Abstract

A numerical approach to nonlinear propagation in waveguides based on real-space Gaussian quadrature integration of the nonlinear polarization during propagation is investigated and compared with the more conventional approach based on expressing the nonlinear polarization by a sum of mode overlap integrals. Using the step-index fiber geometry as an example, it is shown that the Gaussian quadrature approach scales linearly or at most quadratically with the number of guided modes and that it can account for mode profile dispersion without additional computational overhead. These properties make it superior for multimode nonlinear simulations extending over wide frequency ranges.

© 2017 Optical Society of America

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References

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    [Crossref]
  2. R. Ryf, H. Chen, N. K. Fontaine, A. M. Velazquez-Benitez, J. Antonio-Lopez, C. Jin, B. Huang, M. Bigot-Astruc, D. Molin, F. Achten, P. Sillard, and R. Amezcua-Correa, “10-mode mode-multiplexed transmission over 125-km single-span multimode fiber,” in European Conference on Optical Communication (ECOC) (2015), pp. 1–3.
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
  9. K. Krupa, A. Tonello, A. Barthélémy, V. Couderc, B. M. Shalaby, A. Bendahmane, G. Millot, and S. Wabnitz, “Observation of geometric parametric instability induced by the periodic spatial self-imaging of multimode waves,” Phys. Rev. Lett. 116, 183901 (2016).
    [Crossref]
  10. F. Tani, J. C. Travers, and P. S. Russell, “Multimode ultrafast nonlinear optics in optical waveguides: numerical modeling and experiments in kagome photonic-crystal fiber,” J. Opt. Soc. Am. B 31, 311–320 (2014).
    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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2016 (2)

K. Krupa, A. Tonello, A. Barthélémy, V. Couderc, B. M. Shalaby, A. Bendahmane, G. Millot, and S. Wabnitz, “Observation of geometric parametric instability induced by the periodic spatial self-imaging of multimode waves,” Phys. Rev. Lett. 116, 183901 (2016).
[Crossref]

C. Antonelli, M. Shtaif, and A. Mecozzi, “Modeling of nonlinear propagation in space-division multiplexed fiber-optic transmission,” J. Lightwave Technol. 34, 36–54 (2016).
[Crossref]

2015 (5)

2014 (2)

2012 (1)

2009 (1)

2008 (1)

2007 (1)

2004 (1)

M. Kolesik, E. M. Wright, and J. V. Moloney, “Simulation of femtosecond pulse propagation in sub-micron diameter tapered fibers,” Appl. Phys. B 79, 293–300 (2004).
[Crossref]

2000 (1)

H. R. Stuart, “Dispersive multiplexing in multimode optical fiber,” Science 289, 281–283 (2000).
[Crossref]

1995 (1)

N. Akhmediev and M. Karlsson, “Cherenkov radiation emitted by solitons in optical fibers,” Phys. Rev. A 51, 2602–2607 (1995).
[Crossref]

1990 (1)

Achten, F.

R. Ryf, H. Chen, N. K. Fontaine, A. M. Velazquez-Benitez, J. Antonio-Lopez, C. Jin, B. Huang, M. Bigot-Astruc, D. Molin, F. Achten, P. Sillard, and R. Amezcua-Correa, “10-mode mode-multiplexed transmission over 125-km single-span multimode fiber,” in European Conference on Optical Communication (ECOC) (2015), pp. 1–3.

Akhmediev, N.

N. Akhmediev and M. Karlsson, “Cherenkov radiation emitted by solitons in optical fibers,” Phys. Rev. A 51, 2602–2607 (1995).
[Crossref]

Alkeskjold, T. T.

Amezcua-Correa, R.

R. Ryf, H. Chen, N. K. Fontaine, A. M. Velazquez-Benitez, J. Antonio-Lopez, C. Jin, B. Huang, M. Bigot-Astruc, D. Molin, F. Achten, P. Sillard, and R. Amezcua-Correa, “10-mode mode-multiplexed transmission over 125-km single-span multimode fiber,” in European Conference on Optical Communication (ECOC) (2015), pp. 1–3.

Antonelli, C.

Antonio-Lopez, J.

R. Ryf, H. Chen, N. K. Fontaine, A. M. Velazquez-Benitez, J. Antonio-Lopez, C. Jin, B. Huang, M. Bigot-Astruc, D. Molin, F. Achten, P. Sillard, and R. Amezcua-Correa, “10-mode mode-multiplexed transmission over 125-km single-span multimode fiber,” in European Conference on Optical Communication (ECOC) (2015), pp. 1–3.

Barthélémy, A.

K. Krupa, A. Tonello, A. Barthélémy, V. Couderc, B. M. Shalaby, A. Bendahmane, G. Millot, and S. Wabnitz, “Observation of geometric parametric instability induced by the periodic spatial self-imaging of multimode waves,” Phys. Rev. Lett. 116, 183901 (2016).
[Crossref]

Bendahmane, A.

K. Krupa, A. Tonello, A. Barthélémy, V. Couderc, B. M. Shalaby, A. Bendahmane, G. Millot, and S. Wabnitz, “Observation of geometric parametric instability induced by the periodic spatial self-imaging of multimode waves,” Phys. Rev. Lett. 116, 183901 (2016).
[Crossref]

Biancalana, F.

Bigot-Astruc, M.

R. Ryf, H. Chen, N. K. Fontaine, A. M. Velazquez-Benitez, J. Antonio-Lopez, C. Jin, B. Huang, M. Bigot-Astruc, D. Molin, F. Achten, P. Sillard, and R. Amezcua-Correa, “10-mode mode-multiplexed transmission over 125-km single-span multimode fiber,” in European Conference on Optical Communication (ECOC) (2015), pp. 1–3.

Chen, H.

R. Ryf, H. Chen, N. K. Fontaine, A. M. Velazquez-Benitez, J. Antonio-Lopez, C. Jin, B. Huang, M. Bigot-Astruc, D. Molin, F. Achten, P. Sillard, and R. Amezcua-Correa, “10-mode mode-multiplexed transmission over 125-km single-span multimode fiber,” in European Conference on Optical Communication (ECOC) (2015), pp. 1–3.

Chen, Y.

Chernikov, S. V.

Christodoulides, D. N.

L. G. Wright, W. H. Renninger, D. N. Christodoulides, and F. W. Wise, “Spatiotemporal dynamics of multimode optical solitons,” Opt. Express 23, 3492–3506 (2015).
[Crossref]

L. G. Wright, D. N. Christodoulides, and F. W. Wise, “Controllable spatiotemporal nonlinear effects in multimode fibres,” Nat. Photonics 9, 306–310 (2015).
[Crossref]

L. G. Wright, S. Wabnitz, D. N. Christodoulides, and F. W. Wise, “Ultrabroadband dispersive radiation by spatiotemporal oscillation of multimode waves,” Phys. Rev. Lett. 115, 223902 (2015).
[Crossref]

Couderc, V.

K. Krupa, A. Tonello, A. Barthélémy, V. Couderc, B. M. Shalaby, A. Bendahmane, G. Millot, and S. Wabnitz, “Observation of geometric parametric instability induced by the periodic spatial self-imaging of multimode waves,” Phys. Rev. Lett. 116, 183901 (2016).
[Crossref]

Demas, J.

Ding, Y.

Flannery, B. P.

W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes in Fortran 77 (Cambridge University, 2001).

Fontaine, N. K.

R. Ryf, H. Chen, N. K. Fontaine, A. M. Velazquez-Benitez, J. Antonio-Lopez, C. Jin, B. Huang, M. Bigot-Astruc, D. Molin, F. Achten, P. Sillard, and R. Amezcua-Correa, “10-mode mode-multiplexed transmission over 125-km single-span multimode fiber,” in European Conference on Optical Communication (ECOC) (2015), pp. 1–3.

Horak, P.

Huang, B.

R. Ryf, H. Chen, N. K. Fontaine, A. M. Velazquez-Benitez, J. Antonio-Lopez, C. Jin, B. Huang, M. Bigot-Astruc, D. Molin, F. Achten, P. Sillard, and R. Amezcua-Correa, “10-mode mode-multiplexed transmission over 125-km single-span multimode fiber,” in European Conference on Optical Communication (ECOC) (2015), pp. 1–3.

Jin, C.

R. Ryf, H. Chen, N. K. Fontaine, A. M. Velazquez-Benitez, J. Antonio-Lopez, C. Jin, B. Huang, M. Bigot-Astruc, D. Molin, F. Achten, P. Sillard, and R. Amezcua-Correa, “10-mode mode-multiplexed transmission over 125-km single-span multimode fiber,” in European Conference on Optical Communication (ECOC) (2015), pp. 1–3.

Karlsson, M.

N. Akhmediev and M. Karlsson, “Cherenkov radiation emitted by solitons in optical fibers,” Phys. Rev. A 51, 2602–2607 (1995).
[Crossref]

Kolesik, M.

M. Kolesik, E. M. Wright, and J. V. Moloney, “Simulation of femtosecond pulse propagation in sub-micron diameter tapered fibers,” Appl. Phys. B 79, 293–300 (2004).
[Crossref]

Krupa, K.

K. Krupa, A. Tonello, A. Barthélémy, V. Couderc, B. M. Shalaby, A. Bendahmane, G. Millot, and S. Wabnitz, “Observation of geometric parametric instability induced by the periodic spatial self-imaging of multimode waves,” Phys. Rev. Lett. 116, 183901 (2016).
[Crossref]

Lægsgaard, J.

Mafi, A.

Mamyshev, P. V.

Mecozzi, A.

Millot, G.

K. Krupa, A. Tonello, A. Barthélémy, V. Couderc, B. M. Shalaby, A. Bendahmane, G. Millot, and S. Wabnitz, “Observation of geometric parametric instability induced by the periodic spatial self-imaging of multimode waves,” Phys. Rev. Lett. 116, 183901 (2016).
[Crossref]

Molin, D.

R. Ryf, H. Chen, N. K. Fontaine, A. M. Velazquez-Benitez, J. Antonio-Lopez, C. Jin, B. Huang, M. Bigot-Astruc, D. Molin, F. Achten, P. Sillard, and R. Amezcua-Correa, “10-mode mode-multiplexed transmission over 125-km single-span multimode fiber,” in European Conference on Optical Communication (ECOC) (2015), pp. 1–3.

Moloney, J. V.

M. Kolesik, E. M. Wright, and J. V. Moloney, “Simulation of femtosecond pulse propagation in sub-micron diameter tapered fibers,” Appl. Phys. B 79, 293–300 (2004).
[Crossref]

Olausson, C. B.

Ou, H.

Petersen, S. R.

Peucheret, C.

Poletti, F.

Press, W. H.

W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes in Fortran 77 (Cambridge University, 2001).

Ramachandran, S.

Renninger, W. H.

Rishoj, L.

Russell, P. S.

Ryf, R.

R. Ryf, H. Chen, N. K. Fontaine, A. M. Velazquez-Benitez, J. Antonio-Lopez, C. Jin, B. Huang, M. Bigot-Astruc, D. Molin, F. Achten, P. Sillard, and R. Amezcua-Correa, “10-mode mode-multiplexed transmission over 125-km single-span multimode fiber,” in European Conference on Optical Communication (ECOC) (2015), pp. 1–3.

Shalaby, B. M.

K. Krupa, A. Tonello, A. Barthélémy, V. Couderc, B. M. Shalaby, A. Bendahmane, G. Millot, and S. Wabnitz, “Observation of geometric parametric instability induced by the periodic spatial self-imaging of multimode waves,” Phys. Rev. Lett. 116, 183901 (2016).
[Crossref]

Shtaif, M.

Sillard, P.

R. Ryf, H. Chen, N. K. Fontaine, A. M. Velazquez-Benitez, J. Antonio-Lopez, C. Jin, B. Huang, M. Bigot-Astruc, D. Molin, F. Achten, P. Sillard, and R. Amezcua-Correa, “10-mode mode-multiplexed transmission over 125-km single-span multimode fiber,” in European Conference on Optical Communication (ECOC) (2015), pp. 1–3.

Steinvurzel, P.

Stuart, H. R.

H. R. Stuart, “Dispersive multiplexing in multimode optical fiber,” Science 289, 281–283 (2000).
[Crossref]

Tai, B.

Tani, F.

Teukolsky, S. A.

W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes in Fortran 77 (Cambridge University, 2001).

Tonello, A.

K. Krupa, A. Tonello, A. Barthélémy, V. Couderc, B. M. Shalaby, A. Bendahmane, G. Millot, and S. Wabnitz, “Observation of geometric parametric instability induced by the periodic spatial self-imaging of multimode waves,” Phys. Rev. Lett. 116, 183901 (2016).
[Crossref]

Tran, T. X.

Travers, J. C.

Velazquez-Benitez, A. M.

R. Ryf, H. Chen, N. K. Fontaine, A. M. Velazquez-Benitez, J. Antonio-Lopez, C. Jin, B. Huang, M. Bigot-Astruc, D. Molin, F. Achten, P. Sillard, and R. Amezcua-Correa, “10-mode mode-multiplexed transmission over 125-km single-span multimode fiber,” in European Conference on Optical Communication (ECOC) (2015), pp. 1–3.

Vetterling, W. T.

W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes in Fortran 77 (Cambridge University, 2001).

Wabnitz, S.

K. Krupa, A. Tonello, A. Barthélémy, V. Couderc, B. M. Shalaby, A. Bendahmane, G. Millot, and S. Wabnitz, “Observation of geometric parametric instability induced by the periodic spatial self-imaging of multimode waves,” Phys. Rev. Lett. 116, 183901 (2016).
[Crossref]

L. G. Wright, S. Wabnitz, D. N. Christodoulides, and F. W. Wise, “Ultrabroadband dispersive radiation by spatiotemporal oscillation of multimode waves,” Phys. Rev. Lett. 115, 223902 (2015).
[Crossref]

Wise, F. W.

L. G. Wright, D. N. Christodoulides, and F. W. Wise, “Controllable spatiotemporal nonlinear effects in multimode fibres,” Nat. Photonics 9, 306–310 (2015).
[Crossref]

L. G. Wright, S. Wabnitz, D. N. Christodoulides, and F. W. Wise, “Ultrabroadband dispersive radiation by spatiotemporal oscillation of multimode waves,” Phys. Rev. Lett. 115, 223902 (2015).
[Crossref]

L. G. Wright, W. H. Renninger, D. N. Christodoulides, and F. W. Wise, “Spatiotemporal dynamics of multimode optical solitons,” Opt. Express 23, 3492–3506 (2015).
[Crossref]

Wright, E. M.

M. Kolesik, E. M. Wright, and J. V. Moloney, “Simulation of femtosecond pulse propagation in sub-micron diameter tapered fibers,” Appl. Phys. B 79, 293–300 (2004).
[Crossref]

Wright, L. G.

L. G. Wright, S. Wabnitz, D. N. Christodoulides, and F. W. Wise, “Ultrabroadband dispersive radiation by spatiotemporal oscillation of multimode waves,” Phys. Rev. Lett. 115, 223902 (2015).
[Crossref]

L. G. Wright, D. N. Christodoulides, and F. W. Wise, “Controllable spatiotemporal nonlinear effects in multimode fibres,” Nat. Photonics 9, 306–310 (2015).
[Crossref]

L. G. Wright, W. H. Renninger, D. N. Christodoulides, and F. W. Wise, “Spatiotemporal dynamics of multimode optical solitons,” Opt. Express 23, 3492–3506 (2015).
[Crossref]

Xu, J.

Appl. Phys. B (1)

M. Kolesik, E. M. Wright, and J. V. Moloney, “Simulation of femtosecond pulse propagation in sub-micron diameter tapered fibers,” Appl. Phys. B 79, 293–300 (2004).
[Crossref]

J. Lightwave Technol. (2)

J. Opt. Soc. Am. B (2)

Nat. Photonics (1)

L. G. Wright, D. N. Christodoulides, and F. W. Wise, “Controllable spatiotemporal nonlinear effects in multimode fibres,” Nat. Photonics 9, 306–310 (2015).
[Crossref]

Opt. Express (5)

Opt. Lett. (1)

Optica (1)

Phys. Rev. A (1)

N. Akhmediev and M. Karlsson, “Cherenkov radiation emitted by solitons in optical fibers,” Phys. Rev. A 51, 2602–2607 (1995).
[Crossref]

Phys. Rev. Lett. (2)

L. G. Wright, S. Wabnitz, D. N. Christodoulides, and F. W. Wise, “Ultrabroadband dispersive radiation by spatiotemporal oscillation of multimode waves,” Phys. Rev. Lett. 115, 223902 (2015).
[Crossref]

K. Krupa, A. Tonello, A. Barthélémy, V. Couderc, B. M. Shalaby, A. Bendahmane, G. Millot, and S. Wabnitz, “Observation of geometric parametric instability induced by the periodic spatial self-imaging of multimode waves,” Phys. Rev. Lett. 116, 183901 (2016).
[Crossref]

Science (1)

H. R. Stuart, “Dispersive multiplexing in multimode optical fiber,” Science 289, 281–283 (2000).
[Crossref]

Other (2)

R. Ryf, H. Chen, N. K. Fontaine, A. M. Velazquez-Benitez, J. Antonio-Lopez, C. Jin, B. Huang, M. Bigot-Astruc, D. Molin, F. Achten, P. Sillard, and R. Amezcua-Correa, “10-mode mode-multiplexed transmission over 125-km single-span multimode fiber,” in European Conference on Optical Communication (ECOC) (2015), pp. 1–3.

W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes in Fortran 77 (Cambridge University, 2001).

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Figures (7)

Fig. 1.
Fig. 1.

Total number of GQ integration points required to calculate effective areas of all guided modes in a step-index fiber within a specified relative accuracy, versus the number of guided modes. Results are shown for two values of the outer integration boundary, rs=1.3rc and rs=2rc. Zero values indicate that some effective areas could not be converged within the specified accuracy.

Fig. 2.
Fig. 2.

Spectral densities for the six modes of a fiber with core radius rc=7  μm, core-cladding index difference Δn=0.01, after propagation over 2.1 cm of a 50 fs input pulse with 1 MW peak power, evenly distributed over the six modes, with fixed linear polarization. Index labels c and s on the LPm1 modes with m>0 indicate modes with angular dependence of cos(mϕ), sin(mϕ), respectively. The results were found using the OI method with mode profiles calculated at a wavelength of 1.55 μm.

Fig. 3.
Fig. 3.

Total spectral densities, summed over all modes, for the OI calculation shown in Fig. 2, as well as corresponding GQ calculations with rs=1.3rc, n=56.

Fig. 4.
Fig. 4.

Integrated absolute differences in total spectral density between OI and GQ calculations as a function of the number of radial gridpoints n used in the GQ calculations. Results are divided by the integral of the total OI spectral density.

Fig. 5.
Fig. 5.

Final spectral density for the same pulse and fiber parameters as in Fig. 2, when using a GQ approach including mode profile dispersion. rs=1.3rc and n=8.

Fig. 6.
Fig. 6.

Total spectral densities, summed over all modes, for the OI calculation without mode profile dispersion shown in Fig. 2, as well as corresponding GQ calculations with rs=1.3rc, n=58.

Fig. 7.
Fig. 7.

Integrated absolute differences in total spectral density between a GQ calculation with rs=2rc and n=40 and GQ calculations with rs=1.3rc as a function of the number of radial gridpoints n used in the latter case. Results are divided by the integral of the total OI spectral density.

Equations (22)

Equations on this page are rendered with MathJax. Learn more.

E(r,t)=12πmdωG˜m(z,ω)em(r,ω)e[i(βm(ω)zωt)],
G˜m(z,ω)z=ϵ0χ(3)2πiω2Nm(ω)eiβm(ω)zdteiωtdrem*(r,ω)·E(r,t)dtR(tt)|E(r,t)|2.
dr[em×hn*+en*×hm]=2Nm(ω)δmn,
G˜m(z,ω)z=iωϵ04πNm(ω)eiβm(ω)znqpdω1dω2Gn(z,ω1)Gq*(z,ω1+ω2ω)Gp(z,ω2)R(ωω1)Kmnqp(ω,ω1,ω2)
Gm(z,ω)=G˜m(z,ω)eiβm(ω)z
Kmnqp(ω,ω1,ω2)χ(3)drem*(r,ω)·en(r,ω1)eq*(r,ω1+ω2ω)·ep(r,ω2).
G˜m(z,ω)z=iωϵ04πNm(ω)eiβm(ω)zdteiωt,
nqpKmnqpGm(z,t)dtGq*(z,t)Gp(z,t)R(tt)Gm(z,t)=12πdωGm(z,ω)eiωt.
G˜m(z,ω)z=iωπϵ0χ(3)Nm(ω)eiβm(ω)zdteiωt,
k=1Pwkem*(rk)·E(rk,z,t)dt|E(rk,z,t)|2R(tt)E(rk,z,t)=mGm(z,t)em(rk).
E(rk,z,t)=12πdωeiωtmGm(z,ω)em(rk,ω).
Nph=mdω|G˜m(z,ω)|2ω
dNphdz=mdω2Re[G˜m*(z,ω)G˜m(z,ω)z]ω
=Re[mdωG˜m*(z,ω)i2πϵ0χ(3)eiβm(ω)zdteiωtk=1Pwkem*(rk,ω)·E(rk,z,t)dt|E(rk,z,t)|2R(tt)].
dNphdz=Re[i(2π)2ϵ0χ(3)dtk=1Pwk|E(rk,z,t)|2dt|E(rk,z,t)|2R(tt)]=0,
ekl(r,ω)=p^eax2pk1(x)eay2pl1(y);a=n0ω2cr0,
dxe4ax2pm(x)=k=12(lmax1)wkpm(xk)
iωϵ04πNm(ω)eiβm(ω)zdteiωtp=12lmax1q=12lmax1wpqem(rpq)E(rpq,z,t)dt|E(rpq,z,t)|2R(tt)
rpq=(xp,xq);wpq=wpwq.
Ψml(r)=Rml(r)fm(ϕ),
rpq=rp(cos(2πq14M+1),sin(2πq14M+1)).
drΨml4(r),