The estimates in Table 2 indicate a transmission distance of $z=1000$, equivalent to a physical distance of 125 cm. This transmission distance of 125 cm is sufficient to empirically establish the stability of solitons within the system.

Table 2. Dimensionless parameters normalized by units.
Dimensionless parameter	Units
$z=1$	0.125 cm
$x=1$	10 µm
$\|u\|^2=\|v\|^2=1$, $\|w\|^2=1$	20 MV/cm$^{2}$, 40 MV/cm$^{2}$
$P=1$	20 W

In summary, we have introduced the QPM technique to the dipole soliton in the process of $\chi^{(2)}$ nonlinear optical crystal transmission, and the positive polarization and negative polarization each account for half of the modulation period in the transmission direction. The output results of four types are obtained by setting the two wave crests of the dipole soliton with different positive and negative directions, and stable transmission of the dipole soliton is realized in the three-wave mixing case. The solution of the dipole soliton is obtained based on convergence with the ITP method, the stability of the soliton solution is verified with the RTP method. Soliton solutions are thus found to exist in the multistability case. In simulation, we observe a positive correlation between the propagation constant of the dipole soliton and the power, detuning and period, and the propagation constant no longer increases after the period expands past a certain degree. There is a negative correlation between the propagation constant and the proportionality coefficient of the dipole soliton, $P_{1}/P$ is positively correlated with the proportionality coefficient, and $P_{2}/P$ is negatively correlated with the proportionality coefficient, which agrees with our hypotheses. The physical properties of the solitons we obtained are within a range that can be realized with existing experimental techniques. We thus provide a theoretical reference value for studying how solitons are transported in $\chi^{(2)}$ nonlinear optical crystals. Introducing $\chi^{(2)}$ and self-defocusing cubic $\chi^{(3)}$ competition on the basis of this work may allow for the formation of liquid photonic droplets in nonlinear photonic crystals.[89] Acknowledgments. This work was supported by the National Natural Science Foundation of China (Grant Nos. 12274077 and 11874112), the Research Fund of the Guangdong Hong Kong Macao Joint Laboratory for Intelligent Micro-Nano Optoelectronic Technology (Grant No. 2020B1212030010), and the Graduate Innovative Talents Training Program of Foshan University. 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