Abstract

The article concerns an investigation of the Fresnel diffraction characteristics of two types of phase optical elements under Gaussian laser beam illumination. Both elements provide an azimuthal periodicity of the phase retardation. The first element possesses azimuthal cosine-profiled phase changes deposited on a plane base. The second element is a combination of the first element and a thin phase axicon. The cosine profile of the phase retardation of both diffractive elements produces an azimuthal cosine-profiled modulation on their diffractograms. It destroys the vortex characteristics of their diffraction fields.

© 2011 Optical Society of America

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References

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  1. J. Dyson, “Circular and spiral diffraction gratings,” Proc. R. Soc. A 248, 93-106 (1958).
    [CrossRef]
  2. J. Durnin, “Exact solutions for non-diffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A 4, 651-654 (1987).
    [CrossRef]
  3. J. Durnin, J. J. Miceli, Jr., and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499-1501 (1987).
    [CrossRef] [PubMed]
  4. F. Gori, G. Guattari, and C. Padovani, “Bessel-Gauss beams,” Opt. Commun. 64, 491-495 (1987).
    [CrossRef]
  5. D. McGloin and K. Dholakia, “Bessel beams: diffraction in a new light,” Contemp. Phys. 46, 15-28 (2005).
    [CrossRef]
  6. J. H. McLeod, “The axicon: a new type of optical element,” J. Opt. Soc. Am. 44, 592-597 (1954).
    [CrossRef]
  7. R. M. Herman and T. A. Wiggins, “Production and uses of diffractionless beams,” J. Opt. Soc. Am. A 8, 932-942 (1991).
    [CrossRef]
  8. J. Turunen, A. Vasara, and A. T. Friberg, “Holographic generation of diffraction-free beams,” Appl. Opt. 27, 3959-3962 (1988).
    [CrossRef] [PubMed]
  9. A. C. S. van Heel, “Modern alignment devices,” in Vol. 1 of Progress in Optics (Springer-Verlag, 1961), pp. 289-329.
    [CrossRef]
  10. J. Ojeda-Castañeda, P. Andrés, and M. Martínez-Corral, “Zero axial irradiance by annular screens with angular variation,” Appl. Opt. 31, 4600-4602 (1992).
    [CrossRef] [PubMed]
  11. V. V. Kotlyar, A. A. Almazov, S. N. Khonina, V. A. Sofier, H. Elfstrom, and J. Turunen, “Generation of phase singularity through diffracting a plane or Gaussian beam by a spiral phase plate,” J. Opt. Soc. Am. A 22, 849-861 (2005).
    [CrossRef]
  12. 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]
  13. S. N. Khonina, V. V. Kotlyar, V. A. Soifer, K. Jefimovs, and J. Turunen, “Generation and selection of laser beams represented by a superposition of two angular harmonics,” J. Mod. Opt. 51, 761-773 (2004).
  14. E. B. Brown, Modern Optics (Reinhold, 1965).
  15. C. Holzner, M. Feser, S. Vogt, B. Hornberger, S. B. Baines, and C. Jacobsen, “Zernike phase contrast in scanning microscopy with x-rays,” Nat. Phys. 6, 883-887 (2010).
    [CrossRef]
  16. G. Kopitkovas, T. Lippert, C. David, A. Wokaun, and J. Gobrecht, “Fabrication of micro-optical elements in quartz by laser induced backside wet etching,” Microelectron. Eng. 67-68, 438-444 (2003).
    [CrossRef]
  17. J. A. Davis, E. Carcole, and D. M. Cottrell, “Nondiffracting interference patterns generated with programmable spatial light modulators,” Appl. Opt. 35, 599-602 (1996).
    [CrossRef] [PubMed]
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    [CrossRef]
  19. V. V. Kotlyar, S. N. Khonina, R. V. Skidanov, and V. A. Soifer, “Rotation of laser beams with zero of the orbital angular momentum,” Opt. Commun. 274, 8-14 (2007).
    [CrossRef]
  20. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1964).
  21. M. Born and E. Wolf, Principles of Optics (Cambridge University, 1999).
  22. A. P. Prudnikov, Y. A. Brichkov, and O. I. Marichev, Integrals and Series; Special Functions (Nauka, 1983).
    [PubMed]
  23. Lj. Janicijevic and S. Topuzoski, “Fresnel and Fraunhofer diffraction of a Gaussian laser beam by fork-shaped gratings,” J. Opt. Soc. Am. A 25, 2659-2669 (2008).
  24. J. A. Davis, E. Carcole, and D. M. Cottrell, “Intensity and phase measurements of nondiffracting beams generated with a magneto-optic spatial light modulator,” Appl. Opt. 35, 593-598(1996).
    [CrossRef] [PubMed]
  25. S. Topuzoski and Lj. Janicijevic, “Conversion of high-order Laguerre-Gaussian beams into Bessel beams of increased, reduced or zeroth order by use of a helical axicon,” Opt. Commun. 282, 3426-3432 (2009).
    [CrossRef]
  26. R. L. Phillips and L. C. Andrews, “Spot size and divergence for Laguerre-Gaussian beams of any order,” Appl. Opt. 22, 643-644(1983).
    [CrossRef] [PubMed]
  27. K. Dholakia and T. Čižmár, “Shaping the future of manipulation,” Nat. Photon. 5, 335-342 (2011).
    [CrossRef]

2011 (1)

K. Dholakia and T. Čižmár, “Shaping the future of manipulation,” Nat. Photon. 5, 335-342 (2011).
[CrossRef]

2010 (1)

C. Holzner, M. Feser, S. Vogt, B. Hornberger, S. B. Baines, and C. Jacobsen, “Zernike phase contrast in scanning microscopy with x-rays,” Nat. Phys. 6, 883-887 (2010).
[CrossRef]

2009 (1)

S. Topuzoski and Lj. Janicijevic, “Conversion of high-order Laguerre-Gaussian beams into Bessel beams of increased, reduced or zeroth order by use of a helical axicon,” Opt. Commun. 282, 3426-3432 (2009).
[CrossRef]

2008 (1)

2007 (1)

V. V. Kotlyar, S. N. Khonina, R. V. Skidanov, and V. A. Soifer, “Rotation of laser beams with zero of the orbital angular momentum,” Opt. Commun. 274, 8-14 (2007).
[CrossRef]

2005 (2)

2004 (1)

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

2003 (1)

G. Kopitkovas, T. Lippert, C. David, A. Wokaun, and J. Gobrecht, “Fabrication of micro-optical elements in quartz by laser induced backside wet etching,” Microelectron. Eng. 67-68, 438-444 (2003).
[CrossRef]

1999 (1)

M. Born and E. Wolf, Principles of Optics (Cambridge University, 1999).

1996 (2)

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]

1992 (2)

S. N. Khonina, V. V. Kotlyar, V. A. Soifer, M. V. Shinkaryev, and G. V. Uspleniev, “Trochoson,” Opt. Commun. 91, 158-162 (1992).
[CrossRef]

J. Ojeda-Castañeda, P. Andrés, and M. Martínez-Corral, “Zero axial irradiance by annular screens with angular variation,” Appl. Opt. 31, 4600-4602 (1992).
[CrossRef] [PubMed]

1991 (1)

1988 (1)

1987 (3)

J. Durnin, J. J. Miceli, Jr., and J. H. Eberly, “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, 491-495 (1987).
[CrossRef]

J. Durnin, “Exact solutions for non-diffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A 4, 651-654 (1987).
[CrossRef]

1983 (2)

A. P. Prudnikov, Y. A. Brichkov, and O. I. Marichev, Integrals and Series; Special Functions (Nauka, 1983).
[PubMed]

R. L. Phillips and L. C. Andrews, “Spot size and divergence for Laguerre-Gaussian beams of any order,” Appl. Opt. 22, 643-644(1983).
[CrossRef] [PubMed]

1965 (1)

E. B. Brown, Modern Optics (Reinhold, 1965).

1964 (1)

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1964).

1961 (1)

A. C. S. van Heel, “Modern alignment devices,” in Vol. 1 of Progress in Optics (Springer-Verlag, 1961), pp. 289-329.
[CrossRef]

1958 (1)

J. Dyson, “Circular and spiral diffraction gratings,” Proc. R. Soc. A 248, 93-106 (1958).
[CrossRef]

1954 (1)

Abramowitz, M.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1964).

Almazov, A. A.

Andrés, P.

Andrews, L. C.

Baines, S. B.

C. Holzner, M. Feser, S. Vogt, B. Hornberger, S. B. Baines, and C. Jacobsen, “Zernike phase contrast in scanning microscopy with x-rays,” Nat. Phys. 6, 883-887 (2010).
[CrossRef]

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]

Born, M.

M. Born and E. Wolf, Principles of Optics (Cambridge University, 1999).

Brichkov, Y. A.

A. P. Prudnikov, Y. A. Brichkov, and O. I. Marichev, Integrals and Series; Special Functions (Nauka, 1983).
[PubMed]

Brown, E. B.

E. B. Brown, Modern Optics (Reinhold, 1965).

Carcole, E.

Cižmár, T.

K. Dholakia and T. Čižmár, “Shaping the future of manipulation,” Nat. Photon. 5, 335-342 (2011).
[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]

Cottrell, D. M.

David, C.

G. Kopitkovas, T. Lippert, C. David, A. Wokaun, and J. Gobrecht, “Fabrication of micro-optical elements in quartz by laser induced backside wet etching,” Microelectron. Eng. 67-68, 438-444 (2003).
[CrossRef]

Davis, J. A.

Dholakia, K.

K. Dholakia and T. Čižmár, “Shaping the future of manipulation,” Nat. Photon. 5, 335-342 (2011).
[CrossRef]

D. McGloin and K. Dholakia, “Bessel beams: diffraction in a new light,” Contemp. Phys. 46, 15-28 (2005).
[CrossRef]

Durnin, J.

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

J. Durnin, “Exact solutions for non-diffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A 4, 651-654 (1987).
[CrossRef]

Dyson, J.

J. Dyson, “Circular and spiral diffraction gratings,” Proc. R. Soc. A 248, 93-106 (1958).
[CrossRef]

Eberly, J. H.

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

Elfstrom, H.

Feser, M.

C. Holzner, M. Feser, S. Vogt, B. Hornberger, S. B. Baines, and C. Jacobsen, “Zernike phase contrast in scanning microscopy with x-rays,” Nat. Phys. 6, 883-887 (2010).
[CrossRef]

Friberg, A. T.

Gobrecht, J.

G. Kopitkovas, T. Lippert, C. David, A. Wokaun, and J. Gobrecht, “Fabrication of micro-optical elements in quartz by laser induced backside wet etching,” Microelectron. Eng. 67-68, 438-444 (2003).
[CrossRef]

Gori, F.

F. Gori, G. Guattari, and C. Padovani, “Bessel-Gauss beams,” Opt. Commun. 64, 491-495 (1987).
[CrossRef]

Guattari, G.

F. Gori, G. Guattari, and C. Padovani, “Bessel-Gauss beams,” Opt. Commun. 64, 491-495 (1987).
[CrossRef]

Herman, R. M.

Holzner, C.

C. Holzner, M. Feser, S. Vogt, B. Hornberger, S. B. Baines, and C. Jacobsen, “Zernike phase contrast in scanning microscopy with x-rays,” Nat. Phys. 6, 883-887 (2010).
[CrossRef]

Hornberger, B.

C. Holzner, M. Feser, S. Vogt, B. Hornberger, S. B. Baines, and C. Jacobsen, “Zernike phase contrast in scanning microscopy with x-rays,” Nat. Phys. 6, 883-887 (2010).
[CrossRef]

Jacobsen, C.

C. Holzner, M. Feser, S. Vogt, B. Hornberger, S. B. Baines, and C. Jacobsen, “Zernike phase contrast in scanning microscopy with x-rays,” Nat. Phys. 6, 883-887 (2010).
[CrossRef]

Janicijevic, Lj.

S. Topuzoski and Lj. Janicijevic, “Conversion of high-order Laguerre-Gaussian beams into Bessel beams of increased, reduced or zeroth order by use of a helical axicon,” Opt. Commun. 282, 3426-3432 (2009).
[CrossRef]

Lj. Janicijevic and S. Topuzoski, “Fresnel and Fraunhofer diffraction of a Gaussian laser beam by fork-shaped gratings,” J. Opt. Soc. Am. A 25, 2659-2669 (2008).

Jefimovs, K.

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

Khonina, S. N.

V. V. Kotlyar, S. N. Khonina, R. V. Skidanov, and V. A. Soifer, “Rotation of laser beams with zero of the orbital angular momentum,” Opt. Commun. 274, 8-14 (2007).
[CrossRef]

V. V. Kotlyar, A. A. Almazov, S. N. Khonina, V. A. Sofier, H. Elfstrom, and J. Turunen, “Generation of phase singularity through diffracting a plane or Gaussian beam by a spiral phase plate,” J. Opt. Soc. Am. A 22, 849-861 (2005).
[CrossRef]

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

S. N. Khonina, V. V. Kotlyar, V. A. Soifer, M. V. Shinkaryev, and G. V. Uspleniev, “Trochoson,” Opt. Commun. 91, 158-162 (1992).
[CrossRef]

Kopitkovas, G.

G. Kopitkovas, T. Lippert, C. David, A. Wokaun, and J. Gobrecht, “Fabrication of micro-optical elements in quartz by laser induced backside wet etching,” Microelectron. Eng. 67-68, 438-444 (2003).
[CrossRef]

Kotlyar, V. V.

V. V. Kotlyar, S. N. Khonina, R. V. Skidanov, and V. A. Soifer, “Rotation of laser beams with zero of the orbital angular momentum,” Opt. Commun. 274, 8-14 (2007).
[CrossRef]

V. V. Kotlyar, A. A. Almazov, S. N. Khonina, V. A. Sofier, H. Elfstrom, and J. Turunen, “Generation of phase singularity through diffracting a plane or Gaussian beam by a spiral phase plate,” J. Opt. Soc. Am. A 22, 849-861 (2005).
[CrossRef]

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

S. N. Khonina, V. V. Kotlyar, V. A. Soifer, M. V. Shinkaryev, and G. V. Uspleniev, “Trochoson,” Opt. Commun. 91, 158-162 (1992).
[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]

Lippert, T.

G. Kopitkovas, T. Lippert, C. David, A. Wokaun, and J. Gobrecht, “Fabrication of micro-optical elements in quartz by laser induced backside wet etching,” Microelectron. Eng. 67-68, 438-444 (2003).
[CrossRef]

Marichev, O. I.

A. P. Prudnikov, Y. A. Brichkov, and O. I. Marichev, Integrals and Series; Special Functions (Nauka, 1983).
[PubMed]

Martínez-Corral, M.

McGloin, D.

D. McGloin and K. Dholakia, “Bessel beams: diffraction in a new light,” Contemp. Phys. 46, 15-28 (2005).
[CrossRef]

McLeod, J. H.

Miceli, J. J.

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

Ojeda-Castañeda, J.

Padovani, C.

F. Gori, G. Guattari, and C. Padovani, “Bessel-Gauss beams,” Opt. Commun. 64, 491-495 (1987).
[CrossRef]

Phillips, R. L.

Prudnikov, A. P.

A. P. Prudnikov, Y. A. Brichkov, and O. I. Marichev, Integrals and Series; Special Functions (Nauka, 1983).
[PubMed]

Shinkaryev, M. V.

S. N. Khonina, V. V. Kotlyar, V. A. Soifer, M. V. Shinkaryev, and G. V. Uspleniev, “Trochoson,” Opt. Commun. 91, 158-162 (1992).
[CrossRef]

Skidanov, R. V.

V. V. Kotlyar, S. N. Khonina, R. V. Skidanov, and V. A. Soifer, “Rotation of laser beams with zero of the orbital angular momentum,” Opt. Commun. 274, 8-14 (2007).
[CrossRef]

Sofier, V. A.

Soifer, V. A.

V. V. Kotlyar, S. N. Khonina, R. V. Skidanov, and V. A. Soifer, “Rotation of laser beams with zero of the orbital angular momentum,” Opt. Commun. 274, 8-14 (2007).
[CrossRef]

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

S. N. Khonina, V. V. Kotlyar, V. A. Soifer, M. V. Shinkaryev, and G. V. Uspleniev, “Trochoson,” Opt. Commun. 91, 158-162 (1992).
[CrossRef]

Stegun, I. A.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1964).

Topuzoski, S.

S. Topuzoski and Lj. Janicijevic, “Conversion of high-order Laguerre-Gaussian beams into Bessel beams of increased, reduced or zeroth order by use of a helical axicon,” Opt. Commun. 282, 3426-3432 (2009).
[CrossRef]

Lj. Janicijevic and S. Topuzoski, “Fresnel and Fraunhofer diffraction of a Gaussian laser beam by fork-shaped gratings,” J. Opt. Soc. Am. A 25, 2659-2669 (2008).

Turunen, J.

Uspleniev, G. V.

S. N. Khonina, V. V. Kotlyar, V. A. Soifer, M. V. Shinkaryev, and G. V. Uspleniev, “Trochoson,” Opt. Commun. 91, 158-162 (1992).
[CrossRef]

van Heel, A. C. S.

A. C. S. van Heel, “Modern alignment devices,” in Vol. 1 of Progress in Optics (Springer-Verlag, 1961), pp. 289-329.
[CrossRef]

Vasara, A.

Vogt, S.

C. Holzner, M. Feser, S. Vogt, B. Hornberger, S. B. Baines, and C. Jacobsen, “Zernike phase contrast in scanning microscopy with x-rays,” Nat. Phys. 6, 883-887 (2010).
[CrossRef]

Wiggins, T. A.

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]

Wokaun, A.

G. Kopitkovas, T. Lippert, C. David, A. Wokaun, and J. Gobrecht, “Fabrication of micro-optical elements in quartz by laser induced backside wet etching,” Microelectron. Eng. 67-68, 438-444 (2003).
[CrossRef]

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Cambridge University, 1999).

Appl. Opt. (5)

Contemp. Phys. (1)

D. McGloin and K. Dholakia, “Bessel beams: diffraction in a new light,” Contemp. Phys. 46, 15-28 (2005).
[CrossRef]

J. Mod. Opt. (1)

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

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (4)

Microelectron. Eng. (1)

G. Kopitkovas, T. Lippert, C. David, A. Wokaun, and J. Gobrecht, “Fabrication of micro-optical elements in quartz by laser induced backside wet etching,” Microelectron. Eng. 67-68, 438-444 (2003).
[CrossRef]

Nat. Photon. (1)

K. Dholakia and T. Čižmár, “Shaping the future of manipulation,” Nat. Photon. 5, 335-342 (2011).
[CrossRef]

Nat. Phys. (1)

C. Holzner, M. Feser, S. Vogt, B. Hornberger, S. B. Baines, and C. Jacobsen, “Zernike phase contrast in scanning microscopy with x-rays,” Nat. Phys. 6, 883-887 (2010).
[CrossRef]

Opt. Commun. (5)

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]

F. Gori, G. Guattari, and C. Padovani, “Bessel-Gauss beams,” Opt. Commun. 64, 491-495 (1987).
[CrossRef]

S. Topuzoski and Lj. Janicijevic, “Conversion of high-order Laguerre-Gaussian beams into Bessel beams of increased, reduced or zeroth order by use of a helical axicon,” Opt. Commun. 282, 3426-3432 (2009).
[CrossRef]

S. N. Khonina, V. V. Kotlyar, V. A. Soifer, M. V. Shinkaryev, and G. V. Uspleniev, “Trochoson,” Opt. Commun. 91, 158-162 (1992).
[CrossRef]

V. V. Kotlyar, S. N. Khonina, R. V. Skidanov, and V. A. Soifer, “Rotation of laser beams with zero of the orbital angular momentum,” Opt. Commun. 274, 8-14 (2007).
[CrossRef]

Phys. Rev. Lett. (1)

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

Proc. R. Soc. A (1)

J. Dyson, “Circular and spiral diffraction gratings,” Proc. R. Soc. A 248, 93-106 (1958).
[CrossRef]

Other (5)

E. B. Brown, Modern Optics (Reinhold, 1965).

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1964).

M. Born and E. Wolf, Principles of Optics (Cambridge University, 1999).

A. P. Prudnikov, Y. A. Brichkov, and O. I. Marichev, Integrals and Series; Special Functions (Nauka, 1983).
[PubMed]

A. C. S. van Heel, “Modern alignment devices,” in Vol. 1 of Progress in Optics (Springer-Verlag, 1961), pp. 289-329.
[CrossRef]

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Figures (6)

Fig. 1
Fig. 1

Cosine change of the phase transmission function of the CPSS in the azimuthal direction. The spatial frequency p = 11 . (a) Two- dimensional representation in polar coordinates. (b) One-dimensional representation of the phase variation due to the cosine change of the thickness of the CPSS.

Fig. 2
Fig. 2

Transverse intensity distribution calculated on base on Eq. (14) at different distances: (a)  z = 60 mm , (c)  z = 100 mm , (d)  z = 160 mm . Parameters used for the CPSS are p = 9 , n = 1.48 for λ = 1 μm , and c = 10 μm ( k δ = 15 ). The beam waist radius is w 0 = 3.5 mm . The zoomed central part of (a) is shown in (b).

Fig. 3
Fig. 3

Geometry of the problem.

Fig. 4
Fig. 4

Transverse intensity profile of the diffracted beam by the CPSSA, according to Eq. (23), for (a)  p = 3 and (b)  p = 9 .

Fig. 5
Fig. 5

Intensity variation on the bright spots situated in a circle around the central dark spot along the beam propagation direction z according to Eq. (23).

Fig. 6
Fig. 6

Transverse intensity profile of the diffracted beam [Eq. (22)] by the CPSSA when the zeroth-diffraction-order beam is not eliminated. The parameters used are p = 9 , z = 16 cm , γ = 0.0235 rad , n = n = 1.48 for incident beam wavelength λ = 1 μm , k δ = 13 , and w 0 = 3.5 mm .

Equations (31)

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d = ( c / 2 ) + ( c / 2 ) cos ( p φ ) ,
T ( φ ) = exp [ i k ( n 1 ) d ] = exp ( i k δ ) exp [ i k δ cos ( p φ ) ] ,
T ( φ ) = exp ( i k δ ) { J 0 ( k δ ) + m = 1 ( i ) m J m ( k δ ) [ exp ( + i m p φ ) + exp ( i m p φ ) ] } ,
t 0 = exp ( i k δ ) J 0 ( k δ ) , t + m = t m = ( i ) m exp ( i k δ ) J m ( k δ ) .
U ( ρ , θ , z ) = i k 2 π z exp [ i k ( z + ρ 2 2 z ) ] Δ T ( φ ) U ( i ) ( r , φ , 0 ) exp [ i k 2 ( r 2 z 2 r ρ cos ( φ θ ) z ) ] r d r d φ ,
U ( ρ , θ , z ) = U 0 ( ρ , θ , z ) + m = 1 [ U + m ( ρ , θ , z ) + U m ( ρ , θ , z ) ] ,
U 0 ( ρ , θ , z ) = i k t 0 2 π z exp [ i k ( z + ρ 2 2 z ) ] 0 { exp [ i k r 2 2 ( 1 z 2 i k w 0 2 ) ] 0 2 π exp [ i k z r ρ cos ( φ θ ) ] d φ } r d r ,
U ± m ( ρ , θ , z ) = i k t m 2 π z exp [ i k ( z + ρ 2 2 z ) ] × 0 { exp [ i k r 2 2 ( 1 z 2 i k w 0 2 ) ] 0 2 π exp [ i k z r ρ cos ( φ θ ) ] exp ( ± i m p φ ) d φ } r d r .
exp [ ( i k / z ) r ρ cos ( φ θ ) ] = J 0 ( k r ρ / z ) + s = 1 i s J s ( k r ρ / z ) { exp [ + i s ( φ θ ) ] + exp [ i s ( φ θ ) ] } ,
0 exp ( a 0 2 t 2 ) t ν + 1 J ν ( b 0 t ) d t = b 0 ν ( 2 a 0 2 ) ν + 1 exp ( b 0 2 4 a 0 2 ) ( for Re ( ν ) > 1 ; Re ( a 0 2 ) > 0 ) ,
0 exp ( a 0 2 t 2 ) t J ν ( b 0 t ) d t = b 0 π 8 a 0 3 exp ( b 0 2 8 a 0 2 ) [ I ( ν 1 ) / 2 ( b 0 2 8 a 0 2 ) I ( ν + 1 ) / 2 ( b 0 2 8 a 0 2 ) ] ( for     Re ( 2 + ν ) > 0 ; Re ( a 0 2 ) > 0 ) ,
U 0 ( ρ , θ , z ) = J 0 ( k δ ) exp ( i k δ ) w 0 w ( z ) exp { i [ k ( z + ρ 2 2 R ( z ) ) arctan ( z / z 0 ) ] } exp ( ρ 2 w 2 ( z ) ) ,
U ± m ( ρ , θ , z ) = ( i ) m J m ( k δ ) exp ( i k δ ) π 2 w 0 w ( z ) [ 1 w 2 ( z ) i k 2 ( 1 z 1 R ( z ) ) ] 1 / 2 exp [ i ϕ ( z ) ] exp [ ± i m p ( θ + π / 2 ) ] ρ exp ( ρ 2 2 w 2 ( z ) ) [ I ( m p 1 ) / 2 ( x / 2 ) I ( m p + 1 ) / 2 ( x / 2 ) ] .
x = [ 1 w 2 ( z ) i k 2 ( 1 z 1 R ( z ) ) ] ρ 2 ,
k δ = y 0 , j , or     c = 2 y 0 , j / k ( n 1 ) ,
U + m ( ρ , θ , z ) + U m ( ρ , θ , z ) = ( i ) m J m ( k δ ) exp ( i k δ ) π w 0 w ( z ) { 1 w 2 ( z ) i k 2 ( 1 z 1 R ( z ) ) } 1 / 2 × exp [ i ϕ ( z ) ] cos [ m p ( θ + π / 2 ) ] ρ exp ( ρ 2 2 w 2 ( z ) ) [ I ( m p 1 ) / 2 ( x / 2 ) I ( m p + 1 ) / 2 ( x / 2 ) ] ,
U ( ρ , θ , z ) = w 0 w ( z ) exp ( i k δ ) exp { i k [ z + ρ 2 2 R ( z ) 1 k arctan ( z / z 0 ) ] } × { exp ( ρ 2 w 2 ( z ) ) J 0 ( k δ ) + π [ i k 2 ( 1 z 1 R ( z ) ) + 1 w 2 ( z ) ] 1 / 2 exp [ i k ρ 2 4 ( 1 R ( z ) 1 z ) ] × ρ exp ( ρ 2 2 w 2 ( z ) ) m = 1 ( i ) m J m ( k δ ) cos ( m p ( θ + π / 2 ) ) [ I ( m p 1 ) / 2 ( x / 2 ) I ( m p + 1 ) / 2 ( x / 2 ) ] } ,
I ( ρ , θ , z ) = | U ( ρ , θ , z ) | 2 = w 0 2 w 2 ( z ) π [ 1 w 4 ( z ) + k 2 4 ( 1 z 1 R ( z ) ) 2 ] 1 / 2 × ρ 2 exp ( ρ 2 w 2 ( z ) ) | m = 1 ( i ) m J m ( k δ ) cos ( m p ( θ + π / 2 ) ) [ I ( m p 1 ) / 2 ( x / 2 ) I ( m p + 1 ) / 2 ( x / 2 ) ] | 2 .
T ( r , φ ) = A ( r ) exp ( i k α 0 r ) exp { i k [ δ + δ cos ( p φ ) ] } .
A ( r ) = { 1 when     w 0 < R 0 , circ ( r / R 0 ) when     w 0 R 0 ,
circ ( r R 0 ) = { 1 , when     r R 0 0 , when     r > R 0 .
U ( ρ , θ , z ) = i k z exp [ i k ( z + ρ 2 / 2 z + δ ) ] { J 0 ( k δ ) Y 0 ( r ) + m = 1 ( i ) m i m p J m ( k δ ) Y m ( r ) [ exp ( + i m p θ ) + exp ( i m p θ ) ] } ,
Y 0 ( r ) = 0 A ( r ) exp [ i k ( r 2 2 z α 0 r ) ] J 0 ( k r ρ z ) exp ( r 2 w 0 2 ) r d r ,
Y m ( r ) = 0 A ( r ) exp [ i k ( r 2 2 z α 0 r ) ] J m p ( k r ρ z ) exp ( r 2 w 0 2 ) r d r .
Y 0 ( r c ) = α 0 z z k A ( r c ) J 0 ( k α 0 ρ ) exp [ z 2 ( w 0 / α 0 ) 2 ] exp ( i k z α 0 2 2 ) ,
Y m ( r c ) = α 0 z z k A ( r c ) J m p ( k α 0 ρ ) exp [ z 2 ( w 0 / α 0 ) 2 ] exp ( i k z α 0 2 2 ) .
U ( ρ , θ , z ) = i w 0 α 0 k 2 A ( r c ) ( z 2 w 0 / α 0 ) 1 / 2 exp ( z 2 w 0 2 / α 0 2 ) exp [ i k ( z α 0 2 2 z + ρ 2 2 z + δ ) ] { J 0 ( k δ ) J 0 ( k α 0 ρ ) + 2 m = 1 ( 1 ) m i m ( p + 1 ) J m ( k δ ) J m p ( k α 0 ρ ) cos ( m p θ ) } .
U ( ρ , θ , z ) = i w 0 α 0 k 2 A ( r c ) ( z 2 w 0 / α 0 ) 1 / 2 exp ( z 2 w 0 2 / α 0 2 ) exp [ i k ( z α 0 2 2 z + ρ 2 2 z + δ ) ] { J 0 ( k δ ) J 0 ( k α 0 ρ ) + 2 m = 1 ( 1 ) m ( p + 1 ) J m ( k δ ) J m ( 2 p 1 ) ( k α 0 ρ ) cos ( m ( 2 p 1 ) θ ) } .
A ( r c ) = { 1 when    w 0 < R 0 , circ ( r c / R 0 ) = circ ( α 0 z / R 0 ) when    w 0 R 0 .
I ( ρ , θ , z ) = 4 w 0 α 0 k 2 A ( r c ) ( 2 z 2 ( w 0 / α 0 ) 2 ) 1 / 2 exp ( 2 z 2 ( w 0 / α 0 ) 2 ) × [ m = 1 ( 1 ) m ( p + 1 ) J m ( k δ ) J m ( 2 p 1 ) ( k α 0 ρ ) cos ( m ( 2 p 1 ) θ ) ] 2 ,
I ( ρ , θ , z ) = 4 w 0 α 0 k 2 A ( r c ) I 1 / 2 ( z , w 0 / α 0 ) × [ m = 1 ( 1 ) m ( p + 1 ) J m ( k δ ) J m ( 2 p 1 ) ( k α 0 ρ ) cos ( m ( 2 p 1 ) θ ) ] 2 .

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