1 Introduction

In recent decades, Airy beams have attracted more and more attentions, because their exotic characteristics are useful for certain applications [1,2,3,4]. One of the exotic characteristics of Airy beams is the self-healing. Broky et al. demonstrated both theoretically and experimentally that an Airy beam tends to reform during propagation in spite of the severity of the imposed perturbations such as scattering media and turbulent media [5]. Chu et al. derived the analytical expression for the optical field of an Airy beam, partially blocked by an opaque obstacle, and showed that the self-healing process of the Airy beam is affected by many factors, such as the opaque obstacle size, propagation distance, and parameters of the Airy beam [6]. Our group showed that an Airy beam cannot retain its shape and the structure due to the thermal blooming effect, when it propagates in the air [7]. Kong et al. studied the dynamics and collapse of two-dimensional Airy beams in nonlocal, nonlinear media and showed that the self-healing characteristic of Airy beams with angular momentum depends on the location and topological charge of the vortex [8].

Recently, the characteristics of Airy array beams were studied. Qian et al. presented theoretically and experimentally, the generalized and symmetric Airy beam by introducing two rotary angle factors and showed that it still has the ability of self-healing and non-diffraction along arbitrary trajectories [9]. Hwang et al. proposed a “dual” Airy beam which is composed of two symmetrical Airy beams which accelerates mutually in the opposite direction and this “dual” Airy beam has the improved self-regeneration characteristic [10]. Gu et al. showed that the scintillation of Airy array beams (i.e., four back-to-back combined Airy beams) is significantly reduced and close to the theoretical minimum [11], while Eyyuboğlu showed that it is possible to obtain substantial scintillation reductions even for a single Airy beam [12]. Ren et al. generated accelerating quad Airy array beams (i.e., four face-to-face combined Airy beams) using a symmetric 3/2 phase-only pattern, which possess self-construction ability when local areas are blocked [13]. Chen et al. studied the propagation of Airy array beams (i.e., four back-to-back combined Airy beams) through atmospheric turbulence and showed that Airy array beams can have the focusing capability which is impacted by the beamlet-combination type, turbulence strength, and Airy-beamlet parameters [14].

However, these studies are only related to the characteristics of Airy array beams for linear propagation [9,10,11,12,13,14]. The Kerr effect occurs, when a powerful beam propagates in the self-focusing media. There exist the natural self-focusing media (e.g., the air [15]). On the other hand, there also exist many synthetic self-focusing media (e.g., chalcogenide glasses [16,17,18], photorefractive crystal [19). In general, the nonlinear refractive index of synthetic self-focusing media is larger than that of natural self-focusing media. Recently, our group investigated the spherical aberration effect on the Kerr nonlinearity of a laser beam propagating through the atmosphere [20]. In this paper, the focusing and self-healing characteristics of Airy array beams (i.e., four back-to-back combined Airy beams) propagating in self-focusing media are studied, where both coherent and incoherent combinations are considered.

2 Focusing characteristic of Airy array beams in self-focusing media

As shown in Fig. 1, an Airy array beam in rectangular symmetry consists of four individual off-axis Airy beams positioned at the source plane z = 0, whose distance of separation is 2d. The field of the off-axis Airy beam centered at point (d, d) can be written as [21]:

$$ u\left( {x,y,0} \right) = u_{0} {\text{Ai}}\left( {\frac{x + d}{b}} \right)\exp \left( {a\frac{x + d}{b}} \right){\text{Ai}}\left( {\frac{y + d}{b}} \right)\exp \left( {a\frac{y + d}{b}} \right) , $$
(1)

where a is the exponential truncation factor, b is the arbitrary transverse scale, and the coefficient u0 can be derived using the relationship, \( P/4 = \int_{ - \infty }^{\infty } {\int_{ - \infty }^{\infty } {\left| u \right|^{2} } } {\text{d}}x{\text{d}}y = {\text{const}} \), which can be expressed as:

$$ u_{0} = \sqrt {\frac{2\pi aP}{{b^{2} \exp \left( {{{4a^{3} } \mathord{\left/ {\vphantom {{4a^{3} } 3}} \right. \kern-0pt} 3}} \right)}}} , $$
(2)

where P is the power of Airy array beams.

Fig. 1
figure 1

A schematic illustration of an Airy beam array (i.e., four back-to-back combined Airy beams)

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For the coherent combination, it is the coherent superposition of the field and the array beam propagation depends on the total field at z = 0 (i.e., the initial field). The total field of the Airy array beam at the z = 0 plane is expressed as:

$$ U\left( {x,y,0} \right) = u\left( {x,y,0} \right) + u\left( { - x,y,0} \right) + u\left( {x, - y,0} \right) + u\left( { - x, - y,0} \right), $$
(3)

For the incoherent combination, it is the superposition of the intensity and the propagation of individual off-axis beams is independent. Thus, at a propagation distance z, the total intensity is the intensity superposition of four individual off-axis Airy beams at the same propagation distance, i.e.

$$ I(x,y,z) = \left[ {u\left( {x,y,z} \right)} \right]^{2} + \left[ {u\left( { - x,y,z} \right)} \right]^{2} + \left[ {u\left( {x, - y,z} \right)} \right]^{2} + \left[ {u\left( { - x, - y,z} \right)} \right]^{2} , $$
(4)

Under the paraxial approximation, the features of diffraction and Kerr nonlinearity of an Airy array beam propagating in self-focusing media can be described by the nonlinear Schrödinger equation, i.e. [15],

$$ \nabla_{ \bot }^{2} A + 2ik\frac{\partial A}{\partial z} + 2k^{2} \frac{{n_{2} }}{{n_{0} }}\left| A \right|^{2} A = 0, $$
(5)

where the third term on the left side of Eq. (5) describes the Kerr nonlinearity, \( \nabla_{ \bot }^{2} = {{\partial^{2} } \mathord{\left/ {\vphantom {{\partial^{2} } {\partial x^{2} }}} \right. \kern-0pt} {\partial x^{2} }} + {{\partial^{2} } \mathord{\left/ {\vphantom {{\partial^{2} } {\partial y^{2} }}} \right. \kern-0pt} {\partial y^{2} }} \) is the transverse Laplace operator to describe transverse diffraction of the field, \( k = {{2\pi } \mathord{\left/ {\vphantom {{2\pi } \lambda }} \right. \kern-0pt} \lambda } \) is the wave number related to the wave length λ, n0, and n2 are the linear and nonlinear refractive indexes of nonlinear media, respectively. For the coherent combination, A is the total field (i.e., U) of the Airy array beam, while for the incoherent combination; A is the field of an off-axis Airy beam (i.e., u).

The nonlinear Schrödinger Eq. (5) can be solved using the multiphase screen method [15]. Let A(x, y zj) be the solution of Eq. (5) at zj plane and the solution of Eq. (5) at zj+1 = zj+ Δz plane can be written as: [22]

$$ A\left( {x,y,z_{j + 1} } \right) = \exp \left( { - \frac{i}{4k}\Delta z\nabla_{ \bot }^{2} } \right)\exp ( - is)\exp \left( { - \frac{i}{4k}\Delta z\nabla_{ \bot }^{2} } \right)A(x,y,z_{j} ), $$
(6)

where \( s = kn_{2} \left| {A(x,y,z_{j} )} \right|^{2} \Delta z/n_{0} \) is the phase modulation due to the Kerr nonlinearity effect within Δz nonlinear propagation. From Eq. (6), one can see that the propagating field over a distance Δz in self-focusing media consists of three steps, i.e., a linear propagation of the field over a distance Δz/2, then an increment in the phase caused by the Kerr nonlinearity effect within Δz nonlinear propagation, and a linear propagation of the resulting field over a distance Δz/2 again. Therefore, we may design a computer code for the beam propagation in self-focusing media using the multiphase screen method.

In this paper, the air is adopted as the self-focusing medium, the focusing characteristic and self-healing characteristic of Airy array beams are studied using the numerical simulation method, the calculation parameters are taken as: λ = 0.53 μm, a = 0.1, b = 100 μm, n0 = 1, and n2 = 5.6 × 10−19 cm2/W. In addition, the relative propagation distance \( \xi = z/z_{\text{R}} \) is adopted, where zR = kb2 is the Rayleigh range of a Gaussian beam.

The intensity profiles along x axis versus the relative propagation distance ξ for d/b = 1 and 2 are shown in Figs. 2 and 3, respectively. Without Kerr effect (i.e., linear propagation in free space), an Airy array beam will focus only once for both coherent and incoherent combinations, which is independent of the value of d/b (see Figs. 2a, d, 3a, d). This result is essentially in agreement with that shown in Ref. [14]. With Kerr effect, when the value of d/b is small (e.g., d/b = 1), an Airy array beam will still focus once (see Fig. 2b, c, e), but for the incoherent combination case, the focal position will move to a further distance, when the beam power is large enough (see Fig. 2f). However, when the value of d/b is not small (e.g., d/b = 2) and the beam power is large enough; for the coherent combination case, an Airy array beam will focus twice due to the Kerr effect (see Fig. 3c), and for the incoherent combination case an Airy array beam will not focus due to the Kerr effect (see Fig. 3f).

Fig. 2
figure 2

Intensity profiles along x axis versus the relative propagation distance ξ, d/b = 1. a–c coherent combination; d–f: incoherent combination

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Fig. 3
figure 3

Intensity profiles along x axis versus the relative propagation distance ξ, d/b = 2. a–c coherent combination; d–f incoherent combination

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The physical explanation for the “focus twice” is shown as follows. For both without and with Kerr effect, the transverse power flow of an Airy array beam with coherent combination at the different relative propagation distance is shown in Figs. 4 and 5, respectively. As the propagation distance increases, first the power flow moves to the center and then the first focus appears (see Figs. 4a, b, 5a, b). Comparing Fig. 5b with Fig. 4b, one can see that the beam spot of the first focus is compressed by the Kerr effect. As the propagation distance increases further, the power flow diverges from the center and then four main lobes are formed (see Figs. 4c, d, 5c, d). As the propagation distance increases even further, the central main lobe and the four main lobes appear alternately (e.g., Figs. 4e–g, 5e–g), but the intensity of the central main lobe is not large enough to be called the focus (e.g., Figs. 4e, g, 5e, g). In the far field, the four main lobes do not appear again (i.e., only the central main lobe is retained) as the propagation distance increases (see Figs. 4g–i, 5g–i). Without Kerr effect, the central main lobe spreads as the propagation distance increases (see Fig. 4g–i). However, the central main lobe is compressed by the Kerr effect, when the beam power is large enough to approach the critical value and then the second focus is formed (see Fig. 5i).

Fig. 4
figure 4

Transverse power flow of an Airy array beam with coherent combination at the different relative propagation distance ξ, d/b = 2, n2 = 0 (without Kerr effect)

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Fig. 5
figure 5

Transverse power flow of an Airy array beam with coherent combination at the different relative propagation distance ξ, d/b = 2, P = 1.78 × 106 W, n2 = 5.6 × 10−19 cm2/W (with Kerr effect)

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3 Self-healing characteristics of Airy array beams in self-focusing media

For the coherent combination, the self-healing of Airy array beams with main lobes blocked is shown in Fig. 6. It is noted that the self-healing of Airy array beams is studied theoretically in this paper and the blocking is achieved by setting the values of the main lobe intensity as zero. In the experiment, the blocking is achieved using the opaque obstacle [5]. One can see that the main lobes blocked are reborn as the propagation distance increases both with and without Kerr effect (see Fig. 6b, c, e, f). In particular, the intensity of the two regenerating diagonal main lobes may be larger than that of other two main lobes (see Fig. 6b, c). In addition, the propagation distance required for reconstruction of Airy array beams with main lobes blocked decreases due to the Kerr effect (compare Fig. 6c with b, compare Fig. 6f with e) and it decreases further as the Kerr effect becomes stronger (the numerical result is omitted here to save space). The transverse power flow of Airy beams increases due to the Kerr effect [23], which is the physical reason why the propagation distance required for reconstruction of Airy array beams with main lobes blocked decreases.

Fig. 6
figure 6

Self-healing of Airy array beams with coherent combination (d/b = 1, P = 5.94 × 105 W). Contour lines of the intensity at different relative propagation distance ξ. a–c an Airy array beam with two diagonal main lobes blocked; d–f an Airy array beam with four main lobes blocked

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To explain the self-healing characteristic, the transverse power flow of Airy array beams with main lobes blocked is shown in Figs. 7 and 8. As the propagation distance increases, the power flow moves to the center first (see Figs. 7a, 8a) and then form the focus (see Figs. 7b, 8b). As the propagation distance increases further, the power flow diverges from the center (see Figs. 7c, 8c) and then the main lobes blocked are reborn (see Figs. 7d, 8d). In this case, the reconstruction of Airy array beams with main lobes blocked finishes after focusing.

Fig. 7
figure 7

Transverse power flow of an Airy array beam with coherent combination (d/b = 1, P = 5.94 × 105 W) when its two diagonal main lobes are blocked, n2 = 0 (without Kerr effect)

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Fig. 8
figure 8

Transverse power flow of an Airy array beam with coherent combination (d/b = 1, P = 5.94 × 105 W) when its four main lobes are blocked, n2 = 0 (without Kerr effect)

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For the incoherent combination case, the self-healing of Airy array beams with four main lobes blocked is shown in Fig. 9. It can be seen that the main lobes blocked can regenerate as the propagation distance increases both with and without Kerr effect, which is similar to the behavior for the coherent combination case. However, the propagation distance required for reconstruction of Airy array beams with main lobes blocked is shorter for the incoherent combination than that for the coherent combination. On comparing Fig. 9b, c with Fig. 6e, f, one can see that a better reconstruction for the incoherent combination case can be achieved than that for the coherent combination case.

Fig. 9
figure 9

Self-healing of an Airy array beam with incoherent combination (d/b = 1, P = 5.94 × 105 W). Contour lines of the intensity at different relative propagation distance

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For the incoherent combination case, the transverse power flow of Airy array beams with four main lobes blocked is shown in Fig. 10. As the propagation distance increases, the power flow moves to the position of four main lobes (see Fig. 10a) and then the four main lobes blocked are regenerated (see Fig. 10b). As the propagation distance increases further, the power flow moves to the center and then the focus is formed (see Fig. 10c). In this case, the reconstruction of Airy array beams with main lobes blocked finishes before focusing.

Fig. 10
figure 10

Transverse power flow of an Airy array beam with incoherent combination (d/b = 1, P = 5.94 × 105 W) when its four main lobes are blocked, n2 = 0 (without Kerr effect)

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It is mentioned that the reconstruction of an Airy array beam with main lobes blocked may finish before or after focusing, which depends on the intensity distribution and the coherence of the array beam and is independent of the Kerr effect. The different situations and physical reasons are analyzed as follows: when one or two or three main lobes are blocked, the reconstruction of the array beams always finishes after focusing whether for the coherent or incoherent combinations. In this case, the main physical factor is that the array intensity distribution is asymmetric. The power flow of array beam focuses first and then redistributes, finally the reconstruction of the Airy array beam finishes. On the other hand, when four main lobes are blocked (i.e., the array intensity distribution is symmetric), the reconstruction of the array beam may finish before or after focusing. In this case, the main physical factor is the array beam coherence. The reconstruction of Airy array beams with four main lobes blocked always finishes before focusing for the incoherent combination case. However, for the coherent combination case, the reconstruction of the array beam with four main lobes blocked finishes after focusing when the beam separate distance d/b is small (e.g., d/b = 1), while the reconstruction of the array beam with four main lobes blocked finishes before focusing when the beam separate distance d/b is not small (e.g., d/b = 2) because this case is close to the incoherent combination one. It is noted that we have proven the results above numerically, which are omitted here to save space.

To examine whether the focusing characteristic of Airy array beams is affected by blocking main lobes, the on-axis intensity distributions for different cases versus the relative propagation distance are shown in Fig. 11. Figure 11a shows that the focal position (i.e., the position of maximum intensity) moves to a further propagation distance as the number of main lobes blocked increases. It is noted that the change of the focal position is small due to blocking main lobes and it decreases further because of the Kerr effect (see Fig. 11b). In addition, the intensity maximum decreases as the number of main lobes blocked increases and the intensity maximum increases greatly because of the Kerr effect.

Fig. 11
figure 11

On-axis intensity I(0, 0, ξ)of Airy array beams (d/b = 1, P = 5.94 × 105 W) for different main lobes blocked cases versus the relative propagation distance ξ. a without Kerr effect; b with Kerr effect

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4 Conclusions

In this paper, the focusing and self-healing characteristics of Airy array beams propagating in self-focusing media are studied in detail. When the Kerr effect is strong enough and the sub-beam separate distance is not small (e.g., d/b = 2), as the propagation distance increases, an Airy array beam will focus twice for the coherent combination case, while an Airy array beam will not focus for the incoherent combination case. These behaviors are quite different from those without Kerr effect (i.e., an Airy array beam will focus only once for both coherent and incoherent combinations).

With and without Kerr effect, the self-healing of Airy array beams with the main lobes blocked can be achieved. Furthermore, a better reconstruction for the incoherent combination case can be achieved than that for the coherent combination case. The propagation distance required for reconstruction of Airy array beams with the main lobes blocked decreases due to the Kerr effect. The reconstruction of an Airy array beam with main lobes blocked may finish before or after focusing, which depends on the intensity distribution and the coherence of the array beam and is independent of the Kerr effect. The relationship of the position between the reconstruction and the focus is analyzed in detail for different cases. The change of the Focal position is small due to blocking the main lobes, and it decreases further due to the Kerr effect. The results obtained in this paper are useful for various applications of Airy array beams.