Abstract

We report the fabrication of large-area phase masks on thin fused-silica substrates that are suitable for shaping multiterawatt femtosecond laser beams. We apply these phase masks for the generation of intense femtosecond optical vortices. We further quantify distortions of the vortex beam patterns that result from several common types of mask defects.

© 2014 Optical Society of America

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References

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  1. G. Mechain, A. Couairon, M. Franco, B. Prade, and A. Mysyrowicz, “Organizing multiple femtosecond filaments in air,” Phys. Rev. Lett. 93, 035003 (2004).
    [CrossRef]
  2. P. Polynkin, M. Kolesik, A. Roberts, D. Faccio, P. Di Trapani, and J. Moloney, “Generation of extended plasma channels in air using femtosecond Bessel beams,” Opt. Express 16, 15733–15740 (2008).
    [CrossRef]
  3. S. Akturk, B. Zhou, M. Franco, A. Couairon, and A. Mysyrowicz, “Generation of long plasma channels in air by focusing ultrashort laser pulses with an axicon,” Opt. Commun. 282, 129–134 (2009).
    [CrossRef]
  4. P. Polynkin, M. Kolesik, and J. Moloney, “Extended filamentation with temporally chirped femtosecond Bessel-Gauss beams in air,” Opt. Express 17, 575–584 (2009).
    [CrossRef]
  5. P. Polynkin, M. Kolesik, J. Moloney, G. Siviloglou, and D. Christodoulides, “Curved plasma channel generation using ultraintense Airy beams,” Science 324, 229–232 (2009).
    [CrossRef]
  6. I. J. Sola, V. Collados, L. Plaja, C. Mendez, J. San Roman, C. Ruiz, I. Arias, A. Villamarin, J. Atencia, M. Quintanilla, and L. Roso, “High power vortex generation with volume phase holograms and non-linear experiments in gases,” Appl. Phys. B 91, 115–118 (2008).
    [CrossRef]
  7. P. Polynkin, C. Ament, and J. V. Moloney, “Self-focusing of ultraintense femtosecond optical vortices in air,” Phys. Rev. Lett. 111, 023901 (2013).
    [CrossRef]
  8. A. Vincotte and L. Berge, “Femtosecond optical vortices in air,” Phys. Rev. Lett. 95, 193901 (2005).
    [CrossRef]
  9. L. T. Vuong, T. D. Grow, A. Ishaaya, A. L. Gaeta, G. W. ’t Hooft, E. R. Eliel, and G. Fibich, “Collapse of optical vortices,” Phys. Rev. Lett. 96, 133901 (2006).
    [CrossRef]
  10. M. Chateauneuf, S. Payeur, J. Dubuis, and J.-C. Kieffer, “Microwave guiding in air by a cylindrical filament array waveguide,” Appl. Phys. Lett. 92, 091104 (2008).
    [CrossRef]
  11. A. Alshershby, Z. Q. Hao, and J. Q. Lin, “Guiding microwave radiation using laser-induced filaments: the hollow conducting waveguide concept,” J. Phys. D 45, 265401 (2012).
  12. M. S. Soskin and M. V. Vasnetsov, “Singular optics,” Prog. Opt. 42, 219–276 (2001).
    [CrossRef]
  13. B. C. Stuart, M. D. Feit, S. Herman, A. M. Rubenchik, B. W. Shore, and M. D. Perry, “Nanosecond-to-femtosecond laser-induced breakdown in dielectrics,” Phys. Rev. B 53, 1749–1761 (1996).
  14. G. P. Agrawal, Nonlinear Fiber Optics (Academic, 2013).

2013 (1)

P. Polynkin, C. Ament, and J. V. Moloney, “Self-focusing of ultraintense femtosecond optical vortices in air,” Phys. Rev. Lett. 111, 023901 (2013).
[CrossRef]

2012 (1)

A. Alshershby, Z. Q. Hao, and J. Q. Lin, “Guiding microwave radiation using laser-induced filaments: the hollow conducting waveguide concept,” J. Phys. D 45, 265401 (2012).

2009 (3)

S. Akturk, B. Zhou, M. Franco, A. Couairon, and A. Mysyrowicz, “Generation of long plasma channels in air by focusing ultrashort laser pulses with an axicon,” Opt. Commun. 282, 129–134 (2009).
[CrossRef]

P. Polynkin, M. Kolesik, and J. Moloney, “Extended filamentation with temporally chirped femtosecond Bessel-Gauss beams in air,” Opt. Express 17, 575–584 (2009).
[CrossRef]

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

2008 (3)

I. J. Sola, V. Collados, L. Plaja, C. Mendez, J. San Roman, C. Ruiz, I. Arias, A. Villamarin, J. Atencia, M. Quintanilla, and L. Roso, “High power vortex generation with volume phase holograms and non-linear experiments in gases,” Appl. Phys. B 91, 115–118 (2008).
[CrossRef]

P. Polynkin, M. Kolesik, A. Roberts, D. Faccio, P. Di Trapani, and J. Moloney, “Generation of extended plasma channels in air using femtosecond Bessel beams,” Opt. Express 16, 15733–15740 (2008).
[CrossRef]

M. Chateauneuf, S. Payeur, J. Dubuis, and J.-C. Kieffer, “Microwave guiding in air by a cylindrical filament array waveguide,” Appl. Phys. Lett. 92, 091104 (2008).
[CrossRef]

2006 (1)

L. T. Vuong, T. D. Grow, A. Ishaaya, A. L. Gaeta, G. W. ’t Hooft, E. R. Eliel, and G. Fibich, “Collapse of optical vortices,” Phys. Rev. Lett. 96, 133901 (2006).
[CrossRef]

2005 (1)

A. Vincotte and L. Berge, “Femtosecond optical vortices in air,” Phys. Rev. Lett. 95, 193901 (2005).
[CrossRef]

2004 (1)

G. Mechain, A. Couairon, M. Franco, B. Prade, and A. Mysyrowicz, “Organizing multiple femtosecond filaments in air,” Phys. Rev. Lett. 93, 035003 (2004).
[CrossRef]

2001 (1)

M. S. Soskin and M. V. Vasnetsov, “Singular optics,” Prog. Opt. 42, 219–276 (2001).
[CrossRef]

1996 (1)

B. C. Stuart, M. D. Feit, S. Herman, A. M. Rubenchik, B. W. Shore, and M. D. Perry, “Nanosecond-to-femtosecond laser-induced breakdown in dielectrics,” Phys. Rev. B 53, 1749–1761 (1996).

’t Hooft, G. W.

L. T. Vuong, T. D. Grow, A. Ishaaya, A. L. Gaeta, G. W. ’t Hooft, E. R. Eliel, and G. Fibich, “Collapse of optical vortices,” Phys. Rev. Lett. 96, 133901 (2006).
[CrossRef]

Agrawal, G. P.

G. P. Agrawal, Nonlinear Fiber Optics (Academic, 2013).

Akturk, S.

S. Akturk, B. Zhou, M. Franco, A. Couairon, and A. Mysyrowicz, “Generation of long plasma channels in air by focusing ultrashort laser pulses with an axicon,” Opt. Commun. 282, 129–134 (2009).
[CrossRef]

Alshershby, A.

A. Alshershby, Z. Q. Hao, and J. Q. Lin, “Guiding microwave radiation using laser-induced filaments: the hollow conducting waveguide concept,” J. Phys. D 45, 265401 (2012).

Ament, C.

P. Polynkin, C. Ament, and J. V. Moloney, “Self-focusing of ultraintense femtosecond optical vortices in air,” Phys. Rev. Lett. 111, 023901 (2013).
[CrossRef]

Arias, I.

I. J. Sola, V. Collados, L. Plaja, C. Mendez, J. San Roman, C. Ruiz, I. Arias, A. Villamarin, J. Atencia, M. Quintanilla, and L. Roso, “High power vortex generation with volume phase holograms and non-linear experiments in gases,” Appl. Phys. B 91, 115–118 (2008).
[CrossRef]

Atencia, J.

I. J. Sola, V. Collados, L. Plaja, C. Mendez, J. San Roman, C. Ruiz, I. Arias, A. Villamarin, J. Atencia, M. Quintanilla, and L. Roso, “High power vortex generation with volume phase holograms and non-linear experiments in gases,” Appl. Phys. B 91, 115–118 (2008).
[CrossRef]

Berge, L.

A. Vincotte and L. Berge, “Femtosecond optical vortices in air,” Phys. Rev. Lett. 95, 193901 (2005).
[CrossRef]

Chateauneuf, M.

M. Chateauneuf, S. Payeur, J. Dubuis, and J.-C. Kieffer, “Microwave guiding in air by a cylindrical filament array waveguide,” Appl. Phys. Lett. 92, 091104 (2008).
[CrossRef]

Christodoulides, D.

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

Collados, V.

I. J. Sola, V. Collados, L. Plaja, C. Mendez, J. San Roman, C. Ruiz, I. Arias, A. Villamarin, J. Atencia, M. Quintanilla, and L. Roso, “High power vortex generation with volume phase holograms and non-linear experiments in gases,” Appl. Phys. B 91, 115–118 (2008).
[CrossRef]

Couairon, A.

S. Akturk, B. Zhou, M. Franco, A. Couairon, and A. Mysyrowicz, “Generation of long plasma channels in air by focusing ultrashort laser pulses with an axicon,” Opt. Commun. 282, 129–134 (2009).
[CrossRef]

G. Mechain, A. Couairon, M. Franco, B. Prade, and A. Mysyrowicz, “Organizing multiple femtosecond filaments in air,” Phys. Rev. Lett. 93, 035003 (2004).
[CrossRef]

Di Trapani, P.

Dubuis, J.

M. Chateauneuf, S. Payeur, J. Dubuis, and J.-C. Kieffer, “Microwave guiding in air by a cylindrical filament array waveguide,” Appl. Phys. Lett. 92, 091104 (2008).
[CrossRef]

Eliel, E. R.

L. T. Vuong, T. D. Grow, A. Ishaaya, A. L. Gaeta, G. W. ’t Hooft, E. R. Eliel, and G. Fibich, “Collapse of optical vortices,” Phys. Rev. Lett. 96, 133901 (2006).
[CrossRef]

Faccio, D.

Feit, M. D.

B. C. Stuart, M. D. Feit, S. Herman, A. M. Rubenchik, B. W. Shore, and M. D. Perry, “Nanosecond-to-femtosecond laser-induced breakdown in dielectrics,” Phys. Rev. B 53, 1749–1761 (1996).

Fibich, G.

L. T. Vuong, T. D. Grow, A. Ishaaya, A. L. Gaeta, G. W. ’t Hooft, E. R. Eliel, and G. Fibich, “Collapse of optical vortices,” Phys. Rev. Lett. 96, 133901 (2006).
[CrossRef]

Franco, M.

S. Akturk, B. Zhou, M. Franco, A. Couairon, and A. Mysyrowicz, “Generation of long plasma channels in air by focusing ultrashort laser pulses with an axicon,” Opt. Commun. 282, 129–134 (2009).
[CrossRef]

G. Mechain, A. Couairon, M. Franco, B. Prade, and A. Mysyrowicz, “Organizing multiple femtosecond filaments in air,” Phys. Rev. Lett. 93, 035003 (2004).
[CrossRef]

Gaeta, A. L.

L. T. Vuong, T. D. Grow, A. Ishaaya, A. L. Gaeta, G. W. ’t Hooft, E. R. Eliel, and G. Fibich, “Collapse of optical vortices,” Phys. Rev. Lett. 96, 133901 (2006).
[CrossRef]

Grow, T. D.

L. T. Vuong, T. D. Grow, A. Ishaaya, A. L. Gaeta, G. W. ’t Hooft, E. R. Eliel, and G. Fibich, “Collapse of optical vortices,” Phys. Rev. Lett. 96, 133901 (2006).
[CrossRef]

Hao, Z. Q.

A. Alshershby, Z. Q. Hao, and J. Q. Lin, “Guiding microwave radiation using laser-induced filaments: the hollow conducting waveguide concept,” J. Phys. D 45, 265401 (2012).

Herman, S.

B. C. Stuart, M. D. Feit, S. Herman, A. M. Rubenchik, B. W. Shore, and M. D. Perry, “Nanosecond-to-femtosecond laser-induced breakdown in dielectrics,” Phys. Rev. B 53, 1749–1761 (1996).

Ishaaya, A.

L. T. Vuong, T. D. Grow, A. Ishaaya, A. L. Gaeta, G. W. ’t Hooft, E. R. Eliel, and G. Fibich, “Collapse of optical vortices,” Phys. Rev. Lett. 96, 133901 (2006).
[CrossRef]

Kieffer, J.-C.

M. Chateauneuf, S. Payeur, J. Dubuis, and J.-C. Kieffer, “Microwave guiding in air by a cylindrical filament array waveguide,” Appl. Phys. Lett. 92, 091104 (2008).
[CrossRef]

Kolesik, M.

Lin, J. Q.

A. Alshershby, Z. Q. Hao, and J. Q. Lin, “Guiding microwave radiation using laser-induced filaments: the hollow conducting waveguide concept,” J. Phys. D 45, 265401 (2012).

Mechain, G.

G. Mechain, A. Couairon, M. Franco, B. Prade, and A. Mysyrowicz, “Organizing multiple femtosecond filaments in air,” Phys. Rev. Lett. 93, 035003 (2004).
[CrossRef]

Mendez, C.

I. J. Sola, V. Collados, L. Plaja, C. Mendez, J. San Roman, C. Ruiz, I. Arias, A. Villamarin, J. Atencia, M. Quintanilla, and L. Roso, “High power vortex generation with volume phase holograms and non-linear experiments in gases,” Appl. Phys. B 91, 115–118 (2008).
[CrossRef]

Moloney, J.

Moloney, J. V.

P. Polynkin, C. Ament, and J. V. Moloney, “Self-focusing of ultraintense femtosecond optical vortices in air,” Phys. Rev. Lett. 111, 023901 (2013).
[CrossRef]

Mysyrowicz, A.

S. Akturk, B. Zhou, M. Franco, A. Couairon, and A. Mysyrowicz, “Generation of long plasma channels in air by focusing ultrashort laser pulses with an axicon,” Opt. Commun. 282, 129–134 (2009).
[CrossRef]

G. Mechain, A. Couairon, M. Franco, B. Prade, and A. Mysyrowicz, “Organizing multiple femtosecond filaments in air,” Phys. Rev. Lett. 93, 035003 (2004).
[CrossRef]

Payeur, S.

M. Chateauneuf, S. Payeur, J. Dubuis, and J.-C. Kieffer, “Microwave guiding in air by a cylindrical filament array waveguide,” Appl. Phys. Lett. 92, 091104 (2008).
[CrossRef]

Perry, M. D.

B. C. Stuart, M. D. Feit, S. Herman, A. M. Rubenchik, B. W. Shore, and M. D. Perry, “Nanosecond-to-femtosecond laser-induced breakdown in dielectrics,” Phys. Rev. B 53, 1749–1761 (1996).

Plaja, L.

I. J. Sola, V. Collados, L. Plaja, C. Mendez, J. San Roman, C. Ruiz, I. Arias, A. Villamarin, J. Atencia, M. Quintanilla, and L. Roso, “High power vortex generation with volume phase holograms and non-linear experiments in gases,” Appl. Phys. B 91, 115–118 (2008).
[CrossRef]

Polynkin, P.

P. Polynkin, C. Ament, and J. V. Moloney, “Self-focusing of ultraintense femtosecond optical vortices in air,” Phys. Rev. Lett. 111, 023901 (2013).
[CrossRef]

P. Polynkin, M. Kolesik, and J. Moloney, “Extended filamentation with temporally chirped femtosecond Bessel-Gauss beams in air,” Opt. Express 17, 575–584 (2009).
[CrossRef]

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

P. Polynkin, M. Kolesik, A. Roberts, D. Faccio, P. Di Trapani, and J. Moloney, “Generation of extended plasma channels in air using femtosecond Bessel beams,” Opt. Express 16, 15733–15740 (2008).
[CrossRef]

Prade, B.

G. Mechain, A. Couairon, M. Franco, B. Prade, and A. Mysyrowicz, “Organizing multiple femtosecond filaments in air,” Phys. Rev. Lett. 93, 035003 (2004).
[CrossRef]

Quintanilla, M.

I. J. Sola, V. Collados, L. Plaja, C. Mendez, J. San Roman, C. Ruiz, I. Arias, A. Villamarin, J. Atencia, M. Quintanilla, and L. Roso, “High power vortex generation with volume phase holograms and non-linear experiments in gases,” Appl. Phys. B 91, 115–118 (2008).
[CrossRef]

Roberts, A.

Roso, L.

I. J. Sola, V. Collados, L. Plaja, C. Mendez, J. San Roman, C. Ruiz, I. Arias, A. Villamarin, J. Atencia, M. Quintanilla, and L. Roso, “High power vortex generation with volume phase holograms and non-linear experiments in gases,” Appl. Phys. B 91, 115–118 (2008).
[CrossRef]

Rubenchik, A. M.

B. C. Stuart, M. D. Feit, S. Herman, A. M. Rubenchik, B. W. Shore, and M. D. Perry, “Nanosecond-to-femtosecond laser-induced breakdown in dielectrics,” Phys. Rev. B 53, 1749–1761 (1996).

Ruiz, C.

I. J. Sola, V. Collados, L. Plaja, C. Mendez, J. San Roman, C. Ruiz, I. Arias, A. Villamarin, J. Atencia, M. Quintanilla, and L. Roso, “High power vortex generation with volume phase holograms and non-linear experiments in gases,” Appl. Phys. B 91, 115–118 (2008).
[CrossRef]

San Roman, J.

I. J. Sola, V. Collados, L. Plaja, C. Mendez, J. San Roman, C. Ruiz, I. Arias, A. Villamarin, J. Atencia, M. Quintanilla, and L. Roso, “High power vortex generation with volume phase holograms and non-linear experiments in gases,” Appl. Phys. B 91, 115–118 (2008).
[CrossRef]

Shore, B. W.

B. C. Stuart, M. D. Feit, S. Herman, A. M. Rubenchik, B. W. Shore, and M. D. Perry, “Nanosecond-to-femtosecond laser-induced breakdown in dielectrics,” Phys. Rev. B 53, 1749–1761 (1996).

Siviloglou, G.

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

Sola, I. J.

I. J. Sola, V. Collados, L. Plaja, C. Mendez, J. San Roman, C. Ruiz, I. Arias, A. Villamarin, J. Atencia, M. Quintanilla, and L. Roso, “High power vortex generation with volume phase holograms and non-linear experiments in gases,” Appl. Phys. B 91, 115–118 (2008).
[CrossRef]

Soskin, M. S.

M. S. Soskin and M. V. Vasnetsov, “Singular optics,” Prog. Opt. 42, 219–276 (2001).
[CrossRef]

Stuart, B. C.

B. C. Stuart, M. D. Feit, S. Herman, A. M. Rubenchik, B. W. Shore, and M. D. Perry, “Nanosecond-to-femtosecond laser-induced breakdown in dielectrics,” Phys. Rev. B 53, 1749–1761 (1996).

Vasnetsov, M. V.

M. S. Soskin and M. V. Vasnetsov, “Singular optics,” Prog. Opt. 42, 219–276 (2001).
[CrossRef]

Villamarin, A.

I. J. Sola, V. Collados, L. Plaja, C. Mendez, J. San Roman, C. Ruiz, I. Arias, A. Villamarin, J. Atencia, M. Quintanilla, and L. Roso, “High power vortex generation with volume phase holograms and non-linear experiments in gases,” Appl. Phys. B 91, 115–118 (2008).
[CrossRef]

Vincotte, A.

A. Vincotte and L. Berge, “Femtosecond optical vortices in air,” Phys. Rev. Lett. 95, 193901 (2005).
[CrossRef]

Vuong, L. T.

L. T. Vuong, T. D. Grow, A. Ishaaya, A. L. Gaeta, G. W. ’t Hooft, E. R. Eliel, and G. Fibich, “Collapse of optical vortices,” Phys. Rev. Lett. 96, 133901 (2006).
[CrossRef]

Zhou, B.

S. Akturk, B. Zhou, M. Franco, A. Couairon, and A. Mysyrowicz, “Generation of long plasma channels in air by focusing ultrashort laser pulses with an axicon,” Opt. Commun. 282, 129–134 (2009).
[CrossRef]

Appl. Phys. B (1)

I. J. Sola, V. Collados, L. Plaja, C. Mendez, J. San Roman, C. Ruiz, I. Arias, A. Villamarin, J. Atencia, M. Quintanilla, and L. Roso, “High power vortex generation with volume phase holograms and non-linear experiments in gases,” Appl. Phys. B 91, 115–118 (2008).
[CrossRef]

Appl. Phys. Lett. (1)

M. Chateauneuf, S. Payeur, J. Dubuis, and J.-C. Kieffer, “Microwave guiding in air by a cylindrical filament array waveguide,” Appl. Phys. Lett. 92, 091104 (2008).
[CrossRef]

J. Phys. D (1)

A. Alshershby, Z. Q. Hao, and J. Q. Lin, “Guiding microwave radiation using laser-induced filaments: the hollow conducting waveguide concept,” J. Phys. D 45, 265401 (2012).

Opt. Commun. (1)

S. Akturk, B. Zhou, M. Franco, A. Couairon, and A. Mysyrowicz, “Generation of long plasma channels in air by focusing ultrashort laser pulses with an axicon,” Opt. Commun. 282, 129–134 (2009).
[CrossRef]

Opt. Express (2)

Phys. Rev. B (1)

B. C. Stuart, M. D. Feit, S. Herman, A. M. Rubenchik, B. W. Shore, and M. D. Perry, “Nanosecond-to-femtosecond laser-induced breakdown in dielectrics,” Phys. Rev. B 53, 1749–1761 (1996).

Phys. Rev. Lett. (4)

G. Mechain, A. Couairon, M. Franco, B. Prade, and A. Mysyrowicz, “Organizing multiple femtosecond filaments in air,” Phys. Rev. Lett. 93, 035003 (2004).
[CrossRef]

P. Polynkin, C. Ament, and J. V. Moloney, “Self-focusing of ultraintense femtosecond optical vortices in air,” Phys. Rev. Lett. 111, 023901 (2013).
[CrossRef]

A. Vincotte and L. Berge, “Femtosecond optical vortices in air,” Phys. Rev. Lett. 95, 193901 (2005).
[CrossRef]

L. T. Vuong, T. D. Grow, A. Ishaaya, A. L. Gaeta, G. W. ’t Hooft, E. R. Eliel, and G. Fibich, “Collapse of optical vortices,” Phys. Rev. Lett. 96, 133901 (2006).
[CrossRef]

Prog. Opt. (1)

M. S. Soskin and M. V. Vasnetsov, “Singular optics,” Prog. Opt. 42, 219–276 (2001).
[CrossRef]

Science (1)

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

Other (1)

G. P. Agrawal, Nonlinear Fiber Optics (Academic, 2013).

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Figures (7)

Fig. 1.
Fig. 1.

Illustration of the second step of the mask fabrication process. The desired grayscale phase pattern, imprinted into a several-micrometers-thin layer of photoresist atop a fused-silica substrate, is etched into the substrate via RIE.

Fig. 2.
Fig. 2.

Examples of numerically simulated ideal vortex beam patterns of orders 3 and 8. The phase profiles, defined modulo 2π, are shown in the left panels, with the corresponding transverse beam intensity distributions shown in the right panels.

Fig. 3.
Fig. 3.

A and B, intensity distributions of vortex beams of orders 3 and 8 in the linear propagation regime (at low intensity). The 1/e2 E-field radius of the Gaussian beam incident on the masks equals 5 mm, and the focal length of the focusing lens used is 1 m. Distortions of the doughnut-shaped beam intensity patterns are evident. C and D, bottle-like distributions of plasma filaments produced through self-focusing of an intense optical vortex of order 3 in air. Focusing conditions are specified in the text. The energy of the 50-fs-long laser pulse is 150 mJ (C) and 230 mJ (D), corresponding to the highest peak laser power of 4.6 TW.

Fig. 4.
Fig. 4.

A, peak-to-peak intensity modulation of the doughnut intensity feature of the beam as a function of the % error in the maximum phase step of the phase mask, for continuous-wave optical vortices of orders 3 and 8. B, beam intensity pattern in the focal plane of the focusing optic, for an optical vortex of order 3 and for 5% error in the maximum phase step. C, same for a vortex of order 8.

Fig. 5.
Fig. 5.

A, peak-to-peak fluence modulation of the vortex ring as a function of the optical bandwidth of the laser beam. B, fluence pattern for an optical vortex of order 3 and for optical bandwidth equal to 100 nm. C, same for a vortex of order 8.

Fig. 6.
Fig. 6.

A, peak-to-peak intensity modulation of the vortex ring as a function of the diameter of the circular defect on the phase mask. B and C, illustrations of the location of the circular defect on vortex phase masks of orders 3 and 8. D and E, corresponding beam patterns for the particular diameter of the circular defect equal to 1 mm.

Fig. 7.
Fig. 7.

A, peak-to-peak intensity modulation of the vortex ring as a function of the width of the line defect on the phase mask. B and C, illustrations of the location of the linear defect on vortex phase masks of orders 3 and 8. D and E, corresponding beam patterns for the particular width of the linear defect equal to 0.5 mm.

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Φ(r,θ)=imθ,

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