Dynamics of Gaussian beam modeled by fractional Schr&#x000F6;dinger equation with a variable coefficient

Feng Zang; Yan Wang; Lu Li

doi:10.1364/OE.26.023740

1. Introduction

Light propagation in linear and nonlinear media has been studied extensively due to its novel properties and potential applications in all-optical controlling and steering [1,2]. A large number of intriguing properties have been reported, such as self-Fourier beams in parabolic potential [3], non-reciprocal light propagation [4] and optical transparency [5] in PT symmetric potentials, Rabi oscillations [6, 7], optical Bloch oscillation [8–10] and so on. These investigations are based on the models of standard linear Schrödinger equation, also the beam dynamics and their manipulations in the nonlinear regimes have been investigated extensively [11–25].

In the last few years, the fractional Schrödinger equation (FSE) received a lot of attention in various areas of physics [26–28]. In particular, since Longhi suggested a scheme to explore the FSE in optics [29], the propagation of light beams in the framework of the FSE has been investigated extensively. Among the investigations, the dynamics of the Gaussian beams in the FSE with harmonic potential has been studied and the results show that the beam propagates along a zigzag trajectory in real space, which corresponds to a modulated anharmonic oscillation in momentum space [30]. Without harmonic potential, the Gaussian beam splits into two nondiffracting Gaussian beams during propagation [31]. The propagation management of light beams has been reported by introducing a double-barrier potential into the FSE [32]. In addition, the FSE with linear potential has been researched analytically and numerically in optical and mathematical domains [33,34]. With the deepening of the research for this field, more intriguing results have been reported, such as the dynamics of waves in a PT symmetric potential [35], resonant mode conversions and Rabi oscillations [36], potential barrier-induced dynamics of finite energy Airy beams [37], optical Bloch oscillation and Zener tunneling [38] and so on. Meanwhile, the beam dynamics in nonlinear regimes received also some attention. Evolution of super-Gaussian beam has been studied in the FSE under the linear and nonlinear regimes [39]. Gap solitons and double-hump solitons in the nonlinear FSE have been also reported [40,41]. However, the research for the FSE with variable coefficient is rarely involved.

In this article, we investigate the propagation dynamics of the Gaussian beam in the framework of the FSE with a longitudinal periodic modulation. The results show that due to the presence of the longitudinal modulation, the evolution of the beam exhibits a periodically oscillating behaviour, which differs completely from the results without the longitudinal modulation [31]. In addition, the oscillating amplitude and period are investigated in detail. The results show that the oscillating amplitude is dependent on the modulation frequency, the Lévy index and the beam’s chirp, and the oscillating period relates only to the modulation frequency. Therefore, one can manipulate the propagation of the beam by varying the modulation frequency, the Lévy index and the beam’s chirp.

The rest of this paper is organized as follows. In the next section, the model and reductions are presented. The propagation properties of the Gaussian beam without and with the chirp are discussed in detail in Sec. III and IV, respectively. The main results of the paper are summarized in Sec. V.

2. Theoretical model

We consider the propagation dynamics of the optical beams in the framwork of the FSE with a variable coefficient, where the dynamics of the light beam ψ(x, z) is governed by the following dimensionless equation [30,31]

i \frac{\partial ψ}{\partial z} - \frac{1}{2} D (z) {(- \frac{\partial^{2}}{\partial x^{2}})}^{α / 2} ψ (x, z) = 0,

where α is the Lévy index (1 < α ≤ 2) and D (z) denotes the variable coefficient, which is a function of the propagation distance z. We speculate that it may be realized by designing skillfully an optical resonator with lens of alterable focal length in 4f or 2f configuration [29, 31]. Especially, when α = 2 and D (z) = 1, Eq. (1) degenerates to the standard Schrödinger equation. Under the Fourier transform, Eq. (1) can be written as

i \frac{\partial \hat{ψ} (k, z)}{\partial z} - \frac{1}{2} {| k |}^{α} D (z) \hat{ψ} (k, z) = 0,

where

\hat{ψ} (k, z) = \int_{- \infty}^{+ \infty} ψ (x, z) e^{- i k x} d x

is the Fourier transform of ψ(x, z) and k is the spatial frequency. Equation (2) describes the propagation of a beam in a symmetric linear potential in the inverse space, and its solution is of the form

\hat{ψ} (k, z) = \hat{ψ} (k, 0) e^{- \frac{i}{2} {| k |}^{α}} \int_{0}^{z} D (ς) d ς,

where

\hat{ψ} (k, 0)

is the Fourier transform of the initial beam. The general solution of Eq. (1) in real space can be given by

ψ (x, z) = \frac{1}{2 π} \int_{- \infty}^{+ \infty} \hat{ψ} (k, 0) e^{- \frac{i}{2} {| k |}^{α}} \int_{0}^{z} D (ς) d ς e^{i k x} d k .

Thus, for a given initial state, we can acquire the solution of Eq. (1) from the expression (4).

Here, the input is chosen as a chirp Gaussian beam as follows

ψ (x, 0) = e^{- σ {(x - x_{0})}^{2}} + i C (x - x_{0})

with x₀ being the initial incidence location, C being the linear chirp, and σ being related to the beam width. The corresponding Fourier transform of the input beam can be written as

\hat{ψ} (k, 0) = \sqrt{\frac{π}{σ}} e^{- \frac{{(k - C)}^{2}}{4 σ} - i k x_{0}} .

From Eq. (6), one can see that the initial state is also a Gaussian wave packet in k space. In our numerical simulation, Eq. (1) is solved by means of a symmetrized split-step Fourier scheme. We consider the influence of the variable coefficient D (z), the chirp parameter C and the Lévy index α on the propagation dynamics of light beams.

3. The case of C = 0

In this Section, we will discuss the propagation dynamics of the Gaussian beam without the chirp, i.e., C = 0. Firstly, we consider the limiting case of the Lévy index, i.e., α = 1. From Eq. (4), the solution of Eq. (1) is of the form

ψ (x, z) = \frac{1}{2 \sqrt{π σ}} \int_{0}^{+ \infty} e^{- \frac{k^{2}}{4 σ} + i k [x - X_{+} (z)]} d k + \frac{1}{2 \sqrt{π σ}} \int_{- \infty}^{0} e^{- \frac{k^{2}}{4 σ} + i k [x - X_{-} (z)]} d k,

where

X_{\pm} (z) = x_{0} \pm \frac{1}{2} \int_{0}^{z} D (ς) d ς

. According to the approach reported in [31], we assume that x − X₊(z) is negative for k < 0, and is positive for k > 0; x – X₋ (z) is negative for k < 0, and is positive for k > 0. Thus, by mathematically simplifying, the solution (7) for Eq. (1) can be approximately written as

\begin{array}{l} ψ (x, z) \approx \frac{1}{4 \sqrt{π σ}} \int_{- \infty}^{+ \infty} e^{- \frac{k^{2}}{4 σ} - i k | x - X_{+} (z) |} d k + \frac{1}{4 \sqrt{π σ}} \int_{- \infty}^{\infty} e^{- \frac{k^{2}}{4 σ} + i k | x - X_{-} (z) |} d k \\ = \frac{1}{2} e^{- σ {[x - X_{+} (z)]}^{2}} + \frac{1}{2} e^{- σ {[x - X_{-} (z)]}^{2}} . \end{array}

From Eq. (8), one can see that the input Gaussian beam without the chirp splits into two diffraction-free Gaussian beams with the central trajectories x = X_±(z). Especially, when D (z) = 1, it reduces to the result in [31]. To verify the analytical result given by Eq. (8), we performed the numerically simulated evolution of the Gaussian beam with C = 0 by solving directly Eq. (1) with longitudinal periodic modulation D (z) = cos(Ωz), where Ω is the modulation frequency, and the result is presented in Fig. 1(a). For comparison, the corresponding analytical result is shown in Fig. 1(b). The numerical result shows that the input Gaussian beam without the chirp splits into two beams, and due to the longitudinal periodic modulation, each of them exhibits a periodically oscillating behaviour, which are agreement with the analytical result given by Eq. (8) except for the width of the wave packet.

Fig. 1 (a) The numerical evolution and (b) analytical result of the Gaussian beam without the chirp in the system with the longitudinal periodic modulation D (z) = cos(Ωz), where the modulation frequency is set as Ω = 0.1. (c) The oscillating amplitude and (d) period versus the modulation frequency Ω. Here, the other parameters are σ = 0.25 and x₀ = 0.

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In order to understand the influence of the modulation frequency on the oscillating behaviour, the dependences of the oscillating amplitude and period of the beam trajectory on the modulation frequency Ω are also calculated numerically, as shown by the red points in Figs. 1(c) and 1(d). One can see that, with an increasing of the modulation frequency Ω, the oscillating amplitude and period decrease. In fact, from the trajectory equations x = x₀ ± 1/(2Ω) sin (Ωz), they are respectively 1/(2Ω) and 2π/Ω, which predict to be the decreasing functions of Ω, as shown by the black curves in Figs. 1(c) and 1(d). Note that the analytical amplitude and the numerical result have a bit deviation for larger Ω, as shown in Fig. 1(c). This is because the width of splitting beam in the numerical simulation is wider than the analytical result. Thus, after the superposition of the two splitting beams in the numerical simulation, their peaks are closer to the x-axis.

Also, it should be pointed out that in the numerical simulation, we find that the beam power is a conservative quantity during the propagation, as shown by the black curve in Fig. 2. However, from the expression (8), we can easily obtain the beam power as

\begin{array}{l} P = \int_{- \infty}^{+ \infty} {| ψ (x, z) |}^{2} d x = \sqrt{\frac{π}{8 σ}} (1 + e^{- σ \frac{\sin^{2} (Ω z)}{2 Ω^{2}}}), \end{array}

which predicts that the beam power varies periodically, as shown by the red curve in Fig. 2. Thus, we conclude that the approximate solution (8) can only describe the splitting behaviour of the Gaussian beam, but can not keep the conservation of the beam power. In fact, because the width of the splitting beam given by Eq. (8) is narrower than the numerical result, the corresponding power becomes smaller at the splitting region of the beam, but at the half of period the power recovers to the initial level due to the interaction of the two splitting beams, as shown by the red curve in Fig. 2.

Fig. 2 The evolution of the beam power, where the black and red curves represent the numerical and analytical results, respectively. The parameters are the same as in Fig. 1.

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In the following, we will consider the case of 1 < α ≤ 2. Figure 3 presents the evolutions of the Gaussian beam for different Lévy index α, in which the first and second rows represent the results of the numerical simulation for Eq. (1) and the numerical integration for Eq. (4), respectively [37]. Clearly, they agree well with each other. This means that our numerical simulation is reliable. From Fig. 3, one can see that for smaller α, the Gaussian beam splits firstly into two beams, and then under the action of the longitudinal periodic modulation, they exhibit the oscillating evolution behaviour. However, with an increasing of α, the splitting behaviour gradually diminishes. Until α = 2, the splitting behaviour is completely replaced by a periodic diffraction behaviour. In this case, the solution for Eq. (1) can be found as

ψ (x, z) = \frac{1}{\sqrt{B (z)}} e^{- \frac{σ {(x - x_{0})}^{2}}{B (z)}},

where B(z ) = 1 + (2σi/Ω) sin(Ωz). It can be seen that the input Gaussian beam does not split and exhibits a periodic diffraction behaviour due to the presence of the longitudinal modulation, as shown in Fig. 3(d) or 3(d₁). Thus, we conclude that the Lévy index α affects only the splitting behaviour of the beam, and does not disrupt its periodicity. This property differs completely from the results without the longitudinal periodic modulation [31].

Fig. 3 The evolutions of the Gaussian beam in the system with the longitudinal periodic modulation D (z) = cos(Ωz) for different Lévy index. (a–d) The results of the numerical simulation for Eq. (1). (a₁–d₁) The numerical integration for Eq. (4). The Lévy index is respectively α = 1.25, 1.5, 1.75, 2 from left to right. Here, the other parameters are the same as in Fig. 1.

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4. The case of C ≠ 0

Now, we turn to discuss the evolution of the Gaussian beam with the chirp, i.e., C ≠ 0. Similarly, we first consider the limiting case of α = 1. In this case, one can obtain from Eq. (4)

\begin{array}{l} ψ (x, z) = \frac{1}{\sqrt{B (z)}} \int_{- \infty}^{+ \infty} e^{- \frac{{(k - C)}^{2}}{4 σ} + i k (x - x_{0}) - \frac{i}{2} | k | \int_{0}^{z} D (ς) d ς} d k \\ = \frac{e^{i C [x - X_{+} (z)]}}{2 \sqrt{π σ}} {\int_{- C}^{+ \infty} e}^{- \frac{k^{2}}{4 σ} + i k [x - X_{+} (z)]} d k + \frac{e^{i C [x - X_{-} (z)]}}{2 \sqrt{π σ}} {\int_{- \infty}^{- C} e}^{- \frac{k^{2}}{4 σ} + i k [x - X_{-} (z)]} d k \\ = {\begin{matrix} e^{- σ {[x - X_{+} (z)]}^{2} + i C [x - X_{+} (z)]} + g_{+} (x, z), & C > 0, \\ e^{- σ {[x - X_{-} (z)]}^{2} + i C [x - X_{-} (z)]} + g_{-} (x, z), & C < 0, \end{matrix} \end{array}

where

g_{\pm} (x, z) = \mp \frac{e^{i C [x - X_{+} (z)]}}{2 \sqrt{π σ}} \int_{| C |}^{+ \infty} e^{- \frac{k^{2}}{4 σ} \mp i k [x - X_{+} (z)]} d k \pm \frac{e^{i C [x - X_{-} (z)]}}{2 \sqrt{π σ}} \int_{| C |}^{+ \infty} e^{- \frac{k^{2}}{4 σ} \mp i k [x - X_{-} (z)]} d k

with

| g_{\pm} (x, z) | \leq \frac{1}{\sqrt{π σ}} \int_{| C |}^{+ \infty} e^{- \frac{k^{2}}{4 σ}} d k

It is easily seen that when |C| tends to infinity, g_± (x, z) vanishes. Thus, if |C| large enough, g_±(x, z) can be regarded as a small quantity and can be neglected. In this case, the solution of Eq. (1) can be approximately written as

ψ (x, z) \approx e^{- σ {[x - X_{\pm} (z)]}^{2} + i C [x - X_{\pm} (z)]} .

where “+” and “−” correspond to the positive and negative chirp parameter, respectively. From it, one can see that the beam does not split and keeps the profile of the Gaussian beam, where its central trajectory is dominated by the equation x = X₊(z) or x = X₋(z) depending on the sign of the chirp parameter. This means that the chirp of the Gaussian beam can suppress the occurrence of the beam splitting, i.e., the positive chirp suppresses the left splitting beam, and the negative chirp suppresses the right splitting beam.

In order to elaborate the availability of Eq. (12), we introduce

ε = \frac{1}{\sqrt{π σ}} \int_{| C |}^{+ \infty} e^{- \frac{k^{2}}{4 σ}} d k

to describe the strength of the suppressed splitting beam. Figure 4 depicts the dependence of |C | on σ for different strength ε, denoted by |C_ε (σ)|. One can see that the chirp parameter |C| increases with σ for a given ε, and at a fixed σ, the larger the chirp parameter |C| is, the smaller the corresponding ε is. This means that with the increasing of |C|, the beam splitting disappears and the evolution behaviour can be described by a Gaussian beam given by Eq. (12).

Fig. 4 The dependence of the chirp parameter |C| on the beam width σ for different ε.

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As the example, we performed the evolution of the initial Gaussian beam (5) with the chirp at σ = 0.62 for different ε, and the results are summarized in Fig. 5. We set ε = 0.2, 0.1, 0.01, thus the corresponding chirp parameter is |C_0.2(0.62)| = 1.44, |C_0.1 (0.62)| = 1.84 and |C_0.01 (0.62)| = 2.88, respectively, as shown by “a”, “b” and “c” in Fig. 4. The first column in Fig. 5 is the case of C > 0, which suppresses the left splitting beam. It is found that with the increasing of |C|, the corresponding strength " of the left splitting beam decreases, and so the beam on the left is gradually weakened while the beam on the right becomes stronger. Especially, when ε = 0.01, i.e., C = 2.88, the left splitting beam almost disappears, only the right beam remains during the propagation. The situation is similar for C < 0. What is different is that the right splitting beam is suppressed, as shown by the second column in Fig. 5. Therefore, when the chirp parameter is larger enough, for example |C| ≥ |C_0.01 (σ)|, the solution for Eq. (1) can be described by the expression (12), and the conservative beam power can be remained.

Fig. 5 The numerical evolution of the initial Gaussian beam with the chirp for different ε in the system with the longitudinal periodic modulation D (z) = cos(Ωz). (a) ε = 0.2, C = 1.44; (b) ε = 0.2, C = −1.44; (c) ε = 0.1, C = 1.84; (d))ε = 0.(1, C) = 1.84; (e) ε = 0.01, C = 2.88; (f) ε = 0.01, C = −2.88. The other parameters are σ = 0.62−, Ω = 0.1 and x₀ = 0.

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Figures 6(a) and 6(b) display the evolution plots of the numerical result and analytical result given by Eq. (12) as C = 3 and σ = 0.25. From them, one can see that their evolutions are consistent. In fact, for our choice of the parameters, ε = 2.209 × 10⁻⁵, so the left splitting beam is completely suppressed, only the right beam remains. In this case, the approximate solution (12) is tenable. Also, we calculated the oscillating amplitude and period of the beam trajectory versus modulation frequency Ω, as shown in Figs. 6(c) and 6(d), where the black curves and red points correspond to theoretical and numerical results, respectively. One can see that they are the decreasing functions of Ω.

Fig. 6 (a) The numerical evolution and (b) analytical result of the Gaussian beam with the chirp in the system with the longitudinal periodic modulation D(z) = cos(Ωz), where Ω = 0.3. (c) The oscillating amplitude and (d) period versus the modulation frequency Ω. Here, the other parameters are σ = 0.25, C = 3 and x₀ = 0.

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Next, we discuss the situation 1 < α ≤ 2 and C ≠ 0. Figure 7 shows the evolution plots of the Gaussian beam with different chirp for different Lévy index α, where α = 1.25, 1.5, 1.75, 2 from top row to bottom row, and C = 3, 6, 9 from left column to right column, respectively. From it, we can see that the Gaussian beam does not split and exhibits the periodically oscillating evolution behaviour. Particularly, for the given α or C, its amplitude increases with the increasing of C or α, but the period is invariant. This means that the oscillating amplitude in the propagation is related to the Lévy index and the chirp parameter, but the oscillating period only depends on the modulation frequency.

Fig. 7 The evolution plots of the Gaussian beam with the chirp in the system with the longitudinal periodic modulation D(z) = cos(Ωz), where the modulation frequency is Ω = 0.3. Here, α = 1.25, 1.5, 1.75, 2 from top row to bottom row, and C = 3, 6, 9 from left column to right column, respectively, and the other parameters are σ = 0.25 and x₀ = 0.

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In order to have a deeper understanding, we plot the dependences of the oscillating amplitude on the Lévy index α for different chirp parameter C, and on the chirp C for different Lévy index α, as shown in Fig. 8. One can see that the oscillating amplitude increases with α, and the larger C is, the faster the increasing trend is [see Fig. 8(a)]. In addition, the oscillating amplitude linearly increases with C for a fixed α, as shown in Fig. 8(b), in which the limiting case of α = 1 is also presented. Thus, we can manipulate the evolution of the beam by controlling the Lévy index α and the chirp parameter C.

Fig. 8 The oscillating amplitude versus (a) the Lévy index α for different chirp parameter C, and (b) the chirp for different Lévy index α. Here, the other parameters are σ = 0.25, Ω = 0.3 and x₀ = 0.

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Specially, when α = 2, it is easy to give the solution for Eq. (1) as

ψ (x, z) = \frac{1}{\sqrt{B (z)}} e^{- \frac{σ X {(x, z)}^{2}}{B (z)}} e^{i C [X (x, z) + \frac{C}{2 Ω} \sin (Ω z)]},

where X (x, z) = x − x₀ − (C/Ω) sin(Ωz). From it, one can see that the central trajectory of the beam is governed by x = x₀ + (C/Ω sin C) (Ωz). Obviously, the oscillating amplitude is dependent on the chirp parameter C, and is proportional to the chirp parameter C, as shown by the navy dashed curve in Fig. 8(b).

5. Conclusions

In conclusions, we have investigated the evolution of the Gaussian beam in the framework of the FSE with a variable coefficient. In the absence of the beam’s chirp, for smaller Lévy index, the Gaussian beam firstly splits into two beams, and then under the action of the longitudinal periodic modulation, they exhibit the periodically oscillating behaviour. And with the increasing of investigated the evolution of the Gaussian beam in the framework of the FSE with a variable coefficient. In the absence of the beam’s chirp, for smaller Lévy index, the Gaussian beam firstly splits into two beams, and then under the action of the longitudinal periodic modulation, they exhibit the periodically oscillating behaviour. And with the increasing of of the beam’s chirp, one of the splitting beams is gradually suppressed with the increasing of the chirp parameter, while the beam on the opposite direction becomes stronger and exhibits the periodically oscillating behaviour. Also, the oscillating amplitude and period of the beam trajectory are investigated in detail. The results showed that the oscillating amplitude is dependent on the modulation frequency, the Lévy index, and the beam’s chirp, while the oscillating period only depends on the modulation frequency. Thus, we can manipulate the evolution of the Gaussian beam to achieve the beam management in the framework of FSE by controlling the system parameters and the beam’s chirp.

Funding

National Natural Science Foundation of China (61475198 and 11705108); Shanxi Scholarship Council of China (2015-011).

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Dynamics of Gaussian beam modeled by fractional Schrödinger equation with a variable coefficient

Abstract

1. Introduction

2. Theoretical model

3. The case of C = 0

4. The case of C ≠ 0

5. Conclusions

Funding

References

Cited By

Figures (8)

Equations (16)

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