Periodical self-focusing and self-accelerating of a finite-energy Airy beam in a gradient-index fiber

Hanmou Zhang; Wenyu Fu; Xingrong Zheng

doi:10.3788/COL201513.S21406

Abstract

We investigate propagation of a finite energy Airy beam in a gradient-index fiber analytically and numerically, and find that the beam not only repeats its intensity features, but also has the phenomenon of self-focusing and self-accelerating periodically. The numerical results show that the beam centroid position and beam width evolve periodically. The radial gradient of the refractive index determines the propagation period for the beam and the truncated parameter affects the amplitude of both the centroid position and beam width.

A diffraction-free beam is a kind of beam with localized optical wave packets that remain invariant during propagation. As a member of the family of these beams, the Airy beam has been studied extensively for its novel characteristics^[1–5]. Its most distinctive characteristic is the propagation along curved trajectories in free-space, which is termed self-accelerating. Another interesting feature is the ability to reconstruct itself during propagation even though parts of the beam are distorted or obstructed^[5]. Due to its unique properties, the Airy beam has attracted much attention in applications such as the generation of a curved plasma channel, micro-particle manipulation, guiding-Airy plasmons, or slow nondispersive wave packets^[6–9]. Recently, Siviloglou and Christodoulides extended another Airy beam model that consists of Airy function coupled with an exponential term describing the truncation of the beam, which has finite energy^[10]. The Airy–Gaussian beam was subsequently proposed^[11]. It carries finite power, retains the nondiffracting propagation properties within in a finite propagation distance, and can be realized experimentally to a very good approximation. The propagation properties of Airy–Gaussian beams in strongly nonlocal nonlinear media and quadratic-index media were investigated^[12,13]. It is found that Evolution of the intensity distribution for the two dimensionalthe linear momentum, the beam width, and the centroid of the beam change periodically; and the influence of parameters on the beam propagation were analyzed. A laser beam propagates in fiber is widely used in optical communications. As far as we know, the propagation of Airy beams in fibers has not been reported. Hence, it becomes important to research how an Airy beam propagates in a fiber.

In this work, we investigate the propagation properties of finite-energy Airy beams propagating in a gradient-index (GRIN) fiber. We show that the centroid position, width, and intensity distribution of the beam vary periodically. In these changing processes, the beam reveals a clear self-focusing and self-accelerating phenomenon. The radial gradient of the refractive index and the truncated parameter of the beam play important roles in determining the propagating period, the amplitude of the centroid position, and width of the beam.

We began our work by considering two-dimensional finite-energy Airy beams propagating in a GRIN fiber. This kind of fiber is characterized by a parabolic refractive-index profile, symmetric about its axis. The index of refraction varies radially, and the index of refraction at the center of the fiber is higher than that at the boundary. Due to this, a GRIN fiber is an inhomogeneous, isotropic medium^[14]. When light propagates in it, the ray varies sinusoidally along the fiber and never reaches the edge. Furthermore, such fibers exhibit lower pulse dispersion than their counterparts, such that they were used extensively in optical communication systems^[15].

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The electric field distribution of the finite-energy Airy beam propagating in a GRIN fiber at the $z = 0$ plane can be expressed as^[4,10,16] $E (x_{0}, y_{0}, 0) = A i (\frac{x_{0}}{w_{0}}) \exp (a_{x} \frac{x_{0}}{w_{0}}) A i (\frac{y_{0}}{w_{0}}) \exp (a_{y} \frac{y_{0}}{w_{0}}),$ (1)where $(x_{0}, y_{0})$ represents the transverse coordinates of a point at the source plane, $A i (\cdot)$ denotes the Airy function, $w_{0}$ is an arbitrary transverse scale, and $a_{x}$ , $a_{y}$ are positive parameters associated with the truncation of the Airy beam.

The GRIN fiber has an axis of symmetry that is along the $z$ axis. The dependence of the square of the refractive index is given by^[17] $n^{2} (x, y, ω) = {\begin{matrix} n_{0}^{2} (ω) [1 - α^{2} (ω) (x^{2} + y^{2})] & x^{2} + y^{2} \leq R_{0}^{2} \\ n_{0}^{2} (ω) [1 - α^{2} (ω) R_{0}^{2}] & x^{2} + y^{2} \geq R_{0}^{2} \end{matrix},$ (2)where $R_{0}$ is the core radius, $n_{0} (ω)$ is the refractive index at the center of the fiber at frequency $ω$ , and $α (ω)$ is the radial gradient of the refractive index. Terms $n_{0} (ω)$ , $n_{1} (ω)$ are the refractive index at center and the boundary of the core, respectively. The $α (ω)$ is given by $α (ω) = \frac{1}{R_{0}} {[1 - \frac{n_{1}^{2} (ω)}{n_{0}^{2} (ω)}]}^{1 / 2} .$ (3)

The ABCD transfer matrix for paraxial beam propagation through a GRIN fiber from the plane $z = 0$ to plane $z = const. > 0$ is given by $[\begin{matrix} A & B \\ C & D \end{matrix}] = [\begin{matrix} \cos (α z) & \sin (α z) / n α \\ - n α \sin (α z) & \cos (α z) \end{matrix}] .$ (4)From the expression of the elements of the transfer matrix, it can be found that the variety of them are periodic, and the periodic length $L = 2 π / α$ .

The Airy beams propagate close to the $z$ axis and along the positive $z$ direction in the fiber. The field at the output plane $z$ can be performed with the Huygens–Fresnel diffraction integral^[18] $E (x, y, z) = - \frac{i k}{2 π B} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} E (x_{0}, y_{0}, 0) \times \exp {\frac{i k}{2 B} [A (x_{0}^{2} + y_{0}^{2}) - 2 (x x_{0} + y y_{0}) + D (x^{2} + y^{2})]} d x_{0} d y_{0} .$ (5)Substitute Eqs. (2)–(4) into Eq. (5), and solve the integral using formula^[19]; then the result can be written as $W (x, y, z) = \frac{1}{A} \prod_{j = x, y} A i (\frac{j}{A w_{0}} - \frac{B^{2}}{4 k^{2} A^{2} w_{0}^{4}} - \frac{a_{j} B}{i k A w_{0}^{2}}) \times \exp (\frac{i k D}{2 B} j^{2}) \exp (\frac{a_{j} j}{A w_{0}} - \frac{a_{x} B^{2}}{2 k^{2} A^{2} w_{0}^{4}} - \frac{i k j^{2}}{2 A B} + \frac{i a_{j}^{2} B}{2 k A w_{0}^{2}} - \frac{i B^{3}}{12 k^{2} A^{3} w_{0}^{6}}) .$ (6)With Eq. (6), the characteristics of the two-dimensional finite-energy Airy beams can be investigated conveniently. The optical intensity of the beam at a point $(x, y, z)$ is defined as^[20] $I (x, y, z) = E (x, y, z) \cdot E^{*} (x, y, z),$ (7)where the asterisk denotes complex conjugation. With Eq. (7), we can obtain the receive intensity profile of Airy beam. Considering the symmetry of the Airy beam on the $x$ and $y$ axis; we only analyze the beam width and the centroid position of the one-dimensional Airy beams. Taking $y = 0$ , the beam reduced into one-dimensional Airy beams, the beam width, which is also termed the mean radius of an optical beam, is defined as^[21] $W (z) = 2 \sqrt{〈 x^{2} 〉 - {〈 x 〉}^{2}},$ (8)where $〈 X 〉 = \int_{- \infty}^{\infty} X {| E (x, 0, z) |}^{2} d x / \int_{- \infty}^{\infty} {| E (x, 0, z) |}^{2} d x$ is the expected value of $X$ . Term $x_{c} (z) = 〈 x 〉$ ^[22] gives the centroid position of the beam as a function of propagation distance. The beam width and the centroid position were obtained by evaluating numerically employing a Romberg integral method.

In order to obtain practical results, we will consider a specific fiber, whose core is made of doped silica (7.9% ${GeO}_{2}$ at the core center) and a cladding made of pure silica ${SiO}_{2}$ will be considered^[15,23]. The frequency dependence of the refractive index is given by the Sellmeier formula^[15] $n^{2} (ω) = 1 + \sum_{j = 1}^{3} \frac{C_{j} ω_{j}^{2}}{ω_{j}^{2} - ω^{2}} .$ (9)For silica glass doped with 7.9% ${GeO}_{2}$ , the parameters are $C_{1} = 0.7136824$ , $C_{2} = 0.4254807$ , $C_{3} = 0.8964226$ , $λ_{1} = 0.0617167 μm$ , $λ_{2} = 0.1270814 μm$ , and $λ_{3} = 9.896161 μm$ . For pure silica, the parameters are $C_{1} = 0.6964663$ , $C_{2} = 0.4079426$ , $C_{3} = 0.8974794$ , $λ_{1} = 0.0684043 μm$ , $λ_{2} = 0.1162414 μm$ , and $λ_{3} = 9.896161 μm$ , where $λ_{j} = 2 π c / ω_{j}$ . Moreover, the incident beam parameters may be chosen as $λ = 632.8 nm$ and $w_{0} = 50 μm$ . From the parameters, we can deduced that the properties of the beam vary periodically with a longitudinal period $L = 2 π / α = 500 w_{0}$ , which will be taken as the unit of length along the beam propagating direction.

Figure 1 shows the evolution of intensity distribution for the beam when the parameter $a_{x} = a_{y} = 0.2$ . It is evident from Figs. 1(a)–1(d) that with increasing of propagation distance, the main lobe will become smaller and the side lobes will close to the main lobe. When $z = L / 4$ , the main lobe and side lobes focus at one point on the axis, and the intensity increase dramatically [Fig. 1(e)]. However, while $z > L / 4$ , the side lobes located in the other side of main lobe, and they will spread and weaken gradually. We can find that the intensity distribution of $z = 0$ and $z = L / 2$ are centro-symmetric. We also find that when the propagation distance $z$ increase from $L / 2$ to $L$ , the intensity distribution change in the inverted sequence in Fig. 1; that is, when $z = L / 2$ , $9 L / 16$ , $5 L / 8$ , $11 L / 16$ , $3 L / 4$ , $13 L / 16$ , $7 L / 8$ , $15 L / 16$ , and $L$ , the corresponding intensity distributions are as shown in Figs. 1(a)–1(i), respectively. We can see that the intensity distribution changes periodically and focuses but the intensity features remain unchanged when the propagation distance $z$ equals to $L / 4$ and $3 L / 4$ in one period. In general, the entire intensity distribution of the propagating beam is simply expanding and shrinking in the medium.

Figure 1.Evolution of the intensity distribution for the two dimensional Airy beam with $a_{x} = a_{y} = 0.2$ . Propagation distance is as follows: (a) $z = 0$ , (b) $L / 16$ , (c) $L / 8$ , (d) $3 L / 16$ , (e) $L / 4$ , (f) $5 L / 16$ , (g) $3 L / 8$ , (h) $7 L / 16$ , and (i) $L / 2$ .

Figure 2 illustrates the intensity distribution of the beam at the plane $z = L / 16$ for different values of $a_{x}$ or $a_{y}$ . It is shown that with the increase of values for $a_{x}$ or $a_{y}$ , the distribution of the intensity changes from a well-known Airy beam profile to Gaussian-like shape; this means that the side lobes of the Airy beam disappear gradually. Meanwhile, location of the main lobe moves from the third quadrant to the first quadrant, and the magnitude of the beam peak reduces initially, but when $a_{x} > 0.63$ the level of the peaks will begin to rise again.

Figure 2.Intensity distribution of the beam at the plane $z = L / 16$ for different values of $a_{x} = a_{y}$ . Values of $a_{x}$ or $a_{y}$ are chosen as follows: (a) 0.1, (b) 0.25, (c) 0.4, (d) 0.5, (e) 0.63, (f) 0.83, (g) 0.9, (h) 1.2, and (i) 1.3.

It is well-known that the Airy beam still exhibits its most exotic feature, i.e., its trend to freely accelerate^[1]. Here we adopt the centroid position of the beams to describe this behavior. The variation of the centroid position of the beam as a function of beam propagation distance $z / L$ with several values of $a_{x}$ are shown in Fig. 3. It is easy to find that the periodical bends (the transversely self-accelerating) of the beam are obvious in the GRIN fiber; this behavior is also reflected in the terms of $(j / A w_{0}) - (B^{2} / 4 k^{2} A^{2} w_{0}^{4})$ , which appears in the argument of the Airy function in Eq. (6). For a fixed value of $a_{x}$ , the centroid position oscillates back and forth across the optical axis. Additionally, it can be found that the increasing value of the truncated parameter $a_{x}$ will weaken the magnitude of the oscillation. When $a_{x} = 0.63$ , the centroid position keeps a constant. However, when $a_{x} > 0.63$ , the centroid position at the initial plane is positive. During the propagation, its behavior is opposite to the case of $a_{x} < 0.63$ .

Figure 3.Centroid position of one-dimensional finite energy Airy beam with different value of $a_{x}$ . Values of $a_{x} = 0.25$ (solid red line), 0.3 (solid green line), 0.45 (solid blue line), 0.63 (solid black line), $a_{x} = 0.9$ (dashed red line), 1.0 (dashed green line), and 1.2 (dashed blue line).

Figure 4 gives the variation of beam width as a function of propagation distance $z / L$ for different values of $a_{x}$ . As a supplementary description of the self-focusing, this shows that the beam width changes and has the minimum value at the propagation distances of $L / 4$ and $3 L / 4$ when $a_{x}$ is fixed in a period. Furthermore, the maximum value of the beam width located at the points of $z = 0$ , $L / 2$ , $L$ and the beam width at different distance almost unchanged when $a_{x}$ approaches 0.63; these results are consistent with that of Fig. 1.

Figure 4.Beam width of one-dimensional finite energy Airy beam with different value of $a_{x}$ . Values of $a_{x} = 0.25$ (solid red line), 0.3 (solid green line), 0.45 (solid blue line), 0.63 (solid black line), $a_{x} = 0.9$ (dashed red line), 1.0 (dashed green line), and 1.2 (dashed blue line).

In conclusion, we show the periodical self-focusing and self-accelerating of a finite energy Airy beam in a GRIN fiber. The periodicity of self-focusing and self-accelerating depend on the fiber parameter $α$ . Also, the truncated parameter $a$ has important influence on other behavior of the beam. When increasing $a_{x}$ and $a_{y}$ , the beam will transform into a Gaussian beam. The centroid position changes periodically; when $a_{x} = 0.63$ , its trajectory is a straight line. The beam width will reduce and enlarge when $a_{x}$ increases and $a_{x} = 0.63$ corresponds to the inflexion.