Abstract

The nonparaxial corrections for Bessel–Gauss beams were derived recently using two different approaches [ Borghi et al., J. Opt. Soc. Am. A 18, 1618 (2001) and Vaveliuk et al., J. Opt. Soc. Am. A 24, 3297 (2007 )]. However, the two obtained results do not agree, so it is necessary to determine which method is correct. In the most recent of those papers, Vaveliuk et al. claimed that their method is correct while the method described by Borghi et al. is incorrect. In the present work, just by solving the rigorous propagation problem, we show that exactly the converse is true.

© 2010 Optical Society of America

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References

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  1. R. Borghi, S. Santarsiero, and M. A. Porras, “Nonparaxial Bessel-Gauss beams,” J. Opt. Soc. Am. A 18, 1618-1625 (2001).
    [CrossRef]
  2. M. Lax, W. H. Louisell, and W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365-1370 (1975).
    [CrossRef]
  3. A. Wünsche, “Transition from the paraxial approximation to exact solutions of the wave equation and application to Gaussian beams,” J. Opt. Soc. Am. A 9, 765-774 (1992).
    [CrossRef]
  4. P. Vaveliuk, G. F. Zebende, M. A. Moret, and B. Ruiz, “Propagating free-space nonparaxial beams,” J. Opt. Soc. Am. A 24, 3297-3302 (2007).
    [CrossRef]
  5. M. Born and E. Wolf, Principles of Optics, 7th (expanded) ed. (Cambridge University Press, 2001), p. 514.
  6. R. K. Luneburg, Mathematical Theory of Optics (University of California Press, 1964), p. 316.
  7. M. Nieto-Vesperinas, Scattering and Diffraction in Physical Optics, 2nd ed. (World Scientific2006), p. 179.
  8. C. J. R. Sheppard and T. Wilson, “Gaussian-beam theory of lenses with annular aperture,” IEE J. Microwaves, Opt. Acoust. 2, 105-112 (1978).
    [CrossRef]
  9. F. Gori, G. Guattari, and C. Padovani, “Bessel-Gauss beams,” Opt. Commun. 64, 491-495 (1987).
    [CrossRef]
  10. O. El Gawhary and S. Severini, “Lorentz beams and symmetry properties in paraxial optics,” J. Opt A: Pure Appl. Opt. 8, 409-414 (2006).
    [CrossRef]
  11. P. Vaveliuk, B. Ruiz, and A. Lencina, “Limits of the paraxial approximation in laser beams,” Opt. Lett. 32, 927-929 (2007).
    [CrossRef] [PubMed]
  12. O. El Gawhary and S. Severini, “Reply to comment on degree of paraxiality for monochromatic light beams,” Opt. Lett. 33, 3006 (2008).
    [CrossRef]
  13. O. El Gawhary and S. Severini, “Degree of paraxiality for monochromatic light beams,” Opt. Lett. 33, 1360-1362 (2008).
    [CrossRef]
  14. J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 18, 3966-3969 (2000).
    [CrossRef]
  15. S. Kawata, T. Ono, and P. Verma, “Subwavelength colour imaging with a metallic nanolens,” Nature Photon. 2, 438-442 (2008).
    [CrossRef]
  16. F. Simonetti, “Multiple scattering: the key to unravel the subwavelength world from the far-field pattern of a scattered wave,” Phys. Rev. E 73, 036619 (13pp.) (2006).
    [CrossRef]
  17. C. J. R. Sheppard, M. A. Alonso, M. Santarsiero, and R. Borghi, “Comment on 'Do evanescent waves really exist in free space?'” Opt. Commun. 266, 448-449 (2006).
    [CrossRef]

2008 (3)

2007 (2)

2006 (3)

F. Simonetti, “Multiple scattering: the key to unravel the subwavelength world from the far-field pattern of a scattered wave,” Phys. Rev. E 73, 036619 (13pp.) (2006).
[CrossRef]

C. J. R. Sheppard, M. A. Alonso, M. Santarsiero, and R. Borghi, “Comment on 'Do evanescent waves really exist in free space?'” Opt. Commun. 266, 448-449 (2006).
[CrossRef]

O. El Gawhary and S. Severini, “Lorentz beams and symmetry properties in paraxial optics,” J. Opt A: Pure Appl. Opt. 8, 409-414 (2006).
[CrossRef]

2001 (1)

2000 (1)

J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 18, 3966-3969 (2000).
[CrossRef]

1992 (1)

1987 (1)

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

1978 (1)

C. J. R. Sheppard and T. Wilson, “Gaussian-beam theory of lenses with annular aperture,” IEE J. Microwaves, Opt. Acoust. 2, 105-112 (1978).
[CrossRef]

1975 (1)

M. Lax, W. H. Louisell, and W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365-1370 (1975).
[CrossRef]

Alonso, M. A.

C. J. R. Sheppard, M. A. Alonso, M. Santarsiero, and R. Borghi, “Comment on 'Do evanescent waves really exist in free space?'” Opt. Commun. 266, 448-449 (2006).
[CrossRef]

Borghi, R.

C. J. R. Sheppard, M. A. Alonso, M. Santarsiero, and R. Borghi, “Comment on 'Do evanescent waves really exist in free space?'” Opt. Commun. 266, 448-449 (2006).
[CrossRef]

R. Borghi, S. Santarsiero, and M. A. Porras, “Nonparaxial Bessel-Gauss beams,” J. Opt. Soc. Am. A 18, 1618-1625 (2001).
[CrossRef]

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th (expanded) ed. (Cambridge University Press, 2001), p. 514.

El Gawhary, O.

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]

Kawata, S.

S. Kawata, T. Ono, and P. Verma, “Subwavelength colour imaging with a metallic nanolens,” Nature Photon. 2, 438-442 (2008).
[CrossRef]

Lax, M.

M. Lax, W. H. Louisell, and W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365-1370 (1975).
[CrossRef]

Lencina, A.

Louisell, W. H.

M. Lax, W. H. Louisell, and W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365-1370 (1975).
[CrossRef]

Luneburg, R. K.

R. K. Luneburg, Mathematical Theory of Optics (University of California Press, 1964), p. 316.

McKnight, W. B.

M. Lax, W. H. Louisell, and W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365-1370 (1975).
[CrossRef]

Moret, M. A.

Nieto-Vesperinas, M.

M. Nieto-Vesperinas, Scattering and Diffraction in Physical Optics, 2nd ed. (World Scientific2006), p. 179.

Ono, T.

S. Kawata, T. Ono, and P. Verma, “Subwavelength colour imaging with a metallic nanolens,” Nature Photon. 2, 438-442 (2008).
[CrossRef]

Padovani, C.

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

Pendry, J. B.

J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 18, 3966-3969 (2000).
[CrossRef]

Porras, M. A.

Ruiz, B.

Santarsiero, M.

C. J. R. Sheppard, M. A. Alonso, M. Santarsiero, and R. Borghi, “Comment on 'Do evanescent waves really exist in free space?'” Opt. Commun. 266, 448-449 (2006).
[CrossRef]

Santarsiero, S.

Severini, S.

Sheppard, C. J. R.

C. J. R. Sheppard, M. A. Alonso, M. Santarsiero, and R. Borghi, “Comment on 'Do evanescent waves really exist in free space?'” Opt. Commun. 266, 448-449 (2006).
[CrossRef]

C. J. R. Sheppard and T. Wilson, “Gaussian-beam theory of lenses with annular aperture,” IEE J. Microwaves, Opt. Acoust. 2, 105-112 (1978).
[CrossRef]

Simonetti, F.

F. Simonetti, “Multiple scattering: the key to unravel the subwavelength world from the far-field pattern of a scattered wave,” Phys. Rev. E 73, 036619 (13pp.) (2006).
[CrossRef]

Vaveliuk, P.

Verma, P.

S. Kawata, T. Ono, and P. Verma, “Subwavelength colour imaging with a metallic nanolens,” Nature Photon. 2, 438-442 (2008).
[CrossRef]

Wilson, T.

C. J. R. Sheppard and T. Wilson, “Gaussian-beam theory of lenses with annular aperture,” IEE J. Microwaves, Opt. Acoust. 2, 105-112 (1978).
[CrossRef]

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 7th (expanded) ed. (Cambridge University Press, 2001), p. 514.

Wünsche, A.

Zebende, G. F.

IEE J. Microwaves, Opt. Acoust. (1)

C. J. R. Sheppard and T. Wilson, “Gaussian-beam theory of lenses with annular aperture,” IEE J. Microwaves, Opt. Acoust. 2, 105-112 (1978).
[CrossRef]

J. Opt A: Pure Appl. Opt. (1)

O. El Gawhary and S. Severini, “Lorentz beams and symmetry properties in paraxial optics,” J. Opt A: Pure Appl. Opt. 8, 409-414 (2006).
[CrossRef]

J. Opt. Soc. Am. A (3)

Nature Photon. (1)

S. Kawata, T. Ono, and P. Verma, “Subwavelength colour imaging with a metallic nanolens,” Nature Photon. 2, 438-442 (2008).
[CrossRef]

Opt. Commun. (2)

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

C. J. R. Sheppard, M. A. Alonso, M. Santarsiero, and R. Borghi, “Comment on 'Do evanescent waves really exist in free space?'” Opt. Commun. 266, 448-449 (2006).
[CrossRef]

Opt. Lett. (3)

Phys. Rev. A (1)

M. Lax, W. H. Louisell, and W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365-1370 (1975).
[CrossRef]

Phys. Rev. E (1)

F. Simonetti, “Multiple scattering: the key to unravel the subwavelength world from the far-field pattern of a scattered wave,” Phys. Rev. E 73, 036619 (13pp.) (2006).
[CrossRef]

Phys. Rev. Lett. (1)

J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 18, 3966-3969 (2000).
[CrossRef]

Other (3)

M. Born and E. Wolf, Principles of Optics, 7th (expanded) ed. (Cambridge University Press, 2001), p. 514.

R. K. Luneburg, Mathematical Theory of Optics (University of California Press, 1964), p. 316.

M. Nieto-Vesperinas, Scattering and Diffraction in Physical Optics, 2nd ed. (World Scientific2006), p. 179.

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Figures (1)

Fig. 1
Fig. 1

Field modulus plotted on the z axis as a function of the dimensionless variable z L : paraxial approximation (solid curve) and nonparaxial corrections (dotted curve) for a Bessel–Gauss input field distribution of the type exp ( r 2 w 0 2 ) J 0 ( β r ) . w 0 λ = 200 ( 2 π ) and β w 0 = 60 as in [4], Fig. 1. L = π w 0 2 λ is the Rayleigh distance. The nonparaxial corrections are evaluated through the Rayleigh–Sommerfeld formula of the first kind.

Equations (8)

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2 U ( x , y , z ) + k 2 U ( x , y , z ) = 0
U | ( x , y , z ) | z = 0 = U ( x , y , 0 )
U ( x , y , z ) = V ( x , y , z ) exp ( i k z ) ,
2 V ( x , y , z ) + 2 i k z V ( x , y , z ) = 0 ,
J 0 ( β r ) exp ( r 2 w 0 2 ) ,
U ( x , y , z ) = 1 2 π U ( ξ , η , 0 ) z [ exp ( i k r ) r ] d ξ d η ,
U parax ( x , y , z ) = [ 1 1 + i z L ] J 0 [ β r ( 1 + i z L ) ] exp [ r 2 w 0 2 ( 1 + i z L ) ] exp [ i β 2 z 2 k ( 1 + i z L ) ] exp ( i k z ) ,
U ( x , y , z ) = U parax ( x , y , z ) + Δ U ( x , y , z )

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