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 [13]. 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

iϕξ+122ϕs2=0,
where ϕ is the Airy wave function, ξ=z/kx02 is a normalized propagation distance and s=x/x0 is a dimensionless transverse coordinate where x0 is a transverse scaling factor [4,5]. k denotes a wave number which is defined as k=2πn/λ0 where n is the refractive index of the medium and λ0 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:

ϕ(s,ξ)=Ai[as+p(ξ)]exp[q(ξ)s+r(ξ)],
where a=±1 determines the accelerating direction of the Airy beam. By the substitution of Eq. (2) into Eq. (1), we can get the following conditions:
p(ξ)=14a4ξ2+ic1aξ+c2,
q(ξ)=i12a3ξ+c1,
r(ξ)=i112a6ξ312c1a3ξ2+i12(a2c2+c12)ξ+c3,
where c1, c2, c3 are arbitrary complex constants of integration. Since c3 is related to a constant amplitude and phase factor of the Airy beam, we will neglect it throughout this paper.

From now on, let us, in the first place, investigate the role of c1 and c2. We can rearrange Eq. (2) as follows:

ϕ(s,ξ)=Ai[k1(s,ξ)+ik2(ξ)]exp[Re{c1}s+i(12a3ξ+Im{c1})s+k3(ξ)+ik4(ξ)],
where

k1(s,ξ)=as14a4ξ2Im{c1}aξ+Re{c2},
k2(ξ)=Re{c1}aξ+Im{c2},
k3(ξ)=12Re{c1}a3ξ212(Im{c2}a2+2Re{c1}Im{c1})ξ,
k4(ξ)=112a6ξ312Im{c1}a3ξ2+12(Re{c2}a2+Re2{c1}Im2{c1})ξ.

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 |arg(z)|π/3 (in the range π/3<|arg(z)|<π, the Airy function grows exponentially as a function of |z|). Therefore, in order to prevent Airy function from diverging, we have to make sure |tan1[k2(ξ)/k1(s,ξ)]|π/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, c1, and c2.

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

s=14a3ξ2+Im{c1}ξRe{c2}a=14a3(ξ+2Im{c1}a3)2Im2{c1}a3Re{c2}a.
From Eq. (11), we can derive the initial launch angle of the Airy beam [6] as follows (see Fig. 1(a) for the definition):
θi=tan1(Im{c1}kx0).
Thus, we can see that the initial launch angle of the Airy beam is determined only by Im{c1}. From Eq. (11), we can also figure out the initial launch position of the Airy beam, which can be given as
xi=x0Re{c2}a.
Therefore, the initial launch position is determined by a and Re{c2}. But it is mainly controlled by Re{c2} because, as mentioned before, |a|=1.

 figure: Fig. 1

Fig. 1 (a) Definition of (launch) angle and the coordinate system. Dual Airy beam profiles when (b) α=1.45, γ=8 and (c) α=1.65, γ=10.

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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

zp=(kx02a)(Im{c2}Re{c1}).

From the practical point of view, it is easy to consider Re{c1} as a truncation factor which should be positive when a>0 and negative when a<0 (If Re{c1}=0, it produces nondispersive, ideal Airy beams. In this paper, we will set |Re{c1}| to 0.1). Therefore, we always have to make aRe{c1}>0. In addition, Im{c2} should be negative because negative zp is meaningless. Once Re{c1} and ξp=zp/kx02 are given, Im{c2} is determined automatically by Im{c2}=aRe{c1}ξp.

We summarized above discussions in Table 1 . It is notable again that we should check numerically the relation |tan1[k2(ξ)/k1(s,ξ)]|π/3 in our region of interest in order to avoid the diverging Airy beam solution.

Tables Icon

Table 1. Summary of roles played by the constants of integration

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 Re{c2} [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.

Tables Icon

Table 2. Simulation parameters used in Fig. 1(b) and (c)

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 2μm×6μm, 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.

 figure: Fig. 2

Fig. 2 Numerical results for the self-regeneration property: (a) single Airy beam and (b) dual Airy beam.

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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 Im{c1} can be decomposed into two parts. One is a shaping factor α, and the other a directional factor β. Each Airy beam’s Im{c1} can be represented as follows:

Im{c1,1}=+α+β,
Im{c1,2}=α+β.

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

θD=tan1(βkx0),
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 .

 figure: Fig. 3

Fig. 3 Numerical results for the directional property of dual Airy beams: (a) β=2 (Media 1, 1.44 MB) and (b) β=2.

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Tables Icon

Table 3. Simulation parameters used in Fig. 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=80μm) confirm this feature: transverse intensity patterns are independent of the beam propagation direction and they are just shifted.

 figure: Fig. 4

Fig. 4 Intensity patterns are not affected by the beam propagation direction: (a) β=0 and (b) β>0.

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 figure: Fig. 5

Fig. 5 Numerical results of the intensity patterns at z=80μm: (a) β=2 and (b) β=2.

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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:

ϕQ(sx,sy,ξx,ξy)=ϕx,D(sx,ξx)ϕy,D(sy,ξy),
where ϕx,D(sx,ξx) and ϕy,D(sy,ξy) are dual Airy beams. sx=x/x0 and sy=y/y0 denote dimensionless transverse coordinates, and ξx=z/kx02 and ξy=z/ky02 are normalized propagation distances [2]. In Fig. 6 , we plotted numerical simulation results of the propagation of quad Airy beam (λ=532nm, x0=y0=1μm, αx=αy=±1.65, βx=βy=0 and γx=γy=10). After long propagation distance, we can see that it maintains its intensity pattern almost unchanged, only getting wider. In addition, we can find that its intensity envelope approaches that of the Gaussian-shaped beam.

 figure: Fig. 6

Fig. 6 Transverse intensity patterns of quad Airy beams, changing the propagation distance from z=0μm to plane (a) z=40μm (Media 2, 794 KB), (b) z=80μm, (c) z=120μm, and (d) z=160μm.

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(2 + 1)D quad Airy beams also have self-regeneration property as mentioned earlier. And by modifying βx and βy, 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 βx and βy. The angle of propagation direction with respect to the z axis can be calculated explicitly as (see Appendix B)

 figure: Fig. 7

Fig. 7 Directional property of quad Airy beams at z=80μm under the changes in two directional factors (a) βx=1, βy=0, (b) βx=0, βy=1, (c) βx=1, βy=1, and (d) βx=2, βy=2 (Media 3, 873 KB). These results prove the invariant property of transverse intensity patterns of quad Airy beams.

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θQ=tan1(1k(βxx0)2+(βyy0)2).

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].

 figure: Fig. 8

Fig. 8 Magnified image of the transverse intensity pattern of quad Airy beam shown in Fig. 7. In our numerical results, radius of the individual optical beam spot is about 0.5μm. This uniform optical array is almost maintained in some range of propagation distance. In this range, we may use it as a multiple optical tweezer.

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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 .

Tables Icon

Table 4. Parameters for the ideal dual Airy beam

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

ϕD(s,ξ)=ϕ1(s,ξ)+ϕ2(s,ξ),
where ϕ1(s,ξ) is
ϕ1(s,ξ)=Ai[s+p1(ξ)]exp[q1(ξ)s+r1(ξ)],
where
p1(ξ)=14ξ2(α+β)ξ+γ,
q1(ξ)=i12ξ+i(α+β),
r1(ξ)=i112ξ3i12(α+β)ξ2+i12(γα2β22αβ)ξ.
And ϕ2(s,ξ) is
ϕ2(s,ξ)=Ai[s+p2(ξ)]exp[q2(ξ)s+r2(ξ)],
where
p2(ξ)=14ξ2+(α+β)ξ+γ,
q2(ξ)=i12ξ+i(α+β),
r2(ξ)=i112ξ3+i12(α+β)ξ2+i12(γα2β2+2αβ)ξ.
The intensity of dual Airy beam is
|ϕ1+ϕ2|2=|Ai[h1(s,ξ)]|2+|Ai[h2(s,ξ)]|2
+2Ai[h1(s,ξ)]Ai[h2(s,ξ)]cos[(ξ+2α)(sβξ)],
where

h1(s,ξ)=(sβξ)14ξ2αξ+γ,
h2(s,ξ)=(sβξ)14ξ2αξ+γ.

As shown in Eqs. (29)-(31), at the propagation distance ξ, β makes the center of transverse intensity pattern move to the location of s=βξ. 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 (sx,d(ξx) and sy,d(ξy) are functions which indicate the propagation directions of the dual Airy beam along the respective axes):

sx,dξx=βx,
sy,dξy=βy.
In their original coordinates, we have

xdz=βxkx0,
ydz=βyky0.

Now let us define rd=(xd,yd), then the propagation direction of the quad Airy beam with respect to z axis can be calculated from the following relation:

|rd|z=1k(βxx0)2+(βyy0)2,
from which we can obtain Eq. (19).

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).

References and links

1. M. V. Berry and N. L. Balazs, “Nonspreading wave packets,” Am. J. Phys. 47(3), 264–267 (1979). [CrossRef]  

2. J. Broky, G. A. Siviloglou, A. Dogariu, and D. N. Christodoulides, “Self-healing properties of optical Airy beams,” Opt. Express 16(17), 12880–12891 (2008). [CrossRef]   [PubMed]  

3. Y. Kaganovsky and E. Heyman, “Wave analysis of Airy beams,” Opt. Express 18(8), 8440–8452 (2010). [CrossRef]   [PubMed]  

4. G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett. 99(21), 213901 (2007). [CrossRef]  

5. G. A. Siviloglou and D. N. Christodoulides, “Accelerating finite energy Airy beams,” Opt. Lett. 32(8), 979–981 (2007). [CrossRef]   [PubMed]  

6. G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Ballistic dynamics of Airy beams,” Opt. Lett. 33(3), 207–209 (2008). [CrossRef]   [PubMed]  

7. Y. Hu, P. Zhang, C. Lou, S. Huang, J. Xu, and Z. Chen, “Optimal control of the ballistic motion of Airy beams,” Opt. Lett. 35(13), 2260–2262 (2010). [CrossRef]   [PubMed]  

8. A. Chong, W. H. Renninger, D. N. Christodoulides, and F. W. Wise, “Airy-Bessel wave packets as versatile linear light bullets,” Nat. Photonics 4(2), 103–106 (2010). [CrossRef]  

9. A. Salandrino and D. N. Christodoulides, “Airy plasmon: a nondiffracting surface wave,” Opt. Lett. 35(12), 2082–2084 (2010). [CrossRef]   [PubMed]  

10. J. Baumgartl, M. Mazilu, and K. Dholakia, “Optically mediated particle clearing using Airy wavepackets,” Nat. Photonics 2(11), 675–678 (2008). [CrossRef]  

11. D. McGloin and K. Dholakia, “Bessel beams: diffraction in a new light,” Contemp. Phys. 46(1), 15–28 (2005). [CrossRef]  

12. Z. Bouchal, J. Wagner, and M. Chlup, “Self-reconstruction of a distorted nondiffracting beam,” Opt. Commun. 151(4-6), 207–211 (1998). [CrossRef]  

13. J. C. Gutiérrez-Vega, M. D. Iturbe-Castillo, and S. Chávez-Cerda, “Alternative formulation for invariant optical fields: Mathieu beams,” Opt. Lett. 25(20), 1493–1495 (2000). [CrossRef]  

14. H. Cheng, W. Zang, W. Zhou, and J. Tian, “Analysis of optical trapping and propulsion of Rayleigh particles using Airy beam,” Opt. Express 18(19), 20384–20394 (2010). [CrossRef]   [PubMed]  

15. V. Garcés-Chávez, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, “Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam,” Nature 419(6903), 145–147 (2002). [CrossRef]   [PubMed]  

16. D. G. Grier, “A revolution in optical manipulation,” Nature 424(6950), 810–816 (2003). [CrossRef]   [PubMed]  

References

  • View by:

  1. M. V. Berry and N. L. Balazs, “Nonspreading wave packets,” Am. J. Phys. 47(3), 264–267 (1979).
    [Crossref]
  2. J. Broky, G. A. Siviloglou, A. Dogariu, and D. N. Christodoulides, “Self-healing properties of optical Airy beams,” Opt. Express 16(17), 12880–12891 (2008).
    [Crossref] [PubMed]
  3. Y. Kaganovsky and E. Heyman, “Wave analysis of Airy beams,” Opt. Express 18(8), 8440–8452 (2010).
    [Crossref] [PubMed]
  4. G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett. 99(21), 213901 (2007).
    [Crossref]
  5. G. A. Siviloglou and D. N. Christodoulides, “Accelerating finite energy Airy beams,” Opt. Lett. 32(8), 979–981 (2007).
    [Crossref] [PubMed]
  6. G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Ballistic dynamics of Airy beams,” Opt. Lett. 33(3), 207–209 (2008).
    [Crossref] [PubMed]
  7. Y. Hu, P. Zhang, C. Lou, S. Huang, J. Xu, and Z. Chen, “Optimal control of the ballistic motion of Airy beams,” Opt. Lett. 35(13), 2260–2262 (2010).
    [Crossref] [PubMed]
  8. A. Chong, W. H. Renninger, D. N. Christodoulides, and F. W. Wise, “Airy-Bessel wave packets as versatile linear light bullets,” Nat. Photonics 4(2), 103–106 (2010).
    [Crossref]
  9. A. Salandrino and D. N. Christodoulides, “Airy plasmon: a nondiffracting surface wave,” Opt. Lett. 35(12), 2082–2084 (2010).
    [Crossref] [PubMed]
  10. J. Baumgartl, M. Mazilu, and K. Dholakia, “Optically mediated particle clearing using Airy wavepackets,” Nat. Photonics 2(11), 675–678 (2008).
    [Crossref]
  11. D. McGloin and K. Dholakia, “Bessel beams: diffraction in a new light,” Contemp. Phys. 46(1), 15–28 (2005).
    [Crossref]
  12. Z. Bouchal, J. Wagner, and M. Chlup, “Self-reconstruction of a distorted nondiffracting beam,” Opt. Commun. 151(4-6), 207–211 (1998).
    [Crossref]
  13. J. C. Gutiérrez-Vega, M. D. Iturbe-Castillo, and S. Chávez-Cerda, “Alternative formulation for invariant optical fields: Mathieu beams,” Opt. Lett. 25(20), 1493–1495 (2000).
    [Crossref]
  14. H. Cheng, W. Zang, W. Zhou, and J. Tian, “Analysis of optical trapping and propulsion of Rayleigh particles using Airy beam,” Opt. Express 18(19), 20384–20394 (2010).
    [Crossref] [PubMed]
  15. V. Garcés-Chávez, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, “Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam,” Nature 419(6903), 145–147 (2002).
    [Crossref] [PubMed]
  16. D. G. Grier, “A revolution in optical manipulation,” Nature 424(6950), 810–816 (2003).
    [Crossref] [PubMed]

2010 (5)

2008 (3)

2007 (2)

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett. 99(21), 213901 (2007).
[Crossref]

G. A. Siviloglou and D. N. Christodoulides, “Accelerating finite energy Airy beams,” Opt. Lett. 32(8), 979–981 (2007).
[Crossref] [PubMed]

2005 (1)

D. McGloin and K. Dholakia, “Bessel beams: diffraction in a new light,” Contemp. Phys. 46(1), 15–28 (2005).
[Crossref]

2003 (1)

D. G. Grier, “A revolution in optical manipulation,” Nature 424(6950), 810–816 (2003).
[Crossref] [PubMed]

2002 (1)

V. Garcés-Chávez, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, “Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam,” Nature 419(6903), 145–147 (2002).
[Crossref] [PubMed]

2000 (1)

1998 (1)

Z. Bouchal, J. Wagner, and M. Chlup, “Self-reconstruction of a distorted nondiffracting beam,” Opt. Commun. 151(4-6), 207–211 (1998).
[Crossref]

1979 (1)

M. V. Berry and N. L. Balazs, “Nonspreading wave packets,” Am. J. Phys. 47(3), 264–267 (1979).
[Crossref]

Balazs, N. L.

M. V. Berry and N. L. Balazs, “Nonspreading wave packets,” Am. J. Phys. 47(3), 264–267 (1979).
[Crossref]

Baumgartl, J.

J. Baumgartl, M. Mazilu, and K. Dholakia, “Optically mediated particle clearing using Airy wavepackets,” Nat. Photonics 2(11), 675–678 (2008).
[Crossref]

Berry, M. V.

M. V. Berry and N. L. Balazs, “Nonspreading wave packets,” Am. J. Phys. 47(3), 264–267 (1979).
[Crossref]

Bouchal, Z.

Z. Bouchal, J. Wagner, and M. Chlup, “Self-reconstruction of a distorted nondiffracting beam,” Opt. Commun. 151(4-6), 207–211 (1998).
[Crossref]

Broky, J.

Chávez-Cerda, S.

Chen, Z.

Cheng, H.

Chlup, M.

Z. Bouchal, J. Wagner, and M. Chlup, “Self-reconstruction of a distorted nondiffracting beam,” Opt. Commun. 151(4-6), 207–211 (1998).
[Crossref]

Chong, A.

A. Chong, W. H. Renninger, D. N. Christodoulides, and F. W. Wise, “Airy-Bessel wave packets as versatile linear light bullets,” Nat. Photonics 4(2), 103–106 (2010).
[Crossref]

Christodoulides, D. N.

Dholakia, K.

J. Baumgartl, M. Mazilu, and K. Dholakia, “Optically mediated particle clearing using Airy wavepackets,” Nat. Photonics 2(11), 675–678 (2008).
[Crossref]

D. McGloin and K. Dholakia, “Bessel beams: diffraction in a new light,” Contemp. Phys. 46(1), 15–28 (2005).
[Crossref]

V. Garcés-Chávez, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, “Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam,” Nature 419(6903), 145–147 (2002).
[Crossref] [PubMed]

Dogariu, A.

Garcés-Chávez, V.

V. Garcés-Chávez, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, “Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam,” Nature 419(6903), 145–147 (2002).
[Crossref] [PubMed]

Grier, D. G.

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Siviloglou, G. A.

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A. Chong, W. H. Renninger, D. N. Christodoulides, and F. W. Wise, “Airy-Bessel wave packets as versatile linear light bullets,” Nat. Photonics 4(2), 103–106 (2010).
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Contemp. Phys. (1)

D. McGloin and K. Dholakia, “Bessel beams: diffraction in a new light,” Contemp. Phys. 46(1), 15–28 (2005).
[Crossref]

Nat. Photonics (2)

J. Baumgartl, M. Mazilu, and K. Dholakia, “Optically mediated particle clearing using Airy wavepackets,” Nat. Photonics 2(11), 675–678 (2008).
[Crossref]

A. Chong, W. H. Renninger, D. N. Christodoulides, and F. W. Wise, “Airy-Bessel wave packets as versatile linear light bullets,” Nat. Photonics 4(2), 103–106 (2010).
[Crossref]

Nature (2)

V. Garcés-Chávez, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, “Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam,” Nature 419(6903), 145–147 (2002).
[Crossref] [PubMed]

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[Crossref] [PubMed]

Opt. Commun. (1)

Z. Bouchal, J. Wagner, and M. Chlup, “Self-reconstruction of a distorted nondiffracting beam,” Opt. Commun. 151(4-6), 207–211 (1998).
[Crossref]

Opt. Express (3)

Opt. Lett. (5)

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[Crossref]

Supplementary Material (3)

Media 1: AVI (1435 KB)     
Media 2: AVI (795 KB)     
Media 3: AVI (873 KB)     

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Figures (8)

Fig. 1
Fig. 1 (a) Definition of (launch) angle and the coordinate system. Dual Airy beam profiles when (b) α = 1.45 , γ = 8 and (c) α = 1.65 , γ = 10 .
Fig. 2
Fig. 2 Numerical results for the self-regeneration property: (a) single Airy beam and (b) dual Airy beam.
Fig. 3
Fig. 3 Numerical results for the directional property of dual Airy beams: (a) β = 2 (Media 1, 1.44 MB) and (b) β = 2 .
Fig. 4
Fig. 4 Intensity patterns are not affected by the beam propagation direction: (a) β = 0 and (b) β > 0 .
Fig. 5
Fig. 5 Numerical results of the intensity patterns at z = 80 μ m : (a) β = 2 and (b) β = 2 .
Fig. 6
Fig. 6 Transverse intensity patterns of quad Airy beams, changing the propagation distance from z = 0 μ m to plane (a) z = 40 μ m (Media 2, 794 KB), (b) z = 80 μ m , (c) z = 120 μ m , and (d) z = 160 μ m .
Fig. 7
Fig. 7 Directional property of quad Airy beams at z = 80 μ m under the changes in two directional factors (a) β x = 1 , β y = 0 , (b) β x = 0 , β y = 1 , (c) β x = 1 , β y = 1 , and (d) β x = 2 , β y = 2 (Media 3, 873 KB). These results prove the invariant property of transverse intensity patterns of quad Airy beams.
Fig. 8
Fig. 8 Magnified image of the transverse intensity pattern of quad Airy beam shown in Fig. 7. In our numerical results, radius of the individual optical beam spot is about 0.5 μ m . This uniform optical array is almost maintained in some range of propagation distance. In this range, we may use it as a multiple optical tweezer.

Tables (4)

Tables Icon

Table 1 Summary of roles played by the constants of integration

Tables Icon

Table 2 Simulation parameters used in Fig. 1(b) and (c)

Tables Icon

Table 3 Simulation parameters used in Fig. 3

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Table 4 Parameters for the ideal dual Airy beam

Equations (37)

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i ϕ ξ + 1 2 2 ϕ s 2 = 0 ,
ϕ ( s , ξ ) = A i [ a s + p ( ξ ) ] exp [ q ( ξ ) s + r ( ξ ) ] ,
p ( ξ ) = 1 4 a 4 ξ 2 + i c 1 a ξ + c 2 ,
q ( ξ ) = i 1 2 a 3 ξ + c 1 ,
r ( ξ ) = i 1 12 a 6 ξ 3 1 2 c 1 a 3 ξ 2 + i 1 2 ( a 2 c 2 + c 1 2 ) ξ + c 3 ,
ϕ ( s , ξ ) = A i [ k 1 ( s , ξ ) + i k 2 ( ξ ) ] exp [ Re { c 1 } s + i ( 1 2 a 3 ξ + Im { c 1 } ) s + k 3 ( ξ ) + i k 4 ( ξ ) ] ,
k 1 ( s , ξ ) = a s 1 4 a 4 ξ 2 Im { c 1 } a ξ + Re { c 2 } ,
k 2 ( ξ ) = Re { c 1 } a ξ + Im { c 2 } ,
k 3 ( ξ ) = 1 2 Re { c 1 } a 3 ξ 2 1 2 ( Im { c 2 } a 2 + 2 Re { c 1 } Im { c 1 } ) ξ ,
k 4 ( ξ ) = 1 12 a 6 ξ 3 1 2 Im { c 1 } a 3 ξ 2 + 1 2 ( Re { c 2 } a 2 + Re 2 { c 1 } Im 2 { c 1 } ) ξ .
s = 1 4 a 3 ξ 2 + Im { c 1 } ξ Re { c 2 } a = 1 4 a 3 ( ξ + 2 Im { c 1 } a 3 ) 2 Im 2 { c 1 } a 3 Re { c 2 } a .
θ i = tan 1 ( Im { c 1 } k x 0 ) .
x i = x 0 Re { c 2 } a .
z p = ( k x 0 2 a ) ( Im { c 2 } Re { c 1 } ) .
Im { c 1 , 1 } = + α + β ,
Im { c 1 , 2 } = α + β .
θ D = tan 1 ( β k x 0 ) ,
ϕ Q ( s x , s y , ξ x , ξ y ) = ϕ x , D ( s x , ξ x ) ϕ y , D ( s y , ξ y ) ,
θ Q = tan 1 ( 1 k ( β x x 0 ) 2 + ( β y y 0 ) 2 ) .
ϕ D ( s , ξ ) = ϕ 1 ( s , ξ ) + ϕ 2 ( s , ξ ) ,
ϕ 1 ( s , ξ ) = A i [ s + p 1 ( ξ ) ] exp [ q 1 ( ξ ) s + r 1 ( ξ ) ] ,
p 1 ( ξ ) = 1 4 ξ 2 ( α + β ) ξ + γ ,
q 1 ( ξ ) = i 1 2 ξ + i ( α + β ) ,
r 1 ( ξ ) = i 1 12 ξ 3 i 1 2 ( α + β ) ξ 2 + i 1 2 ( γ α 2 β 2 2 α β ) ξ .
ϕ 2 ( s , ξ ) = A i [ s + p 2 ( ξ ) ] exp [ q 2 ( ξ ) s + r 2 ( ξ ) ] ,
p 2 ( ξ ) = 1 4 ξ 2 + ( α + β ) ξ + γ ,
q 2 ( ξ ) = i 1 2 ξ + i ( α + β ) ,
r 2 ( ξ ) = i 1 12 ξ 3 + i 1 2 ( α + β ) ξ 2 + i 1 2 ( γ α 2 β 2 + 2 α β ) ξ .
| ϕ 1 + ϕ 2 | 2 = | A i [ h 1 ( s , ξ ) ] | 2 + | A i [ h 2 ( s , ξ ) ] | 2
+ 2 A i [ h 1 ( s , ξ ) ] A i [ h 2 ( s , ξ ) ] cos [ ( ξ + 2 α ) ( s β ξ ) ] ,
h 1 ( s , ξ ) = ( s β ξ ) 1 4 ξ 2 α ξ + γ ,
h 2 ( s , ξ ) = ( s β ξ ) 1 4 ξ 2 α ξ + γ .
s x , d ξ x = β x ,
s y , d ξ y = β y .
x d z = β x k x 0 ,
y d z = β y k y 0 .
| r d | z = 1 k ( β x x 0 ) 2 + ( β y y 0 ) 2 ,

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