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

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  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]
  5. 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]
  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]
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  14. 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]
  15. 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]
  16. M. A. Bandres, “Accelerating beams,” Opt. Lett. 34(24), 3791–3793 (2009).
    [Crossref] [PubMed]
  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 27(12), 2574–2582 (2010).
    [Crossref] [PubMed]
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    [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).
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  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).
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  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]
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  23. C. Bellver-Cebreros and M. Rodriguez-Danta, “Caustics and the Legendre transform,” Opt. Commun. 92(4-6), 187–192 (1992).
    [Crossref]
  24. A. V. Gitin, “Legendre transformation: Connection between transverse aberration of an optical system and its caustic,” Opt. Commun. 281, 3062–3066 (2008).
    [Crossref]
  25. C. E. Gutiérrez, “Reflection, refraction, and the Legendre transform,” J. Opt. Soc. Am. A 28(2), 284–289 (2011).
    [Crossref] [PubMed]
  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]
  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]
  28. K. Dholakia and T. Čižmár, “Shaping the future of manipulation,” Nat. Photonics 5(6), 335–342 (2011).
    [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]

2011 (8)

2010 (5)

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]

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]

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]

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

2009 (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]

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]

M. A. Bandres, “Accelerating beams,” Opt. Lett. 34(24), 3791–3793 (2009).
[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]

2008 (6)

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]

A. V. Gitin, “Legendre transformation: Connection between transverse aberration of an optical system and its caustic,” Opt. Commun. 281, 3062–3066 (2008).
[Crossref]

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

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]

2007 (2)

1995 (1)

1992 (1)

C. Bellver-Cebreros and M. Rodriguez-Danta, “Caustics and the Legendre transform,” Opt. Commun. 92(4-6), 187–192 (1992).
[Crossref]

1979 (1)

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

Alfano, R. R.

Alonso, M. A.

Balazs, N. L.

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

Bandres, M. A.

Baumgartl, J.

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

Bellver-Cebreros, C.

C. Bellver-Cebreros and M. Rodriguez-Danta, “Caustics and the Legendre transform,” Opt. Commun. 92(4-6), 187–192 (1992).
[Crossref]

Berry, M. V.

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

Bhuyan, M. K.

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]

Broky, J.

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]

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]

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]

Chremmos, I.

Christodoulides, D. N.

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]

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

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

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]

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]

Cižmár, T.

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

Cottrell, D. M.

Couairon, A.

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]

Courvoisier, F.

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]

Davis, J. A.

Dholakia, K.

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

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

Di Trapani, P.

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]

Dogariu, A.

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]

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]

Dudley, J. M.

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]

Efremidis, N. K.

Faccio, D.

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]

Fan, X.-L.

Franco, M.

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]

Fuerschbach, K.

Furfaro, L.

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]

Gitin, A. V.

A. V. Gitin, “Legendre transformation: Connection between transverse aberration of an optical system and its caustic,” Opt. Commun. 281, 3062–3066 (2008).
[Crossref]

Greenfield, E.

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]

Gutiérrez, C. E.

Gutiérrez-Vega, J. C.

Hazard, T. M.

Heyman, E.

Jacquot, M.

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]

Kaganovsky, Y.

Kaminer, I.

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

Kivshar, Y. S.

Kolesik, M.

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]

Lacourt, P. A.

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]

Li, J.-X.

Liu, W.

Mazilu, M.

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

Miroshnichenko, A. E.

Moloney, J.

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

Moloney, J. V.

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]

Neshev, D. N.

Papazoglou, D. G.

Polesana, P.

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]

Polynkin, P.

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]

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

Raz, O.

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]

Renninger, W. H.

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]

Rodriguez-Danta, M.

C. Bellver-Cebreros and M. Rodriguez-Danta, “Caustics and the Legendre transform,” Opt. Commun. 92(4-6), 187–192 (1992).
[Crossref]

Rolland, J. P.

Rosen, J.

Saari, P.

Salut, R.

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]

Segev, M.

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

Shadrivov, I. V.

Siviloglou, G. A.

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]

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]

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]

Sztul, H. I.

Thompson, K. P.

Tian, J.-G.

Tzortzakis, S.

Vo, S.

Walasik, W.

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]

Wise, F. W.

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]

Yariv, A.

Zang, W. P.

Am. J. Phys. (1)

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

Appl. Phys. Lett. (1)

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]

J. Opt. Soc. Am. A (2)

Nat. Photonics (3)

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]

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

Opt. Commun. (2)

C. Bellver-Cebreros and M. Rodriguez-Danta, “Caustics and the Legendre transform,” Opt. Commun. 92(4-6), 187–192 (1992).
[Crossref]

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Phys. Rev. A (1)

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]

Phys. Rev. Lett. (4)

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Science (1)

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

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Figures (8)

Fig. 1
Fig. 1

(a) Geometrical construction of a generating phase profile from the properties of an arbitrary desired caustic. The key concepts are (i) to identify the phase derivative dϕ/dy along the plane of the phase mask with the tangent slope to a point on the caustic and (ii) to use the parameterization of the slope c’(z) in terms of y (the Legendre transform) to allow integration to yield ϕ(y). The figure is drawn for the exact non paraxial case. In the paraxial case the approximation sinθ ≈ tanθ ≈θ allows some explicit results to be obtained as described in the text. (b) shows results of this approach comparing target parabolic caustics (white dashed line) with the propagated field using a Gaussian beam to which is applied the calculated phase profile (false color image). The approach is shown to work in both the paraxial (left) and non-paraxial (right) regimes with maximum ray angles to the caustic of 11° and 60° respectively.

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Fig. 2
Fig. 2

Schematic of our experimental setup. The spatial profile of incident 100 fs pulses is shaped with an appropriately-designed phase mask to directly generate a desired convex acceleration trajectory. A beam reduction telescope images the profile to micron-size dimensions where its evolution can be characterized, or where it can be applied to material processing. The figure illustrates in particular the creation of a two dimensional caustic sheet.

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Fig. 3
Fig. 3

Comparing numerical beam propagation as described in the text with experimental measurements for (a) parabolic; (b) quartic; (c) logarithmic beam profiles as indicated. The position z=0 corresponds to the image plane of the SLM after the ×10 demagnification system used in this configuration.

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Fig. 4
Fig. 4

Line profile comparisons in the vicinity of the trajectory minima shown in Fig. 3. The solid black lines are the numerically propagated beam intensity profiles. The red lines with markers show the experimental measurements.

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Fig. 5
Fig. 5

Characterisation of a volume sheet caustic bounded by (a) two parabolic acceleration trajectories and (b) two quartic acceleration trajectories. In each case, the left panel shows the applied phase profile, the middle panel the intensity distribution measured from the SLM image plane (z = 0) after demagnification, and the right panels show the extracted intensity line profiles at the points A and B indicated. The white dashed lines in the centre panel shows the target acceleration trajectories. The intensity line profiles are plotted in arbitrary units but there is an order of magnitude increase in peak intensity from points A to B in both cases because of the autofocussing due to caustic recombination.

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Fig. 6
Fig. 6

Combining an engineered acceleration trajectory with (a) rotational symmetry and (b) an imposed spiral structure. The left panels show the applied phase profiles; the right panels show tomographic representations of the shaped fields in both cases as discussed in the text.

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Fig. 7
Fig. 7

Intensity slice of the center of rotationally-symmetric beams bounded by (a) quartic and (b) parabolic caustic accelerating beams. The intensity line profiles shown to the right in each case correspond to the intensity maximum of the focal region transverse to the propagation direction (top) and along the propagation direction (bottom).

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Fig. 8
Fig. 8

Results showing application of our setup to femtosecond waveguide writing. (a) shows a slice through a double Airy profile synthesized as described in the text. (b) shows the curved region of refractive index modification.

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Tables (1)

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Table 1 In the Paraxial Approximation, this Table gives the Calculated Phase for Desired Acceleration Profiles as Shown.

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Equations (1)

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d ϕ ( y ) d y   = k   sin θ   =   k   c ′ ( z ) 1 + [ c ′ ( z ) ] 2

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