Abstract

We demonstrate the generation of a nondiffracting double helical beam using axicons and ±1 vortex phase plates in a common-path interferometric system. Using linear diffraction theory, a simple analytical expression describing beam propagation is shown to agree with both experiments and Fresnel-diffraction-based simulations. Experiments are performed using continuous laser light in addition to ultrafast pulses, demonstrating that the common-path arrangement and the diffraction theory work equally well for both cases.

© 2011 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. J. Durnin and J. J. Miceli, Jr., “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
    [CrossRef] [PubMed]
  2. F. Gori, G. Guattari, and C. Padovani, “Bessel-Gauss beams,” Opt. Commun. 64, 491–494 (1987).
    [CrossRef]
  3. K. Ait-Ameur and F. Sanchez, “Gaussian beam conversion using an axicon,” J. Mod. Opt. 46, 1537–1548 (1999).
    [CrossRef]
  4. J. Pu, H. Zhang, and S. Nemoto, “Lens axicons illuminated by Gaussian beams for generation of uniform-axial intensity Bessel fields,” Opt. Eng. 39, 803–807 (2000).
    [CrossRef]
  5. R. Vasilyeu, A. Dudley, N. Khilo, and A. Forbes, “Generating superpositions of higher order Bessel beams,” in Frontiers in Optics, OSA Technical Digest (CD) (Optical Society of America, 2009), paper FThB6.
  6. J. Turunen, A. Vasara, and A. T. Friberg, “Holographic generation of diffraction-free beams,” Appl. Opt. 27, 3959–3962 (1988).
    [CrossRef] [PubMed]
  7. M. M. Mendez Otero, G. C. Martinez Jimenez, M. L. Arroyo Carrasco, M. D. Iturbe Catillo, and E. Marti Panameno, “Generation of Bessel-Gauss beams by means of computer-generated holograms for Bessel beams,” in Frontiers in Optics, OSA Technical Digest (CD) (Optical Society of America, 2006), paper JWD129.
  8. W.-X. Cong and N.-X. Chen, “Generation of nondiffracting beams by diffractive phase elements,” J. Opt. Soc. Am. A 15, 2362–2364(1998).
    [CrossRef]
  9. S. Orlov, K. Regelskis, V. Smilgevicius, and A. Stabinis, “Propagation of Bessel beams carrying optical vortices, “Opt. Commun. 209, 155–165 (2002).
    [CrossRef]
  10. M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. P. Woerdman, “Helical-wavefront laser beams produced with a spiral phase plate, Opt. Commun. 112, 321–327 (1994).
    [CrossRef]
  11. J. Arlt and K. Dholakia, “Generation of high-order Bessel beams by use of an axicon,” Opt. Commun. 177, 297–301(2000).
    [CrossRef]
  12. V. Jarutis, R. Paskauskas, and A. Stabinis, “Focusing of Laguerre–Gaussian beams by axicon,” Opt. Commun. 184, 105–112 (2000).
    [CrossRef]
  13. S. Chavez-Cerda, G. S. McDonald, and G. H. C. New, “Nondiffracting beams: travelling, standing, rotating and spiral waves,” Opt. Commun. 123, 225–233 (1996).
    [CrossRef]
  14. S. Chavez-Cerda, M. A. Meneses-Nava, and J. Miguel Hickmann, “Interference of traveling nondiffracting beams,” Opt. Lett. 23, 1871–1873 (1998).
    [CrossRef]
  15. T. Cizmar, V. Kollarova, X. Tsampoula, F. Gunn-Moore, Z. Bouchal, and K. Dholakia, “Generation and control of multiple Bessel beams for optical micromanipulation,” Proc. SPIE 7038, 70380Q (2008).
    [CrossRef]
  16. T. Cizmar, V. Kollarova, X. Tsampoula, F. Gunn-Moore, W. Sibbett, Z. Bouchal, and K. Dholakia, “Generation of multiple Bessel beams for a biophotonics workstation,” Opt. Express 16, 14024–14035 (2008).
    [CrossRef] [PubMed]
  17. C. Paterson and R. Smith, “Helicon waves: propagation-invariant waves in a rotating coordinate system,” Opt. Commun. 124, 131–140 (1996).
    [CrossRef]
  18. S. A. Baluyot and N. Hermosa II, “Intensity profiles and propagation of optical beams with bored helical phase,” Opt. Express 17, 16244–16254 (2009).
    [CrossRef] [PubMed]
  19. V. Jarutis, A. Matijosius, P. Di Trapani, and A. Piskarskas, “Spiraling zero-order Bessel beam,” Opt. Lett. 34, 2129–2131 (2009).
    [CrossRef] [PubMed]
  20. A. Matijosius, V. Jarutis, and A. Piskarskas, “Generation and control of the spiraling zero-order Bessel beam,” Opt. Express 18, 8767–8771 (2010).
    [CrossRef] [PubMed]
  21. S. A. Baluyot and N. Hermosa II, “Controllable rotation of optical beams with bored helical phases,” Appl. Opt. 49, 673–677 (2010).
    [CrossRef] [PubMed]
  22. L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia, “Controlled rotation of optically trapped microscopic particles,” Science 292, 912–914 (2001).
    [CrossRef] [PubMed]
  23. M. P. MacDonald, L. Paterson, K. Volke-Sepulveda, J. Arlt, W. Sibbett, and K. Dholakia, “Creation and manipulation of three-dimensional optically trapped structures,” Science 296, 1101–1103 (2002).
    [CrossRef] [PubMed]
  24. S.-H. Lee, Y. Roichman, and D. G. Grier, “Optical solenoid beams,” Opt. Express 18, 6988–6993 (2010).
    [CrossRef] [PubMed]
  25. K. W. Madison, F. Chevy, W. Wohlleben, and J. Dalibard, “Vortex formation in a stirred Bose-Einstein condensate,” Phys. Rev. Lett. 84, 806–809 (2000).
    [CrossRef] [PubMed]
  26. A. Greengard, Y. Y. Schechner, and R. Piestun, “Depth from diffracted rotation,” Opt. Lett. 31, 181–183 (2006).
    [CrossRef] [PubMed]
  27. S. R. P. Pavani, A. Greengard, and R. Piestun, “Three-dimensional localization with nanometer accuracy using a detector-limited double-helix point spread function system,” Appl. Phys. Lett. 95, 021103 (2009).
    [CrossRef]
  28. S. R. P. Pavani, M. A. Thompson, J. S. Biteen, S. J. Lord, N. Liu, R. J. Twieg, R. Piestun, and W. E. Moerner, “Three-dimensional, single-molecule fluorescence imaging beyond the diffraction limit by using a double-helix point spread function,” Proc. Natl. Acad. Sci. USA 106, 2995–2999 (2009).
    [CrossRef] [PubMed]
  29. C. Rotschild, M. Saraf, A. Barak, R. A. El-Ganainy, E. Lifshitz, D. N. Christodoulides, and M. Segev, “Complex nonlinear opto-fluidity,” in Conference on Lasers and Electro-Optics/Quantum Electronics and Laser Science Conference and Photonic Applications Systems Technologies, OSA Technical Digest (CD) (2008), paper CPDA4.
  30. T. Carmon, R. Uzdin, C. Pigier, Z. H. Musslimani, M. Segev, and A. Nepomnyashchy, “Rotating propeller solitons,” Phys. Rev. Lett. 87, 143901 (2001).
    [CrossRef] [PubMed]
  31. A. S. Desyatnikov and Y. S. Kivshar, “Rotating optical soliton clusters,” Phys. Rev. Lett. 88, 053901 (2002).
    [CrossRef] [PubMed]
  32. A. S. Desyatnikov, D. Buccoliero, M. R. Dennis, and Y. S. Kivshar, “Suppression of collapse for spiraling elliptic solitons,” Phys. Rev. Lett. 104, 053902 (2010).
    [CrossRef] [PubMed]
  33. A. Ya. Bekshaev, M. S. Soskin, and M. V. Vasnetsov, “Angular momentum of a rotating light beam,” Opt. Commun. 249, 367–378 (2005).
    [CrossRef]
  34. M. Pitchumani, H. Hockel, W. Mohammed, and E. G. Johnson, “Additive lithography for fabrication of diffractive optics,” Appl. Opt. 41, 6176–6181 (2002).
    [CrossRef] [PubMed]

2010 (4)

2009 (4)

S. R. P. Pavani, A. Greengard, and R. Piestun, “Three-dimensional localization with nanometer accuracy using a detector-limited double-helix point spread function system,” Appl. Phys. Lett. 95, 021103 (2009).
[CrossRef]

S. R. P. Pavani, M. A. Thompson, J. S. Biteen, S. J. Lord, N. Liu, R. J. Twieg, R. Piestun, and W. E. Moerner, “Three-dimensional, single-molecule fluorescence imaging beyond the diffraction limit by using a double-helix point spread function,” Proc. Natl. Acad. Sci. USA 106, 2995–2999 (2009).
[CrossRef] [PubMed]

S. A. Baluyot and N. Hermosa II, “Intensity profiles and propagation of optical beams with bored helical phase,” Opt. Express 17, 16244–16254 (2009).
[CrossRef] [PubMed]

V. Jarutis, A. Matijosius, P. Di Trapani, and A. Piskarskas, “Spiraling zero-order Bessel beam,” Opt. Lett. 34, 2129–2131 (2009).
[CrossRef] [PubMed]

2008 (2)

T. Cizmar, V. Kollarova, X. Tsampoula, F. Gunn-Moore, Z. Bouchal, and K. Dholakia, “Generation and control of multiple Bessel beams for optical micromanipulation,” Proc. SPIE 7038, 70380Q (2008).
[CrossRef]

T. Cizmar, V. Kollarova, X. Tsampoula, F. Gunn-Moore, W. Sibbett, Z. Bouchal, and K. Dholakia, “Generation of multiple Bessel beams for a biophotonics workstation,” Opt. Express 16, 14024–14035 (2008).
[CrossRef] [PubMed]

2006 (1)

2005 (1)

A. Ya. Bekshaev, M. S. Soskin, and M. V. Vasnetsov, “Angular momentum of a rotating light beam,” Opt. Commun. 249, 367–378 (2005).
[CrossRef]

2002 (4)

M. Pitchumani, H. Hockel, W. Mohammed, and E. G. Johnson, “Additive lithography for fabrication of diffractive optics,” Appl. Opt. 41, 6176–6181 (2002).
[CrossRef] [PubMed]

A. S. Desyatnikov and Y. S. Kivshar, “Rotating optical soliton clusters,” Phys. Rev. Lett. 88, 053901 (2002).
[CrossRef] [PubMed]

M. P. MacDonald, L. Paterson, K. Volke-Sepulveda, J. Arlt, W. Sibbett, and K. Dholakia, “Creation and manipulation of three-dimensional optically trapped structures,” Science 296, 1101–1103 (2002).
[CrossRef] [PubMed]

S. Orlov, K. Regelskis, V. Smilgevicius, and A. Stabinis, “Propagation of Bessel beams carrying optical vortices, “Opt. Commun. 209, 155–165 (2002).
[CrossRef]

2001 (2)

L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia, “Controlled rotation of optically trapped microscopic particles,” Science 292, 912–914 (2001).
[CrossRef] [PubMed]

T. Carmon, R. Uzdin, C. Pigier, Z. H. Musslimani, M. Segev, and A. Nepomnyashchy, “Rotating propeller solitons,” Phys. Rev. Lett. 87, 143901 (2001).
[CrossRef] [PubMed]

2000 (4)

K. W. Madison, F. Chevy, W. Wohlleben, and J. Dalibard, “Vortex formation in a stirred Bose-Einstein condensate,” Phys. Rev. Lett. 84, 806–809 (2000).
[CrossRef] [PubMed]

J. Arlt and K. Dholakia, “Generation of high-order Bessel beams by use of an axicon,” Opt. Commun. 177, 297–301(2000).
[CrossRef]

V. Jarutis, R. Paskauskas, and A. Stabinis, “Focusing of Laguerre–Gaussian beams by axicon,” Opt. Commun. 184, 105–112 (2000).
[CrossRef]

J. Pu, H. Zhang, and S. Nemoto, “Lens axicons illuminated by Gaussian beams for generation of uniform-axial intensity Bessel fields,” Opt. Eng. 39, 803–807 (2000).
[CrossRef]

1999 (1)

K. Ait-Ameur and F. Sanchez, “Gaussian beam conversion using an axicon,” J. Mod. Opt. 46, 1537–1548 (1999).
[CrossRef]

1998 (2)

1996 (2)

S. Chavez-Cerda, G. S. McDonald, and G. H. C. New, “Nondiffracting beams: travelling, standing, rotating and spiral waves,” Opt. Commun. 123, 225–233 (1996).
[CrossRef]

C. Paterson and R. Smith, “Helicon waves: propagation-invariant waves in a rotating coordinate system,” Opt. Commun. 124, 131–140 (1996).
[CrossRef]

1994 (1)

M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. P. Woerdman, “Helical-wavefront laser beams produced with a spiral phase plate, Opt. Commun. 112, 321–327 (1994).
[CrossRef]

1988 (1)

1987 (2)

J. Durnin and J. J. Miceli, Jr., “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[CrossRef] [PubMed]

F. Gori, G. Guattari, and C. Padovani, “Bessel-Gauss beams,” Opt. Commun. 64, 491–494 (1987).
[CrossRef]

Ait-Ameur, K.

K. Ait-Ameur and F. Sanchez, “Gaussian beam conversion using an axicon,” J. Mod. Opt. 46, 1537–1548 (1999).
[CrossRef]

Arlt, J.

M. P. MacDonald, L. Paterson, K. Volke-Sepulveda, J. Arlt, W. Sibbett, and K. Dholakia, “Creation and manipulation of three-dimensional optically trapped structures,” Science 296, 1101–1103 (2002).
[CrossRef] [PubMed]

L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia, “Controlled rotation of optically trapped microscopic particles,” Science 292, 912–914 (2001).
[CrossRef] [PubMed]

J. Arlt and K. Dholakia, “Generation of high-order Bessel beams by use of an axicon,” Opt. Commun. 177, 297–301(2000).
[CrossRef]

Arroyo Carrasco, M. L.

M. M. Mendez Otero, G. C. Martinez Jimenez, M. L. Arroyo Carrasco, M. D. Iturbe Catillo, and E. Marti Panameno, “Generation of Bessel-Gauss beams by means of computer-generated holograms for Bessel beams,” in Frontiers in Optics, OSA Technical Digest (CD) (Optical Society of America, 2006), paper JWD129.

Baluyot, S. A.

Barak, A.

C. Rotschild, M. Saraf, A. Barak, R. A. El-Ganainy, E. Lifshitz, D. N. Christodoulides, and M. Segev, “Complex nonlinear opto-fluidity,” in Conference on Lasers and Electro-Optics/Quantum Electronics and Laser Science Conference and Photonic Applications Systems Technologies, OSA Technical Digest (CD) (2008), paper CPDA4.

Beijersbergen, M. W.

M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. P. Woerdman, “Helical-wavefront laser beams produced with a spiral phase plate, Opt. Commun. 112, 321–327 (1994).
[CrossRef]

Bekshaev, A. Ya.

A. Ya. Bekshaev, M. S. Soskin, and M. V. Vasnetsov, “Angular momentum of a rotating light beam,” Opt. Commun. 249, 367–378 (2005).
[CrossRef]

Biteen, J. S.

S. R. P. Pavani, M. A. Thompson, J. S. Biteen, S. J. Lord, N. Liu, R. J. Twieg, R. Piestun, and W. E. Moerner, “Three-dimensional, single-molecule fluorescence imaging beyond the diffraction limit by using a double-helix point spread function,” Proc. Natl. Acad. Sci. USA 106, 2995–2999 (2009).
[CrossRef] [PubMed]

Bouchal, Z.

T. Cizmar, V. Kollarova, X. Tsampoula, F. Gunn-Moore, Z. Bouchal, and K. Dholakia, “Generation and control of multiple Bessel beams for optical micromanipulation,” Proc. SPIE 7038, 70380Q (2008).
[CrossRef]

T. Cizmar, V. Kollarova, X. Tsampoula, F. Gunn-Moore, W. Sibbett, Z. Bouchal, and K. Dholakia, “Generation of multiple Bessel beams for a biophotonics workstation,” Opt. Express 16, 14024–14035 (2008).
[CrossRef] [PubMed]

Bryant, P. E.

L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia, “Controlled rotation of optically trapped microscopic particles,” Science 292, 912–914 (2001).
[CrossRef] [PubMed]

Buccoliero, D.

A. S. Desyatnikov, D. Buccoliero, M. R. Dennis, and Y. S. Kivshar, “Suppression of collapse for spiraling elliptic solitons,” Phys. Rev. Lett. 104, 053902 (2010).
[CrossRef] [PubMed]

Carmon, T.

T. Carmon, R. Uzdin, C. Pigier, Z. H. Musslimani, M. Segev, and A. Nepomnyashchy, “Rotating propeller solitons,” Phys. Rev. Lett. 87, 143901 (2001).
[CrossRef] [PubMed]

Chavez-Cerda, S.

S. Chavez-Cerda, M. A. Meneses-Nava, and J. Miguel Hickmann, “Interference of traveling nondiffracting beams,” Opt. Lett. 23, 1871–1873 (1998).
[CrossRef]

S. Chavez-Cerda, G. S. McDonald, and G. H. C. New, “Nondiffracting beams: travelling, standing, rotating and spiral waves,” Opt. Commun. 123, 225–233 (1996).
[CrossRef]

Chen, N.-X.

Chevy, F.

K. W. Madison, F. Chevy, W. Wohlleben, and J. Dalibard, “Vortex formation in a stirred Bose-Einstein condensate,” Phys. Rev. Lett. 84, 806–809 (2000).
[CrossRef] [PubMed]

Christodoulides, D. N.

C. Rotschild, M. Saraf, A. Barak, R. A. El-Ganainy, E. Lifshitz, D. N. Christodoulides, and M. Segev, “Complex nonlinear opto-fluidity,” in Conference on Lasers and Electro-Optics/Quantum Electronics and Laser Science Conference and Photonic Applications Systems Technologies, OSA Technical Digest (CD) (2008), paper CPDA4.

Cizmar, T.

T. Cizmar, V. Kollarova, X. Tsampoula, F. Gunn-Moore, W. Sibbett, Z. Bouchal, and K. Dholakia, “Generation of multiple Bessel beams for a biophotonics workstation,” Opt. Express 16, 14024–14035 (2008).
[CrossRef] [PubMed]

T. Cizmar, V. Kollarova, X. Tsampoula, F. Gunn-Moore, Z. Bouchal, and K. Dholakia, “Generation and control of multiple Bessel beams for optical micromanipulation,” Proc. SPIE 7038, 70380Q (2008).
[CrossRef]

Coerwinkel, R. P. C.

M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. P. Woerdman, “Helical-wavefront laser beams produced with a spiral phase plate, Opt. Commun. 112, 321–327 (1994).
[CrossRef]

Cong, W.-X.

Dalibard, J.

K. W. Madison, F. Chevy, W. Wohlleben, and J. Dalibard, “Vortex formation in a stirred Bose-Einstein condensate,” Phys. Rev. Lett. 84, 806–809 (2000).
[CrossRef] [PubMed]

Dennis, M. R.

A. S. Desyatnikov, D. Buccoliero, M. R. Dennis, and Y. S. Kivshar, “Suppression of collapse for spiraling elliptic solitons,” Phys. Rev. Lett. 104, 053902 (2010).
[CrossRef] [PubMed]

Desyatnikov, A. S.

A. S. Desyatnikov, D. Buccoliero, M. R. Dennis, and Y. S. Kivshar, “Suppression of collapse for spiraling elliptic solitons,” Phys. Rev. Lett. 104, 053902 (2010).
[CrossRef] [PubMed]

A. S. Desyatnikov and Y. S. Kivshar, “Rotating optical soliton clusters,” Phys. Rev. Lett. 88, 053901 (2002).
[CrossRef] [PubMed]

Dholakia, K.

T. Cizmar, V. Kollarova, X. Tsampoula, F. Gunn-Moore, W. Sibbett, Z. Bouchal, and K. Dholakia, “Generation of multiple Bessel beams for a biophotonics workstation,” Opt. Express 16, 14024–14035 (2008).
[CrossRef] [PubMed]

T. Cizmar, V. Kollarova, X. Tsampoula, F. Gunn-Moore, Z. Bouchal, and K. Dholakia, “Generation and control of multiple Bessel beams for optical micromanipulation,” Proc. SPIE 7038, 70380Q (2008).
[CrossRef]

M. P. MacDonald, L. Paterson, K. Volke-Sepulveda, J. Arlt, W. Sibbett, and K. Dholakia, “Creation and manipulation of three-dimensional optically trapped structures,” Science 296, 1101–1103 (2002).
[CrossRef] [PubMed]

L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia, “Controlled rotation of optically trapped microscopic particles,” Science 292, 912–914 (2001).
[CrossRef] [PubMed]

J. Arlt and K. Dholakia, “Generation of high-order Bessel beams by use of an axicon,” Opt. Commun. 177, 297–301(2000).
[CrossRef]

Di Trapani, P.

Dudley, A.

R. Vasilyeu, A. Dudley, N. Khilo, and A. Forbes, “Generating superpositions of higher order Bessel beams,” in Frontiers in Optics, OSA Technical Digest (CD) (Optical Society of America, 2009), paper FThB6.

Durnin, J.

J. Durnin and J. J. Miceli, Jr., “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[CrossRef] [PubMed]

El-Ganainy, R. A.

C. Rotschild, M. Saraf, A. Barak, R. A. El-Ganainy, E. Lifshitz, D. N. Christodoulides, and M. Segev, “Complex nonlinear opto-fluidity,” in Conference on Lasers and Electro-Optics/Quantum Electronics and Laser Science Conference and Photonic Applications Systems Technologies, OSA Technical Digest (CD) (2008), paper CPDA4.

Forbes, A.

R. Vasilyeu, A. Dudley, N. Khilo, and A. Forbes, “Generating superpositions of higher order Bessel beams,” in Frontiers in Optics, OSA Technical Digest (CD) (Optical Society of America, 2009), paper FThB6.

Friberg, A. T.

Gori, F.

F. Gori, G. Guattari, and C. Padovani, “Bessel-Gauss beams,” Opt. Commun. 64, 491–494 (1987).
[CrossRef]

Greengard, A.

S. R. P. Pavani, A. Greengard, and R. Piestun, “Three-dimensional localization with nanometer accuracy using a detector-limited double-helix point spread function system,” Appl. Phys. Lett. 95, 021103 (2009).
[CrossRef]

A. Greengard, Y. Y. Schechner, and R. Piestun, “Depth from diffracted rotation,” Opt. Lett. 31, 181–183 (2006).
[CrossRef] [PubMed]

Grier, D. G.

Guattari, G.

F. Gori, G. Guattari, and C. Padovani, “Bessel-Gauss beams,” Opt. Commun. 64, 491–494 (1987).
[CrossRef]

Gunn-Moore, F.

T. Cizmar, V. Kollarova, X. Tsampoula, F. Gunn-Moore, Z. Bouchal, and K. Dholakia, “Generation and control of multiple Bessel beams for optical micromanipulation,” Proc. SPIE 7038, 70380Q (2008).
[CrossRef]

T. Cizmar, V. Kollarova, X. Tsampoula, F. Gunn-Moore, W. Sibbett, Z. Bouchal, and K. Dholakia, “Generation of multiple Bessel beams for a biophotonics workstation,” Opt. Express 16, 14024–14035 (2008).
[CrossRef] [PubMed]

Hermosa, N.

Hockel, H.

Iturbe Catillo, M. D.

M. M. Mendez Otero, G. C. Martinez Jimenez, M. L. Arroyo Carrasco, M. D. Iturbe Catillo, and E. Marti Panameno, “Generation of Bessel-Gauss beams by means of computer-generated holograms for Bessel beams,” in Frontiers in Optics, OSA Technical Digest (CD) (Optical Society of America, 2006), paper JWD129.

Jarutis, V.

Johnson, E. G.

Khilo, N.

R. Vasilyeu, A. Dudley, N. Khilo, and A. Forbes, “Generating superpositions of higher order Bessel beams,” in Frontiers in Optics, OSA Technical Digest (CD) (Optical Society of America, 2009), paper FThB6.

Kivshar, Y. S.

A. S. Desyatnikov, D. Buccoliero, M. R. Dennis, and Y. S. Kivshar, “Suppression of collapse for spiraling elliptic solitons,” Phys. Rev. Lett. 104, 053902 (2010).
[CrossRef] [PubMed]

A. S. Desyatnikov and Y. S. Kivshar, “Rotating optical soliton clusters,” Phys. Rev. Lett. 88, 053901 (2002).
[CrossRef] [PubMed]

Kollarova, V.

T. Cizmar, V. Kollarova, X. Tsampoula, F. Gunn-Moore, W. Sibbett, Z. Bouchal, and K. Dholakia, “Generation of multiple Bessel beams for a biophotonics workstation,” Opt. Express 16, 14024–14035 (2008).
[CrossRef] [PubMed]

T. Cizmar, V. Kollarova, X. Tsampoula, F. Gunn-Moore, Z. Bouchal, and K. Dholakia, “Generation and control of multiple Bessel beams for optical micromanipulation,” Proc. SPIE 7038, 70380Q (2008).
[CrossRef]

Kristensen, M.

M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. P. Woerdman, “Helical-wavefront laser beams produced with a spiral phase plate, Opt. Commun. 112, 321–327 (1994).
[CrossRef]

Lee, S.-H.

Lifshitz, E.

C. Rotschild, M. Saraf, A. Barak, R. A. El-Ganainy, E. Lifshitz, D. N. Christodoulides, and M. Segev, “Complex nonlinear opto-fluidity,” in Conference on Lasers and Electro-Optics/Quantum Electronics and Laser Science Conference and Photonic Applications Systems Technologies, OSA Technical Digest (CD) (2008), paper CPDA4.

Liu, N.

S. R. P. Pavani, M. A. Thompson, J. S. Biteen, S. J. Lord, N. Liu, R. J. Twieg, R. Piestun, and W. E. Moerner, “Three-dimensional, single-molecule fluorescence imaging beyond the diffraction limit by using a double-helix point spread function,” Proc. Natl. Acad. Sci. USA 106, 2995–2999 (2009).
[CrossRef] [PubMed]

Lord, S. J.

S. R. P. Pavani, M. A. Thompson, J. S. Biteen, S. J. Lord, N. Liu, R. J. Twieg, R. Piestun, and W. E. Moerner, “Three-dimensional, single-molecule fluorescence imaging beyond the diffraction limit by using a double-helix point spread function,” Proc. Natl. Acad. Sci. USA 106, 2995–2999 (2009).
[CrossRef] [PubMed]

MacDonald, M. P.

M. P. MacDonald, L. Paterson, K. Volke-Sepulveda, J. Arlt, W. Sibbett, and K. Dholakia, “Creation and manipulation of three-dimensional optically trapped structures,” Science 296, 1101–1103 (2002).
[CrossRef] [PubMed]

L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia, “Controlled rotation of optically trapped microscopic particles,” Science 292, 912–914 (2001).
[CrossRef] [PubMed]

Madison, K. W.

K. W. Madison, F. Chevy, W. Wohlleben, and J. Dalibard, “Vortex formation in a stirred Bose-Einstein condensate,” Phys. Rev. Lett. 84, 806–809 (2000).
[CrossRef] [PubMed]

Marti Panameno, E.

M. M. Mendez Otero, G. C. Martinez Jimenez, M. L. Arroyo Carrasco, M. D. Iturbe Catillo, and E. Marti Panameno, “Generation of Bessel-Gauss beams by means of computer-generated holograms for Bessel beams,” in Frontiers in Optics, OSA Technical Digest (CD) (Optical Society of America, 2006), paper JWD129.

Martinez Jimenez, G. C.

M. M. Mendez Otero, G. C. Martinez Jimenez, M. L. Arroyo Carrasco, M. D. Iturbe Catillo, and E. Marti Panameno, “Generation of Bessel-Gauss beams by means of computer-generated holograms for Bessel beams,” in Frontiers in Optics, OSA Technical Digest (CD) (Optical Society of America, 2006), paper JWD129.

Matijosius, A.

McDonald, G. S.

S. Chavez-Cerda, G. S. McDonald, and G. H. C. New, “Nondiffracting beams: travelling, standing, rotating and spiral waves,” Opt. Commun. 123, 225–233 (1996).
[CrossRef]

Mendez Otero, M. M.

M. M. Mendez Otero, G. C. Martinez Jimenez, M. L. Arroyo Carrasco, M. D. Iturbe Catillo, and E. Marti Panameno, “Generation of Bessel-Gauss beams by means of computer-generated holograms for Bessel beams,” in Frontiers in Optics, OSA Technical Digest (CD) (Optical Society of America, 2006), paper JWD129.

Meneses-Nava, M. A.

Miceli, J. J.

J. Durnin and J. J. Miceli, Jr., “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[CrossRef] [PubMed]

Miguel Hickmann, J.

Moerner, W. E.

S. R. P. Pavani, M. A. Thompson, J. S. Biteen, S. J. Lord, N. Liu, R. J. Twieg, R. Piestun, and W. E. Moerner, “Three-dimensional, single-molecule fluorescence imaging beyond the diffraction limit by using a double-helix point spread function,” Proc. Natl. Acad. Sci. USA 106, 2995–2999 (2009).
[CrossRef] [PubMed]

Mohammed, W.

Musslimani, Z. H.

T. Carmon, R. Uzdin, C. Pigier, Z. H. Musslimani, M. Segev, and A. Nepomnyashchy, “Rotating propeller solitons,” Phys. Rev. Lett. 87, 143901 (2001).
[CrossRef] [PubMed]

Nemoto, S.

J. Pu, H. Zhang, and S. Nemoto, “Lens axicons illuminated by Gaussian beams for generation of uniform-axial intensity Bessel fields,” Opt. Eng. 39, 803–807 (2000).
[CrossRef]

Nepomnyashchy, A.

T. Carmon, R. Uzdin, C. Pigier, Z. H. Musslimani, M. Segev, and A. Nepomnyashchy, “Rotating propeller solitons,” Phys. Rev. Lett. 87, 143901 (2001).
[CrossRef] [PubMed]

New, G. H. C.

S. Chavez-Cerda, G. S. McDonald, and G. H. C. New, “Nondiffracting beams: travelling, standing, rotating and spiral waves,” Opt. Commun. 123, 225–233 (1996).
[CrossRef]

Orlov, S.

S. Orlov, K. Regelskis, V. Smilgevicius, and A. Stabinis, “Propagation of Bessel beams carrying optical vortices, “Opt. Commun. 209, 155–165 (2002).
[CrossRef]

Padovani, C.

F. Gori, G. Guattari, and C. Padovani, “Bessel-Gauss beams,” Opt. Commun. 64, 491–494 (1987).
[CrossRef]

Paskauskas, R.

V. Jarutis, R. Paskauskas, and A. Stabinis, “Focusing of Laguerre–Gaussian beams by axicon,” Opt. Commun. 184, 105–112 (2000).
[CrossRef]

Paterson, C.

C. Paterson and R. Smith, “Helicon waves: propagation-invariant waves in a rotating coordinate system,” Opt. Commun. 124, 131–140 (1996).
[CrossRef]

Paterson, L.

M. P. MacDonald, L. Paterson, K. Volke-Sepulveda, J. Arlt, W. Sibbett, and K. Dholakia, “Creation and manipulation of three-dimensional optically trapped structures,” Science 296, 1101–1103 (2002).
[CrossRef] [PubMed]

L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia, “Controlled rotation of optically trapped microscopic particles,” Science 292, 912–914 (2001).
[CrossRef] [PubMed]

Pavani, S. R. P.

S. R. P. Pavani, M. A. Thompson, J. S. Biteen, S. J. Lord, N. Liu, R. J. Twieg, R. Piestun, and W. E. Moerner, “Three-dimensional, single-molecule fluorescence imaging beyond the diffraction limit by using a double-helix point spread function,” Proc. Natl. Acad. Sci. USA 106, 2995–2999 (2009).
[CrossRef] [PubMed]

S. R. P. Pavani, A. Greengard, and R. Piestun, “Three-dimensional localization with nanometer accuracy using a detector-limited double-helix point spread function system,” Appl. Phys. Lett. 95, 021103 (2009).
[CrossRef]

Piestun, R.

S. R. P. Pavani, A. Greengard, and R. Piestun, “Three-dimensional localization with nanometer accuracy using a detector-limited double-helix point spread function system,” Appl. Phys. Lett. 95, 021103 (2009).
[CrossRef]

S. R. P. Pavani, M. A. Thompson, J. S. Biteen, S. J. Lord, N. Liu, R. J. Twieg, R. Piestun, and W. E. Moerner, “Three-dimensional, single-molecule fluorescence imaging beyond the diffraction limit by using a double-helix point spread function,” Proc. Natl. Acad. Sci. USA 106, 2995–2999 (2009).
[CrossRef] [PubMed]

A. Greengard, Y. Y. Schechner, and R. Piestun, “Depth from diffracted rotation,” Opt. Lett. 31, 181–183 (2006).
[CrossRef] [PubMed]

Pigier, C.

T. Carmon, R. Uzdin, C. Pigier, Z. H. Musslimani, M. Segev, and A. Nepomnyashchy, “Rotating propeller solitons,” Phys. Rev. Lett. 87, 143901 (2001).
[CrossRef] [PubMed]

Piskarskas, A.

Pitchumani, M.

Pu, J.

J. Pu, H. Zhang, and S. Nemoto, “Lens axicons illuminated by Gaussian beams for generation of uniform-axial intensity Bessel fields,” Opt. Eng. 39, 803–807 (2000).
[CrossRef]

Regelskis, K.

S. Orlov, K. Regelskis, V. Smilgevicius, and A. Stabinis, “Propagation of Bessel beams carrying optical vortices, “Opt. Commun. 209, 155–165 (2002).
[CrossRef]

Roichman, Y.

Rotschild, C.

C. Rotschild, M. Saraf, A. Barak, R. A. El-Ganainy, E. Lifshitz, D. N. Christodoulides, and M. Segev, “Complex nonlinear opto-fluidity,” in Conference on Lasers and Electro-Optics/Quantum Electronics and Laser Science Conference and Photonic Applications Systems Technologies, OSA Technical Digest (CD) (2008), paper CPDA4.

Sanchez, F.

K. Ait-Ameur and F. Sanchez, “Gaussian beam conversion using an axicon,” J. Mod. Opt. 46, 1537–1548 (1999).
[CrossRef]

Saraf, M.

C. Rotschild, M. Saraf, A. Barak, R. A. El-Ganainy, E. Lifshitz, D. N. Christodoulides, and M. Segev, “Complex nonlinear opto-fluidity,” in Conference on Lasers and Electro-Optics/Quantum Electronics and Laser Science Conference and Photonic Applications Systems Technologies, OSA Technical Digest (CD) (2008), paper CPDA4.

Schechner, Y. Y.

Segev, M.

T. Carmon, R. Uzdin, C. Pigier, Z. H. Musslimani, M. Segev, and A. Nepomnyashchy, “Rotating propeller solitons,” Phys. Rev. Lett. 87, 143901 (2001).
[CrossRef] [PubMed]

C. Rotschild, M. Saraf, A. Barak, R. A. El-Ganainy, E. Lifshitz, D. N. Christodoulides, and M. Segev, “Complex nonlinear opto-fluidity,” in Conference on Lasers and Electro-Optics/Quantum Electronics and Laser Science Conference and Photonic Applications Systems Technologies, OSA Technical Digest (CD) (2008), paper CPDA4.

Sibbett, W.

T. Cizmar, V. Kollarova, X. Tsampoula, F. Gunn-Moore, W. Sibbett, Z. Bouchal, and K. Dholakia, “Generation of multiple Bessel beams for a biophotonics workstation,” Opt. Express 16, 14024–14035 (2008).
[CrossRef] [PubMed]

M. P. MacDonald, L. Paterson, K. Volke-Sepulveda, J. Arlt, W. Sibbett, and K. Dholakia, “Creation and manipulation of three-dimensional optically trapped structures,” Science 296, 1101–1103 (2002).
[CrossRef] [PubMed]

L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia, “Controlled rotation of optically trapped microscopic particles,” Science 292, 912–914 (2001).
[CrossRef] [PubMed]

Smilgevicius, V.

S. Orlov, K. Regelskis, V. Smilgevicius, and A. Stabinis, “Propagation of Bessel beams carrying optical vortices, “Opt. Commun. 209, 155–165 (2002).
[CrossRef]

Smith, R.

C. Paterson and R. Smith, “Helicon waves: propagation-invariant waves in a rotating coordinate system,” Opt. Commun. 124, 131–140 (1996).
[CrossRef]

Soskin, M. S.

A. Ya. Bekshaev, M. S. Soskin, and M. V. Vasnetsov, “Angular momentum of a rotating light beam,” Opt. Commun. 249, 367–378 (2005).
[CrossRef]

Stabinis, A.

S. Orlov, K. Regelskis, V. Smilgevicius, and A. Stabinis, “Propagation of Bessel beams carrying optical vortices, “Opt. Commun. 209, 155–165 (2002).
[CrossRef]

V. Jarutis, R. Paskauskas, and A. Stabinis, “Focusing of Laguerre–Gaussian beams by axicon,” Opt. Commun. 184, 105–112 (2000).
[CrossRef]

Thompson, M. A.

S. R. P. Pavani, M. A. Thompson, J. S. Biteen, S. J. Lord, N. Liu, R. J. Twieg, R. Piestun, and W. E. Moerner, “Three-dimensional, single-molecule fluorescence imaging beyond the diffraction limit by using a double-helix point spread function,” Proc. Natl. Acad. Sci. USA 106, 2995–2999 (2009).
[CrossRef] [PubMed]

Tsampoula, X.

T. Cizmar, V. Kollarova, X. Tsampoula, F. Gunn-Moore, Z. Bouchal, and K. Dholakia, “Generation and control of multiple Bessel beams for optical micromanipulation,” Proc. SPIE 7038, 70380Q (2008).
[CrossRef]

T. Cizmar, V. Kollarova, X. Tsampoula, F. Gunn-Moore, W. Sibbett, Z. Bouchal, and K. Dholakia, “Generation of multiple Bessel beams for a biophotonics workstation,” Opt. Express 16, 14024–14035 (2008).
[CrossRef] [PubMed]

Turunen, J.

Twieg, R. J.

S. R. P. Pavani, M. A. Thompson, J. S. Biteen, S. J. Lord, N. Liu, R. J. Twieg, R. Piestun, and W. E. Moerner, “Three-dimensional, single-molecule fluorescence imaging beyond the diffraction limit by using a double-helix point spread function,” Proc. Natl. Acad. Sci. USA 106, 2995–2999 (2009).
[CrossRef] [PubMed]

Uzdin, R.

T. Carmon, R. Uzdin, C. Pigier, Z. H. Musslimani, M. Segev, and A. Nepomnyashchy, “Rotating propeller solitons,” Phys. Rev. Lett. 87, 143901 (2001).
[CrossRef] [PubMed]

Vasara, A.

Vasilyeu, R.

R. Vasilyeu, A. Dudley, N. Khilo, and A. Forbes, “Generating superpositions of higher order Bessel beams,” in Frontiers in Optics, OSA Technical Digest (CD) (Optical Society of America, 2009), paper FThB6.

Vasnetsov, M. V.

A. Ya. Bekshaev, M. S. Soskin, and M. V. Vasnetsov, “Angular momentum of a rotating light beam,” Opt. Commun. 249, 367–378 (2005).
[CrossRef]

Volke-Sepulveda, K.

M. P. MacDonald, L. Paterson, K. Volke-Sepulveda, J. Arlt, W. Sibbett, and K. Dholakia, “Creation and manipulation of three-dimensional optically trapped structures,” Science 296, 1101–1103 (2002).
[CrossRef] [PubMed]

Woerdman, J. P.

M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. P. Woerdman, “Helical-wavefront laser beams produced with a spiral phase plate, Opt. Commun. 112, 321–327 (1994).
[CrossRef]

Wohlleben, W.

K. W. Madison, F. Chevy, W. Wohlleben, and J. Dalibard, “Vortex formation in a stirred Bose-Einstein condensate,” Phys. Rev. Lett. 84, 806–809 (2000).
[CrossRef] [PubMed]

Zhang, H.

J. Pu, H. Zhang, and S. Nemoto, “Lens axicons illuminated by Gaussian beams for generation of uniform-axial intensity Bessel fields,” Opt. Eng. 39, 803–807 (2000).
[CrossRef]

Appl. Opt. (3)

Appl. Phys. Lett. (1)

S. R. P. Pavani, A. Greengard, and R. Piestun, “Three-dimensional localization with nanometer accuracy using a detector-limited double-helix point spread function system,” Appl. Phys. Lett. 95, 021103 (2009).
[CrossRef]

J. Mod. Opt. (1)

K. Ait-Ameur and F. Sanchez, “Gaussian beam conversion using an axicon,” J. Mod. Opt. 46, 1537–1548 (1999).
[CrossRef]

J. Opt. Soc. Am. A (1)

Opt. Commun. (8)

S. Orlov, K. Regelskis, V. Smilgevicius, and A. Stabinis, “Propagation of Bessel beams carrying optical vortices, “Opt. Commun. 209, 155–165 (2002).
[CrossRef]

M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. P. Woerdman, “Helical-wavefront laser beams produced with a spiral phase plate, Opt. Commun. 112, 321–327 (1994).
[CrossRef]

J. Arlt and K. Dholakia, “Generation of high-order Bessel beams by use of an axicon,” Opt. Commun. 177, 297–301(2000).
[CrossRef]

V. Jarutis, R. Paskauskas, and A. Stabinis, “Focusing of Laguerre–Gaussian beams by axicon,” Opt. Commun. 184, 105–112 (2000).
[CrossRef]

S. Chavez-Cerda, G. S. McDonald, and G. H. C. New, “Nondiffracting beams: travelling, standing, rotating and spiral waves,” Opt. Commun. 123, 225–233 (1996).
[CrossRef]

F. Gori, G. Guattari, and C. Padovani, “Bessel-Gauss beams,” Opt. Commun. 64, 491–494 (1987).
[CrossRef]

C. Paterson and R. Smith, “Helicon waves: propagation-invariant waves in a rotating coordinate system,” Opt. Commun. 124, 131–140 (1996).
[CrossRef]

A. Ya. Bekshaev, M. S. Soskin, and M. V. Vasnetsov, “Angular momentum of a rotating light beam,” Opt. Commun. 249, 367–378 (2005).
[CrossRef]

Opt. Eng. (1)

J. Pu, H. Zhang, and S. Nemoto, “Lens axicons illuminated by Gaussian beams for generation of uniform-axial intensity Bessel fields,” Opt. Eng. 39, 803–807 (2000).
[CrossRef]

Opt. Express (4)

Opt. Lett. (3)

Phys. Rev. Lett. (5)

J. Durnin and J. J. Miceli, Jr., “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[CrossRef] [PubMed]

T. Carmon, R. Uzdin, C. Pigier, Z. H. Musslimani, M. Segev, and A. Nepomnyashchy, “Rotating propeller solitons,” Phys. Rev. Lett. 87, 143901 (2001).
[CrossRef] [PubMed]

A. S. Desyatnikov and Y. S. Kivshar, “Rotating optical soliton clusters,” Phys. Rev. Lett. 88, 053901 (2002).
[CrossRef] [PubMed]

A. S. Desyatnikov, D. Buccoliero, M. R. Dennis, and Y. S. Kivshar, “Suppression of collapse for spiraling elliptic solitons,” Phys. Rev. Lett. 104, 053902 (2010).
[CrossRef] [PubMed]

K. W. Madison, F. Chevy, W. Wohlleben, and J. Dalibard, “Vortex formation in a stirred Bose-Einstein condensate,” Phys. Rev. Lett. 84, 806–809 (2000).
[CrossRef] [PubMed]

Proc. Natl. Acad. Sci. USA (1)

S. R. P. Pavani, M. A. Thompson, J. S. Biteen, S. J. Lord, N. Liu, R. J. Twieg, R. Piestun, and W. E. Moerner, “Three-dimensional, single-molecule fluorescence imaging beyond the diffraction limit by using a double-helix point spread function,” Proc. Natl. Acad. Sci. USA 106, 2995–2999 (2009).
[CrossRef] [PubMed]

Proc. SPIE (1)

T. Cizmar, V. Kollarova, X. Tsampoula, F. Gunn-Moore, Z. Bouchal, and K. Dholakia, “Generation and control of multiple Bessel beams for optical micromanipulation,” Proc. SPIE 7038, 70380Q (2008).
[CrossRef]

Science (2)

L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia, “Controlled rotation of optically trapped microscopic particles,” Science 292, 912–914 (2001).
[CrossRef] [PubMed]

M. P. MacDonald, L. Paterson, K. Volke-Sepulveda, J. Arlt, W. Sibbett, and K. Dholakia, “Creation and manipulation of three-dimensional optically trapped structures,” Science 296, 1101–1103 (2002).
[CrossRef] [PubMed]

Other (3)

C. Rotschild, M. Saraf, A. Barak, R. A. El-Ganainy, E. Lifshitz, D. N. Christodoulides, and M. Segev, “Complex nonlinear opto-fluidity,” in Conference on Lasers and Electro-Optics/Quantum Electronics and Laser Science Conference and Photonic Applications Systems Technologies, OSA Technical Digest (CD) (2008), paper CPDA4.

R. Vasilyeu, A. Dudley, N. Khilo, and A. Forbes, “Generating superpositions of higher order Bessel beams,” in Frontiers in Optics, OSA Technical Digest (CD) (Optical Society of America, 2009), paper FThB6.

M. M. Mendez Otero, G. C. Martinez Jimenez, M. L. Arroyo Carrasco, M. D. Iturbe Catillo, and E. Marti Panameno, “Generation of Bessel-Gauss beams by means of computer-generated holograms for Bessel beams,” in Frontiers in Optics, OSA Technical Digest (CD) (Optical Society of America, 2006), paper JWD129.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (6)

Fig. 1
Fig. 1

Serial optical configuration for helical beam synthesis.

Fig. 2
Fig. 2

Beam profile generated from the superposition of two first-order Bessel beams.

Fig. 3
Fig. 3

Simulation flowchart.

Fig. 4
Fig. 4

Experimental setup for the generation of helical beams.

Fig. 5
Fig. 5

Transverse beam profiles of a helical beam at various points along the optical axis. The dot indicates rotation angle. Top, obtained from MATLAB simulation. Middle, obtained from experiments using continuous light. Bottom, obtained from experiments using pulses.

Fig. 6
Fig. 6

Rotation of the irradiance peaks of the double helical beam (see Fig. 5) as a function of propagation distance.

Equations (23)

Equations on this page are rendered with MathJax. Learn more.

U 2 ( r 2 , φ 2 ) = exp ( i k 0 z ) i λ 0 z exp [ i k 0 r 2 2 2 z ] · 0 d r 1 0 2 π d φ 1 [ U 1 ( r 1 , φ 1 ) r 1 exp ( i k 0 r 1 2 2 z ) exp ( i k 0 r 1 r 2 z cos ( φ 2 φ 1 ) ) ] ,
U inner ( r , φ ) = exp ( i φ ) × exp ( i k 0 β f 1 r ) × exp ( i k 0 β s ) = exp ( i φ ) exp ( i k 0 β 1 r ) ,
U outer ( r , φ ) = exp ( i φ ) × exp ( i k 0 β f 1 r ) × exp ( i k 0 β s r ) = exp ( i φ ) exp ( i k 0 β 1 r ) ,
U ( r , φ ) = I 0 exp ( r 2 w 0 2 ) exp ( i k 0 r 2 2 f ) [ u ( r r i ) U outer + u ( r r i ) U inner ] = I 0 exp ( r 2 w 0 2 ) exp ( i k 0 r 2 2 f ) [ u ( r r i ) exp ( i φ ) exp ( i k 0 β 1 r ) + u ( r i r ) e i φ exp ( i k 0 β 1 r ) ] ,
exp [ i k 0 r 1 r 2 z cos ( φ 2 φ 1 ) ] = m = ( i ) m J m ( k 0 r 1 r 2 z ) exp ( i m φ 1 ) exp ( i m φ 2 ) ,
1 2 π 0 2 π d r 1 m = exp ( i m φ 1 ) exp ( i m φ 2 ) [ f 1 ( r 1 , r 2 ) exp ( i φ 1 ) + f 1 ( r 1 , r 2 ) exp ( i φ 1 ) ] = f 1 ( r 1 , r 2 ) exp ( i φ 2 ) + f 1 ( r 1 , r 2 ) exp ( i φ 2 ) .
U 2 ( r 2 , φ 2 ) = I 0 2 π exp ( i k 0 z ) λ 0 z exp [ i k 0 r 2 2 2 z ] 0 d r 1 r 1 exp ( r 1 2 w 0 2 ) exp [ i k 0 r 1 2 2 ( 1 z 1 f ) ] G ( r 1 , r 2 , φ 2 ) , G = u ( r 1 r i ) J 1 ( k 0 r 1 r 2 z ) exp ( i k 0 β 1 r 1 ) exp ( i φ 2 ) u ( r i r 1 ) J 1 ( k 0 r 1 r 2 z ) exp ( i k 0 β 1 r 1 ) exp ( i φ 2 ) .
exp [ i ( k 0 r 1 2 2 z ( 1 z f ) k 0 β x r 1 ) ] = exp [ i φ ( r 1 ) ] .
φ r 1 = k 0 r 1 z ( 1 z f ) k 0 β x = 0 , r 1 = β x z ( 1 z f ) 1 = F β x z ,
F = ( 1 z f ) 1
U 2 ( r 2 , φ 2 ) = I 0 2 π   exp ( i k 0 z ) λ 0 exp [ i k 0 r 2 2 2 z ] [ G 1 ( r 2 , z ) exp ( i 2 ) + G 1 ( r 2 , z ) exp ( i φ 2 ) ] , G 1 ( r 2 , z ) = β 1 u ( F β 1 z r i ) exp ( F 2 β 1 2 z 2 w 0 2 ) exp ( i F β 1 2 k 0 z 2 ) J 1 ( F β 1 k 0 r 2 ) 0 d r 1 exp [ i 2 2 1 2 ( r 1 F β 1 z ) 2 ] , G 1 ( r 2 , z ) = β 1 u ( r i F β 1 ) exp ( F 2 β 1 2 z 2 w 0 2 ) exp ( i F β 1 2 k 0 z 2 ) J 1 ( F β 1 k 0 r 2 ) 0 d r 1 exp [ i 2 2 r 1 2 ( r 1 F β 1 z ) 2 ] ,
0 d r exp [ i 2 2 r 1 2 ( r 1 F β x z ) 2 ] = 0 d r exp [ i k 0 2 F z ( r 1 F β x z ) 2 ] = F λ 0 z 2 exp ( i π 4 ) .
U ( r , φ ) = U 1 ( r ) exp ( i φ ) + U 1 ( r ) exp ( i φ ) ,
U 1 ( r ) = I 0 e i 3 π 4 k 1 u ( z z 1 ) F λ 0 z 2 exp ( z 2 l d f 1 2 ) J 1 ( F k 1 r ) exp ( i k 0 2 z r 2 ) exp [ i ( k 0 F k 1 2 2 k 0 ) z ] , U 1 ( r ) = I 0 e i 3 π 4 k 1 u ( z 1 z ) F λ 0 z 2 exp ( z 2 l d f 1 2 ) J 1 ( F k 1 r ) exp ( i k 0 2 z r 2 ) exp [ i ( k 0 F k ( 1 ) 2 2 k 0 ) z ] ,
z 1 = r i F β 1 , z 1 = r i F β 1 ,
z 1 = r i β 1 + r i f , z 1 = r i β 1 + r i f .
J 1 ( F k 1 r ) k 1 exp ( z 2 l d f 1 2 ) = J 1 ( F k 1 r ) k 1 exp ( z 2 l d f 1 2 ) + [ J 1 ( F k 1 r ) k 1 exp ( z 2 l d f 1 2 ) J 1 ( F k 1 r ) k 1 exp ( z 2 l d f 1 2 ) ] = J 1 ( F k 1 r ) k 1 exp ( z 2 l d f 1 2 ) + B ( r , z ) ,
exp [ i ( k 0 F k 1 2 2 k 0 ) z ] exp ( i φ ) + exp [ i ( k 0 F k 1 2 2 k 0 ) z ] exp ( i φ ) = 2 cos ( φ F k 1 2 k 1 2 4 k 0 ) exp [ i ( k 0 F k 1 2 k 1 2 4 k 0 ) ] ,
U ( r ) = I 0 exp ( i 3 π 4 ) k 1 F λ 0 z exp ( z 2 l d f 1 2 ) J 1 ( F k 1 r ) cos ( φ F k 1 2 k 1 2 4 k 0 z ) exp ( i k 0 2 z r 2 ) exp [ i ( k 0 F k 1 2 + k 1 2 4 k 0 ) z ] + I 0 exp ( i 3 π 4 ) F λ 0 z 2 B ( r , z ) exp ( i φ ) exp ( i k 0 2 z r 2 ) exp [ i ( k 0 F k 1 2 2 k 0 ) z ] ,
I ( r ) = I 0 F λ 0 z k 1 2 exp ( 2 z 2 l d f 1 2 ) J 1 2 ( F k 1 r ) cos 2 ( φ F k 1 2 k 1 2 4 k 0 z ) + I 0 4 F λ 0 B 2 ( r , z ) + I 0 F λ 0 z k 1 B ( r , z ) exp ( z 2 l d f 1 2 ) J 1 ( F k 1 r ) cos 2 ( φ F k 1 2 k 1 2 4 k 0 z ) = C ( z ) J 1 2 ( F k 1 r ) cos 2 [ θ ( φ , z ) ] + D ( r , z ) J 1 ( F k 1 r ) cos 2 [ θ ( φ , z ) ] + E ( r , z ) ,
θ ( φ , z ) = φ F k 1 2 k 1 2 4 k 0 z .
θ z = [ F ( 1 + z f F ) k 1 2 k 1 2 4 k 0 ] = π 2 F ( 1 + z f F ) ( β 1 2 β 1 2 λ 0 ) .
d max = 3.8317 F k = 1.22 β 1 + β 1 λ 0 F .

Metrics