Controllable transmission of chirped cosh-Gaussian beams in parabolic potential

doi:10.21203/rs.3.rs-2254336/v1

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Controllable transmission of chirped cosh-Gaussian beams in parabolic potential

https://doi.org/10.21203/rs.3.rs-2254336/v1

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In our study, based on the normalized linear Schrödinger equation, we have analytically and numerically investigated the propagation dynamics of chirped cosh-Gaussian beams in a medium with parabolic potential. The obtained results show that cosh-Gaussian beams perform a periodic auto-focusing behavior and the parabolic potential determines the focusing ability, including the focal distance as well as peak intensity at the focus. Especially, the intensity distributions and waveform of cosh-Gaussian beam are related to the initial parameter of cosh function. Furthermore, we also demonstrate the effect of chirp factors on the beam and find that the periodic oscillating behavior caused by linear chirp can be used to manipulated the propagation trajectory of beam, but linear chirp does not affect the focal intensity. While the quadratic chirp can enhance the focusing ability and peak intensity of beam on the axis, which indicates that the quadratic chirp factor plays a significant role in the modulation of the energy localization.

chirped cosh-Gaussian beams

parabolic potential

auto-focusing

self-Fourier transform

In recent years, how to realize effective modulation of optical beam and control the dynamics of beam in optics has become a key research topic for researchers. As a helpful method, the photonic potential is extensively investigated and employed in linear and nonlinear system. It is well known that different external potentials can be achieved by appropriately adjusting the refractive index of the material and various forms of potentials have been reported recently, such as linear potential [1, 2], parabolic potential [3–5], symmetric potential barrier [6], Gaussian potential [7], dynamic potentials [8], etc. However, among these potentials, the parabolic potential, extensively applied in laser-plasma physics [9], Bose-Einstein condensates [10], and ultracold atoms [11], is one of the most attractive and concerned potential mode. For example, the dynamics of finite-energy Airy beams characterized by non-diffraction and self-acceleration in a linear medium with a parabolic potential are discussed, whose results show that the beam undergoes periodic inversion, phase transition and unusual oscillation during propagation [12]. Meanwhile, the same team studied an anharmonic propagation of two-dimensional beams in a parabolic potential, in which periodic inversion and variable rotation are still there [13]. It should be noted that the parabolic potential has various impacts on the beam propagation, including Fourier transform operator [3, 14–16], modulation of a symmetric beam [17] and so on [18, 19]. Since self-Fourier transform can be associated with gradient-index (GRIN) media, the propagation of pulses in graded-index fibers can be achieved by introducing external parabolic potential in optics. In 2016, the propagation of Airy–Gaussian vortex beams through the gradient-index medium was studied with the transfer matrix method [20]. Besides, researchers investigated the propagation of the Pearcey Gaussian (PeG) beams in a medium with a parabolic refractive index, in which the parabolic refractive index is responsible for the focusing effect [21]. Subsequently, plenty of special beams in a medium with parabolic refractive index were studied, including cosh-Airy beam [17], Hermite–Gaussian vortex beam [22], chirped Pearcey Gaussian vortex beam [23] and so on [24]. Especially, strongly nonlocal nonlinear model proposed by Snyder and Mitchell [25] can be converted into the linear Schrödinger equation with a parabolic potential, thus we can conveniently simplify the nonlinear problem to a linear problem, which is why more and more related intriguing results were reported one after another [26–31].

On the other hand, due to its auto-focusing properties, the cosh-Gaussian beams can be used in particle manipulation and many references have studied the propagation properties of such beams in different optical systems [31–33]. Even though there are plentiful reports dealing with the parabolic potential, the propagation dynamics of the cosh-Gaussian beam with embedded first and second order chirped factors in a parabolic potential has not been discussed yet. Hence, we demonstrate the influence of the parabolic potential and initial chirp on the propagation properties of the cosh-Gaussian beam theoretically and numerically. We believe that these results obtained in this paper will provide effective and diverse ways for some specific optical applications, such as energy focusing, beam manipulation, signal processing and other fields.

The organization of this paper is as follows. In Section 2, we present a theoretical model of the propagation of cosh-Gaussian beams in a parabolic potential and derive the corresponding analytical solution. In Section 3, we give the properties analysis based on the numerical calculation results for the propagation of cosh-Gaussian beams in the parabolic potential medium. In Section 4, we discuss the effect of initial chirp on the dynamics of the optical beam. Finally, the paper is concluded in Section 5.

In the paraxial optical system, a beam propagating in a parabolic potential satisfies the normalized (1 + 1) D dimensionless linear Schrödinger equation [12]

$$i\frac{{\partial \psi }}{{\partial z}}+\frac{1}{2}\frac{{{\partial ^2}}}{{\partial {x^2}}}\psi \left( {x,z} \right) - \frac{1}{2}{\alpha ^2}{x^2}\psi \left( {x,z} \right)=0$$, (1)

where $\psi \left( {x,z} \right)$ is the complex amplitude envelope of beam, x and z represent the transverse coordinate and propagation distance, respectively, $\alpha$ is the depth of parabolic potential, which corresponds to the strong nonlocal nonlinearity. According to the Refs. [3, 12], the general solution of Eq. (1) with an arbitrary initial beam $\psi \left( {x,0} \right)$ in real space can be written as $\psi \left( {x,z} \right)=f\left( {x,z} \right)\int_{{ - \infty }}^{{+\infty }} {\left[ {\psi \left( {\xi ,0} \right){e^{ib{\xi ^2}}}} \right]} {e^{ - iK\xi }}d\xi$, (2)

where $b=\frac{\alpha }{2}\cot \left( {\alpha z} \right)$, $K=\alpha x\csc \left( {\alpha z} \right)$ and $f\left( {x,z} \right)={e^{ib{x^2}}}\sqrt { - \frac{i}{{2\pi }}\frac{K}{x}}$.

One can see that the integral in Eq. (2) is a Fourier transform of $\psi \left( {x,0} \right)\exp \left( {ib{x^2}} \right)$, with K being the spatial frequency. Then, according to the convolution property of Fourier transform, Eq. (2) can be rewritten as

$$\psi \left( {x,z} \right)=f\left( {x,z} \right)\frac{1}{{2\pi }}\int_{{ - \infty }}^{{+\infty }} {\widehat {\psi }} \left( {k,0} \right)\widehat {G}\left( {K - k} \right)dk$$, (3)

where $\psi \left( {x,0} \right)$ and $\widehat {G}\left( {k,0} \right)$ are the Fourier transform of $\psi \left( {x,0} \right)$ and ${e^{ib{x^2}}}$, respectively. If we can calculate the Fourier transform of the initial beam, the analytical solution for arbitrary beams in a medium with parabolic potential can be easily obtained. To study the dynamic behavior of one-dimensional cosh-Gaussian beam, the input beam is taken as

$$\psi \left( {x,0} \right)={A_0}\cosh \left( {\Omega x} \right)\exp \left( { - \sigma {x^2}} \right)$$, (4)

where ${A_0}$ is a normalized coefficient to ensure that the input power is equal to 1. $\cosh \left( \bullet \right)$ is the hyperbolic cosine function, $\Omega$ is an initial parameter of the hyperbolic cosine function. $\sigma$ is a parameter related to the width of Gaussian function.

Then, we can obtain the Fourier transforms of the input beam $\psi \left( {x,0} \right)$ and $G\left( x \right)=\exp \left( {ib{x^2}} \right)$ as

$$\widehat {\psi }\left( {k,0} \right)=\sqrt {\frac{\pi }{\sigma }} \exp \left( {\frac{{{\Omega ^2} - {k^2}}}{{4\sigma }}} \right)\cosh \left( {\frac{{i\Omega k}}{{2\sigma }}} \right)$$, (5)

$$\widehat {G}\left( {K,0} \right)=\sqrt {i\frac{\pi }{b}} \exp \left( { - \frac{i}{{4b}}{k^2}} \right)$$, (6)

Substituting Eqs. (5) and (6) into Eq. (3) for integral calculation, the analytical solution of Eq. (1) corresponding to the initial beam (4) can be derived as

$$\psi \left( {x,z} \right)=\sqrt {\frac{{i\pi }}{{b+i\sigma }}} f\left( {x,z} \right)\cosh \left[ {\frac{{\Omega K}}{{2\left( {b+i\sigma } \right)}}} \right]\exp \left[ {\frac{{i\left( {{\Omega ^2} - {K^2}} \right)}}{{4\left( {b+i\sigma } \right)}}} \right]$$, (7)

From Eq. (5), one can observe that the initial state is also a cosh-Gaussian wave packet in k space. In other words, the propagation of the beam in a parabolic potential is equivalent to a kind of automatic Fourier transform. In our numerical simulation, Eq. (1) is solved by means of a symmetrized split-step Fourier method, which assumes that the diffraction step and the nonlinear step separately. Generally speaking, the propagation properties of optical beam can be controlled by choosing appropriate parameters such as the depth of parabolic potential $\alpha$, the linear chirp parameter and quadratic chirp parameter $\beta$. Therefore, it is meaningful to discuss how these parameters affect the propagation properties of the cosh-Gaussian beam. Next, we will investigate the influence of the parabolic potential, the linear chirp parameter and quadratic chirp parameter on the dynamics of optical beams in details.

In this section, we mainly focus on the propagation properties of cosh-Gaussian beams with different parabolic potential parameter $\alpha$. In the following numerical simulations, the parameter $\sigma =0.25$ is fixed.

According to Eq. (7), we plot the evolutions of the cosh-Gaussian beam in parabolic potential with different parameters $\alpha$ and the corresponding analytical results are depicted in Fig. 1(a1)-(c1). It is found that the two sub-beams attract each other at first, then focus at $x=0$, finally defocus with the increase of propagation distance, exhibiting a periodic focusing and defocusing behavior. Furthermore, with the increase of the depth of parabolic potential, the beam width and evolution period gradually decrease, as shown in Figs. 1(a), 1(b) and 1(c). Besides, it’s easy to observe that the energy seems to be scattered in some focal points, thus the peak intensity of beam increases with the increasing of $\alpha$, which indicates that the auto-focusing ability of beam can be adjusted by controlling the depth of parabolic potential, as shown in Fig. 1(d). In order to verify the analytical results given by Eq. (7), we performed the numerical simulation evolution of the cosh-Gaussian beam by solving Eq. (1) directly. Figures 1(a2)-(c2) present the numerical evolutions of cosh-Gaussian beam, respectively. We can find that the auto-focusing always exists and the beam exhibits periodic evolution. When the parameter α increases, the auto-focusing arises more frequently and the focusing ability of beam is determined by the parabolic parameters $\alpha$, thus the focal distance and beam width can also be influenced, which is agreement with the analytical results.

Figure 2 shows the intensity distribution of cosh-Gaussian beams with different initial parameters $\Omega$ at different transmission distances, from which we can see that the optical beam keeps focusing towards the center and the waveform undergoes a process of transformation from double peaks or triple peaks to single peak with the increasing of transmission distance, when $z=3$, the intensity reaches to the maximum value, which corresponds to the first focusing position. In addition, no matter what value the initial parameters $\Omega$ take, the first focusing position remains unchanged. In other words, the focusing position is only related to the depth of parabolic potential.

Figure 3 displays the numerical evolution of the cosh-Gaussian beam for different initial parameters $\Omega$ within a certain distance, from which we can clearly see that the smaller $\Omega$ is, the more stable the evolution of beam is, as depicted in Fig. 3(a1). In fact, this is because Gaussian beam is a stable solution of Eq. (1) in quantum mechanics and when the value of $\Omega$ is small, Gaussian function plays a dominant role and the cosh-Gaussian beam tends to Gaussian distribution. Furthermore, when the value of $\Omega$ becomes larger, the cosh-Gaussian beam is composed of two sub-beams whose oscillation amplitude is dependent on the initial parameter $\Omega$. In other words, the oscillating amplitude increases with the increasing of $\Omega$. Similarly, if we keep the initial parameter unchanged and increase the value of $\alpha$, the period and the first focusing position decrease, which is identical to the conclusion drawn in Fig. 1.

In this section, we discuss the effect of initial chirp on the dynamics of the optical beam and consider the following two cases: the linear chirped beam and the quadratic chirped beam.

4.1 Linear chirped beam

The initial beam is taken as a cosh-Gaussian beam with linear chirp:

$$\psi \left( {x,0} \right)={A_0}\cosh \left( {\Omega x} \right)\exp \left( { - \sigma {x^2}} \right)\exp \left( {iCx} \right)$$, (8)

where is linear chirp parameter related to the incident angle of the optical beam. In the following, we use the split-step Fourier method to simulate the evolution of linear chirped cosh-Gaussian beam in parabolic potential media.

Figure 4 displays the evolution of cosh-Gaussian beam with linear chirp for $\Omega =0.2$. From Figs. 4(b1)-(d1), one can see that the beam propagation trajectory changes significantly due to the introduction of linear chirp. The beam exhibits a periodic oscillating behavior and the oscillating amplitude is proportional to the linear chirp parameter, which is quite different from the optical beam without chirp, as shown in Fig. 4(a1), (b1), (c1) and (e). More interestingly, in the case of $C<0$, we can see from Fig. 4(d1) that the beam periodically oscillates in the opposite direction and the oscillating amplitude depends on the absolute value of chirp. However, the beam’s peak intensity is not affected by the linear chirp parameter, as shown in Fig. 4(f). Moreover, the first focusing position is still $z=3$, which means that the linear chirp has no influence on the longitudinal coordinate of the beam self-focusing. Considering the case of increasing $\alpha$, it is worth mentioning that the oscillating amplitude of beam for $\alpha =1$ is smaller than that for $\alpha =0.5$, which indicates that the parabolic potential parameter $\alpha$ not only alters the evolution period of beam, but also hinders the oscillation behavior to some extent, as shown in Figs. 4(a2)-(d2). Therefore, we can conclude that linear chirp only works for the oscillating amplitude of beam during propagation, while the depth of parabolic potential has great impact on both the oscillation and periodic focusing behavior of the beam.

Figure 5 demonstrates the evolution of cosh-Gaussian beam with linear chirp for $\Omega =2$. As expected, the beam’s transmission trajectory is also modulated by the linear chirp. When the value of chirp parameter is small, the two sub-beams oscillate in smaller amplitude, which leads to an irregular shape as shown in Fig. 5(a1). With the increasing of the chirp, the two sub-beams’ oscillating amplitude become larger, as shown in Fig. 5(b) and (c), which presents the parallelogram-like trajectory in a period. Especially, by comparing Fig. (a1)-(c1) and Fig. (a2)-(c2), we can clearly see that the beam oscillation amplitude decreases with the increasing of the depth of parabolic potential, which is similar to the case without chirp.

4.2 Quadratic chirped beam

It is universally known that the quadratic chirp parameter is physically equivalent to the ratio of the Rayleigh length to the focal length of a spherical lens and plays a significant role in controlling the beam propagation dynamics. In this subsection, we consider the effect of initial quadratic chirp on the beam propagation properties. If the initial cosh-Gaussian beam carries a quadratic chirp, it can be written in the form:

$$\psi \left( {x,0} \right)={A_0}\cosh \left( {\Omega x} \right)\exp \left( { - \sigma {x^2}} \right)\exp \left( {i\beta {x^2}} \right)$$, (9)

where $\beta$ is quadratic chirp parameter.

Figure 6 depicts the evolution of cosh-Gaussian beam with different quadratic chirp parameters for $\Omega =0.2$. The results in Fig. 6 (a1)-(c1) suggest that cosh-Gaussian beam exhibits a periodically focusing behavior and the first focusing position increases a little as the absolute value of the quadratic chirp factor increases, while the breathing period keep unchanged. As it can be seen from Figs. 6(d) that the maximum peak intensity becomes larger when the absolute value of $\beta$ increases, which means that the quadratic chirp factor enhances the focusing ability of the beam and plays an essential role in the modulation of the beam energy localized. By comparing Fig. 6(a1)-(c1), (e) and Fig. 6(a2)-(c2), we can see that the beam peak intensity and the breathing frequency increase with the increasing of parameter $\alpha$, which is similar to the conclusion drawn in unchirped beam. When the initial parameter takes $\Omega =2$, the evolution of cosh-Gaussian beam with different quadratic chirp are shown in Fig. 7. It can be seen that the evolution is similar to that shown in Fig. 6 except for the oscillating amplitude.

In summary, based on the normalized linear Schrödinger equation, we investigate the propagation dynamics of the chirped cosh-Gaussian beams in a parabolic potential analytically and numerically. The analytical solution of the cosh-Gaussian beams propagating in a parabolic potential is derived and the result has been verified by corresponding numerical simulations. It is found that the optical beam exhibits a periodic auto-focusing behavior under the impact of parabolic potential. What’s more, the evolution period, beam width, focal distance and the peak intensity can be manipulated by choosing appropriate parabolic potential parameters $\alpha$. The beam first focusing position decreases and the focus ability is enhanced with the increases of $\alpha$. In addition, the initial parameter $\Omega$ has impact on the intensity distributions and waveform. The beam’s oscillating amplitude becomes larger with the increase of $\Omega$. Finally, chirp factors are taken into consideration and through adjusting the linear chirp factor, the beam transmission trajectory can be changed significantly, which leads to beam oscillate periodically. Besides, the oscillating amplitude increases with the increasing of chirp parameter, while both the peak intensity and oscillating period are not influenced by the linear chirp. In the case of quadratic chirped beam, the modulation of energy localizing and the peak intensity increasing can be achieved by adjusting the quadratic chirp properly, which indicates the quadratic chirp parameter strongly contributes to enrich the beam propagation dynamics. We do believe that the results obtained in this paper will provide theoretical references for some specific optical applications, such as energy focusing, particles manipulation, signal processing, information storage and other fields.

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No competing interests reported.

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Controllable transmission of chirped cosh-Gaussian beams in parabolic potential

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Abstract

Figures

1. Introduction

2. Theoretical Model

3. Properties Analysis

4. Chirped Cosh-gaussian Beams

4.1 Linear chirped beam

4.2 Quadratic chirped beam

5. Conclusion

References

Additional Declarations

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