Abstract

A full-vectorial nonlinear propagation equation for short pulses in tapered optical fibers is developed. Specific emphasis is placed on the importance of the field normalization convention for the structure of the equations, and the interpretation of the resulting field amplitudes. Different numerical schemes for interpolation of fiber parameters along the taper are discussed and tested in numerical simulations on soliton propagation and generation of continuum radiation in short photonic-crystal fiber tapers.

© 2012 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. T. A. Birks, W. J. Wadsworth, and P. S. J. Russell, “Supercontinuum generation in tapered fibers,” Opt. Lett. 25, 1415–1417 (2000).
    [CrossRef]
  2. L. Tong, R. R. Gattass, J. B. Ashcom, S. He, J. Lou, M. Shen, I. Maxwell, and E. Mazur, “Subwavelength-diameter silica wires for low-loss optical wave guiding,” Nature 426, 816–819 (2003).
    [CrossRef]
  3. G. Brambilla, V. Finazzi, and D. Richardson, “Ultra-low-loss optical fiber nanotapers,” Opt. Express 12, 2258–2263 (2004).
    [CrossRef]
  4. S. Leon-Saval, T. Birks, W. Wadsworth, P. S. J. Russell, and M. Mason, “Supercontinuum generation in submicron fibre waveguides,” Opt. Express 12, 2864–2869 (2004).
    [CrossRef]
  5. M. A. Foster, J. M. Dudley, B. Kibler, Q. Cao, D. Lee, R. Trebino, and A. L. Gaeta, “Nonlinear pulse propagation and supercontinuum generation in photonic nanowires: experiment and simulation,” Appl. Phys. B 81, 363–367 (2005).
    [CrossRef]
  6. R. R. Gattass, G. T. Svacha, L. Tong, and E. Mazur, “Supercontinuum generation in submicrometer diameter silica fibers,” Opt. Express 14, 9408–9414(2006).
    [CrossRef]
  7. A. Kudlinski, A. George, J. Knight, J. Travers, A. Rulkov, S. Popov, and J. Taylor, “Zero-dispersion wavelength decreasing photonic crystal fibers for ultraviolet-extended supercontinuum generation,” Opt. Express 14, 5715–5722 (2006).
    [CrossRef]
  8. J. C. Travers and J. R. Taylor, “Soliton trapping of dispersive waves in tapered optical fibers,” Opt. Lett. 34, 115–117(2009).
    [CrossRef]
  9. A. Kudlinski, M. Lelek, B. Barviau, L. Audry, and A. Mussot, “Efficient blue conversion from a 1064 nm microchip laser in long photonic crystal fiber tapers for fluorescence microscopy,” Opt. Express 18, 16640–16645 (2010).
    [CrossRef]
  10. S. T. Sørensen, A. Judge, C. L. Thomsen, and O. Bang, “Optimum fiber tapers for increasing the power in the blue edge of a supercontinuum—group-acceleration matching,” Opt. Lett. 36, 816–818 (2011).
    [CrossRef]
  11. J. M. Dudley, G. Genty, and S. Coen, “Supercontinuum generation in photonic crystal fiber,” Rev. Mod. Phys. 78, 1135–1184 (2006).
    [CrossRef]
  12. F. W. Wise, A. Chong, and W. H. Renninger, “High-energy femtosecond fiber lasers based on pulse propagation at normal dispersion,” Laser Photon. Rev. 2, 58–73 (2008).
    [CrossRef]
  13. X. Liu, J. Lægsgaard, and D. Turchinovich, “Self-stabilization of a mode-locked femtosecond fiber laser using a photonic bandgap fiber,” Opt. Lett. 35, 913–915 (2010).
    [CrossRef]
  14. X. Liu, J. Lægsgaard, and D. Turchinovich, “Highly-stable monolithic femtosecond yb-fiber laser system based on photonic crystal fibers,” Opt. Express 18, 15475–15483 (2010).
    [CrossRef]
  15. X. Liu, J. Lægsgaard, U. Møller, H. Tu, S. A. Boppart, and D. Turchinovich, “All-fiber femtosecond Cherenkov radiation source,” Opt. Lett. 37, 2769–2771 (2012).
    [CrossRef]
  16. O. Vanvincq, J. C. Travers, and A. Kudlinski, “Conservation of the photon number in the generalized nonlinear Schrödinger equation in axially varying optical fibers,” Phys. Rev. A 84, 063820 (2011).
    [CrossRef]
  17. 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]
  18. S. A. Vahid and T. M. Monro, “A full vectorial model for pulse propagation in emerging waveguides with subwavelength structures part I: Kerr nonlinearity,” Opt. Express 17, 2298–2318 (2009).
    [CrossRef]
  19. M. D. Turner, T. M. Monro, and S. A. Vahid, “A full vectorial model for pulse propagation in emerging waveguides with subwavelength structures part II: stimulated Raman scattering,” Opt. Express 17, 11565–11581 (2009).
    [CrossRef]
  20. R. W. Hellwarth, “3rd-order optical susceptibilities of liquids and solids,” Prog. Quantum Electron. 5, 1–68 (1977).
    [CrossRef]
  21. J. Lægsgaard, “Mode profile dispersion in the generalised nonlinear Schrödinger equation,” Opt. Express 15, 16110–16123 (2007).
    [CrossRef]
  22. J. Lægsgaard and P. J. Roberts, “Dispersive pulse compression in hollow-core photonic bandgap fibers,” Opt. Express 16, 9628–9644 (2008).
    [CrossRef]
  23. K. J. Blow and D. Wood, “Theoretical description of transient stimulated “Raman scattering in optical fibers,” IEEE J. Quantum Electron. 25, 2665–2673 (1989).
    [CrossRef]
  24. G. P. Agrawal, Nonlinear Fiber Optics (Academic, 2007).
  25. J. Lægsgaard, A. Bjarklev, and S. E. B. Libori, “Chromatic dispersion in photonic crystal fibers: fast and accurate scheme for calculation,” J. Opt. Soc. Am. B 20, 443–448 (2003).
    [CrossRef]
  26. A. V. Gorbach and D. V. Skryabin, “Theory of radiation trapping by the accelerating solitons in optical fibers,” Phys. Rev. A 76, 053803 (2007).
    [CrossRef]

2012

2011

S. T. Sørensen, A. Judge, C. L. Thomsen, and O. Bang, “Optimum fiber tapers for increasing the power in the blue edge of a supercontinuum—group-acceleration matching,” Opt. Lett. 36, 816–818 (2011).
[CrossRef]

O. Vanvincq, J. C. Travers, and A. Kudlinski, “Conservation of the photon number in the generalized nonlinear Schrödinger equation in axially varying optical fibers,” Phys. Rev. A 84, 063820 (2011).
[CrossRef]

2010

2009

2008

F. W. Wise, A. Chong, and W. H. Renninger, “High-energy femtosecond fiber lasers based on pulse propagation at normal dispersion,” Laser Photon. Rev. 2, 58–73 (2008).
[CrossRef]

J. Lægsgaard and P. J. Roberts, “Dispersive pulse compression in hollow-core photonic bandgap fibers,” Opt. Express 16, 9628–9644 (2008).
[CrossRef]

2007

J. Lægsgaard, “Mode profile dispersion in the generalised nonlinear Schrödinger equation,” Opt. Express 15, 16110–16123 (2007).
[CrossRef]

A. V. Gorbach and D. V. Skryabin, “Theory of radiation trapping by the accelerating solitons in optical fibers,” Phys. Rev. A 76, 053803 (2007).
[CrossRef]

2006

2005

M. A. Foster, J. M. Dudley, B. Kibler, Q. Cao, D. Lee, R. Trebino, and A. L. Gaeta, “Nonlinear pulse propagation and supercontinuum generation in photonic nanowires: experiment and simulation,” Appl. Phys. B 81, 363–367 (2005).
[CrossRef]

2004

2003

J. Lægsgaard, A. Bjarklev, and S. E. B. Libori, “Chromatic dispersion in photonic crystal fibers: fast and accurate scheme for calculation,” J. Opt. Soc. Am. B 20, 443–448 (2003).
[CrossRef]

L. Tong, R. R. Gattass, J. B. Ashcom, S. He, J. Lou, M. Shen, I. Maxwell, and E. Mazur, “Subwavelength-diameter silica wires for low-loss optical wave guiding,” Nature 426, 816–819 (2003).
[CrossRef]

2000

1989

K. J. Blow and D. Wood, “Theoretical description of transient stimulated “Raman scattering in optical fibers,” IEEE J. Quantum Electron. 25, 2665–2673 (1989).
[CrossRef]

1977

R. W. Hellwarth, “3rd-order optical susceptibilities of liquids and solids,” Prog. Quantum Electron. 5, 1–68 (1977).
[CrossRef]

Agrawal, G. P.

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

Ashcom, J. B.

L. Tong, R. R. Gattass, J. B. Ashcom, S. He, J. Lou, M. Shen, I. Maxwell, and E. Mazur, “Subwavelength-diameter silica wires for low-loss optical wave guiding,” Nature 426, 816–819 (2003).
[CrossRef]

Audry, L.

Bang, O.

Barviau, B.

Birks, T.

Birks, T. A.

Bjarklev, A.

Blow, K. J.

K. J. Blow and D. Wood, “Theoretical description of transient stimulated “Raman scattering in optical fibers,” IEEE J. Quantum Electron. 25, 2665–2673 (1989).
[CrossRef]

Boppart, S. A.

Brambilla, G.

Cao, Q.

M. A. Foster, J. M. Dudley, B. Kibler, Q. Cao, D. Lee, R. Trebino, and A. L. Gaeta, “Nonlinear pulse propagation and supercontinuum generation in photonic nanowires: experiment and simulation,” Appl. Phys. B 81, 363–367 (2005).
[CrossRef]

Chong, A.

F. W. Wise, A. Chong, and W. H. Renninger, “High-energy femtosecond fiber lasers based on pulse propagation at normal dispersion,” Laser Photon. Rev. 2, 58–73 (2008).
[CrossRef]

Coen, S.

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

Dudley, J. M.

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

M. A. Foster, J. M. Dudley, B. Kibler, Q. Cao, D. Lee, R. Trebino, and A. L. Gaeta, “Nonlinear pulse propagation and supercontinuum generation in photonic nanowires: experiment and simulation,” Appl. Phys. B 81, 363–367 (2005).
[CrossRef]

Finazzi, V.

Foster, M. A.

M. A. Foster, J. M. Dudley, B. Kibler, Q. Cao, D. Lee, R. Trebino, and A. L. Gaeta, “Nonlinear pulse propagation and supercontinuum generation in photonic nanowires: experiment and simulation,” Appl. Phys. B 81, 363–367 (2005).
[CrossRef]

Gaeta, A. L.

M. A. Foster, J. M. Dudley, B. Kibler, Q. Cao, D. Lee, R. Trebino, and A. L. Gaeta, “Nonlinear pulse propagation and supercontinuum generation in photonic nanowires: experiment and simulation,” Appl. Phys. B 81, 363–367 (2005).
[CrossRef]

Gattass, R. R.

R. R. Gattass, G. T. Svacha, L. Tong, and E. Mazur, “Supercontinuum generation in submicrometer diameter silica fibers,” Opt. Express 14, 9408–9414(2006).
[CrossRef]

L. Tong, R. R. Gattass, J. B. Ashcom, S. He, J. Lou, M. Shen, I. Maxwell, and E. Mazur, “Subwavelength-diameter silica wires for low-loss optical wave guiding,” Nature 426, 816–819 (2003).
[CrossRef]

Genty, G.

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

George, A.

Gorbach, A. V.

A. V. Gorbach and D. V. Skryabin, “Theory of radiation trapping by the accelerating solitons in optical fibers,” Phys. Rev. A 76, 053803 (2007).
[CrossRef]

He, S.

L. Tong, R. R. Gattass, J. B. Ashcom, S. He, J. Lou, M. Shen, I. Maxwell, and E. Mazur, “Subwavelength-diameter silica wires for low-loss optical wave guiding,” Nature 426, 816–819 (2003).
[CrossRef]

Hellwarth, R. W.

R. W. Hellwarth, “3rd-order optical susceptibilities of liquids and solids,” Prog. Quantum Electron. 5, 1–68 (1977).
[CrossRef]

Judge, A.

Kibler, B.

M. A. Foster, J. M. Dudley, B. Kibler, Q. Cao, D. Lee, R. Trebino, and A. L. Gaeta, “Nonlinear pulse propagation and supercontinuum generation in photonic nanowires: experiment and simulation,” Appl. Phys. B 81, 363–367 (2005).
[CrossRef]

Knight, J.

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]

Kudlinski, A.

Lægsgaard, J.

Lee, D.

M. A. Foster, J. M. Dudley, B. Kibler, Q. Cao, D. Lee, R. Trebino, and A. L. Gaeta, “Nonlinear pulse propagation and supercontinuum generation in photonic nanowires: experiment and simulation,” Appl. Phys. B 81, 363–367 (2005).
[CrossRef]

Lelek, M.

Leon-Saval, S.

Libori, S. E. B.

Liu, X.

Lou, J.

L. Tong, R. R. Gattass, J. B. Ashcom, S. He, J. Lou, M. Shen, I. Maxwell, and E. Mazur, “Subwavelength-diameter silica wires for low-loss optical wave guiding,” Nature 426, 816–819 (2003).
[CrossRef]

Mason, M.

Maxwell, I.

L. Tong, R. R. Gattass, J. B. Ashcom, S. He, J. Lou, M. Shen, I. Maxwell, and E. Mazur, “Subwavelength-diameter silica wires for low-loss optical wave guiding,” Nature 426, 816–819 (2003).
[CrossRef]

Mazur, E.

R. R. Gattass, G. T. Svacha, L. Tong, and E. Mazur, “Supercontinuum generation in submicrometer diameter silica fibers,” Opt. Express 14, 9408–9414(2006).
[CrossRef]

L. Tong, R. R. Gattass, J. B. Ashcom, S. He, J. Lou, M. Shen, I. Maxwell, and E. Mazur, “Subwavelength-diameter silica wires for low-loss optical wave guiding,” Nature 426, 816–819 (2003).
[CrossRef]

Møller, U.

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]

Monro, T. M.

Mussot, A.

Popov, S.

Renninger, W. H.

F. W. Wise, A. Chong, and W. H. Renninger, “High-energy femtosecond fiber lasers based on pulse propagation at normal dispersion,” Laser Photon. Rev. 2, 58–73 (2008).
[CrossRef]

Richardson, D.

Roberts, P. J.

Rulkov, A.

Russell, P. S. J.

Shen, M.

L. Tong, R. R. Gattass, J. B. Ashcom, S. He, J. Lou, M. Shen, I. Maxwell, and E. Mazur, “Subwavelength-diameter silica wires for low-loss optical wave guiding,” Nature 426, 816–819 (2003).
[CrossRef]

Skryabin, D. V.

A. V. Gorbach and D. V. Skryabin, “Theory of radiation trapping by the accelerating solitons in optical fibers,” Phys. Rev. A 76, 053803 (2007).
[CrossRef]

Sørensen, S. T.

Svacha, G. T.

Taylor, J.

Taylor, J. R.

Thomsen, C. L.

Tong, L.

R. R. Gattass, G. T. Svacha, L. Tong, and E. Mazur, “Supercontinuum generation in submicrometer diameter silica fibers,” Opt. Express 14, 9408–9414(2006).
[CrossRef]

L. Tong, R. R. Gattass, J. B. Ashcom, S. He, J. Lou, M. Shen, I. Maxwell, and E. Mazur, “Subwavelength-diameter silica wires for low-loss optical wave guiding,” Nature 426, 816–819 (2003).
[CrossRef]

Travers, J.

Travers, J. C.

O. Vanvincq, J. C. Travers, and A. Kudlinski, “Conservation of the photon number in the generalized nonlinear Schrödinger equation in axially varying optical fibers,” Phys. Rev. A 84, 063820 (2011).
[CrossRef]

J. C. Travers and J. R. Taylor, “Soliton trapping of dispersive waves in tapered optical fibers,” Opt. Lett. 34, 115–117(2009).
[CrossRef]

Trebino, R.

M. A. Foster, J. M. Dudley, B. Kibler, Q. Cao, D. Lee, R. Trebino, and A. L. Gaeta, “Nonlinear pulse propagation and supercontinuum generation in photonic nanowires: experiment and simulation,” Appl. Phys. B 81, 363–367 (2005).
[CrossRef]

Tu, H.

Turchinovich, D.

Turner, M. D.

Vahid, S. A.

Vanvincq, O.

O. Vanvincq, J. C. Travers, and A. Kudlinski, “Conservation of the photon number in the generalized nonlinear Schrödinger equation in axially varying optical fibers,” Phys. Rev. A 84, 063820 (2011).
[CrossRef]

Wadsworth, W.

Wadsworth, W. J.

Wise, F. W.

F. W. Wise, A. Chong, and W. H. Renninger, “High-energy femtosecond fiber lasers based on pulse propagation at normal dispersion,” Laser Photon. Rev. 2, 58–73 (2008).
[CrossRef]

Wood, D.

K. J. Blow and D. Wood, “Theoretical description of transient stimulated “Raman scattering in optical fibers,” IEEE J. Quantum Electron. 25, 2665–2673 (1989).
[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]

Appl. Phys. B

M. A. Foster, J. M. Dudley, B. Kibler, Q. Cao, D. Lee, R. Trebino, and A. L. Gaeta, “Nonlinear pulse propagation and supercontinuum generation in photonic nanowires: experiment and simulation,” Appl. Phys. B 81, 363–367 (2005).
[CrossRef]

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]

IEEE J. Quantum Electron.

K. J. Blow and D. Wood, “Theoretical description of transient stimulated “Raman scattering in optical fibers,” IEEE J. Quantum Electron. 25, 2665–2673 (1989).
[CrossRef]

J. Opt. Soc. Am. B

Laser Photon. Rev.

F. W. Wise, A. Chong, and W. H. Renninger, “High-energy femtosecond fiber lasers based on pulse propagation at normal dispersion,” Laser Photon. Rev. 2, 58–73 (2008).
[CrossRef]

Nature

L. Tong, R. R. Gattass, J. B. Ashcom, S. He, J. Lou, M. Shen, I. Maxwell, and E. Mazur, “Subwavelength-diameter silica wires for low-loss optical wave guiding,” Nature 426, 816–819 (2003).
[CrossRef]

Opt. Express

S. A. Vahid and T. M. Monro, “A full vectorial model for pulse propagation in emerging waveguides with subwavelength structures part I: Kerr nonlinearity,” Opt. Express 17, 2298–2318 (2009).
[CrossRef]

M. D. Turner, T. M. Monro, and S. A. Vahid, “A full vectorial model for pulse propagation in emerging waveguides with subwavelength structures part II: stimulated Raman scattering,” Opt. Express 17, 11565–11581 (2009).
[CrossRef]

X. Liu, J. Lægsgaard, and D. Turchinovich, “Highly-stable monolithic femtosecond yb-fiber laser system based on photonic crystal fibers,” Opt. Express 18, 15475–15483 (2010).
[CrossRef]

A. Kudlinski, M. Lelek, B. Barviau, L. Audry, and A. Mussot, “Efficient blue conversion from a 1064 nm microchip laser in long photonic crystal fiber tapers for fluorescence microscopy,” Opt. Express 18, 16640–16645 (2010).
[CrossRef]

G. Brambilla, V. Finazzi, and D. Richardson, “Ultra-low-loss optical fiber nanotapers,” Opt. Express 12, 2258–2263 (2004).
[CrossRef]

S. Leon-Saval, T. Birks, W. Wadsworth, P. S. J. Russell, and M. Mason, “Supercontinuum generation in submicron fibre waveguides,” Opt. Express 12, 2864–2869 (2004).
[CrossRef]

A. Kudlinski, A. George, J. Knight, J. Travers, A. Rulkov, S. Popov, and J. Taylor, “Zero-dispersion wavelength decreasing photonic crystal fibers for ultraviolet-extended supercontinuum generation,” Opt. Express 14, 5715–5722 (2006).
[CrossRef]

R. R. Gattass, G. T. Svacha, L. Tong, and E. Mazur, “Supercontinuum generation in submicrometer diameter silica fibers,” Opt. Express 14, 9408–9414(2006).
[CrossRef]

J. Lægsgaard, “Mode profile dispersion in the generalised nonlinear Schrödinger equation,” Opt. Express 15, 16110–16123 (2007).
[CrossRef]

J. Lægsgaard and P. J. Roberts, “Dispersive pulse compression in hollow-core photonic bandgap fibers,” Opt. Express 16, 9628–9644 (2008).
[CrossRef]

Opt. Lett.

Phys. Rev. A

A. V. Gorbach and D. V. Skryabin, “Theory of radiation trapping by the accelerating solitons in optical fibers,” Phys. Rev. A 76, 053803 (2007).
[CrossRef]

O. Vanvincq, J. C. Travers, and A. Kudlinski, “Conservation of the photon number in the generalized nonlinear Schrödinger equation in axially varying optical fibers,” Phys. Rev. A 84, 063820 (2011).
[CrossRef]

Prog. Quantum Electron.

R. W. Hellwarth, “3rd-order optical susceptibilities of liquids and solids,” Prog. Quantum Electron. 5, 1–68 (1977).
[CrossRef]

Rev. Mod. Phys.

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

Other

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

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (6)

Fig. 1.
Fig. 1.

(a) Aeff and Aeff as a function of normalized wavelengths for a silica nanowire and a PCF. (b) Relative difference between Aeff and Aeff in the two cases. The wavelength is normalized to nanowire diameter d, and PCF pitch Λ, respectively.

Fig. 2.
Fig. 2.

Dispersion curves for the PCF investigated at small values of Λ. The vertical line indicates the wavelength of the initial pulse, λ=1.03μm.

Fig. 3.
Fig. 3.

(a) Soliton peak power versus pitch (Λ) of the tapered fiber for different taper rates. (b) Soliton peak power versus distance for a 16 cm taper, using either perturbative calculation of the dispersion properties, or linear interpolation between parameters for a finite number of Λ-values.

Fig. 4.
Fig. 4.

Final spectra for soliton propagation in a tapered fiber of 16 cm, for the different fiber interpolation schemes.

Fig. 5.
Fig. 5.

Spectrogram of fs pulse propagated in 5 cm fiber taper, from Λ=4μm to Λ=1μm. The main features of the broken-up pulse are indicated with “DW” denoting dispersive waves.

Fig. 6.
Fig. 6.

(a) Total spectrum of a 20 kW 50 fs gaussian input pulse after a 5 cm taper from Λ=4μm to Λ=1μm, with fiber parameters calculated along the taper by the perturbative approach. (b) Short-wavelength dispersive radiation, propagating behind main fundamental soliton, calculated either in the perturbative approach, or with linear interpolation between different Λ-values. (c) Intermediate-wavelength dispersive waves, propagating in front of soliton. (d) Spectrum of main soliton and long-wavelength dispersive waves.

Equations (50)

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

×E=μ0Ht,
×H=ε0ε(r,ω)Et+δPt.
H(r,t)=12πmdωAm(z,ω)hm(r,ω;t),
E(r,t)=12πmdωAm(z,ω)em(r,ω;t),
hm(r,ω;t)=hm(r,z,ω)Φm(ω,z,t)em(r,ω;t)=em(r,z,ω)Φm(ω,z,t),
Am(z,ω)=Am*(z,ω);em(r,z,ω)=em*(r,z,ω),
Φm(ω,z,t)=ei(ωtBm(z,ω));Bm(z,ω)=0zdzβm(ω,z),
dr[em(r,z,ω)×hn*(r,z,ω)hm(r,z,ω)×en*(r,z,ω)n]z=Nm(ω,z)δmn.
×hm*(r,ω;t)=iωε0ε(r,ω)em*(r,ω;t)+δem*(r,ω;t),
×em*(r,ω;t)=iωμ0hm*(r,ω;t)+δhm*(r,ω;t),
δem(r,ω;t)=Φm(ω,z,t)z^×hm(r,z,ω)z,
δhm(r,ω;t)=Φm(ω,z,t)z^×em(r,z,ω)z.
em*(r,ω;t)·[δPt+Dt]=em*(r,ω;t)·×H,
μ0hm*(r,ω;t)·Ht=hm*(r,ω;t)·×E,
em*(r,ω;t)·[δPt+Dt]=·[H×em*(r,ω;t)]+H·×em*(r,ω;t),
μ0hm*(r,ω;t)·Ht=·[E×hm*(r,ω;t)]+E·×hm*(r,ω;t).
em*(r,ω;t)·[δPt+Dt]=·[H×em*(r,ω;t)]+H·[iωμ0hm*(r,ω;t)+δhm*(r,ω;t)],
μ0hm*(r,ω;t)·Ht=·[E×hm*(r,ω;t)]+E·[iωε0ε(r,ω)em*(r,ω;t)+δem*(r,ω;t)].
iωdrdtem*(r,ω;t)[δP+D]=zdrdt[H×em*(r,ω;t)]z+drdtH·[iωμ0hm*(r,ω;t)+δhm*(r,ω;t)],
iωμ0drdthm*(r,ω;t)·H=zdrdt[E×hm*(r,ω;t)]z+drdtE·[iωε0ε(r,ω)em*(r,ω;t)+δem*(r,ω;t)].
iωdrdtem*(r,ω;t)δP+iωε02πndrAn(z,ω)ε(r,ω)en(r,ω;t)·em*(r,ω;t)=2πzndr[An(z,ω)hn(r,ω;t)×em*(r,ω;t)]z+2πndr[iωμ0An(z,ω)hn(r,ω;t)·hm*(r,ω;t)+hn(r,ω;t)·δhm*(r,ω;t)],
iωμ02πndrAn(z,ω)hn(r,ω;t)·hm*(r,ω;t)=2πzndrAn(z,ω)[en(r,ω;t)×hm*(r,ω;t)]z+2πndrAn(z,ω)[iωε0ε(r,ω)en(r,ω;t)·em*(r,ω;t)+en(r,ω;t)·δem*(r,ω;t)].
iω2πdrdtem*(r,ω;t)δP=zAm(z,ω)Nm(ω,z)+nAn(z,ω)dr(hn(r,ω;t)·δhm*(r,ω;t)en(r,ω;t)·δem*(r,ω;t)).
Am(z,ω)dr[hm(r,ω;t)·δhm*(r,ω;t)em(r,ω;t)·δem*(r,ω;t)]=Am(z,ω)drz^·[em(r,z,ω)z×hm*(r,z,ω)hm(r,z,ω)z×em*(r,z,ω)]=12Am(z,ω)Nmz,
Nm(ω,z)Amz=iω2πdrdtem*(r,ω;t)δP12Am(z,ω)Nmz+nmAn(z,ω)ei(Bm(ω,z)Bn(ω,z))dr[en(r,z,ω)×hm(r,z,ω)z+hn(r,z,ω)×em(r,z,ω)z]z.
Nm(ω,z)Amz=iω2πdrdtem*(r,ω;t)δP12NmzAm(z,ω).
drE(r)2=1,
P(t)=12dr[E(r,t)×H*(r,t)H(r,t)×E*(r,t)]z=14πmndω1dω2Am(z,ω1)An*(z,ω2)×Φm(ω1,z,t)Φn*(ω2,z,t)dr[em(r,z,ω1)×hn*(r,z,ω2)hm(r,z,ω1)×en*(r,z,ω2)]z.
P(t)=mNm(z)A˜m(z,t)2;A˜m(z,t)=12π0dωeiωtA˜m(z,ω);A˜m(z,ω)=Am(z,ω)eiBm(ω,z).
dr[em(r,z,ω1)×hm*(r,z,ω2)hm(r,z,ω1)×em*(r,z,ω2)]z=dr[em(r,z,ω1)×hm(r,z,ω2)hm(r,z,ω1)×em(r,z,ω2)]z=Nm(ω1,z)+dr[em(r,z,ω1)×hm(r,z,ω1)ω1hm(r,z,ω1)×em(r,z,ω1)ω1]z(ω2ω1)+O((ω2ω1)2)=Nm(ω1,z)+12Nm(ω1,z)ω1(ω2ω1)+O((ω2ω1)2).
Ep=dtP(t)=12mdωA˜m(z,ω)2Nm(ω,z),
Pi(r)=jklEj(r,t)χ(3)(r)dtrijkl(tt)Ek(r,t)El(r,t).
rijklR(t)=a(t)δijδkl+12b(t)(δikδjl+δilδjk).
δP(r,t)=ε0χ(3)(r)E(r,t)dtR˜(r,tt)E(r,t)2.
iω2πdrdtem*(r,ω;t)·δP(r,t)=iωε02πnpqdω12A˜n(z,ω1)A˜p*(z,ω2)A˜q(z,ωω1+ω2)eiBm(ω,z)×drχ(3)(r)R˜(r,ωω1)em*(r,z,ω)·en(r,z,ω1)ep*(r,z,ω2)·eq(r,z,ωω1+ω2),
iω2πdrdtem*(r,ω;t)·δP(r,t)iωε02πnpq+dω12A˜n(z,ω1)A˜p*(z,ω2)A˜q(z,ωω1+ω2)eiBm(ω,z)×drχ(3)(r)[2R˜(r,ωω1)em*(r,z,ω)·en(r,z,ω1)ep*(r,z,ω2)·eq(r,z,ωω1+ω2)+R˜(r,ω+ω2)em*(r,z,ω)·en*(r,z,ω2)ep(r,z,ω1)·eq(r,z,ωω1+ω2)].
R˜(r,ω)=(1fR)+fRR(r,ω).
iω2πdrdtem*(r,ω;t)·δP(r,t)iωε02πnpq+dω12A˜n(z,ω1)A˜p*(z,ω2)A˜q(z,ωω1+ω2)eiBm(ω,z)×[(1fR)(2Kmnpq(1)R˜(r,ωω1)+Kmnpq(2))+2fRR(r,ωω1)Kmnpq(1)],
Kmnpq(1)=drχ(3)(r)em*(r,z,ω)·en(r,z,ω1)ep*(r,z,ω2)·eq(r,z,ωω1+ω2),
Kmnpq(2)=drχ(3)(r)em*(r,z,ω)·en*(r,z,ω2)ep(r,z,ω1)·eq(r,z,ωω1+ω2).
A(z,ω)z=iωε02πdω12A˜(z,ω1)A˜*(z,ω2)A˜(z,ωω1+ω2)eiB(ω,z)×[(1fR)(2K(1)R˜(r,ωω1)+K(2))+2fRR(r,ωω1)K(1)].
K(1)μ0χ(3)4nm2ε0[Aeff(z,ω)Aeff(z,ω1)Aeff(z,ω2)Aeff(z,ωω1+ω2)]1/4,
K(2)μ0χ(3)4nm2ε0[Aeff(z,ω)Aeff(z,ω1)Aeff(z,ω2)Aeff(z,ωω1+ω2)]1/4,
Aeff(z,ω)=μ0[Re(dre(r,z,ω)×h*(r,z,ω))]2nm2ε0mdre*(r,z,ω)4,
Aeff(z,ω)=μ0[Re(dre(r,z,ω)×h*(r,z,ω))]2nm2ε0mdre*(r,z,ω)·e*(r,z,ω)e(r,z,ω)·e(r,z,ω).
A(z,ω)z=iωμ0χ(3)4nm22πdω12{2[Aeff(z,ω)]1/4A¯(z,ω1)A¯*(z,ω2)A¯(z,ωω1+ω2)eiB(ω,z)[(1fR)+fRR(r,ωω1)]+[Aeff(z,ω)]1/4A¯(z,ω1)A¯*(z,ω2)A¯(z,ωω1+ω2)eiB(ω,z)(1fR)},
A¯(z,ω)=A˜(z,ω)[Aeff(z,ω)]1/4;A¯(z,ω)=A˜(z,ω)[Aeff(z,ω)]1/4.
A(z,ω)z=iω3μ0χ(3)4nm22πdω12[Aeff(z,ω)]1/4A¯(z,ω1)A¯*(z,ω2)A¯(z,ωω1+ω2)eiB(ω,z)×[(1fR)+23fRR(r,ωω1)]=iωcn22πdω12[Aeff(z,ω)]1/4A¯(z,ω1)A¯*(z,ω2)A¯(z,ωω1+ω2)eiB(ω,z)[(1fR)+23fRR(r,ωω1)]=iωcn˜22πdω12[Aeff(z,ω)]1/4A¯(z,ω1)A¯*(z,ω2)A¯(z,ωω1+ω2)eiB(ω,z)[(1f˜R)+f˜RR(r,ωω1)],
n2=3χ(3)4ε0cnm2;n˜2=n2(1fR3);f˜R=2fR3(1fR3).
dβdω=β1(ω);dβ1dω=β2(ω),

Metrics