Abstract

The validity of the paraxial approximation for laser beams in free space is studied via an integral criterion based on the propagation invariants of Helmholtz and paraxial wave equations. This approach allows one to determine the paraxial limit for beams with nondefined spot size and for beams described by more parameters in addition to typical longitudinal wavelength and transverse waist. As examples, the paraxiality of higher-order Hermite, Laguerre, and Bessel–Gaussian beams was completely determined. This method could be extended to nonlinear optics and Bose condensates.

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References

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  1. A. Yariv, Quantum Elecronics (Wiley, 1989).
  2. J. T. Verdeyen, Laser Electronics (Prentice-Hall, 1981).
  3. A. E. Siegman, Lasers (University Science, 1986).
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    [CrossRef]
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  9. N. Akhmediev, A. Ankiewicz, and J. M. Soto-Crespo, Opt. Lett. 18, 411 (1993).
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  10. A. P. Sheppard and M. Haelterman, Opt. Lett. 23, 1820 (1998).
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    [CrossRef]
  12. A. Lencina, P. Vaveliuk, B. Ruiz, M. Tebaldi, and N. Bolognini, Phys. Rev. E 74, 056614 (2006).
    [CrossRef]
  13. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products (Academic, 1980).
  14. J. Durnin, J. J. Miceli, and J. H. Eberly, Phys. Rev. Lett. 58, 1499 (1987).
    [CrossRef] [PubMed]

2006 (1)

A. Lencina, P. Vaveliuk, B. Ruiz, M. Tebaldi, and N. Bolognini, Phys. Rev. E 74, 056614 (2006).
[CrossRef]

2005 (1)

A. Lencina and P. Vaveliuk, Phys. Rev. E 71, 056614 (2005).
[CrossRef]

2004 (1)

2001 (1)

1998 (1)

1996 (1)

E. V. Goldstein, K. Plättner, and P. Meystre, J. Res. Natl. Inst. Stand. Technol. 101, 583 (1996).

1993 (1)

1987 (2)

F. Gori, G. Guattari, and C. Padovani, Opt. Commun. 64, 491 (1987).
[CrossRef]

J. Durnin, J. J. Miceli, and J. H. Eberly, Phys. Rev. Lett. 58, 1499 (1987).
[CrossRef] [PubMed]

1975 (1)

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

Akhmediev, N.

Ankiewicz, A.

Bandres, M. A.

Bolognini, N.

A. Lencina, P. Vaveliuk, B. Ruiz, M. Tebaldi, and N. Bolognini, Phys. Rev. E 74, 056614 (2006).
[CrossRef]

Borghi, R.

Durnin, J.

J. Durnin, J. J. Miceli, and J. H. Eberly, Phys. Rev. Lett. 58, 1499 (1987).
[CrossRef] [PubMed]

Eberly, J. H.

J. Durnin, J. J. Miceli, and J. H. Eberly, Phys. Rev. Lett. 58, 1499 (1987).
[CrossRef] [PubMed]

Goldstein, E. V.

E. V. Goldstein, K. Plättner, and P. Meystre, J. Res. Natl. Inst. Stand. Technol. 101, 583 (1996).

Gori, F.

F. Gori, G. Guattari, and C. Padovani, Opt. Commun. 64, 491 (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, Opt. Commun. 64, 491 (1987).
[CrossRef]

Gutiérrez-Vega, J. C.

Haelterman, M.

Lax, M.

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

Lencina, A.

A. Lencina, P. Vaveliuk, B. Ruiz, M. Tebaldi, and N. Bolognini, Phys. Rev. E 74, 056614 (2006).
[CrossRef]

A. Lencina and P. Vaveliuk, Phys. Rev. E 71, 056614 (2005).
[CrossRef]

Louisell, W. H.

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

McKnight, W. B.

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

Meystre, P.

E. V. Goldstein, K. Plättner, and P. Meystre, J. Res. Natl. Inst. Stand. Technol. 101, 583 (1996).

Miceli, J. J.

J. Durnin, J. J. Miceli, and J. H. Eberly, Phys. Rev. Lett. 58, 1499 (1987).
[CrossRef] [PubMed]

Padovani, C.

F. Gori, G. Guattari, and C. Padovani, Opt. Commun. 64, 491 (1987).
[CrossRef]

Plättner, K.

E. V. Goldstein, K. Plättner, and P. Meystre, J. Res. Natl. Inst. Stand. Technol. 101, 583 (1996).

Porras, M. A.

Ruiz, B.

A. Lencina, P. Vaveliuk, B. Ruiz, M. Tebaldi, and N. Bolognini, Phys. Rev. E 74, 056614 (2006).
[CrossRef]

Ryzhik, I. M.

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

Santarsiero, M.

Sheppard, A. P.

Siegman, A. E.

A. E. Siegman, Lasers (University Science, 1986).

Soto-Crespo, J. M.

Tebaldi, M.

A. Lencina, P. Vaveliuk, B. Ruiz, M. Tebaldi, and N. Bolognini, Phys. Rev. E 74, 056614 (2006).
[CrossRef]

Vaveliuk, P.

A. Lencina, P. Vaveliuk, B. Ruiz, M. Tebaldi, and N. Bolognini, Phys. Rev. E 74, 056614 (2006).
[CrossRef]

A. Lencina and P. Vaveliuk, Phys. Rev. E 71, 056614 (2005).
[CrossRef]

Verdeyen, J. T.

J. T. Verdeyen, Laser Electronics (Prentice-Hall, 1981).

Yariv, A.

A. Yariv, Quantum Elecronics (Wiley, 1989).

J. Opt. Soc. Am. A (1)

J. Res. Natl. Inst. Stand. Technol. (1)

E. V. Goldstein, K. Plättner, and P. Meystre, J. Res. Natl. Inst. Stand. Technol. 101, 583 (1996).

Opt. Commun. (1)

F. Gori, G. Guattari, and C. Padovani, Opt. Commun. 64, 491 (1987).
[CrossRef]

Opt. Lett. (3)

Phys. Rev. A (1)

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

Phys. Rev. E (2)

A. Lencina and P. Vaveliuk, Phys. Rev. E 71, 056614 (2005).
[CrossRef]

A. Lencina, P. Vaveliuk, B. Ruiz, M. Tebaldi, and N. Bolognini, Phys. Rev. E 74, 056614 (2006).
[CrossRef]

Phys. Rev. Lett. (1)

J. Durnin, J. J. Miceli, and J. H. Eberly, Phys. Rev. Lett. 58, 1499 (1987).
[CrossRef] [PubMed]

Other (4)

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

A. Yariv, Quantum Elecronics (Wiley, 1989).

J. T. Verdeyen, Laser Electronics (Prentice-Hall, 1981).

A. E. Siegman, Lasers (University Science, 1986).

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Figures (2)

Fig. 1
Fig. 1

Curves P ̃ versus k ̃ for the first five HG and LG modes. The shaded region indicates where the paraxial approximation is fulfilled. The inset depicts the profiles of E E * (thick curve) and Im { 2 E * z ̃ E k ̃ 2 } (thin curve) for N = 0 at z ̃ = 0 , 0.25, 0.75 and 1. r ̃ [ 2 , 2 ] .

Fig. 2
Fig. 2

Density plot of P ̃ m ( BG ) versus ( k ̃ , β ̃ ) for (a) m = 0 and (b) m = 1 with ( k ̃ , β ̃ ) [ 10 1 , 10 3 ] . The paraxial region is indicated by PR.

Equations (12)

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2 E + k 2 E = 0 ,
t 2 A + 2 i k z A = 0 ,
z [ + Im { E h * z E h } d x d y ] = 0 ,
z [ + E p E p * d x d y ] = 0 ,
+ Im { E h * z E h } d x d y + E p E p * k d x d y = + Ψ h 2 ( k + z ϕ h ) d x d y + Ψ p 2 k d x d y .
P ̃ = + Im { E p * z E p } d x d y + E p E p * k d x d y .
A m , n ( HG ) E 0 = 1 1 + i z ̃ H m [ 2 x ̃ 1 + z ̃ 2 ] H n [ 2 y ̃ 1 + z ̃ 2 ] × ( 1 i z ̃ 1 + z ̃ 2 ) m + n exp [ r ̃ 2 1 + i z ̃ ] ,
A l , p ( LG ) E 0 = 2 l r ̃ l ( 1 i z ̃ ) p ( 1 + i z ̃ ) p + l + 1 L p l [ 2 r ̃ 2 1 + z ̃ 2 ] e ( i l ϕ r ̃ 2 1 + i z ̃ ) ,
P ̃ N ( HG ) ( k ̃ ) = P ̃ N ( LG ) ( k ̃ ) = 1 ( N + 1 ) k ̃ 2 .
A m ( BG ) E 0 = e i m ϕ ( 1 + i z ̃ ) exp [ i β ̃ 2 z ̃ + 4 r ̃ 2 4 ( 1 + i z ̃ ) ] J m [ β ̃ r ̃ 1 + i z ̃ ] ,
P ̃ 0 ( BG ) ( k ̃ , β ̃ ) = 1 1 k ̃ 2 ( β ̃ 2 + 8 4 k ̃ 2 ) I 1 [ β ̃ 2 4 ] I 0 [ β ̃ 2 4 ] ,
P ̃ 1 ( BG ) ( k ̃ , β ̃ ) = 1 ( β ̃ 2 k ̃ ) 2 ( 1 + I 0 [ β ̃ 2 4 ] I 1 [ β ̃ 2 4 ] ) ,

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