## Arbitrary accelerating micron-scale caustic beams in two and three dimensions |

Optics Express, Vol. 19, Issue 17, pp. 16455-16465 (2011)

http://dx.doi.org/10.1364/OE.19.016455

Acrobat PDF (2001 KB)

### Abstract

We generate arbitrary convex accelerating beams by direct application of an appropriate spatial phase profile on an incident Gaussian beam. The spatial phase calculation exploits the geometrical properties of optical caustics and the Legendre transform. Using this technique, accelerating sheet caustic beams with parabolic profiles (i.e. Airy beams), as well as quartic and logarithmic profiles are experimentally synthesized from an incident Gaussian beam, and we show compatibility with material processing applications using an imaging system to reduce the main intensity lobe at the caustic to sub-10 micron transverse dimension. By applying additional and rotational spatial phase, we generate caustic-bounded sheet and volume beams, which both show evidence of the recently predicted effect of abrupt autofocussing. In addition, an engineered accelerating profile with femtosecond pulses is applied to generate a curved zone of refractive index modification in glass. These latter results provide proof of principle demonstration of how this technique may yield new degrees of freedom in both nonlinear optics and femtosecond micromachining.

© 2011 OSA

## 1. Introduction

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

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

4. P. Polynkin, M. Kolesik, and J. Moloney, “Filamentation of femtosecond laser Airy beams in water,” Phys. Rev. Lett. **103**(12), 123902 (2009). [CrossRef] [PubMed]

9. 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]

10. I. Kaminer, M. Segev, and D. N. Christodoulides, “Self-accelerating self-trapped optical beams,” Phys. Rev. Lett. **106**(21), 213903 (2011). [CrossRef] [PubMed]

18. Y. Kaganovsky and E. Heyman, “Wave analysis of Airy beams,” Opt. Express **18**(8), 8440–8452 (2010), http://www.opticsinfobase.org/abstract.cfm?URI=oe-18-8-8440. [CrossRef] [PubMed]

19. W. Liu, D. N. Neshev, I. V. Shadrivov, A. E. Miroshnichenko, and Y. S. Kivshar, “Plasmonic Airy beam manipulation in linear optical potentials,” Opt. Lett. **36**(7), 1164–1166 (2011). [CrossRef] [PubMed]

20. J.-X. Li, X.-L. Fan, W. P. Zang, and J.-G. Tian, “Vacuum electron acceleration driven by two crossed Airy beams,” Opt. Lett. **36**(5), 648–650 (2011). [CrossRef] [PubMed]

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

*engineering*of a much wider class of paraxial acceleration profiles through the design of an appropriate generating spatial phase mask [21

21. E. Greenfield, M. Segev, W. Walasik, and O. Raz, “Accelerating Light Beams along Arbitrary Convex Trajectories,” Phys. Rev. Lett. **106**(21), 213902 (2011). [CrossRef] [PubMed]

22. J. Rosen and A. Yariv, “Snake beam: a paraxial arbitrary focal line,” Opt. Lett. **20**(20), 2042–2044 (1995). [CrossRef] [PubMed]

## 2. Acceleration profile engineering via caustics

26. 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**(17), 2634–2636 (2009). [CrossRef] [PubMed]

21. E. Greenfield, M. Segev, W. Walasik, and O. Raz, “Accelerating Light Beams along Arbitrary Convex Trajectories,” Phys. Rev. Lett. **106**(21), 213902 (2011). [CrossRef] [PubMed]

## 3. Experimental setup and sheet caustics

27. M. K. Bhuyan, F. Courvoisier, P. A. Lacourt, M. Jacquot, R. Salut, L. Furfaro, and J. M. Dudley, “High aspect ratio nanochannel machining using single shot femtosecond Bessel beams,” Appl. Phys. Lett. **97**(8), 081102 (2010). [CrossRef]

_{1}=1000 mm and f

_{2}=100 mm. Note that imaging in a 4f configuration (see Fig. 2) preserves the acceleration profile imposed by the phase mask at the SLM and allows a wide range of spatial frequencies to be used in constructing the desired acceleration profile. Also note that with the 100 fs pulses in our experiments (10 nm spectral FWHM), the effect of material dispersion in our system has negligible effect on the characteristics of the generated caustic profile. Our experiments therefore represent an important confirmation that arbitrary acceleration profile design technique [21

21. E. Greenfield, M. Segev, W. Walasik, and O. Raz, “Accelerating Light Beams along Arbitrary Convex Trajectories,” Phys. Rev. Lett. **106**(21), 213902 (2011). [CrossRef] [PubMed]

**106**(21), 213902 (2011). [CrossRef] [PubMed]

## 4. Volume caustics and extension to sub-10 μm dimensions

_{c}(y)∙H(y)+φ

_{c}(-y)∙H(-y) where φ

_{c}(y) is the calculated single caustic phase and H(y) is the Heaviside step function. The applied phase functions are shown in the left panels of the figure. The caustic-bounded beam profiles obtained experimentally are shown in the central panels, and were measured after ×120 demagnification. We also show here as the white dashed line the target acceleration profile from which the phase function was calculated. As with the results in Fig. 4, the experimental measurements confirm the ability of this technique to generate the target acceleration trajectories. The figure also shows line profiles at different points along the direction of propagation as indicated. The profiles at the points A illustrate the near zero intensity in the central region and the Airy fringes at the exterior of the beam. At the points of convergence labeled B in the figure, our results show signatures of the abrupt autofocussing phenomena as reported in Refs. 6

6. N. K. Efremidis and D. N. Christodoulides, “Abruptly autofocusing waves,” Opt. Lett. **35**(23), 4045–4047 (2010). [CrossRef] [PubMed]

8. I. Chremmos, N. K. Efremidis, and D. N. Christodoulides, “Pre-engineered abruptly autofocusing beams,” Opt. Lett. **36**(10), 1890–1892 (2011). [CrossRef] [PubMed]

*ϕ*as a function of radial distance

_{c}(r)*r*from the centre of the SLM as shown. As with the two dimensional results shown above, we measure the intensity distribution beyond the image plane of the SLM but in order to highlight the three-dimensional nature of the imaged field and the clear volume boundary associated with the caustic, we use a tomographic representation in the figure with partial transparency. The left subfigure is a transparent slice through the beam centre whilst the perspective selected in the right subfigure illustrates clearly the three dimensional nature of the caustic volume. The particular case chosen for illustration here is that of a rotationally-symmetric quartic caustic but similar results are obtained for other caustic forms and we present additional results for both quartic and parabolic caustics in Fig. 7 below. The demagnification factor here is again ×120 so that we obtain intensity lobes of sub-10 μm dimension as we shall see.

28. K. Dholakia and T. Čižmár, “Shaping the future of manipulation,” Nat. Photonics **5**(6), 335–342 (2011). [CrossRef]

*ϕ(r,θ) = φ*

_{c}

*(r) B(r,θ)*where

*φ*

_{c}

*(r)*is a rotationally symmetric quartic phase as in Fig. 6(a), and

*Β*(

*r,θ*) = rect[(

*r*-

*aθ*)/

*Δr*] describes the binary spiral function. Here,

*θ*varies between 0 and

*n*2π and the parameters are

*a*=

*r*

_{max}/2π

*n*where

*n*is the number of spiral turns of zeroing width

*Δr*, and

*r*

_{max}is the maximal exterior radius of the spiral. In this case, the spiralling nature of the caustic surface means that tomographic visualisation in a plane as in the centre panel is not effective in illustrating the nature of the accelerating field, but this is shown very clearly with the perspective chosen in the right panel of Fig. 6(b).

6. N. K. Efremidis and D. N. Christodoulides, “Abruptly autofocusing waves,” Opt. Lett. **35**(23), 4045–4047 (2010). [CrossRef] [PubMed]

7. D. G. Papazoglou, N. K. Efremidis, D. N. Christodoulides, and S. Tzortzakis, “Observation of abruptly autofocusing waves,” Opt. Lett. **36**(10), 1842–1844 (2011). [CrossRef] [PubMed]

8. I. Chremmos, N. K. Efremidis, and D. N. Christodoulides, “Pre-engineered abruptly autofocusing beams,” Opt. Lett. **36**(10), 1890–1892 (2011). [CrossRef] [PubMed]

7. D. G. Papazoglou, N. K. Efremidis, D. N. Christodoulides, and S. Tzortzakis, “Observation of abruptly autofocusing waves,” Opt. Lett. **36**(10), 1842–1844 (2011). [CrossRef] [PubMed]

## 5. Applications to femtosecond micromachining

27. M. K. Bhuyan, F. Courvoisier, P. A. Lacourt, M. Jacquot, R. Salut, L. Furfaro, and J. M. Dudley, “High aspect ratio nanochannel machining using single shot femtosecond Bessel beams,” Appl. Phys. Lett. **97**(8), 081102 (2010). [CrossRef]

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

4. P. Polynkin, M. Kolesik, and J. Moloney, “Filamentation of femtosecond laser Airy beams in water,” Phys. Rev. Lett. **103**(12), 123902 (2009). [CrossRef] [PubMed]

27. M. K. Bhuyan, F. Courvoisier, P. A. Lacourt, M. Jacquot, R. Salut, L. Furfaro, and J. M. Dudley, “High aspect ratio nanochannel machining using single shot femtosecond Bessel beams,” Appl. Phys. Lett. **97**(8), 081102 (2010). [CrossRef]

*c*(

*z*)=(

*a*/n)(

*z*-

*z*)

_{0}^{2}, where n is the index of refraction of glass and

*z*is the extremum of the parabola. More generally, to obtain a caustic

_{0}*c*(

*z*) in a medium of refractive index

*n*, the phase to be applied at the interface requires simply an n-fold increase in the phase required to produce the same caustic trajectory in air.

## 6. Conclusions

**106**(21), 213902 (2011). [CrossRef] [PubMed]

**97**(8), 081102 (2010). [CrossRef]

29. P. Polesana, M. Franco, A. Couairon, D. Faccio, and P. Di Trapani, “Filamentation in Kerr media from pulsed Bessel beams,” Phys. Rev. A **77**(4), 043814 (2008). [CrossRef]

## Acknowledgements

## References and links

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

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

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

4. | P. Polynkin, M. Kolesik, and J. Moloney, “Filamentation of femtosecond laser Airy beams in water,” Phys. Rev. Lett. |

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

6. | N. K. Efremidis and D. N. Christodoulides, “Abruptly autofocusing waves,” Opt. Lett. |

7. | D. G. Papazoglou, N. K. Efremidis, D. N. Christodoulides, and S. Tzortzakis, “Observation of abruptly autofocusing waves,” Opt. Lett. |

8. | I. Chremmos, N. K. Efremidis, and D. N. Christodoulides, “Pre-engineered abruptly autofocusing beams,” Opt. Lett. |

9. | A. Chong, W. H. Renninger, D. N. Christodoulides, and F. W. Wise, “Airy-Bessel wave packets as versatile linear light bullets,” Nat. Photonics |

10. | I. Kaminer, M. Segev, and D. N. Christodoulides, “Self-accelerating self-trapped optical beams,” Phys. Rev. Lett. |

11. | M. A. Bandres and J. C. Gutiérrez-Vega, “Airy-Gauss beams and their transformation by paraxial optical systems,” Opt. Express |

12. | H. E. Hernandez-Figueroa, M. Zamboni-Rached, and E. Recami, eds., Localized Waves: Theory and Applications, (J. Wiley, New York, 2008) |

13. | P. Saari, “Laterally accelerating airy pulses,” Opt. Express |

14. | G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Ballistic dynamics of Airy beams,” Opt. Lett. |

15. | H. I. Sztul and R. R. Alfano, “The Poynting vector and angular momentum of Airy beams,” Opt. Express |

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

17. | S. Vo, K. Fuerschbach, K. P. Thompson, M. A. Alonso, and J. P. Rolland, “Airy beams: a geometric optics perspective,” J. Opt. Soc. Am. A |

18. | Y. Kaganovsky and E. Heyman, “Wave analysis of Airy beams,” Opt. Express |

19. | W. Liu, D. N. Neshev, I. V. Shadrivov, A. E. Miroshnichenko, and Y. S. Kivshar, “Plasmonic Airy beam manipulation in linear optical potentials,” Opt. Lett. |

20. | J.-X. Li, X.-L. Fan, W. P. Zang, and J.-G. Tian, “Vacuum electron acceleration driven by two crossed Airy beams,” Opt. Lett. |

21. | E. Greenfield, M. Segev, W. Walasik, and O. Raz, “Accelerating Light Beams along Arbitrary Convex Trajectories,” Phys. Rev. Lett. |

22. | J. Rosen and A. Yariv, “Snake beam: a paraxial arbitrary focal line,” Opt. Lett. |

23. | C. Bellver-Cebreros and M. Rodriguez-Danta, “Caustics and the Legendre transform,” Opt. Commun. |

24. | A. V. Gitin, “Legendre transformation: Connection between transverse aberration of an optical system and its caustic,” Opt. Commun. |

25. | C. E. Gutiérrez, “Reflection, refraction, and the Legendre transform,” J. Opt. Soc. Am. A |

26. | 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. |

27. | M. K. Bhuyan, F. Courvoisier, P. A. Lacourt, M. Jacquot, R. Salut, L. Furfaro, and J. M. Dudley, “High aspect ratio nanochannel machining using single shot femtosecond Bessel beams,” Appl. Phys. Lett. |

28. | K. Dholakia and T. Čižmár, “Shaping the future of manipulation,” Nat. Photonics |

29. | P. Polesana, M. Franco, A. Couairon, D. Faccio, and P. Di Trapani, “Filamentation in Kerr media from pulsed Bessel beams,” Phys. Rev. A |

**OCIS Codes**

(190.7110) Nonlinear optics : Ultrafast nonlinear optics

(320.2250) Ultrafast optics : Femtosecond phenomena

(350.3390) Other areas of optics : Laser materials processing

(070.6120) Fourier optics and signal processing : Spatial light modulators

**ToC Category:**

Physical Optics

**History**

Original Manuscript: June 15, 2011

Revised Manuscript: July 28, 2011

Manuscript Accepted: August 1, 2011

Published: August 11, 2011

**Citation**

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 **19**, 16455-16465 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-17-16455

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### References

- M. V. Berry and N. L. Balazs, “Nonspreading wave packets,” Am. J. Phys. 47(3), 264–267 (1979). [CrossRef]
- G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett. 99(21), 213901 (2007). [CrossRef] [PubMed]
- J. Baumgartl, M. Mazilu, and K. Dholakia, “Optically mediated particle clearing using Airy wavepackets,” Nat. Photonics 2(11), 675–678 (2008). [CrossRef]
- P. Polynkin, M. Kolesik, and J. Moloney, “Filamentation of femtosecond laser Airy beams in water,” Phys. Rev. Lett. 103(12), 123902 (2009). [CrossRef] [PubMed]
- P. Polynkin, M. Kolesik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, “Curved plasma channel generation using ultraintense Airy beams,” Science 324(5924), 229–232 (2009). [CrossRef] [PubMed]
- N. K. Efremidis and D. N. Christodoulides, “Abruptly autofocusing waves,” Opt. Lett. 35(23), 4045–4047 (2010). [CrossRef] [PubMed]
- D. G. Papazoglou, N. K. Efremidis, D. N. Christodoulides, and S. Tzortzakis, “Observation of abruptly autofocusing waves,” Opt. Lett. 36(10), 1842–1844 (2011). [CrossRef] [PubMed]
- I. Chremmos, N. K. Efremidis, and D. N. Christodoulides, “Pre-engineered abruptly autofocusing beams,” Opt. Lett. 36(10), 1890–1892 (2011). [CrossRef] [PubMed]
- 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]
- I. Kaminer, M. Segev, and D. N. Christodoulides, “Self-accelerating self-trapped optical beams,” Phys. Rev. Lett. 106(21), 213903 (2011). [CrossRef] [PubMed]
- M. A. Bandres and J. C. Gutiérrez-Vega, “Airy-Gauss beams and their transformation by paraxial optical systems,” Opt. Express 15(25), 16719–16728 (2007), http://www.opticsinfobase.org/abstract.cfm?URI=oe-15-25-16719 . [CrossRef] [PubMed]
- H. E. Hernandez-Figueroa, M. Zamboni-Rached, and E. Recami, eds., Localized Waves: Theory and Applications, (J. Wiley, New York, 2008)
- P. Saari, “Laterally accelerating airy pulses,” Opt. Express 16(14), 10303–10308 (2008), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-14-10303 . [CrossRef] [PubMed]
- 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]
- H. I. Sztul and R. R. Alfano, “The Poynting vector and angular momentum of Airy beams,” Opt. Express 16(13), 9411–9416 (2008), http://www.opticsinfobase.org/abstract.cfm?URI=oe-16-13-9411 . [CrossRef] [PubMed]
- M. A. Bandres, “Accelerating beams,” Opt. Lett. 34(24), 3791–3793 (2009). [CrossRef] [PubMed]
- S. Vo, K. Fuerschbach, K. P. Thompson, M. A. Alonso, and J. P. Rolland, “Airy beams: a geometric optics perspective,” J. Opt. Soc. Am. A 27(12), 2574–2582 (2010). [CrossRef] [PubMed]
- Y. Kaganovsky and E. Heyman, “Wave analysis of Airy beams,” Opt. Express 18(8), 8440–8452 (2010), http://www.opticsinfobase.org/abstract.cfm?URI=oe-18-8-8440 . [CrossRef] [PubMed]
- W. Liu, D. N. Neshev, I. V. Shadrivov, A. E. Miroshnichenko, and Y. S. Kivshar, “Plasmonic Airy beam manipulation in linear optical potentials,” Opt. Lett. 36(7), 1164–1166 (2011). [CrossRef] [PubMed]
- J.-X. Li, X.-L. Fan, W. P. Zang, and J.-G. Tian, “Vacuum electron acceleration driven by two crossed Airy beams,” Opt. Lett. 36(5), 648–650 (2011). [CrossRef] [PubMed]
- E. Greenfield, M. Segev, W. Walasik, and O. Raz, “Accelerating Light Beams along Arbitrary Convex Trajectories,” Phys. Rev. Lett. 106(21), 213902 (2011). [CrossRef] [PubMed]
- J. Rosen and A. Yariv, “Snake beam: a paraxial arbitrary focal line,” Opt. Lett. 20(20), 2042–2044 (1995). [CrossRef] [PubMed]
- C. Bellver-Cebreros and M. Rodriguez-Danta, “Caustics and the Legendre transform,” Opt. Commun. 92(4-6), 187–192 (1992). [CrossRef]
- A. V. Gitin, “Legendre transformation: Connection between transverse aberration of an optical system and its caustic,” Opt. Commun. 281, 3062–3066 (2008). [CrossRef]
- C. E. Gutiérrez, “Reflection, refraction, and the Legendre transform,” J. Opt. Soc. Am. A 28(2), 284–289 (2011). [CrossRef] [PubMed]
- 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(17), 2634–2636 (2009). [CrossRef] [PubMed]
- M. K. Bhuyan, F. Courvoisier, P. A. Lacourt, M. Jacquot, R. Salut, L. Furfaro, and J. M. Dudley, “High aspect ratio nanochannel machining using single shot femtosecond Bessel beams,” Appl. Phys. Lett. 97(8), 081102 (2010). [CrossRef]
- K. Dholakia and T. Čižmár, “Shaping the future of manipulation,” Nat. Photonics 5(6), 335–342 (2011). [CrossRef]
- P. Polesana, M. Franco, A. Couairon, D. Faccio, and P. Di Trapani, “Filamentation in Kerr media from pulsed Bessel beams,” Phys. Rev. A 77(4), 043814 (2008). [CrossRef]

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