Self-induced periodic interfering behavior of dual Airy beam in strongly nonlocal medium

Feng Zang; Yan Wang; Lu Li

doi:10.1364/OE.27.015079

1. Introduction

It is well known that Schrödinger equation describing a free particle in quantum mechanics can support a nonspreading Airy wave packet solution [1]. Although several works on Airy beam have been reported [2, 3], it has not attracted much attention due to its infinite energy, which is unrealizable in experiment. A decade ago, finite energy Airy beam in optical system was proposed [4], and exhibited some novel properties, such as weak diffraction, self-acceleration and self-healing [5–10]. Due to the unique characteristics, the researches are further focused on its generation [11,12], propagation and interaction [13,14] in different systems. Finite energy Airy beam was also involved in plasma channel [15], particle clearing [16], spatiotemporal light bullet [17], atmospheric communication [18] and so on. In other physical settings, spatiotemporal wavepackets and self-accelerating wave were also investigated [19–25].

Among these investigations, the dynamics of Airy beam in optical systems has been widely concerned by researchers. Recently, the periodic inversion and phase transition of Airy beam in the medium with parabolic potential were firstly found [26]. Subsequently, Hermite–Gauss, Bessel–Gauss and finite energy Airy beams in such medium have been deeply studied, and the results show that the beams perform an automatic Fourier transform oscillating behaviour [27]. Meanwhile, the dynamics of two-dimensional circular Airy beam has been studied extensively, such as abruptly autofocusing property of blocked circular Airy beams [28], dual abruptly focus of modulated circular Airy beams [29], propagation dynamics of a circular Airy beam in a uniaxial crystal [30]. Very recently, the dynamics of circular Airy beam in parabolic potential has been also studied in depth, and the results show that the circular Airy beam exhibits a periodic abruptly autofocusing and autodefocusing behavior [31]. In fact, the parabolic potential can be realized in strongly nonlocal medium provided that the characteristic response length of the nonlocal medium is much broader than the beamwidth [32–35]. Inspired by the results of these researches, we will consider the evolution of one-dimensional dual Airy beam in strongly nonlocal medium, which is helpful for us to better understand the dynamics of circular Airy beam in parabolic potential due to the comparability between dual Airy beam and circular Airy beam.

In this article, we focus on the evolution of dual Airy beam in strongly nonlocal medium. The analytical and numerical results show that the dual Airy beam exhibits a periodic focusing and defocusing behavior, and forms the interference fringes between the focusing and defocusing positions. Moreover, we systematically investigate the characteristics of the interference fringes, and based on these properties, we propose a method for measuring the system parameters and a scheme to generate dual Airy beam in strongly nonlocal medium, which are rarely involved in the literatures.

The rest of this article is structured as follows. The model and its reductions are presented in the next section. In Sec. III, based on the linear Schrödinger equation with the parabolic potential, the analytical results describing the evolution of dual Airy beam are presented. The numerical verification in strongly nonlocal medium and the characteristics of the interference fringes are discussed in Sec. IV. Finally, the main results of the article are summarized in Sec. V.

2. The model and reductions

Let us consider the dynamics of beam propagation in a nonlocal nonlinear medium, which can be described by the nonlocal nonlinear Schrödinger equation [35]

i 2 k \frac{\partial ψ}{\partial z} + \frac{\partial^{2} ψ}{\partial x^{2}} + 2 k^{2} η Δ n (x, z) ψ = 0,

where ψ(x, z) is the complex amplitude envelope of beam, x and z represent the transverse and longitudinal coordinates, respectively. k = ωn₀/c is the wave number, n₀ is the linear refractive index of the medium, and c is the light speed in vacuum. η is a constant related to the medium with η > 0 and η < 0 denoting the focusing and defocusing nonlinearity, respectively. Here, we only consider the case of η > 0, i.e., the focusing nonlinearity. Δn is the refractive index change induced by optical beam with intensity I(x, z) = |ψ(x, z)|², which is given by

Δ n (x, z) = \int_{- \infty}^{+ \infty} R (x^{'} - x) I (x^{'}, z) d x^{'},

where R(x) is the response function of nonlocal medium and is normalized to unit, i.e.,

\int_{- \infty}^{+ \infty} R (x) d x = 1

, whose characteristic width determines the degree of nonlocality. For the strongly nonlocal medium, i.e., the characteristic width of the response function is much larger than that of the optical beam, Eq. (1) can be simplified as the well-known Snyder-Mitchell linear model [32,35]

i 2 k \frac{\partial ψ}{\partial z} + \frac{\partial^{2} ψ}{\partial x^{2}} - k^{2} γ^{2} P x^{2} ψ = 0,

where γ² = −ηR_xx(0) with R_xx(0) < 0 is a constant related to the medium and

P = \int_{- \infty}^{+ \infty} {| ψ |}^{2} d x

is the power of the beam, which is conserved for Eq. (3). By introducing the normalized transformation

ψ \to \sqrt{P / ρ_{0}} ϕ

and scaling the transverse width and propagation distance by ρ₀ and

k ρ_{0}^{2}

, respectively, Eq. (3) can be transformed into the dimensionless form [26,27]

i \frac{\partial ϕ}{\partial z} + \frac{1}{2} \frac{\partial^{2} ϕ}{\partial x^{2}} - \frac{1}{2} α^{2} x^{2} ϕ = 0

with

\int_{- \infty}^{+ \infty} {| ϕ |}^{2} = 1

, where α² = P/P_c with critical power

P_{c} = 1 / (k^{2} γ^{2} ρ_{0}^{4})

, ρ₀ is the normalized width. Equation (4) is the linear Schrödinger equation with the parabolic potential, where α stands for the depth of parabolic potential. Thus, we can obtain the dynamics of the beam in the strongly nonlocal medium by solving the linear Eq. (4).

3. The analytical descriptions for the evolution of dual Airy beam

In this Section, we will exhibit the evolution behaviors of the beam governed by Eq. (4). Here, the initial beam is taken as dual Airy beam [36,37]

ϕ (x, 0) = Ai [- (| x | - x_{0})] exp [- a (| x | - x_{0})],

where

Ai (x) = 1 / (2 π) \int_{- \infty}^{+ \infty} exp [i (x t + t^{3} / 3)] d t

represents the Airy function, x₀ is the input transverse position of each branch of dual Airy beam, and a (0 < a < 1) is the truncation coefficient which ensures that the energy of Airy beam is finite. Here, the truncation coefficient is taken as a = 0.16 throughout. We performed the propagation dynamics of the dual Airy beam by solving numerically Eq. (4). Figure 1 presents its numerical evolutions for different α and x₀. One can see that its evolution exhibits a periodic focusing and defocusing behavior, and forms the interference fringes between the focusing position and the defocusing position. Besides, it can be also found that for a fixed x₀, with the increasing of α, the first focusing position, the first defocusing position and the distance between them are gradually decreased, as shown in Figs. 1(a) and 1(b). When α is fixed, increasing x₀ will result in the increasing of the number of the interference fringes and the narrowing of the spacing of the interference fringes, as shown in Figs. 1(b) and 1(c).

Fig. 1 The self-induced periodic interfering behavior of the dual Airy beam governed by Eq. (4). (a) α = 0.1, x₀ = 5; (b) α = 0.2, x₀ = 5; (c) α = 0.2, x₀ = 8, where the white and yellow dashed lines are the focusing position and the defocusing position, respectively. As comparison, the green curves show the trajectories of two main lobes given by Eq. (11).

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In the following, we try to explain the dynamic behavior by an analytical method. In fact, the solution for Eq. (4) with initial condition ϕ(x, 0) can be written as [26,27,38]

ϕ (x, z) = f (x, z) \int_{- \infty}^{+ \infty} [ϕ (ξ, 0) e^{i b ξ^{2}}] e^{- i K ξ} d ξ,

where b = α cot(αz)/2, K = αx csc(αz) and

f (x, z) = e^{i b x^{2}} \sqrt{- \frac{i}{2 π} \frac{K}{x}} .

Thus, for a given initial state ϕ(x, 0), we can get the analytical solution for Eq. (4) by calculating the integral in Eq. (6). Note that the integral indeed is the Fourier transform of ϕ(x, 0) exp(ibx²), with K being the corresponding spatial frequency, which can be obtained by the convolution of the Fourier transforms of ϕ(x, 0) and exp(ibx²). Here, we need to calculate the Fourier transforms of the input beam (5) and exp(ibx²), respectively. But it is difficult to calculate directly the Fourier transform of the dual Airy beam (5), so we have to consider a linear superposition of two Airy beams, i.e., ϕ₁(x, 0) = Ai(x + x₀) exp[a(x + x₀)] and ϕ₂(x, 0) = Ai[−(x − x₀)] exp[−a(x − x₀)]. The numerical results show that when x₀ is large enough, for example x₀ ≥ 3, the intensity distribution and power of the dual Airy beam are in agreement with that of ϕ₁(x, 0) + ϕ₂(x, 0), as shown in Fig. 2. This means that using the linear superposition to replace the initial dual Airy beam is reasonable. Thus, we can study the dynamics of the dual Airy beam by its replacement ϕ₁(x, z) + ϕ₂(x, z), where ϕ_1,2(x, z) is the solution corresponding the initial state ϕ_1,2(x, 0).

Fig. 2 Intensity distribution of the dual Airy beam (5) and ϕ₁(x, 0) + ϕ₂(x, 0) for different x₀. (a) x₀ = 1, (b) x₀ = 5, (c) x₀ = 8. (d) Powers of the dual Airy beam and ϕ₁(x, 0) + ϕ₂(x, 0) versus x₀. The black curve and red points correspond to the dual Airy beam and ϕ₁(x, 0) + ϕ₂(x, 0), respectively.

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It is easy to give the Fourier transforms of ϕ_1,2(x, 0) and G(x) = exp(ibx²) as

{\hat{ϕ}}_{1, 2} (k, 0) = exp (\pm i k x_{0}) exp [- a k^{2}] exp [\frac{a^{3}}{3} \pm \frac{i}{3} (k^{3} - 3 a^{2} k)],

and

\hat{G} (k) = \sqrt{i \frac{π}{b}} exp (- \frac{i}{4 b} k^{2}),

respectively. Thus, employing the convolution property of Fourier transform, we have

\begin{array}{l} ϕ_{1, 2} (x, z) & = f (x, z) \int_{- \infty}^{+ \infty} [ϕ_{1, 2} (ξ, 0) e^{i b ξ^{2}}] e^{- i K ξ} d ξ \\ = f (x, z) \frac{1}{2 π} \int_{- \infty}^{+ \infty} {\hat{ϕ}}_{1, 2} (k, 0) \hat{G} (K - k) d k . \end{array}

Substituting Eqs. (7) and (8) into Eq. (9) and employing the integral formula

\int_{- \infty}^{+ \infty} exp [i (t^{3} / 3 + M t^{2} + N t)] d t = 2 π

Ai(N − M²) exp[iM(2M²/3 − N)] [39], we can easily obtain the solution ϕ_1,2(x, z) as follows

\begin{array}{l} ϕ_{1, 2} (x, z) & = f (x, z) \sqrt{i \frac{π}{b}} exp (\frac{a^{3}}{3}) Ai (\pm \frac{K}{2 b} - \frac{1}{16 b^{2}} + x_{0} + i \frac{a}{2 b}) \\ \times exp [(a + \frac{i}{4 b}) (\pm \frac{K}{2 b} - \frac{1}{16 b^{2}} + x_{0} + i \frac{a}{2 b})] \\ \times exp [- \frac{i}{4 b} K^{2} - \frac{1}{3} {(a + \frac{i}{4 b})}^{3}] . \end{array}

From Eq. (10), one can find that the trajectories of the main lobe of the two Airy beams are of the form

x_{\pm} (z) = \pm \frac{{sin}^{2} (α z)}{4 α^{2} cos (α z)} \mp x_{0} cos (α z) .

They are periodic function of z with the period T = 2π/α and z ≠ z_m ≡ (2m + 1)T/4, m = 0, 1, 2, · · ·. In order to verify the above analytical result, the trajectories of the two main lobes given by Eq. (11) are shown by the green curves in Fig. 1. From them, one can see that the analytical result is consistent with the numerical result. Moreover, from Eq. (11), one can obtain their intersecting point at z axis as

\begin{array}{l} z_{f}^{n} = \frac{arctan (2 α \sqrt{x_{0}})}{α} + \frac{(n - 1) π}{α}, \\ z_{d}^{n} = - \frac{arctan (2 α \sqrt{x_{0}})}{α} + \frac{n}{α} π, \end{array}

which represent the focusing and defocusing positions, as shown by the white and yellow dashed lines in Fig. 1. Here, n is a nonzero positive integer. When n = 1,

z_{f}^{1}

and

z_{d}^{1}

denote the first focusing and defocusing positions. From Eq. (12), we can get the range of the interference fringes along propagation direction

r = z_{d}^{n} - z_{f}^{n} = \frac{π}{α} - \frac{2 arctan (2 α \sqrt{x_{0}})}{α},

Obviously,

z_{f}^{n}

,

z_{d}^{n}

and r are functions of α and x₀, where the dependences of

z_{f}^{1}

,

z_{d}^{1}

and r on α and x₀ are shown by the black curves in Fig. 3. From it, one can see that for a fixed x₀, as α increases,

z_{f}^{1}

,

z_{d}^{1}

and r decrease. When α is fixed, with the increasing of x₀,

z_{f}^{1}

increases, while

z_{d}^{1}

and r decrease. These results match very well with the numerical results shown by the red points in Fig. 3.

Fig. 3 (a, b) The first focusing position $z_{f}^{1}$ , (c, d) the first defocusing position $z_{d}^{1}$ , and (e, f) the range of interference fringes r versus the parameters α and x₀, respectively. Here, x₀ = 5 in (a, c, e) and α = 0.1 in (b, d, f).

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Note that Eq. (10) is invalid at z = z_m due to b = 0 in Eq. (8), so we need to discuss specially the intensity distribution at z = z_m. In this case, we can directly employ Eq. (6) and obtain

\begin{array}{l} ϕ_{1, 2} (x, z_{m}) & = \sqrt{- \frac{i s α}{2 π}} exp (\frac{a^{3}}{3}) exp (- a α^{2} x^{2}) \\ \times exp {\pm i s [\frac{1}{3} α^{3} x^{3} - (a^{2} - x_{0}) α x]}, \end{array}

where s = +1 (−1) when m is even (odd). From Eq. (14), one can see that the two Airy beams evolve into two Gaussian beams with opposite chirp at z = z_m. After superposing, the intensity distribution is expressed as

\begin{array}{l} {| ϕ (x, z_{m}) |}^{2} & \approx {| ϕ_{1} (x, z_{m}) + ϕ_{2} (x, z_{m}) |}^{2} \\ = \frac{α}{π} exp (\frac{2 a^{3}}{3}) exp (- 2 a α^{2} x^{2}) \\ \times {1 + cos [\frac{2 α^{3}}{3} x^{3} - 2 α (a^{2} - x_{0}) x]}, \end{array}

which describes the interference pattern at z = z_m. Figure 4 depicts intensity distributions at z₀ = T/4 for different α and x₀. From it, one can see that when x₀ is fixed, with the increasing of α, the central peak p_c of interference pattern increases, while the spacing between the central fringe and its adjacent fringe d and the width of the envelope of the interference fringes W decrease, as shown in Figs. 4(a) and 4(b). For a fixed α, as x₀ increases, the interference fringes become more closer, as shown in Figs. 4(b) and 4(c). The corresponding numerical results are also shown in Fig. 4, and are consistent with the analytical results given by Eq. (15). Also, we find that a semi-analytical dynamic solution for Eq. (4) can be obtained by the expansion of the input (5) over a truncated set of eigenmodes of the parabolic potential reported in [40], and the results are consistent with our results given by Eqs. (10) and (14).

Fig. 4 Intensity distribution at z₀ = T/4. (a) α = 0.1, x₀ = 5, (b) α = 0.2, x₀ = 5, and (c) α = 0.2, x₀ = 8.

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4. The numerical verifications for the analytical results

Until now, we have analytically described the dynamic behaviors of the dual Airy beam (5) by Eqs. (10) and (14). It should be pointed out that the above results are based on the Schrödinger equation with the parabolic potential, which is an approximation of Eq. (1) when the characteristic width of the response function is broader than the width of the beam.

To verify our results in strongly nonlocal medium, we consider directly Eq. (1) with the Gaussian response function

R (x) = \frac{1}{\sqrt{2 π} ρ_{m}} exp (- \frac{x^{2}}{2 ρ_{m}^{2}}),

where ρ_m is the characteristic width of the response function. By the same normalized transformation as Eq. (4), Eq. (1) can be written as [35]

i \frac{\partial ϕ}{\partial z} + \frac{1}{2} \frac{\partial^{2} ϕ}{\partial x^{2}} + α^{2} δ^{2} \int_{- \infty}^{+ \infty} e^{- \frac{{(x^{'} - x)}^{2}}{2 δ^{2}}} {| ϕ (x^{'}, z) |}^{2} d x^{'} ϕ = 0,

where δ = ρ_m/ρ₀. Note that δ is the rate of the characteristic response width ρ_m to the normalized width ρ₀. To simulate the dynamic behavior in strongly nonlocal regime, we need to calculate the width ρ for the initial dual Airy beam (5), which is dependent on x₀. Thus, the relative rate of the characteristic response width ρ_m to the beamwidth ρ can be expressed as Δ = ρ_m/ρ = (ρ₀/ρ)δ. Figure 5 shows the evolutions of the dual Airy beam and the coreesponding intensity distribution at z = T/4 for different Δ at x₀ = 5, where the width of the initial dual Airy beam ρ = 9.7749ρ₀ as x₀ = 5. We find that when Δ is large enough, the numerical results match very well with the analytical results given by Eqs. (11) and (15). This means that the approximate solution given by Eqs. (10) and (14) can be used to describe the dynamics of the dual Airy beam in strongly nonlocal medium, provided that the characteristic width is much broader than that of the initial beam, for example, Δ ≥ 15. Thus, we can investigate the dynamical properties of the dual Airy beam in strongly nonlocal medium by the analytical results in above section.

Fig. 5 The numerical evolution of the dual Airy beam governed by Eq. (16). (a) Δ = 5, (b) Δ = 10, (c) Δ = 15, where the green curves are the trajectories of the two main lobes of the dual Airy beams given by Eq. (11). (d) The intensity distribution at z = T/4, where the red, yellow and green dashed curves correspond to Δ = 5, 10, and 15, respectively. As comparison, the analytical result given by Eq. (15) is shown by the black curve. Here, the other parameters are α = 0.3 and x₀ = 5.

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In the following, we discuss the characteristics of the interference fringes induced by the dual Airy beam. Firstly, from Eq. (15), one can find that the maximum and minimum of the intensity at z = z_m are reached at 2α³x³/3−2α(a² − x₀)x = ±2nπ and 2α³x³/3−2α(a² − x₀)x = ±(2n+1)π, respectively, where n = 0, 1, 2, · · ·. For simplicity, we only discuss the case of the maximum with n = 0 and 1, i.e., 2α³x³/3 − 2α(a² − x₀)x = 0 and 2π. From them, one can obtain x₁ = 0 and x₂ = [(3π + X₀)/(2α³)]^1/3 + [(3π − X₀)/(2α³)]^1/3, which correspond to the positions of the central fringe and the adjacent fringe in the interference pattern given by Eq. (15), where X₀ = [9π² + 4(x₀ − a²)³]^1/2. Thus, we can obtain the spacing between the central fringe and its adjacent fringe as

d = x_{2} - x_{1} = \sqrt[3]{(3 π + X_{0}) / (2 α^{3})} + \sqrt[3]{(3 π - X_{0}) / (2 α^{3})} .

Also, from Eq. (15), the envelope of the interference fringes can be expressed as 2αe^2a³/3e^−2aα²x²/π. Its central peak and width are of the form

p_{c} = \frac{2 α}{π} e^{2 a^{3} / 3},

and

W = \frac{1}{\sqrt{2 a} α} .

From Eqs. (17), (18) and (19), one can see that d depends on α and x₀, and p_c and W are independent of x₀.

Figure 6 presents the dependences of d, p_c and W on α and x₀, respectively. One can see that for a fixed x₀, as α increases, the spacing between the central fringe and its adjacent fringe d and the width of the envelope W are decreasing, while the central peak p_c increases, as shown in Figs. 6(a), 6(c) and 6(e). When α is fixed, with the increasing of x₀, d decreases, while p_c and W remain unchanged, as shown in Figs. 6(b), 6(d) and 6(f). These results agree with the numerical results shown by the red points in Fig. 6. These properties can be used for the measurement of the system parameter. For example, according to Fig. 6(a), we can obtain the system parameter α by measuring the spacing between the central fringe and its adjacent fringe d in the interference pattern at z = T/4.

Fig. 6 (a, b) The spacing between the central fringe and its adjacent fringe d, (c, d) the central peak p_c, and (e, f) the width of the envelope W versus α and x₀, respectively. Here, x₀ = 5 in (a, c, e); α = 0.1 in (b, d, f). The black curves and red points correspond to the analytical results and the numerical results, respectively.

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Finally, we discuss the generation of the dual Airy beam in strongly nonlocal medium. Intuitively, if we take ϕ (x, 0) = ϕ₁(x, z_m) + ϕ₂(x, z_m) as an initial input, then the dual Airy beam can be obtained by ϕ(x, 3T/4) because of the periodic feature of the evolution dynamics. As an example, we consider an initial input

ϕ (x, 0) = ϕ_{1} (x, z_{0}) + ϕ_{2} (x, z_{0}) .

From Eq. (14), one can see that it is the superposition of two Gaussian beams with opposite linear and cubic chirp. Thus, from Eq. (6), we can obtain

ϕ (x, 3 T / 4) = Ai (x + x_{0}) exp [a (x + x_{0})] + Ai [- (x - x_{0})] exp [- a (x - x_{0})],

which are the sum of two Airy beams. As shown in above section, as x₀ ≥ 3, it can be replaced by a dual Airy beam, i.e.,

ϕ (x / 3 T / 4) \approx Ai [- (| x | - x_{0})] exp [- a (| x | - x_{0})] .

The numerical simulation shows that, as predicted by Eq. (21), at z = 3T/4 the initial state (20) is transformed into the dual Airy beam that agrees with the standard dual Airy beam (5), as shown in Fig. 7. Thus, we can use the superposition of two Gaussian beams with opposite chirp to generate the dual Airy beam in strongly nonlocal medium.

Fig. 7 The intensity distribution at z = 0 and z = 3T/4, where the black dashed curve is the profile of the dual Airy beam given by Eq. (5). The parameters are x₀ = 5 and α = 0.3.

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5. Conclusions

In summary, based on a reduction of nonlocal nonlinear Schrödinger equation in strongly nonlocal regime, the linear Scgrödinger equation with the parabolic potential, we presented the analytical description of the evolution of the dual Airy beam in strongly nonlocal medium. The results have shown that the evolution of the dual Airy beam in strongly nonlocal medium exhibits a periodic focusing and defocusing behavior, and forms the interference fringes between the focusing and defocusing positions. The results have been verified by numerically simulating nonlocal nonlinear Schrödinger equation, and have shown that they are reasonable when the characteristic width of the response function is broader than the width of the dual Airy beam. Furthermore, the characteristics of the interference fringes induced by the dual Airy beam were also investigated in detail, and are useful for the measurement of the system parameters. In addition, we proposed a scheme to generate the dual Airy beam in strongly nonlocal medium. On the strongly nonlocal medium, the related experiments have been reported around 2000 [41–43], so we speculate that the results may be realized experimentally.

Funding

National Natural Science Foundation of China (NSFC) (61475198, 11705108).

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Self-induced periodic interfering behavior of dual Airy beam in strongly nonlocal medium

Abstract

1. Introduction

2. The model and reductions

3. The analytical descriptions for the evolution of dual Airy beam

4. The numerical verifications for the analytical results

5. Conclusions

Funding

References

Cited By

Figures (7)

Equations (24)

Optics Express