Abstract

The well-established generalized nonlinear Schrödinger equation (GNLSE) to simulate nonlinear pulse propagation in optical fibers and waveguides becomes inefficient if only narrow spectral bands are occupied that are widely separated in frequency/wavelength, for example in parametric amplifiers. Here we present a solution to this in the form of a coupled frequency-banded nonlinear Schrödinger equation (BNLSE) that only simulates selected narrow frequency bands while still including all dispersive and nonlinear effects, in particular the inter-band Raman and Kerr nonlinearities. This allows for high accuracy spectral resolution in regions of interest while omitting spectral ranges between the selected frequency bands, thus providing an efficient and accurate way for simulating the nonlinear interaction of pulses at widely different carrier frequencies. We derive and test our BNLSE by comparison with the GNLSE. We finally demonstrate the accuracy of the BNLSE and compare the computational execution times for the different models.

Published by The Optical Society under the terms of the Creative Commons Attribution 4.0 License. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

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References

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2016 (3)

2014 (1)

2013 (2)

H. Cheng, Z. Luo, C. Ye, Y. Huang, C. Liu, and Z. Cai, “Numerical modeling of mid-infrared fiber optical parametric oscillator based on the degenerated FWM of tellurite photonic crystal fiber,” Appl. Opt. 52(3), 525–529 (2013).
[Crossref] [PubMed]

R. J. Essiambre, M. A. Mestre, R. Ryf, A. H. Gnauck, R. W. Tkach, A. R. Chraplyvy, Yi. Sun, X. Jiang, and R. Lingle, “Experimental investigation of inter-modal four-wave mixing in few-mode fibers,” IEEE Photon. Technol. Lett. 25(6), 539–5429 (2013).
[Crossref]

2012 (1)

2011 (1)

2007 (3)

T. E. Oliphant, “Python for scientific computing,” Comput. Sci. Eng. 9(3), 10–20 (2007).
[Crossref]

J. D Hunter, “Matplotlib: A 2D graphics environment,” Comput. Sci. Eng. 9(3), 90–95 (2007).
[Crossref]

J.-P Fève, P. E. Schrader, R. L. Farrow, and D. A. V. Kliner, “Four-wave mixing in nanosecond pulsed fiber amplifiers,” Opt. Express 15(8), 4647–4662 (2007).
[Crossref] [PubMed]

2006 (1)

2004 (2)

2003 (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(3), 506–520 (2002).
[Crossref]

2001 (1)

M. C. Ho, K. Uesaka, M. Marhic, Y. Akasaka, and L. G. Kazovsky, “200-nm-bandwidth fiber optical amplifier combining parametric and Raman gain,” J. Light. Technol. 19(7), 977–981 (2001).
[Crossref]

2000 (1)

T. Sylvestre, P. T. Dinda, H. Maillotte, E. Lantz, A. B. Moubissi, and S. Pitois, “Wavelength conversion from 1.3 µm to 1.5 µm in single-mode optical fibres using Raman-assisted three-wave mixing,” J. Opt. A: Pure Appl. Opt. 2(1), 132–141 (2000).
[Crossref]

1999 (1)

1994 (1)

1992 (1)

1973 (1)

R. H. Stolen and E. P. Ippen, “Raman gain in glass optical waveguides,” Appl. Phys. Lett. 22(6), 276–278 (1973).
[Crossref]

Agrawal, G. P.

Akasaka, Y.

M. C. Ho, K. Uesaka, M. Marhic, Y. Akasaka, and L. G. Kazovsky, “200-nm-bandwidth fiber optical amplifier combining parametric and Raman gain,” J. Light. Technol. 19(7), 977–981 (2001).
[Crossref]

Anderson, T. A.

O. Pavlyk, D. Nagorny, A. Guzman-Ballen, A. Malakhov, H. Liu, E. Totoni, T. A. Anderson, and S. Maidanov, “Accelerating Scientific Python with Intel Optimizations,” in Proceedings of the 15th Python in Science Conf. (SciPy 2017), (2007).

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(3), 506–520 (2002).
[Crossref]

Babin, S. A.

Bai, H.

Begleris, I.

S. M. M. Friis, I. Begleris, Y. Jung, K. Rottwitt, P. Petropoulos, D. J. Richardson, P. Horak, and F. Parmigiani, “Inter-modal four-wave mixing study in a two-mode fiber,” Opt. Express 24(36), 30338–30349 (2016).
[Crossref]

I. Begleris and P. Horak, “Cascade simulations of unidirectional fiber optical parametric oscillators,” in Proceedings of the 2017 International Conference on Numerical Simulation of Optoelectronic Devices (NUSOD), (IEEE, 2017).

Butcher, J. C.

J. C. Butcher, The Numerical Analysis of Ordinary Differential Equations: Runge-Kutta and General Linear Methods (Wiley-Interscience, 1987).

Cai, Z.

Cheng, H.

Chraplyvy, A. R.

R. J. Essiambre, M. A. Mestre, R. Ryf, A. H. Gnauck, R. W. Tkach, A. R. Chraplyvy, Yi. Sun, X. Jiang, and R. Lingle, “Experimental investigation of inter-modal four-wave mixing in few-mode fibers,” IEEE Photon. Technol. Lett. 25(6), 539–5429 (2013).
[Crossref]

Coen, S.

Corredera, P.

Corrons, A.

Debaes, C.

Dinda, P. T.

T. Sylvestre, P. T. Dinda, H. Maillotte, E. Lantz, A. B. Moubissi, and S. Pitois, “Wavelength conversion from 1.3 µm to 1.5 µm in single-mode optical fibres using Raman-assisted three-wave mixing,” J. Opt. A: Pure Appl. Opt. 2(1), 132–141 (2000).
[Crossref]

T. Sylvestre, H. Maillotte, E. Lantz, and P. T. Dinda, “Raman-assisted parametric frequency conversion in a normally dispersive single-mode fiber,” Opt. Lett. 24(22), 1561–1563 (1999).
[Crossref]

Emplit, P.

Essiambre, R. J.

R. J. Essiambre, M. A. Mestre, R. Ryf, A. H. Gnauck, R. W. Tkach, A. R. Chraplyvy, Yi. Sun, X. Jiang, and R. Lingle, “Experimental investigation of inter-modal four-wave mixing in few-mode fibers,” IEEE Photon. Technol. Lett. 25(6), 539–5429 (2013).
[Crossref]

Farrow, R. L.

Fauchet, P. M.

Fève, J.-P

Fried, A.

F. K. Tittel, D. Richter, and A. Fried, “Mid-Infrared Laser Applications in Spectroscopy,” in Solid-State Mid-Infrared Laser Sources. Topics in Applied Physics, vol 89, I.T. Sorokina and K. L. Vodopyanoveds., (Springer, 2003)
[Crossref]

Friis, S. M. M.

Gao, S.

Gnauck, A. H.

R. J. Essiambre, M. A. Mestre, R. Ryf, A. H. Gnauck, R. W. Tkach, A. R. Chraplyvy, Yi. Sun, X. Jiang, and R. Lingle, “Experimental investigation of inter-modal four-wave mixing in few-mode fibers,” IEEE Photon. Technol. Lett. 25(6), 539–5429 (2013).
[Crossref]

Golovchenko, E. A.

Guzman-Ballen, A.

O. Pavlyk, D. Nagorny, A. Guzman-Ballen, A. Malakhov, H. Liu, E. Totoni, T. A. Anderson, and S. Maidanov, “Accelerating Scientific Python with Intel Optimizations,” in Proceedings of the 15th Python in Science Conf. (SciPy 2017), (2007).

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(3), 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(3), 506–520 (2002).
[Crossref]

Hernanz, M. L.

Herraez, M. G.

Ho, M. C.

M. C. Ho, K. Uesaka, M. Marhic, Y. Akasaka, and L. G. Kazovsky, “200-nm-bandwidth fiber optical amplifier combining parametric and Raman gain,” J. Light. Technol. 19(7), 977–981 (2001).
[Crossref]

Höök, A.

Horak, P.

S. M. M. Friis, I. Begleris, Y. Jung, K. Rottwitt, P. Petropoulos, D. J. Richardson, P. Horak, and F. Parmigiani, “Inter-modal four-wave mixing study in a two-mode fiber,” Opt. Express 24(36), 30338–30349 (2016).
[Crossref]

I. Begleris and P. Horak, “Cascade simulations of unidirectional fiber optical parametric oscillators,” in Proceedings of the 2017 International Conference on Numerical Simulation of Optoelectronic Devices (NUSOD), (IEEE, 2017).

Huang, Y.

Hunter, J. D

J. D Hunter, “Matplotlib: A 2D graphics environment,” Comput. Sci. Eng. 9(3), 90–95 (2007).
[Crossref]

Ippen, E. P.

R. H. Stolen and E. P. Ippen, “Raman gain in glass optical waveguides,” Appl. Phys. Lett. 22(6), 276–278 (1973).
[Crossref]

Jiang, X.

R. J. Essiambre, M. A. Mestre, R. Ryf, A. H. Gnauck, R. W. Tkach, A. R. Chraplyvy, Yi. Sun, X. Jiang, and R. Lingle, “Experimental investigation of inter-modal four-wave mixing in few-mode fibers,” IEEE Photon. Technol. Lett. 25(6), 539–5429 (2013).
[Crossref]

Jung, Y.

Kablukov, S. I.

Kazovsky, L. G.

M. C. Ho, K. Uesaka, M. Marhic, Y. Akasaka, and L. G. Kazovsky, “200-nm-bandwidth fiber optical amplifier combining parametric and Raman gain,” J. Light. Technol. 19(7), 977–981 (2001).
[Crossref]

Khubchandani, B. L.

Kliner, D. A. V.

Kumar, P.

Lam, S. K.

S. K. Lam, A. Pitrou, and S. Seibert, “Numba: A LLVM-based Python JIT Compiler,” in Proceedings of the Second Workshop on the LLVM Compiler Infrastructure in HPC - LLVM ’15, (ACM, 2007).

Lantz, E.

T. Sylvestre, P. T. Dinda, H. Maillotte, E. Lantz, A. B. Moubissi, and S. Pitois, “Wavelength conversion from 1.3 µm to 1.5 µm in single-mode optical fibres using Raman-assisted three-wave mixing,” J. Opt. A: Pure Appl. Opt. 2(1), 132–141 (2000).
[Crossref]

T. Sylvestre, H. Maillotte, E. Lantz, and P. T. Dinda, “Raman-assisted parametric frequency conversion in a normally dispersive single-mode fiber,” Opt. Lett. 24(22), 1561–1563 (1999).
[Crossref]

Lefevre, Y.

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(3), 506–520 (2002).
[Crossref]

Lin, Q.

Lingle, R.

R. J. Essiambre, M. A. Mestre, R. Ryf, A. H. Gnauck, R. W. Tkach, A. R. Chraplyvy, Yi. Sun, X. Jiang, and R. Lingle, “Experimental investigation of inter-modal four-wave mixing in few-mode fibers,” IEEE Photon. Technol. Lett. 25(6), 539–5429 (2013).
[Crossref]

Liu, C.

Liu, H.

O. Pavlyk, D. Nagorny, A. Guzman-Ballen, A. Malakhov, H. Liu, E. Totoni, T. A. Anderson, and S. Maidanov, “Accelerating Scientific Python with Intel Optimizations,” in Proceedings of the 15th Python in Science Conf. (SciPy 2017), (2007).

Lopez, S. M.

Luo, Z.

Maidanov, S.

O. Pavlyk, D. Nagorny, A. Guzman-Ballen, A. Malakhov, H. Liu, E. Totoni, T. A. Anderson, and S. Maidanov, “Accelerating Scientific Python with Intel Optimizations,” in Proceedings of the 15th Python in Science Conf. (SciPy 2017), (2007).

Maillotte, H.

T. Sylvestre, P. T. Dinda, H. Maillotte, E. Lantz, A. B. Moubissi, and S. Pitois, “Wavelength conversion from 1.3 µm to 1.5 µm in single-mode optical fibres using Raman-assisted three-wave mixing,” J. Opt. A: Pure Appl. Opt. 2(1), 132–141 (2000).
[Crossref]

T. Sylvestre, H. Maillotte, E. Lantz, and P. T. Dinda, “Raman-assisted parametric frequency conversion in a normally dispersive single-mode fiber,” Opt. Lett. 24(22), 1561–1563 (1999).
[Crossref]

Malakhov, A.

O. Pavlyk, D. Nagorny, A. Guzman-Ballen, A. Malakhov, H. Liu, E. Totoni, T. A. Anderson, and S. Maidanov, “Accelerating Scientific Python with Intel Optimizations,” in Proceedings of the 15th Python in Science Conf. (SciPy 2017), (2007).

Marhic, M.

M. C. Ho, K. Uesaka, M. Marhic, Y. Akasaka, and L. G. Kazovsky, “200-nm-bandwidth fiber optical amplifier combining parametric and Raman gain,” J. Light. Technol. 19(7), 977–981 (2001).
[Crossref]

Marhic, M.E.

M.E. Marhic, Fiber Optical Parametric Amplifiers, Oscillators and Related Devices (Cambridge University, 2008).

Mestre, M. A.

R. J. Essiambre, M. A. Mestre, R. Ryf, A. H. Gnauck, R. W. Tkach, A. R. Chraplyvy, Yi. Sun, X. Jiang, and R. Lingle, “Experimental investigation of inter-modal four-wave mixing in few-mode fibers,” IEEE Photon. Technol. Lett. 25(6), 539–5429 (2013).
[Crossref]

Moubissi, A. B.

T. Sylvestre, P. T. Dinda, H. Maillotte, E. Lantz, A. B. Moubissi, and S. Pitois, “Wavelength conversion from 1.3 µm to 1.5 µm in single-mode optical fibres using Raman-assisted three-wave mixing,” J. Opt. A: Pure Appl. Opt. 2(1), 132–141 (2000).
[Crossref]

Nagorny, D.

O. Pavlyk, D. Nagorny, A. Guzman-Ballen, A. Malakhov, H. Liu, E. Totoni, T. A. Anderson, and S. Maidanov, “Accelerating Scientific Python with Intel Optimizations,” in Proceedings of the 15th Python in Science Conf. (SciPy 2017), (2007).

Oliphant, T. E.

T. E. Oliphant, “Python for scientific computing,” Comput. Sci. Eng. 9(3), 10–20 (2007).
[Crossref]

Parmigiani, F.

Pavlyk, O.

O. Pavlyk, D. Nagorny, A. Guzman-Ballen, A. Malakhov, H. Liu, E. Totoni, T. A. Anderson, and S. Maidanov, “Accelerating Scientific Python with Intel Optimizations,” in Proceedings of the 15th Python in Science Conf. (SciPy 2017), (2007).

Petropoulos, P.

Pilipetskii, A. N.

Pitois, S.

T. Sylvestre, P. T. Dinda, H. Maillotte, E. Lantz, A. B. Moubissi, and S. Pitois, “Wavelength conversion from 1.3 µm to 1.5 µm in single-mode optical fibres using Raman-assisted three-wave mixing,” J. Opt. A: Pure Appl. Opt. 2(1), 132–141 (2000).
[Crossref]

Pitrou, A.

S. K. Lam, A. Pitrou, and S. Seibert, “Numba: A LLVM-based Python JIT Compiler,” in Proceedings of the Second Workshop on the LLVM Compiler Infrastructure in HPC - LLVM ’15, (ACM, 2007).

Richardson, D. J.

Richter, D.

F. K. Tittel, D. Richter, and A. Fried, “Mid-Infrared Laser Applications in Spectroscopy,” in Solid-State Mid-Infrared Laser Sources. Topics in Applied Physics, vol 89, I.T. Sorokina and K. L. Vodopyanoveds., (Springer, 2003)
[Crossref]

Rottwitt, K.

Ryf, R.

R. J. Essiambre, M. A. Mestre, R. Ryf, A. H. Gnauck, R. W. Tkach, A. R. Chraplyvy, Yi. Sun, X. Jiang, and R. Lingle, “Experimental investigation of inter-modal four-wave mixing in few-mode fibers,” IEEE Photon. Technol. Lett. 25(6), 539–5429 (2013).
[Crossref]

Schrader, P. E.

Seibert, S.

S. K. Lam, A. Pitrou, and S. Seibert, “Numba: A LLVM-based Python JIT Compiler,” in Proceedings of the Second Workshop on the LLVM Compiler Infrastructure in HPC - LLVM ’15, (ACM, 2007).

Stolen, R. H.

R. H. Stolen and E. P. Ippen, “Raman gain in glass optical waveguides,” Appl. Phys. Lett. 22(6), 276–278 (1973).
[Crossref]

Sun, Yi.

R. J. Essiambre, M. A. Mestre, R. Ryf, A. H. Gnauck, R. W. Tkach, A. R. Chraplyvy, Yi. Sun, X. Jiang, and R. Lingle, “Experimental investigation of inter-modal four-wave mixing in few-mode fibers,” IEEE Photon. Technol. Lett. 25(6), 539–5429 (2013).
[Crossref]

Sylvestre, T.

T. Sylvestre, P. T. Dinda, H. Maillotte, E. Lantz, A. B. Moubissi, and S. Pitois, “Wavelength conversion from 1.3 µm to 1.5 µm in single-mode optical fibres using Raman-assisted three-wave mixing,” J. Opt. A: Pure Appl. Opt. 2(1), 132–141 (2000).
[Crossref]

T. Sylvestre, H. Maillotte, E. Lantz, and P. T. Dinda, “Raman-assisted parametric frequency conversion in a normally dispersive single-mode fiber,” Opt. Lett. 24(22), 1561–1563 (1999).
[Crossref]

Thienpont, H.

Tittel, F. K.

F. K. Tittel, D. Richter, and A. Fried, “Mid-Infrared Laser Applications in Spectroscopy,” in Solid-State Mid-Infrared Laser Sources. Topics in Applied Physics, vol 89, I.T. Sorokina and K. L. Vodopyanoveds., (Springer, 2003)
[Crossref]

Tkach, R. W.

R. J. Essiambre, M. A. Mestre, R. Ryf, A. H. Gnauck, R. W. Tkach, A. R. Chraplyvy, Yi. Sun, X. Jiang, and R. Lingle, “Experimental investigation of inter-modal four-wave mixing in few-mode fibers,” IEEE Photon. Technol. Lett. 25(6), 539–5429 (2013).
[Crossref]

Totoni, E.

O. Pavlyk, D. Nagorny, A. Guzman-Ballen, A. Malakhov, H. Liu, E. Totoni, T. A. Anderson, and S. Maidanov, “Accelerating Scientific Python with Intel Optimizations,” in Proceedings of the 15th Python in Science Conf. (SciPy 2017), (2007).

Uesaka, K.

M. C. Ho, K. Uesaka, M. Marhic, Y. Akasaka, and L. G. Kazovsky, “200-nm-bandwidth fiber optical amplifier combining parametric and Raman gain,” J. Light. Technol. 19(7), 977–981 (2001).
[Crossref]

Vanholsbeeck, F.

Vermeulen, N.

Voss, P. L.

Wei, Y.

Westlund, M.

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Figures (6)

Fig. 1
Fig. 1 Nonlinear phase of the probe at a frequency separation of F = 60 THz (outside the Raman gain bandwidth of the pump) throughout propagation calculated using the GNLSE (left) and the relative error of the CNLSE (middle) and BNLSE (right) compared to the GNLSE. All parameters are given in the text.
Fig. 2
Fig. 2 Nonlinear phase of the probe at a frequency separation of F = 15.2 THz (inside the Raman gain bandwidth of the pump) throughout propagation calculated using the GNLSE (left) and the relative error of the CNLSE (middle) and BNLSE (right) compared to the GNLSE.
Fig. 3
Fig. 3 Average power of the idler wave with respect to the propagation distance with probe inside, at F = 15.5 THz, (left) and outside, at F = 42.5 THz, (right) the Raman gain bandwidth.
Fig. 4
Fig. 4 Temporal (left) and spectral (right, on a logarithmic scale) distribution of the idler wave with probe inside the Raman gain bandwidth, at F = 15.5 THz.
Fig. 5
Fig. 5 Temporal (left) and spectral (right, on a logarithmic scale) distribution of the idler wave with probe outside the Raman gain bandwidth, at F = 42.5 THz.
Fig. 6
Fig. 6 Execution times of the GNLSE (at fr = 0 and 0.18), CNLSE and BNLSE for a varying frequency grid.

Equations (8)

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d A d z = [ i n 2 β n n ! ( i t ) n + i γ ( 1 + i ω 0 t ) ( ( 1 f r ) | A ( z , t ) | 2 + f r h ( τ ) | A ( z , t τ ) | 2 d τ ) ] A ( z , t ) ,
A ( z , t ) = l = 1 1 A l ( z , t ) exp ( Ω l t ) .
h ( τ ) A m ( t τ ) A n * ( t τ ) exp ( i Ω l τ ) d τ A m ( t ) A n * ( t ) h ( τ ) exp ( i Ω l τ ) d τ = A m ( t ) A n * ( t ) h ˜ ( Ω l ) .
d A l d z = i n 2 ( β n n ! ( i t Ω l ) n ) A l + i γ l ( 1 + i ω l t ) ( N l A l + M l ) ,
( N 1 N 0 N 1 ) = K ( | A 1 | 2 | A 0 | 2 | A 1 | 2 ) ,
K = ( 1 2 + f r ( h ˜ ( Ω 1 ) 1 ) 2 + f r ( h ˜ ( Ω 1 Ω + 1 ) 1 ) 2 + f r ( h ˜ ( Ω 1 ) 1 ) 1 2 + f r ( h ˜ ( Ω + 1 ) 1 ) 2 + f r ( h ˜ ( Ω + 1 Ω 1 ) 1 ) 2 + f r ( h ˜ ( Ω + 1 ) 1 ) 1 ) ,
( M 1 M 0 M 1 ) = ( ( 1 f r ( 1 h ˜ ( Ω + 1 ) ) ) A 0 2 A 1 * exp ( i( Ω + 1 + Ω 1 ) t ) ( 1 f r ( 1 h ˜ ( Ω 1 ) + h ˜ ( Ω + 1 ) 2 ) ) 2 A 1 A 1 A 0 * exp ( i ( Ω + 1 + Ω 1 ) t ) ( 1 f r ( 1 h ˜ ( Ω 1 ) ) ) A 0 2 A 1 * exp ( i ( Ω + 1 + Ω 1 ) t ) ) ,
Δ β ( Ω ) + 2 γ P p = 0 ,

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