## Abstract

We derive a general form of Airy wave function which satisfies paraxial equation of diffraction. Based on this, we propose a new form of Airy beam, which is composed of two symmetrical Airy beams which accelerate mutually in the opposite directions. This ‘dual’ Airy beam shows several distinguishing features: it has a symmetric transverse intensity pattern and improved self-regeneration property. In addition, we can easily control the propagation direction. We also propose ‘quad’ Airy beam, which forms a rectangular shaped optical array of narrow beams that travel along a straight line. We can control its propagation direction without changing transverse intensity patterns. These kinds of superposed optical beams are expected to be useful for various applications with their unique properties.

©2010 Optical Society of America

## 1. Introduction

Airy beam is well-known for its nondiffracting and transversely accelerating features. An interesting phenomenon is that Airy beam is regenerated after the propagation of some distance when it encounters an obstruction object [1–3]. Finite Airy beam which is truncated for physical realization also shows unique properties of ideal Airy beam [4,5]. By analyzing ballistic dynamics of Airy beam, we can control its initial launch condition [6], and also its location of intensity peak [7]. Recently, as an application of these Airy wave packets, Airy-Bessel linear light bullets were introduced [8]. Another potential application of Airy beams is the realization of Airy surface plasmonic waves that show diffraction-free propagation characteristic [9]. It was also shown that using the accelerating property, Airy beams can be used for microparticle clearing [10]. Airy beams have distinguishing optical trapping features compared with other families of nondiffracting beams such as Bessel beams [11,12] or Mathieu beams [13] and recently, trapping of Rayleigh particles by Airy beams was quantitatively analyzed [14].

Our study is motivated from the fact that single Airy beams show asymmetric transverse intensity patterns. Although this characteristic is interesting and quite useful in some applications, it is somewhat improper in some other applications from the practical point of view. In this paper, we want to compensate this demerit of Airy beam and consider its improvement by considering the superposition of two or four single Airy beams. These ‘dual’ or ‘quad’ Airy beams will be suitable for, for example, the optical manipulation of microparticles because their symmetric beam intensity patterns do not require any specific (or preferred) directions in alignment between the beams and the particles and thus, can simplify the experimental configurations significantly, especially in the case of simultaneous manipulation of multiple particles [15,16].

We also note that these dual and quad Airy beams have enhanced self-regeneration property. This property will be very useful in plasmonic devices [9] because the beams are less affected by surface roughness, resulting in an increase in the practical propagation length.

We first derive a general form of Airy wave packet solution and investigate its physical implication. Then, we will introduce a new form of ‘superposed’ Airy beams, which shows a symmetric transverse intensity pattern and intriguing directional property appropriate for the various applications.

## 2. General form of the Airy wave function

Airy beams propagating in free space are the solution of normalized paraxial equation of diffraction, which can be written as follows in case of (1 + 1)D

*ϕ*is the Airy wave function, $\xi =z/k{x}_{0}^{2}$ is a normalized propagation distance and $s=x/{x}_{0}$ is a dimensionless transverse coordinate where ${x}_{0}$ is a transverse scaling factor [4,5].

*k*denotes a wave number which is defined as $k=2\pi n/{\lambda}_{0}$ where

*n*is the refractive index of the medium and ${\lambda}_{0}$ is the free space wavelength.

To discuss the properties of Airy beams, let us assume a general form of (1 + 1)D Airy beam wave function:

where $a=\pm 1$ determines the accelerating direction of the Airy beam. By the substitution of Eq. (2) into Eq. (1), we can get the following conditions:From now on, let us, in the first place, investigate the role of ${c}_{1}$ and ${c}_{2}$. We can rearrange Eq. (2) as follows:

Let us first consider the Airy function part. First, we have to mention that the Airy function with a complex argument *z* is physically meaningful only when we have $\left|\mathrm{arg}\left(z\right)\right|\le \pi /3$ (in the range $\pi /3<\left|\mathrm{arg}\left(z\right)\right|<\pi $, the Airy function grows exponentially as a function of $\left|z\right|$). Therefore, in order to prevent Airy function from diverging, we have to make sure $\left|{\mathrm{tan}}^{-1}\left[{k}_{2}\left(\xi \right)/{k}_{1}\left(s,\xi \right)\right]\right|\le \pi /3$ for all range of propagation region. We will assume throughout this paper that in our region of interest, this condition can be satisfied numerically by selecting appropriate values of *a*, ${c}_{1}$, and ${c}_{2}$.

We can discern the trajectory of the Airy beam by analyzing Eq. (7). It is given by

*a*and $\mathrm{Re}\left\{{c}_{2}\right\}$. But it is mainly controlled by $\mathrm{Re}\left\{{c}_{2}\right\}$ because, as mentioned before, $\left|a\right|=1$.

From the imaginary part of the argument of the Airy function, that is Eq. (8), we can determine the intensity peak position [7]. It is given as

From the practical point of view, it is easy to consider $\mathrm{Re}\left\{{c}_{1}\right\}$ as a truncation factor which should be positive when $a>0$ and negative when $a<0$ (If $\mathrm{Re}\left\{{c}_{1}\right\}=0$, it produces nondispersive, ideal Airy beams. In this paper, we will set $\left|\mathrm{Re}\left\{{c}_{1}\right\}\right|$ to 0.1). Therefore, we always have to make $a\mathrm{Re}\left\{{c}_{1}\right\}>0$. In addition, $\mathrm{Im}\left\{{c}_{2}\right\}$ should be negative because negative ${z}_{p}$ is meaningless. Once $\mathrm{Re}\left\{{c}_{1}\right\}$ and ${\xi}_{p}={z}_{p}/k{x}_{0}^{2}$ are given, $\mathrm{Im}\left\{{c}_{2}\right\}$ is determined automatically by $\mathrm{Im}\left\{{c}_{2}\right\}=-a\mathrm{Re}\left\{{c}_{1}\right\}{\xi}_{p}$.

We summarized above discussions in Table 1 . It is notable again that we should check numerically the relation $\left|{\mathrm{tan}}^{-1}\left[{k}_{2}\left(\xi \right)/{k}_{1}\left(s,\xi \right)\right]\right|\le \pi /3$ in our region of interest in order to avoid the diverging Airy beam solution.

The Fourier transform spectrum of the Airy beam with $a=-1$ is nearly identical (the magnitude spectrum is exactly the same) to that of the Airy beam with $a=1$. Their only difference lies in the phase of the spectrum: Briefly speaking, the phase spectrums of these two Airy beams are almost the same in absolute value but opposite in sign.

## 3. Formulation and numerical results of dual Airy beam

Using the property that we can control the initial launch position of Airy beams by changing $\mathrm{Re}\left\{{c}_{2}\right\}$ [see Eq. (13)], we can launch two Airy beam tails simultaneously, which accelerate in the mutually opposite directions. From now on we will call this new kind of beam ‘dual Airy beam’. Figures 1(b) and (c) show the intensity profiles of the dual Airy beam. Detailed parameters used in the calculations are summarized in Table 2 . From the results, we can find several straight lines generated by the interference of two Airy beams. It is notable that they maintain the nearly constant intensity values for relatively long propagation distances.

It is well-known that Airy beams have ‘nondiffracting’ property like Bessel beams, although their nondiffracting mechanism is different from that of the Bessel beams [11,12]. This nondiffraction enables the self-regeneration of Airy beam when it encounters obstructions: the unobstructed caustic of rays departing from side regions that are away from the position of obstructions can reconstruct the main lobe of the Airy beam [3]. In case of the dual Airy beam, we can expect improved self-regeneration property compared with that of the single Airy beam, since the dual Airy beam is regenerated by ray contributions arriving side-ways not only from side regions of Airy beam 1 (with $a=1$) but also from the side regions of Airy beam 2 (with $a=-1$). To check this, we calculated the beam propagation profiles when an obstruction object, whose size is about $2\mu m\times 6\mu m$, hinders the Airy beams. Figure 2 is the numerical results using a finite element method (FEM)-based commercial solver COMSOL on this self-regeneration property of single and dual Airy beams. From the figure, we can find the improved self-regeneration property of dual Airy beam.

Dual Airy beams have an interesting property: we can control their propagation directions. We can say that single Airy beams also have this property, but it has limited applications because of its asymmetric transverse intensity patterns. In the case of dual Airy beams, initial launch angle factor $\mathrm{Im}\left\{{c}_{1}\right\}$ can be decomposed into two parts. One is a shaping factor *α*, and the other a directional factor *β*. Each Airy beam’s $\mathrm{Im}\left\{{c}_{1}\right\}$ can be represented as follows:

Subscript numbers located behind the comma indicate the respective beam numbers. It can be shown after some algebra (see Appendix A) that the direction of dual Airy beams, which we defined as the direction of center peaks of the interference pattern is given by

where we can find that the direction of dual Airy beams are determined only by*β*, independent of

*α*. Figure 3 confirms this property: the added parameter

*β*determines the propagation direction of dual Airy beams. Calculation specifications are shown in Table 3 .

There is another interesting property of dual Airy beams. In fact, this property can also be found in the single Airy beam, but it is more meaningful in the case of dual Airy beams having transversely-symmetrical intensity patterns. Transverse intensity patterns are not affected by the change in beam propagation directions as is illustrated schematically in Fig. 4 . Numerical simulation results shown in Fig. 5 (which denote the intensity patterns at $z=80\mu m$) confirm this feature: transverse intensity patterns are independent of the beam propagation direction and they are just shifted.

## 4. Numerical results for (2 + 1)D case - quad Airy beam

We can easily expand previous works which are related to (1 + 1)D Airy beams to (2 + 1)D ones. In the latter case, we can also superpose four Airy beams to generate ‘quad Airy beam’, which can be expressed as follows:

(2 + 1)D quad Airy beams also have self-regeneration property as mentioned earlier. And by modifying ${\beta}_{x}$ and ${\beta}_{y}$, we can control the propagation directions. Of course, in this case its transverse intensity pattern is also invariant under the change of the propagation direction as can be expected. Figure 7
shows that the transverse intensity patterns are moved transversely by changing ${\beta}_{x}$ and ${\beta}_{y}$. The angle of propagation direction with respect to the *z* axis can be calculated explicitly as (see Appendix B)

Figure 6 suggests that quad Airy beams can be applied to the optical trapping for microparticles. As can be seen in the figure, quad Airy beams have the shape of rectangular array. This optical array can be used for simultaneous trapping of multiple microparticles. Each straight line of superposed beams can trap microparticles by dragging them toward its center position having a maximum intensity value.

Transverse intensity pattern of the superposed beams at the center of optical array is shown in more detail in Fig. 8 . Because the intensity pattern of quad Airy beam is formed by four symmetric single Airy beams, its nondiffracting phenomenon occurs by the principle mentioned previously. Using the shift property of the transverse intensity pattern, we might move particles transversely in some ranges. In addition, because the Airy beam has a self-regeneration property, it would also be possible to implement optical manipulation of multiple planes [15].

## 5. Conclusion

We have derived a general form of Airy wave solution, and proposed ‘dual’ Airy beam which is the superposition of two Airy beams, and numerically simulated them under various conditions. Dual Airy beams have symmetric transverse intensity profiles and show the directional property which can be controlled (easily in theory but might be challenging in practice). Furthermore, they show good self-regeneration property. We also proposed quad Airy beam, which has a rectangular-shaped array of narrow nondiffracting beams. We expect these new forms of beams will be used for various purposes such as optical trapping, sorting and manipulation of microparticles. Especially, the rectangular array of straight optical beams generated by the superposition of two Airy beams is strongly localized in the area which is surrounded by each single Airy beam’s main lobe, and each straight beam can be used for simultaneous trapping or manipulation of microparticles. In addition, by using their superior directional property and improved self-regeneration feature, the superposed Airy beams will find lots of applications in various fields of optics including plasmonics.

## Appendix A: Proof of Eq. (17)

For the simplicity of calculation, let us assume an ideal dual Airy beam. Parameters for this ideal dual Airy beam are shown in Table 4 .

In this case, the dual Airy beam can be represented as follows:

where ${\varphi}_{1}(s,\xi )$ isAs shown in Eqs. (29)-(31), at the propagation distance *ξ*, *β* makes the center of transverse intensity pattern move to the location of $s=\beta \xi $. Thus the propagation angle of dual Airy beam can be expressed as Eq. (17).

## Appendix B: Proof of Eq. (19)

From the previous discussions (for example, see the last paragraph of Appendix A), we can see that the propagation direction of the quad Airy beam with respect to each transverse axis in its normalized coordinates can be derived from the following relations (${s}_{x,d}({\xi}_{x})$ and ${s}_{y,d}({\xi}_{y})$ are functions which indicate the propagation directions of the dual Airy beam along the respective axes):

In their original coordinates, we haveNow let us define $\overrightarrow{{r}_{d}}=\left({x}_{d},{y}_{d}\right)$, then the propagation direction of the quad Airy beam with respect to *z* axis can be calculated from the following relation:

## Acknowledgment

The authors acknowledge the support of the National Research Foundation and the Ministry of Education, Science and Technology of Korea through the Creative Research Initiative Program (Active Plasmonics Application Systems).

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