Abstract

We present an optimized split-step method for solving nonlinear coupled-mode equations that model wave propagation in nonlinear fiber Bragg gratings. By separately controlling the spatial and the temporal step size of the solution, we could significantly decrease the run time duration without significantly affecting the result accuracy. The accuracy of the method and the dependence of the error on the algorithm parameters are studied in several examples. Physical considerations are given to determine the required resolution.

© 2008 Optical Society of America

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References

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  1. H. G. Winful, “Pulse compression in optical fiber filters,” Appl. Phys. Lett. 46, 527-529 (1985).
    [CrossRef]
  2. A. Rosenthal and M. Horowitz, “Bragg-soliton formation and pulse compression in a one-dimensional periodic structure,” Phys. Rev. E 74, 066611 (2006).
    [CrossRef]
  3. C. M. de Sterke and J. E. Sipe, “Switching dynamics of finite periodic nonlinear media: a numerical study,” Phys. Rev. A 42, 2858-2869 (1990).
    [CrossRef] [PubMed]
  4. S. Pereira and J. E. Sipe, “Nonlinear pulse propagation in birefringent fiber Bragg gratings,” Opt. Express 3, 418-432 (1998).
    [CrossRef] [PubMed]
  5. D. Taverner, N. G. R. Broderick, D. J. Richardson, M. Ibsen, and R. I. Laming, “All-optical AND gate based on coupled gap-soliton formation in a fiber Bragg grating,” Opt. Lett. 23, 259-261 (1998).
    [CrossRef]
  6. Y. P. Shapira and M. Horowitz, “Optical AND gate based on soliton interaction in a fiber Bragg grating,” Opt. Lett. 32, 1211-1213 (2007).
    [CrossRef] [PubMed]
  7. J. T. Mok, C. M. de Sterke, I. C. M. Littler, and B. J. Eggleton, “Dispersionless slow light using gap solitons,” Nat. Phys. 2, 775-780 (2006).
    [CrossRef]
  8. J. T. Mok, C. M. de Sterke, and B. J. Eggleton, “Delay-tunable gap soliton-based slow-light system,” Opt. Express 14, 11987-11996 (2006).
    [CrossRef] [PubMed]
  9. A. B. Aceves and S. Wabnitz, “Self-induced transparency solitons in nonlinear refractive periodic media,” Phys. Lett. A 141, 37-42 (1989).
    [CrossRef]
  10. D. N. Christodoulides and R. I. Joseph, “Slow Bragg solitons in nonlinear periodic structures,” Phys. Rev. Lett. 62, 1746-1749 (1989).
    [CrossRef] [PubMed]
  11. C. M. de Sterke and J. E. Sipe, “Gap solitons,” in Progress in Optics XXXIII, E.Wolf, ed. (Elsevier, 1994), pp. 203-260.
  12. W. C. K. Mak, B. A. Malomed, and P. L. Chu, “Formation of a standing light pulse through collision of gap solitons,” Phys. Rev. E 68, 26609 (2003).
    [CrossRef]
  13. C. M. de Sterke, K. R. Jackson, and B. D. Robert, “Nonlinear coupled-mode equations on a finite interval: a numerical procedure,” J. Opt. Soc. Am. B 8, 403-412 (1991).
    [CrossRef]
  14. A. Rosenthal and M. Horowitz, “Analysis and design of nonlinear fiber Bragg gratings and their application for optical compression of reflected pulses,” Opt. Lett. 31, 1334-1336 (2006).
    [CrossRef] [PubMed]
  15. O. V. Sinkin, J. Zweck, and C. R. Menyuk, “Optimization of the split-step Fourier method in modeling optical-fiber communications systems,” J. Lightwave Technol. 21, 61-68 (2003).
    [CrossRef]
  16. G. P. Agrawal, “Numerical methods,” in Nonlinear Fiber Optics (Academic, 1995), pp. 50-55.
  17. C. E. Shannon, “Communication in the presence of noise,” Proc. IRE 37, 10-21 (1949).
    [CrossRef]
  18. J. Saijonmaa and D. Yevick, “Beam-propagation analysis of loss in bent optical waveguides and fibers,” J. Opt. Soc. Am. 73, 1785-1791 (1983).
    [CrossRef]
  19. M. D. Feit and J. A. Fleck, Jr., “Light propagation in graded-index optical fibers,” Appl. Opt. 17, 3990-3998 (1978).
    [CrossRef] [PubMed]
  20. C. Vassallo and F. Collino, “Highly efficient absorbing boundary conditions for the beam propagation method,” J. Lightwave Technol. 14, 1570-1577 (1996).
    [CrossRef]
  21. J. P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114, 185-200 (1994).
    [CrossRef]
  22. T. Dohnal and T. Hagstrom, “Perfectly matched layers in photonics computations: 1D and 2D nonlinear coupled mode equations,” J. Comput. Phys. 223, 690-710 (2007).
    [CrossRef]
  23. R. L. Burden and J. D. Faires, “Richardson's extrapolation,” in Numerical Analysis (ITP, 1997), pp. 180-183.

2007

T. Dohnal and T. Hagstrom, “Perfectly matched layers in photonics computations: 1D and 2D nonlinear coupled mode equations,” J. Comput. Phys. 223, 690-710 (2007).
[CrossRef]

Y. P. Shapira and M. Horowitz, “Optical AND gate based on soliton interaction in a fiber Bragg grating,” Opt. Lett. 32, 1211-1213 (2007).
[CrossRef] [PubMed]

2006

A. Rosenthal and M. Horowitz, “Analysis and design of nonlinear fiber Bragg gratings and their application for optical compression of reflected pulses,” Opt. Lett. 31, 1334-1336 (2006).
[CrossRef] [PubMed]

J. T. Mok, C. M. de Sterke, and B. J. Eggleton, “Delay-tunable gap soliton-based slow-light system,” Opt. Express 14, 11987-11996 (2006).
[CrossRef] [PubMed]

A. Rosenthal and M. Horowitz, “Bragg-soliton formation and pulse compression in a one-dimensional periodic structure,” Phys. Rev. E 74, 066611 (2006).
[CrossRef]

J. T. Mok, C. M. de Sterke, I. C. M. Littler, and B. J. Eggleton, “Dispersionless slow light using gap solitons,” Nat. Phys. 2, 775-780 (2006).
[CrossRef]

2003

W. C. K. Mak, B. A. Malomed, and P. L. Chu, “Formation of a standing light pulse through collision of gap solitons,” Phys. Rev. E 68, 26609 (2003).
[CrossRef]

O. V. Sinkin, J. Zweck, and C. R. Menyuk, “Optimization of the split-step Fourier method in modeling optical-fiber communications systems,” J. Lightwave Technol. 21, 61-68 (2003).
[CrossRef]

1998

1996

C. Vassallo and F. Collino, “Highly efficient absorbing boundary conditions for the beam propagation method,” J. Lightwave Technol. 14, 1570-1577 (1996).
[CrossRef]

1994

J. P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114, 185-200 (1994).
[CrossRef]

1991

1990

C. M. de Sterke and J. E. Sipe, “Switching dynamics of finite periodic nonlinear media: a numerical study,” Phys. Rev. A 42, 2858-2869 (1990).
[CrossRef] [PubMed]

1989

A. B. Aceves and S. Wabnitz, “Self-induced transparency solitons in nonlinear refractive periodic media,” Phys. Lett. A 141, 37-42 (1989).
[CrossRef]

D. N. Christodoulides and R. I. Joseph, “Slow Bragg solitons in nonlinear periodic structures,” Phys. Rev. Lett. 62, 1746-1749 (1989).
[CrossRef] [PubMed]

1985

H. G. Winful, “Pulse compression in optical fiber filters,” Appl. Phys. Lett. 46, 527-529 (1985).
[CrossRef]

1983

1978

1949

C. E. Shannon, “Communication in the presence of noise,” Proc. IRE 37, 10-21 (1949).
[CrossRef]

Appl. Opt.

Appl. Phys. Lett.

H. G. Winful, “Pulse compression in optical fiber filters,” Appl. Phys. Lett. 46, 527-529 (1985).
[CrossRef]

J. Comput. Phys.

J. P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114, 185-200 (1994).
[CrossRef]

T. Dohnal and T. Hagstrom, “Perfectly matched layers in photonics computations: 1D and 2D nonlinear coupled mode equations,” J. Comput. Phys. 223, 690-710 (2007).
[CrossRef]

J. Lightwave Technol.

C. Vassallo and F. Collino, “Highly efficient absorbing boundary conditions for the beam propagation method,” J. Lightwave Technol. 14, 1570-1577 (1996).
[CrossRef]

O. V. Sinkin, J. Zweck, and C. R. Menyuk, “Optimization of the split-step Fourier method in modeling optical-fiber communications systems,” J. Lightwave Technol. 21, 61-68 (2003).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. B

Nat. Phys.

J. T. Mok, C. M. de Sterke, I. C. M. Littler, and B. J. Eggleton, “Dispersionless slow light using gap solitons,” Nat. Phys. 2, 775-780 (2006).
[CrossRef]

Opt. Express

Opt. Lett.

Phys. Lett. A

A. B. Aceves and S. Wabnitz, “Self-induced transparency solitons in nonlinear refractive periodic media,” Phys. Lett. A 141, 37-42 (1989).
[CrossRef]

Phys. Rev. A

C. M. de Sterke and J. E. Sipe, “Switching dynamics of finite periodic nonlinear media: a numerical study,” Phys. Rev. A 42, 2858-2869 (1990).
[CrossRef] [PubMed]

Phys. Rev. E

A. Rosenthal and M. Horowitz, “Bragg-soliton formation and pulse compression in a one-dimensional periodic structure,” Phys. Rev. E 74, 066611 (2006).
[CrossRef]

W. C. K. Mak, B. A. Malomed, and P. L. Chu, “Formation of a standing light pulse through collision of gap solitons,” Phys. Rev. E 68, 26609 (2003).
[CrossRef]

Phys. Rev. Lett.

D. N. Christodoulides and R. I. Joseph, “Slow Bragg solitons in nonlinear periodic structures,” Phys. Rev. Lett. 62, 1746-1749 (1989).
[CrossRef] [PubMed]

Proc. IRE

C. E. Shannon, “Communication in the presence of noise,” Proc. IRE 37, 10-21 (1949).
[CrossRef]

Other

R. L. Burden and J. D. Faires, “Richardson's extrapolation,” in Numerical Analysis (ITP, 1997), pp. 180-183.

C. M. de Sterke and J. E. Sipe, “Gap solitons,” in Progress in Optics XXXIII, E.Wolf, ed. (Elsevier, 1994), pp. 203-260.

G. P. Agrawal, “Numerical methods,” in Nonlinear Fiber Optics (Academic, 1995), pp. 50-55.

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Figures (11)

Fig. 1
Fig. 1

Comparison between the intensity calculated using the explicit one-soliton solution in FBGs [9] (solid curve) and the numerical solution obtained using nonsymmetrized OSSM with a spatial step size of Δ Z = W S 40 (dashed curve) and Δ Z = W S 2400 (dashed–dotted curve), where W S = 9.72 mm is the spatial FWHM of the soliton. The temporal step size was equal in both cases to Δ T = W S ( 2400 V g ) and the result was calculated after the soliton propagated a distance of 13.5 cm inside the grating.

Fig. 2
Fig. 2

Comparison between the intensity of the explicit one-soliton solution (solid curve) and the numerical solution that was calculated using nonsymmetrized OSSM with a temporal step size of Δ T = W S ( 1600 V g ) (dashed curve), Δ T = W S ( 800 V g ) (dashed–dotted curve), Δ T = W S ( 4000 V g ) (dotted curve), and Δ T = W S ( 200 V g ) (left-pointing triangle marker). The comparison was performed after the soliton had propagated a distance of 13.5 cm along the grating. The spatial step size was equal to Δ Z = W S 40 .

Fig. 3
Fig. 3

Relative error, defined in Eq. (23), between the explicit one-soliton solution and the numerical results as a function of the temporal step size, Δ T V g W S , calculated after the soliton has propagated a distance of 13.5 cm along the grating. The spatial step size was equal to Δ Z = W S 40 , where W S is the spatial FWHM of the soliton. The solid line is a least-square mean error linear fit: ln ( ε ) = 1.9708 ln ( Δ T V g W S ) + 15.228 .

Fig. 4
Fig. 4

Relative error between the explicit one-soliton solution and the numerical results of the output soliton amplitude (solid curve) and the output soliton position (dashed curve) as a function of the temporal step size, Δ T , calculated after the soliton has propagated a distance of 13.5 cm along the grating. The spatial step size was equal to Δ Z = W S 40 .

Fig. 5
Fig. 5

Collision between two solitons calculated using nonsymmetrized OSSM shown in (a) a three-dimensional plot and in (b) a two-dimensional plot. The simulation parameters were Δ Z = W S 80 , V g Δ T = W S 800 , where W S = 8.86 mm is the spatial FWHM of the shorter soliton. The peak power of the two input solitons and their frequency offset relative to the local Bragg frequency are equal to 582 W , 478.8 W , 297.78 GHz , and 298.46 GHz , respectively.

Fig. 6
Fig. 6

Intensity of the two solitons after their interaction calculated at t = 25.52 ns . The spatial step size was equal to Δ Z = W S 40 and the temporal step size was equal to Δ T = W S ( 8000 V g ) (solid curve), Δ T = W S ( 1600 V g ) (dashed–dotted curve), and Δ T = W S ( 800 V g ) (dashed curve).

Fig. 7
Fig. 7

Intensity of two solitons, calculated at the end of their interaction at t = 25.52 ns , by using symmetrized OSSM (solid curve) and by using nonsymmetrized OSSM (dashed curve). The spatial and temporal resolution are Δ Z = W S 800 and Δ T = W S ( 800 V g ) , respectively. The solitons' parameters are the same as in Fig. 5.

Fig. 8
Fig. 8

Intensity of two solitons after their interaction, calculated at t = 25.52 ns using nonsymmetrized OSSM for a spatial step size of Δ Z = W S 800 (solid curve) and Δ Z = W S 80 (dashed curve). The temporal step size was equal to Δ T = W S ( 800 V g ) in both cases.

Fig. 9
Fig. 9

Transmissivity versus the incoming amplitude of a bistable device formed by a uniform FBG.

Fig. 10
Fig. 10

Output intensity after launching an input hyperbolic-secant pulse through an apodization section and 19 cm of uniform grating. The solid curve gives the result calculated using a uniform spatial step size with Δ Z = V g Δ T = 0.005 mm , and the dashed curve gives the result obtained using nonsymmetrized OSSM with a nonuniform spatial step size with Δ Z = V g Δ T = 0.005 mm in the apodized grating region and Δ Z = 1 mm , V g Δ T = 0.005 mm in the uniform region.

Fig. 11
Fig. 11

Peak power as a function of the propagation duration obtained by using a uniform spatial step size (solid curve) and by using nonsymmetrized OSSM with a nonuniform spatial step size (dashed curve). The simulation parameters are the same as used in Fig. 10.

Equations (32)

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± i z u ± + i V g 1 t u ± + κ ( z ) u + Γ ( u ± 2 + 2 u 2 ) u ± + δ ( z ) u ± = 0 ,
t w = ( D ̂ + N ̂ ) w ,
D ̂ ( z ) = V g ( i δ ( z ) i κ ( z ) i κ ( z ) i δ ( z ) ) ,
N ̂ ( z , t ) = V g ( z + N ( z , t ) 0 0 z + N + ( z , t ) ) ,
N ( z , t ) = i Γ ( u 2 + 2 u ± 2 ) ,
w = [ u ( z , t ) u + ( z , t ) ] .
w ( z , t + Δ T ) e Δ T D ̂ e t t + Δ T N ̂ d t w ( z , t ) .
exp ( Δ T D ̂ ) = e i δ h ( cos [ κ ( z ) h ] i sin [ κ ( z ) h ] i sin [ κ ( z ) h ] cos [ κ ( z ) h ] ) ,
τ ± = 1 2 ( V g t ± z ) .
N ̂ ( τ + , τ ) = ( N ( τ + , τ ) 0 0 N + ( τ + , τ ) ) ,
u ( τ + , τ + h ) = e τ τ + h N ( τ + , τ ) d τ u ( τ + , τ ) u + ( τ + + h , τ ) = e τ + τ + + h N + ( τ + , τ ) d τ + u + ( τ + , τ ) .
τ τ + h N ( τ + , τ ) d τ h N ( τ + , τ ) ,
τ + τ + + h N + ( τ + , τ ) d τ + h N + ( τ + , τ ) .
u ( z h , t + Δ T ) = exp [ h N ( z , t ) ] u ( z , t ) .
u ( z , t + Δ T ) = e i δ h { cos ( κ h ) exp [ h N ( z + h , t ) ] u ( z + h , t ) + i sin ( κ h ) exp [ h N + ( z h , t ) ] u + ( z h , t ) } ,
u + ( z , t + Δ T ) = e i δ h { cos ( κ h ) exp [ h N + ( z h , t ) ] u + ( z h , t ) + i sin ( κ h ) exp [ h N ( z + h , t ) ] u ( z + h , t ) } .
w ( z , t + Δ T ) e Δ T D ̂ 2 e t t + Δ T N ̂ d t e Δ T D ̂ 2 w ( z , t ) .
τ τ + h N ( τ + , τ ) d τ 1 2 h [ N ( τ + , τ ) + N ( τ + , τ + h ) ] ,
τ + τ + + h N + ( τ + , τ ) d τ + 1 2 h [ N + ( τ + , τ ) + N + ( τ + + h , τ ) ] .
u ( z h , t + Δ T ) = e ( h 2 ) [ N ( z , t ) + N ( z h , t + Δ T ) ] u ( z , t ) .
u ( z , t + Δ T ) = e i δ h [ cos ( κ h ) exp { h 2 [ N ( z + h , t ) + N ( z , t + Δ T ) ] } u ( z + h , t ) + i sin ( κ h ) exp { h 2 [ N + ( z h , t ) + N + ( z , t + Δ T ) ] } u + ( z h , t ) ] ,
u + ( z , t + Δ T ) = e i δ h [ cos ( κ h ) exp { h 2 [ N + ( z h , t ) + N + ( z , t + Δ T ) ] } u + ( z h , t ) + i sin ( κ h ) exp { h 2 [ N ( z + h , t ) + N ( z , t + Δ T ) ] } u ( z + h , t ) ] .
u ± ( z n + h , t ) = { u ± ( z n + 1 , t ) n = 1 , , N 1 0 n = N .
u ± ( z n h , t ) = { 0 n = 1 u ± ( z n 1 , t ) n = 2 , , N .
u ( z ± h , t + Δ T ) = IFT { e ± j k h FT [ u ( z , t ) ] } ,
u ± ( z m , t ) = n = 1 M u ± ( z n , t ) sinc [ ω c ( z m n Δ z o ) ] ,
W ( z ) = { sin 1 3 [ π ( z + L 2 ) ( 2 L a ) ] L 2 z L 2 + L a 1 L 2 + L a z L 2 L a sin 1 3 [ π ( z L 2 ) ( 2 L a ) ] L 2 L a z L 2 ) ,
ε = I 1 I 2 I 1 ,
ε a = P 1 P 2 P 1 ,
ε z = Z 1 Z 2 W S ,
ε v = 1 v V g Δ Z ( t 2 ) Δ Z ( t 1 ) t 2 t 1 ,
u ± ( z , t + 2 Δ T ) = ( 8 7 ) u ± f ( z , t + 2 Δ T ) ( 1 7 ) u ± c ( z , t + 2 Δ T ) + O [ ( Δ T ) 4 ] ,

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