October  2011, 4(5): 957-973. doi: 10.3934/dcdss.2011.4.957

Analysis of supercontinuum generation under general dispersion characteristics and beyond the slowly varying envelope approximation

1. 

Department of Mathematics, Southern Methodist University, Dallas TX 75275, United States, United States, United States

2. 

Department of Mathematics and Statistics, The University of New Mexico, Albuquerque, NM 87131, United States, United States

Received  August 2009 Revised  January 2010 Published  December 2010

The generation of broadband supercontinua (SC) in air-silica microstructured fibers results from a delicate balance of dispersion and nonlinearity. We analyze two models aimed at better understanding SC. In the first one, we characterize linear dispersion in the Fourier domain from the calculated group velocity dispersion (GVD) without using a Taylor approximation for the propagation constant. Results of our numerical simulations are in good agreement with experiments. A novel relevant length scale, namely the length for shock formation, is introduced and its role is discussed. The second part shows similar dynamics for a model that goes beyond the slowly varying approximation for optical pulse propagation.
Citation: Alejandro B. Aceves, Rondald Chen, Yeojin Chung, Thomas Hagstrom, Michelle Hummel. Analysis of supercontinuum generation under general dispersion characteristics and beyond the slowly varying envelope approximation. Discrete and Continuous Dynamical Systems - S, 2011, 4 (5) : 957-973. doi: 10.3934/dcdss.2011.4.957
References:
[1]

G. P. Agrawal, "Nonlinear Fiber Optics," 3rd edition, Academic Press, 2001.

[2]

R. Alfano, "The Supercontinuum Laser Source," New York, Springer-Verlag, 1989.

[3]

D. Anderson and M. Lisak, Nonlinear asymmetric self-phase modulation and self-steepening of pulses in long optical waveguides, Phys. Rev. A, 27 (1983), 1393-1398. doi: doi:10.1103/PhysRevA.27.1393.

[4]

B. Barviau, B. Kibler, S. Coen and A. Picozzi, Toward a thermodynamic description of supercontinuum generation, Opt. Lett., 33 (2008), 2833-2835. doi: doi:10.1364/OL.33.002833.

[5]

B. Barviau, B. Kibler, A. Kudlinski, A. Mussot, G. Millot and A. Picozzi, Experimental signature of optical wave thermalization through supercontinuum generation in photonic crystal fiber, Opt. Exp., 17 (2009), 7392-7406. doi: doi:10.1364/OE.17.007392.

[6]

B. Barviau, B. Kibler and A. Picozzi, Wave-turbulence approach of supercontinuum generation: Influence of self-steepening and higher order dispersion, Phys. Rev. A, 79 (2009), 063840. doi: doi:10.1103/PhysRevA.79.063840.

[7]

T. A. Birks, W. J. Wadsworth and P. S. J. Russell, Supercontinuum generation in tapered fibers, Opt. Lett., 25 (2000), 1415-1417. doi: doi:10.1364/OL.25.001415.

[8]

A. Biswas and A. B. Aceves, Dynamics of solitons in optical fibres, Journ. of Modern Optics, 48 (2001), 1135-1150.

[9]

Y. Chung and T. Schäfer, Stabilization of ultra-short pulse in cubic nonlinear media, Phys. Lett. A., 361 (2007), 63-69. doi: doi:10.1016/j.physleta.2006.08.087.

[10]

J. Dudley and S. Coen, Numerical simulations and coherence properties of supercontinuum generation in photonic crystal and tapered optical fibers, IEEE J. Sel. Top. Quantum Electron, 8 (2002), 651-659. doi: doi:10.1109/JSTQE.2002.1016369.

[11]

J. Dudley, G. Genty and S. Coen, Supercontinuum generation in photonic crystal fiber, Rev. of Mod. Phys., 78 (2006), 1135-1184. doi: doi:10.1103/RevModPhys.78.1135.

[12]

J. M. Dudley, X. Gu, L. Xu, M. Kimmel, E. Zeek, P. O'Shea, R. Trebino, S. Coen and R. S. Windeler, Cross-correlation frequency resolved optical gating analysis of broadband continuum generation in photonic crystal fiber: Simulations and experiments, Opt. Exp., 10 (2002), 1215-1221

[13]

J. M. Dudley, L. Provino, N. Grossard, H. Maillotte, R. S. Windeler, B. J. Eggleton and S. Coen, Supercontinuum generation in air-silica microstructured fibers with nanosecond and femtosecond pulse pumping, JOSA B, 19 (2002), 765-771. doi: doi:10.1364/JOSAB.19.000765.

[14]

G. Genty, S. Cohen and J. M. Dudley, Fiber supercontinuum sources (Invited), JOSA B, 24 (2007), 1771-1785. doi: doi:10.1364/JOSAB.24.001771.

[15]

G. Genty, P. Kinsler, B. Kibler and J. M. Dudley, Nonlinear envelope equation modeling of sub-cycle dynamics and harmonic generation in nonlinear waveguides, Opt. Exp., 15 (2007) 5382-5387. doi: doi:10.1364/OE.15.005382.

[16]

I. Hartl, X. D. Li, C. Chudoba, R. K. Ghanta, T. H. Ko, J. G. Fujimoto, J. K. Ranka and R. S. Windeler, Ultrahigh-resolution optical coherence tomography using continuum generation in an air-silica microstructure optical fiber, Opt. Lett., 26 (2001), 608-610. doi: doi:10.1364/OL.26.000608.

[17]

J. Herrmann, U. Griebner, N. Zhavoronkov, A. V. Husakou, D. Nickel, J. C. Knight, W. J. Wadsworth, P. S. J. Russell and G. Korn, Experimental evidence for supercontinuum generation by fission of higher-order solitons in photonic fibers, Phys. Rev. Lett., 88 (2002), 173901. doi: doi:10.1103/PhysRevLett.88.173901.

[18]

A. V. Husakou and J. Herrmann, Supercontinuum Generation of Higher-Order Solitons by Fission in Photonic Crystal Fibers, Phys. Rev. Lett., 87 (2001), 203901. doi: doi:10.1103/PhysRevLett.87.203901.

[19]

J. C. Knight, T. A. Birks, D. M. Atkin and P. S. J. Russell, Pure silica single-mode fiber with hexagonal photonic crystal cladding, in "Proc. Opt. Fiber Commun. Conf. No. PD3," San Jose, California, Feb. (1996).

[20]

M. Kolesik and J. V. Moloney, Nonlinear optical pulse propagation simulation: From Maxwell's to unidirectional equations, Phys. Rev. E, 70 (2004), 036604. doi: doi:10.1103/PhysRevE.70.036604.

[21]

M. Kolesik, E. M. Wright and J. V. Moloney, Supercontinuum and third-harmonic generation accompanying optical filamentation as first order scattering processes, Opt. Lett., 342 (2007), 2816-2818. doi: doi:10.1364/OL.32.002816.

[22]

V. V. R. K. Kumar, A. K. George, W. H. Reeves, J. C. Knight, P. S. J. Russell, F. G. Omenetto and A. J. Taylor, Extruded soft glass photonic crystal fiber for ultrabroad supercontinuum generation, Opt. Exp., 10 (2002), 1520-1525.

[23]

K. Lindfors, T. Kalkbrenner, P. Stoller and V. Sandoghdar, Detection and spectroscopy of gold nanoparticles using supercontinuum white light confocal microscopy, Phys. Rev. Lett., 93 (2004), 037401. doi: doi:10.1103/PhysRevLett.93.037401.

[24]

F. G. Omenetto, N. A. Wolchover, M. R. Wehner, M. Ross, A. Efimov, A. J. Taylor, V. V. R. K. Kumar, A. K. George, J. C. Knight, N. Y. Joly and P. S. J. Russell, Spectrally smooth supercontinuum from 350 nm to 3 $\mu$m in sub-centimeter lengths of soft-glass photonic crystal fibers, Opt. Exp., 14 (2006), 4928-4934. doi: doi:10.1364/OE.14.004928.

[25]

K. Shi, P. Li, S. Yin and Z. Liu, Chromatic confocal microscopy using supercontinuum light, Opt. Exp., 12 (2004), 2096-2101. doi: doi:10.1364/OPEX.12.002096.

[26]

K. Shi, F. G. Omenetto and Z. Liu, Supercontinuum generation in an imaging fiber taper, Opt. Exp., 14 (2006), 12359-12364. doi: doi:10.1364/OE.14.012359.

[27]

J. C. A. Tyrrell, P. Kinsler and G. H. C. New, Pseudospectral spatial-domain: A new method for nonlinear pulse propagation in a few-cycle regime with arbitrary dispersion, Journ of Mod. Opt., 52 (2005), 973-986. doi: doi:10.1080/09500340512331334086.

[28]

T. Udem, R. Holzwarth and T. W. Hänsch, Optical frequency metrology, Nature, 416 (2002), 233-237. doi: doi:10.1038/416233a.

[29]

W. J. Wadsworth, A. Ortigosa-Blanch, J. C. Knight, T. A. Birks, T-P. M. Man and P. S. J. Russell, Supercontinuum generation in photonic crystal fibers and optical fiber tapers: A novel light source, JOSA B, Special issue on nonlinear optics in photonic crystals and fibres, 19 (2002), 2148-2155.

[30]

P. K. A. Wai, C. R. Menyuk, Y. C. Lee and H. H. Chen, Nonlinear pulse propagation in the neighborhood of the zero-dispersion wavelength of monomode optical fibers, Opt. Lett., 11 (1986), 464-466. doi: doi:10.1364/OL.11.000464.

[31]

G. B. Whitham, "Linear and Nonlinear Waves," New York, Wiley Ed., 1974.

[32]

G. Yang and Y. R. Shen, Spectral broadening of ultrashort pulses in a nonlinear medium, Opt. Lett., 9 (1984), 510-512. doi: doi:10.1364/OL.9.000510.

[33]

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

show all references

References:
[1]

G. P. Agrawal, "Nonlinear Fiber Optics," 3rd edition, Academic Press, 2001.

[2]

R. Alfano, "The Supercontinuum Laser Source," New York, Springer-Verlag, 1989.

[3]

D. Anderson and M. Lisak, Nonlinear asymmetric self-phase modulation and self-steepening of pulses in long optical waveguides, Phys. Rev. A, 27 (1983), 1393-1398. doi: doi:10.1103/PhysRevA.27.1393.

[4]

B. Barviau, B. Kibler, S. Coen and A. Picozzi, Toward a thermodynamic description of supercontinuum generation, Opt. Lett., 33 (2008), 2833-2835. doi: doi:10.1364/OL.33.002833.

[5]

B. Barviau, B. Kibler, A. Kudlinski, A. Mussot, G. Millot and A. Picozzi, Experimental signature of optical wave thermalization through supercontinuum generation in photonic crystal fiber, Opt. Exp., 17 (2009), 7392-7406. doi: doi:10.1364/OE.17.007392.

[6]

B. Barviau, B. Kibler and A. Picozzi, Wave-turbulence approach of supercontinuum generation: Influence of self-steepening and higher order dispersion, Phys. Rev. A, 79 (2009), 063840. doi: doi:10.1103/PhysRevA.79.063840.

[7]

T. A. Birks, W. J. Wadsworth and P. S. J. Russell, Supercontinuum generation in tapered fibers, Opt. Lett., 25 (2000), 1415-1417. doi: doi:10.1364/OL.25.001415.

[8]

A. Biswas and A. B. Aceves, Dynamics of solitons in optical fibres, Journ. of Modern Optics, 48 (2001), 1135-1150.

[9]

Y. Chung and T. Schäfer, Stabilization of ultra-short pulse in cubic nonlinear media, Phys. Lett. A., 361 (2007), 63-69. doi: doi:10.1016/j.physleta.2006.08.087.

[10]

J. Dudley and S. Coen, Numerical simulations and coherence properties of supercontinuum generation in photonic crystal and tapered optical fibers, IEEE J. Sel. Top. Quantum Electron, 8 (2002), 651-659. doi: doi:10.1109/JSTQE.2002.1016369.

[11]

J. Dudley, G. Genty and S. Coen, Supercontinuum generation in photonic crystal fiber, Rev. of Mod. Phys., 78 (2006), 1135-1184. doi: doi:10.1103/RevModPhys.78.1135.

[12]

J. M. Dudley, X. Gu, L. Xu, M. Kimmel, E. Zeek, P. O'Shea, R. Trebino, S. Coen and R. S. Windeler, Cross-correlation frequency resolved optical gating analysis of broadband continuum generation in photonic crystal fiber: Simulations and experiments, Opt. Exp., 10 (2002), 1215-1221

[13]

J. M. Dudley, L. Provino, N. Grossard, H. Maillotte, R. S. Windeler, B. J. Eggleton and S. Coen, Supercontinuum generation in air-silica microstructured fibers with nanosecond and femtosecond pulse pumping, JOSA B, 19 (2002), 765-771. doi: doi:10.1364/JOSAB.19.000765.

[14]

G. Genty, S. Cohen and J. M. Dudley, Fiber supercontinuum sources (Invited), JOSA B, 24 (2007), 1771-1785. doi: doi:10.1364/JOSAB.24.001771.

[15]

G. Genty, P. Kinsler, B. Kibler and J. M. Dudley, Nonlinear envelope equation modeling of sub-cycle dynamics and harmonic generation in nonlinear waveguides, Opt. Exp., 15 (2007) 5382-5387. doi: doi:10.1364/OE.15.005382.

[16]

I. Hartl, X. D. Li, C. Chudoba, R. K. Ghanta, T. H. Ko, J. G. Fujimoto, J. K. Ranka and R. S. Windeler, Ultrahigh-resolution optical coherence tomography using continuum generation in an air-silica microstructure optical fiber, Opt. Lett., 26 (2001), 608-610. doi: doi:10.1364/OL.26.000608.

[17]

J. Herrmann, U. Griebner, N. Zhavoronkov, A. V. Husakou, D. Nickel, J. C. Knight, W. J. Wadsworth, P. S. J. Russell and G. Korn, Experimental evidence for supercontinuum generation by fission of higher-order solitons in photonic fibers, Phys. Rev. Lett., 88 (2002), 173901. doi: doi:10.1103/PhysRevLett.88.173901.

[18]

A. V. Husakou and J. Herrmann, Supercontinuum Generation of Higher-Order Solitons by Fission in Photonic Crystal Fibers, Phys. Rev. Lett., 87 (2001), 203901. doi: doi:10.1103/PhysRevLett.87.203901.

[19]

J. C. Knight, T. A. Birks, D. M. Atkin and P. S. J. Russell, Pure silica single-mode fiber with hexagonal photonic crystal cladding, in "Proc. Opt. Fiber Commun. Conf. No. PD3," San Jose, California, Feb. (1996).

[20]

M. Kolesik and J. V. Moloney, Nonlinear optical pulse propagation simulation: From Maxwell's to unidirectional equations, Phys. Rev. E, 70 (2004), 036604. doi: doi:10.1103/PhysRevE.70.036604.

[21]

M. Kolesik, E. M. Wright and J. V. Moloney, Supercontinuum and third-harmonic generation accompanying optical filamentation as first order scattering processes, Opt. Lett., 342 (2007), 2816-2818. doi: doi:10.1364/OL.32.002816.

[22]

V. V. R. K. Kumar, A. K. George, W. H. Reeves, J. C. Knight, P. S. J. Russell, F. G. Omenetto and A. J. Taylor, Extruded soft glass photonic crystal fiber for ultrabroad supercontinuum generation, Opt. Exp., 10 (2002), 1520-1525.

[23]

K. Lindfors, T. Kalkbrenner, P. Stoller and V. Sandoghdar, Detection and spectroscopy of gold nanoparticles using supercontinuum white light confocal microscopy, Phys. Rev. Lett., 93 (2004), 037401. doi: doi:10.1103/PhysRevLett.93.037401.

[24]

F. G. Omenetto, N. A. Wolchover, M. R. Wehner, M. Ross, A. Efimov, A. J. Taylor, V. V. R. K. Kumar, A. K. George, J. C. Knight, N. Y. Joly and P. S. J. Russell, Spectrally smooth supercontinuum from 350 nm to 3 $\mu$m in sub-centimeter lengths of soft-glass photonic crystal fibers, Opt. Exp., 14 (2006), 4928-4934. doi: doi:10.1364/OE.14.004928.

[25]

K. Shi, P. Li, S. Yin and Z. Liu, Chromatic confocal microscopy using supercontinuum light, Opt. Exp., 12 (2004), 2096-2101. doi: doi:10.1364/OPEX.12.002096.

[26]

K. Shi, F. G. Omenetto and Z. Liu, Supercontinuum generation in an imaging fiber taper, Opt. Exp., 14 (2006), 12359-12364. doi: doi:10.1364/OE.14.012359.

[27]

J. C. A. Tyrrell, P. Kinsler and G. H. C. New, Pseudospectral spatial-domain: A new method for nonlinear pulse propagation in a few-cycle regime with arbitrary dispersion, Journ of Mod. Opt., 52 (2005), 973-986. doi: doi:10.1080/09500340512331334086.

[28]

T. Udem, R. Holzwarth and T. W. Hänsch, Optical frequency metrology, Nature, 416 (2002), 233-237. doi: doi:10.1038/416233a.

[29]

W. J. Wadsworth, A. Ortigosa-Blanch, J. C. Knight, T. A. Birks, T-P. M. Man and P. S. J. Russell, Supercontinuum generation in photonic crystal fibers and optical fiber tapers: A novel light source, JOSA B, Special issue on nonlinear optics in photonic crystals and fibres, 19 (2002), 2148-2155.

[30]

P. K. A. Wai, C. R. Menyuk, Y. C. Lee and H. H. Chen, Nonlinear pulse propagation in the neighborhood of the zero-dispersion wavelength of monomode optical fibers, Opt. Lett., 11 (1986), 464-466. doi: doi:10.1364/OL.11.000464.

[31]

G. B. Whitham, "Linear and Nonlinear Waves," New York, Wiley Ed., 1974.

[32]

G. Yang and Y. R. Shen, Spectral broadening of ultrashort pulses in a nonlinear medium, Opt. Lett., 9 (1984), 510-512. doi: doi:10.1364/OL.9.000510.

[33]

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

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