Abstract

We derive a set of new unidirectional evolution equations for photonic nanowires, i.e. waveguides with sub-wavelength core diameter. Contrary to previous approaches, our formulation simultaneously takes into account both the vector nature of the electromagnetic field and the full variations of the effective modal profiles with wavelength. This leads to the discovery of new, previously unexplored nonlinear effects which have the potential to affect soliton propagation considerably. We specialize our theoretical considerations to the case of perfectly circular silica strands in air, and we support our analysis with detailed numerical simulations.

© 2009 Optical Society of America

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References

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  1. G. P. Agrawal, Nonlinear Fiber Optics, 4th ed. (Academic Press, San Diego, 2007).
  2. A. Taflove and S. C. Hagness, Computational Electrodynamics, 3rd ed. (Artech House, London, 2005).
  3. A. Hasegawa and Y. Kodama, "Nonlinear pulse propagation in a monomode dielectric guide," IEEE J. Quantum Electron. 23, 510-524 (1987).
    [CrossRef]
  4. F. Biancalana, D. V. Skryabin, and P. St. J. Russell, "Four-wave mixing instabilities in photonic crystal and tapered fibers," Phys. Rev. E 68, 046603 (2003).
    [CrossRef]
  5. J. P. Gordon, "Theory of the soliton self-frequency shift," Opt. Lett. 11, 662-664 (1986).
    [CrossRef] [PubMed]
  6. A. L. Gaeta, "Nonlinear propagation and continuum generation in microstructured optical fibers," Opt. Lett. 27, 924-926 (2002).
    [CrossRef]
  7. K. Shi, F. G. Omenetto, and Z. Liu, "Supercontinuum generation in an imaging fiber taper," Opt. Express 14, 12359-12364 (2006).
  8. P. St. J. Russell, "Photonic crystal fibers," Science 299, 358-362 (2003).
    [CrossRef] [PubMed]
  9. J. M. Dudley, G. Genty and S. Coen, "Supercontinuum generation in photonic crystal fibers," Rev. Mod. Phys. 78, 1135-1184 (2006).
    [CrossRef]
  10. N. Akhmediev and M. Karlsson, "Cherenkov radiation emitted by solitons in optical fibers," Phys. Rev. A 51, 2602-2607 (1995).
    [CrossRef] [PubMed]
  11. D. V. Skryabin, F. Luan, J. C. Knight, and P. St. J. Russell, "Soliton self-frequency shift cancellation in photonic crystal fibers," Science 301, 1705-1707 (2003).
    [CrossRef] [PubMed]
  12. F. Biancalana, D. V. Skryabin, and A. V. Yulin, "Theory of the soliton self-frequency shift compensation by the resonant radiation in photonic crystal fibers," Phys. Rev. E 70, 016615 (2004).
    [CrossRef]
  13. T. Brabec and F. Krausz, "Nonlinear optical pulse propagation in the single-cycle regime," Phys. Rev. Lett. 78, 3282-3285 (1997).
    [CrossRef]
  14. T. G. Euser, J. S. Y. Chen, M. Scharrer, P. St. J. Russell, N. J. Farrer and P. J. Sadler, "Quantitative broadband chemical sensing in air-suspended solid-core fibers," J. Appl. Phys. 103, 103108 (2008).
    [CrossRef]
  15. F. Benabid, F. Biancalana, P. S. Light, F. Couny, A. Luiten, P. J. Roberts, J. Peng, A. V. Sokolov, "Fourth-order dispersion mediated solitonic radiations in HC-PCF cladding," Opt. Lett. 33,2680-2682 (2008).
    [CrossRef] [PubMed]
  16. S. Afshar V. 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] [PubMed]
  17. M. D. Turner, T. M. Monro and S. Afshar V., "A full vectorial model for pulse propagation in emerging waveguides with sub-wavelength structures part II: Stimulated Raman Scattering," Opt. Express 17, 11565-11581 (2009).
    [CrossRef] [PubMed]
  18. M. A. Foster, A. C. Turner, M. Lipson, and A. L. Gaeta, "Nonlinear optics in photonic nanowires," Opt. Express 16, 1300-1320 (2008).
    [CrossRef] [PubMed]
  19. M. A. Foster, K. D. Moll, and A. L. Gaeta, "Optimal waveguide dimensions for nonlinear interactions," Opt. Express 12,2880-2887 (2004).
    [CrossRef] [PubMed]
  20. A. Zheltikov, "Gaussian-mode analysis of waveguide-enhanced Kerr-type nonlinearity of optical fibers and photonic wires," J. Opt. Soc. Am. B 22,1100-1104 (2005).
    [CrossRef]
  21. S. Afshar V., W. Zhang and T. M. Monro, "Experimental confirmation of a generalized definition of the effective nonlinear coefficient in emerging waveguides with sub-wavelength structures," CThBB6, CLEO Conference, Baltimore, USA (2009).
  22. P. V. Mamyshev and S. V. Chernikov, "Ultrashort-pulse propagation in optical fibers," Opt. Lett. 15,1076-1078 (1990).
    [CrossRef] [PubMed]
  23. M. Kolesik and J. V. Moloney, "Nonlinear optical pulse propagation simulation: From Maxwell’s to unidirectional equations," Phys. Rev. E 70, 036604 (2004).
    [CrossRef]
  24. J. Lægsgaard, "Mode profile dispersion in the generalised nonlinear Schrdinger equation," Opt. Express 15,16110-16123 (2007).
    [CrossRef] [PubMed]
  25. J. D. Jackson, Classical Electrodynamics (Wiley & Sons, New York, 1998).
  26. A. W. Snyder and J. Love, Optical Waveguide Theory (Kluwer, Boston, 1983).
  27. B. Kibler, J. M. Dudley, S. Coen, "Supercontinuum generation and nonlinear pulse propagation in photonic crystal fiber: influence of the frequency-dependent effective mode area," Appl. Phys. B 81, 337-342 (2005).
    [CrossRef]
  28. R. W. Boyd, Nonlinear Optics, 3rd ed. (Academic Press, San Diego, 2008).
  29. R. W. Hellwarth, "Third-order optical susceptibilities of liquids ans solids," Prog. Quantum Electron. 5,1-68 (1977).
    [CrossRef]
  30. F. Poletti and P. Horak, "Description of ultrashort pulse propagation in multimode optical fibers," J. Opt. Soc. Am. B 25, 1645-1654 (2008).
    [CrossRef]
  31. J. Santhanam and G. P. Agrawal, "Raman-induced spectral shifts in optical fibers: general theory based on the moment method," Opt. Commun. 222, 413-420 (2003).
    [CrossRef]
  32. F. Biancalana and Tr. X. Tran, in preparation.

2009 (2)

2008 (4)

2007 (1)

2006 (2)

K. Shi, F. G. Omenetto, and Z. Liu, "Supercontinuum generation in an imaging fiber taper," Opt. Express 14, 12359-12364 (2006).

J. M. Dudley, G. Genty and S. Coen, "Supercontinuum generation in photonic crystal fibers," Rev. Mod. Phys. 78, 1135-1184 (2006).
[CrossRef]

2005 (2)

B. Kibler, J. M. Dudley, S. Coen, "Supercontinuum generation and nonlinear pulse propagation in photonic crystal fiber: influence of the frequency-dependent effective mode area," Appl. Phys. B 81, 337-342 (2005).
[CrossRef]

A. Zheltikov, "Gaussian-mode analysis of waveguide-enhanced Kerr-type nonlinearity of optical fibers and photonic wires," J. Opt. Soc. Am. B 22,1100-1104 (2005).
[CrossRef]

2004 (3)

M. Kolesik and J. V. Moloney, "Nonlinear optical pulse propagation simulation: From Maxwell’s to unidirectional equations," Phys. Rev. E 70, 036604 (2004).
[CrossRef]

F. Biancalana, D. V. Skryabin, and A. V. Yulin, "Theory of the soliton self-frequency shift compensation by the resonant radiation in photonic crystal fibers," Phys. Rev. E 70, 016615 (2004).
[CrossRef]

M. A. Foster, K. D. Moll, and A. L. Gaeta, "Optimal waveguide dimensions for nonlinear interactions," Opt. Express 12,2880-2887 (2004).
[CrossRef] [PubMed]

2003 (4)

P. St. J. Russell, "Photonic crystal fibers," Science 299, 358-362 (2003).
[CrossRef] [PubMed]

F. Biancalana, D. V. Skryabin, and P. St. J. Russell, "Four-wave mixing instabilities in photonic crystal and tapered fibers," Phys. Rev. E 68, 046603 (2003).
[CrossRef]

D. V. Skryabin, F. Luan, J. C. Knight, and P. St. J. Russell, "Soliton self-frequency shift cancellation in photonic crystal fibers," Science 301, 1705-1707 (2003).
[CrossRef] [PubMed]

J. Santhanam and G. P. Agrawal, "Raman-induced spectral shifts in optical fibers: general theory based on the moment method," Opt. Commun. 222, 413-420 (2003).
[CrossRef]

2002 (1)

1997 (1)

T. Brabec and F. Krausz, "Nonlinear optical pulse propagation in the single-cycle regime," Phys. Rev. Lett. 78, 3282-3285 (1997).
[CrossRef]

1995 (1)

N. Akhmediev and M. Karlsson, "Cherenkov radiation emitted by solitons in optical fibers," Phys. Rev. A 51, 2602-2607 (1995).
[CrossRef] [PubMed]

1990 (1)

1987 (1)

A. Hasegawa and Y. Kodama, "Nonlinear pulse propagation in a monomode dielectric guide," IEEE J. Quantum Electron. 23, 510-524 (1987).
[CrossRef]

1986 (1)

1977 (1)

R. W. Hellwarth, "Third-order optical susceptibilities of liquids ans solids," Prog. Quantum Electron. 5,1-68 (1977).
[CrossRef]

Afshar V., S.

Agrawal, G. P.

J. Santhanam and G. P. Agrawal, "Raman-induced spectral shifts in optical fibers: general theory based on the moment method," Opt. Commun. 222, 413-420 (2003).
[CrossRef]

Akhmediev, N.

N. Akhmediev and M. Karlsson, "Cherenkov radiation emitted by solitons in optical fibers," Phys. Rev. A 51, 2602-2607 (1995).
[CrossRef] [PubMed]

Benabid, F.

Biancalana, F.

F. Benabid, F. Biancalana, P. S. Light, F. Couny, A. Luiten, P. J. Roberts, J. Peng, A. V. Sokolov, "Fourth-order dispersion mediated solitonic radiations in HC-PCF cladding," Opt. Lett. 33,2680-2682 (2008).
[CrossRef] [PubMed]

F. Biancalana, D. V. Skryabin, and A. V. Yulin, "Theory of the soliton self-frequency shift compensation by the resonant radiation in photonic crystal fibers," Phys. Rev. E 70, 016615 (2004).
[CrossRef]

F. Biancalana, D. V. Skryabin, and P. St. J. Russell, "Four-wave mixing instabilities in photonic crystal and tapered fibers," Phys. Rev. E 68, 046603 (2003).
[CrossRef]

F. Biancalana and Tr. X. Tran, in preparation.

Brabec, T.

T. Brabec and F. Krausz, "Nonlinear optical pulse propagation in the single-cycle regime," Phys. Rev. Lett. 78, 3282-3285 (1997).
[CrossRef]

Chen, J. S. Y.

T. G. Euser, J. S. Y. Chen, M. Scharrer, P. St. J. Russell, N. J. Farrer and P. J. Sadler, "Quantitative broadband chemical sensing in air-suspended solid-core fibers," J. Appl. Phys. 103, 103108 (2008).
[CrossRef]

Chernikov, S. V.

Coen, S.

J. M. Dudley, G. Genty and S. Coen, "Supercontinuum generation in photonic crystal fibers," Rev. Mod. Phys. 78, 1135-1184 (2006).
[CrossRef]

B. Kibler, J. M. Dudley, S. Coen, "Supercontinuum generation and nonlinear pulse propagation in photonic crystal fiber: influence of the frequency-dependent effective mode area," Appl. Phys. B 81, 337-342 (2005).
[CrossRef]

Couny, F.

Dudley, J. M.

J. M. Dudley, G. Genty and S. Coen, "Supercontinuum generation in photonic crystal fibers," Rev. Mod. Phys. 78, 1135-1184 (2006).
[CrossRef]

B. Kibler, J. M. Dudley, S. Coen, "Supercontinuum generation and nonlinear pulse propagation in photonic crystal fiber: influence of the frequency-dependent effective mode area," Appl. Phys. B 81, 337-342 (2005).
[CrossRef]

Euser, T. G.

T. G. Euser, J. S. Y. Chen, M. Scharrer, P. St. J. Russell, N. J. Farrer and P. J. Sadler, "Quantitative broadband chemical sensing in air-suspended solid-core fibers," J. Appl. Phys. 103, 103108 (2008).
[CrossRef]

Farrer, N. J.

T. G. Euser, J. S. Y. Chen, M. Scharrer, P. St. J. Russell, N. J. Farrer and P. J. Sadler, "Quantitative broadband chemical sensing in air-suspended solid-core fibers," J. Appl. Phys. 103, 103108 (2008).
[CrossRef]

Foster, M. A.

Gaeta, A. L.

Genty, G.

J. M. Dudley, G. Genty and S. Coen, "Supercontinuum generation in photonic crystal fibers," Rev. Mod. Phys. 78, 1135-1184 (2006).
[CrossRef]

Gordon, J. P.

Hasegawa, A.

A. Hasegawa and Y. Kodama, "Nonlinear pulse propagation in a monomode dielectric guide," IEEE J. Quantum Electron. 23, 510-524 (1987).
[CrossRef]

Hellwarth, R. W.

R. W. Hellwarth, "Third-order optical susceptibilities of liquids ans solids," Prog. Quantum Electron. 5,1-68 (1977).
[CrossRef]

Horak, P.

Karlsson, M.

N. Akhmediev and M. Karlsson, "Cherenkov radiation emitted by solitons in optical fibers," Phys. Rev. A 51, 2602-2607 (1995).
[CrossRef] [PubMed]

Kibler, B.

B. Kibler, J. M. Dudley, S. Coen, "Supercontinuum generation and nonlinear pulse propagation in photonic crystal fiber: influence of the frequency-dependent effective mode area," Appl. Phys. B 81, 337-342 (2005).
[CrossRef]

Knight, J. C.

D. V. Skryabin, F. Luan, J. C. Knight, and P. St. J. Russell, "Soliton self-frequency shift cancellation in photonic crystal fibers," Science 301, 1705-1707 (2003).
[CrossRef] [PubMed]

Kodama, Y.

A. Hasegawa and Y. Kodama, "Nonlinear pulse propagation in a monomode dielectric guide," IEEE J. Quantum Electron. 23, 510-524 (1987).
[CrossRef]

Kolesik, M.

M. Kolesik and J. V. Moloney, "Nonlinear optical pulse propagation simulation: From Maxwell’s to unidirectional equations," Phys. Rev. E 70, 036604 (2004).
[CrossRef]

Krausz, F.

T. Brabec and F. Krausz, "Nonlinear optical pulse propagation in the single-cycle regime," Phys. Rev. Lett. 78, 3282-3285 (1997).
[CrossRef]

Lægsgaard, J.

Light, P. S.

Lipson, M.

Liu, Z.

Luan, F.

D. V. Skryabin, F. Luan, J. C. Knight, and P. St. J. Russell, "Soliton self-frequency shift cancellation in photonic crystal fibers," Science 301, 1705-1707 (2003).
[CrossRef] [PubMed]

Luiten, A.

Mamyshev, P. V.

Moll, K. D.

Moloney, J. V.

M. Kolesik and J. V. Moloney, "Nonlinear optical pulse propagation simulation: From Maxwell’s to unidirectional equations," Phys. Rev. E 70, 036604 (2004).
[CrossRef]

Monro, T. M.

Omenetto, F. G.

Peng, J.

Poletti, F.

Roberts, P. J.

Russell, P. St. J.

T. G. Euser, J. S. Y. Chen, M. Scharrer, P. St. J. Russell, N. J. Farrer and P. J. Sadler, "Quantitative broadband chemical sensing in air-suspended solid-core fibers," J. Appl. Phys. 103, 103108 (2008).
[CrossRef]

D. V. Skryabin, F. Luan, J. C. Knight, and P. St. J. Russell, "Soliton self-frequency shift cancellation in photonic crystal fibers," Science 301, 1705-1707 (2003).
[CrossRef] [PubMed]

P. St. J. Russell, "Photonic crystal fibers," Science 299, 358-362 (2003).
[CrossRef] [PubMed]

F. Biancalana, D. V. Skryabin, and P. St. J. Russell, "Four-wave mixing instabilities in photonic crystal and tapered fibers," Phys. Rev. E 68, 046603 (2003).
[CrossRef]

Sadler, P. J.

T. G. Euser, J. S. Y. Chen, M. Scharrer, P. St. J. Russell, N. J. Farrer and P. J. Sadler, "Quantitative broadband chemical sensing in air-suspended solid-core fibers," J. Appl. Phys. 103, 103108 (2008).
[CrossRef]

Santhanam, J.

J. Santhanam and G. P. Agrawal, "Raman-induced spectral shifts in optical fibers: general theory based on the moment method," Opt. Commun. 222, 413-420 (2003).
[CrossRef]

Scharrer, M.

T. G. Euser, J. S. Y. Chen, M. Scharrer, P. St. J. Russell, N. J. Farrer and P. J. Sadler, "Quantitative broadband chemical sensing in air-suspended solid-core fibers," J. Appl. Phys. 103, 103108 (2008).
[CrossRef]

Shi, K.

Skryabin, D. V.

F. Biancalana, D. V. Skryabin, and A. V. Yulin, "Theory of the soliton self-frequency shift compensation by the resonant radiation in photonic crystal fibers," Phys. Rev. E 70, 016615 (2004).
[CrossRef]

F. Biancalana, D. V. Skryabin, and P. St. J. Russell, "Four-wave mixing instabilities in photonic crystal and tapered fibers," Phys. Rev. E 68, 046603 (2003).
[CrossRef]

D. V. Skryabin, F. Luan, J. C. Knight, and P. St. J. Russell, "Soliton self-frequency shift cancellation in photonic crystal fibers," Science 301, 1705-1707 (2003).
[CrossRef] [PubMed]

Sokolov, A. V.

Turner, A. C.

Turner, M. D.

Yulin, A. V.

F. Biancalana, D. V. Skryabin, and A. V. Yulin, "Theory of the soliton self-frequency shift compensation by the resonant radiation in photonic crystal fibers," Phys. Rev. E 70, 016615 (2004).
[CrossRef]

Zheltikov, A.

Appl. Phys. B (1)

B. Kibler, J. M. Dudley, S. Coen, "Supercontinuum generation and nonlinear pulse propagation in photonic crystal fiber: influence of the frequency-dependent effective mode area," Appl. Phys. B 81, 337-342 (2005).
[CrossRef]

IEEE J. Quantum Electron. (1)

A. Hasegawa and Y. Kodama, "Nonlinear pulse propagation in a monomode dielectric guide," IEEE J. Quantum Electron. 23, 510-524 (1987).
[CrossRef]

J. Appl. Phys. (1)

T. G. Euser, J. S. Y. Chen, M. Scharrer, P. St. J. Russell, N. J. Farrer and P. J. Sadler, "Quantitative broadband chemical sensing in air-suspended solid-core fibers," J. Appl. Phys. 103, 103108 (2008).
[CrossRef]

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

Opt. Commun. (1)

J. Santhanam and G. P. Agrawal, "Raman-induced spectral shifts in optical fibers: general theory based on the moment method," Opt. Commun. 222, 413-420 (2003).
[CrossRef]

Opt. Express (6)

Opt. Lett. (4)

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] [PubMed]

Phys. Rev. E (3)

F. Biancalana, D. V. Skryabin, and A. V. Yulin, "Theory of the soliton self-frequency shift compensation by the resonant radiation in photonic crystal fibers," Phys. Rev. E 70, 016615 (2004).
[CrossRef]

F. Biancalana, D. V. Skryabin, and P. St. J. Russell, "Four-wave mixing instabilities in photonic crystal and tapered fibers," Phys. Rev. E 68, 046603 (2003).
[CrossRef]

M. Kolesik and J. V. Moloney, "Nonlinear optical pulse propagation simulation: From Maxwell’s to unidirectional equations," Phys. Rev. E 70, 036604 (2004).
[CrossRef]

Phys. Rev. Lett. (1)

T. Brabec and F. Krausz, "Nonlinear optical pulse propagation in the single-cycle regime," Phys. Rev. Lett. 78, 3282-3285 (1997).
[CrossRef]

Prog. Quantum Electron. (1)

R. W. Hellwarth, "Third-order optical susceptibilities of liquids ans solids," Prog. Quantum Electron. 5,1-68 (1977).
[CrossRef]

Rev. Mod. Phys. (1)

J. M. Dudley, G. Genty and S. Coen, "Supercontinuum generation in photonic crystal fibers," Rev. Mod. Phys. 78, 1135-1184 (2006).
[CrossRef]

Science (2)

D. V. Skryabin, F. Luan, J. C. Knight, and P. St. J. Russell, "Soliton self-frequency shift cancellation in photonic crystal fibers," Science 301, 1705-1707 (2003).
[CrossRef] [PubMed]

P. St. J. Russell, "Photonic crystal fibers," Science 299, 358-362 (2003).
[CrossRef] [PubMed]

Tran (1)

F. Biancalana and Tr. X. Tran, in preparation.

Other (6)

R. W. Boyd, Nonlinear Optics, 3rd ed. (Academic Press, San Diego, 2008).

J. D. Jackson, Classical Electrodynamics (Wiley & Sons, New York, 1998).

A. W. Snyder and J. Love, Optical Waveguide Theory (Kluwer, Boston, 1983).

S. Afshar V., W. Zhang and T. M. Monro, "Experimental confirmation of a generalized definition of the effective nonlinear coefficient in emerging waveguides with sub-wavelength structures," CThBB6, CLEO Conference, Baltimore, USA (2009).

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

A. Taflove and S. C. Hagness, Computational Electrodynamics, 3rd ed. (Artech House, London, 2005).

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Figures (5)

Fig. 1.
Fig. 1.

(Color online) (a,b) Respectively transverse and longitudinal components of electric field in a silica strand in air of diameter d=2 µm, for a pump wavelength λ=1 µm. (c,d) The same as (a,b), but for diameter d=0.6 µm. The silica core is indicated with a dashed black line. Note the expulsion of the total field from the core towards the low-index medium. Also, the magnitude of the longitudinal component in case (c,d) is more than 30% of the transverse component.

Fig. 2.
Fig. 2.

(Color online) (a) Plot of the dimensionless coefficient rγ 1/γ 010000000 as a function of wavelength, for silica strands in air of diameters d=0.5,0.6,0.8,1,1.6 and 4 µm in the fundamental mode of one polarization (m=1). This function is predominantly positive in a broad range of wavelengths. Coefficient 4r/(ω 0 t 0) quantifies the relative importance between the new nonlinear term T3 in Eq. (28) and the Kerr effect term T1. (b) Plot of the dimensionless coefficient α S ≡4Γ1000/[Γ0000 ω 0 TR ]=4 γ1/[γ 0 ω 0 TR ] as a function of wavelength. α S directly quantifies the relative importance between the new nonlinear term T3 in Eq. (28) and the Raman effect term T2. Both r and α S tend to zero at all wavelengths when progressively increasing the core diameter.

Fig. 3.
Fig. 3.

(Color online) (a) Plots of calculated values of α 1≡4γ 1/(ω 0 t 0) in [m-1W-1] describing the dispersion of the first-order nonlinear coefficient as a function of wavelength, for silica strands in air of diameters d=0.5,0.6,0.8,1,1.6 and 4 µm in the fundamental mode of one polarization (m=1). Pulse duration is t 0=100 fs. For a given diameter d, the largest value of |α 1| is located at approximately λCd, with C≃1.6. (b) Plots of calculated GVD for the same diameters as in (a). Red dashed line indicates the GVD for bulk silica. A comparison between (a) and (b) shows that, for silica strands, the maximum of |α 1| is always located in the normal GVD region.

Fig. 4.
Fig. 4.

(Color online) Simulation of light propagation in a photonic nanowire of diameter d=0.6 µm, for only one polarization state (m=1). Input pulse duration is t 0=100 fs, and input pulse power is always N=7. (a) Output spectra of the simulated Eq. (19) in normal dispersion regime after a propagation length of ξ=1, by using different truncations of the Taylor series contained into Eq. (28). Input pulse is centered at λ 0=1.2 µm (black solid line). Blue solid line indicates the solution of Eq. (19) when truncating the sum up to the first-order in the Taylor expansion. Green dashed line gives the same as the blue solid line, but when the sum in Eq. (19) is truncated at the zero-th order of expansion, i.e. when neglecting the influence of the new first-order nonlinear terms described in section 4. (b) Same as in (a), but for a pump wavelength λ 0≃0.8 µm. (c) Same as (a), but pumping at a wavelength λ 0=0.55 µm in the anomalous dispersion, for ξ=1. (d) Same as (c), but for ξ=1.4.

Fig. 5.
Fig. 5.

(Color online) (a) Final spectrum of a linearly polarized input pulse (95% on the m=1 axis and 5% along the m=2 axis), as obtained by simulating Eq. (19) truncated at the first-order of the Taylor series. All other parameters for the pulse and the fiber are the same as in Fig. 4(c). Black solid and red dashed lines are respectively the polarization components of the input pulse along m=1 and m=2 respectively. Blue solid line is output pulse along m=1, while green dashed line is along m=2. (b) Time domain picture of the propagation shown in (a). The formation of several vector solitons of different amplitudes is observed, each of them subjected to RSFS according to their individual intensities.

Equations (40)

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

×H=1ctD+1ctPNL
×E=1ctH
·H=0,·D=0,
Em(r,t,ω)=em(r,ω)Nm(ω) eiβm(ω)ziωt
Hm(r,t,ω)=hm(r,ω)Nm(ω) eiβm(ω)ziωt ,
E(r,t)=12π m d ω Am (z,ω) em(r,ω)Nm(ω) eiβm(ω)ziωt
H(r,t)=12π m d ω Am (z,ω) hm(r,ω)Nm(ω) eiβm(ω)ziωt ,
12[ej(r,ω)×hk*(r,ω)]·ẑdr=12[ek*(r,ω)×hj(r,ω)]·ẑdr=δjkNj(ω)
12[êj(r,ω)×ĥk*(r,ω)]·ẑdr=δjk .
Sm(r,t,ω)=12 ε0 c [Em(r,t,ω)×H*(r,t)+E*(r,t)×Hm(r,t,ω)] .
·Sm(ω)=12 ε0 {iωH*·Hm(ω)Em(ω)·tD*Em(ω)·tPNL*(r,t)Hm(ω)·tH*+iωε(ω)E*·Em(ω)}
zAm(z,ω)=iω4ceiβm(ω)zêm*(r,ω)·PNL(r,ω)dr.
PNL=14 χxxxx(3) (r) {E(t)R(tt1)[E(t1)·E*(t1)]dt1+R(tt1)E(t1)[E(t)·E*(t1)]dt1+
+R(tt1)E*(t1)[E(t)·E(t1)]dt1} ,
R(t)=(1θ)δ(t)+θΘ(t)h(t),
zAm(ω)=iω4c141(2π)2χxxxx(3)dr(kdldωdωR(ωω)Ak(ω)Al(ω)Ad*(ω+ωω)×
ei[βk(ω)+βl(ω)βd(ω+ωω)βm(ω)]z [[êm*(ω)·êk(ω)][êl(ω)·êd*(ω+ωω)]+
+[êm*(ω)·êl(ω)][êk(ω)·êd*(ω+ωω)]+[êm*(ω)·êd*(ω+ωω)][êk(ω)·êl(ω)]) .
êm(r,ω)=j01j!fm(j)(r,ω0)(ωω0ω0)j=j01j![ω0jjêm(r,ω)ωj]ω=ω0(Δωω0)j ,
Γmkldjhpv(ω0)116 d r χxxxx(3)(r)j!h!p!v!×
{[fm*(j)(ω0)·fk(h)(ω0)][fl(p)(ω0)·fd*(v)(ω0)]+[fm*(j)(ω0)·fl(p)(ω0)][fk(h)(ω0)·fd*(v)(ω0)]+
[fm*(j)(ω0)·fd*(v)(ω0)][fk(h)(ω0)·fl(p)(ω0)]}
Am(ω)Qm(Δω)ei[βm(ω0)βm(ω)]zeiω0t .
izQm+D̂m(it)Qm+k0kldjhpvΓmkldjhpv[Gj(t)Φkldhpv(t)]=0 ,
Gj(Δω)(1+Δωω0) (Δωω0)j
Φkldhpv(t)[(it)hQk(t)]{R(t)([(it)pQl(t)][(it)vQd*(t)])}ω0h+p+v ,
izQm+D̂(it)Qm+k0kldΓmkld0000[G0Φkld000]=0.
izQm+D̂(it)Qm+k0Γmmmm0000[G0{Qm(RQm2)}]+k0Γmnmm0000[G0{Qn(RQm2)}]+
k0Γmmnm0000[G0{Qm(RQnQm*)}]+k0Γmmmn0000[G0{Qm(RQmQn*)}]+
k0Γmnnn0000[G0{Qn(RQn2)}]+k0Γmmnn0000[G0{Qm(RQn2)}]+
k0Γmnmn0000[G0{Qn(RQmQn*)}]+k0Γmnnm0000[G0{Qn(RQnQm*)}]0.
izQ+D̂(it)Q+k0Γ0000Φ000+iω0k0Γ0000tΦ000+k0Γ1000[iω0tΦ000+Φ100+Φ010+Φ001]0 .
izQ+D̂(it)Q+γ0[1+iω0(1+2γ1γ0)t]Φ000+2γ1Φ0010 ,
Φ000=Q(z,t)+R(t)Q(z,tt)2dtQ(z,t)+R(t){Q(z,t)2ttQ(z,t)2}dt=
=Q2QTRQtQ2,
Φ001=iω0Q(z,t)+R(tt)Q(z,t)tQ*(z,t)dt=
iω0Q(z,t)+R(t)Q(z,tt)[Q*(z,τ)τ]τ=ttdt
iω0Q(z,t)+R(t)[Q(z,t)ttQ(t)][Q*(t)tt'2Q*(t)t2]dt
iω0[Q2tQ*TRQ2t2Q*TRQtQ2+TRQ(tQ)(t2Q*)],
izQ+D̂(it)Q+γ0[(1+iω0tT0)Q2QT1TRQtQ2T2]+4iγ1ω0Q2tQT30 .

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