Abstract

The study of the nonparaxial propagation of optical beams has received considerable attention. In particular, the so-called complex-source/sink model can be used to describe strongly focused beams near the beam waist, but this method has not yet been applied to the Bessel–Gauss (BG) beam. In this paper, the complex-source/sink solution for the nonparaxial BG beam is expressed as a superposition of nonparaxial elegant Laguerre–Gaussian beams. This provides a direct way to write the explicit expression for a tightly focused BG beam that is an exact solution of the Helmholtz equation. It reduces correctly to the paraxial BG beam, the nonparaxial Gaussian beam, and the Bessel beam in the appropriate limits. The analytical expression can be used to calculate the field of a BG beam near its waist, and it may be useful in investigating the features of BG beams under tight focusing conditions.

© 2011 Optical Society of America

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  1. J. Durnin, J. J. Miceli, Jr., and J. H. Eberly, “Nondiffracting beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
    [CrossRef] [PubMed]
  2. J. Durnin, “Exact solutions for nondiffracting beams. I. the scalar theory,” J. Opt. Soc. Am. A 4, 651–654 (1987).
    [CrossRef]
  3. C. J. R. Sheppard and T. Wilson, “Gaussian-beam theory of lenses with annular aperture,” IEE J. Microw. Opt. Acoust. 2, 105–112 (1978).
    [CrossRef]
  4. C. J. R. Sheppard, “Beam duality, with application to generalized Bessel–Gaussian, and Hermite– and Laguerre–Gaussian beams,” Opt. Express 17, 3690–3697 (2009).
    [CrossRef] [PubMed]
  5. F. Gori, G. Guattari, and C. Padovani, “Bessel–Gauss beams,” Opt. Commun. 64, 491–495 (1987).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  14. C. J. R. Sheppard and S. Saghafi, “Beam modes beyond the paraxial approximation: a scalar treatment,” Phys. Rev. A 57, 2971–2979 (1998).
    [CrossRef]
  15. Z. Ulanowski and I. K. Ludlow, “Scalar field of nonparaxial Gaussian beams,” Opt. Lett. 25, 1792–1794 (2000).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]

2010 (2)

2009 (1)

2008 (3)

2007 (1)

2004 (1)

2002 (1)

2001 (4)

2000 (2)

1998 (1)

C. J. R. Sheppard and S. Saghafi, “Beam modes beyond the paraxial approximation: a scalar treatment,” Phys. Rev. A 57, 2971–2979 (1998).
[CrossRef]

1996 (2)

M. Santarsiero, “Propagation of generalized Bessel–Gauss beams through ABCD optical systems,” Opt. Commun. 132, 1–7 (1996).
[CrossRef]

V. Bagini, F. Frezza, M. Santarsiero, G. Schettini, and G. S. Spagnolo, “Generalized Bessel–Gauss beams,” J. Mod. Opt. 43, 1155–1166 (1996).

1987 (3)

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

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

J. Durnin, “Exact solutions for nondiffracting beams. I. the scalar theory,” J. Opt. Soc. Am. A 4, 651–654 (1987).
[CrossRef]

1981 (1)

M. Couture and P. A. Bélanger, “From Gaussian beam to complex-source-point spherical wave,” Phys. Rev. A 24, 355–359(1981).
[CrossRef]

1979 (1)

L. W. Davis, “Theory of electromagnetic beams,” Phys. Rev. A 19, 1177–1779 (1979).
[CrossRef]

1978 (1)

C. J. R. Sheppard and T. Wilson, “Gaussian-beam theory of lenses with annular aperture,” IEE J. Microw. 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]

1971 (1)

G. A. Deschamps, “Gaussian beam as a bundle of complex rays,” Electron. Lett. 7, 684–685 (1971).
[CrossRef]

1959 (1)

B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems. II. structure of the image field in an aplanatic system,” Proc. R. Soc. London. Ser. A 253, 358–379 (1959).
[CrossRef]

April, A.

Bagini, V.

V. Bagini, F. Frezza, M. Santarsiero, G. Schettini, and G. S. Spagnolo, “Generalized Bessel–Gauss beams,” J. Mod. Opt. 43, 1155–1166 (1996).

Bélanger, P. A.

M. Couture and P. A. Bélanger, “From Gaussian beam to complex-source-point spherical wave,” Phys. Rev. A 24, 355–359(1981).
[CrossRef]

Bokor, N.

N. Bokor and N. Davidson, “4π Focusing with single paraboloid mirror,” Opt. Commun. 281, 5499–5503 (2008).
[CrossRef]

Borghi, R.

Born, M.

M. Born and E. Wolf, Principles of Optics Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 5th ed. (Pergamon, 1975), Sec. 2.2.2.
[PubMed]

Chávez-Cerda, S.

Couture, M.

M. Couture and P. A. Bélanger, “From Gaussian beam to complex-source-point spherical wave,” Phys. Rev. A 24, 355–359(1981).
[CrossRef]

Davidson, N.

N. Bokor and N. Davidson, “4π Focusing with single paraboloid mirror,” Opt. Commun. 281, 5499–5503 (2008).
[CrossRef]

Davis, L. W.

L. W. Davis, “Theory of electromagnetic beams,” Phys. Rev. A 19, 1177–1779 (1979).
[CrossRef]

Deschamps, G. A.

G. A. Deschamps, “Gaussian beam as a bundle of complex rays,” Electron. Lett. 7, 684–685 (1971).
[CrossRef]

Durnin, J.

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

J. Durnin, “Exact solutions for nondiffracting beams. I. the scalar theory,” J. Opt. Soc. Am. A 4, 651–654 (1987).
[CrossRef]

Eberly, J. H.

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

Felsen, L. B.

Frezza, F.

V. Bagini, F. Frezza, M. Santarsiero, G. Schettini, and G. S. Spagnolo, “Generalized Bessel–Gauss beams,” J. Mod. Opt. 43, 1155–1166 (1996).

Gawhary, O. E.

Gori, F.

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

Gradshteyn, I. S.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, 1980).

Guattari, G.

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

Heyman, E.

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]

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]

Ludlow, I. K.

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]

Miceli, J. J.

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

Moret, M. A.

Padovani, C.

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

Piché, M.

Porras, M. A.

Richards, B.

B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems. II. structure of the image field in an aplanatic system,” Proc. R. Soc. London. Ser. A 253, 358–379 (1959).
[CrossRef]

Rodríguez-Morales, G.

Ruiz, B.

Ryzhik, I. M.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, 1980).

Saghafi, S.

C. J. R. Sheppard and S. Saghafi, “Beam modes beyond the paraxial approximation: a scalar treatment,” Phys. Rev. A 57, 2971–2979 (1998).
[CrossRef]

Santarsiero, M.

R. Borghi, M. Santarsiero, and M. A. Porras, “Nonparaxial Bessel–Gauss beams,” J. Opt. Soc. Am. A 18, 1618–1624(2001).
[CrossRef]

M. A. 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. S. Spagnolo, “Generalized Bessel–Gauss beams,” J. Mod. Opt. 43, 1155–1166 (1996).

M. Santarsiero, “Propagation of generalized Bessel–Gauss beams through ABCD optical systems,” Opt. Commun. 132, 1–7 (1996).
[CrossRef]

Schettini, G.

V. Bagini, F. Frezza, M. Santarsiero, G. Schettini, and G. S. Spagnolo, “Generalized Bessel–Gauss beams,” J. Mod. Opt. 43, 1155–1166 (1996).

Seshadri, S. R.

Severini, S.

Sheppard, C. J. R.

C. J. R. Sheppard, “Beam duality, with application to generalized Bessel–Gaussian, and Hermite– and Laguerre–Gaussian beams,” Opt. Express 17, 3690–3697 (2009).
[CrossRef] [PubMed]

C. J. R. Sheppard, “High-aperture beams,” J. Opt. Soc. Am. A 18, 1579–1587 (2001).
[CrossRef]

C. J. R. Sheppard, “Polarization of almost-plane waves,” J. Opt. Soc. Am. A 17, 335–341 (2000).
[CrossRef]

C. J. R. Sheppard and S. Saghafi, “Beam modes beyond the paraxial approximation: a scalar treatment,” Phys. Rev. A 57, 2971–2979 (1998).
[CrossRef]

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

Siegman, A. E.

A. E. Siegman, Lasers (University Science Books, 1986), p. 628.

Spagnolo, G. S.

V. Bagini, F. Frezza, M. Santarsiero, G. Schettini, and G. S. Spagnolo, “Generalized Bessel–Gauss beams,” J. Mod. Opt. 43, 1155–1166 (1996).

Ulanowski, Z.

Vaveliuk, P.

Wilson, T.

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

Wolf, E.

B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems. II. structure of the image field in an aplanatic system,” Proc. R. Soc. London. Ser. A 253, 358–379 (1959).
[CrossRef]

M. Born and E. Wolf, Principles of Optics Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 5th ed. (Pergamon, 1975), Sec. 2.2.2.
[PubMed]

Zebende, G. F.

Electron. Lett. (1)

G. A. Deschamps, “Gaussian beam as a bundle of complex rays,” Electron. Lett. 7, 684–685 (1971).
[CrossRef]

IEE J. Microw. Opt. Acoust. (1)

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

J. Mod. Opt. (1)

V. Bagini, F. Frezza, M. Santarsiero, G. Schettini, and G. S. Spagnolo, “Generalized Bessel–Gauss beams,” J. Mod. Opt. 43, 1155–1166 (1996).

J. Opt. Soc. Am. A (8)

Opt. Commun. (3)

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

N. Bokor and N. Davidson, “4π Focusing with single paraboloid mirror,” Opt. Commun. 281, 5499–5503 (2008).
[CrossRef]

M. Santarsiero, “Propagation of generalized Bessel–Gauss beams through ABCD optical systems,” Opt. Commun. 132, 1–7 (1996).
[CrossRef]

Opt. Express (2)

Opt. Lett. (5)

Phys. Rev. A (4)

L. W. Davis, “Theory of electromagnetic beams,” Phys. Rev. A 19, 1177–1779 (1979).
[CrossRef]

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

M. Couture and P. A. Bélanger, “From Gaussian beam to complex-source-point spherical wave,” Phys. Rev. A 24, 355–359(1981).
[CrossRef]

C. J. R. Sheppard and S. Saghafi, “Beam modes beyond the paraxial approximation: a scalar treatment,” Phys. Rev. A 57, 2971–2979 (1998).
[CrossRef]

Phys. Rev. Lett. (1)

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

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

B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems. II. structure of the image field in an aplanatic system,” Proc. R. Soc. London. Ser. A 253, 358–379 (1959).
[CrossRef]

Other (3)

A. E. Siegman, Lasers (University Science Books, 1986), p. 628.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, 1980).

M. Born and E. Wolf, Principles of Optics Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 5th ed. (Pergamon, 1975), Sec. 2.2.2.
[PubMed]

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Figures (6)

Fig. 1
Fig. 1

The Bessel beam can be viewed as the superposition of plane waves whose wave vectors k lie on a cone whose half-aperture angle is α 0 and whose axis of symmetry is the z axis.

Fig. 2
Fig. 2

The BG beam can be seen as the superposition of Gaussian beams whose mean directions of propagation k ¯ are distributed on a cone of angular half-aperture α 0 whose axis of symmetry is the z axis.

Fig. 3
Fig. 3

The confocal vector a, which is aligned with the propagation axis of the Gaussian beam, is defined by its magnitude a = | a | and two spherical angular coordinates, α 0 and β 0 . The radial and longitudinal components of the vector a are a r and a z , respectively.

Fig. 4
Fig. 4

Square modulus at the beam waist (at z = 0 ) of the nonparaxial BG beam, the paraxial BG beam, and the Bessel beam for which m = 0 , with (a)  k a = 20 and α 0 = 10 ° , (b)  k a = 20 and α 0 = 90 ° , and (c)  k a = 1 and α 0 = 90 ° .

Fig. 5
Fig. 5

Square modulus in the r z plane of beams for which m = 0 , with k a = 1 and α 0 = 90 ° . (a) Nonparaxial BG beam, (b) paraxial BG beam, and (c) Bessel beam. Minimum and maximum intensities correspond to black and white, respectively.

Fig. 6
Fig. 6

Square modulus in the r z plane of beams for which m = 0 , with k a = 1 and α 0 = 90 ° . (a) Type 3 nonparaxial BG beam defined by Eq. (13) and (b) Type 2 nonparaxial BG beam. Minimum and maximum intensities correspond to black and white, respectively.

Equations (27)

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b ˜ m e ( r ) = j z R q ˜ ( z ) J m ( j z R h r q ˜ ( z ) ) exp ( h 2 z R z 2 k q ˜ ( z ) j k r 2 2 q ˜ ( z ) j k z ) cos ( m ϕ ) ,
0 exp ( u k r 2 ) I m ( 2 v k r ) J m ( 2 w k r ) k r d k r = 1 2 u exp ( v 2 u w 2 u ) J m ( 2 v w u ) ,
b ˜ m e ( r ) = z R k exp ( h 2 z R 2 k j k z ) cos ( m ϕ ) 0 I m ( z R h k k r ) exp ( j q ˜ ( z ) 2 k k r 2 ) J m ( k r r ) k r d k r .
B ˜ m e ( r ) = 2 a exp ( k a ) cos ( m ϕ ) 0 I m ( k r a r ) cos [ k z ( z + j a z ) ] k z J m ( k r r ) k r d k r ,
lim k a 1 α 0 1 { 2 k exp ( k a ) cos [ k z ( z + j a z ) ] k z } = exp ( h 2 z R 2 k + j q ˜ ( z ) 2 k k r 2 j k z ) .
lim k a 1 α 0 1 B ˜ m e ( r ) = b ˜ m e ( r ) .
lim α 0 0 B ˜ 0 e ( r ) = 2 k a U ˜ 0 , 0 e ( r ; a ) .
B ˜ m e ( r ) = k a exp ( k a ) cos ( m ϕ ) 0 π I m ( k a sin α 0 sin α ) × exp ( k a cos α 0 cos α j k z cos α ) J m ( k r sin α ) sin α d α .
lim k a 1 { k a I m ( k a sin α 0 sin α ) exp [ k a ( cos α 0 cos α 1 ) ] } = ( k a 2 π ) 1 / 2 exp { k a [ cos ( α α 0 ) 1 ] } ( sin α 0 sin α ) 1 / 2 .
lim k a { k a I m ( k a sin α 0 sin α ) exp [ k a ( cos α 0 cos α 1 ) ] } = lim k a ( k a 2 π ) 1 / 2 exp [ 1 2 k a ( α α 0 ) 2 ] sin α 0 = 1 sin α 0 δ ( α α 0 ) .
lim k a B ˜ m e ( r ) = J m ( k r sin α 0 ) exp ( j k z cos α 0 ) cos ( m ϕ ) ,
I m ( k r a r ) = p = 0 ( 1 2 k r a r ) 2 p + m p ! ( p + m ) ! .
B ˜ m e ( r ) = 2 k a exp ( k a z k a ) p = 0 ( 1 2 k a r ) 2 p + m p ! ( p + m ) ! U ˜ p , m e ( r ; a z ) ,
a = a ^ x a x + a ^ y a y + a ^ z a z = a ( a ^ x cos β 0 sin α 0 + a ^ y sin β 0 sin α 0 + a ^ z cos α 0 ) .
R ˜ = [ r 2 + 2 j r a r cos ( ϕ β 0 ) a r 2 + ( z + j a z ) 2 ] 1 / 2 .
B ˜ m e ( r ) = 2 k a j m 2 π 0 2 π U ˜ 0 , 0 e ( r ; a ) cos ( m β 0 ) d β 0 ,
R ˜ Q ˜ ( z ) + r 2 + 2 j r z R sin α 0 cos ( ϕ β 0 ) 2 Q ˜ ( z ) ,
Q ˜ ( z ) [ q ˜ 2 ( z ) 4 j z R z sin 2 ( α 0 / 2 ) ] 1 / 2 ,
U ˜ 0 , 0 e ( r ; a ) j 2 k Q ˜ ( z ) exp { j k [ Q ˜ ( z ) j z R + r 2 2 Q ˜ ( z ) + j r z R sin α 0 cos ( ϕ β 0 ) Q ˜ ( z ) ] } .
0 2 π exp [ j ζ cos ( ϕ β 0 ) ] cos ( m β 0 ) d β 0 = 2 π j m J m ( ζ ) cos ( m ϕ ) .
lim k a 1 B ˜ m e ( r ) = j z R Q ˜ ( z ) exp [ k z R j k Q ˜ ( z ) j k r 2 2 Q ˜ ( z ) ] J m ( j z R k r sin α 0 Q ˜ ( z ) ) cos ( m ϕ ) .
U ˜ p , m e ( r ; a ) = exp ( k a ) k 2 p + m + 1 cos ( m ϕ ) 0 k r 2 p + m cos [ k z ( z + j a ) ] k z J m ( k r r ) k r d k r .
U ˜ p , m e ( r ; a ) = ( 2 p ) ! ! s = 0 p ( p + m s + m ) ( 4 s + 2 m + 1 ) ( 2 s 1 ) !! ( 2 p + 2 s + 2 m + 1 ) !! ψ ˜ 2 s + m , m e ( r ; a ) ,
U ˜ 0 , 0 e ( r ; a ) = exp ( k a ) k 0 cos [ k z ( z + j a ) ] k z J 0 ( k r r ) k r d k r = exp ( k a ) sin ( k R ˜ ) k R ˜ .
B ˜ m e ( r ) = a exp ( k a ) cos ( m ϕ ) { 0 I m ( k r a r ) exp [ j k z ( z + j a z ) ] k z J m ( k r r ) k r d k r + 0 I m ( k r a r ) exp [ j k z ( z + j a z ) ] k z J m ( k r r ) k r d k r } ,
B ˜ m e ( r ) = k a exp ( k a ) cos ( m ϕ ) { 0 π / 2 j I m ( k a r sin α ) exp [ j k ( z + j a z ) cos α ] J m ( k r sin α ) sin α d α + 0 π / 2 + j I m ( k a r sin α ) exp [ j k ( z + j a z ) cos α ] J m ( k r sin α ) sin α d α } .
B ˜ m e ( r ) = k a exp ( k a ) cos ( m ϕ ) { π / 2 + j π I m ( k a r sin α ) exp [ j k ( z + j a z ) cos α ] J m ( k r sin α ) sin α d α + 0 π / 2 + j I m ( k a r sin α ) exp [ j k ( z + j a z ) cos α ] J m ( k r sin α ) sin α d α } .

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