Abstract

The concept of the dual of a beam is discussed. The duals of Bessel-Gauss beams, elegant Hermite- or Laguerre-Gaussian beams and generalized Hermite- or Laguerre-Gauss beams are described. Duality is considered within the framework of hypergeometric beams in Cartesian and polar coordinates. The connection with the “modified Laguerre-Gauss” beam is discussed.

© 2009 Optical Society of America

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  1. S. A. Collins, "Lens-system diffraction integral written in terms of matrix optics," J. Opt. Soc. Am. A 60, 1168-1177 (1970).
    [CrossRef]
  2. A. W. Lohmann, "Ein neues Dualitatsprinzip in der Optik," Optik 11, 478-488 (1954).
  3. A. W. Lohmann, "Duality in optics," Optik 89, 93-97 (1992).
  4. F. Riesz and B. Szökefalvi-Nagy, Functional Analysis (Dover, New York, 1990).
  5. M. J. Caola, "Self-Fourier functions," J. Phys. A 24, L1143-L1144 (1991).
    [CrossRef]
  6. A. W. Lohmann and D. Mendlovic, "Self-Fourier objects and other self-transform objects," J. Opt. Soc. Am. A 9, 2009-2012 (1992).
    [CrossRef]
  7. C. J. R. Sheppard and S. Saghafi, "Flattened light beams," Opt. Comm. 132, 144-152 (1996).
    [CrossRef]
  8. C. J. R. Sheppard and T. Wilson, "Gaussian-beam theory of lenses with annular aperture," IEE J. Microwaves Opt. Acoustics 2, 105-112 (1978).
    [CrossRef]
  9. F. Gori, G. Guatteri, and C. Padovani, "Bessel-Gauss beams," Opt. Comm. 64, 491-495 (1987).
    [CrossRef]
  10. J. Durnin, J. J. Miceli, Jr., and J. H. Eberly, "Diffraction-free beams," Phys. Rev. Lett. 58, 1499-1501 (1987).
    [CrossRef] [PubMed]
  11. V. Bagini, F. Frezza, M. Santarsiero, G. Schettini, and G. Shirripa Spagnolo, "Generalized Bessel-Gauss beams," J. Mod. Optics 43, 1155-1166 (1996).
  12. H. Kogelnik and T. Li, "Laser beams and resonators," App. Opt. 5, 1550-1567 (1966).
    [CrossRef]
  13. A. Wünsche, "Analogien zwischen ausserordentlichen und ordentlichen Wellen nach nichtorthogonaler Koordinatentransformation und die parabolischen Näherungsgleichungen," Ann. Phys. 25, 113-135 (1970).
    [CrossRef]
  14. A. Wünsche, "Generalized Gaussian beam solutions of paraxial optics and their connections to a hidden symmetry," J. Opt. Soc. Am. A 6, 1320-1329 (1989).
    [CrossRef]
  15. C. J. R. Sheppard, "High aperture beams," Journal of the Optical Society of America A 18, 1579-1587 (2001).
    [CrossRef]
  16. A. E. Siegman, "Hermite-Gaussian functions of complex arguments as optical-beam eigenfunctions," J. Opt. Soc. Am. 63, 1093-1094 (1973).
    [CrossRef]
  17. R. Pratesi and L. Ronchi, "Generalized Gaussian beams in free space," J. Opt. Soc. Am. 67, 1274-1276 (1977).
    [CrossRef]
  18. I. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series, and Products (Academic Press, New York, 1994).
  19. S. Saghafi and C. J. R. Sheppard, "Near field and far field of elegant Hermite-Gaussian and Laguerre-Gaussian modes," J. Mod. Optics 45, 1999-2009 (1998).
    [CrossRef]
  20. M. Porras, R. Borghi, and M. Santarsiero, "Relationship between elegant Laguerre-Gauss and Bessel-Gauss beams," J. Opt. Soc. Am. A 18, 177-184 (2001).
    [CrossRef]
  21. M. A. Bandres and J. C. Gutierrez-Vega, "Circular beams," Opt. Lett. 33, 177-179 (2008).
    [CrossRef] [PubMed]
  22. V. V. Kotlyar, R. V. Skidanov, S. N. Khonina, and V. A. Soifer, "Hypergeometric modes," Opt. Lett. 32, 742-744 (2007).
    [CrossRef] [PubMed]
  23. E. Karimi, G. Zito, B. Piccirillo, L. Marrucci, and E. Santamato, "Hypergeometric-Gaussian beams," Opt. Lett. 32, 3053-3055 (2007).
    [CrossRef] [PubMed]
  24. M. A. Bandres and J. C. Gutierrez-Vega, "Cartesian beams," Opt. Lett. 32, 3459-3461 (2007).25.
    [CrossRef] [PubMed]
  25. J. C. Gutiérrez-Vega, "Fractionalization of optical beams: I. Planar analysis," Opt. Lett. 11, 1521-1523 (2007).
    [CrossRef]
  26. J. C. Gutiérrez-Vega, "Fractionalization of optical beams: II. Elegant Laguerre-Gaussian modes," Optics Express 15, 6300-6313 (2007).
    [CrossRef] [PubMed]
  27. C. F. R. Caron and R. M. Potvliege, "Bessel-modulated Gaussian beams with quadratic radial dependence," Opt. Comm. 164, 83-93 (1999).
    [CrossRef]

2008 (1)

2007 (4)

V. V. Kotlyar, R. V. Skidanov, S. N. Khonina, and V. A. Soifer, "Hypergeometric modes," Opt. Lett. 32, 742-744 (2007).
[CrossRef] [PubMed]

E. Karimi, G. Zito, B. Piccirillo, L. Marrucci, and E. Santamato, "Hypergeometric-Gaussian beams," Opt. Lett. 32, 3053-3055 (2007).
[CrossRef] [PubMed]

J. C. Gutiérrez-Vega, "Fractionalization of optical beams: I. Planar analysis," Opt. Lett. 11, 1521-1523 (2007).
[CrossRef]

J. C. Gutiérrez-Vega, "Fractionalization of optical beams: II. Elegant Laguerre-Gaussian modes," Optics Express 15, 6300-6313 (2007).
[CrossRef] [PubMed]

2001 (2)

1999 (1)

C. F. R. Caron and R. M. Potvliege, "Bessel-modulated Gaussian beams with quadratic radial dependence," Opt. Comm. 164, 83-93 (1999).
[CrossRef]

1998 (1)

S. Saghafi and C. J. R. Sheppard, "Near field and far field of elegant Hermite-Gaussian and Laguerre-Gaussian modes," J. Mod. Optics 45, 1999-2009 (1998).
[CrossRef]

1996 (2)

C. J. R. Sheppard and S. Saghafi, "Flattened light beams," Opt. Comm. 132, 144-152 (1996).
[CrossRef]

V. Bagini, F. Frezza, M. Santarsiero, G. Schettini, and G. Shirripa Spagnolo, "Generalized Bessel-Gauss beams," J. Mod. Optics 43, 1155-1166 (1996).

1992 (2)

1991 (1)

M. J. Caola, "Self-Fourier functions," J. Phys. A 24, L1143-L1144 (1991).
[CrossRef]

1989 (1)

1987 (2)

F. Gori, G. Guatteri, and C. Padovani, "Bessel-Gauss beams," Opt. Comm. 64, 491-495 (1987).
[CrossRef]

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

1978 (1)

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

1977 (1)

1973 (1)

1970 (2)

A. Wünsche, "Analogien zwischen ausserordentlichen und ordentlichen Wellen nach nichtorthogonaler Koordinatentransformation und die parabolischen Näherungsgleichungen," Ann. Phys. 25, 113-135 (1970).
[CrossRef]

S. A. Collins, "Lens-system diffraction integral written in terms of matrix optics," J. Opt. Soc. Am. A 60, 1168-1177 (1970).
[CrossRef]

1966 (1)

H. Kogelnik and T. Li, "Laser beams and resonators," App. Opt. 5, 1550-1567 (1966).
[CrossRef]

1954 (1)

A. W. Lohmann, "Ein neues Dualitatsprinzip in der Optik," Optik 11, 478-488 (1954).

Bagini, V.

V. Bagini, F. Frezza, M. Santarsiero, G. Schettini, and G. Shirripa Spagnolo, "Generalized Bessel-Gauss beams," J. Mod. Optics 43, 1155-1166 (1996).

Bandres, M. A.

Borghi, R.

Caola, M. J.

M. J. Caola, "Self-Fourier functions," J. Phys. A 24, L1143-L1144 (1991).
[CrossRef]

Caron, C. F. R.

C. F. R. Caron and R. M. Potvliege, "Bessel-modulated Gaussian beams with quadratic radial dependence," Opt. Comm. 164, 83-93 (1999).
[CrossRef]

Collins, S. A.

S. A. Collins, "Lens-system diffraction integral written in terms of matrix optics," J. Opt. Soc. Am. A 60, 1168-1177 (1970).
[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]

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]

Frezza, F.

V. Bagini, F. Frezza, M. Santarsiero, G. Schettini, and G. Shirripa Spagnolo, "Generalized Bessel-Gauss beams," J. Mod. Optics 43, 1155-1166 (1996).

Gori, F.

F. Gori, G. Guatteri, and C. Padovani, "Bessel-Gauss beams," Opt. Comm. 64, 491-495 (1987).
[CrossRef]

Guatteri, G.

F. Gori, G. Guatteri, and C. Padovani, "Bessel-Gauss beams," Opt. Comm. 64, 491-495 (1987).
[CrossRef]

Gutierrez-Vega, J. C.

Gutiérrez-Vega, J. C.

J. C. Gutiérrez-Vega, "Fractionalization of optical beams: II. Elegant Laguerre-Gaussian modes," Optics Express 15, 6300-6313 (2007).
[CrossRef] [PubMed]

J. C. Gutiérrez-Vega, "Fractionalization of optical beams: I. Planar analysis," Opt. Lett. 11, 1521-1523 (2007).
[CrossRef]

Karimi, E.

Khonina, S. N.

Kogelnik, H.

H. Kogelnik and T. Li, "Laser beams and resonators," App. Opt. 5, 1550-1567 (1966).
[CrossRef]

Kotlyar, V. V.

Li, T.

H. Kogelnik and T. Li, "Laser beams and resonators," App. Opt. 5, 1550-1567 (1966).
[CrossRef]

Lohmann, A. W.

A. W. Lohmann and D. Mendlovic, "Self-Fourier objects and other self-transform objects," J. Opt. Soc. Am. A 9, 2009-2012 (1992).
[CrossRef]

A. W. Lohmann, "Duality in optics," Optik 89, 93-97 (1992).

A. W. Lohmann, "Ein neues Dualitatsprinzip in der Optik," Optik 11, 478-488 (1954).

Marrucci, L.

Mendlovic, D.

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]

Padovani, C.

F. Gori, G. Guatteri, and C. Padovani, "Bessel-Gauss beams," Opt. Comm. 64, 491-495 (1987).
[CrossRef]

Piccirillo, B.

Porras, M.

Potvliege, R. M.

C. F. R. Caron and R. M. Potvliege, "Bessel-modulated Gaussian beams with quadratic radial dependence," Opt. Comm. 164, 83-93 (1999).
[CrossRef]

Pratesi, R.

Ronchi, L.

Saghafi, S.

S. Saghafi and C. J. R. Sheppard, "Near field and far field of elegant Hermite-Gaussian and Laguerre-Gaussian modes," J. Mod. Optics 45, 1999-2009 (1998).
[CrossRef]

C. J. R. Sheppard and S. Saghafi, "Flattened light beams," Opt. Comm. 132, 144-152 (1996).
[CrossRef]

Santamato, E.

Santarsiero, M.

M. Porras, R. Borghi, and M. Santarsiero, "Relationship between elegant Laguerre-Gauss and Bessel-Gauss beams," J. Opt. Soc. Am. A 18, 177-184 (2001).
[CrossRef]

V. Bagini, F. Frezza, M. Santarsiero, G. Schettini, and G. Shirripa Spagnolo, "Generalized Bessel-Gauss beams," J. Mod. Optics 43, 1155-1166 (1996).

Schettini, G.

V. Bagini, F. Frezza, M. Santarsiero, G. Schettini, and G. Shirripa Spagnolo, "Generalized Bessel-Gauss beams," J. Mod. Optics 43, 1155-1166 (1996).

Sheppard, C. J. R.

C. J. R. Sheppard, "High aperture beams," Journal of the Optical Society of America A 18, 1579-1587 (2001).
[CrossRef]

S. Saghafi and C. J. R. Sheppard, "Near field and far field of elegant Hermite-Gaussian and Laguerre-Gaussian modes," J. Mod. Optics 45, 1999-2009 (1998).
[CrossRef]

C. J. R. Sheppard and S. Saghafi, "Flattened light beams," Opt. Comm. 132, 144-152 (1996).
[CrossRef]

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

Shirripa Spagnolo, G.

V. Bagini, F. Frezza, M. Santarsiero, G. Schettini, and G. Shirripa Spagnolo, "Generalized Bessel-Gauss beams," J. Mod. Optics 43, 1155-1166 (1996).

Siegman, A. E.

Skidanov, R. V.

Soifer, V. A.

Wilson, T.

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

Wünsche, A.

A. Wünsche, "Generalized Gaussian beam solutions of paraxial optics and their connections to a hidden symmetry," J. Opt. Soc. Am. A 6, 1320-1329 (1989).
[CrossRef]

A. Wünsche, "Analogien zwischen ausserordentlichen und ordentlichen Wellen nach nichtorthogonaler Koordinatentransformation und die parabolischen Näherungsgleichungen," Ann. Phys. 25, 113-135 (1970).
[CrossRef]

Zito, G.

Ann. Phys. (1)

A. Wünsche, "Analogien zwischen ausserordentlichen und ordentlichen Wellen nach nichtorthogonaler Koordinatentransformation und die parabolischen Näherungsgleichungen," Ann. Phys. 25, 113-135 (1970).
[CrossRef]

App. Opt. (1)

H. Kogelnik and T. Li, "Laser beams and resonators," App. Opt. 5, 1550-1567 (1966).
[CrossRef]

J. Mod. Optics (2)

V. Bagini, F. Frezza, M. Santarsiero, G. Schettini, and G. Shirripa Spagnolo, "Generalized Bessel-Gauss beams," J. Mod. Optics 43, 1155-1166 (1996).

S. Saghafi and C. J. R. Sheppard, "Near field and far field of elegant Hermite-Gaussian and Laguerre-Gaussian modes," J. Mod. Optics 45, 1999-2009 (1998).
[CrossRef]

J. Opt. Soc. Am. (2)

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

J. Phys. A (1)

M. J. Caola, "Self-Fourier functions," J. Phys. A 24, L1143-L1144 (1991).
[CrossRef]

Journal of the Optical Society of America A (1)

C. J. R. Sheppard, "High aperture beams," Journal of the Optical Society of America A 18, 1579-1587 (2001).
[CrossRef]

Opt. Comm. (3)

C. J. R. Sheppard and S. Saghafi, "Flattened light beams," Opt. Comm. 132, 144-152 (1996).
[CrossRef]

F. Gori, G. Guatteri, and C. Padovani, "Bessel-Gauss beams," Opt. Comm. 64, 491-495 (1987).
[CrossRef]

C. F. R. Caron and R. M. Potvliege, "Bessel-modulated Gaussian beams with quadratic radial dependence," Opt. Comm. 164, 83-93 (1999).
[CrossRef]

Opt. Lett. (4)

Optics and Acoustics (1)

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

Optics Express (1)

J. C. Gutiérrez-Vega, "Fractionalization of optical beams: II. Elegant Laguerre-Gaussian modes," Optics Express 15, 6300-6313 (2007).
[CrossRef] [PubMed]

Optik (2)

A. W. Lohmann, "Ein neues Dualitatsprinzip in der Optik," Optik 11, 478-488 (1954).

A. W. Lohmann, "Duality in optics," Optik 89, 93-97 (1992).

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]

Other (3)

F. Riesz and B. Szökefalvi-Nagy, Functional Analysis (Dover, New York, 1990).

M. A. Bandres and J. C. Gutierrez-Vega, "Cartesian beams," Opt. Lett. 32, 3459-3461 (2007).25.
[CrossRef] [PubMed]

I. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series, and Products (Academic Press, New York, 1994).

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Figures (4)

Fig. 1.
Fig. 1.

The intensity in the focal region for BG beams for b = 10 (a, b), and for the dual of BG, dBG, for b’ = 10 (c).

Fig. 2.
Fig. 2.

The intensity in the waist for HG beams for n = 5 for different real values of the parameter c.

Fig. 3.
Fig. 3.

The intensity in the focal region for HG beams with n = 5 for different real values of the parameter c.

Fig. 4.
Fig. 4.

The phase in the focal region for HG beams for the case when kz 0 = 5 and n = 5 . Phase is shown wrapped, between -π (black) and π (white), with zero as mid gray. For c = -1 (deHG) there is a phase singularity at the origin.

Equations (27)

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

B G m ( r , z ) = 1 μ J m ( br μ w 0 ) exp ( r 2 μ w 0 2 b 2 4 + b 2 4 μ + imϕ ) ,
dB G m ( r , z ) = 1 μ { I m ( b r μ w 0 ) exp ( b r μ w 0 ) } exp [ 1 μ ( r w 0 b 2 ) 2 + b 2 4 + imϕ ] .
gH G n ( x , z ) = v n / 2 μ n + 1 / 2 H n [ x w 0 ( 2 ν ) 1 / 2 ] exp ( x 2 w 0 2 μ )
gH G n ( x , z ) = ( c ) n / 2 ( n / 2 ) ! 1 μ 1 / 2 exp ( x 2 w 0 2 μ ) , n even
= ( c ) n / 2 [ ( n 2 ) / 2 ] ! 1 μ 3 / 2 2 2 x w 0 exp ( x 2 w 0 2 μ ) , n odd .
gH G n ( x , z ) = 1 i n + 1 / 2 ( 1 c ) n / 2 k w 0 2 2 z H n [ k w 0 ( x / z ) 2 ( 1 c ) 1 / 2 ] exp { [ k w 0 ( x / z ) 2 ] 2 + ik x 2 2 z } .
lim ε 0 ε n H n ( y ε ) = 2 n y n ,
eH G n ( x , z ) = 1 i n + 1 / 2 k w 0 2 2 z [ 2 k w 0 ( x / z ) ] n exp { [ k w 0 ( x / z ) 2 ] 2 + ik x 2 2 z } .
( 1 c ) n / 2 H n [ k w 0 ( x / z ) 2 ( 1 c ) 1 / 2 ]
gH G n ( x , 0 ) = ( 1 + c ) n / 2 H n [ 2 x w 0 ( 1 + c ) 1 / 2 ] exp ( x 2 w 0 2 ) .
deH G n ( x , 0 ) = ( 2 2 x w 0 ) n exp ( x 2 w 0 2 ) .
deH G n ( x , z ) = ( 2 iz ) n / 2 μ ( n + 1 ) / 2 H n [ x w 0 ( i Z μ ) 1 / 2 ] exp ( x 2 w 0 2 μ ) .
gH G nm ( r , ϕ , z ) = v ( 2 n + m ) / 2 μ 2 n + m + 1 ( 2 r 2 w 0 2 ν ) m / 2 L n m ( 2 r 2 w 0 2 ν ) exp ( r 2 w 0 2 μ + imϕ ) .
gH G nm ( r , ϕ , 0 ) = ( 1 + c ) n ( 2 r 2 w 0 2 ) m / 2 L n m ( 2 r 2 w 0 2 ( 1 + c ) ) exp ( r 2 w 0 2 + imϕ ) .
gH G nm ( r , ϕ , 0 ) = ( i ) 2 n + m + 1 k w 0 2 2 z ( 1 c ) n ( k 2 r 2 / z 2 2 w 0 2 ) m / 2
× L n m [ k 2 r 2 / z 2 2 w 0 2 ( 1 c ) ] exp { [ k w 0 ( r / z ) 2 ] 2 + ik r 2 2 z + imϕ } .
lim ε 0 ε n L n m ( y 4 ε ) = ( 1 ) n y 2 n 2 2 n n !
eL G nm ( r , ϕ , z ) = ( i ) 2 n + m + 1 n ! k w 0 2 2 z ( k 2 r 2 / z 2 2 w 0 2 ) n + m / 2 exp { [ k w 0 ( r / z ) 2 ] 2 + ik r 2 2 z + imϕ } .
deL G nm ( r , ϕ , z ) = ( 2 iZ μ ) n + m / 2 1 μ ( i r 2 w 0 2 ) m / 2 L n m ( i r 2 w 0 2 ) exp ( r 2 w 0 2 μ + imϕ ) .
deL G nm ( r , ϕ , 0 ) = 1 n ! ( 2 r 2 w 0 2 ) n + m / 2 exp ( r 2 w 0 2 + imϕ ) .
H 2 n ( x ) exp ( x 2 2 ) ( ) n 4 n n ! πn cos ( 2 nx )
H 2 n + 1 ( x ) exp ( x 2 2 ) ( ) n 4 n n ! π sin ( 2 nx )
ν d = ν + c ( ν 4 μ ) 1 c ,
χ d 2 χ 2 = 1 4 ( 1 + c ) ν .
gH G nm ( x , z ) = ν n / 2 μ m + 1 / 2 { 1 Γ [ ( 1 2 ) / 2 ] 1 F 1 ( n 2 , 1 2 ; k x 2 ν z 0 )
2 Γ [ ( n / 2 ) ] ( k x 2 ν z 0 ) 1 / 2 1 F 1 ( 1 n 2 , 3 2 ; k x 2 ν z 0 ) } exp ( k x 2 2 μ z 0 ) ,
gH G nm ( x , z , ϕ , ) = ν n + m / 2 μ 2 n + m + 1 ( k r 2 ν z 0 ) m / 2 1 F 1 ( n , 1 + m ; k r 2 ν z 0 ) exp ( k r 2 2 μ z 0 + imϕ ) ,

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