Abstract

Based on the perturbative series representation of a complex-source-point spherical wave an expression for cylindrically symmetrical complex-argument Laguerre–Gauss beams of radial order n is derived. This description acquires the accuracy up to any order of diffraction angle, and its first three corrected terms are in accordance with those given by Seshadri [Opt. Lett. 27, 1872 (2002)] based on the virtual source method. Numerical results show that on the beam axis the number of orders of nonvanishing nonparaxial corrections is equal to n. Meanwhile a higher radial mode number n leads to a smaller convergent domain of radius.

© 2008 Optical Society of America

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References

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  1. Q. Zhan, Opt. Express 12, 3377 (2004).
    [CrossRef] [PubMed]
  2. R. Dorn, S. Quabis, and G. Leuchs, Phys. Rev. Lett. 91, 233901 (2003).
    [CrossRef] [PubMed]
  3. Y. I. Salamin and C. H. Keitel, Phys. Rev. Lett. 88, 095005 (2002).
    [CrossRef] [PubMed]
  4. S. Yan and B. Yao, Phys. Rev. A 76, 053836 (2007).
    [CrossRef]
  5. H. Kawauchi, K. Yonezawa, Y. Kozawa, and S. Sato, Opt. Lett. 32, 1839 (2007).
    [CrossRef] [PubMed]
  6. N. Hayazawa, Y. Saito, and S. Kawata, Appl. Phys. Lett. 85, 6239 (2004).
    [CrossRef]
  7. M. Lax, W. H. Louisell, and W. B. McKnight, Phys. Rev. A 11, 1365 (1975).
    [CrossRef]
  8. S. R. Seshadri, Opt. Lett. 27, 1872 (2002).
    [CrossRef]
  9. S. Yan and B. Yao, Opt. Lett. 32, 3367 (2007).
    [CrossRef] [PubMed]
  10. M. Couture and P. Belanger, Phys. Rev. A 24, 355 (1981).
    [CrossRef]
  11. Z. Wang and D. Guo, Special Functions (World Scientific, 1996), formula 6.14.18.
  12. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series and Products (Academic, 1996), p. 853, formula 7.421.4.
  13. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995), pp. 287-290.

2007 (3)

2004 (2)

Q. Zhan, Opt. Express 12, 3377 (2004).
[CrossRef] [PubMed]

N. Hayazawa, Y. Saito, and S. Kawata, Appl. Phys. Lett. 85, 6239 (2004).
[CrossRef]

2003 (1)

R. Dorn, S. Quabis, and G. Leuchs, Phys. Rev. Lett. 91, 233901 (2003).
[CrossRef] [PubMed]

2002 (2)

Y. I. Salamin and C. H. Keitel, Phys. Rev. Lett. 88, 095005 (2002).
[CrossRef] [PubMed]

S. R. Seshadri, Opt. Lett. 27, 1872 (2002).
[CrossRef]

1996 (2)

Z. Wang and D. Guo, Special Functions (World Scientific, 1996), formula 6.14.18.

S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series and Products (Academic, 1996), p. 853, formula 7.421.4.

1995 (1)

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

1981 (1)

M. Couture and P. Belanger, Phys. Rev. A 24, 355 (1981).
[CrossRef]

1975 (1)

M. Lax, W. H. Louisell, and W. B. McKnight, Phys. Rev. A 11, 1365 (1975).
[CrossRef]

Belanger, P.

M. Couture and P. Belanger, Phys. Rev. A 24, 355 (1981).
[CrossRef]

Couture, M.

M. Couture and P. Belanger, Phys. Rev. A 24, 355 (1981).
[CrossRef]

Dorn, R.

R. Dorn, S. Quabis, and G. Leuchs, Phys. Rev. Lett. 91, 233901 (2003).
[CrossRef] [PubMed]

Gradshteyn, S.

S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series and Products (Academic, 1996), p. 853, formula 7.421.4.

Guo, D.

Z. Wang and D. Guo, Special Functions (World Scientific, 1996), formula 6.14.18.

Hayazawa, N.

N. Hayazawa, Y. Saito, and S. Kawata, Appl. Phys. Lett. 85, 6239 (2004).
[CrossRef]

Kawata, S.

N. Hayazawa, Y. Saito, and S. Kawata, Appl. Phys. Lett. 85, 6239 (2004).
[CrossRef]

Kawauchi, H.

Keitel, C. H.

Y. I. Salamin and C. H. Keitel, Phys. Rev. Lett. 88, 095005 (2002).
[CrossRef] [PubMed]

Kozawa, Y.

Lax, M.

M. Lax, W. H. Louisell, and W. B. McKnight, Phys. Rev. A 11, 1365 (1975).
[CrossRef]

Leuchs, G.

R. Dorn, S. Quabis, and G. Leuchs, Phys. Rev. Lett. 91, 233901 (2003).
[CrossRef] [PubMed]

Louisell, W. H.

M. Lax, W. H. Louisell, and W. B. McKnight, Phys. Rev. A 11, 1365 (1975).
[CrossRef]

Mandel, L.

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

McKnight, W. B.

M. Lax, W. H. Louisell, and W. B. McKnight, Phys. Rev. A 11, 1365 (1975).
[CrossRef]

Quabis, S.

R. Dorn, S. Quabis, and G. Leuchs, Phys. Rev. Lett. 91, 233901 (2003).
[CrossRef] [PubMed]

Ryzhik, I. M.

S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series and Products (Academic, 1996), p. 853, formula 7.421.4.

Saito, Y.

N. Hayazawa, Y. Saito, and S. Kawata, Appl. Phys. Lett. 85, 6239 (2004).
[CrossRef]

Salamin, Y. I.

Y. I. Salamin and C. H. Keitel, Phys. Rev. Lett. 88, 095005 (2002).
[CrossRef] [PubMed]

Sato, S.

Seshadri, S. R.

Wang, Z.

Z. Wang and D. Guo, Special Functions (World Scientific, 1996), formula 6.14.18.

Wolf, E.

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

Yan, S.

S. Yan and B. Yao, Opt. Lett. 32, 3367 (2007).
[CrossRef] [PubMed]

S. Yan and B. Yao, Phys. Rev. A 76, 053836 (2007).
[CrossRef]

Yao, B.

S. Yan and B. Yao, Phys. Rev. A 76, 053836 (2007).
[CrossRef]

S. Yan and B. Yao, Opt. Lett. 32, 3367 (2007).
[CrossRef] [PubMed]

Yonezawa, K.

Zhan, Q.

Appl. Phys. Lett. (1)

N. Hayazawa, Y. Saito, and S. Kawata, Appl. Phys. Lett. 85, 6239 (2004).
[CrossRef]

Opt. Express (1)

Opt. Lett. (3)

Phys. Rev. A (3)

M. Couture and P. Belanger, Phys. Rev. A 24, 355 (1981).
[CrossRef]

S. Yan and B. Yao, Phys. Rev. A 76, 053836 (2007).
[CrossRef]

M. Lax, W. H. Louisell, and W. B. McKnight, Phys. Rev. A 11, 1365 (1975).
[CrossRef]

Phys. Rev. Lett. (2)

R. Dorn, S. Quabis, and G. Leuchs, Phys. Rev. Lett. 91, 233901 (2003).
[CrossRef] [PubMed]

Y. I. Salamin and C. H. Keitel, Phys. Rev. Lett. 88, 095005 (2002).
[CrossRef] [PubMed]

Other (3)

Z. Wang and D. Guo, Special Functions (World Scientific, 1996), formula 6.14.18.

S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series and Products (Academic, 1996), p. 853, formula 7.421.4.

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

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Figures (2)

Fig. 1
Fig. 1

Intensity distribution in the focal plane ( z = 0 ) of cylindrically symmetrical Laguerre–Gauss beams with radial mode number n = 5 evaluated by Eq. (16) with m = 3 (dashed curve), m = 10 (square symbols), and m = 13 (solid curve). The diffraction angle ε is chosen to be equal to 0.4, and all intensities are normalized to the maximum of m = 13 .

Fig. 2
Fig. 2

Intensity distribution in the focal plane ( z = 0 ) of cylindrically symmetrical Laguerre–Gauss beams with radial mode numbers n = 0 and 3: solid curve, zeroth-order Laguerre–Gauss beam with five correction terms; square symbols, zeroth-order Laguerre–Gauss beam with six correction terms; dashed curve, third-order Laguerre–Gauss beam with seven correction terms; dashed-dotted curve, third-order Laguerre–Gauss beam with eight correction terms. The diffraction angle ε is chosen to be equal to 0.6, and each intensity distribution is normalized to its value on the optical axis.

Equations (17)

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1 ρ ρ ( ρ ψ ρ ) 4 i ψ ζ = ε 2 2 ψ ζ 2 .
ψ = ψ 0 + ε 2 ψ 2 + ε 4 ψ 4 + .
1 ρ ρ ( ρ ψ 0 ρ ) 4 i ψ 0 ζ = 0 ,
1 ρ ρ ( ρ ψ 2 m ρ ) 4 i ψ 2 m ζ = 2 ψ 2 m 2 ζ 2 , m = 1 , 2 , .
ψ 0 ( 0 ) = Q exp ( Q ρ 2 ) ,
ψ ( 0 ) = Q exp ( x ) [ 1 + m = 1 f 2 m Q m x m L m m ( x ) ] ,
F ( n ) = A ( z ) [ 1 ρ ρ ( ρ ρ ) ] n ψ ( 0 ) ,
A ( z ) = ( 2 k i ) n w 0 2 n exp ( i k z ) .
x s L m μ ( x ) = r = 0 m + s α r s L m + s r ( x ) .
ψ 2 m ( 0 ) = Q m + 1 k = 0 m α k m T m , k ( x ) , α k m = ( 1 ) m + k Γ ( m + 1 ) ( m k ) ( 2 m m ) ,
ψ 2 m ( n ) = Q m + 1 ( 2 ρ 2 + 1 ρ ρ ) n k = 0 m α k m T m , k ( x ) .
T m , k ( ρ ) = 0 d y [ y T ¯ m , k ( y ) J 0 ( y ρ ) ] ,
T ¯ m , k ( y ) = 0 d ρ [ ρ T m , k ( ρ ) J 0 ( y ρ ) ] .
0 y d y [ y 2 n exp ( y 2 4 Q ) J 0 ( y ρ ) ] = n ! 2 ( 4 Q ) n + 1 exp ( Q ρ 2 ) L n ( Q ρ 2 ) ,
ψ ¯ 2 m ( n ) = ( 1 ) n Q m + 1 exp ( y 2 4 Q ) k = 0 m 2 α k m y 2 ( 2 m k + n ) ( 2 m k ) ! ( 4 Q ) 2 m k + 1 .
ψ 2 m ( n ) = ( 1 ) n 4 n Q n + m + 1 exp ( x ) k = 0 m α k m ( 2 m k + n ) ! ( 2 m k ) ! L 2 m k + n ( x ) .
F ( n ) ( ρ , z ) = ( 4 ) n A ( z ) Q n + 1 exp ( x ) m = 0 [ f 2 m Q m k = 0 m ( 2 m k + n ) ! ( 2 m k ) ! α k m L 2 m k + n ( x ) ] .

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