American Institute of Mathematical Sciences

Advanced
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

Alejandro B. Aceves 1, , Rondald Chen 2, , Yeojin Chung 1, , Thomas Hagstrom 1, and  Michelle Hummel 2,

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.
Keywords: photonic crystal fibers, Nonlinear optics, pulse propagation and solitons., supercontinuum generation.
Mathematics Subject Classification: Primary: 35Q55, 74J30, 78A60, 35L6.
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.

[1]

Tukur Abdulkadir Sulaiman, Hasan Bulut, Haci Mehmet Baskonus. Optical solitons to the fractional perturbed NLSE in nano-fibers. Discrete and Continuous Dynamical Systems - S, 2020, 13 (3) : 925-936. doi: 10.3934/dcdss.2020054
[2]

Patricia Bauman, Daniel Phillips. Analysis and stability of bent-core liquid crystal fibers. Discrete and Continuous Dynamical Systems - B, 2012, 17 (6) : 1707-1728. doi: 10.3934/dcdsb.2012.17.1707
[3]

Gang Bao. Mathematical modeling of nonlinear diffracvtive optics. Conference Publications, 1998, 1998 (Special) : 89-99. doi: 10.3934/proc.1998.1998.89
[4]

Daomin Cao, Ezzat S. Noussair, Shusen Yan. On the profile of solutions for an elliptic problem arising in nonlinear optics. Discrete and Continuous Dynamical Systems, 2004, 11 (2&3) : 649-666. doi: 10.3934/dcds.2004.11.649
[5]

Eric P. Choate, Hong Zhou. Optimization of electromagnetic wave propagation through a liquid crystal layer. Discrete and Continuous Dynamical Systems - S, 2015, 8 (2) : 303-312. doi: 10.3934/dcdss.2015.8.303
[6]

Alexander Komech, Elena Kopylova, David Stuart. On asymptotic stability of solitons in a nonlinear Schrödinger equation. Communications on Pure and Applied Analysis, 2012, 11 (3) : 1063-1079. doi: 10.3934/cpaa.2012.11.1063
[7]

Fabrice Delbary, Kim Knudsen. Numerical nonlinear complex geometrical optics algorithm for the 3D Calderón problem. Inverse Problems and Imaging, 2014, 8 (4) : 991-1012. doi: 10.3934/ipi.2014.8.991
[8]

Walid K. Abou Salem, Xiao Liu, Catherine Sulem. Numerical simulation of resonant tunneling of fast solitons for the nonlinear Schrödinger equation. Discrete and Continuous Dynamical Systems, 2011, 29 (4) : 1637-1649. doi: 10.3934/dcds.2011.29.1637
[9]

Andrey Shishkov, Laurent Véron. Propagation of singularities of nonlinear heat flow in fissured media. Communications on Pure and Applied Analysis, 2013, 12 (4) : 1769-1782. doi: 10.3934/cpaa.2013.12.1769
[10]

Willem M. Schouten-Straatman, Hermen Jan Hupkes. Nonlinear stability of pulse solutions for the discrete FitzHugh-Nagumo equation with infinite-range interactions. Discrete and Continuous Dynamical Systems, 2019, 39 (9) : 5017-5083. doi: 10.3934/dcds.2019205
[11]

Manuel Gutiérrez. Lorentz geometry technique in nonimaging optics. Conference Publications, 2003, 2003 (Special) : 386-392. doi: 10.3934/proc.2003.2003.386
[12]

M. D. Todorov, C. I. Christov. Collision dynamics of circularly polarized solitons in nonintegrable coupled nonlinear Schrödinger system. Conference Publications, 2009, 2009 (Special) : 780-789. doi: 10.3934/proc.2009.2009.780
[13]

A. Pankov. Gap solitons in periodic discrete nonlinear Schrödinger equations II: A generalized Nehari manifold approach. Discrete and Continuous Dynamical Systems, 2007, 19 (2) : 419-430. doi: 10.3934/dcds.2007.19.419
[14]

Zhong Wang. Stability of Hasimoto solitons in energy space for a fourth order nonlinear Schrödinger type equation. Discrete and Continuous Dynamical Systems, 2017, 37 (7) : 4091-4108. doi: 10.3934/dcds.2017174
[15]

Christopher Chong, Dmitry Pelinovsky. Variational approximations of bifurcations of asymmetric solitons in cubic-quintic nonlinear Schrödinger lattices. Discrete and Continuous Dynamical Systems - S, 2011, 4 (5) : 1019-1031. doi: 10.3934/dcdss.2011.4.1019
[16]

Vianney Combet. Multi-existence of multi-solitons for the supercritical nonlinear Schrödinger equation in one dimension. Discrete and Continuous Dynamical Systems, 2014, 34 (5) : 1961-1993. doi: 10.3934/dcds.2014.34.1961
[17]

Wouter Rogiest, Koen De Turck, Koenraad Laevens, Dieter Fiems, Sabine Wittevrongel, Herwig Bruneel. On the optimality of packet-oriented scheduling in photonic switches with delay lines. Numerical Algebra, Control and Optimization, 2011, 1 (4) : 727-747. doi: 10.3934/naco.2011.1.727
[18]

Antonio Fasano, Angiolo Farina. Modeling high flux hollow fibers dialyzers. Discrete and Continuous Dynamical Systems - B, 2012, 17 (6) : 1903-1937. doi: 10.3934/dcdsb.2012.17.1903
[19]

Hao Wu. Long-time behavior for nonlinear hydrodynamic system modeling the nematic liquid crystal flows. Discrete and Continuous Dynamical Systems, 2010, 26 (1) : 379-396. doi: 10.3934/dcds.2010.26.379
[20]

Hongyu Liu, Ting Zhou. Two dimensional invisibility cloaking via transformation optics. Discrete and Continuous Dynamical Systems, 2011, 31 (2) : 525-543. doi: 10.3934/dcds.2011.31.525

2020 Impact Factor: 2.425

Tools

Metrics

  • PDF downloads (65)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy