Abstract

For modified Bessel–Gauss beams, the modulating function for the Gaussian, instead of a Bessel function of real argument, is a Bessel function of imaginary argument. The modified Bessel–Gauss beams and their full wave generalizations are treated with particular attention to the spreading properties on propagation for the azimuthal mode numbers m=0 and m=1. The spreading on propagation of the peak and the null in the radiation pattern obtained in the propagation direction for m=0 and m=1, respectively, is substantially less for the modified Bessel–Gauss waves than that for the corresponding Bessel–Gauss waves. The total power transported by the waves is determined and compared with that of the corresponding paraxial beam to assess the quality of the paraxial beam approximation for the wave. The powers in the Bessel–Gauss wave and the modified Bessel–Gauss wave are finite in contrast to that in the Bessel wave. With respect to both the spreading properties and the quality of the paraxial beam approximation, the modified Bessel–Gauss beam is an improvement over the Bessel–Gauss beam.

© 2007 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef]
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2007 (1)

2006 (1)

2004 (1)

2002 (1)

1998 (2)

1997 (1)

1987 (2)

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, "Diffraction-free beams," Phys. Rev. Lett. 58, 1499-1501 (1987).
[CrossRef] [PubMed]

1985 (1)

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]

1966 (1)

Bandres, M. A.

Casperson, L. W.

Chen, Z.

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]

Fukumitsu, 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]

Gutierrez-Vega, J. C.

Hall, D. G.

Ji, J.

Kogelnik, H.

Li, T.

Magnus, W.

W. Magnus and F. Oberhettinger, Functions of Mathematical Physics (Chelsea, 1954), pp. 16-20.

Mandel, L.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995), pp. 287-290.

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]

Oberhettinger, F.

W. Magnus and F. Oberhettinger, Functions of Mathematical Physics (Chelsea, 1954), pp. 16-20.

Padovani, C.

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

Seshadri, S. R.

Sheppard, C. J. R.

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]

Shi, Z.

Song, Y.

Takenaka, T.

Tovar, A. A.

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.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995), pp. 287-290.

Yokota, M.

Zhang, Y.

Appl. Opt. (2)

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. Soc. Am. A (4)

Opt. Commun. (1)

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

Opt. Lett. (3)

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 (2)

W. Magnus and F. Oberhettinger, Functions of Mathematical Physics (Chelsea, 1954), pp. 16-20.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995), pp. 287-290.

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Figures (6)

Fig. 1
Fig. 1

Radiation patterns for m = 0 and k w 0 = 4.0 . Curve a, the Laguerre–Gauss wave of radial mode number n = 0 ; curve b, the Bessel–Gauss wave, β w 0 = 2.4 ; curve c, the modified Bessel–Gauss wave, α w 0 = 2.4 . The radiation patterns do not vary in the ϕ direction.

Fig. 2
Fig. 2

Radiation patterns for m = 1 , k w 0 = 4.0 and ϕ = 0 . Curves a, b, and c are the same as in Fig. 1. The radiation patterns have cos 2 ϕ variation in the ϕ direction.

Fig. 3
Fig. 3

Radiation patterns for the modified Bessel–Gauss wave for m = 0 and k w 0 = 4.0 . Curve a, α w 0 = 0 ; curve b, α w 0 = 1 ; curve c, α w 0 = 2 ; and curve d, α w 0 = 3 . The radiation patterns do not vary in the ϕ direction.

Fig. 4
Fig. 4

Radiation patterns for the modified Bessel–Gauss wave for m = 1 , k w 0 = 4.0 , and ϕ = 0 . Curves a, b, c, and d are the same as in Fig. 3. The radiation patterns have cos 2 ϕ variation in the ϕ direction.

Fig. 5
Fig. 5

1 P 0 as functions of w 0 λ for 0.05 < w 0 λ < 1.0 . P 0 , power transported by the m = 0 wave. Curves a, b, and c are the same as in Fig. 1. The power transported by the paraxial beam P 0 ( 0 ) = 1 . The straight line corresponding to 1 P 0 ( 0 ) is included for comparison purposes.

Fig. 6
Fig. 6

1 P 1 as functions of w 0 λ for 0.05 < w 0 λ < 1.0 . P 1 , power transported by the m = 1 wave. Curves a, b, and c are the same as in Fig. 1. The power transported by the paraxial beam P 1 ( 0 ) = 1 . The straight line corresponding to 1 P 1 ( 0 ) is included for comparison purposes.

Equations (22)

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F m ( 0 ) ( ρ , ϕ , 0 ) = N M B cos m ϕ exp ( ν ) I m ( α ρ ) exp ( ρ 2 w 0 2 ) ,
F m ( 0 ) ( ρ , ϕ , z ) = exp ( i k z ) N M B cos m ϕ q 2 exp ( q 2 ν ) I m ( q 2 α ρ ) exp ( q 2 ρ 2 w 0 2 ) ,
Π z , m ( 0 ) ( ρ , ϕ , z ) = 1 2 Re [ i ω F m ( 0 ) * ( ρ , ϕ , z ) z F m ( 0 ) ( ρ , ϕ , z ) ] ,
P m ( 0 ) = 0 2 π d ϕ 0 d ρ ρ ( ω k 2 ) F m ( 0 ) ( ρ , ϕ , z ) 2 .
P m ( 0 ) = N M B 2 ( ω k π ε m w 0 2 8 ) exp ( ν ) I m ( ν ) ,
N M B = [ 8 ω k π ε m w 0 2 exp ( ν ) I m ( ν ) ] 1 2 .
ρ e x = d = α w 0 2 2
S e x = i 4 π b exp ( k b ) N M B cos m ϕ .
z e x = i b .
( 2 ρ 2 + 1 ρ ρ + 1 ρ 2 2 ϕ 2 + 2 z 2 + k 2 ) F m ( ρ , ϕ , z ) = i 4 π b exp ( k b ) N M B cos m ϕ δ ( ρ d ) 2 π ρ δ ( z i b ) .
F m ( ρ , ϕ , z ) = b exp ( k b ) N M B cos m ϕ 0 d η η J m ( η ρ ) J m ( η d ) ζ 1 exp [ i ζ ( z i b ) ] ,
ζ = ( k 2 η 2 ) 1 2 .
P m M B = 0 2 π d ϕ 0 d ρ ρ Π z , m ( ρ , ϕ , z ) ,
P m M B = 0 2 π d ϕ 0 π 2 d θ sin θ Φ m M B ( θ , ϕ ) ,
Φ m M B ( θ , ϕ ) = cos 2 m ϕ J m 2 ( 0.5 k α w 0 2 sin θ ) ε m π f 0 2 exp ( ν ) I m ( ν ) exp [ k 2 w 0 2 ( 1 cos θ ) ]
Lim α w 0 0 F m ( 0 ) ( ρ , ϕ , z ) = L m , 0 ( 0 ) ( ρ , ϕ , z ) = exp ( i k z ) ( 8 ε m π ω k w 0 2 m ! ) 1 2 2 m 2 q 2 ( q 2 ρ w 0 ) m cos m ϕ exp ( q 2 ρ 2 w 0 2 ) .
Lim α w 0 0 Φ m M B ( θ , ϕ ) = Φ m , 0 L G ( θ , ϕ ) = cos 2 m ϕ sin 2 m θ ε m π f 0 2 ( m + 1 ) m ! 2 m exp [ k 2 w 0 2 ( 1 cos θ ) ] ,
P 0 , 0 L G = 1 exp ( k 2 w 0 2 ) ,
P 1 , 0 L G = 1 f 0 2 ( 0.5 k 2 w 0 2 f 0 2 ) exp ( k 2 w 0 2 ) .
G m ( 0 ) ( ρ , ϕ , z ) = exp ( i k z ) N B cos m ϕ q 2 exp ( q 2 γ ) J m ( q 2 β ρ ) exp ( q 2 ρ 2 w 0 2 ) ,
N B = [ 8 ω k π ε m w 0 2 exp ( γ ) I m ( γ ) ] 1 2 ,
Φ m B ( θ , ϕ ) = cos 2 m ϕ I m 2 ( 0.5 k β w 0 2 sin θ ) ε m π f 0 2 exp ( γ ) I m ( γ ) exp [ k 2 w 0 2 ( 1 cos θ ) ] ,

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