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Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 20, Iss. 24 — Nov. 19, 2012
  • pp: 26913–26921
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Negative propagation effect in nonparaxial Airy beams

Pablo Vaveliuk and Oscar Martinez-Matos  »View Author Affiliations


Optics Express, Vol. 20, Issue 24, pp. 26913-26921 (2012)
http://dx.doi.org/10.1364/OE.20.026913


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Abstract

Negative propagation is an unusual effect concerning the local sign change in the Poynting vector components of an optical beam under free propagation. We report this effect for finite-energy Airy beams in a subwavelength nonparaxial regime. This effect is due to a coupling process between propagating and evanescent plane waves forming the beam in the spectral domain and it is demonstrated for a single TE or TM mode. This is contrary to what happens for vector Bessel beams and vector X-waves, for which a complex superposition of TE and TM modes is mandatory. We also show that evanescent waves cannot contribute to the energy flux density by themselves such that a pure evanescent Airy beam is not physically realizable. The break of the shape-preserving and diffraction-free properties of Airy beams in a nonparaxial regime is exclusively caused by the propagating waves. The negative propagation effect in subwavelength nonparaxial Airy beams opens new capabilities in optical traps and tweezers, optical detection of invisibility cloacks and selective on-chip manipulation of nanoparticles.

© 2012 OSA

1. Introduction

A non-square-integrable wave-packet in terms of Airy functions was firstly reported in quantum mechanics [1

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

]. Such a wavelike solution possesses peculiar features: lack of parity symmetry about the origin and an ideal diffraction-free propagation associated with self-bending dynamics. In optics, a solution of the paraxial wave equation (PEq) was derived giving rise to the so-called finite-energy Airy beam (AiB) [2

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

]. This beam maintains the quoted properties of the quantum wave packet but with the propagation being quasi-nondiffracting due to an exponential apodization in the field profile transforming it in a square-integrable beam. The AiB was experimentally realized by diverse phase mask encoded methods [3

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

6

6. J. A. Davis, M. J. Mitry, M. A. Bandres, I. Ruiz, K. P. McAuley, and D. M. Cottrell, “Generation of accelerating Airy and accelerating parabolic beams using phase-only patterns,” Appl. Opt. 48, 3170–3176 (2009). [CrossRef] [PubMed]

] and numerous Airylike beams, variants of that original finite-energy Airy beam, were widely analyzed from theoretical and experimental viewpoint [6

6. J. A. Davis, M. J. Mitry, M. A. Bandres, I. Ruiz, K. P. McAuley, and D. M. Cottrell, “Generation of accelerating Airy and accelerating parabolic beams using phase-only patterns,” Appl. Opt. 48, 3170–3176 (2009). [CrossRef] [PubMed]

10

10. M. I. Carvalho and M. Facão, “Propagation of Airy-related beams,” Opt. Express 18, 21938–21949 (2010). [CrossRef] [PubMed]

]. In any case, the AiB and its generalizations have physical meaning only within a paraxial framework since the PEq is a first-order approximation of the full Helmholtz equation (HEq) [11

11. P. Vaveliuk, G. F. Zebende, M. A. Moret, and B. Ruiz, “Propagating free-space nonparaxial beams,” J. Opt. Soc. Am. A 24, 3297–3302 (2007). [CrossRef]

]. Under nonparaxial conditions, the features of these paraxial beams should suffer strong disturbances. Nowadays, these disturbances are viewed as undesirable effects since they inhibit the self-bending and diffracting-free properties that are present in the paraxial regime. Reference [12

12. A. V. Novitsky and D. V. Novitsky, “Nonparaxial Airy beams: role of evanescent waves,” Opt. Lett. 34, 3430–3432 (2009). [CrossRef] [PubMed]

] confirmed the breaking of those properties. The evanescent waves seem to play a key role in this, to the extent that the so-called evanescent Airy beam, consisting exclusively of its own evanescent waves, was proposed as a fundamental result under strong nonparaxial conditions [12

12. A. V. Novitsky and D. V. Novitsky, “Nonparaxial Airy beams: role of evanescent waves,” Opt. Lett. 34, 3430–3432 (2009). [CrossRef] [PubMed]

]. These detrimental effects in nonparaxial AiBs encouraged the research on Airylike beams that maintain the self-bending and shape-preserving properties in that regime. Hence, the great importance of the recent derivation of nonparaxial self-bending Bessel-like beams [13

13. I. Kaminer, R. Bekenstein, J. Nemirovsky, and M. Segev, “Nondiffracting accelerating wave packets of Maxwell’s equations,” Phys. Rev. Lett. 108, 163901 (2012). [CrossRef] [PubMed]

] that not only conserves but also enhances those features [13

13. I. Kaminer, R. Bekenstein, J. Nemirovsky, and M. Segev, “Nondiffracting accelerating wave packets of Maxwell’s equations,” Phys. Rev. Lett. 108, 163901 (2012). [CrossRef] [PubMed]

, 14

14. Z. Chen, “Viewpoint: light bends itself into an arcs,” Phys. 5, 44 (2012). [CrossRef]

]. From this, one can see that the nonparaxial region is a rich source of emergence of novel unconventional features for this class of beams.

In this work, the negative propagation effect is reported under a strong nonparaxial regime for finite-energy AiBs. This is an unusual effect concerning the local sign change in the time-averaged Poynting vector components along the propagation direction of an optical beam in free space. The negative propagation was recently demonstrated for other kinds of nondiffracting structures such as vector Bessel beams [15

15. A. V. Novitsky and D. V. Novitsky, “Negative propagation of vector Bessel beams,” J. Opt. Soc. Am. A 24, 2844–2849 (2007). [CrossRef]

] and vector X-waves [16

16. M. A. Salem and H. Bağci, “Energy flow characteristics of vector X-waves,” Opt. Express 19, 8526–8532 (2011). [CrossRef] [PubMed]

]. The origin of this peculiar phenomenon for Bessel beams and X-waves is a collinear weighted superposition of TE and TM-polarization components in the real space. But the cause is quite different for AiBs. The negative propagation does occur in a single TE or TM mode and is due to a coupling process (interference-like process) between propagating and evanescent plane wave components in the spectral domain under strong nonparaxial conditions. Furthermore, this work shows that evanescent waves by themselves have null contribution to the energy flux density even under the most extreme nonparaxial conditions. Thereby, the evanescent Airy beam [12

12. A. V. Novitsky and D. V. Novitsky, “Nonparaxial Airy beams: role of evanescent waves,” Opt. Lett. 34, 3430–3432 (2009). [CrossRef] [PubMed]

] is not physically realizable. At this instance, a required question naturally arises: Is it feasible to generate highly nonparaxial beams? Recent significant advances have been made in one such direction. The dielectric-metal surfaces have been demonstrated to be promising systems since Ref. [17

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

] suggested a new class of diffraction-free surface plasmonic waves: the Airy plasmon. From that work, Airy plasmons possessing a subwavelength size of the central lobe have been generated by several groups [18

18. A. Minovich, A. E. Klein, N. Janunts, T. Pertsch, D. N. Neshev, and Yu. S. Kivshar, “Generation and near-field imaging of Airy surface plasmons,” Phys. Rev. Lett. 107, 116802 (2011). [CrossRef] [PubMed]

20

20. P. Zhang, S. Wang, Y. Liu, X. Yin, C. Lu, Z. Chen, and X. Zhang, “Plasmonic Airy beams with dynamically controlled trajectories,” Opt. Lett. 36, 3191–3193 (2011). [CrossRef] [PubMed]

]. The experiments performed in [18

18. A. Minovich, A. E. Klein, N. Janunts, T. Pertsch, D. N. Neshev, and Yu. S. Kivshar, “Generation and near-field imaging of Airy surface plasmons,” Phys. Rev. Lett. 107, 116802 (2011). [CrossRef] [PubMed]

] on Airy Plasmons were in agreement with the nonparaxial theory confirming that these beams lie out of the paraxial region. The quick development in subwavelength nonparaxial AiBs open real opportunities for detection of the negative propagation effect in these dielectric-metal surfaces [21

21. A. Salandrino and D. N. Christodoulides, “Viewpoint: Airy plasmons defeat diffraction on the surface,” Phys. 4, 69 (2011). [CrossRef]

] and other class of materials [22

22. T. Schneider, A. A. Serga, A. V. Chumak, C. W. Sandweg, S. Trudel, S. Wolff, M. P. Kostylev, V. S. Tiberkevich, A. N. Slavin, and B. Hillebrands, “Nondiffractive subwavelength wave beams in a medium with externally controlled anisotropy,” Phys. Rev. Lett. 104, 197203 (2010). [CrossRef] [PubMed]

], in particular, for the near- and mid-infrared range (1 – 100μm) [23

23. W. Liu, D. N. Neshev, I. V. Shadrivov, A. E. Miroshnichenko, and Yu. S. Kivshar, “Plasmonic Airy beam manipulation in linear optical potentials,” Opt. Lett. 36, 1164–1166 (2011). [CrossRef] [PubMed]

].

The paper is structured as follows: Section 2 develops the theoretical framework based on the angular spectrum theory and sets the paraxial-nonparaxial limits in terms of the main beam parameters. Section 3 tackles the propagation dynamics of Airy beam under a nonparaxial regime, highlighting the conditions under which the negative propagation effect takes place. Finally, Section 4 gives the final considerations.

2. Theoretical background and paraxial-nonparaxial limit for finite-energy Airy beams

We start from a monochromatic TE-polarized solution of Maxwell’s equations E⃗ = E0U (x, z)eiωtŷ where ω is the frequency, E0 is a constant with electric field dimensions and U is the dimensionless field obeying the HEq in a medium of permittivity ε:
[2/x2+2/z2+ε(2π/λ2)]U(x,z)=0,
(1)
with λ being the wavelength. We introduce the spacial dimensionless variables x̃ = ε1/2x/λ and z̃ = ε1/2x/λ. This normalization simplifies the mathematics and allows for easier interpretation of the results. The field U can be analyzed by employing the angular spectrum formalism [24

24. J. W. Goodman, Introduction to Fourier Optics, (McGraw-Hill, New York, 1968).

].

Fig. 1 Map of 𝒫 vs. (x̃0, a). The points set the different configurations of Airy beams in this paraxial-nonparaxial map with the following values of the paraxial estimator. For Ai1: 𝒫 = 0.999996; For Ai2: 𝒫 = 0.92; For Ai3: 𝒫 = 0.75; For Ai4: 𝒫 = −0.58. The dashing line corresponds to 𝒫 = 0.

3. Nonparaxial Airy beams and negative propagation effect

Fig. 2 Features of a nonparaxial Airy beam for (a)–(b) configuration Ai2 and (c)–(d) configuration Ai3. (a) and (c) Energy flow density as a function of normalized spatial coordinates. (b) and (d) Profiles of both the total energy flow density and the energy flow density due to the propagating plane waves. Clearly, the contribution of evanescent waves is null for both configurations in the overall spatial range.

The numerical results for the configuration Ai4 are illustrated in Fig. 3. Part (a) of this figure shows that the self-bending and shape-preserving properties are completely lost. The beam propagates approximately a wavelength before a strong spreading. The paraxial-nonparaxial breaking region is located at the beginning of the propagation and all the beam dynamics happens in a subwavelength region. In spite of the full break on accelerating and diffraction-free properties, a peculiar new nonparaxial effect appears: The secondary peaks of the z-component of the time-averaged Poynting vector Spresents negative values along the propagation coordinates. The spatial range of this phenomenon (up to z̃ ≈ 0.25) is approximately 25% of the total propagation range of the beam as indicates the green region in Fig. 3(a). The amplitude of the major of negative peaks is greater than 10% of the principal peak. These data suggest that this effect would be highly detectable if this configuration would be carried out experimentally. Of course, this effect will be even greater if the beam size decreases further. The coupling term is fully responsible by such a phenomenon as shown by the comparison between Sand Spr [Fig. 3(b)]. The magnitudes are very different in the near field region where the evanescent waves exert influence. As the beam propagates away from the near field region, the dynamics is exclusively governed by the propagating waves and both profiles begin to be practically equivalents after z̃ = 0.25. A great advantage is that this negative propagation takes place for a single TE or TM mode contrary to what happens for Bessel beams and X-waves [15

15. A. V. Novitsky and D. V. Novitsky, “Negative propagation of vector Bessel beams,” J. Opt. Soc. Am. A 24, 2844–2849 (2007). [CrossRef]

,16

16. M. A. Salem and H. Bağci, “Energy flow characteristics of vector X-waves,” Opt. Express 19, 8526–8532 (2011). [CrossRef] [PubMed]

], which require a complex superposition of TE and TM modes, hard to be performed in a experimental way. From this finding, the subwavelength nonparaxial Airy beam may not be viewed as an undesirable one since this novel effect opens promising opportunities in certain applications. For instance, in optical tweezers it can offers a novel way of optically manipulating micro and nanoparticles. In fact, the variation in sign of the Poynting vector could create multiple traps for confining particles in the vicinity of the positive-negative intensity lobes. This trapping pattern could be more stable than that created by focused Airy beams [31

31. Z. Zheng, B.-F. Zhang, H. Chen, J. Ding, and H.-T. Wang, “Optical trapping with focused Airy beams,” Appl. Opt. 50, 43–49 (2011). [CrossRef] [PubMed]

]. Furthermore, the transition from 3D to 2D tweezers is possible by exploiting evanescent fields bound at interfaces to achieve subwavelength trapping volumes. The nonparaxial AiBs opens a huge potential towards the elaboration of future lab-on-a-chip devices entirely operated with light [32

32. M. Righini, A. S. Zelenina, C. Girard, and R. Quidant, “Parallel and selective trapping in a patterned plasmonic landscape,” Nature Phys. 3, 477–480 (2007). [CrossRef]

]. On the other hand, the negative propagation effect could also be used as an effective tool in the detection of invisibility cloacks. This nonparaxial AiB could serve as an additional bridge from the flat mechanical space to the curved electromagnetic space by means of the interaction with a charged particle through a perfect invisibility cloak [33

33. B. Zhang and B.-I. Wu, “Electromagnetic detection of a perfect invisibility cloak,” Phys. Rev. Lett. 103, 243901 (2009). [CrossRef]

].

Fig. 3 z-component of the time-averaged Poynting vector for a Airy beam in the configuration Ai4. (a) Such a magnitude as a function of normalized spatial coordinates. Sis negative at the green regions. (b) Profiles of both the total energy flow density and the energy flow density due to the propagating plane waves at z̃ = 0 and z̃ = 0.25 (inset). Clearly, both magnitudes differs due to a non-null coupling term between propagating and evanescent waves. This last term is the responsible by the negative propagation. Out of the region of influence of evanescent waves, Sand Spr are equivalent.

Fig. 4 Squared-field amplitude modulus of an Airy beam for the configuration Ai4. (a) Such a magnitude as a function of normalized spatial coordinates. (b) Profiles of both the total squared-field amplitude modulus and that due to the propagating plane waves at z̃ = 0 and z̃ = 0.25 (inset). If 2π|U|2 is taken as the energy flux density, the negative propagation effect does not take place.

4. Concluding remarks

To summarize, we have reported the negative propagation effect in subwavelength nonparaxial Airy beams that is due to the coupling between propagating and evanescent plane waves. This effect can occur for a single TE(TM)-mode contrary to what happens for Bessel beams and X-waves. We also show that the evanescent waves cannot contribute to the energy flux density by themselves such that a pure evanescent Airy beam is physically forbidden. The breaking of the shape-preserving and diffraction-free properties of Airy beams in the nonparaxial regime are caused exclusively by the propagating waves. This work could have profound implications in several applications since it brings the usefulness of (before undesired) nonparaxial Airy beams to a subwavelength regime. This opens new opportunities in optical traps and tweezers for manipulating micro- and nanoparticles, optical detection of invisibility cloacks and selective on-chip manipulation of nanoparticles.

Acknowledgments

The Authors thank Valéria Loureiro da Silva for valuable advice. Financial support from the Conselho Nacional de Desenvolvimento Cientifico e Tecnologico (CNPq), Brazil, under project 477260/2010-1 and Spanish Ministry of Science and Innovation under projects TEC 2008-04105 and TEC 2011-23629 is acknowledged. P.V. acknowledges a PQ fellowship of CNPq. The publication of this work was supported by Servicio Nacional de Aprendizagem Industrial (SENAI)-DR/Bahia, Brazil.

References and links

1.

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

2.

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

3.

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

4.

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

5.

D. M. Cottrell, J. A. Davis, and T. M. Hazard, “Direct generation of accelerating Airy beams using a 3/2 phase-only pattern,” Opt. Lett. 34, 2634–2636 (2009). [CrossRef] [PubMed]

6.

J. A. Davis, M. J. Mitry, M. A. Bandres, I. Ruiz, K. P. McAuley, and D. M. Cottrell, “Generation of accelerating Airy and accelerating parabolic beams using phase-only patterns,” Appl. Opt. 48, 3170–3176 (2009). [CrossRef] [PubMed]

7.

M. A. Bandres and J. C. Gutierrez-Vega, “Airy-Gauss beams and their transformation by paraxial optical systems,” Opt. Express 15, 16719–16728 (2007). [CrossRef] [PubMed]

8.

J. A. Davis, M. J. Mintry, M. A. Bandres, and D. M. Cottrell, “Observation of accelerating parabolic beams,” Opt. Express 16, 12866–12871 (2008). [CrossRef] [PubMed]

9.

J. E. Morris, M. Mazilu, J. Baumgartl, T. Ciz̃már, and K. Dholakia, “Propagation characteristics of Airy beams: dependence upon spatial coherence and wavelength,” Opt. Express 17, 13236–13245 (2009). [CrossRef] [PubMed]

10.

M. I. Carvalho and M. Facão, “Propagation of Airy-related beams,” Opt. Express 18, 21938–21949 (2010). [CrossRef] [PubMed]

11.

P. Vaveliuk, G. F. Zebende, M. A. Moret, and B. Ruiz, “Propagating free-space nonparaxial beams,” J. Opt. Soc. Am. A 24, 3297–3302 (2007). [CrossRef]

12.

A. V. Novitsky and D. V. Novitsky, “Nonparaxial Airy beams: role of evanescent waves,” Opt. Lett. 34, 3430–3432 (2009). [CrossRef] [PubMed]

13.

I. Kaminer, R. Bekenstein, J. Nemirovsky, and M. Segev, “Nondiffracting accelerating wave packets of Maxwell’s equations,” Phys. Rev. Lett. 108, 163901 (2012). [CrossRef] [PubMed]

14.

Z. Chen, “Viewpoint: light bends itself into an arcs,” Phys. 5, 44 (2012). [CrossRef]

15.

A. V. Novitsky and D. V. Novitsky, “Negative propagation of vector Bessel beams,” J. Opt. Soc. Am. A 24, 2844–2849 (2007). [CrossRef]

16.

M. A. Salem and H. Bağci, “Energy flow characteristics of vector X-waves,” Opt. Express 19, 8526–8532 (2011). [CrossRef] [PubMed]

17.

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

18.

A. Minovich, A. E. Klein, N. Janunts, T. Pertsch, D. N. Neshev, and Yu. S. Kivshar, “Generation and near-field imaging of Airy surface plasmons,” Phys. Rev. Lett. 107, 116802 (2011). [CrossRef] [PubMed]

19.

L. Li, T. Li, S. M. Wang, C. Zhang, and S. N. Zhu, “Plasmonic Airy beam generated by in-plane diffraction,” Phys. Rev. Lett. 107, 126804 (2011). [CrossRef] [PubMed]

20.

P. Zhang, S. Wang, Y. Liu, X. Yin, C. Lu, Z. Chen, and X. Zhang, “Plasmonic Airy beams with dynamically controlled trajectories,” Opt. Lett. 36, 3191–3193 (2011). [CrossRef] [PubMed]

21.

A. Salandrino and D. N. Christodoulides, “Viewpoint: Airy plasmons defeat diffraction on the surface,” Phys. 4, 69 (2011). [CrossRef]

22.

T. Schneider, A. A. Serga, A. V. Chumak, C. W. Sandweg, S. Trudel, S. Wolff, M. P. Kostylev, V. S. Tiberkevich, A. N. Slavin, and B. Hillebrands, “Nondiffractive subwavelength wave beams in a medium with externally controlled anisotropy,” Phys. Rev. Lett. 104, 197203 (2010). [CrossRef] [PubMed]

23.

W. Liu, D. N. Neshev, I. V. Shadrivov, A. E. Miroshnichenko, and Yu. S. Kivshar, “Plasmonic Airy beam manipulation in linear optical potentials,” Opt. Lett. 36, 1164–1166 (2011). [CrossRef] [PubMed]

24.

J. W. Goodman, Introduction to Fourier Optics, (McGraw-Hill, New York, 1968).

25.

I. M. Besieris and A. M. Shaarawi, “A note on an accelerating finite energy Airy beam,” Opt. Lett. 32, 2447–2449 (2007). [CrossRef] [PubMed]

26.

A. Lencina and P. Vaveliuk, “Squared-field amplitude modulus and radiation intensity nonequivalence within nonlinear slabs,” Phys. Rev. E 71, 056614 (2005). [CrossRef]

27.

P. Vaveliuk, B. Ruiz, and A. Lencina, “Limits of the paraxial aproximation in laser beams,” Opt. Lett. 32, 927–929 (2007). [CrossRef] [PubMed]

28.

P. Vaveliuk and O. Martinez-Matos, “Physical interpretation of the paraxial estimator,” Opt. Commun. 285, 4816–4820 (2012). [CrossRef]

29.

P. Vaveliuk, “Quantifying the paraxiality for laser beams from the M2-factor,” Opt. Lett. 34, 340–342 (2009). [CrossRef] [PubMed]

30.

P. Vaveliuk and O. Martinez-Matos, “Effect of ABCD transformations on beam paraxiality,” Opt. Express 19, 25944–25953 (2011). [CrossRef]

31.

Z. Zheng, B.-F. Zhang, H. Chen, J. Ding, and H.-T. Wang, “Optical trapping with focused Airy beams,” Appl. Opt. 50, 43–49 (2011). [CrossRef] [PubMed]

32.

M. Righini, A. S. Zelenina, C. Girard, and R. Quidant, “Parallel and selective trapping in a patterned plasmonic landscape,” Nature Phys. 3, 477–480 (2007). [CrossRef]

33.

B. Zhang and B.-I. Wu, “Electromagnetic detection of a perfect invisibility cloak,” Phys. Rev. Lett. 103, 243901 (2009). [CrossRef]

OCIS Codes
(070.2580) Fourier optics and signal processing : Paraxial wave optics
(260.2110) Physical optics : Electromagnetic optics
(350.5500) Other areas of optics : Propagation

ToC Category:
Physical Optics

History
Original Manuscript: August 9, 2012
Revised Manuscript: October 17, 2012
Manuscript Accepted: October 19, 2012
Published: November 14, 2012

Citation
Pablo Vaveliuk and Oscar Martinez-Matos, "Negative propagation effect in nonparaxial Airy beams," Opt. Express 20, 26913-26921 (2012)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-24-26913


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References

  1. M. V. Berry and N. L. Balazs, “Nonspreading wave packets,” Am. J. Phys.47, 264 –267 (1979). [CrossRef]
  2. G. A. Siviloglou and D. N. Christodoulides, “Accelerating finite energy Airy beams,” Opt. Lett.32, 979–981 (2007). [CrossRef] [PubMed]
  3. G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett.99, 213901 (2007). [CrossRef]
  4. G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Ballistic dynamics of Airy beams,” Opt. Lett.33, 207–209 (2008). [CrossRef] [PubMed]
  5. D. M. Cottrell, J. A. Davis, and T. M. Hazard, “Direct generation of accelerating Airy beams using a 3/2 phase-only pattern,” Opt. Lett.34, 2634–2636 (2009). [CrossRef] [PubMed]
  6. J. A. Davis, M. J. Mitry, M. A. Bandres, I. Ruiz, K. P. McAuley, and D. M. Cottrell, “Generation of accelerating Airy and accelerating parabolic beams using phase-only patterns,” Appl. Opt.48, 3170–3176 (2009). [CrossRef] [PubMed]
  7. M. A. Bandres and J. C. Gutierrez-Vega, “Airy-Gauss beams and their transformation by paraxial optical systems,” Opt. Express15, 16719–16728 (2007). [CrossRef] [PubMed]
  8. J. A. Davis, M. J. Mintry, M. A. Bandres, and D. M. Cottrell, “Observation of accelerating parabolic beams,” Opt. Express16, 12866–12871 (2008). [CrossRef] [PubMed]
  9. J. E. Morris, M. Mazilu, J. Baumgartl, T. Ciz̃már, and K. Dholakia, “Propagation characteristics of Airy beams: dependence upon spatial coherence and wavelength,” Opt. Express17, 13236–13245 (2009). [CrossRef] [PubMed]
  10. M. I. Carvalho and M. Facão, “Propagation of Airy-related beams,” Opt. Express18, 21938–21949 (2010). [CrossRef] [PubMed]
  11. P. Vaveliuk, G. F. Zebende, M. A. Moret, and B. Ruiz, “Propagating free-space nonparaxial beams,” J. Opt. Soc. Am. A24, 3297–3302 (2007). [CrossRef]
  12. A. V. Novitsky and D. V. Novitsky, “Nonparaxial Airy beams: role of evanescent waves,” Opt. Lett.34, 3430–3432 (2009). [CrossRef] [PubMed]
  13. I. Kaminer, R. Bekenstein, J. Nemirovsky, and M. Segev, “Nondiffracting accelerating wave packets of Maxwell’s equations,” Phys. Rev. Lett.108, 163901 (2012). [CrossRef] [PubMed]
  14. Z. Chen, “Viewpoint: light bends itself into an arcs,” Phys.5, 44 (2012). [CrossRef]
  15. A. V. Novitsky and D. V. Novitsky, “Negative propagation of vector Bessel beams,” J. Opt. Soc. Am. A24, 2844–2849 (2007). [CrossRef]
  16. M. A. Salem and H. Bağci, “Energy flow characteristics of vector X-waves,” Opt. Express19, 8526–8532 (2011). [CrossRef] [PubMed]
  17. A. Salandrino and D. N. Christodoulides, “Airy plasmon: a nondiffracting surface wave,” Opt. Lett.35, 2082–2084 (2010). [CrossRef] [PubMed]
  18. A. Minovich, A. E. Klein, N. Janunts, T. Pertsch, D. N. Neshev, and Yu. S. Kivshar, “Generation and near-field imaging of Airy surface plasmons,” Phys. Rev. Lett.107, 116802 (2011). [CrossRef] [PubMed]
  19. L. Li, T. Li, S. M. Wang, C. Zhang, and S. N. Zhu, “Plasmonic Airy beam generated by in-plane diffraction,” Phys. Rev. Lett.107, 126804 (2011). [CrossRef] [PubMed]
  20. P. Zhang, S. Wang, Y. Liu, X. Yin, C. Lu, Z. Chen, and X. Zhang, “Plasmonic Airy beams with dynamically controlled trajectories,” Opt. Lett.36, 3191–3193 (2011). [CrossRef] [PubMed]
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