Abstract

An efficient algorithm, which exhibits a fourth-order global accuracy, for the numerical solution of the normal and generalized nonlinear Schrödinger equations is presented. It has applications for studies of nonlinear pulse propagation and spectral broadening in optical fibers. Simulation of supercontinuum generation processes, in particular, places high demands on numerical accuracy, which makes efficient high-order schemes attractive. The algorithm that is presented here is an adaptation for use in the nonlinear optics field of the fourth-order Runge–Kutta in the Interaction Picture (RK4IP) method, which was originally developed for studies on Bose–Einstein condensates. The performance of the RK4IP method is validated and compared to a number of conventional methods by modeling both the propagation of a second-order soliton and the generation of supercontinuum radiation. It exhibits the expected global fourth-order accuracy for both problems studied and is the most accurate and efficient of the methods tested.

© 2007 IEEE

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  5. C. Lin, R. H. Stolen, "New nanosecond continuum for excited-state spectroscopy," Appl. Phys. Lett. 28, 216-218 (1976).
  6. G. P. Agrawal, Nonlinear Fiber Optics (Academic, 2007).
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2007 (1)

R. G. Scott, C. W. Gardiner, D. A. W. Hutchinson, "Nonequilibrium dynamics: Studies of the reflection of Bose–Einstein condensates ," Laser Phys. 17, 527-532 (2007).

2006 (2)

J. M. Dudley, G. Genty, S. Coen, "Supercontinuum generation in photonic crystal fiber," Rev. Mod. Phys. 78, 1135-1184 (2006).

T. H. Z. Siederdissen, N. C. Nielsen, J. Kuhl, H. Giessen, "Influence of near-resonant self-phase modulation on pulse propagation in semiconductors ," J. Opt. Soc. Amer. B, Opt. Phys. 23, 1360-1370 (2006).

2005 (2)

W. Wüster, J. J. Hope, C. M. Savage, "Collapsing Bose–Einstein condensates beyond the Gross–Pitaevskii approximation ," Phys. Rev. A, Gen. Phys. 71, 033 604 (2005).

G. M. Muslu, H. A. Erbay, "Higher-order split-step Fourier schemes for the generalized nonlinear Schrödinger equation ," Math. Comput. Simul. 67, 581-595 (2005).

2003 (6)

O. V. Sinkin, R. Holzlöhner, J. Zweck, C. R. Menyuk, "Optimization of the split-step Fourier method in modeling optical-fiber communications systems ," J. Lightw. Technol. 21, 61-68 (2003).

C. M. Savage, N. P. Robins, J. J. Hope, "Bose–Einstein condensate collapse: A comparison between theory and experiment ," Phys. Rev. A, Gen. Phys. 67, 014 304 (2003).

J. W. Nicholson, A. K. Abeeluck, C. Headley, M. F. Yan, C. G. Jørgensen, "Pulsed and continuous-wave supercontinuum generation in highly nonlinear dispersion-shifted fibers ," Appl. Phys. B, Photophys. Laser Chem. 77, 211-218 (2003).

W. H. Reeves, D. V. Skyabin, F. Biancalana, J. C. Knight, F. G. Omenetto, A. Efimov, A. J. Taylor, "Transformation and control of ultra-short pulses in dispersion-engineered photonic crystal fibres ," Nature 424, 511-515 (2003).

K. M. Hillingsøe, H. N. Paulsen, J. Thøgersen, S. R. Keiding, J. J. Larsen, "Initial steps of supercontinuum generation in photonic crystal fibers," J. Opt. Soc. Amer. B, Opt. Phys. 20, 1887-1893 (2003).

T. A. Bartels, K. L. Corwin, N. R. Newbury, L. Hollberg, S. A. Diddams, J. W. Nicholson, M. F. Yan, "420-MHz Cr:forsterite femtosecond ring laser and continuum generation in the 1–2- ${\rm \mu}\hbox{m}$ range," Opt. Lett. 28, 1368-1370 (2003).

2002 (2)

J. M. Dudley, S. Coen, "Numerical simulations and coherence properties of supercontinuum generation in photonic crystal and tapered optical fibers," IEEE J. Sel. Topics Quantum Electron. 8, 651-659 (2002).

D. Hollenbeck, C. D. Cantrell, "Multiple-vibrational-mode model for fiber-optic Raman gain spectrum and response function ," J. Opt. Soc. Amer. B, Opt. Phys. 19, 2886-2892 (2002).

2000 (2)

J. K. Ranka, R. S. Windeler, A. J. Stentz, "Visible continuum generation in air–silica microstructure optical fibers with anomalous dispersion at 800 nm," Opt. Lett. 25, 25-27 (2000).

G. Bosco, A. Carena, V. Curri, R. Gaudino, P. Poggiolini, S. Benedetto, "Suppression of spurious tones induced by the split-step method in fiber systems simulation ," IEEE Photon. Technol. Lett. 12, 489-491 (2000).

1993 (1)

1989 (2)

K. J. Blow, D. Wood, "Theoretical description of transient stimulated Raman scattering in optical fibers," IEEE J. Quantum Electron. 25, 2665-2673 (1989).

R. H. Stolen, J. P. Gordon, W. J. Tomlinson, H. A. Haus, "Raman response function of silica-core fibers," J. Opt. Soc. Amer. B, Opt. Phys. 6, 1159-1166 (1989).

1987 (1)

Y. Kodoma, A. Hasegawa, "Nonlinear pulse propagation in a monomode dielectric guide," IEEE J. Quantum Electron. QE-23, 510-524 (1987).

1976 (1)

C. Lin, R. H. Stolen, "New nanosecond continuum for excited-state spectroscopy," Appl. Phys. Lett. 28, 216-218 (1976).

1972 (1)

V. E. Zakharov, A. B. Shabat, "Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media," Sov. Phys.—JETP 34, 62-69 (1972).

Appl. Phys. B, Photophys. Laser Chem. (1)

J. W. Nicholson, A. K. Abeeluck, C. Headley, M. F. Yan, C. G. Jørgensen, "Pulsed and continuous-wave supercontinuum generation in highly nonlinear dispersion-shifted fibers ," Appl. Phys. B, Photophys. Laser Chem. 77, 211-218 (2003).

Appl. Phys. Lett. (1)

C. Lin, R. H. Stolen, "New nanosecond continuum for excited-state spectroscopy," Appl. Phys. Lett. 28, 216-218 (1976).

IEEE J. Quantum Electron. (2)

K. J. Blow, D. Wood, "Theoretical description of transient stimulated Raman scattering in optical fibers," IEEE J. Quantum Electron. 25, 2665-2673 (1989).

Y. Kodoma, A. Hasegawa, "Nonlinear pulse propagation in a monomode dielectric guide," IEEE J. Quantum Electron. QE-23, 510-524 (1987).

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

J. M. Dudley, S. Coen, "Numerical simulations and coherence properties of supercontinuum generation in photonic crystal and tapered optical fibers," IEEE J. Sel. Topics Quantum Electron. 8, 651-659 (2002).

IEEE Photon. Technol. Lett. (1)

G. Bosco, A. Carena, V. Curri, R. Gaudino, P. Poggiolini, S. Benedetto, "Suppression of spurious tones induced by the split-step method in fiber systems simulation ," IEEE Photon. Technol. Lett. 12, 489-491 (2000).

J. Lightw. Technol. (1)

O. V. Sinkin, R. Holzlöhner, J. Zweck, C. R. Menyuk, "Optimization of the split-step Fourier method in modeling optical-fiber communications systems ," J. Lightw. Technol. 21, 61-68 (2003).

J. Opt. Soc. Amer. B, Opt. Phys. (4)

R. H. Stolen, J. P. Gordon, W. J. Tomlinson, H. A. Haus, "Raman response function of silica-core fibers," J. Opt. Soc. Amer. B, Opt. Phys. 6, 1159-1166 (1989).

D. Hollenbeck, C. D. Cantrell, "Multiple-vibrational-mode model for fiber-optic Raman gain spectrum and response function ," J. Opt. Soc. Amer. B, Opt. Phys. 19, 2886-2892 (2002).

K. M. Hillingsøe, H. N. Paulsen, J. Thøgersen, S. R. Keiding, J. J. Larsen, "Initial steps of supercontinuum generation in photonic crystal fibers," J. Opt. Soc. Amer. B, Opt. Phys. 20, 1887-1893 (2003).

T. H. Z. Siederdissen, N. C. Nielsen, J. Kuhl, H. Giessen, "Influence of near-resonant self-phase modulation on pulse propagation in semiconductors ," J. Opt. Soc. Amer. B, Opt. Phys. 23, 1360-1370 (2006).

Laser Phys. (1)

R. G. Scott, C. W. Gardiner, D. A. W. Hutchinson, "Nonequilibrium dynamics: Studies of the reflection of Bose–Einstein condensates ," Laser Phys. 17, 527-532 (2007).

Math. Comput. Simul. (1)

G. M. Muslu, H. A. Erbay, "Higher-order split-step Fourier schemes for the generalized nonlinear Schrödinger equation ," Math. Comput. Simul. 67, 581-595 (2005).

Nature (1)

W. H. Reeves, D. V. Skyabin, F. Biancalana, J. C. Knight, F. G. Omenetto, A. Efimov, A. J. Taylor, "Transformation and control of ultra-short pulses in dispersion-engineered photonic crystal fibres ," Nature 424, 511-515 (2003).

Opt. Lett. (3)

Phys. Rev. A, Gen. Phys. (2)

C. M. Savage, N. P. Robins, J. J. Hope, "Bose–Einstein condensate collapse: A comparison between theory and experiment ," Phys. Rev. A, Gen. Phys. 67, 014 304 (2003).

W. Wüster, J. J. Hope, C. M. Savage, "Collapsing Bose–Einstein condensates beyond the Gross–Pitaevskii approximation ," Phys. Rev. A, Gen. Phys. 71, 033 604 (2005).

Rev. Mod. Phys. (1)

J. M. Dudley, G. Genty, S. Coen, "Supercontinuum generation in photonic crystal fiber," Rev. Mod. Phys. 78, 1135-1184 (2006).

Sov. Phys.—JETP (1)

V. E. Zakharov, A. B. Shabat, "Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media," Sov. Phys.—JETP 34, 62-69 (1972).

Other (4)

B. M. Caradoc-Davies, Vortex dynamics in Bose–Einstein condensates Ph.D. dissertation Univ. OtagoDunedinNew Zealand (2000) http://www.physics.otago.ac.nz/bec2/bmcd/phdthesis/.

G. P. Agrawal, Nonlinear Fiber Optics (Academic, 2007).

T. Hohage, F. Schmidt, On the numerical solution of nonlinear Schrödinger type equations in fiber optics Konrad-Zuse-Zentrum für InformationstechnikBerlinGermany Tech. Rep. ZIB-Report 02-04 (2002).

The Supercontinuum Laser Source (Springer-Verlag, 2006).

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