Abstract

We investigate numerically the supercontinuum generation (SCG) phenomenon, using femtosecond pulses in the subnanoscale of energies through the generalized nonlinear Schrödinger equation that includes non-Kerr terms. Our results with 50 fs pulses in the anomalous dispersion regime show that, in comparison to the single cubic Kerr nonlinearity (CKN) case, the cooperative nonlinearities improve the spectral broadening, while the competing ones compress the spectral SCG bandwidth. Surprisingly, with the reduction of the pulse width, the cooperative nonlinearities induce a spectral compression while the competing ones keep the SCG bandwidth nearly constant from the input to the output of the considered waveguide. The increase of both the energy and the nonlinearity confirms this feature, showing that spectral compression is also obtained in the single CKN case, but less than in the case of cooperative nonlinearities.

© 2013 Optical Society of America

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References

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  1. R. R. Alfano, The Supercontinuum Laser Source (Springer-Verlag, 1989).
  2. J. M. Dudley, G. Genty, and S. Coen, “Supercontinuum generation in photonics crystal fiber,” Rev. Mod. Phys. 78, 1135–1184 (2006).
    [CrossRef]
  3. G. P. Agrawal, Nonlinear Fiber Optics, 4th ed. (Academic, 2007).
  4. J. M. Dudley and J. R. Taylor, “Ten years of nonlinear optics in photonic crystal fiber,” Nat. Photonics 3, 85–90 (2009).
    [CrossRef]
  5. J. M. Dudley and J. R. Taylor, Supercontinuum Generation in Optical Fibers, 1st ed. (Cambridge University Press, 2010).
  6. J. C. Travers, “Blue extension of optical fibre supercontinuum generation,” J. Opt. 12, 113001 (2010).
    [CrossRef]
  7. G. P. Agrawal, “Supercontinuum generation,” in Applications of Nonlinear Fiber Optics (Academic, 2008), pp. 414–431.
  8. R. Radhakrishnan, A. Kundu, and M. Lakshmanan, “Coupled nonlinear Schrödinger equations with cubic-quintic nonlinearity: integrability and soliton interaction in non-Kerr media,” Phys. Rev. E 60, 3314–3323 (1999).
    [CrossRef]
  9. R. V. J. Raja, K. Porsezian, S. K. Varshney, and S. Sivabalan, “Modeling photonic crystal fiber for efficient soliton pulse propagation at 850 nm,” Opt. Commun. 283, 5000–5006 (2010).
    [CrossRef]
  10. A. Choudhuri and K. Porsezian, “Dark-in-the-bright solitary wave solution of higher-order nonlinear Schrödinger equation with non-Kerr terms,” Opt. Commun. 285, 364–367 (2012).
    [CrossRef]
  11. A. Choudhuri and K. Porsezian, “Impact of dispersion and non-Kerr nonlinearity on the modulational instability of the higher-order nonlinear Schrödinger equation,” Phys. Rev. A 85, 033820 (2012).
    [CrossRef]
  12. K. Senthilnathan, Q. Li, K. Nakkeeran, and P. K. A. Wai, “Robust pedestal-free pulse compression in cubic-quintic nonlinear media,” Phys. Rev. A 78, 033835 (2008).
    [CrossRef]
  13. J. C. Travers, M. H. Frosz, and J. M. Dudley, “Nonlinear fibre optics overview,” in Handbook of Supercontinuum Generation in Optical Fibers, J. M. Dudley and J. R. Taylor, eds. (Cambridge University, 2010), Chap. 3, pp. 46–49.

2012 (2)

A. Choudhuri and K. Porsezian, “Dark-in-the-bright solitary wave solution of higher-order nonlinear Schrödinger equation with non-Kerr terms,” Opt. Commun. 285, 364–367 (2012).
[CrossRef]

A. Choudhuri and K. Porsezian, “Impact of dispersion and non-Kerr nonlinearity on the modulational instability of the higher-order nonlinear Schrödinger equation,” Phys. Rev. A 85, 033820 (2012).
[CrossRef]

2010 (2)

R. V. J. Raja, K. Porsezian, S. K. Varshney, and S. Sivabalan, “Modeling photonic crystal fiber for efficient soliton pulse propagation at 850 nm,” Opt. Commun. 283, 5000–5006 (2010).
[CrossRef]

J. C. Travers, “Blue extension of optical fibre supercontinuum generation,” J. Opt. 12, 113001 (2010).
[CrossRef]

2009 (1)

J. M. Dudley and J. R. Taylor, “Ten years of nonlinear optics in photonic crystal fiber,” Nat. Photonics 3, 85–90 (2009).
[CrossRef]

2008 (1)

K. Senthilnathan, Q. Li, K. Nakkeeran, and P. K. A. Wai, “Robust pedestal-free pulse compression in cubic-quintic nonlinear media,” Phys. Rev. A 78, 033835 (2008).
[CrossRef]

2006 (1)

J. M. Dudley, G. Genty, and S. Coen, “Supercontinuum generation in photonics crystal fiber,” Rev. Mod. Phys. 78, 1135–1184 (2006).
[CrossRef]

1999 (1)

R. Radhakrishnan, A. Kundu, and M. Lakshmanan, “Coupled nonlinear Schrödinger equations with cubic-quintic nonlinearity: integrability and soliton interaction in non-Kerr media,” Phys. Rev. E 60, 3314–3323 (1999).
[CrossRef]

Agrawal, G. P.

G. P. Agrawal, “Supercontinuum generation,” in Applications of Nonlinear Fiber Optics (Academic, 2008), pp. 414–431.

G. P. Agrawal, Nonlinear Fiber Optics, 4th ed. (Academic, 2007).

Alfano, R. R.

R. R. Alfano, The Supercontinuum Laser Source (Springer-Verlag, 1989).

Choudhuri, A.

A. Choudhuri and K. Porsezian, “Dark-in-the-bright solitary wave solution of higher-order nonlinear Schrödinger equation with non-Kerr terms,” Opt. Commun. 285, 364–367 (2012).
[CrossRef]

A. Choudhuri and K. Porsezian, “Impact of dispersion and non-Kerr nonlinearity on the modulational instability of the higher-order nonlinear Schrödinger equation,” Phys. Rev. A 85, 033820 (2012).
[CrossRef]

Coen, S.

J. M. Dudley, G. Genty, and S. Coen, “Supercontinuum generation in photonics crystal fiber,” Rev. Mod. Phys. 78, 1135–1184 (2006).
[CrossRef]

Dudley, J. M.

J. M. Dudley and J. R. Taylor, “Ten years of nonlinear optics in photonic crystal fiber,” Nat. Photonics 3, 85–90 (2009).
[CrossRef]

J. M. Dudley, G. Genty, and S. Coen, “Supercontinuum generation in photonics crystal fiber,” Rev. Mod. Phys. 78, 1135–1184 (2006).
[CrossRef]

J. M. Dudley and J. R. Taylor, Supercontinuum Generation in Optical Fibers, 1st ed. (Cambridge University Press, 2010).

J. C. Travers, M. H. Frosz, and J. M. Dudley, “Nonlinear fibre optics overview,” in Handbook of Supercontinuum Generation in Optical Fibers, J. M. Dudley and J. R. Taylor, eds. (Cambridge University, 2010), Chap. 3, pp. 46–49.

Frosz, M. H.

J. C. Travers, M. H. Frosz, and J. M. Dudley, “Nonlinear fibre optics overview,” in Handbook of Supercontinuum Generation in Optical Fibers, J. M. Dudley and J. R. Taylor, eds. (Cambridge University, 2010), Chap. 3, pp. 46–49.

Genty, G.

J. M. Dudley, G. Genty, and S. Coen, “Supercontinuum generation in photonics crystal fiber,” Rev. Mod. Phys. 78, 1135–1184 (2006).
[CrossRef]

Kundu, A.

R. Radhakrishnan, A. Kundu, and M. Lakshmanan, “Coupled nonlinear Schrödinger equations with cubic-quintic nonlinearity: integrability and soliton interaction in non-Kerr media,” Phys. Rev. E 60, 3314–3323 (1999).
[CrossRef]

Lakshmanan, M.

R. Radhakrishnan, A. Kundu, and M. Lakshmanan, “Coupled nonlinear Schrödinger equations with cubic-quintic nonlinearity: integrability and soliton interaction in non-Kerr media,” Phys. Rev. E 60, 3314–3323 (1999).
[CrossRef]

Li, Q.

K. Senthilnathan, Q. Li, K. Nakkeeran, and P. K. A. Wai, “Robust pedestal-free pulse compression in cubic-quintic nonlinear media,” Phys. Rev. A 78, 033835 (2008).
[CrossRef]

Nakkeeran, K.

K. Senthilnathan, Q. Li, K. Nakkeeran, and P. K. A. Wai, “Robust pedestal-free pulse compression in cubic-quintic nonlinear media,” Phys. Rev. A 78, 033835 (2008).
[CrossRef]

Porsezian, K.

A. Choudhuri and K. Porsezian, “Dark-in-the-bright solitary wave solution of higher-order nonlinear Schrödinger equation with non-Kerr terms,” Opt. Commun. 285, 364–367 (2012).
[CrossRef]

A. Choudhuri and K. Porsezian, “Impact of dispersion and non-Kerr nonlinearity on the modulational instability of the higher-order nonlinear Schrödinger equation,” Phys. Rev. A 85, 033820 (2012).
[CrossRef]

R. V. J. Raja, K. Porsezian, S. K. Varshney, and S. Sivabalan, “Modeling photonic crystal fiber for efficient soliton pulse propagation at 850 nm,” Opt. Commun. 283, 5000–5006 (2010).
[CrossRef]

Radhakrishnan, R.

R. Radhakrishnan, A. Kundu, and M. Lakshmanan, “Coupled nonlinear Schrödinger equations with cubic-quintic nonlinearity: integrability and soliton interaction in non-Kerr media,” Phys. Rev. E 60, 3314–3323 (1999).
[CrossRef]

Raja, R. V. J.

R. V. J. Raja, K. Porsezian, S. K. Varshney, and S. Sivabalan, “Modeling photonic crystal fiber for efficient soliton pulse propagation at 850 nm,” Opt. Commun. 283, 5000–5006 (2010).
[CrossRef]

Senthilnathan, K.

K. Senthilnathan, Q. Li, K. Nakkeeran, and P. K. A. Wai, “Robust pedestal-free pulse compression in cubic-quintic nonlinear media,” Phys. Rev. A 78, 033835 (2008).
[CrossRef]

Sivabalan, S.

R. V. J. Raja, K. Porsezian, S. K. Varshney, and S. Sivabalan, “Modeling photonic crystal fiber for efficient soliton pulse propagation at 850 nm,” Opt. Commun. 283, 5000–5006 (2010).
[CrossRef]

Taylor, J. R.

J. M. Dudley and J. R. Taylor, “Ten years of nonlinear optics in photonic crystal fiber,” Nat. Photonics 3, 85–90 (2009).
[CrossRef]

J. M. Dudley and J. R. Taylor, Supercontinuum Generation in Optical Fibers, 1st ed. (Cambridge University Press, 2010).

Travers, J. C.

J. C. Travers, “Blue extension of optical fibre supercontinuum generation,” J. Opt. 12, 113001 (2010).
[CrossRef]

J. C. Travers, M. H. Frosz, and J. M. Dudley, “Nonlinear fibre optics overview,” in Handbook of Supercontinuum Generation in Optical Fibers, J. M. Dudley and J. R. Taylor, eds. (Cambridge University, 2010), Chap. 3, pp. 46–49.

Varshney, S. K.

R. V. J. Raja, K. Porsezian, S. K. Varshney, and S. Sivabalan, “Modeling photonic crystal fiber for efficient soliton pulse propagation at 850 nm,” Opt. Commun. 283, 5000–5006 (2010).
[CrossRef]

Wai, P. K. A.

K. Senthilnathan, Q. Li, K. Nakkeeran, and P. K. A. Wai, “Robust pedestal-free pulse compression in cubic-quintic nonlinear media,” Phys. Rev. A 78, 033835 (2008).
[CrossRef]

J. Opt. (1)

J. C. Travers, “Blue extension of optical fibre supercontinuum generation,” J. Opt. 12, 113001 (2010).
[CrossRef]

Nat. Photonics (1)

J. M. Dudley and J. R. Taylor, “Ten years of nonlinear optics in photonic crystal fiber,” Nat. Photonics 3, 85–90 (2009).
[CrossRef]

Opt. Commun. (2)

R. V. J. Raja, K. Porsezian, S. K. Varshney, and S. Sivabalan, “Modeling photonic crystal fiber for efficient soliton pulse propagation at 850 nm,” Opt. Commun. 283, 5000–5006 (2010).
[CrossRef]

A. Choudhuri and K. Porsezian, “Dark-in-the-bright solitary wave solution of higher-order nonlinear Schrödinger equation with non-Kerr terms,” Opt. Commun. 285, 364–367 (2012).
[CrossRef]

Phys. Rev. A (2)

A. Choudhuri and K. Porsezian, “Impact of dispersion and non-Kerr nonlinearity on the modulational instability of the higher-order nonlinear Schrödinger equation,” Phys. Rev. A 85, 033820 (2012).
[CrossRef]

K. Senthilnathan, Q. Li, K. Nakkeeran, and P. K. A. Wai, “Robust pedestal-free pulse compression in cubic-quintic nonlinear media,” Phys. Rev. A 78, 033835 (2008).
[CrossRef]

Phys. Rev. E (1)

R. Radhakrishnan, A. Kundu, and M. Lakshmanan, “Coupled nonlinear Schrödinger equations with cubic-quintic nonlinearity: integrability and soliton interaction in non-Kerr media,” Phys. Rev. E 60, 3314–3323 (1999).
[CrossRef]

Rev. Mod. Phys. (1)

J. M. Dudley, G. Genty, and S. Coen, “Supercontinuum generation in photonics crystal fiber,” Rev. Mod. Phys. 78, 1135–1184 (2006).
[CrossRef]

Other (5)

G. P. Agrawal, Nonlinear Fiber Optics, 4th ed. (Academic, 2007).

J. M. Dudley and J. R. Taylor, Supercontinuum Generation in Optical Fibers, 1st ed. (Cambridge University Press, 2010).

R. R. Alfano, The Supercontinuum Laser Source (Springer-Verlag, 1989).

G. P. Agrawal, “Supercontinuum generation,” in Applications of Nonlinear Fiber Optics (Academic, 2008), pp. 414–431.

J. C. Travers, M. H. Frosz, and J. M. Dudley, “Nonlinear fibre optics overview,” in Handbook of Supercontinuum Generation in Optical Fibers, J. M. Dudley and J. R. Taylor, eds. (Cambridge University, 2010), Chap. 3, pp. 46–49.

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Figures (4)

Fig. 1.
Fig. 1.

(a) SCG output spectra. Contour plots of SCG pulse spectral propagation: (b) case of single CKN γ¯2=0, (c) case of cooperative nonlinearities γ¯2=0.05W2m1, and (d) case of competing nonlinearities γ¯2=0.05W2m1.

Fig. 2.
Fig. 2.

Row 2, contour plots of SCG pulse spectral propagation and Row 1, SCG input and output spectra: (a.1), (a.2) for the case of cooperative nonlinearities; (b.1), (b.2) for the case of competing nonlinearities; (c.1), (c.2) for the case of single CKN.

Fig. 3.
Fig. 3.

SCG spectra.

Fig. 4.
Fig. 4.

SCG 20dB bandwidths corresponding to the cases plotted in Fig. 3. (I) For input bandwidths of approximately, case (a), 1144.57 nm; case (b), 1130.95 nm; and case (c), 1195.65 nm. (II) For case (d), input bandwidth of approximately 3026.58 nm.

Equations (8)

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iuz+k=2M(i)kβkk!kuTk+γ(1+iω0T)(u(z,T)tR(TT)|u(z,T)|2dt)=iα2u,
u˜z=iγ¯1ωδ1exp(L^(ω)z)F[u(z,T)R(T)|u(z,TT)|2dT]
γ¯1=n2n0ω0cneffAeff,
iuz+k=2M(i)kβkk!kuTk+γ1|u|2uγ2|u|4u=iα2u+iδ1(|u|2u)T+δ2(|u|2)Tu+iδ3(|u|4u)T+δ4(|u|4)Tu,
iuz+k=2M(i)kβkk!kuTk=iα2uγ1(1+iδ1T)[u(z,T)R(T)|u(z,TT)|2dT]γ2(1+iδ3T)[u(z,T)R(T)|u(z,TT)|4dT].
u˜z=iωγ¯1δ1exp(L^(ω)z)F[u(z,T)R(T)|u(z,TT)|2dT]+iωγ¯2δ3exp(L^(ω)z)F[u(z,T)R(T)|u(z,TT)|4dT],
βk(λ0)(1)kβ2(λ0)t0k2,
u(z,T)=P0sech(T/t0).

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