Abstract

We use a method based on the simultaneous combination of the propagation operator and the Fourier transform with arbitrary index in propagating the transverse component of a nonparaxial beam in free space from an arbitrary initial transverse field structure. Being an iterative method, this approach can easily be implemented computationally. As an example of its efficiency, we derive the closed-form nonparaxial corrections to a Bessel–Gaussian beam, showing that our results differ strongly from those reported previously. The validity of our approach is supported by an analysis of the paraxiality estimator recently introduced in the literature.

© 2007 Optical Society of America

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  1. A. Lencina and P. Vaveliuk, "Squared-field amplitude modulus and radiation intensity nonequivalence within nonlinear slabs," Phys. Rev. E 71, 056614 (2005).
    [CrossRef]
  2. G. P. Agrawal and D. N. Pattanayak, "Gaussian beam propagation beyond the paraxial approximation," J. Opt. Soc. Am. 69, 575-578 (1979).
    [CrossRef]
  3. G. P. Agrawal and M. Lax, "Free-space propagation beyond the paraxial approxiamtion," Phys. Rev. A 27, 1693-1695 (1983).
    [CrossRef]
  4. T. Takenaka, M. Yokota, and O. Fukumitsu, "Propagation of light beams beyond the paraxial approximation," J. Opt. Soc. Am. A 2, 826-829 (1985).
    [CrossRef]
  5. A. Wünsche, "Transition from the paraxial to the exact solutions of the wave equation and application to Gaussian beams," J. Opt. Soc. Am. A 9, 765-774 (1992).
    [CrossRef]
  6. Q. Cao and X. Deng, "Corrections to the paraxial approximation of an arbitrary free-propagation beam," J. Opt. Soc. Am. A 15, 1144-1148 (1998).
    [CrossRef]
  7. H. Laabs, "Propagation of Hermite-Gaussian beams beyond the paraxial approximation," Opt. Commun. 147, 1-4 (1998).
    [CrossRef]
  8. R. Borghi, M. Santarsiero, and M. A. Porras, "Nonparaxial Bessel-Gauss beams," J. Opt. Soc. Am. A 18, 1618-1626 (2001).
    [CrossRef]
  9. S. R. Seshadri, "Nonparaxial corrections for the fundamental Gaussian beam," J. Opt. Soc. Am. A 19, 2134-2141 (2002).
    [CrossRef]
  10. L. W. Davis, "Theory of electromagnetic beams," Phys. Rev. A 19, 1177-1179 (1979).
    [CrossRef]
  11. H. C. Kim and Y. H. Lee, "Hermite-Gaussian and Laguerre-Gaussian beams beyond the paraxial approximation," Opt. Commun. 169, 9-16 (1999).
    [CrossRef]
  12. C. G. Chen, P. T. Konkola, J. Ferrera, R. K. Heilmann, and M. L. Schattenburg, "Analyses of vector Gaussian beam propagation and the validity of paraxial and spherical approximations," J. Opt. Soc. Am. A 19, 404-412 (2002).
    [CrossRef]
  13. K. Duan, B. Wang, and B. Lü, "Propagation of Hermite-Gaussian and Laguerre-Gaussian beams beyond the paraxial approximation," J. Opt. Soc. Am. A 22, 1976-1980 (2005).
    [CrossRef]
  14. A. H. Lohmann, J. Ojeda Castañeda, and N. Streibl, "Differential operator for three-dimensional imaging," Proc. SPIE 402, 186-191 (1983).
  15. B. Ruiz and H. Rabal, "Differential operators, the Fourier transform and its applications to optics," Optik (Stuttgart) 103, 171-178 (1996).
  16. P. Vaveliuk, B. Ruiz, and A. Lencina, "Limits of the paraxial approximation in laser beams," Opt. Lett. 32, 927-929 (2007).
    [CrossRef] [PubMed]
  17. J. C. Gutiérrez-Vega and M. A. Bandres, "Helmholtz-Gauss waves," J. Opt. Soc. Am. A 22, 289-298 (2005).
    [CrossRef]
  18. M. A. Bandres and J. C. Gutiérrez-Vega, "Ince-Gaussian modes of the paraxial wave equation and stable resonators," J. Opt. Soc. Am. A 21, 873-880 (2004).
    [CrossRef]
  19. M. Lax, W. H. Louisell, and W. B. McKnight, "From Maxwell to paraxial wave optics," Phys. Rev. A 11, 1365-1370 (1975).
    [CrossRef]
  20. J. Durnin, "Exact solutions for nondiffracting beams. I. The scalar theory," J. Opt. Soc. Am. A 4, 651-654 (1987).
    [CrossRef]
  21. F. Gori, G. Guattari, and C. Padovani, "Bessel-Gauss beams," Opt. Commun. 64, 491-495 (1987).
    [CrossRef]
  22. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968).
  23. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products (Academic, 1980), pp. 716, 718, 1037, 1058.
  24. The coordinate z was normalized in terms of L instead of D=kw0/β (see ) because L is not dependant on β.
  25. A. Lencina, B. Ruiz, and P. Vaveliuk, "Alternative method for wave propagation within bounded linear media: conceptual and practical implications," Optik (Stuttgart) (to be published) doi:10.1016/j.ijleo.2006.11.006.

2007 (1)

2005 (3)

2004 (1)

2002 (2)

2001 (1)

1999 (1)

H. C. Kim and Y. H. Lee, "Hermite-Gaussian and Laguerre-Gaussian beams beyond the paraxial approximation," Opt. Commun. 169, 9-16 (1999).
[CrossRef]

1998 (2)

Q. Cao and X. Deng, "Corrections to the paraxial approximation of an arbitrary free-propagation beam," J. Opt. Soc. Am. A 15, 1144-1148 (1998).
[CrossRef]

H. Laabs, "Propagation of Hermite-Gaussian beams beyond the paraxial approximation," Opt. Commun. 147, 1-4 (1998).
[CrossRef]

1996 (1)

B. Ruiz and H. Rabal, "Differential operators, the Fourier transform and its applications to optics," Optik (Stuttgart) 103, 171-178 (1996).

1992 (1)

1987 (2)

J. Durnin, "Exact solutions for nondiffracting beams. I. The scalar theory," J. Opt. Soc. Am. A 4, 651-654 (1987).
[CrossRef]

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

1985 (1)

1983 (2)

G. P. Agrawal and M. Lax, "Free-space propagation beyond the paraxial approxiamtion," Phys. Rev. A 27, 1693-1695 (1983).
[CrossRef]

A. H. Lohmann, J. Ojeda Castañeda, and N. Streibl, "Differential operator for three-dimensional imaging," Proc. SPIE 402, 186-191 (1983).

1979 (2)

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]

Agrawal, G. P.

G. P. Agrawal and M. Lax, "Free-space propagation beyond the paraxial approxiamtion," Phys. Rev. A 27, 1693-1695 (1983).
[CrossRef]

G. P. Agrawal and D. N. Pattanayak, "Gaussian beam propagation beyond the paraxial approximation," J. Opt. Soc. Am. 69, 575-578 (1979).
[CrossRef]

Bandres, M. A.

Borghi, R.

Cao, Q.

Chen, C. G.

Davis, L. W.

L. W. Davis, "Theory of electromagnetic beams," Phys. Rev. A 19, 1177-1179 (1979).
[CrossRef]

Deng, X.

Duan, K.

Durnin, J.

Ferrera, J.

Fukumitsu, O.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968).

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), pp. 716, 718, 1037, 1058.

Guattari, G.

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

Gutiérrez-Vega, J. C.

Heilmann, R. K.

Kim, H. C.

H. C. Kim and Y. H. Lee, "Hermite-Gaussian and Laguerre-Gaussian beams beyond the paraxial approximation," Opt. Commun. 169, 9-16 (1999).
[CrossRef]

Konkola, P. T.

Laabs, H.

H. Laabs, "Propagation of Hermite-Gaussian beams beyond the paraxial approximation," Opt. Commun. 147, 1-4 (1998).
[CrossRef]

Lax, M.

G. P. Agrawal and M. Lax, "Free-space propagation beyond the paraxial approxiamtion," Phys. Rev. A 27, 1693-1695 (1983).
[CrossRef]

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

Lee, Y. H.

H. C. Kim and Y. H. Lee, "Hermite-Gaussian and Laguerre-Gaussian beams beyond the paraxial approximation," Opt. Commun. 169, 9-16 (1999).
[CrossRef]

Lencina, A.

P. Vaveliuk, B. Ruiz, and A. Lencina, "Limits of the paraxial approximation in laser beams," Opt. Lett. 32, 927-929 (2007).
[CrossRef] [PubMed]

A. Lencina and P. Vaveliuk, "Squared-field amplitude modulus and radiation intensity nonequivalence within nonlinear slabs," Phys. Rev. E 71, 056614 (2005).
[CrossRef]

A. Lencina, B. Ruiz, and P. Vaveliuk, "Alternative method for wave propagation within bounded linear media: conceptual and practical implications," Optik (Stuttgart) (to be published) doi:10.1016/j.ijleo.2006.11.006.

Lohmann, A. H.

A. H. Lohmann, J. Ojeda Castañeda, and N. Streibl, "Differential operator for three-dimensional imaging," Proc. SPIE 402, 186-191 (1983).

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]

Lü, B.

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]

Ojeda Castañeda, J.

A. H. Lohmann, J. Ojeda Castañeda, and N. Streibl, "Differential operator for three-dimensional imaging," Proc. SPIE 402, 186-191 (1983).

Padovani, C.

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

Pattanayak, D. N.

Porras, M. A.

Rabal, H.

B. Ruiz and H. Rabal, "Differential operators, the Fourier transform and its applications to optics," Optik (Stuttgart) 103, 171-178 (1996).

Ruiz, B.

P. Vaveliuk, B. Ruiz, and A. Lencina, "Limits of the paraxial approximation in laser beams," Opt. Lett. 32, 927-929 (2007).
[CrossRef] [PubMed]

B. Ruiz and H. Rabal, "Differential operators, the Fourier transform and its applications to optics," Optik (Stuttgart) 103, 171-178 (1996).

A. Lencina, B. Ruiz, and P. Vaveliuk, "Alternative method for wave propagation within bounded linear media: conceptual and practical implications," Optik (Stuttgart) (to be published) doi:10.1016/j.ijleo.2006.11.006.

Ryzhik, I. M.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products (Academic, 1980), pp. 716, 718, 1037, 1058.

Santarsiero, M.

Schattenburg, M. L.

Seshadri, S. R.

Streibl, N.

A. H. Lohmann, J. Ojeda Castañeda, and N. Streibl, "Differential operator for three-dimensional imaging," Proc. SPIE 402, 186-191 (1983).

Takenaka, T.

Vaveliuk, P.

P. Vaveliuk, B. Ruiz, and A. Lencina, "Limits of the paraxial approximation in laser beams," Opt. Lett. 32, 927-929 (2007).
[CrossRef] [PubMed]

A. Lencina and P. Vaveliuk, "Squared-field amplitude modulus and radiation intensity nonequivalence within nonlinear slabs," Phys. Rev. E 71, 056614 (2005).
[CrossRef]

A. Lencina, B. Ruiz, and P. Vaveliuk, "Alternative method for wave propagation within bounded linear media: conceptual and practical implications," Optik (Stuttgart) (to be published) doi:10.1016/j.ijleo.2006.11.006.

Wang, B.

Wünsche, A.

Yokota, M.

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (10)

R. Borghi, M. Santarsiero, and M. A. Porras, "Nonparaxial Bessel-Gauss beams," J. Opt. Soc. Am. A 18, 1618-1626 (2001).
[CrossRef]

S. R. Seshadri, "Nonparaxial corrections for the fundamental Gaussian beam," J. Opt. Soc. Am. A 19, 2134-2141 (2002).
[CrossRef]

T. Takenaka, M. Yokota, and O. Fukumitsu, "Propagation of light beams beyond the paraxial approximation," J. Opt. Soc. Am. A 2, 826-829 (1985).
[CrossRef]

A. Wünsche, "Transition from the paraxial to the exact solutions of the wave equation and application to Gaussian beams," J. Opt. Soc. Am. A 9, 765-774 (1992).
[CrossRef]

Q. Cao and X. Deng, "Corrections to the paraxial approximation of an arbitrary free-propagation beam," J. Opt. Soc. Am. A 15, 1144-1148 (1998).
[CrossRef]

C. G. Chen, P. T. Konkola, J. Ferrera, R. K. Heilmann, and M. L. Schattenburg, "Analyses of vector Gaussian beam propagation and the validity of paraxial and spherical approximations," J. Opt. Soc. Am. A 19, 404-412 (2002).
[CrossRef]

K. Duan, B. Wang, and B. Lü, "Propagation of Hermite-Gaussian and Laguerre-Gaussian beams beyond the paraxial approximation," J. Opt. Soc. Am. A 22, 1976-1980 (2005).
[CrossRef]

J. C. Gutiérrez-Vega and M. A. Bandres, "Helmholtz-Gauss waves," J. Opt. Soc. Am. A 22, 289-298 (2005).
[CrossRef]

M. A. Bandres and J. C. Gutiérrez-Vega, "Ince-Gaussian modes of the paraxial wave equation and stable resonators," J. Opt. Soc. Am. A 21, 873-880 (2004).
[CrossRef]

J. Durnin, "Exact solutions for nondiffracting beams. I. The scalar theory," J. Opt. Soc. Am. A 4, 651-654 (1987).
[CrossRef]

Opt. Commun. (3)

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

H. C. Kim and Y. H. Lee, "Hermite-Gaussian and Laguerre-Gaussian beams beyond the paraxial approximation," Opt. Commun. 169, 9-16 (1999).
[CrossRef]

H. Laabs, "Propagation of Hermite-Gaussian beams beyond the paraxial approximation," Opt. Commun. 147, 1-4 (1998).
[CrossRef]

Opt. Lett. (1)

Optik (Stuttgart) (2)

B. Ruiz and H. Rabal, "Differential operators, the Fourier transform and its applications to optics," Optik (Stuttgart) 103, 171-178 (1996).

A. Lencina, B. Ruiz, and P. Vaveliuk, "Alternative method for wave propagation within bounded linear media: conceptual and practical implications," Optik (Stuttgart) (to be published) doi:10.1016/j.ijleo.2006.11.006.

Phys. Rev. A (3)

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

L. W. Davis, "Theory of electromagnetic beams," Phys. Rev. A 19, 1177-1179 (1979).
[CrossRef]

G. P. Agrawal and M. Lax, "Free-space propagation beyond the paraxial approxiamtion," Phys. Rev. A 27, 1693-1695 (1983).
[CrossRef]

Phys. Rev. E (1)

A. Lencina and P. Vaveliuk, "Squared-field amplitude modulus and radiation intensity nonequivalence within nonlinear slabs," Phys. Rev. E 71, 056614 (2005).
[CrossRef]

Proc. SPIE (1)

A. H. Lohmann, J. Ojeda Castañeda, and N. Streibl, "Differential operator for three-dimensional imaging," Proc. SPIE 402, 186-191 (1983).

Other (3)

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968).

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products (Academic, 1980), pp. 716, 718, 1037, 1058.

The coordinate z was normalized in terms of L instead of D=kw0/β (see ) because L is not dependant on β.

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Figures (3)

Fig. 1
Fig. 1

Graph of BG 0 on-axis paraxial field modulus u 1 ( r = 0 , z ) together with the two first corrections u 1 u 2 and u 2 u 3 plotted against the normalized propagation coordinate z L . Panel (a) is a reproduction of Fig. 2(c) in [8]; panel (b) depicts the results obtained in this paper. Although a direct comparison of the corrections one by one between both methods is not strictly valid, the depicted results show a paraxial solution equivalent to the exact Helmholtz solution via this paper’s approach that is contrary to the results in [8], where the exact solution cannot be approximated by the paraxial solution.

Fig. 2
Fig. 2

Density plot of P versus ( k w 0 , β w 0 ) for BG 0 . The paraxial region is indicated by PR. The “⋆” symbol marks the parameters used in Fig. 1: ( k w 0 = 200 , β w 0 = 60 ), showing that the propagation setup is clearly into the paraxial region. The “+” symbol marks the parameters used in Fig. 3: ( k w 0 = 200 , β w 0 = 160 ), already out of the paraxial region.

Fig. 3
Fig. 3

Graph of BG 0 on-axis paraxial field modulus u 1 together with the first two corrections u 1 u 2 and u 2 u 3 plotted against the normalized propagation coordinate z L . The inset depicts the first two approximations u 2 and u 3 . Clearly, the nonnegligible first-order correction, u 1 u 2 , indicates the propagation of a nonparaxial beam and is in agreement with Fig. 2.

Equations (26)

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( 2 + k 2 ) u ( r , z ) = 0 ,
u ( r , 0 ) = u 0 ( r ) ,
u ( r , z ) = exp ( i k z O ̂ ) u 0 ( r ) ,
O ̂ ( 1 + 2 k 2 ) 1 2 ,
O ̂ 1 + 2 2 k 2 m = 2 ( 2 m 3 ) ! ! ( 2 m ) ! ! ( 2 k 2 ) m ,
exp ( i k z O ̂ ) m = 0 ( i k z ) m m ! O ̂ m .
u 1 = exp ( i k z + i z 2 2 k ) u 0 = e i k z A ( r , z ) .
( 2 i k z + 2 ) A = 0 ,
u 2 = exp ( i k z + i z 2 2 k i z 4 2 3 k 3 ) u 0 ,
u n = exp ( i k z ) × exp [ i z 2 2 k i k z m = 2 n ( 2 m 3 ) ! ! ( 2 m ) ! ! ( 2 k 2 ) m ] u 0 ,
U ( x ̃ , y ̃ ) F p { u ( x , y ) } = i p 2 π u ( x , y ) e i p ( x x ̃ + y y ̃ ) d x d y ,
U ( r ̃ , θ ̃ ) = F p { u ( r , θ ) } = i p 2 π 0 2 π 0 u ( r , θ ) e i p r r ̃ cos ( θ ̃ θ ) r d r d θ ,
F p { exp ( i q 2 ) f ( r ) } = exp ( i q p 2 r ̃ 2 ) F p { f } ,
U 1 = exp ( i k z ) exp ( i z 2 k p 2 r ̃ 2 ) U 0 ,
exp ( i k z + i z 2 2 k i z 4 2 3 k 3 + )
F p { ( 2 ) m u 0 } = ( p 2 r ̃ 2 ) m U ̃ 0
U 2 = exp ( i k z ) exp [ i k z ( p 2 r ̃ 2 2 k 2 + p 4 r ̃ 4 ( 2 k 2 ) 2 2 ! ) ] U 0
U n = exp ( i k z ) exp [ i k z ( p 2 r ̃ 2 2 k 2 + m = 2 n ( p 2 r ̃ 2 k 2 ) m ( 2 m 3 ) ! ! ( 2 m ) ! ! ) ] U 0 .
U = exp ( i k z 1 p 2 r ̃ 2 k 2 ) U 0 .
u n = F p 1 { U n } , u = F p 1 { U } .
u 0 ( r , θ ) = exp ( i l θ ) exp ( r 2 k 2 L ) J l ( β r ) ,
U 0 = ( i ) l + 1 p L k I l ( β p r ̃ L k ) × exp ( i l θ ̃ ) exp [ L ( β 2 + p 2 r ̃ 2 ) 2 k ] ,
u 1 = exp ( i k z ) F p 1 { U 0 exp ( i z p 2 r ̃ 2 2 k ) } = exp ( i k z ) exp ( i l θ ) ( 1 + i z L ) J l ( β r 1 + i z L ) exp [ ( i β 2 z k + r 2 k L ) 1 2 ( 1 + i z L ) ] .
u 2 = exp ( i k z ) F p 1 { U 0 exp [ i z p 2 r ̃ 2 2 k ( 1 + p 2 r ̃ 2 4 k 2 ) ] } = exp ( i k z ) ( 1 + i z L ) [ β r 2 ( 1 + i z L ) ] l exp [ i l θ β 2 L 2 k r 2 k 2 L ( 1 + i z L ) ] × m , j = 0 [ i z 2 k L 2 ( 1 + i z L ) 2 ] j [ β 2 L 2 k ( 1 + i z L ) ] m × ( 2 j + m ) ! m ! j ! ( m + l ) ! L m + 2 j l [ r 2 k 2 L ( 1 + i z L ) ] ,
u n + 1 = exp ( i k z ) 1 + i z L exp ( i l θ ) exp ( β 2 L 2 k ) exp [ r 2 k 2 L ( 1 + i z L ) ] [ β r 2 ( 1 + i z L ) ] l × j 0 , j 1 , , j n = 0 ( i z k L 2 ) m = 1 n j m ( 2 1 + i z L ) m = 1 n m j m × ( 2 1 + i z L ) m = 0 n ( m + 1 ) j m j 0 = 1 n [ ( 2 j 0 1 ) ! ! ( 2 j 0 + 2 ) ! ! ] j 0 × ( β 2 L 2 k ( 1 + i z L ) ) j 0 m = 0 n ( m + 1 ) j m j 0 ! ( j 0 + l ) ! j 1 ! j n ! × L j 0 + 2 j 1 + + ( n + 1 ) j n l [ r 2 k 2 L ( 1 + i z L ) ] .
P = 1 1 ( k w 0 ) 2 [ ( β w 0 ) 2 + 8 4 ( k w 0 ) 2 ] I 1 [ ( β w 0 ) 2 4 ] I 0 [ ( β w 0 ) 2 4 ] ,

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