Abstract

We present a numerical procedure for solving a set of nonlinear coupled-mode equations on a finite interval. These equations originally arose in the study of the dynamics of gap solitons in nonlinear periodic media. Our procedure, which uses an implicit fourth-order Runge–Kutta method, is easy to implement, versatile, and quite well suited for vectorization or parallelization.

© 1991 Optical Society of America

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References

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  1. A. Yariv, “Coupled mode theory for guided wave optics,” IEEE J. Quantum Electron. QE-9, 919–933 (1973).
    [CrossRef]
  2. D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, New York, 1974).
  3. S. M. Jensen, “The nonlinear coherent coupler,” IEEE J. Quantum Electron. QE-9, 1580–1583 (1982).
    [CrossRef]
  4. C. Martijn de Sterke and J. E. Sipe, “Switching behavior of finite periodic nonlinear media,” Phys. Rev. A 42, 2558–2569 (1990).
  5. H. G. Winful and G. D. Cooperman, “Self-pulsing and chaos in distributed feedback bistable optical devices,” Appl. Phys. Lett. 40, 298–300 (1982).
    [CrossRef]
  6. B. D. Robert and J. E. Sipe, “Instability in an illuminated nonlinear waveguide: a phase conjugation effect,” Opt. Lett. 15, 261–263 (1990).
    [CrossRef] [PubMed]
  7. G. P. Agrawal, Nonlinear Fiber Optics (Academic, San Diego, Calif., 1989), Chap. 7.
  8. For a recent overview, see, e.g., G. I. Stegeman and R. H. Stolen, eds., feature on nonlinear guided-wave phenomena, J. Opt. Soc. Am. B 5, 263–574 (1988).
    [CrossRef]
  9. H. G. Winful, J. H. Marburger, and E. Garmire, “Theory of bistability in nonlinear distributed feedback structures,” Appl. Phys. Lett. 35, 379–381 (1979).
    [CrossRef]
  10. N. Bleistein, Mathematical Methods for Wave Phenomena (Academic, Orlando, Fla., 1984).
  11. W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes (Cambridge U. Press, Cambridge, 1986).
  12. E. Hairer, S. P. Nørsett, and G. Wanner, Solving Ordinary Differential Equations I (Springer-Verlag, Berlin, 1987).
  13. K. Dekker and J. G. Verwer, Stability of Runge–Kutta Methods for Stiff Nonlinear Differential Equations (North-Holland, Amsterdam, 1984).
  14. E. Isaacson and H. B. Keller, Analysis of Numerical Methods (Wiley, New York, 1966).
  15. J. M. Ortega and W. C. Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables (Academic, New York, 1970).

1990 (2)

C. Martijn de Sterke and J. E. Sipe, “Switching behavior of finite periodic nonlinear media,” Phys. Rev. A 42, 2558–2569 (1990).

B. D. Robert and J. E. Sipe, “Instability in an illuminated nonlinear waveguide: a phase conjugation effect,” Opt. Lett. 15, 261–263 (1990).
[CrossRef] [PubMed]

1988 (1)

For a recent overview, see, e.g., G. I. Stegeman and R. H. Stolen, eds., feature on nonlinear guided-wave phenomena, J. Opt. Soc. Am. B 5, 263–574 (1988).
[CrossRef]

1982 (2)

H. G. Winful and G. D. Cooperman, “Self-pulsing and chaos in distributed feedback bistable optical devices,” Appl. Phys. Lett. 40, 298–300 (1982).
[CrossRef]

S. M. Jensen, “The nonlinear coherent coupler,” IEEE J. Quantum Electron. QE-9, 1580–1583 (1982).
[CrossRef]

1979 (1)

H. G. Winful, J. H. Marburger, and E. Garmire, “Theory of bistability in nonlinear distributed feedback structures,” Appl. Phys. Lett. 35, 379–381 (1979).
[CrossRef]

1973 (1)

A. Yariv, “Coupled mode theory for guided wave optics,” IEEE J. Quantum Electron. QE-9, 919–933 (1973).
[CrossRef]

Agrawal, G. P.

G. P. Agrawal, Nonlinear Fiber Optics (Academic, San Diego, Calif., 1989), Chap. 7.

Bleistein, N.

N. Bleistein, Mathematical Methods for Wave Phenomena (Academic, Orlando, Fla., 1984).

Cooperman, G. D.

H. G. Winful and G. D. Cooperman, “Self-pulsing and chaos in distributed feedback bistable optical devices,” Appl. Phys. Lett. 40, 298–300 (1982).
[CrossRef]

Dekker, K.

K. Dekker and J. G. Verwer, Stability of Runge–Kutta Methods for Stiff Nonlinear Differential Equations (North-Holland, Amsterdam, 1984).

Flannery, B. P.

W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes (Cambridge U. Press, Cambridge, 1986).

Garmire, E.

H. G. Winful, J. H. Marburger, and E. Garmire, “Theory of bistability in nonlinear distributed feedback structures,” Appl. Phys. Lett. 35, 379–381 (1979).
[CrossRef]

Hairer, E.

E. Hairer, S. P. Nørsett, and G. Wanner, Solving Ordinary Differential Equations I (Springer-Verlag, Berlin, 1987).

Isaacson, E.

E. Isaacson and H. B. Keller, Analysis of Numerical Methods (Wiley, New York, 1966).

Jensen, S. M.

S. M. Jensen, “The nonlinear coherent coupler,” IEEE J. Quantum Electron. QE-9, 1580–1583 (1982).
[CrossRef]

Keller, H. B.

E. Isaacson and H. B. Keller, Analysis of Numerical Methods (Wiley, New York, 1966).

Marburger, J. H.

H. G. Winful, J. H. Marburger, and E. Garmire, “Theory of bistability in nonlinear distributed feedback structures,” Appl. Phys. Lett. 35, 379–381 (1979).
[CrossRef]

Marcuse, D.

D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, New York, 1974).

Martijn de Sterke, C.

C. Martijn de Sterke and J. E. Sipe, “Switching behavior of finite periodic nonlinear media,” Phys. Rev. A 42, 2558–2569 (1990).

Nørsett, S. P.

E. Hairer, S. P. Nørsett, and G. Wanner, Solving Ordinary Differential Equations I (Springer-Verlag, Berlin, 1987).

Ortega, J. M.

J. M. Ortega and W. C. Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables (Academic, New York, 1970).

Press, W. H.

W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes (Cambridge U. Press, Cambridge, 1986).

Rheinboldt, W. C.

J. M. Ortega and W. C. Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables (Academic, New York, 1970).

Robert, B. D.

Sipe, J. E.

B. D. Robert and J. E. Sipe, “Instability in an illuminated nonlinear waveguide: a phase conjugation effect,” Opt. Lett. 15, 261–263 (1990).
[CrossRef] [PubMed]

C. Martijn de Sterke and J. E. Sipe, “Switching behavior of finite periodic nonlinear media,” Phys. Rev. A 42, 2558–2569 (1990).

Teukolsky, S. A.

W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes (Cambridge U. Press, Cambridge, 1986).

Verwer, J. G.

K. Dekker and J. G. Verwer, Stability of Runge–Kutta Methods for Stiff Nonlinear Differential Equations (North-Holland, Amsterdam, 1984).

Vetterling, W. T.

W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes (Cambridge U. Press, Cambridge, 1986).

Wanner, G.

E. Hairer, S. P. Nørsett, and G. Wanner, Solving Ordinary Differential Equations I (Springer-Verlag, Berlin, 1987).

Winful, H. G.

H. G. Winful and G. D. Cooperman, “Self-pulsing and chaos in distributed feedback bistable optical devices,” Appl. Phys. Lett. 40, 298–300 (1982).
[CrossRef]

H. G. Winful, J. H. Marburger, and E. Garmire, “Theory of bistability in nonlinear distributed feedback structures,” Appl. Phys. Lett. 35, 379–381 (1979).
[CrossRef]

Yariv, A.

A. Yariv, “Coupled mode theory for guided wave optics,” IEEE J. Quantum Electron. QE-9, 919–933 (1973).
[CrossRef]

Appl. Phys. Lett. (2)

H. G. Winful, J. H. Marburger, and E. Garmire, “Theory of bistability in nonlinear distributed feedback structures,” Appl. Phys. Lett. 35, 379–381 (1979).
[CrossRef]

H. G. Winful and G. D. Cooperman, “Self-pulsing and chaos in distributed feedback bistable optical devices,” Appl. Phys. Lett. 40, 298–300 (1982).
[CrossRef]

IEEE J. Quantum Electron. (2)

S. M. Jensen, “The nonlinear coherent coupler,” IEEE J. Quantum Electron. QE-9, 1580–1583 (1982).
[CrossRef]

A. Yariv, “Coupled mode theory for guided wave optics,” IEEE J. Quantum Electron. QE-9, 919–933 (1973).
[CrossRef]

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

For a recent overview, see, e.g., G. I. Stegeman and R. H. Stolen, eds., feature on nonlinear guided-wave phenomena, J. Opt. Soc. Am. B 5, 263–574 (1988).
[CrossRef]

Opt. Lett. (1)

Phys. Rev. A (1)

C. Martijn de Sterke and J. E. Sipe, “Switching behavior of finite periodic nonlinear media,” Phys. Rev. A 42, 2558–2569 (1990).

Other (8)

G. P. Agrawal, Nonlinear Fiber Optics (Academic, San Diego, Calif., 1989), Chap. 7.

D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, New York, 1974).

N. Bleistein, Mathematical Methods for Wave Phenomena (Academic, Orlando, Fla., 1984).

W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes (Cambridge U. Press, Cambridge, 1986).

E. Hairer, S. P. Nørsett, and G. Wanner, Solving Ordinary Differential Equations I (Springer-Verlag, Berlin, 1987).

K. Dekker and J. G. Verwer, Stability of Runge–Kutta Methods for Stiff Nonlinear Differential Equations (North-Holland, Amsterdam, 1984).

E. Isaacson and H. B. Keller, Analysis of Numerical Methods (Wiley, New York, 1966).

J. M. Ortega and W. C. Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables (Academic, New York, 1970).

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Figures (3)

Fig. 1
Fig. 1

Schematic of our geometry. The envelope of the incoming radiation is denoted by A(t), while the reflected and transmitted radiation is designated by R(t) and T(t), respectively. L is length.

Fig. 2
Fig. 2

Integration domain in the coordinates ζ and τ. The boundaries of this domain are given by thick solid lines. The dotted lines indicate points of equal time. In a finite-difference scheme the integration takes place along the dashed lines. Note that in the figure the envelopes are sampled at N + 1 = 5 positions only inside the system. In our calculations N is taken to be much larger. The parameter Tr designates a single round-trip time.

Fig. 3
Fig. 3

Numerical results for a calculation in which our method is applied to Eqs. (9). Shown is the magnitude of the transmitted radiation |(L, t)| as a function of time. Eventually (after approximately t = 100) the amplitude of the signal settles, and the output is then truly periodic. The parameters are κ = 5.0, δ = 4.5, Γ = 0.1, L = 1.0; in the rotating frame a constant incoming signal of A(t) = A0 = 2.0. This driving signal is taken to increase smoothly from A(0) = 0 to A(2) = A0 in order to avoid shocks. A(t) = A0 for t ≥ 2.

Tables (1)

Tables Icon

Table 1 Values of Coefficients Used in Integration and Extrapolation Procedures in Eqs. (7) and (8)a

Equations (33)

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+ i F + z + i F + t + κ F - + Γ F + 2 F + + 2 Γ F - 2 F + = 0 , - i F - z + i F - t + κ F + + Γ F - 2 F - + 2 Γ F + 2 F - = 0 ,
F + ( 0 , t ) = A ( t ) , F - ( L , t ) = 0.
F ± ( z , 0 ) = 0.
z = ζ - τ ,             t = ζ + τ ,
i F + ζ = - κ F - - Γ F + 2 F + - 2 Γ F - 2 F + ,
i F - τ = - κ F + - Γ F - 2 F - - 2 Γ F + 2 F - ,
k 1 = h f ( x n , y n ) , k 2 = h f ( x n + h / 2 , y n + k 1 / 2 ) , k 3 = h f ( x n + h / 2 , y n + k 2 / 2 ) , k 4 = h f ( x n + h , y n + k 3 ) , y n + 1 = y n + ( k 1 + 2 k 2 + 2 k 3 + k 4 ) / 6 ,
y n + 1 / 2 = y n + h [ a 21 f ( x n , y n ) + a 22 f ( x n + 1 / 2 , y n + 1 / 2 ) + a 23 f ( x n + 1 , y n + 1 ) ] , y n + 1 = y n + h [ a 31 f ( x n , y n ) + a 32 f ( x n + 1 / 2 , y n + 1 / 2 ) + a 33 f ( x n + 1 , y n + 1 ) ] ,
y n + 3 / 2 ( 0 ) = y n + h [ a ^ 21 f ( x n , y n ) + a ^ 22 f ( x n + 1 / 2 , y n + 1 / 2 ) + a ^ 23 f ( x n + 1 , y n + 1 ) ] , y n + 2 ( 0 ) = y n + h [ a ^ 31 f ( x n , y n ) + a ^ 32 f ( x n + 1 / 2 , y n + 1 / 2 ) + a ^ 33 f ( x n + 1 , y n + 1 ) ] ,
+ i G + z + i G + t + δ G + + κ G - + Γ G + 2 G + + 2 Γ G - 2 G + = 0 , - i G - z + i G - t + δ G - + κ G + + Γ G - 2 G - + 2 Γ G + 2 G - = 0.
y ( x ) = f ( x , y ( x ) ) for x [ a , b ] , y : R C m , f : R × C m C m , y ( a ) = y 0 .
u n ( x n + c i h ) = f ( x n + c i h , u n ( x n + c i h ) ) ,             i = 1 , , s , u n ( x n ) = y n ,
u n ( x ) = i = 1 s l i ( x ) f ( x n + c i h , y n + c i ) ,
l i ( x ) = j = 1 j 1 s x - ( x n + c j h ) h ( c i - c j )
u n ( x ) = u n ( x n ) + x n x u n ( x ^ ) d x ^ = y n + j = 1 s [ x n x l j ( x ^ ) d x ^ ] f ( x n + c j h , y n + c j ) .
y n + c i = u n ( x n + c i h ) = y n + h j = 1 s a i j f ( x n + c j h , y n + c j ) ,
a i j = 1 h x n x n + c i h l j ( x ^ ) d x ^ = 0 c i ( k = 1 k j s x ˜ - c k c j - c k ) d x ˜ .
y n + 1 = u n ( x n + h ) = y n + h j = 1 s b j f ( x n + c j h , y n + c j ) ,
k i = h f ( x n + c i h , y n + h j = 1 s a i j f ( x n + c j h , y n + c j ) )
k i = h f ( x n + c i h , y n + j = 1 s a i j k j ) ,             i = 1 , , s , y n + 1 = y n + j = 1 s b j k j .
y n + 1 - y n = h j = 1 s b j f ( x n + c j h ) = x n x n + 1 u n ( x ) d x x n x n + 1 f ( x ) d x .
y n ( x ) = f ( x , y n ( x ) ) ,             y n ( x n ) = y n .
u n ( x ) = f ( x , u n ( x ) ) + { u n ( x ) - f ( x , u n ( x ) ) } , u n ( x n ) = y n ,
y n ( x ) - u n ( x ) = O ( h s + 1 )             for x [ x n , x n + 2 ] .
y n + 1 + c i ( 0 ) = u n ( x n + 1 + c i h ) = y n + h i = 1 s a ^ i j f ( x n + c i h , y n + c i ) ,
a ^ i j = 1 h x n x n + 1 + c i h l j ( x ^ ) d x ^ = 0 1 + c i ( k = 1 k j s x ˜ - c k c j - c k ) d x ˜ .
y n ( x n + 1 ) - y n + 1 ( x n + 1 ) = y n ( x n + 1 ) - y n + 1 = O ( h p + 1 )
y n + 1 + c i ( 0 ) - y n + 1 + c i = u n ( x n + 1 + c i h ) - u n + 1 ( x n + 1 + c i h ) = [ u n ( x n + 1 + c i h ) - y n ( x n + 1 + c i h ) ] + [ y n ( x n + 1 + c i h ) - y n + 1 ( x n + 1 + c i h ) ] + [ y n + 1 ( x n + 1 + c i h ) - u n + 1 ( x n + 1 + c i h ) ] = O ( h s + 1 ) .
y n + c i ( k ) = y n + h j = 1 s a i j f ( x n + c j h , y n + c j ( k - 1 ) )
y n + c i - y n + c i ( k ) = h j = 1 s a i j f ( x n + c j h , y n + c j ) - f ( x n + c j h , y n + c j ( k - 1 ) ) h L j = 1 s a i j y n + c j - y n + c j ( k - 1 ) = O ( h q + k + 1 ) .
y n + 1 - y n + 1 ( k ) = O ( h q + k + 1 ) .
h L j = 1 s a i j < 1             for i = 1 , , s ,
j = 1 s b j = 1 ,             j = 1 s a i j = c i ,             j = 1 s a ^ i j = 1 + c i             for i = 1 , , s .

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