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Abstract
In this paper, complex-variable sine-Gaussian cross-phase (CVSGCP) beams are proposed, and the transmission dynamics properties of the CVSGCP beams in strongly nonlocal nonlinear media are investigated. CVSGCP beams can produce a variety of mode transformation characteristics during transmission. The roles of parameters in the sine and cross-phase terms of the initial light field expression in the evolution of light intensity modes, phase, and beam width are analyzed in detail, and it is proved that the effect of cross phase is to cause the beams to rotate. The control of different modes can be achieved by selecting suitable parameters, which have certain advantages in the practical application of CVSGCP beams. CVSGCP beams can be regarded as generalized high-order breathers because light intensity modes and beam width show periodic oscillation distribution during transmission. The typical evolution characteristics of the CVSGCP beams are verified by numerical simulation. Strongly nonlocal nonlinear optical media can be mathematically equivalent to a variety of optical systems, such as gradient index potential wells and resonant potential wells, so the conclusions in this paper can also be extended to these equivalent optical systems.
© 2024 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
In the transmission process, the beams are inevitably affected by the diffraction effect and generate the broadening phenomenon, but in nonlocal media, the beams are also affected by the nonlinear self-focusing effect, which can accurately offset the diffraction effect on the beams to form spatial optical solitons. In recent years, it has been found that when the nonlinear effect of optical media has strongly spatial nonlocal characteristics, it is easier to suppress the collapse of the beams during transmission, so it is more conducive to the generation of spatial optical solitons [1]. Therefore, more and more attention has been paid to beam propagation in various types of nonlocal nonlinear media, such as nematic liquid crystal [2], lead glass [3,4], and ionic gas [5], etc. Optical solitons have been widely used in quantum communication [6] and optical control [7,8] because of their properties, such as suppressing collapse [9], mutual attraction between antiphase solitons [10], and self-reconstruction of the optical field [11]. Many solitons can be observed in nonlocal media, such as Airy solitons [12], hollow solitons [13], vortex solitons [14,15], etc.
It is well known that vortex beams have a wide application value due to the fact that they carry orbital angular momentum (OAM) and their phase changes helically along the propagation distance [16]. At present, in addition to vortex beams, a type of vortex-free beam with OAM has been found, i.e., cross-phase beams. Cross-phase beams carrying OAM can exert force on particles, resulting in rotation and anisotropic diffraction [17], and can act as “optical wrenches” to trap and rotate colloidal particles or even living cells [18]. Due to this advantage, the cross-phase beams with OAM have received widespread attention. The importance of cross-phase has been proven through research. Hermite-Gaussian beams with cross-phase can be transformed into Laguerre-Gaussian beams, which provides a new method for detecting the topological kernel of beams [19]. Gaussian Schell model beams with cross-phase can alleviate the turbulence-induced scintillation, and can be applied to free-space optical communication [20]. Piercey beams can flexibly adjust the focal length, focusing ability, and beam direction only by using the cross-phase structure, which may find potential applications in optical tweezers and material processing [21]. By introducing the cross-phase structure into the coherence function, the self-reconstruction capability of the coherence function can be significantly enhanced [22].
Since Siegman first proposed complex-variable Hermite-Gaussian beams [23], complex-variable beams have gradually become a hot spot for their non-diffraction and self-healing properties [24]. In recent years, a series of valuable conclusions have been obtained in the study of complex-variable beams. Deng et al. proved that Gaussian beams with complex-variable have a self-trapping property and can form solitons or breathers depending on the input power [25]. Rado$\dot {z}$ycki demonstrated that Gaussian beams with complex-variable are closer to the real experimental situation than ideal beams, showing beautiful mathematical symmetry [26]. In addition, sinusoidal beams are also the focus of research. Zhang et al. introduced laser beams into a sinusoidal modulated Gaussian temperature field, which can produce second harmonic photoacoustic signals [27]. Huang et al. investigated the propagation characteristics of partially coherent electromagnetic hyperbolic sinusoidal Gaussian vortex beams in atmospheric turbulence [28]. The complex-variable sine-Gaussian cross-phase (CVSGCP) beams introduced are composite beams with complex-variable, sine and cross-phase properties. They are solutions to the paraxial wave equation in Cartesian coordinates. As far as we know, such beams have not yet been studied. In this paper, the transmission evolution of CVSGCP beams in strongly nonlocal nonlinear media is studied by means of parameter control. CVSGCP beams can achieve mode control, and can become generalized high-order breathers with periodic changes in light intensity modes and beam width. Through mathematical transformation, strongly nonlocal nonlinear media can be equivalent to many types of optical systems, such as gradient refractive index potential wells and resonant potential wells [29–31], fractional Fourier optical systems [32], quadratic nonlinear systems [33], etc. Therefore, the research results of this paper have broad potential application value.
The structure of this paper is as follows. In Section 2, the nonlocal nonlinear Schrödinger equation is given to describe the beam propagation process. The evolution expression of CVSGCP beams during transmission is obtained. In Section 3, the influences of the parameters in the initial light field expression on the propagation characteristics are studied in detail. In Section 4, the transmission rules are summarized.
2. Theoretical model and analytical solutions
The transmission of paraxial beams can be described by the nonlocal nonlinear Schrödinger equation (NNLSE) [1,34,35]
The initial light field expression (ILFE) of CVSGCP beams at the incident plane $z=0$ is
It can be seen from Eq. (8) that the parameter $\theta$ describing the transmission position always exists in the periodic function, so the CVSGCP beams are transmitted in a periodic form. The transmission period is $\Delta z_{T}=\pi z_{p}=\pi z_{r}\sqrt {\frac {U_{gc}}{U_{0}}}$, and $U_{gc}$ is the Gaussian beam critical power [35].
Beam width is an important attribute of CVSGCP beams. Because the mode transformation of CVSGCP beams is very complex, second-moment statistical beam widths
and can show the evolution process of CVSGCP beam width for convenience [41].The mode control of beams can be used in optical communication [42] and optical tweezers [43], which has certain research value. The CVSGCP beams can also achieve mode control by controlling parameters in ILFE. As can be seen from Fig. 1, the light intensity mode of Fig. 1(a2) is the same as that of Fig. 1(b1), that is, the light intensity mode at $z/z_{p}=0.17\pi$ when parameters $m=5$ and $n=2$ is the same as that at $z/z_{p}=0$ when parameters $m=4$ and $n=4$. Therefore, the light intensity mode during propagation can be propagated as the light intensity mode at $z/z_{p}=0$. In addition, the light intensity modes in Figs. 1(a2)–1(a7) and Figs. 1(b1)–1(b6) are the same, so the propagation process of light intensity modes under these two sets of parameters is the same. Therefore, by selecting the appropriate parameters, the light intensity mode of the initial position can be controlled, and the evolution process of the light intensity modes during the transmission process does not change, which can prove that mode control has been achieved. This is an important advantage of the CVSGCP beams in potential applications.
Fig. 1. Evolution of light intensity of CVSGCP beams. Parameters $m$ and $n$: $m=5$, $n=2$ for (a); $m=4$, $n=4$ for (b). Other parameters: $p=0$, $\omega _{x0}=\omega _{y0}=1$, $U_{0}=2U_{gc}$.
3. Propagation characteristics
When the intensity peaks cross each other, interference occurs, resulting in a rich variety of light intensity modes. In the following, the propagation laws of CVSGCP beams are first described by introducing evolutions of light intensity and phase. As can be seen in Fig. 2, the light intensity changes abruptly at $x=0$ and $y=0$ in the initial position, as the direction of beam energy movement is the same as the phase gradient, resulting in phase misalignment. The energy of the beams is distributed in four quadrants at the initial position, and then the four parts of the energy diffuse outward, forming a parallelogram intensity pattern consisting of four circular bright spots [as shown in Fig. 2(a4)]. As the spots gradually move towards the center, the spots in the first and third quadrants interfere, forming multiple interference fringes [as shown in Fig. 2(a5)]. With the further concentration of energy, the degree of interference gradually increases, forming a light intensity mode similar to water ripples [as shown in Fig. 2(a6)].
Fig. 2. (a1)-(a7): The light intensity distribution of CVSGCP beams. (b1)-(b14): The phase distribution of CVSGCP beams. Parameters: $m=5$, $n=2$, $p=1$, $\omega _{x0}=\omega _{y0}=1$, $U_{0}=5U_{gc}$.
The parameters in ILFE control the evolution of beams. Figures 3–6 will study the effects of parameters $m$, $n$, and $p$ in ILFE on beam transmission. As can be seen from Figs. 3(a4)–3(c4), increasing parameter $m$ can make the distance between the spots at $z/z_{p}=0.5\pi$ larger. This is because the greater the parameter $m$, the smaller the distance between spots at the initial position, resulting in the larger distance between the interference fringes. Since the parameters of the sine term that determine the evolution trend in the $x$ and $y$ directions are the same, the evolution process of the beam width in the $x$ and $y$ directions is exactly the same. With the increase of parameter $m$, the beam width corresponding to the maximum (minimum) gradually increases (decreases), the difference between the maximum and minimum of the beam width gradually increases, and the positions at which the beam width reaches the maximum and minimum values are gradually delayed. The change of beam width in traditional beams changes only once [40]. The beam width evolution trend of CVSGCP beams changes twice, and the turning point is not at the $z/z_{p}=0.5\pi$ position, so CVSGCP beams are very different from traditional beams. When the diffraction effect is strong, the beams are first broadened (compressed) and then compressed (broadened), thus forming a periodic evolution, which is the result of the competition between the diffraction effect and the nonlinear self-focusing effect. Figure 3(c2) presents four elliptically distributed intensity peaks, similar to elliptical Gaussian beams, with possible applications for particle capture and rotation [44]. The beams can produce the four-leaf clover light intensity mode shown in Fig. 3(b1), which may be applied to fiber optic sensing [45].
Fig. 3. (a)-(c): Light intensity evolution under different parameter $m$. (d)-(f): Evolution of beam width in the $x$ direction ($y$ direction) with different parameter $m$. Parameter $m$: $m=1$ for (a) and (d); $m=3$ for (b) and (e); $m=5$ for (c) and (f). Other parameters: $n=1$, $p=1$, $\omega _{x0}=\omega _{y0}=1$, $U_{0}=2U_{gc}$.
Fig. 4. (a)-(c): Light intensity evolution under different parameter $n$. (d)-(f): Evolution of beam width in $x$ direction ($y$ direction) with different parameter $n$. Parameter $n$: $n=1$ for (a) and (d); $n=3$ for (b) and (e); $n=5$ for (c) and (f). Other parameters: $m=1$, $p=1$, $\omega _{x0}=\omega _{y0}=1$, $U_{0}=1.5U_{gc}$.
Fig. 5. (a1)-(a7): Transverse light intensity evolution when parameter $p$ is not zero. (b1)-(b7): Transverse light intensity evolution when parameter $p$ is zero. (c) and (d): Evolution of beam width in the $x$ direction ($y$ direction) of CVSGCP beams. Parameter $p$: $p=2$ for (a) and (c); $p=0$ for (b) and (d). Other parameters: $m=3$, $n=4$, $\omega _{x0}=\omega _{y0}=1$, $U_{0}=1.5U_{gc}$.
Fig. 6. (a)-(c): Light intensity evolution under different parameters $p$. (d)-(f): Evolution of beam width in the $x$ direction ($y$ direction) with different parameters $p$. Parameter $p$: $p=1$ for (a) and (d); $p=3$ for (b) and (e); $p=5$ for (c) and (f). Other parameters: $m=2$, $n=2$, $\omega _{x0}=\omega _{y0}=1$, $U_{0}=2U_{gc}$.
Figure 3 explores the effect of the real part coefficient $m$ of the sine term in ILFE on the beam evolution process. Figure 4 shows the evolution process as the imaginary part coefficient $n$ of the sine term in ILFE increases. Comparing the evolution process of light intensity modes in Figs. 3(a)–3(c) and Figs. 4(a)–4(c), when the value of parameter $m$ (parameter $n$) is large, the energy at the initial position is concentrated in the center (around), then gradually diffuses to the four sides (converges to the center). Therefore, by controlling parameters $m$ and $n$, the convergence and diffusion of energy can be controlled. Comparing the evolution process of beam width in Figs. 3(d)–3(f) and Figs. 4(d)–4(f), as the parameter $n$ increases, the maximum (minimum) of the beam width gradually decreases (increases), and the positions where the beam width reaches its maximum and minimum values gradually advance, which is completely opposite to the influence of parameter $m$ on beam width. The similarity between the effects of parameters $m$ and $n$ on beam width is that as parameters $m$ and $n$ increase, the difference between the maximum and minimum of the beam width will gradually increase.
Figure 3 and Fig. 4 introduce the influence of parameters $m$ and $n$ in the sine term of ILFE on the beam evolution, and Fig. 5 and Fig. 6 will explore the impact of parameter $p$ in cross-phase term on a beam evolution. As can be seen from Fig. 5, when parameter $p$ is not zero, the beams rotate, but when parameter $p$ is zero, the beams do not rotate, so the function of parameter $p$ is to control the rotation of the beams. As shown in Figs. 5(c) and 5(d), the cross phase term makes the difference between the maximum and minimum of the beam width larger. As can be seen from Fig. 6, the beams at the initial position show a square intensity mode consisting of four bright spots, then gradually form a parallelogram intensity mode, and the distance between the bright spots gradually decreases. The cross phase parameter $p$ can control OAM, so as the parameter $p$ increases, the orbital angular momentum will also increase. When the initial incident power is insufficient to provide the centripetal force to maintain the light intensity mode, the beams will gradually widen. As the parameter $p$ increases, the maximum (minimum) of the beam width increases (decreases), and the positions at which the beam width reaches the maximum and minimum values are gradually delayed, which is the same as the influence of parameter $m$ on the beam width.
At the half cycle position ($z/z_{p}=0.5\pi$), the beams can present a parallelogram intensity distribution [as shown in Fig. 3(c4)] and interference pattern [as shown in Fig. 4(c4)]. Figure 7 will study the evolution of the light intensity modes and beam width at the $z/z_{p}=0.5\pi$ position. As the parameter $m$ (parameter $n$) increases, energy gradually diffuses from the center position to the surrounding areas, forming four bright spots. These four spots gradually rotate clockwise around the center and approach a square distribution [as shown in Figs. 7(a1)–7(a7)]. With the increase of parameter $m$, the beam width at $z/z_{p}=0.5\pi$ gradually increases, and the rate gradually increases. However, with the increase of parameter $n$, the beam width at $z/z_{p}=0.5\pi$ first slightly decreases and then gradually increases. From the slope of the lines in Figs. 7(c) and 7(d), it can be seen that the change rate of the beam width with parameter $m$ is greater than the change rate of the beam width with parameter $n$. When the parameters $m$ and $n$ are small, the center has energy. As the gradual increase of the sine term parameters, the spots gradually spread around, and the energy at the central position becomes zero.
Fig. 7. (a1)-(a7): As the parameter $m$ increases, the transverse light intensity evolution process at the half cycle position ($z/z_{p}=0.5\pi$). (b1)-(b7): As the parameter $n$ increases, the transverse light intensity evolution process at the half cycle position. (c): The evolution process of beam width in the $x$ direction ($y$ direction) at half cycle position with parameter $m$. (d): The evolution process of beam width in the $x$ direction ($y$ direction) at half cycle position with parameter $n$. The values of parameter $(m, n)$ have been labeled in the figure. Other parameters: $p=1$, $\omega _{x0}=\omega _{y0}=1$, $U_{0}=4U_{gc}$.
Figure 7 presents the propagation laws of beams as parameter $m$ or parameter $n$ increase, and Fig. 8 will explore the evolution process of the on-axis light intensity (the light intensity when $\textbf {r}=0$), the $\tau$ direction (the direction indicated by the dashed line in Fig. 7(a1)) light intensity, and the beam width when parameters $m$ and $n$ are increased simultaneously. As can be seen from Figs. 8(a1)–8(a3), with the increase of parameters $m$ and $n$, the distribution of on-axis light intensity gradually changes from “$n$” type to “$m$” type, and the energy distribution gradually concentrates in the second half cycle. The larger the parameters $m$ and $n$, the faster the energy increases and decreases [as shown in Figs. 8(a1)–8(a3)]. From the $\tau$ direction light intensity distribution and beam width distribution, it can be seen that the larger the parameters $m$ and $n$, the wider the range of energy distribution. The reason is that the $U_{0}$ is certain, then the input energy is certain, and the lower the energy value, the wider the energy distribution.
Fig. 8. (a1)-(a3): The transformation of on-axis light intensity as parameters $m$ and $n$ increase simultaneously. (b1)-(b3): With the simultaneous increase of the parameters $m$ and $n$, the transformation of the light intensity in $\tau$ direction. (c1)-(c3): When the parameters $m$ and $n$ increase simultaneously, the beam width evolution in the $x$ direction ($y$ direction) of CVSGCP beams. Parameters $m$ and $n$: $m=1$, $n=2$ for (a1)-(c1); $m=2$, $n=3$ for (a2)-(c2); $m=3$, $n=4$ for (a3)-(c3). Other parameters: $p=1$, $\omega _{x0}=\omega _{y0}=1$, $U_{0}=2U_{gc}$.
The above discusses the evolution characteristics of ILFE are both positive. Figure 9 will explore the influences of the negative parameter value on the transmission. By comparing Figs. 9(a)–9(c), it is found that the light intensity evolution process in Figs. 9(b) and 9(c) is identical, indicating that the effect of taking negative values of the real part coefficient $m$ and the imaginary part coefficient $n$ in the sine term is to rotate the light intensity modes with positive parameters by 90 degrees and then transmit them in the reverse direction. Comparing Fig. 9(a) with Fig. 9(d), we can see that the effect of taking a negative value of parameter $p$ is to rotate the light intensity modes by 90 degrees when parameter $p$ is positive.
Fig. 9. (a1)-(a7): Transverse light intensity evolution when the parameters in ILFE are both positive. (b1)-(b7): Transverse light intensity evolution when the real part parameter $m$ of the sine term is negative. (c1)-(c7): Transverse light intensity evolution when the imaginary part parameter $n$ of the sine term is negative. (d1)-(d7): Transverse light intensity evolution when the parameter $p$ of the cross-phase term is negative. Parameters $m$, $n$ and $p$: $m=2$, $n=3$, $p=1$ for (a1)-(a7); $m=-2$, $n=3$, $p=1$ for (b1)-(b7); $m=2$, $n=-3$, $p=1$ for (c1)-(c7); $m=2$, $n=3$, $p=-1$ for (d1)-(d7). Other parameters: $\omega _{x0}=\omega _{y0}=1$, $U_{0}=U_{gc}$.
4. Conclusion
In summary, the evolution expression of CVSGCP beams during transmission was obtained. A series of meaningful conclusions are obtained by analyzing the light intensity modes, phase, and beam width. It is found that the parameters $m$, $n$, and $p$ in ILFE control the evolution process of the CVSGCP beams, and it has been proven that the effect of the cross-phase is to cause the beams to rotate. CVSGCP beams can achieve mode control, which is an important advantage of the beams in potential applications. The light intensity modes and beam width of the CVSGCP beams exhibit a periodic oscillation distribution during transmission, so the CVSGCP beams propagate in the form of breathers, which can be referred to as generalized high-order breathers. As the parameter $m$ (parameter $n$) increases, the maximum value of the beam width gradually increases (decreases), the minimum value gradually decreases (increases), and the difference between the maximum and minimum values of the beam width gradually increases. With the increase of parameter $m$, the beam width at $z/z_{p}=0.5\pi$ gradually increases, and the rate gradually increases. However, with the increase of parameter $n$, the beam width at $z/z_{p}=0.5\pi$ first slightly decreases and then gradually increases. The negative values of the real part coefficient $m$ of the sine term and the imaginary part coefficient $n$ of the sine term rotate the light intensity modes with positive parameters by 90 degrees and then reverse evolve. The negative value of the parameter $p$ has the effect of rotating the light intensity modes by 90 degrees when parameter $p$ is positive.
Funding
National Natural Science Foundation of China (12304320); Science Research Project of Hebei Education Department (ZD2022036, QN2023038); Science Foundation of Hebei Normal University (L2023B07).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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