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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    [CrossRef]
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    [CrossRef] [PubMed]
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    [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).
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2005

2004

1999

1998

1996

1995

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

1991

1981

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

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.

IEEE Trans. Electromagn. Compat.

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.

J. Opt. Soc. Am. A

Opt. Lett.

Other

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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