Abstract

A two-dimensional optical field paraxial propagation scheme, in Cartesian and cylindrical coordinate systems, is proposed. This is achieved by extending the method originally proposed by Ladouceur [Opt. Lett. 21, 4 (1996) ] for boundaryless beam propagation to two-dimensional optical wave fields. With this formulation the arbitrary choice of physical window size is avoided by mapping the infinite transverse dimensions into a finite-size domain with an appropriate change of variables, thus avoiding the energy loss through the artificial physical boundary that is usually required for the absorbing or the transparent boundary approach.

© 2006 Optical Society of America

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References

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  1. M. D. Feit and J. A. Fleck, "Light propagation in graded-index optical fibers," Appl. Opt. 17, 3990-3998 (1978).
    [CrossRef] [PubMed]
  2. G. Mur, "Absorbing boundary conditions for finite-difference approximation of the time-domain electromagnetic field equations," IEEE Trans. Electromagn. Compat. 23, 377-382 (1981).
    [CrossRef]
  3. G. R. Hadley, "Transparent boundary condition for beam propagation," Opt. Lett. 16, 624-626 (1991).
    [CrossRef] [PubMed]
  4. Y. Arai, A. Maruta, and M. Matsuhara, "Transparent boundary for the finite-element beam-propagation method," Opt. Lett. 18, 765-767 (1993).
    [CrossRef] [PubMed]
  5. F. Ladouceur, "Boundaryless beam propagation," Opt. Lett. 21, 4-5 (1996).
    [CrossRef] [PubMed]
  6. J. Shibayama, K. Matsubara, M. Sekiguchi, J. Yamauchi, and H. Nakano, "Efficient nonuniform schemes for paraxial and wide-angle finite-difference beam propagation methods," J. Lightwave Technol. 17, 677-683 (1999).
    [CrossRef]
  7. S. J. Hewlett and F. Ladouceur, "Fourier decomposition method applied to mapped infinite domains: scalar analysis of dielectric waveguides down to modal cutoff," J. Lightwave Technol. 13, 375-383 (1995).
    [CrossRef]
  8. K. M. Lo and E. H. Li, "Solutions of the quasi-vector wave equation for optical waveguides in a mapped infinite domains by the Galerkin's method," J. Lightwave Technol. 16, 937-944 (1998).
    [CrossRef]
  9. C. Pozrikidis, Numerical Computation in Science and Engineering (Oxford U. Press, 1998).
  10. J. C. Gutiérrez-Vega and M. A. Bandres, "Helmholtz-Gauss waves," J. Opt. Soc. Am. A 22, 289-298 (2005).
    [CrossRef]
  11. M. Guizar-Sicairos and J. C. Gutiérrez-Vega, "Computation of quasi-discrete Hankel transforms of integer order for propagating optical wavefields," J. Opt. Soc. Am. A 21, 53-58 (2004).
    [CrossRef]

2005 (1)

2004 (1)

1999 (1)

1998 (1)

1996 (1)

1995 (1)

S. J. Hewlett and F. Ladouceur, "Fourier decomposition method applied to mapped infinite domains: scalar analysis of dielectric waveguides down to modal cutoff," J. Lightwave Technol. 13, 375-383 (1995).
[CrossRef]

1993 (1)

1991 (1)

1981 (1)

G. Mur, "Absorbing boundary conditions for finite-difference approximation of the time-domain electromagnetic field equations," IEEE Trans. Electromagn. Compat. 23, 377-382 (1981).
[CrossRef]

1978 (1)

Arai, Y.

Bandres, M. A.

Feit, M. D.

Fleck, J. A.

Guizar-Sicairos, M.

Gutiérrez-Vega, J. C.

Hadley, G. R.

Hewlett, S. J.

S. J. Hewlett and F. Ladouceur, "Fourier decomposition method applied to mapped infinite domains: scalar analysis of dielectric waveguides down to modal cutoff," J. Lightwave Technol. 13, 375-383 (1995).
[CrossRef]

Ladouceur, F.

F. Ladouceur, "Boundaryless beam propagation," Opt. Lett. 21, 4-5 (1996).
[CrossRef] [PubMed]

S. J. Hewlett and F. Ladouceur, "Fourier decomposition method applied to mapped infinite domains: scalar analysis of dielectric waveguides down to modal cutoff," J. Lightwave Technol. 13, 375-383 (1995).
[CrossRef]

Li, E. H.

Lo, K. M.

Maruta, A.

Matsubara, K.

Matsuhara, M.

Mur, G.

G. Mur, "Absorbing boundary conditions for finite-difference approximation of the time-domain electromagnetic field equations," IEEE Trans. Electromagn. Compat. 23, 377-382 (1981).
[CrossRef]

Nakano, H.

Pozrikidis, C.

C. Pozrikidis, Numerical Computation in Science and Engineering (Oxford U. Press, 1998).

Sekiguchi, M.

Shibayama, J.

Yamauchi, J.

Appl. Opt. (1)

IEEE Trans. Electromagn. Compat. (1)

G. Mur, "Absorbing boundary conditions for finite-difference approximation of the time-domain electromagnetic field equations," IEEE Trans. Electromagn. Compat. 23, 377-382 (1981).
[CrossRef]

J. Lightwave Technol. (3)

J. Opt. Soc. Am. A (2)

Opt. Lett. (3)

Other (1)

C. Pozrikidis, Numerical Computation in Science and Engineering (Oxford U. Press, 1998).

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Figures (2)

Fig. 1
Fig. 1

(a) Cosine-Gauss field intensity at z = 0 . (b) Comparison between analytical cosine-Gauss field intensity at z = 4 m (solid curve) and that obtained with the proposed algorithm (dots). (c) Propagating field intensity in the ( x , z ) plane. (d) Percentage error of computed energy as a function of the propagated distance for the proposed algorithm (solid curve) and a typical absorbing boundary method (dashed curve).

Fig. 2
Fig. 2

(a) Bessel–Gauss field intensity at z = 0 . (b) Comparison between an analytical Bessel–Gauss field at z = 5 m (solid curve) and that obtained with the proposed algorithm (dashed–dotted curve). (c) Propagating field intensity in the ( r , z ) plane. (d) Percentage error of computed energy as a function of the propagated distance for the proposed algorithm (solid curve) and a typical absorbing boundary method (dashed curve).

Equations (32)

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2 U ( r , z ) x 2 + 2 U ( r , z ) y 2 + 2 i k ¯ U ( r , z ) z + [ k 2 k ¯ 2 ] U ( r , z ) = 0 ,
E ( r , z ) = U ( r , z ) exp ( i k ¯ z ) ,
x = α tan u ,
y = β tan v ,
U ( u , v , z ) = ψ ( u , v , z ) cos u cos v ,
cos 4 u α 2 2 ψ u 2 + cos 4 v β 2 2 ψ v 2 + [ k 2 k ¯ 2 + cos 4 u α 2 + cos 4 v β 2 ] ψ + 2 i k ¯ ψ z = 0 .
u p p Δ u , p = 0 , 1 , , N u 1 ,
v j j Δ v , j = 0 , 1 , , N v 1 ,
ψ p , j ( m ) ψ ( u p , v j , m Δ z ) , m = 0 , 1 , 2 , .
2 i k ¯ Δ z 2 [ ψ p , j ( m + 1 2 ) ψ p , j ( m ) ] + cos 4 u p Δ u 2 α 2 [ ψ p + 1 , j ( m + 1 2 ) 2 ψ p , j ( m + 1 2 ) + ψ p 1 , j ( m + 1 2 ) ] + cos 4 v j Δ v 2 β 2 [ ψ p , j + 1 ( m ) 2 ψ p , j ( m ) + ψ p , j 1 ( m ) ] + [ cos 4 u p α 2 k ¯ 2 2 ] ψ p , j ( m + 1 2 ) + [ cos 4 v j β 2 k ¯ 2 2 ] ψ p , j ( m ) = 0 ,
2 i k ¯ Δ z 2 [ ψ p , j ( m + 1 ) ψ p , j ( m + 1 2 ) ] + cos 4 u p Δ u 2 α 2 [ ψ p + 1 , j ( m + 1 2 ) 2 ψ p , j ( m + 1 2 ) + ψ p 1 , j ( m + 1 2 ) ] + cos 4 v j Δ v 2 β 2 [ ψ p , j + 1 ( m + 1 ) 2 ψ p , j ( m + 1 ) + ψ p , j 1 ( m + 1 ) ] + [ cos 4 u p α 2 k ¯ 2 2 ] ψ p , j ( m + 1 2 ) + [ cos 4 v j β 2 k ¯ 2 2 ] ψ p , j ( m + 1 ) + 2 ( k p , j ( m + 1 2 ) ) 2 ψ p , j ( m + 1 2 ) = 0 ,
[ 4 i k ¯ Δ z + b p ] ψ p , j ( m + 1 2 ) + a p [ ψ p + 1 , j ( m + 1 2 ) + ψ p 1 , j ( m + 1 2 ) ] = [ 4 i k ¯ Δ z d j ] ψ p , j ( m ) c j [ ψ p , j + 1 ( m ) + ψ p , j 1 ( m ) ] ,
[ 4 i k ¯ Δ z + d j ] ψ p , j ( m + 1 ) + c j [ ψ p , j + 1 ( m + 1 ) + ψ p , j 1 ( m + 1 ) ] = [ 4 i k ¯ Δ z b p 2 ( k p , j ( m + 1 2 ) ) 2 ] ψ p , j ( m + 1 2 ) a p [ ψ p + 1 , j ( m + 1 2 ) + ψ p 1 , j ( m + 1 2 ) ] ,
a p cos 4 u p Δ u 2 α 2 ,
b p a p ( Δ u 2 2 ) k ¯ 2 2 ,
c j cos 4 v j Δ v 2 β 2 ,
d j c j ( Δ v 2 2 ) k ¯ 2 2 .
U ( r , 0 ) = exp ( r 2 w 0 2 ) cos ( k t x ) ,
ψ p , j ( 0 ) = exp ( x p 2 + y j 2 w 0 2 ) cos ( k t x p ) cos u p cos v j ,
e ( z ) = Ω ( z ) Ω 0 Ω 0 ( 100 % ) ,
cos [ ( 2 p + 1 ) Δ u ] = 2 α k t sin Δ u π cos Δ u .
1 r r [ r f ( r , z ) r ] + 2 i k ¯ f ( r , z ) z + [ k 2 k ¯ 2 l 2 r 2 ] f ( r , z ) = 0 ,
U ( r , z ) = f ( r , z ) exp ( i l ϕ ) .
r = γ tan ρ ,
cos 4 ρ γ 2 2 f ρ 2 + cos 3 ρ γ 2 sin ρ [ 2 cos 2 ρ 1 ] f ρ + 2 i k ¯ f z + [ k 2 k ¯ 2 l 2 γ 2 tan 2 ρ ] f = 0 .
[ 2 i k ¯ Δ z + a p ] f p ( m + 1 ) + b p f p + 1 ( m + 1 ) + c p f p 1 ( m + 1 ) = [ 2 i k ¯ Δ z a p ] f p ( m ) b p f p + 1 ( m ) c p f p 1 ( m ) ,
a p 1 2 [ ( k p ( m + 1 2 ) ) 2 k ¯ 2 l 2 γ 2 tan 2 ρ p ] cos 4 ρ p γ 2 Δ ρ 2 ,
b p cos 4 ρ p 2 γ 2 Δ ρ 2 + cos 3 ρ p 4 γ 2 Δ ρ sin ρ p [ 2 cos 2 ρ p 1 ] ,
c p cos 4 ρ p 2 γ 2 Δ ρ 2 cos 3 ρ p 4 γ 2 Δ ρ sin ρ p [ 2 cos 2 ρ p 1 ] ,
k p ( m ) k ( ρ p , m Δ z ) .
U ( r , 0 ) = J l ( k t r ) exp ( i l ϕ ) exp ( r 2 w 0 2 ) ,
f p ( 0 ) = J l ( k t γ tan ρ p ) exp ( γ 2 tan 2 ρ p w 0 2 ) .

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