Evolution of Cos–Gaussian Beams in the Periodic Potential Optical Lattice

Abstract

The evolution of Cos−Gaussian beams in periodic potential optical lattices is theoretically and numerically investigated. By theoretical analysis, a breathing soliton solution of the Gross–Pitaevskii equation with periodic potential is obtained, and the period of the breathing soliton is solved. In addition, the evolution of Cos−Gaussian beams in periodic potential optical lattices is numerically simulated. It is found that breathing solitons generate by appropriately choosing initial medium and beam parameters. Firstly, the effects of the initial parameters of Cos−Gaussian beams (initial phase and width) on its initial waveform and the propagation characteristics of breathing soliton are discussed in detail. Then, the influence of the initial parameters (modulation intensity and modulation frequency) of a photonic lattice on the propagation characteristics of breathing solitons is investigated. Finally, the effects of modulation intensity and modulation frequency on the width and period of the breathing soliton are analyzed. The results show that the number of breathing solitons is manipulated by controlling the initial parameters of Cos−Gaussian beams. The period and width of a breathing soliton are controlled by manipulating the initial parameters of a periodic photonic lattice. The results provide some theoretical basis for the generation and manipulation of breathing solitons.

1. Introduction

Wave packet diffusion due to dispersion or diffraction is a general phenomenon and a limiting factor in many practical applications. Therefore, it becomes very important to study ways to overcome or reduce this phenomenon. Non-diffraction waves, also known as localized waves, are able to resist diffraction, which has attracted increasing research interest [1,2,3]. Breathing soliton is a brand new non-diffraction wave with self-local structures with temporal or spatial periodic oscillations and is widely found in nonlinear systems, such as Bose–Einstein condensates [4], grain lattice [5], hydromechanics [6], optics [7], etc. Unlike the conventional case where fundamental solitons are unchanged in time or spatial shape during transport, breathing solitons undergo regular compression and stretching in the time domain or airspace, periodically exchanging energy between the central, anterior, and posterior spectra [8]. To date, the kinetic properties of breathing solitons have been studied and mainly explored in single-mode fibers, multimode fibers [9,10], and optical microresonators [11,12,13]. The three-dimensional spatial characteristics increase the complexity of the simulation model, and the scientific principles and physical properties of respiratory solitons are more elusive [14]. Therefore, the expansion of nonlinear systems to produce and manipulate different forms of breathing solitons can refine the basic theory of breathing soliton dynamics.

Recently, with the development of laser technology, various different forms of laser beams have been discovered, which greatly enriches in the soliton forms. The cosine beam is a diffraction-free beam that is an exact solution to the cosine function form of the Helmholtz equation, whose strength shape is independent of the propagation distance [15,16]. However, it is not physically possible to achieve Cos-beams (CBs), because its total energy is infinite in any transverse plane. To overcome this problem, a finite-energy Cos−Gaussian beam is introduced by adding a truncation factor to CBs [17]. In contrast to the ideal CBs, the Cos−Gaussian beam still exhibits unique diffraction-free and self-healing abilities over a limited propagation distance, beyond which dispersion necessarily occurs [18]. Due to the self-healing and diffraction-free qualities of the Cos−Gaussian beam, its propagation dynamics in different systems are gaining increasing attention, including free space [19], turbulent atmosphere [20], uniaxial crystals [21], associated beams [22], etc.

Breathing solitons can be produced from materials with general nonlinear and periodic potentials [23]. The Bose–Einstein condensate in the mean-field approximation framework is this material [24,25]. The periodic potential intermittently regulates the waveform and spectrum of the initial input beam, leading to periodic changes in the energy and width of the beam [26,27]. On the other hand, the general nonlinearity makes it possible for the accumulation and transfer of energy in the time domain and the frequency domain, and can lead to energy localization. In the cosine potential photonic lattice, the periodic action of the cosine potential inevitably causes the periodic change in the width and strength of the beam to form breathing solitons. However, there are still many scientific problems to further address concerning the production, propagation, and manipulation of breathing solitons when various different forms of beams are propagated in the cosine potential.

As an effective method to control beam propagation, an external potential such as the linear and parabolic potential is introduced. Linear and dynamic linear potentials are used to manipulate the trajectory of the beam during the propagation process [28,29]. Changfu Liu et al. have reported the exact periodic solution of matter waves in optical lattices with periodic potential [23]. At present, there is little research on the evolution of Cos−Gaussian beams with higher degrees of freedom in periodic potential optical lattices. Cos−Gaussian beams also form breathing solitons in periodic potential optical lattices, but their initial parameters will enrich the shape of breathing solitons. This paper mainly studies the influence of the initial parameters of the Cos−Gaussian beam on the shape of breathing solitons and the influence of a periodic potential optical lattice on the width and period of respiratory solitons.

2. Theoretical Model

Considering the one-dimensional case, the propagation of the laser in the optical lattice of the cosine potential can be described using the Schrodinger equation in the paraxial approximation. The equation is normalized and then expressed as [23]:

$$i\frac{\partial \psi (x,z)}{\partial z}=-\frac{1}{2}\frac{{\partial }^2\psi (x,z)}{\partial x^2}+V(x)\psi (x,z)+g{\left|\psi (x,z)\right|}^2\psi (x,z)$$

(1)

In the formula, $\psi (x,z)$ is the optical field, x represent the transverse dimensionless coordinates, z represents the longitudinal dimensionless coordinates, V(x) is the potential coefficient, and g = 1 is the Kerr nonlinear coefficient. Equation (1) is often called the Gross–Pitaevskii equation (GPE). Changfu Liu et al. have studied the analytical solution of (2 + 1) dimensional GPE [23]. However, the exact stable soliton solution of GPE only exists in the periodic potential optical lattice with fixed modulation frequency, and the exact derivation of the precisely stable soliton solution and the soliton period in periodic potential with varying modulation frequency is unknown. When the beam propagates in the periodic potential photonic lattice, the nonlinear potential can be expressed as:

$$V(x)=P_0\cos ({\omega }_{0}x)$$

(2)

where P₀ is the modulation intensity and ω₀ is the modulation frequency. Then, the normalized form for 1D GPE is as follows:

$$i\frac{\partial \psi (x,z)}{\partial z}=-\frac{1}{2}\frac{{\partial }^2\psi (x,z)}{\partial x^2}+P_0\cos ({\omega }_{0}x)\psi (x,z)+g{\left|\psi (x,z)\right|}^2\psi (x,z)$$

(3)

As far as we know, the exact solution of Equation (3) with potential function is still a big challenge. In this work, we study the periodic solution in Equation (3) and find the relation between its period and optical lattice characteristic parameters. The results will play an important role in guiding the propagation of Cos−Gaussian beams in periodic optical lattices.

Firstly, using the transformation $\psi (x,z)=e^{i\mu z}u(x)$, Equation (3) is transformed into the following form

$$\left[\frac{{\partial }^{2}u(x)}{\partial x^2}+{\left(\frac{{\omega }_0}{2}\right)}^{2}u\right]+2P_0\cos ({\omega }_{0}x)-2\mu -2g{\left|u\right|}^2-{\left(\frac{{\omega }_0}{2}\right)}^2=0$$

(4)

where µ is a modulation factor related to the transmission distance. We can obtain two types of the exact periodic solutions for Equation (4)

$$\frac{{\partial }^{2}u}{\partial x^2}+{\left(\frac{{\omega }_0}{2}\right)}^{2}u=0$$

(5)

$$2P_0\cos ({\omega }_{0}x)-2\mu -2g{\left|u\right|}^2-{\left(\frac{{\omega }_0}{2}\right)}^2=0$$

(6)

when g = 1, we have another type of periodic solutions for the Equations (7) and (8)

$$u(x)=\sqrt{2P_0}\left[\sqrt{1-a^2}\cos (\frac{{\omega }_0}{2}x)+ia\cos \left(\frac{{\omega }_0}{2}x\right)\right]$$

(7)

$$\mu =-P_0-\frac{1}{2}{\left(\frac{{\omega }_0}{2}\right)}^2$$

(8)

where $1\ll a\ll 1$. Therefore, the special periodic solutions of Equation (4) with g = 1 are obtained, which read

$$\psi (x,z)=\sqrt{2P_0}\left[\sqrt{1-a^2}\cos (\frac{{\omega }_0}{2}x)+ia\cos \left(\frac{{\omega }_0}{2}x\right)\right]e^{-i\left(P_0+\frac{1}{2}{\left(\frac{{\omega }_0}{2}\right)}^2\right)z}$$

(9)

From Equation (9), we can interpret the period of the wave function in the mode of transmission into the following form

$$\lambda =\frac{2\pi }{P_0+\frac{1}{2}{\left(\frac{{\omega }_0}{2}\right)}^2}$$

(10)

Through the above theoretical analysis, it can be known that the periodic soliton solutions exist in periodic potential optical lattices. The law described in Formula (10) is applicable to the evolution of beams of other shapes in the periodic potential optical lattice. When the shape of the incident beam is different, the shape of the breathing soliton is different, but the breathing law is similar. Therefore, some evolution rules of the beam in the periodic potential optical lattice are obtained from Formula (10). Therefore, it is predicted that the period λ decreases with P₀ and ω₀. The prediction results are consistent with the numerical simulation results

The input initial Cos−Gaussian beam is generated by the interference of two complex conjugate Gaussian beams with initial chirp. Its spatial domain form can be expressed as [30]:

$$\psi (x,z=0)=\exp \left[-\frac{{\cos }^2{\phi }_0}{4}{\left(\frac{x}{x_0}\right)}^2\right]·\cos \left[\frac{\sin 2{\phi }_0}{8x_0^2}{x}^2+\left(\phi -{\phi }_0\right)\right]$$

(11)

In the Formula (11), $\psi (x,0)$ describes the input field, x₀ represents the initial width of each Gaussian beam, φ₀ is the phase difference between the two Gaussian beams, while φ is the phase of the Cos−Gaussian beam. The two free parameters φ₀ and φ provide the flexibility to change the shape of the Cos−Gaussian beam. In the extreme case, when φ₀ is 0, the Cos−Gaussian beam becomes a pure Gaussian beam. According to Euler’s formula, the Cos−Gaussian beam can be obtained by the superposition of two interfering Gaussian beams, while φ₀ corresponds to the phase difference in the two Gaussian beams.

Mathematically, its spectral distribution is expressed according to reference [30]. In the formula, $\tilde{U}(0,\omega )$ is the Fourier transform of $U(0,\omega )$.

$$\begin{array}{ll}\tilde{\psi }(w,z=0)& ={\displaystyle \int \psi (0,w)\exp \left(i\frac{{\omega }^{2}z}{2}-i\omega x\right)d\omega }\\ & =\exp \left[-{\omega }^2{\left(\frac{x}{x_0}\right)}^2\right]·\cos \left(\frac{x^2}{x_0}{\omega }^2\tan {\phi }_0-\phi \right)\end{array}$$

(12)

3. Analysis of Numerical Calculation Results

According to Formula (4), the distribution of the incident light field of the Cos−Gaussian beam can be obtained. Since the initial phase φ₀ and the initial width x₀ determine the initial waveform of the Cos−Gaussian beam, the influence of the initial phase φ₀ and the initial width x₀ on the light field intensity of the beam is discussed in detail. In addition, when Cos−Gaussian beams of different shapes are propagated in the cosine potential optical lattices, different forms of breathing solitons are generated. The influence of the initial phase φ₀ and initial width x₀ on the shape of the breathing solitons is analyzed. Meanwhile, the influence of the modulation intensity and modulation width of the cosine potential optical lattice on the breathing period and intensity of the breathing soliton is studied.

Figure 1 shows a waveform diagram of the initial Cos−Gaussian beam with different phases. At φ₀ = π/6, it corresponds to a Gaussian beam whose waveform is a symmetrically distributed bell-type structure (Figure 1a). As φ₀ increases, the Cos−Gaussian beam is strengthened by hyperbolic cosine function modulation, symmetrical side lobes modulated by the cosine function gradually appear on the left and right sides of the beam, and the main peak begins to split (Figure 1b). When φ₀ is further increased, the waveform of the Cos−Gaussian beam is a symmetric trimodal shape with a larger intermediate peak (Figure 1c). The reason why the initial phase energy affects the waveform of the Cos−Gaussian beam and generates the multi-peak structure is that the Cos−Gaussian beam is a superposition of two Gaussian beams with different phases, and multiple Cos−Gaussian beams can be reshaped by controlling the phase factor.

Phase φ₀ is a degree of freedom that controls the initial waveform of the Cos−Gaussian beam. In order to investigate the effect of phase on the breathing soliton of the Cos−Gaussian beam, numerical simulation is carried out. Figure 2 shows the space domain waveform and peak power evolution diagram of the Cos−Gaussian beam with different initial phases φ₀. It is found that the number and peak power of breathing solitons depend on the initial phase of the Cos−Gaussian beam. When φ₀ = π/6, the Cos−Gaussian beam has a single-peak structure. At the initial stage of propagation, diffraction plays a leading role. After a very short compression distance, the energy will be rapidly radiated and broadened. Meanwhile, due to the increase in the cosine potential, the diffraction cannot continue to broaden the light beam for a long distance (Figure 2a). Under the action of the cosine potential and Kerr nonlinearity, the light beam will be compressed. When the light beam is compressed to the minimum width, it will be widened again as the diffraction plays a leading role due to the smaller cosine potential. This process causes the beam to expand and compress periodically, eventually forming stable breathers. When the phase is 3π/8, side lobes appear at the front and rear of the Cos−Gaussian beam, and the side lobes also demonstrate respiration during propagation (Figure 2b). It is worth noting that when φ₀ = 2π/3, the Cos−Gaussian beam is compressed for a short distance, and the Cos−Gaussian beam composed of two Gaussian beams with different phases is split into two solitons by the cosine potential. When two breathers collide effectively in the plane, a higher-order strange wave is generated at the collision position, and each breather still keeps its original size, direction, and period after the collision. Under the combined action of cosine potential and diffraction effect, breathing solitons exhibit intertwined double-breathing solitons with DNA structure (Figure 2c). In addition, the peak power evolution diagram of breathing soliton shows that with the increase in the phase, the respiration rate of breathing soliton does not change significantly, but the maximum peak power of respiration increases first and then decreases due to soliton splitting (Figure 2d). Meanwhile, the maximum peak power of breathing solitons remains stable, indicating that the radiated energy can rebound well without energy loss.

Figure 3 shows the changes in the Cos−Gaussian beam waveform with different x₀. When the initial phase φ₀ = π/6, the Cos−Gaussian beam demonstrates a multi-peak structure with center symmetry. Compared with the energy of a single pump, the side lobes located on the wings of the Cos−Gaussian beam can share part of the energy of the central part (Figure 1a). With the increase in x₀, not only does the width of the Cos−Gaussian beam increase, but the distance between its main peak and the two side lobes also increase (Figure 1b,c). While the nonlinearity of the optical lattice is the near-field interaction force, the distance between the optical waves will greatly affect the interaction force between the main peak and the side lobes. Moreover, with different interval sizes of adjacent light solitons, the interactions between light solitons during propagation will trigger a series of distortion, leading the Cos−Gaussian beam to produce many new singular structures.

The initial width x₀ is a degree of freedom controlling the spacing between the peaks of the multipeak Cos−Gaussian beam, and numerical simulation studies are performed to explore the effects of the initial width on the breathing solitons produced by the Cos−Gaussian beam. Figure 4 shows the evolution of the space domain waveform and peak power of the Cos−Gaussian beam with different widths. The Cos−Gaussian beam demonstrates a multipeak structure at φ₀ = 2π/3, whose energy is mainly distributed in the main lobe while the side lobe energy is relatively weak. Although the initial width of the beam cannot change its shape, it can change the spacing between the different peaks of the Cos−Gaussian beam. While the nonlinearity is a near-field interaction, the magnitude of the interaction force depends on the spacing of the peaks. Different peaks of the Cos−Gaussian beam will attract each other. At x₀ = 1, the breathing solitons exhibit an intertwined DNA structure under the combination of the cosine potential and the diffraction effects (Figure 4a). When the initial width increases, the energy of the main lobe converges to the central position to form a breathing soliton, and the side lobe radiates outward due to the weak attraction. Meanwhile, the energy of the side lobe rebounds after encountering the potential well, forming three breathing solitons (Figure 4b,c). In addition, the peak power evolution of the breathing solitons shows that as the beam width increases, the breathing frequency of the breathing solitons do not change significantly, but their maximum peak power of breathing gradually increases (Figure 4d); the maximum peak power of the breathing solitons remains stable, indicating that the outward energy can rebound well without energy loss.

The lattice modulation intensity P₀ is an important parameter affecting the nonlinear action of the optical lattice of the cosine potential, and numerical simulation studies are conducted to explore the effects of the modulation intensity on breathing solitons from the Cos−Gaussian beam. Since the number of the cosine potential determines the ability of the optical lattice to bind the optical soliton, the higher the value is, the stronger the binding ability will be. For an optical lattice with cosine nonlinear modulation, it can relatively strongly bind the light solitons in the initial stage, and the light beam is compressed; thus, with the increase in the propagation distance, the binding capacity of the optical lattice gradually decreases. When the binding ability of the cosine potential is less than the diffraction power of the beam, the light beam begins to widen (Figure 5a), then it is again squeezed and spread, demonstrating a circulating phenomenon. Therefore, such a periodic breathing behavior is formed. Since the magnitude of the cosine potential varies periodically, the diffraction and the cosine potential of the optical lattice cannot reach strict equilibrium, and the beam width and the light field intensity of the solitons produced by the Cos−Gaussian beam change periodically with the propagation distance, that is, the breathing change. It is found that the peak of the light field intensity at the widest beam width is the smallest, while the peak of the light field intensity at the narrowest beam width is the largest. As the lattice modulation intensity increases, the distance between the two maximum peaks (also called the breathing cycle) decreases, the maximum breathing amplitude of the breathing soliton gradually decreases, and the breathing frequency gradually increases (Figure 5b,c). Notably, modulation intensity does not change the maximum intensity of breathing solitons (Figure 5d), indicating that breathing solitons show energy conservation throughout the propagation progress. From the above analysis, it is found that the breathing magnitude and frequency of spatial solitons can be controlled by varying the strength of the lattice modulation. It is concluded that the numerical simulation results are basically consistent with the theoretical analysis from Equation (10).

Lattice modulation frequency ω₀ is an important parameter affecting the breathing speed of the cosine potential optical lattice. In order to explore the effects of modulation intensity on the breathing soliton of the Cos−Gaussian beam, numerical simulation is carried out. For the optical lattice system with cosine nonlinear modulation, the Cos−Gaussian beam self-focuses for a short distance at the initial stage of propagation, and then the nonlinear self-focusing effect weakens and radiates outward. The energy of the sidelobes rebounds after meeting the potential well, forming a double-breathing soliton with an intertwined DNA structure (Figure 6a). The breathing solitons exhibit local characteristics in the space distribution direction and periodic respiration characteristics in the propagation direction. With the increase in modulation frequency ω₀, for the same longitudinal propagation distance (z = 100), the pulse period gradually decreases, while the frequency gradually increases. The energy of the beam is gradually bound to the central waveguide and the breathing phenomenon gradually weakens (Figure 6b,c). It is noteworthy that the modulation frequency ω₀ can slightly change the maximum intensity of the breathing soliton (Figure 6d). The energy of breathing solitons is conserved throughout the transport process. Through the above analysis, it is known that the maximum breath of width and frequency of the spatial solitons can be controlled by changing the lattice modulation frequency. It is concluded that the numerical simulation results are consistent with the theoretical analysis from Equation (10).

As can be seen from the diagram of the relation between the period and width of the breathing solitons formed by the Cos−Gaussian beam and the modulation intensity and frequency, the breathing cycle of the breathing soliton increases monotonically with the increase in the modulation intensity and modulation frequency (Figure 7a); meanwhile, the maximum width of the breathing solitons also has the same pattern (Figure 7b). Constructing different optical lattice structures with different modulation intensities and modulation frequencies can modulate the period and maximum width of breathing solitons, which plays a guiding role in the study of optical communication devices such as optical switching.

4. Conclusions

By changing the initial parameters of the Cos−Gaussian beam and the cosine potential optical lattices, the breathing solitons with local characteristics in the direction of space distribution and periodic breathing characteristics in the direction of propagation can be obtained. The numbers, periods, and widths of breathing solitons are manipulated by appropriately choosing the initial parameters. Firstly, the effects of the initial phase and width of the Cos−Gaussian beam on its initial waveform are analyzed in detail, and it is found that choosing different parameters generates breathing solitons as single-breathing solitons, double-breathing solitons, and three-breath solitons. Secondly, the effects of the modulation intensity and modulation frequency of the periodic potential optical lattice on breathing solitons are analyzed in detail, and it is found that the period and width of the breathing soliton decrease with the increment of the modulation intensity and modulation frequency, while the peak power of the breathing soliton increases slightly with increasing modulation intensity and modulation frequency. Finally, when the modulation intensity and modulation frequency increase, the period and width of the breathing solitons increase significantly. Our results will provide useful evidence for the investigation of the evolution of light waves in complex physical systems.