Abstract

We deduce and study an analytical expression for Fresnel diffraction of a plane wave by a spiral phase plate (SPP) that imparts an arbitrary-order phase singularity on the light field. Estimates for the optical vortex radius that depends on the singularity’s integer order n (also termed topological charge, or order of the dislocation) have been derived. The near-zero vortex intensity is shown to be proportional to ρ2n, where ρ is the radial coordinate. Also, an analytical expression for Fresnel diffraction of the Gaussian beam by a SPP with nth-order singularity is analyzed. The far-field intensity distribution is derived. The radius of maximal intensity is shown to depend on the singularity number. The behavior of the Gaussian beam intensity after a SPP with second-order singularity (n=2) is studied in more detail. The parameters of the light beams generated numerically with the Fresnel transform and via analytical formulas are in good agreement. In addition, the light fields with first- and second-order singularities were generated by a 32-level SPP fabricated on the resist by use of the electron-beam lithography technique.

© 2005 Optical Society of America

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References

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  1. J. F. Nye, M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. London Ser. A 336, 165–190 (1974).
    [CrossRef]
  2. S. N. Khonina, V. V. Kotlyar, M. V. Shinkaryev, V. A. Soifer, G. V. Uspleniev, “The phase rotor filter,” J. Mod. Opt. 39, 1147–1154 (1992).
    [CrossRef]
  3. V. Yu. Bazhenov, M. S. Soskin, M. V. Vasnetsov, “Screw dislocations in light wavefronts,” J. Mod. Opt. 39, 985–990 (1992).
    [CrossRef]
  4. I. V. Basistiy, V. Yu. Bazhenov, M. S. Soskin, M. V. Vasnetsov, “Optics of light beams with screw dislocations,” Opt. Commun. 103, 422–428 (1993).
    [CrossRef]
  5. G. Indebetouw, “Optical vortices and their propagation,” J. Mod. Opt. 40, 73–87 (1993).
    [CrossRef]
  6. M. S. Soskin, V. N. Gorshkov, M. V. Vasnetsov, “Topological charge and angular momentum of light beams carrying optical vortices,” Phys. Rev. A 56, 4064–4075 (1997).
    [CrossRef]
  7. D. Rozas, C. T. Law, G. A. Swartzlander, “Propagation dynamics of optical vortices,” J. Opt. Soc. Am. B 14, 3054–3065 (1997).
    [CrossRef]
  8. Z. S. Sacks, D. Rozas, G. A. Swartzlander, “Holographic formation of optical-vortex filaments,” J. Opt. Soc. Am. B 15, 2226–2234 (1998).
    [CrossRef]
  9. G. Peele, K. A. Nugent, “X-ray vortex beams: a theoretical analysis,” Opt. Express 11, 2315–2322 (2003).
    [CrossRef] [PubMed]
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    [CrossRef]
  11. P. Prudnikov, Y. A. Brichkov, O. I. Marichev, Integrals and Series. Special Functions (Nauka, Moscow, 1983).
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    [CrossRef]
  13. E. Abramochkin, N. Losevsky, V. Volostnikov, “Generation of spiral-type laser beams,” Opt. Commun. 141, 59–64 (1997).
    [CrossRef]
  14. N. R. Heckenberg, R. McDuff, C. P. Smith, A. G. White, “Generation of optical phase singularities by computer-generated holograms,” Opt. Lett. 17, 221–223 (1992).
    [CrossRef] [PubMed]
  15. R. Oron, N. Davidson, A. A. Friesem, “Continuous-phase elements can improve laser beam quality,” Opt. Lett. 25, 939–941 (2000).
    [CrossRef]
  16. S. N. Khonina, V. V. Kotlyar, V. A. Soifer, P. Paakkonen, J. Simonen, J. Turunen, “An analysis of the angular momentum of a light field in terms of angular harmonics,” J. Mod. Opt. 48, 1543–1557 (2001).
    [CrossRef]
  17. S. N. Khonina, V. V. Kotlyar, V. A. Soifer, M. Honkanen, J. Lautanen, J. Turunen, “Generation of rotating Gauss–Laguerre modes with binary-phase diffractive optics,” J. Mod. Opt. 46, 227–238 (1999).
  18. M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristiensen, J. P. Woerdman, “Helical-wave front laser beams produced with a spiral phase plate,” Opt. Commun. 112, 321–327 (1994).
    [CrossRef]
  19. S. S. R. Oemrawsingh, J. A. W. van Houwelingen, E. R. Eliel, J. P. Woerdman, E. J. K. Verstegen, J. G. Kloosterboer, G. W. Hooft, “Production and characterization of spiral phase plates for optical wavelengths,” Appl. Opt. 43, 688–694 (2004).
    [CrossRef] [PubMed]

2004 (2)

S. N. Khonina, V. V. Kotlyar, V. A. Soifer, K. Jefimovs, J. Turunen, “Generation and selection of laser beams represented by a superposition of two angular harmonics,” J. Mod. Opt. 51, 761–773 (2004).
[CrossRef]

S. S. R. Oemrawsingh, J. A. W. van Houwelingen, E. R. Eliel, J. P. Woerdman, E. J. K. Verstegen, J. G. Kloosterboer, G. W. Hooft, “Production and characterization of spiral phase plates for optical wavelengths,” Appl. Opt. 43, 688–694 (2004).
[CrossRef] [PubMed]

2003 (1)

2001 (1)

S. N. Khonina, V. V. Kotlyar, V. A. Soifer, P. Paakkonen, J. Simonen, J. Turunen, “An analysis of the angular momentum of a light field in terms of angular harmonics,” J. Mod. Opt. 48, 1543–1557 (2001).
[CrossRef]

2000 (1)

1999 (1)

S. N. Khonina, V. V. Kotlyar, V. A. Soifer, M. Honkanen, J. Lautanen, J. Turunen, “Generation of rotating Gauss–Laguerre modes with binary-phase diffractive optics,” J. Mod. Opt. 46, 227–238 (1999).

1998 (1)

1997 (3)

M. S. Soskin, V. N. Gorshkov, M. V. Vasnetsov, “Topological charge and angular momentum of light beams carrying optical vortices,” Phys. Rev. A 56, 4064–4075 (1997).
[CrossRef]

D. Rozas, C. T. Law, G. A. Swartzlander, “Propagation dynamics of optical vortices,” J. Opt. Soc. Am. B 14, 3054–3065 (1997).
[CrossRef]

E. Abramochkin, N. Losevsky, V. Volostnikov, “Generation of spiral-type laser beams,” Opt. Commun. 141, 59–64 (1997).
[CrossRef]

1994 (1)

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

1993 (3)

M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96, 123–132 (1993).
[CrossRef]

I. V. Basistiy, V. Yu. Bazhenov, M. S. Soskin, M. V. Vasnetsov, “Optics of light beams with screw dislocations,” Opt. Commun. 103, 422–428 (1993).
[CrossRef]

G. Indebetouw, “Optical vortices and their propagation,” J. Mod. Opt. 40, 73–87 (1993).
[CrossRef]

1992 (3)

S. N. Khonina, V. V. Kotlyar, M. V. Shinkaryev, V. A. Soifer, G. V. Uspleniev, “The phase rotor filter,” J. Mod. Opt. 39, 1147–1154 (1992).
[CrossRef]

V. Yu. Bazhenov, M. S. Soskin, M. V. Vasnetsov, “Screw dislocations in light wavefronts,” J. Mod. Opt. 39, 985–990 (1992).
[CrossRef]

N. R. Heckenberg, R. McDuff, C. P. Smith, A. G. White, “Generation of optical phase singularities by computer-generated holograms,” Opt. Lett. 17, 221–223 (1992).
[CrossRef] [PubMed]

1974 (1)

J. F. Nye, M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. London Ser. A 336, 165–190 (1974).
[CrossRef]

Abramochkin, E.

E. Abramochkin, N. Losevsky, V. Volostnikov, “Generation of spiral-type laser beams,” Opt. Commun. 141, 59–64 (1997).
[CrossRef]

Allen, L.

M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96, 123–132 (1993).
[CrossRef]

Basistiy, I. V.

I. V. Basistiy, V. Yu. Bazhenov, M. S. Soskin, M. V. Vasnetsov, “Optics of light beams with screw dislocations,” Opt. Commun. 103, 422–428 (1993).
[CrossRef]

Bazhenov, V. Yu.

I. V. Basistiy, V. Yu. Bazhenov, M. S. Soskin, M. V. Vasnetsov, “Optics of light beams with screw dislocations,” Opt. Commun. 103, 422–428 (1993).
[CrossRef]

V. Yu. Bazhenov, M. S. Soskin, M. V. Vasnetsov, “Screw dislocations in light wavefronts,” J. Mod. Opt. 39, 985–990 (1992).
[CrossRef]

Beijersbergen, M. W.

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

M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96, 123–132 (1993).
[CrossRef]

Berry, M. V.

J. F. Nye, M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. London Ser. A 336, 165–190 (1974).
[CrossRef]

Brichkov, Y. A.

P. Prudnikov, Y. A. Brichkov, O. I. Marichev, Integrals and Series. Special Functions (Nauka, Moscow, 1983).

Coerwinkel, R. P. C.

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

Davidson, N.

Eliel, E. R.

Friesem, A. A.

Gorshkov, V. N.

M. S. Soskin, V. N. Gorshkov, M. V. Vasnetsov, “Topological charge and angular momentum of light beams carrying optical vortices,” Phys. Rev. A 56, 4064–4075 (1997).
[CrossRef]

Heckenberg, N. R.

Honkanen, M.

S. N. Khonina, V. V. Kotlyar, V. A. Soifer, M. Honkanen, J. Lautanen, J. Turunen, “Generation of rotating Gauss–Laguerre modes with binary-phase diffractive optics,” J. Mod. Opt. 46, 227–238 (1999).

Hooft, G. W.

Indebetouw, G.

G. Indebetouw, “Optical vortices and their propagation,” J. Mod. Opt. 40, 73–87 (1993).
[CrossRef]

Jefimovs, K.

S. N. Khonina, V. V. Kotlyar, V. A. Soifer, K. Jefimovs, J. Turunen, “Generation and selection of laser beams represented by a superposition of two angular harmonics,” J. Mod. Opt. 51, 761–773 (2004).
[CrossRef]

Khonina, S. N.

S. N. Khonina, V. V. Kotlyar, V. A. Soifer, K. Jefimovs, J. Turunen, “Generation and selection of laser beams represented by a superposition of two angular harmonics,” J. Mod. Opt. 51, 761–773 (2004).
[CrossRef]

S. N. Khonina, V. V. Kotlyar, V. A. Soifer, P. Paakkonen, J. Simonen, J. Turunen, “An analysis of the angular momentum of a light field in terms of angular harmonics,” J. Mod. Opt. 48, 1543–1557 (2001).
[CrossRef]

S. N. Khonina, V. V. Kotlyar, V. A. Soifer, M. Honkanen, J. Lautanen, J. Turunen, “Generation of rotating Gauss–Laguerre modes with binary-phase diffractive optics,” J. Mod. Opt. 46, 227–238 (1999).

S. N. Khonina, V. V. Kotlyar, M. V. Shinkaryev, V. A. Soifer, G. V. Uspleniev, “The phase rotor filter,” J. Mod. Opt. 39, 1147–1154 (1992).
[CrossRef]

Kloosterboer, J. G.

Kotlyar, V. V.

S. N. Khonina, V. V. Kotlyar, V. A. Soifer, K. Jefimovs, J. Turunen, “Generation and selection of laser beams represented by a superposition of two angular harmonics,” J. Mod. Opt. 51, 761–773 (2004).
[CrossRef]

S. N. Khonina, V. V. Kotlyar, V. A. Soifer, P. Paakkonen, J. Simonen, J. Turunen, “An analysis of the angular momentum of a light field in terms of angular harmonics,” J. Mod. Opt. 48, 1543–1557 (2001).
[CrossRef]

S. N. Khonina, V. V. Kotlyar, V. A. Soifer, M. Honkanen, J. Lautanen, J. Turunen, “Generation of rotating Gauss–Laguerre modes with binary-phase diffractive optics,” J. Mod. Opt. 46, 227–238 (1999).

S. N. Khonina, V. V. Kotlyar, M. V. Shinkaryev, V. A. Soifer, G. V. Uspleniev, “The phase rotor filter,” J. Mod. Opt. 39, 1147–1154 (1992).
[CrossRef]

Kristiensen, M.

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

Lautanen, J.

S. N. Khonina, V. V. Kotlyar, V. A. Soifer, M. Honkanen, J. Lautanen, J. Turunen, “Generation of rotating Gauss–Laguerre modes with binary-phase diffractive optics,” J. Mod. Opt. 46, 227–238 (1999).

Law, C. T.

Losevsky, N.

E. Abramochkin, N. Losevsky, V. Volostnikov, “Generation of spiral-type laser beams,” Opt. Commun. 141, 59–64 (1997).
[CrossRef]

Marichev, O. I.

P. Prudnikov, Y. A. Brichkov, O. I. Marichev, Integrals and Series. Special Functions (Nauka, Moscow, 1983).

McDuff, R.

Nugent, K. A.

Nye, J. F.

J. F. Nye, M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. London Ser. A 336, 165–190 (1974).
[CrossRef]

Oemrawsingh, S. S. R.

Oron, R.

Paakkonen, P.

S. N. Khonina, V. V. Kotlyar, V. A. Soifer, P. Paakkonen, J. Simonen, J. Turunen, “An analysis of the angular momentum of a light field in terms of angular harmonics,” J. Mod. Opt. 48, 1543–1557 (2001).
[CrossRef]

Peele, G.

Prudnikov, P.

P. Prudnikov, Y. A. Brichkov, O. I. Marichev, Integrals and Series. Special Functions (Nauka, Moscow, 1983).

Rozas, D.

Sacks, Z. S.

Shinkaryev, M. V.

S. N. Khonina, V. V. Kotlyar, M. V. Shinkaryev, V. A. Soifer, G. V. Uspleniev, “The phase rotor filter,” J. Mod. Opt. 39, 1147–1154 (1992).
[CrossRef]

Simonen, J.

S. N. Khonina, V. V. Kotlyar, V. A. Soifer, P. Paakkonen, J. Simonen, J. Turunen, “An analysis of the angular momentum of a light field in terms of angular harmonics,” J. Mod. Opt. 48, 1543–1557 (2001).
[CrossRef]

Smith, C. P.

Soifer, V. A.

S. N. Khonina, V. V. Kotlyar, V. A. Soifer, K. Jefimovs, J. Turunen, “Generation and selection of laser beams represented by a superposition of two angular harmonics,” J. Mod. Opt. 51, 761–773 (2004).
[CrossRef]

S. N. Khonina, V. V. Kotlyar, V. A. Soifer, P. Paakkonen, J. Simonen, J. Turunen, “An analysis of the angular momentum of a light field in terms of angular harmonics,” J. Mod. Opt. 48, 1543–1557 (2001).
[CrossRef]

S. N. Khonina, V. V. Kotlyar, V. A. Soifer, M. Honkanen, J. Lautanen, J. Turunen, “Generation of rotating Gauss–Laguerre modes with binary-phase diffractive optics,” J. Mod. Opt. 46, 227–238 (1999).

S. N. Khonina, V. V. Kotlyar, M. V. Shinkaryev, V. A. Soifer, G. V. Uspleniev, “The phase rotor filter,” J. Mod. Opt. 39, 1147–1154 (1992).
[CrossRef]

Soskin, M. S.

M. S. Soskin, V. N. Gorshkov, M. V. Vasnetsov, “Topological charge and angular momentum of light beams carrying optical vortices,” Phys. Rev. A 56, 4064–4075 (1997).
[CrossRef]

I. V. Basistiy, V. Yu. Bazhenov, M. S. Soskin, M. V. Vasnetsov, “Optics of light beams with screw dislocations,” Opt. Commun. 103, 422–428 (1993).
[CrossRef]

V. Yu. Bazhenov, M. S. Soskin, M. V. Vasnetsov, “Screw dislocations in light wavefronts,” J. Mod. Opt. 39, 985–990 (1992).
[CrossRef]

Swartzlander, G. A.

Turunen, J.

S. N. Khonina, V. V. Kotlyar, V. A. Soifer, K. Jefimovs, J. Turunen, “Generation and selection of laser beams represented by a superposition of two angular harmonics,” J. Mod. Opt. 51, 761–773 (2004).
[CrossRef]

S. N. Khonina, V. V. Kotlyar, V. A. Soifer, P. Paakkonen, J. Simonen, J. Turunen, “An analysis of the angular momentum of a light field in terms of angular harmonics,” J. Mod. Opt. 48, 1543–1557 (2001).
[CrossRef]

S. N. Khonina, V. V. Kotlyar, V. A. Soifer, M. Honkanen, J. Lautanen, J. Turunen, “Generation of rotating Gauss–Laguerre modes with binary-phase diffractive optics,” J. Mod. Opt. 46, 227–238 (1999).

Uspleniev, G. V.

S. N. Khonina, V. V. Kotlyar, M. V. Shinkaryev, V. A. Soifer, G. V. Uspleniev, “The phase rotor filter,” J. Mod. Opt. 39, 1147–1154 (1992).
[CrossRef]

van der Veen, H. E. L. O.

M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96, 123–132 (1993).
[CrossRef]

van Houwelingen, J. A. W.

Vasnetsov, M. V.

M. S. Soskin, V. N. Gorshkov, M. V. Vasnetsov, “Topological charge and angular momentum of light beams carrying optical vortices,” Phys. Rev. A 56, 4064–4075 (1997).
[CrossRef]

I. V. Basistiy, V. Yu. Bazhenov, M. S. Soskin, M. V. Vasnetsov, “Optics of light beams with screw dislocations,” Opt. Commun. 103, 422–428 (1993).
[CrossRef]

V. Yu. Bazhenov, M. S. Soskin, M. V. Vasnetsov, “Screw dislocations in light wavefronts,” J. Mod. Opt. 39, 985–990 (1992).
[CrossRef]

Verstegen, E. J. K.

Volostnikov, V.

E. Abramochkin, N. Losevsky, V. Volostnikov, “Generation of spiral-type laser beams,” Opt. Commun. 141, 59–64 (1997).
[CrossRef]

White, A. G.

Woerdman, J. P.

S. S. R. Oemrawsingh, J. A. W. van Houwelingen, E. R. Eliel, J. P. Woerdman, E. J. K. Verstegen, J. G. Kloosterboer, G. W. Hooft, “Production and characterization of spiral phase plates for optical wavelengths,” Appl. Opt. 43, 688–694 (2004).
[CrossRef] [PubMed]

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

M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96, 123–132 (1993).
[CrossRef]

Appl. Opt. (1)

J. Mod. Opt. (6)

S. N. Khonina, V. V. Kotlyar, V. A. Soifer, P. Paakkonen, J. Simonen, J. Turunen, “An analysis of the angular momentum of a light field in terms of angular harmonics,” J. Mod. Opt. 48, 1543–1557 (2001).
[CrossRef]

S. N. Khonina, V. V. Kotlyar, V. A. Soifer, M. Honkanen, J. Lautanen, J. Turunen, “Generation of rotating Gauss–Laguerre modes with binary-phase diffractive optics,” J. Mod. Opt. 46, 227–238 (1999).

S. N. Khonina, V. V. Kotlyar, M. V. Shinkaryev, V. A. Soifer, G. V. Uspleniev, “The phase rotor filter,” J. Mod. Opt. 39, 1147–1154 (1992).
[CrossRef]

V. Yu. Bazhenov, M. S. Soskin, M. V. Vasnetsov, “Screw dislocations in light wavefronts,” J. Mod. Opt. 39, 985–990 (1992).
[CrossRef]

G. Indebetouw, “Optical vortices and their propagation,” J. Mod. Opt. 40, 73–87 (1993).
[CrossRef]

S. N. Khonina, V. V. Kotlyar, V. A. Soifer, K. Jefimovs, J. Turunen, “Generation and selection of laser beams represented by a superposition of two angular harmonics,” J. Mod. Opt. 51, 761–773 (2004).
[CrossRef]

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

Opt. Commun. (4)

I. V. Basistiy, V. Yu. Bazhenov, M. S. Soskin, M. V. Vasnetsov, “Optics of light beams with screw dislocations,” Opt. Commun. 103, 422–428 (1993).
[CrossRef]

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

M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96, 123–132 (1993).
[CrossRef]

E. Abramochkin, N. Losevsky, V. Volostnikov, “Generation of spiral-type laser beams,” Opt. Commun. 141, 59–64 (1997).
[CrossRef]

Opt. Express (1)

Opt. Lett. (2)

Phys. Rev. A (1)

M. S. Soskin, V. N. Gorshkov, M. V. Vasnetsov, “Topological charge and angular momentum of light beams carrying optical vortices,” Phys. Rev. A 56, 4064–4075 (1997).
[CrossRef]

Proc. R. Soc. London Ser. A (1)

J. F. Nye, M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. London Ser. A 336, 165–190 (1974).
[CrossRef]

Other (1)

P. Prudnikov, Y. A. Brichkov, O. I. Marichev, Integrals and Series. Special Functions (Nauka, Moscow, 1983).

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Figures (11)

Fig. 1
Fig. 1

Intensity distribution for diffraction of a plane wave by a SPP with (a) first- and (b) second-order phase singularities at a distance of 10 mm from the plane z = 0 .

Fig. 2
Fig. 2

Intensity distribution for diffraction of a plane wave by a SPP with singularity of n = 1 ,3,9 orders at a distance of 10 mm.

Fig. 3
Fig. 3

Intensity distribution for diffraction of the Gaussian beam by a SPP with second-order singularity at distances of 10, 65, and 150 mm.

Fig. 4
Fig. 4

Intensity distribution for diffraction of the Gaussian beam by a SPP with second-order singularity at distances of 3000, 5000, and 10,000 mm.

Fig. 5
Fig. 5

Intensity distribution for diffraction of the Gaussian beam ( w = 0.391 mm ) by a SPP with singularity of n = 1 ,3,9 orders at a distance of 10 mm.

Fig. 6
Fig. 6

Cross-section intensity distributions (negative) and phases ( 2 π = black , 0 = white ) of the beams diffracted by a SPP with second-order singularity in the input plane ( z = 0 , two left columns), for far-field diffraction ( z = 5000 mm , two center columns), and in the lens focal plane (two right columns), the image size being 5 mm × 5 mm .

Fig. 7
Fig. 7

Cross-section intensity distribution (negative) for the beams diffracted by a SPP with phase singularity n = 2 at different distances from the input plane, the image size being 3.5 mm × 3.5 mm .

Fig. 8
Fig. 8

Cross-section intensity distributions (negative) and phases ( 2 π = black , 0 = white ) of the beams diffracted by a SPP with two phase singularities, n = 2 and n 2 = 1 , in the input plane ( z = 0 , two left columns), for far-field diffraction ( z = 5000 mm , two center columns), and in the lens focal plane (two right columns), the image size being 5 mm × 5 mm .

Fig. 9
Fig. 9

Cross-section intensity distribution (negative) for the beams diffracted by a SPP with two phase singularities, n = 2 and n 2 = 1 , at different distances from the input plane, the image size being 3.5 mm × 3.5 mm .

Fig. 10
Fig. 10

Generation of the laser field with first-order phase singularity: (a) estimated phase distribution and (b) central part of the SPP microrelief. The field intensity distributions (negative) registered by the CCD camera at different distances from the SPP: (c) z = 120 mm , (d) z = 300 mm , and (e) z = 520 mm .

Fig. 11
Fig. 11

Generation of the laser field with second-order phase singularity: (a) estimated phase distribution and (b) central part of the SPP microrelief. The field intensity distributions (negative) registered by the CCD camera at different distances from the SPP: (c) z = 110 mm , (d) z = 300 mm , and (e) z = 480 mm .

Equations (83)

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f n ( r , φ ) = exp ( i n φ ) ,
F n ( ρ , θ , z ) = ( i ) n + 1 k z exp ( i n θ + i k 2 z ρ 2 ) 0 exp ( i k 2 z r 2 ) J n ( k z r ρ ) r d r ,
F n ( ρ , θ , z ) = ( i ) n + 1 ( π 2 ) 1 2 ( k ρ 2 4 z ) 1 2 exp ( i n θ i π n 4 + i k 4 z ρ 2 ) × [ i J ( n 1 ) 2 ( k ρ 2 4 z ) + J ( n + 1 ) 2 ( k ρ 2 4 z ) ] .
J ( n ± 1 ) 2 ( x ) exp ( i n θ ) .
I ̂ n ( ρ , z ) = F n ( ρ , θ , z ) 2 = π 2 x [ J ( n 1 ) 2 2 ( x ) + J ( n + 1 ) 2 2 ( x ) ] ,
d I ̂ n ( ρ , z ) d ρ = π k ρ n 4 z [ J ( n 1 ) 2 2 ( k ρ 2 4 z ) J ( n + 1 ) 2 2 ( k ρ 2 4 z ) ] .
J ( n 1 ) 2 ( x ) = J ( n + 1 ) 2 ( x ) .
J ν ( x ) ( 2 π x ) 1 2 cos ( x ν π 2 π 4 ) ,
tan [ x ( n 1 ) π 4 π 4 ] = 1 ,
ρ n [ ( n + 1 ) λ z 2 ] 1 2 .
γ ν , 1 ν + 1.86 ν 1 3 + 1.03 ν 1 3 , J ν ( γ ν , 1 ) = 0 , ν 1 .
z < 2 λ ( n + 1 ) ,
J 0 ( x ) = J 1 ( x ) ,
ρ 1 0.94 λ z .
cot ( x ) = 1 x x ,
ρ 2 1.13 λ z .
ρ 1 λ z ,
ρ 2 1.22 λ z .
ρ n ( 2 α n λ z π ) 1 2 ,
J v ( x ) ( x 2 ) ν Γ 1 ( ν + 1 ) ,
I ̂ n ( ρ , z ) π 2 x ( 2 π x { cos 2 [ x ( n 1 ) π 4 π 4 ] + cos 2 [ x ( n + 1 ) π 4 π 4 ] } ) = 1 .
I ̂ n ( ρ , z ) π Γ [ ( n + 1 ) 2 ] ( k ρ 2 8 z ) n .
f n ( r , φ ) = circ ( r R ) exp ( i n φ ) ,
circ ( r R ) = { 1 , r R 0 , r > R } .
E n 0 ( r , φ ) = exp [ ( r w ) 2 + i n φ ] .
E n ( ρ , θ , z ) = ( i ) k 2 π z 0 0 2 π E n 0 ( r , φ ) exp { i k 2 z [ r 2 + ρ 2 2 r ρ cos ( φ θ ) ] } r d r d φ = ( i ) n + 1 π 2 ( z 0 z ) 2 ( ρ w ) [ 1 + ( z 0 z ) 2 ] 3 4 exp [ i 3 2 tan 1 ( z 0 z ) i k ρ 2 2 R 0 ( z ) + i k ρ 2 2 z ρ 2 w 2 ( z ) + i n θ ] ( I ( n 1 ) 2 { ρ 2 [ 1 w 2 ( z ) + i k 2 R 0 ( z ) ] } I ( n + 1 ) 2 { ρ 2 [ 1 w 2 ( z ) + i k 2 R 0 ( z ) ] } ) ,
w 2 ( z ) = 2 w 2 [ 1 + ( z z 0 ) 2 ] ,
R 0 ( z ) = 2 z [ 1 + ( z z 0 ) 2 ] , z 0 = k w 2 2 ,
[ 1 + ( z 0 z ) 2 ] 3 4 ( z 0 z ) 3 2 , exp [ i 3 2 tan 1 ( z 0 z ) ] exp ( i 3 π 4 ) , R 0 ( z ) 2 z ,
ρ 2 [ 1 w 2 ( z ) + i k 2 R 0 ( z ) ] i k ρ 2 4 z , exp [ i k ρ 2 2 z i k ρ 2 2 R 0 ( z ) ] exp ( i k ρ 2 4 z )
E n ( ρ 0 , θ , z 0 )
( i ) n + 1 π 2 ( k ρ 2 4 z ) 1 2 exp ( i 3 π 4 + i k ρ 2 4 z + i n θ ) i ( n 3 ) 2 [ i J ( n 1 ) 2 ( k ρ 2 4 z ) + J ( n + 1 ) 2 ( k ρ 2 4 z ) ] .
I n ( i x ) = i n J n ( x ) .
E n ( ρ 0 , θ , z 0 ) = i n F n ( ρ , θ , z ) .
w 2 ( z ) 2 w 2 z 2 z 0 2 , R 0 ( z ) 2 z 3 z 0 2 ,
ρ 2 [ 1 w 2 ( z ) + i k 2 R 0 ( z ) ] ρ 2 w 2 ( z ) ,
exp [ i k ρ 2 2 z i k ρ 2 2 R 0 ( z ) ] exp ( i k ρ 2 2 z ) .
E n ( ρ , θ , z )
= ( i ) n + 1 π 2 ( z 0 z ) 2 ( ρ w ) exp ( i n θ ) exp [ ρ 2 w 2 ( z ) + i k ρ 2 2 z ] { I ( n 1 ) 2 [ ρ 2 w 2 ( z ) ] I ( n + 1 ) 2 [ ρ 2 w 2 ( z ) ] } .
I ̂ n ( ρ , z ) π 4 ( z 0 z ) 4 ( ρ w ) 2 exp [ 2 ρ 2 w 2 ( z ) ] { I ( n 1 ) 2 [ ρ 2 w 2 ( z ) ] I ( n + 1 ) 2 [ ρ 2 w 2 ( z ) ] } 2 = π 2 ( z 0 z ) 2 x exp ( 2 x ) [ I ( n 1 ) 2 ( x ) I ( n + 1 ) 2 ( x ) ] 2 ,
d I ̂ n ( x , z ) d x = π 2 ( z 0 z ) 2 exp ( 2 x ) [ I ( n 1 ) 2 ( x ) I ( n + 1 ) 2 ( x ) ] [ ( n 4 x ) I ( n 1 ) 2 ( x ) + ( n + 4 x ) I ( n + 1 ) 2 ( x ) ] .
( n 4 x ) I ( n 1 ) 2 ( x ) + ( n + 4 x ) I ( n + 1 ) 2 ( x ) = 0 .
I 1 2 ( x ) = ( 2 π x ) 1 2 sh ( x ) ,
I 3 2 ( x ) = ( 2 π x ) 1 2 [ ch ( x ) sh ( x ) x ] .
( 1 + 2 x ) ch ( x ) ( 1 + 2 x + 1 x ) sh ( x ) = 0 .
exp ( 2 x ) = 1 + 2 x + 4 x 2 ,
ρ 2 0.947 w ( z ) .
E 2 ( ρ , θ , z ) = A ( θ , z ) ρ 2 exp ( i k ρ 2 2 z ) { 1 [ 1 + 2 ρ 2 w 2 ( z ) + i k ρ 2 R 0 ( z ) ] exp [ 2 ρ 2 w 2 ( z ) i k ρ 2 R 0 ( z ) ] } ,
A ( θ , z ) = i 2 2 w ( z 0 z ) 2 [ 1 + ( z 0 z ) 2 ] 3 4 [ 1 w 2 ( z ) + i k 2 R 0 ( z ) ] 3 2 × exp [ 2 i θ + i 3 2 tan 1 ( z 0 z ) ] .
( 2 ρ 2 + 1 ρ ρ + 1 ρ 2 2 θ 2 + 2 i k z ) E 2 ( ρ , θ , z ) = 0 ,
R 0 ( z ) 2 z ; tan 1 ( z 0 z ) π 2 ; w ( z ) 2 w ;
exp ( i k ρ 2 2 z ) 0 ; [ 1 + 2 ρ 2 w 2 ( z ) + i k ρ 2 R 0 ( z ) ] i k ρ 2 2 z .
E 2 ( ρ , θ , z 0 ) = exp ( i 2 θ ρ 2 w 2 ) .
R 0 ( z ) 2 z 3 z 0 2 ; tan 1 ( z 0 z ) 0 ; w ( z ) 2 w z z 0 ;
exp ( i k ρ 2 2 z ) 1 ; [ 1 + 2 ρ 2 w 2 ( z ) + i k ρ 2 R 0 ( z ) ] 1 + 2 ρ 2 w 2 ( z ) ; [ 1 w 2 ( z ) + i k 2 R 0 ( z ) ] 1 w 2 ( z ) .
E 2 ( ρ , θ , z ) i w w ( z ) 2 ρ 2 exp ( i 2 θ + i k ρ 2 2 z ) { 1 [ 1 + 2 ρ 2 w 2 ( z ) ] exp [ 2 ρ 2 w 2 ( z ) ] } .
I ̂ 2 ( ρ , z ) 1 2 [ w w ( z ) ] 2 [ w ( z ) ρ ] 4 { 1 [ 1 + 2 ρ 2 w 2 ( z ) ] exp [ 2 ρ 2 w 2 ( z ) ] } 2 .
I ̂ 2 ( ρ , z ) w 4 2 ρ 4 .
I ̂ 2 ( ρ 0 , z ) 2 w 2 ρ 4 w 6 ( z ) .
d I ̂ 2 ( ρ , z ) d ρ = 2 w 2 x 5 2 w 3 ( z ) [ 1 ( 1 + 2 x ) exp ( 2 x ) ] [ 1 ( 1 + 2 x + 4 x 2 ) exp ( 2 x ) ] ,
exp ( 2 x ) = 1 + 2 x + 4 x 2 .
ρ 2 0.426 λ z w .
w ( z ) 2 w , R 0 ( z ) 2 z , exp ( 2 ρ 2 w 2 ) 1 ,
E 2 ( ρ 0 , θ , z 0 ) = 2 i z k ρ 2 exp ( i 2 θ ) { exp [ i k ρ 2 2 z ] i k ρ 2 2 z 1 } .
I ̂ 2 ( ρ , θ , z ) y 2 ( 2 + y 2 2 cos y 2 y sin y ) ,
2 2 cos y 2 y sin y + y 2 cos y = 0 ,
y 1.3 π .
ρ 2 1.14 λ z .
M 1 r [ f ( r , φ ) ] = 0 0 2 π f ( r , φ ) r 2 d r d φ 0 0 2 π f ( r , φ ) r d r d φ .
Ψ 0 n ( r , φ ) = 1 σ π n ! ( r σ ) n exp [ ( r 2 σ ) 2 ] exp ( i n φ ) ,
I 0 n GL ( r ) = 1 σ 2 π n ! ( r σ ) 2 n exp [ ( r σ ) 2 ] ,
M 1 r [ I 0 n GL ( r ) ] = σ [ π n ! 2 ( n + 1 ) p = 0 n ( 2 p + 1 ) ] .
f n g ( r , φ ) = exp [ ( r 2 σ 1 ) 2 ] exp ( i n φ ) ,
M 1 r [ I n g ( r ) ] = π 2 σ 1 ,
M 1 r [ I n c ( r ) ] = 2 3 R .
σ 1 = 2 π σ [ π n ! 2 ( n + 1 ) p = 0 n ( 2 p + 1 ) ] ,
R = 3 2 σ [ π n ! 2 ( n + 1 ) p = 0 n ( 2 p + 1 ) ] .
T ( φ ) = k = 1 K C k exp ( i n k φ ) .
d max = 31 λ 32 ( n r 1 ) ,
( x + i y ) 2 + a + i b = 0
x 1 , 2 = [ a ± ( a 2 + b 2 ) 1 2 2 ] 1 2 ,
y 1 , 2 = b 2 x .
I max , ν π v J v 2 ( v ) 1.45 v 1 3 ,

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