Abstract

We present an investigation of accumulated numerical error in the simulation of four-wave mixing, which is frequently used to model a variety of nonlinear optical devices. The Dormand-Prince method (commonly used by commercial solving software such as MATLAB) has been found to be susceptible to numerical error, which manifests itself in a 3.8% increase in the total power over 200 m of nonlinear interaction length. This numerical error can lead to qualitatively mistaken physical interpretations of simulation results, which are similar to those found in previously published materials. We use a home-built Adams-Bashforth solver to simulate four-wave mixing, which produces results that do not lead to unphysical results, even for simulation over a large nonlinear interaction length. By comparing the results of these two methods we were able to illustrate the qualitative effects of the accumulated numerical error in the former. The source of this cumulative power error is traced to the solutions provided by the Dormand-Prince method for the self- and cross-phase modulation terms of the coupled mode equations; this error increases for larger nonlinearities or if step size increases. Even when this power accumulates from infinitesimal per-step errors, significant changes occur that could lead to qualitative differences in generated power values and conversion efficiencies. This reveals the potential danger of applying commonly used numerical solvers in simulating nonlinear optical processes.

© 2012 Optical Society of America

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References

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  1. A. Vatarescu, “Light conversion in nonlinear monomode optical fibers,” J. Lightwave Technol. 5, 1652–1659 (1987).
    [CrossRef]
  2. Z. Vujicic, N. B. Pavlovic, G. Parca, and A. Teixeira, “Effect of four-wave mixing spatial dependence on idler residual modulation,” Phys. Scr. T149, 014045 (2012).
    [CrossRef]
  3. X. Xu, Y. Yao, X. Zhao, and D. Chen, “Multiple four-wave-mixing processes and their application to multiwavelength erbium-doped fiber lasers,” J. Lightwave Technol. 27, 2876–2885 (2009).
    [CrossRef]
  4. P. Parolari, L. Marazzi, E. Rognoni, and M. Martinelli, “Influence of pump parameters on two-pump optical parametric amplification,” J. Lightwave Technol. 23, 2524–2530 (2005).
    [CrossRef]
  5. G. P. Agrawal, Nonlinear Fiber Optics 4E (Academic, 2007), Chap. 10.
  6. J. Hansryd, P. A. Andrekson, M. Westlund, J. Li, and P. O. Hedekvist, “Fiber-based optical parametric amplifiers and their applications,” IEEE J. Sel. Top. Quantum Electron. 8, 506–520 (2002).
    [CrossRef]
  7. H. Itoh, G. M. Davis, and S. Sudo, “Continuous-wave-pumped modulation instability in an optical fiber,” Opt. Lett. 14, 1368–1370 (1989).
    [CrossRef]
  8. G. Cappellini and S. Trillo, “Third order three-wave mixing in single-mode fibers: Exact solutions and spatial instability effects,” J. Opt. Soc. Am. B 8, 824–838 (1991).
    [CrossRef]
  9. N. Shibata, R. P. Braun, and R. G. Waarts, “Phase-mismatch dependence of efficiency of wave generation through four-wave mixing in a single-mode optical fiber,” IEEE J. Quantum Electron. 23, 1205–1210 (1987).
    [CrossRef]
  10. Thompson Reuters, “ISI Web of knowledge,” http://www.webofknowledge.com .
  11. R. H. Stolen and J. E. Bjorkholm, “Parametric amplification and frequency conversion in optical fibers,” IEEE J. Quantum Electron. 18, 1062–1072 (1982).
    [CrossRef]
  12. Y. Chen and A. W. Snyder, “Four-photon parametric mixing in optical fibers: effect of pump depletion,” Opt. Lett. 14, 87–89 (1989).
    [CrossRef]
  13. L. F. Shampine, “Some practical Runge-Kutta formulas,” Math. Comput. 46, 135–150 (1986).
    [CrossRef]
  14. J. C. Butcher, Numerical Methods for Ordinary Differential Equations (Wiley, 2003), Chap. 2.
  15. K. E. Atkinson, An Introduction to Numerical Analysis (Wiley, 1989), Chap. 6.

2012 (1)

Z. Vujicic, N. B. Pavlovic, G. Parca, and A. Teixeira, “Effect of four-wave mixing spatial dependence on idler residual modulation,” Phys. Scr. T149, 014045 (2012).
[CrossRef]

2009 (1)

2005 (1)

2002 (1)

J. Hansryd, P. A. Andrekson, M. Westlund, J. Li, and P. O. Hedekvist, “Fiber-based optical parametric amplifiers and their applications,” IEEE J. Sel. Top. Quantum Electron. 8, 506–520 (2002).
[CrossRef]

1991 (1)

1989 (2)

1987 (2)

N. Shibata, R. P. Braun, and R. G. Waarts, “Phase-mismatch dependence of efficiency of wave generation through four-wave mixing in a single-mode optical fiber,” IEEE J. Quantum Electron. 23, 1205–1210 (1987).
[CrossRef]

A. Vatarescu, “Light conversion in nonlinear monomode optical fibers,” J. Lightwave Technol. 5, 1652–1659 (1987).
[CrossRef]

1986 (1)

L. F. Shampine, “Some practical Runge-Kutta formulas,” Math. Comput. 46, 135–150 (1986).
[CrossRef]

1982 (1)

R. H. Stolen and J. E. Bjorkholm, “Parametric amplification and frequency conversion in optical fibers,” IEEE J. Quantum Electron. 18, 1062–1072 (1982).
[CrossRef]

Agrawal, G. P.

G. P. Agrawal, Nonlinear Fiber Optics 4E (Academic, 2007), Chap. 10.

Andrekson, P. A.

J. Hansryd, P. A. Andrekson, M. Westlund, J. Li, and P. O. Hedekvist, “Fiber-based optical parametric amplifiers and their applications,” IEEE J. Sel. Top. Quantum Electron. 8, 506–520 (2002).
[CrossRef]

Atkinson, K. E.

K. E. Atkinson, An Introduction to Numerical Analysis (Wiley, 1989), Chap. 6.

Bjorkholm, J. E.

R. H. Stolen and J. E. Bjorkholm, “Parametric amplification and frequency conversion in optical fibers,” IEEE J. Quantum Electron. 18, 1062–1072 (1982).
[CrossRef]

Braun, R. P.

N. Shibata, R. P. Braun, and R. G. Waarts, “Phase-mismatch dependence of efficiency of wave generation through four-wave mixing in a single-mode optical fiber,” IEEE J. Quantum Electron. 23, 1205–1210 (1987).
[CrossRef]

Butcher, J. C.

J. C. Butcher, Numerical Methods for Ordinary Differential Equations (Wiley, 2003), Chap. 2.

Cappellini, G.

Chen, D.

Chen, Y.

Davis, G. M.

Hansryd, J.

J. Hansryd, P. A. Andrekson, M. Westlund, J. Li, and P. O. Hedekvist, “Fiber-based optical parametric amplifiers and their applications,” IEEE J. Sel. Top. Quantum Electron. 8, 506–520 (2002).
[CrossRef]

Hedekvist, P. O.

J. Hansryd, P. A. Andrekson, M. Westlund, J. Li, and P. O. Hedekvist, “Fiber-based optical parametric amplifiers and their applications,” IEEE J. Sel. Top. Quantum Electron. 8, 506–520 (2002).
[CrossRef]

Itoh, H.

Li, J.

J. Hansryd, P. A. Andrekson, M. Westlund, J. Li, and P. O. Hedekvist, “Fiber-based optical parametric amplifiers and their applications,” IEEE J. Sel. Top. Quantum Electron. 8, 506–520 (2002).
[CrossRef]

Marazzi, L.

Martinelli, M.

Parca, G.

Z. Vujicic, N. B. Pavlovic, G. Parca, and A. Teixeira, “Effect of four-wave mixing spatial dependence on idler residual modulation,” Phys. Scr. T149, 014045 (2012).
[CrossRef]

Parolari, P.

Pavlovic, N. B.

Z. Vujicic, N. B. Pavlovic, G. Parca, and A. Teixeira, “Effect of four-wave mixing spatial dependence on idler residual modulation,” Phys. Scr. T149, 014045 (2012).
[CrossRef]

Rognoni, E.

Shampine, L. F.

L. F. Shampine, “Some practical Runge-Kutta formulas,” Math. Comput. 46, 135–150 (1986).
[CrossRef]

Shibata, N.

N. Shibata, R. P. Braun, and R. G. Waarts, “Phase-mismatch dependence of efficiency of wave generation through four-wave mixing in a single-mode optical fiber,” IEEE J. Quantum Electron. 23, 1205–1210 (1987).
[CrossRef]

Snyder, A. W.

Stolen, R. H.

R. H. Stolen and J. E. Bjorkholm, “Parametric amplification and frequency conversion in optical fibers,” IEEE J. Quantum Electron. 18, 1062–1072 (1982).
[CrossRef]

Sudo, S.

Teixeira, A.

Z. Vujicic, N. B. Pavlovic, G. Parca, and A. Teixeira, “Effect of four-wave mixing spatial dependence on idler residual modulation,” Phys. Scr. T149, 014045 (2012).
[CrossRef]

Trillo, S.

Vatarescu, A.

A. Vatarescu, “Light conversion in nonlinear monomode optical fibers,” J. Lightwave Technol. 5, 1652–1659 (1987).
[CrossRef]

Vujicic, Z.

Z. Vujicic, N. B. Pavlovic, G. Parca, and A. Teixeira, “Effect of four-wave mixing spatial dependence on idler residual modulation,” Phys. Scr. T149, 014045 (2012).
[CrossRef]

Waarts, R. G.

N. Shibata, R. P. Braun, and R. G. Waarts, “Phase-mismatch dependence of efficiency of wave generation through four-wave mixing in a single-mode optical fiber,” IEEE J. Quantum Electron. 23, 1205–1210 (1987).
[CrossRef]

Westlund, M.

J. Hansryd, P. A. Andrekson, M. Westlund, J. Li, and P. O. Hedekvist, “Fiber-based optical parametric amplifiers and their applications,” IEEE J. Sel. Top. Quantum Electron. 8, 506–520 (2002).
[CrossRef]

Xu, X.

Yao, Y.

Zhao, X.

IEEE J. Quantum Electron. (2)

N. Shibata, R. P. Braun, and R. G. Waarts, “Phase-mismatch dependence of efficiency of wave generation through four-wave mixing in a single-mode optical fiber,” IEEE J. Quantum Electron. 23, 1205–1210 (1987).
[CrossRef]

R. H. Stolen and J. E. Bjorkholm, “Parametric amplification and frequency conversion in optical fibers,” IEEE J. Quantum Electron. 18, 1062–1072 (1982).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron. (1)

J. Hansryd, P. A. Andrekson, M. Westlund, J. Li, and P. O. Hedekvist, “Fiber-based optical parametric amplifiers and their applications,” IEEE J. Sel. Top. Quantum Electron. 8, 506–520 (2002).
[CrossRef]

J. Lightwave Technol. (3)

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

Math. Comput. (1)

L. F. Shampine, “Some practical Runge-Kutta formulas,” Math. Comput. 46, 135–150 (1986).
[CrossRef]

Opt. Lett. (2)

Phys. Scr. (1)

Z. Vujicic, N. B. Pavlovic, G. Parca, and A. Teixeira, “Effect of four-wave mixing spatial dependence on idler residual modulation,” Phys. Scr. T149, 014045 (2012).
[CrossRef]

Other (4)

G. P. Agrawal, Nonlinear Fiber Optics 4E (Academic, 2007), Chap. 10.

Thompson Reuters, “ISI Web of knowledge,” http://www.webofknowledge.com .

J. C. Butcher, Numerical Methods for Ordinary Differential Equations (Wiley, 2003), Chap. 2.

K. E. Atkinson, An Introduction to Numerical Analysis (Wiley, 1989), Chap. 6.

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Figures (5)

Fig. 1.
Fig. 1.

Evolution of idler power (a) and relative phase (b) over the 200 m length of fiber when solved by the DP method. Input parameters: 70 W pump power, 0.1 μW idler power, 0.1 μW signal power for generation case (solid line), and 6 mW signal power for amplification case (dashed line). The dotted line traces the buildup of power that is inconsistent with power conservation, and the gray ellipse highlights the gradual increase of successive minima and consequent narrowing of the low-idler region.

Fig. 2.
Fig. 2.

Evolution of idler power (a) and relative phase (b) over the 200 m length of fiber when solved by the AB method. Input parameters: 70 W pump power, 0.1 μW idler power, 0.1 μW signal power for generation case (solid line), and 6 mW signal power for amplification case (dashed line).

Fig. 3.
Fig. 3.

Idler power (a) and relative phase (b) over a range of artificial numerical gain values.

Fig. 4.
Fig. 4.

Idler power (a) and relative phase (b) over a range of artificial numerical loss values.

Fig. 5.
Fig. 5.

Amplitude changes in the various approximate solutions to Eq. (10).

Equations (14)

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dA1dz=in2ω1cAeff[(|A1|2+2|A2|2+2|A3|2+2|A4|2)A1+2A2*A3A4exp(iΔkz)],dA2dz=in2ω2cAeff[(2|A1|2+|A2|2+2|A3|2+2|A4|2)A2+2A1*A3A4exp(iΔkz)],dA3dz=in2ω3cAeff[(2|A1|2+2|A2|2+|A3|2+2|A4|2)A3+2A1A2A4*exp(iΔkz)],dA4dz=in2ω4cAeff[(2|A1|2+2|A2|2+2|A3|2+|A4|2)A4+2A1A2A3*exp(iΔkz)],
θ=φ1+φ2φ3φ4,
yn=yn1+h(32f(yn1)12f(yn2)).
dAjdz=γ·Aj.
dAjdz=α·Aj.
yn=yn1+j=1sbj·kj,
kj=h·f(yn1+l=1j1ajlkl)k1h·f(yn1).
|a21as1as,s1b1bs1bs.
15340940444556153291937265612536021876444865612127299017316835533467325247491765103186563538405001113125192218767841184353840500111312519221876784118405179576000757116695393640920973392001872100140,
dydz=iΓyΓ{n2ωcAeffP(SPM)2n2ωcAeffP(XPM).
yn=yn1exp(iΓh).
yn=yn1(1+i(hΓ)(hΓ)22i241216(hΓ)3+(hΓ)424+i(hΓ)5120(hΓ)6600),
yn=yn1(1+i(hΓ)(hΓ)22i498469(hΓ)3+(hΓ)424i1936487(hΓ)54130559(hΓ)6i(hΓ)72400).
yn=yn1(1+i(hΓ)(hΓ)22i(hΓ)36+(hΓ)424+i(hΓ)5120(hΓ)6720i(hΓ)75040),

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