## Accelerating light beams with arbitrarily transverse shapes |

Optics Express, Vol. 22, Issue 3, pp. 3490-3500 (2014)

http://dx.doi.org/10.1364/OE.22.003490

Acrobat PDF (3875 KB)

### Abstract

Accelerating beams are wave packets that preserve their shape while propagating along curved trajectories. Their unique characteristics have opened the door to applications that range from optical micromanipulation and plasma-channel generation to laser micromachining. Here, we demonstrate, theoretically and experimentally, that accelerating beams can be generated with a variety of arbitrarily chosen transverse shapes. We present a general method to construct such beams in the paraxial and nonparaxial regime and demonstrate experimentally their propagation in the paraxial case. The key ingredient of our method is the use of the spectral representation of the accelerating beams, which offers a unique and compact description of these beams. The on-demand accelerating light patterns described here are likely to give rise to new applications and add versatility to the current ones.

© 2014 Optical Society of America

## 1. Introduction

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]

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. M. A. Bandres, “Accelerating parabolic beams,” Opt. Lett. **33**, 1678–1680 (2008). [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, “Accelerating beams,” Opt. Lett. **34**, 3791–3793 (2009). [CrossRef] [PubMed]

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

10. P. Zhang, Y. Hu, D. Cannan, A. Salandrino, T. Li, R. Morandotti, X. Zhang, and Z. Chen, “Generation of linear and nonlinear nonparaxial accelerating beams,” Opt. Lett. **37**, 2820–2822 (2012). [CrossRef] [PubMed]

11. M. A. Bandres and B. M. Rodríguez-Lara, “Nondiffracting accelerating waves: Weber waves and parabolic momentum,” New Journal of Physics **15**, 013054 (2013). [CrossRef]

12. P. Zhang, Y. Hu, T. Li, D. Cannan, X. Yin, R. Morandotti, Z. Chen, and X. Zhang, “Nonparaxial Mathieu and Weber accelerating beams,” Phys. Rev. Lett. **109**, 193901 (2012). [CrossRef] [PubMed]

12. P. Zhang, Y. Hu, T. Li, D. Cannan, X. Yin, R. Morandotti, Z. Chen, and X. Zhang, “Nonparaxial Mathieu and Weber accelerating beams,” Phys. Rev. Lett. **109**, 193901 (2012). [CrossRef] [PubMed]

13. P. Aleahmad, M.-A. Miri, M. S. Mills, I. Kaminer, M. Segev, and D. N. Christodoulides, “Fully vectorial accelerating diffraction-free Helmholtz beams,” Phys. Rev. Lett. **109**, 203902 (2012). [CrossRef] [PubMed]

13. P. Aleahmad, M.-A. Miri, M. S. Mills, I. Kaminer, M. Segev, and D. N. Christodoulides, “Fully vectorial accelerating diffraction-free Helmholtz beams,” Phys. Rev. Lett. **109**, 203902 (2012). [CrossRef] [PubMed]

15. M. A. Bandres, M. A. Alonso, I. Kaminer, and M. Segev, “Three-dimensional accelerating electromagnetic waves,” Opt. Express **21**, 13917–13929 (2013). [CrossRef] [PubMed]

15. M. A. Bandres, M. A. Alonso, I. Kaminer, and M. Segev, “Three-dimensional accelerating electromagnetic waves,” Opt. Express **21**, 13917–13929 (2013). [CrossRef] [PubMed]

16. P. Polynkin, M. Kolesik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, “Curved plasma channel generation using ultraintense Airy beams,” Science **324**, 229–232 (2009). [CrossRef] [PubMed]

17. I. Dolev, I. Kaminer, A. Shapira, M. Segev, and A. Arie, “Experimental observation of self-accelerating beams in quadratic nonlinear media,” Phys. Rev. Lett. **108**, 113903 (2012). [CrossRef] [PubMed]

18. N. Voloch-Bloch, Y. Lereah, Y. Lilach, A. Gover, and A. Arie, “Generation of electron Airy beams,” Nature **494**, 331–335 (2013). [CrossRef] [PubMed]

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

20. A. Mathis, F. Courvoisier, L. Froehly, L. Furfaro, M. Jacquot, P. A. Lacourt, and J. M. Dudley, “Micromachining along a curve: Femtosecond laser micromachining of curved profiles in diamond and silicon using accelerating beams,” Appl. Phys. Lett. **101**, 071110 (2012). [CrossRef]

21. A. Mathis, L. Froehly, L. Furfaro, M. Jacquot, J. Dudley, and F. Courvoisier, “Direct machining of curved trenches in silicon with femtosecond accelerating beams,” J. Euro. Opt. Soc. Rapid publications8(2013). [CrossRef]

7. M. A. Bandres, “Accelerating beams,” Opt. Lett. **34**, 3791–3793 (2009). [CrossRef] [PubMed]

15. M. A. Bandres, M. A. Alonso, I. Kaminer, and M. Segev, “Three-dimensional accelerating electromagnetic waves,” Opt. Express **21**, 13917–13929 (2013). [CrossRef] [PubMed]

## 2. Shaped accelerating beams

22. M. A. Bandres, J. C. Gutiérrez-Vega, and S. Chávez-Cerda, “Parabolic nondiffracting optical wave fields,” Opt. Lett. **29**, 44–46 (2004). [CrossRef] [PubMed]

23. C. López-Mariscal and K. Helmerson, “Shaped nondiffracting beams,” Opt. Lett. **35**, 1215–1217 (2010). [CrossRef] [PubMed]

*ψ*(

*x*,

*y*, 0)| = |

*ψ*(

*x*,

*y*,

*z*)|. For example, Bessel, Mathieu and parabolic (Weber) beams, represent three families of separable solutions [22

22. M. A. Bandres, J. C. Gutiérrez-Vega, and S. Chávez-Cerda, “Parabolic nondiffracting optical wave fields,” Opt. Lett. **29**, 44–46 (2004). [CrossRef] [PubMed]

24. J. C. Gutiérrez-Vega and M. A. Bandres, “Helmholtz-Gauss waves,” J. Opt. Soc. Am. A **22**, 289–298 (2005). [CrossRef]

23. C. López-Mariscal and K. Helmerson, “Shaped nondiffracting beams,” Opt. Lett. **35**, 1215–1217 (2010). [CrossRef] [PubMed]

25. T. Čižmár and K. Dholakia, “Tunable Bessel light modes: engineering the axial propagation,” Opt. Express **17**, 15558–15570 (2009). [CrossRef] [PubMed]

*ψ*(

*x*,

*y*,

*z*)| = |

*ψ*(

*x*+

*Az*

^{2},

*y*, 0)|. Researchers in [7

7. M. A. Bandres, “Accelerating beams,” Opt. Lett. **34**, 3791–3793 (2009). [CrossRef] [PubMed]

*L*(

*k*), and vice versa. For example, the 2D Airy beams have harmonic functions as line spectra, while the accelerating parabolic beams are characterized by the quartic oscillator functions and the Airy-plane-wave beams have a Dirac-delta function as line spectra [7

_{v}7. M. A. Bandres, “Accelerating beams,” Opt. Lett. **34**, 3791–3793 (2009). [CrossRef] [PubMed]

^{2}

*ψ*+

*i∂*= 0, where ∇

_{s}ψ^{2}=

*∂*+

_{uu}*∂*, (

_{vv}*u*,

*v*) = (

*x*,

*y*) /

*κ*represents the dimensionless transverse coordinates,

*κ*is an arbitrary transverse scale factor,

*s*=

*z*/2

*kκ*

^{2}is the normalized propagation distance, and

*k*is the wave number. Here it is important to remember, that although a beam satisfies the paraxial equation it does not necessarily satisfy the paraxial condition that assumes that the angle between the optical axis and the wave vectors of the plane waves that constitute a beam is small enough to ensure that the wave does not deviate too much from its propagation direction. For example, the exact infinite energy Airy beams satisfy the paraxial wave equation. However, despite the fact that they carry infinite power they are not paraxial beams because their Fourier spectrum is not confined around the optical axis. In this case, the arbitrary transverse scale factor

*κ*plays a crucial role. For example by increasing

*κ*we can get a beam into the paraxial regime, since this will compress its Fourier spectrum by the scaling properties of the Fourier transform. Note also that

*s*=

*z*/2

*kκ*

^{2}depends on

*κ*and therefore changing the transverse scale of the beam also affects its propagation behavior and in the case of accelerating beams their degree of bending. Finally, we can also make the accelerating beams finite and paraxial by introducing an exponential or a Gaussian apodization function. This can be readily done by using the results of [26

26. M. A. Bandres and M. Guizar-Sicairos, “Paraxial group,” Opt. Lett. **34**, 13–15 (2009). [CrossRef]

*ABCD*optical system (including misalignments) if one knows the closed form solution in free space.

7. M. A. Bandres, “Accelerating beams,” Opt. Lett. **34**, 3791–3793 (2009). [CrossRef] [PubMed]

*γ*,

*k*∈ ℜ are the family parameters. The orthogonally relation of Airy-plane-wave beams at the transverse plane

_{v}*s*= 0 is given by All beams with the same

*γ*belong to a subfamily of accelerating beams that maintain their relative phase upon propagation, i.e., the terms exp(

*is*

*γ*) is the same for the whole subfamily. Since different subfamilies at

*s*= 0 just differ in a transverse translation in the

*x*-axis we can set

*γ*= 0 for the rest of this work without loss of generality.

*ℱ*

^{−1}is the inverse Fourier transform operator. The last part of this equation shows that the

*k*-spectrum of any accelerating beam is a one dimensional modulation of a cubic phase. The middle part of Eq. 3 shows that any accelerating beam is composed by a superposition of Airy-plane-waves with different frequency

*k*and modulation

_{v}*L*(

*k*). It is important to note the behavior of the Airy-plane-wave basis. For example, Airy-plane-waves with different

_{v}*k*have different frequencies in the

_{v}*v*-axis but also their Airy functions in the

*u*-axis

*ψ*(

*u*,

*v*, 0) ≈

*X*(

*u*)

*Y*(

*v*) perfectly by just setting

*L*(

*k*) =

_{v}*ℱ*[

*Y*(

*v*)] where

*ℱ*is the Fourier transform operator. However this is not possible because removing the translation implies changing the subfamily of the basis and breaking the accelerating properties of the beam.

*Y*(

*v*). Therefore, by using the scale parameter

*α*, we can write

*Y*(

*v*) →

*Y*(

*v/α*) and by the Fourier scaling theorem we have

*L*(

*k*) →

_{v}*L*(

*αk*)/|

_{v}*α*|. Consequently by increasing

*α*we can compress

*L*(

*αk*) and confine it to small values of

_{v}*k*. This implies a small translation of the Airy component of the Airy-plane-wave basis and therefore the accelerating beam will have the desired shape

_{v}*ψ*(

*u*,

*v*, 0) ≈

*X*(

*u*)

*Y*(

*v*).

*k*that contains 95% of

_{v}*L*(

*k*). We will call this parameter

_{v}*k*

_{95%}and it is defined as We found that if

*k*

_{95%}≲ 0.25, the engineered accelerating beam satisfies

*ψ*≈

*X*(

*u*)

*Y*(

*v/α*) almost perfectly. As

*k*

_{95%}increases to reach the range 0.25 ≲

*k*

_{95%}≲ 0.6, the accelerating beam will start losing the desired modulation but at the

*x*position of the first local maximum the modulation

*Y*(

*v/α*) will prevail qualitative to a good approximation at least until 0.6 ≲

*k*

_{95%}.

*Y*(

*v*): (a) triangle function, (b) a cosine-Gauss function, and (c) a complex function with the triangle function in the real part (red line) and the cosine-Gauss in the imaginary part (blue line). The black lines depict the

*y*-line profile of the engineered beam at the first 20 local

*x*-maxima; these lines closely overlap with the desired modulation showing that our method truly encodes the desired function in the engineered accelerating beam. The spectrum of the modulation function is depicted in the second row of Fig. 1 and their extent is

*k*

_{95%}≈ 0.26. As mentioned above we need to have functions with a small spectral extent in order for our method to work. The third row shows the invariant intensity structure of the constructed accelerating beams; one can clearly appreciate that the beams have the desired form

*ψ*≈

*X*(

*x*)

*Y*(

*y*) where

*Y*(

*y*) is the arbitrary on-demand modulation depicted in the first row. The last row of Fig. 1 depicts the propagation of our shaped accelerating beams. For the triangular profile in (a), we did not use any apodization function. For the cosine-Gauss profile in (b), we used an exponential apodization function and we used a Gaussian apodization function for the complex profile in (c). As one can appreciate from the last row of Fig. 1, the accelerating characteristics of our on-demand accelerating beams are exactly the same as the Airy beam or any other two-dimensional paraxial accelerating beam.

*Y*(

*y/α*). The red lines show the desired target modulation while the blue lines are the engineered beam

*y*-line profiles at different local

*x*-maxima of the beam. The darkest tone of blue represents the first

*x*-maximum cut while the lighter blue tones represent subsequent

*x*-maxima cuts. From the first to the last row we change the extent of the desired modulation from

*k*

_{95%}= 0.2 to

*k*

_{95%}= 0.8. The first row with

*k*

_{95%}= 0.2 has a spectrum confined between −0.2 <

*k*< 0.2 and, in each subsequent row, we increased the extent of the spectral function by stretching it and consequently compressing the desired modulation

*Y*(

*y/α*), i.e.,

*α*decreases. Notice that for the first and second rows of Fig. 2, i.e., modulations with small spectral extent

*k*

_{95%}≲ 0.4, the

*y*-line profiles of the beam (blue lines) perfectly overlap with the desired modulation (red lines) showing that our engineered accelerating beam accurately encodes the desired modulation. As we increase the extent of the spectrum to 0.4 ≲

*k*

_{95%}, as in the last two rows of Fig. 2, the beam

*y*-line profile (blue lines) starts to deviate from the desired modulation (red line); the main

*x*-maximum cut of the beam retains its

*y*-modulation for the longest extent. For this reason the light blue lines that correspond to the

*y*-line profile at subsequent

*x*-maxima deviate more from the target modulation. It is important to have in mind that for most practical applications what is important is a qualitative agreement with a desired shape function. Hence, although for modulations with bigger spectral extent the quantitative agreement with the target modulation is not perfect, like the one in the last row of Fig. 2, the desired shape is qualitatively the same.

*y*-coordinate in amplitude and phase as shown in Fig. 1(c). If one wants to impose only an amplitude

*y*-modulation, then it is possible to use the phase of the modulation as another degree of freedom to try to improve the beam shaping, for example using analogous iterative methods similar to the ones used in [23

23. C. López-Mariscal and K. Helmerson, “Shaped nondiffracting beams,” Opt. Lett. **35**, 1215–1217 (2010). [CrossRef] [PubMed]

25. T. Čižmár and K. Dholakia, “Tunable Bessel light modes: engineering the axial propagation,” Opt. Express **17**, 15558–15570 (2009). [CrossRef] [PubMed]

**35**, 1215–1217 (2010). [CrossRef] [PubMed]

25. T. Čižmár and K. Dholakia, “Tunable Bessel light modes: engineering the axial propagation,” Opt. Express **17**, 15558–15570 (2009). [CrossRef] [PubMed]

## 3. Experimental realization

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]

27. J. A. Davis, D. M. Cottrell, J. Campos, M. J. Yzuel, and I. Moreno, “Encoding amplitude information onto phase-only filters,” Appl. Opt. **38**, 5004–5013 (1999). [CrossRef]

*π*radians as a function of gray level at the Argon laser wavelength of 514.5 nm. Amplitude information is encoded by spatially varying the phase depth of a grating that multiplies the desired phase pattern [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]

27. J. A. Davis, D. M. Cottrell, J. Campos, M. J. Yzuel, and I. Moreno, “Encoding amplitude information onto phase-only filters,” Appl. Opt. **38**, 5004–5013 (1999). [CrossRef]

**48**, 3170–3176 (2009). [CrossRef] [PubMed]

^{2}. The direction of the transverse shift can be reversed using the complex conjugate of the phase mask. The deflection of the accelerating beam can be measured for instance by comparing the results obtained with a given pattern and with the complex conjugated pattern. The origin could then be found as the common center point [5

5. 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]

**48**, 3170–3176 (2009). [CrossRef] [PubMed]

*Y*(

*y*) for each case; notice the good agreement between the desired modulation and the beam structure. Also, the experimental (second row) and the simulated (first row) optical intensities at the initial plane are in good agreement with each other. The second to last row shows the experimentally measured optical beam at

*z*= 0, 15, 30 and 45 cm measured from the back focal plane. Notice that the beams do not diffract during this 45 cm range of propagation and that they transversely shift (“accelerate”) under propagation. The last column of Fig. 3 shows the propagation of an 2D Airy beam for comparison; as one can see, our engineered accelerating beams have the same accelerating properties as an 2D Airy beam.

*ax*

^{2}+

*b*, to calculate the transverse constant shift, “

*b*” for each beam. The experimental transverse translation data (relative to the constant shift) are represented in Fig. 4. The results there show that all beams follow the same parabolic trajectory.

## 4. Shaped nonparaxial accelerating beams

*Y*(

*y*) in the angular spectral

*g*(

*θ*) of the nonparaxial accelerating beam by using the following relation, where

*L*(

*k*) is the spectrum of

_{y}*Y*(

*y*) as before. In this nonparaxial case we found that we can achieve a good shaping of the beam if

*g*(

*θ*) is confined to an angle of less than 10° around

*θ*=

*π*/2, for

*m*≲ 200. The encoding is better if the angular confinement is small and beams with larger

*m*requires more confinement. In this way, the

*k*components of these beams are paraxial and control the vertical modulation of the beam; but the

_{y}*k*components are nonparaxial and control the steep bending of the beam. Several works [28

_{x}28. E. Greenfield, M. Segev, W. Walasik, and O. Raz, “Accelerating light beams along arbitrary convex trajectories,” Phys. Rev. Lett. **106**, 213902 (2011). [CrossRef] [PubMed]

30. A. Mathis, F. Courvoisier, R. Giust, L. Furfaro, M. Jacquot, L. Froehly, and J. M. Dudley, “Arbitrary nonparaxial accelerating periodic beams and spherical shaping of light,” Opt. Lett. **38**, 2218–2220 (2013). [CrossRef] [PubMed]

*imϕ*) in Eq. 5 by an engineered phase base on optical caustics, however, this is at the expense of losing the shape-preserving property of the beams.

*Y*(

*v*): (a) Mexican hat shape, (b) transverse apodized Bessel shape, and (c) saw shape, for different values of

*m*. The black lines depict the vertical profile of the engineered beam at the first 5 local horizontal maxima; these lines closely overlap with the desired modulation showing that our method truly encodes the desired function in the engineered nonparaxial accelerating beam. The angular spectrum

*g*(

*θ*) is depicted in the second row of Fig. 5 and their extent around

*θ*=

*π*/2 is

*θ*

_{95%}≲ 10°. The third row shows the invariant intensity structure of the constructed nonparaxial accelerating beams; one can clearly appreciate that the beams have the desired form

*ψ*≈

*X*(

*x*)

*Y*(

*y*) where

*Y*(

*y*) is the arbitrary on-demand modulation depicted in the first row. The last row of Fig. 5 depicts the quasi-invariant propagation of our shaped accelerating beams along a semicircular path or radius

*m/k*.

## Acknowledgments

## References and links

1. | M. V. Berry and N.L. Balazs, “Nonspreading wave packets,” Am. J. Phys. |

2. | G. A. Siviloglou and D. N. Christodoulides, “Accelerating finite energy Airy beams,” Opt. Lett. |

3. | G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett. |

4. | M. A. Bandres, “Accelerating parabolic beams,” Opt. Lett. |

5. | J. A. Davis, M. J. Mintry, M. A. Bandres, and D. M. Cottrell, “Observation of accelerating parabolic beams,” Opt. Express |

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. |

7. | M. A. Bandres, “Accelerating beams,” Opt. Lett. |

8. | I. Kaminer, R. Bekenstein, J. Nemirovsky, and M. Segev, “Nondiffracting accelerating wave packets of Maxwell’s equations,” Phys. Rev. Lett. |

9. | F. Courvoisier, A. Mathis, L. Froehly, R. Giust, L. Furfaro, P. A. Lacourt, M. Jacquot, and J. M. Dudley, “Sending femtosecond pulses in circles: highly nonparaxial accelerating beams,” Opt. Lett. |

10. | P. Zhang, Y. Hu, D. Cannan, A. Salandrino, T. Li, R. Morandotti, X. Zhang, and Z. Chen, “Generation of linear and nonlinear nonparaxial accelerating beams,” Opt. Lett. |

11. | M. A. Bandres and B. M. Rodríguez-Lara, “Nondiffracting accelerating waves: Weber waves and parabolic momentum,” New Journal of Physics |

12. | P. Zhang, Y. Hu, T. Li, D. Cannan, X. Yin, R. Morandotti, Z. Chen, and X. Zhang, “Nonparaxial Mathieu and Weber accelerating beams,” Phys. Rev. Lett. |

13. | P. Aleahmad, M.-A. Miri, M. S. Mills, I. Kaminer, M. Segev, and D. N. Christodoulides, “Fully vectorial accelerating diffraction-free Helmholtz beams,” Phys. Rev. Lett. |

14. | M. A. Alonso and M. A. Bandres, “Spherical fields as nonparaxial accelerating waves,” Opt. Lett. |

15. | M. A. Bandres, M. A. Alonso, I. Kaminer, and M. Segev, “Three-dimensional accelerating electromagnetic waves,” Opt. Express |

16. | P. Polynkin, M. Kolesik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, “Curved plasma channel generation using ultraintense Airy beams,” Science |

17. | I. Dolev, I. Kaminer, A. Shapira, M. Segev, and A. Arie, “Experimental observation of self-accelerating beams in quadratic nonlinear media,” Phys. Rev. Lett. |

18. | N. Voloch-Bloch, Y. Lereah, Y. Lilach, A. Gover, and A. Arie, “Generation of electron Airy beams,” Nature |

19. | J. Baumgartl, M. Mazilu, and K. Dholakia, “Optically mediated particle clearing using Airy wavepackets,” Nat. Photonics |

20. | A. Mathis, F. Courvoisier, L. Froehly, L. Furfaro, M. Jacquot, P. A. Lacourt, and J. M. Dudley, “Micromachining along a curve: Femtosecond laser micromachining of curved profiles in diamond and silicon using accelerating beams,” Appl. Phys. Lett. |

21. | A. Mathis, L. Froehly, L. Furfaro, M. Jacquot, J. Dudley, and F. Courvoisier, “Direct machining of curved trenches in silicon with femtosecond accelerating beams,” J. Euro. Opt. Soc. Rapid publications8(2013). [CrossRef] |

22. | M. A. Bandres, J. C. Gutiérrez-Vega, and S. Chávez-Cerda, “Parabolic nondiffracting optical wave fields,” Opt. Lett. |

23. | C. López-Mariscal and K. Helmerson, “Shaped nondiffracting beams,” Opt. Lett. |

24. | J. C. Gutiérrez-Vega and M. A. Bandres, “Helmholtz-Gauss waves,” J. Opt. Soc. Am. A |

25. | T. Čižmár and K. Dholakia, “Tunable Bessel light modes: engineering the axial propagation,” Opt. Express |

26. | M. A. Bandres and M. Guizar-Sicairos, “Paraxial group,” Opt. Lett. |

27. | J. A. Davis, D. M. Cottrell, J. Campos, M. J. Yzuel, and I. Moreno, “Encoding amplitude information onto phase-only filters,” Appl. Opt. |

28. | E. Greenfield, M. Segev, W. Walasik, and O. Raz, “Accelerating light beams along arbitrary convex trajectories,” Phys. Rev. Lett. |

29. | L. Froehly, F. Courvoisier, A. Mathis, M. Jacquot, L. Furfaro, R. Giust, P. A. Lacourt, and J. M. Dudley, “Arbitrary accelerating micron-scale caustic beams in two and three dimensions,” Opt. Express |

30. | A. Mathis, F. Courvoisier, R. Giust, L. Furfaro, M. Jacquot, L. Froehly, and J. M. Dudley, “Arbitrary nonparaxial accelerating periodic beams and spherical shaping of light,” Opt. Lett. |

**OCIS Codes**

(050.1970) Diffraction and gratings : Diffractive optics

(140.3300) Lasers and laser optics : Laser beam shaping

(230.6120) Optical devices : Spatial light modulators

(350.5500) Other areas of optics : Propagation

(070.3185) Fourier optics and signal processing : Invariant optical fields

(070.7345) Fourier optics and signal processing : Wave propagation

**ToC Category:**

Physical Optics

**History**

Original Manuscript: October 22, 2013

Revised Manuscript: December 20, 2013

Manuscript Accepted: December 27, 2013

Published: February 6, 2014

**Citation**

Adrian Ruelas, Jeffrey A. Davis, Ignacio Moreno, Don M. Cottrell, and Miguel A. Bandres, "Accelerating light beams with arbitrarily transverse shapes," Opt. Express **22**, 3490-3500 (2014)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-22-3-3490

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### References

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