Abstract

Description of pulse propagation in waveguides with subwavelength features and high refractive index contrasts requires an accurate account of the dispersion of nonlinearity due to the considerable mode profile variation with the wavelength. The corresponding model derived from asymptotic expansion of Maxwell equations contains a complicated network of interactions between different harmonics of the pulse, and therefore is not convenient for analysis. We demonstrate that this model can be reduced to the generalized nonlinear Schrödinger-type pulse propagation equation under the assumption of factorization of the four-frequency dependence of nonlinear coefficients. We analyze two different semiconductor waveguide geometries and find that the factorization works reasonably well within large wavelength windows. This allows us to utilize the pulse propagation equation for the description of a broadband signal evolution. We study the mechanism of modulational instability induced by the dispersion of nonlinearity and find that the power threshold predicted by the simple model with three interacting harmonics is effectively removed when using pulses, while the efficiency of this process grows for shorter pulse durations. Also, we identify the effects of geometrical and material dispersion of nonlinearity on spectral broadening of short pulses in semiconductor waveguides.

© 2013 Optical Society of America

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    [CrossRef]
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    [CrossRef]
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2012 (1)

2011 (4)

A. Marini, R. Hartley, A. V. Gorbach, and D. V. Skryabin, “Surface-induced nonlinearity enhancement in subwavelength rod waveguides,” Phys. Rev. A 84, 063839 (2011).
[CrossRef]

S. Amiranashvili, U. Bandelow, and N. Akhmediev, “Dispersion of nonlinear group velocity determines shortest envelope solitons,” Phys. Rev. A 84, 043834 (2011).
[CrossRef]

D. V. Skryabin, A. V. Gorbach, and A. Marini, “Surface-induced nonlinearity enhancement of TM modes in planar subwavelength waveguides,” J. Opt. Soc. Am. B 28, 109–114 (2011).
[CrossRef]

A. V. Gorbach, X. Zhao, and D. V. Skryabin, “Dispersion of nonlinearity and modulation instability in subwavelength semiconductor waveguides,” Opt. Express 19, 9345–9351 (2011).
[CrossRef]

2010 (3)

J. Leuthold, C. Koos, and W. Freude, “Nonlinear silicon photonics,” Nat. Photonics 4, 535–544 (2010).
[CrossRef]

D. V. Skryabin and A. V. Gorbach, “Looking at a soliton through the prism of optical supercontinuum,” Rev. Mod. Phys. 82, 1287–1299 (2010).
[CrossRef]

A. V. Gorbach, W. Ding, O. K. Staines, C. E. de Nobriga, G. D. Hobbs, W. J. Wadsworth, J. C. Knight, D. V. Skryabin, A. Samarelli, M. Sorel, and R. M. De La Rue, “Spatiotemporal nonlinear optics in arrays of subwavelength waveguides,” Phys. Rev. A 82, 041802 (2010).
[CrossRef]

2009 (6)

2008 (1)

2007 (5)

2006 (1)

J. Gomez-Gardeñes, B. A. Malomed, L. M. Flora, and A. R. Bishop, “Solitons in the Salerno model with competing nonlinearities,” Phys. Rev. E 73, 036608 (2006).
[CrossRef]

2004 (1)

P. Kinsler and G. H. C. New, “Few-cycle soliton propagation,” Phys. Rev. A 69, 013805 (2004).
[CrossRef]

Afshar V., S.

Agrawal, G. P.

Q. Lin, J. Zhang, G. Piredda, R. W. Boyd, P. M. Fauchet, and G. P. Agrawal, “Dispersion of silicon nonlinearities in the near infrared region,” Appl. Phys. Lett. 91, 21111 (2007).
[CrossRef]

Q. Lin, O. J. Painter, and G. P. Agrawal, “Nonlinear optical phenomena in silicon waveguides: modeling and applications,” Opt. Express 15, 16604–16644 (2007).
[CrossRef]

G. P. Agrawal, Nonlinear Fiber Optics, 3rd ed. (Academic, 2001).

Aitchison, J. S.

Akhmediev, N.

S. Amiranashvili, U. Bandelow, and N. Akhmediev, “Dispersion of nonlinear group velocity determines shortest envelope solitons,” Phys. Rev. A 84, 043834 (2011).
[CrossRef]

Amiranashvili, S.

S. Amiranashvili, U. Bandelow, and N. Akhmediev, “Dispersion of nonlinear group velocity determines shortest envelope solitons,” Phys. Rev. A 84, 043834 (2011).
[CrossRef]

Bandelow, U.

S. Amiranashvili, U. Bandelow, and N. Akhmediev, “Dispersion of nonlinear group velocity determines shortest envelope solitons,” Phys. Rev. A 84, 043834 (2011).
[CrossRef]

Biancalana, F.

Bishop, A. R.

J. Gomez-Gardeñes, B. A. Malomed, L. M. Flora, and A. R. Bishop, “Solitons in the Salerno model with competing nonlinearities,” Phys. Rev. E 73, 036608 (2006).
[CrossRef]

Boyd, R. W.

Q. Lin, J. Zhang, G. Piredda, R. W. Boyd, P. M. Fauchet, and G. P. Agrawal, “Dispersion of silicon nonlinearities in the near infrared region,” Appl. Phys. Lett. 91, 21111 (2007).
[CrossRef]

Chen, X.

Chou, C.-Y.

Dadap, J. I.

Dai, X.

Davoyan, A. R.

De La Rue, R. M.

A. V. Gorbach, W. Ding, O. K. Staines, C. E. de Nobriga, G. D. Hobbs, W. J. Wadsworth, J. C. Knight, D. V. Skryabin, A. Samarelli, M. Sorel, and R. M. De La Rue, “Spatiotemporal nonlinear optics in arrays of subwavelength waveguides,” Phys. Rev. A 82, 041802 (2010).
[CrossRef]

de Nobriga, C.

de Nobriga, C. E.

A. V. Gorbach, W. Ding, O. K. Staines, C. E. de Nobriga, G. D. Hobbs, W. J. Wadsworth, J. C. Knight, D. V. Skryabin, A. Samarelli, M. Sorel, and R. M. De La Rue, “Spatiotemporal nonlinear optics in arrays of subwavelength waveguides,” Phys. Rev. A 82, 041802 (2010).
[CrossRef]

Ding, W.

W. Ding, O. K. Staines, G. D. Hobbs, A. V. Gorbach, C. de Nobriga, W. J. Wadsworth, J. C. Knight, D. V. Skryabin, M. J. Strain, and M. Sorel, “Modulational instability in a silicon-on-insulator directional coupler: role of the coupling-induced group velocity dispersion,” Opt. Lett. 37, 668–670 (2012).
[CrossRef]

A. V. Gorbach, W. Ding, O. K. Staines, C. E. de Nobriga, G. D. Hobbs, W. J. Wadsworth, J. C. Knight, D. V. Skryabin, A. Samarelli, M. Sorel, and R. M. De La Rue, “Spatiotemporal nonlinear optics in arrays of subwavelength waveguides,” Phys. Rev. A 82, 041802 (2010).
[CrossRef]

Dulkeith, E.

Fan, D.

Fauchet, P. M.

Q. Lin, J. Zhang, G. Piredda, R. W. Boyd, P. M. Fauchet, and G. P. Agrawal, “Dispersion of silicon nonlinearities in the near infrared region,” Appl. Phys. Lett. 91, 21111 (2007).
[CrossRef]

Flora, L. M.

J. Gomez-Gardeñes, B. A. Malomed, L. M. Flora, and A. R. Bishop, “Solitons in the Salerno model with competing nonlinearities,” Phys. Rev. E 73, 036608 (2006).
[CrossRef]

Freude, W.

J. Leuthold, C. Koos, and W. Freude, “Nonlinear silicon photonics,” Nat. Photonics 4, 535–544 (2010).
[CrossRef]

Gomez-Gardeñes, J.

J. Gomez-Gardeñes, B. A. Malomed, L. M. Flora, and A. R. Bishop, “Solitons in the Salerno model with competing nonlinearities,” Phys. Rev. E 73, 036608 (2006).
[CrossRef]

Gorbach, A. V.

W. Ding, O. K. Staines, G. D. Hobbs, A. V. Gorbach, C. de Nobriga, W. J. Wadsworth, J. C. Knight, D. V. Skryabin, M. J. Strain, and M. Sorel, “Modulational instability in a silicon-on-insulator directional coupler: role of the coupling-induced group velocity dispersion,” Opt. Lett. 37, 668–670 (2012).
[CrossRef]

A. V. Gorbach, X. Zhao, and D. V. Skryabin, “Dispersion of nonlinearity and modulation instability in subwavelength semiconductor waveguides,” Opt. Express 19, 9345–9351 (2011).
[CrossRef]

D. V. Skryabin, A. V. Gorbach, and A. Marini, “Surface-induced nonlinearity enhancement of TM modes in planar subwavelength waveguides,” J. Opt. Soc. Am. B 28, 109–114 (2011).
[CrossRef]

A. Marini, R. Hartley, A. V. Gorbach, and D. V. Skryabin, “Surface-induced nonlinearity enhancement in subwavelength rod waveguides,” Phys. Rev. A 84, 063839 (2011).
[CrossRef]

D. V. Skryabin and A. V. Gorbach, “Looking at a soliton through the prism of optical supercontinuum,” Rev. Mod. Phys. 82, 1287–1299 (2010).
[CrossRef]

A. V. Gorbach, W. Ding, O. K. Staines, C. E. de Nobriga, G. D. Hobbs, W. J. Wadsworth, J. C. Knight, D. V. Skryabin, A. Samarelli, M. Sorel, and R. M. De La Rue, “Spatiotemporal nonlinear optics in arrays of subwavelength waveguides,” Phys. Rev. A 82, 041802 (2010).
[CrossRef]

A. V. Gorbach and D. V. Skryabin, “Spatial solitons in periodic nanostructures,” Phys. Rev. A 79, 053812 (2009).
[CrossRef]

Green, W. M.

Hartley, R.

A. Marini, R. Hartley, A. V. Gorbach, and D. V. Skryabin, “Surface-induced nonlinearity enhancement in subwavelength rod waveguides,” Phys. Rev. A 84, 063839 (2011).
[CrossRef]

Hobbs, G. D.

W. Ding, O. K. Staines, G. D. Hobbs, A. V. Gorbach, C. de Nobriga, W. J. Wadsworth, J. C. Knight, D. V. Skryabin, M. J. Strain, and M. Sorel, “Modulational instability in a silicon-on-insulator directional coupler: role of the coupling-induced group velocity dispersion,” Opt. Lett. 37, 668–670 (2012).
[CrossRef]

A. V. Gorbach, W. Ding, O. K. Staines, C. E. de Nobriga, G. D. Hobbs, W. J. Wadsworth, J. C. Knight, D. V. Skryabin, A. Samarelli, M. Sorel, and R. M. De La Rue, “Spatiotemporal nonlinear optics in arrays of subwavelength waveguides,” Phys. Rev. A 82, 041802 (2010).
[CrossRef]

Hsieh, I.-W.

Jiang, X.

Joannopoulos, J. D.

Jugessur, A.

Kinsler, P.

P. Kinsler and G. H. C. New, “Few-cycle soliton propagation,” Phys. Rev. A 69, 013805 (2004).
[CrossRef]

Kivshar, Y. S.

Knight, J. C.

W. Ding, O. K. Staines, G. D. Hobbs, A. V. Gorbach, C. de Nobriga, W. J. Wadsworth, J. C. Knight, D. V. Skryabin, M. J. Strain, and M. Sorel, “Modulational instability in a silicon-on-insulator directional coupler: role of the coupling-induced group velocity dispersion,” Opt. Lett. 37, 668–670 (2012).
[CrossRef]

A. V. Gorbach, W. Ding, O. K. Staines, C. E. de Nobriga, G. D. Hobbs, W. J. Wadsworth, J. C. Knight, D. V. Skryabin, A. Samarelli, M. Sorel, and R. M. De La Rue, “Spatiotemporal nonlinear optics in arrays of subwavelength waveguides,” Phys. Rev. A 82, 041802 (2010).
[CrossRef]

Koos, C.

J. Leuthold, C. Koos, and W. Freude, “Nonlinear silicon photonics,” Nat. Photonics 4, 535–544 (2010).
[CrossRef]

Leuthold, J.

J. Leuthold, C. Koos, and W. Freude, “Nonlinear silicon photonics,” Nat. Photonics 4, 535–544 (2010).
[CrossRef]

Lin, Q.

Q. Lin, J. Zhang, G. Piredda, R. W. Boyd, P. M. Fauchet, and G. P. Agrawal, “Dispersion of silicon nonlinearities in the near infrared region,” Appl. Phys. Lett. 91, 21111 (2007).
[CrossRef]

Q. Lin, O. J. Painter, and G. P. Agrawal, “Nonlinear optical phenomena in silicon waveguides: modeling and applications,” Opt. Express 15, 16604–16644 (2007).
[CrossRef]

Liu, X.

Maes, B.

Malomed, B. A.

J. Gomez-Gardeñes, B. A. Malomed, L. M. Flora, and A. R. Bishop, “Solitons in the Salerno model with competing nonlinearities,” Phys. Rev. E 73, 036608 (2006).
[CrossRef]

Marini, A.

A. Marini, R. Hartley, A. V. Gorbach, and D. V. Skryabin, “Surface-induced nonlinearity enhancement in subwavelength rod waveguides,” Phys. Rev. A 84, 063839 (2011).
[CrossRef]

D. V. Skryabin, A. V. Gorbach, and A. Marini, “Surface-induced nonlinearity enhancement of TM modes in planar subwavelength waveguides,” J. Opt. Soc. Am. B 28, 109–114 (2011).
[CrossRef]

Meier, J.

Mohammed, W. S.

Mojahedi, M.

Monro, T. M.

New, G. H. C.

P. Kinsler and G. H. C. New, “Few-cycle soliton propagation,” Phys. Rev. A 69, 013805 (2004).
[CrossRef]

Osgood, J.

Osgood, R. M.

Painter, O. J.

Panoiu, N. C.

Piredda, G.

Q. Lin, J. Zhang, G. Piredda, R. W. Boyd, P. M. Fauchet, and G. P. Agrawal, “Dispersion of silicon nonlinearities in the near infrared region,” Appl. Phys. Lett. 91, 21111 (2007).
[CrossRef]

Qian, L.

Samarelli, A.

A. V. Gorbach, W. Ding, O. K. Staines, C. E. de Nobriga, G. D. Hobbs, W. J. Wadsworth, J. C. Knight, D. V. Skryabin, A. Samarelli, M. Sorel, and R. M. De La Rue, “Spatiotemporal nonlinear optics in arrays of subwavelength waveguides,” Phys. Rev. A 82, 041802 (2010).
[CrossRef]

Shadrivov, I. V.

Skryabin, D. V.

W. Ding, O. K. Staines, G. D. Hobbs, A. V. Gorbach, C. de Nobriga, W. J. Wadsworth, J. C. Knight, D. V. Skryabin, M. J. Strain, and M. Sorel, “Modulational instability in a silicon-on-insulator directional coupler: role of the coupling-induced group velocity dispersion,” Opt. Lett. 37, 668–670 (2012).
[CrossRef]

A. V. Gorbach, X. Zhao, and D. V. Skryabin, “Dispersion of nonlinearity and modulation instability in subwavelength semiconductor waveguides,” Opt. Express 19, 9345–9351 (2011).
[CrossRef]

D. V. Skryabin, A. V. Gorbach, and A. Marini, “Surface-induced nonlinearity enhancement of TM modes in planar subwavelength waveguides,” J. Opt. Soc. Am. B 28, 109–114 (2011).
[CrossRef]

A. Marini, R. Hartley, A. V. Gorbach, and D. V. Skryabin, “Surface-induced nonlinearity enhancement in subwavelength rod waveguides,” Phys. Rev. A 84, 063839 (2011).
[CrossRef]

A. V. Gorbach, W. Ding, O. K. Staines, C. E. de Nobriga, G. D. Hobbs, W. J. Wadsworth, J. C. Knight, D. V. Skryabin, A. Samarelli, M. Sorel, and R. M. De La Rue, “Spatiotemporal nonlinear optics in arrays of subwavelength waveguides,” Phys. Rev. A 82, 041802 (2010).
[CrossRef]

D. V. Skryabin and A. V. Gorbach, “Looking at a soliton through the prism of optical supercontinuum,” Rev. Mod. Phys. 82, 1287–1299 (2010).
[CrossRef]

A. V. Gorbach and D. V. Skryabin, “Spatial solitons in periodic nanostructures,” Phys. Rev. A 79, 053812 (2009).
[CrossRef]

Soljacic, M.

Sorel, M.

W. Ding, O. K. Staines, G. D. Hobbs, A. V. Gorbach, C. de Nobriga, W. J. Wadsworth, J. C. Knight, D. V. Skryabin, M. J. Strain, and M. Sorel, “Modulational instability in a silicon-on-insulator directional coupler: role of the coupling-induced group velocity dispersion,” Opt. Lett. 37, 668–670 (2012).
[CrossRef]

A. V. Gorbach, W. Ding, O. K. Staines, C. E. de Nobriga, G. D. Hobbs, W. J. Wadsworth, J. C. Knight, D. V. Skryabin, A. Samarelli, M. Sorel, and R. M. De La Rue, “Spatiotemporal nonlinear optics in arrays of subwavelength waveguides,” Phys. Rev. A 82, 041802 (2010).
[CrossRef]

Staines, O. K.

W. Ding, O. K. Staines, G. D. Hobbs, A. V. Gorbach, C. de Nobriga, W. J. Wadsworth, J. C. Knight, D. V. Skryabin, M. J. Strain, and M. Sorel, “Modulational instability in a silicon-on-insulator directional coupler: role of the coupling-induced group velocity dispersion,” Opt. Lett. 37, 668–670 (2012).
[CrossRef]

A. V. Gorbach, W. Ding, O. K. Staines, C. E. de Nobriga, G. D. Hobbs, W. J. Wadsworth, J. C. Knight, D. V. Skryabin, A. Samarelli, M. Sorel, and R. M. De La Rue, “Spatiotemporal nonlinear optics in arrays of subwavelength waveguides,” Phys. Rev. A 82, 041802 (2010).
[CrossRef]

Strain, M. J.

Su, W.

Tang, Z.

Torner, L.

Tran, T. X.

Vlasov, Y. A.

Wadsworth, W. J.

W. Ding, O. K. Staines, G. D. Hobbs, A. V. Gorbach, C. de Nobriga, W. J. Wadsworth, J. C. Knight, D. V. Skryabin, M. J. Strain, and M. Sorel, “Modulational instability in a silicon-on-insulator directional coupler: role of the coupling-induced group velocity dispersion,” Opt. Lett. 37, 668–670 (2012).
[CrossRef]

A. V. Gorbach, W. Ding, O. K. Staines, C. E. de Nobriga, G. D. Hobbs, W. J. Wadsworth, J. C. Knight, D. V. Skryabin, A. Samarelli, M. Sorel, and R. M. De La Rue, “Spatiotemporal nonlinear optics in arrays of subwavelength waveguides,” Phys. Rev. A 82, 041802 (2010).
[CrossRef]

Wen, S.

Xia, F.

Xiang, Y.

Xu, Z.

Zhang, J.

Q. Lin, J. Zhang, G. Piredda, R. W. Boyd, P. M. Fauchet, and G. P. Agrawal, “Dispersion of silicon nonlinearities in the near infrared region,” Appl. Phys. Lett. 91, 21111 (2007).
[CrossRef]

Zhao, X.

Adv. Opt. Photon. (1)

Appl. Phys. Lett. (1)

Q. Lin, J. Zhang, G. Piredda, R. W. Boyd, P. M. Fauchet, and G. P. Agrawal, “Dispersion of silicon nonlinearities in the near infrared region,” Appl. Phys. Lett. 91, 21111 (2007).
[CrossRef]

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

Nat. Photonics (1)

J. Leuthold, C. Koos, and W. Freude, “Nonlinear silicon photonics,” Nat. Photonics 4, 535–544 (2010).
[CrossRef]

Opt. Express (7)

Opt. Lett. (3)

Phys. Rev. A (5)

A. V. Gorbach and D. V. Skryabin, “Spatial solitons in periodic nanostructures,” Phys. Rev. A 79, 053812 (2009).
[CrossRef]

A. Marini, R. Hartley, A. V. Gorbach, and D. V. Skryabin, “Surface-induced nonlinearity enhancement in subwavelength rod waveguides,” Phys. Rev. A 84, 063839 (2011).
[CrossRef]

P. Kinsler and G. H. C. New, “Few-cycle soliton propagation,” Phys. Rev. A 69, 013805 (2004).
[CrossRef]

S. Amiranashvili, U. Bandelow, and N. Akhmediev, “Dispersion of nonlinear group velocity determines shortest envelope solitons,” Phys. Rev. A 84, 043834 (2011).
[CrossRef]

A. V. Gorbach, W. Ding, O. K. Staines, C. E. de Nobriga, G. D. Hobbs, W. J. Wadsworth, J. C. Knight, D. V. Skryabin, A. Samarelli, M. Sorel, and R. M. De La Rue, “Spatiotemporal nonlinear optics in arrays of subwavelength waveguides,” Phys. Rev. A 82, 041802 (2010).
[CrossRef]

Phys. Rev. E (1)

J. Gomez-Gardeñes, B. A. Malomed, L. M. Flora, and A. R. Bishop, “Solitons in the Salerno model with competing nonlinearities,” Phys. Rev. E 73, 036608 (2006).
[CrossRef]

Rev. Mod. Phys. (1)

D. V. Skryabin and A. V. Gorbach, “Looking at a soliton through the prism of optical supercontinuum,” Rev. Mod. Phys. 82, 1287–1299 (2010).
[CrossRef]

Other (1)

G. P. Agrawal, Nonlinear Fiber Optics, 3rd ed. (Academic, 2001).

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Figures (13)

Fig. 1.
Fig. 1.

Geometry and profile of the dominant electric field component (ex) of the quasi-TE guided mode in subwavelength semiconductor waveguides: (a) AlGaAs waveguide, λ0=1.665μm and (b) SOI waveguide, λ0=2.3μm.

Fig. 2.
Fig. 2.

Relative factorization error Δ for the AlGaAs waveguide as in Fig. 1(a) and the fixed frequency ωn=2πc/λn, λn=1.665μm; see text for details.

Fig. 3.
Fig. 3.

Modified nonlinear coefficient γ˜ for AlGaAs waveguide; see Eq. (12). Dashed curve shows the conventional nonlinear coefficient γ=ωΓω for the same waveguide.

Fig. 4.
Fig. 4.

Dispersion of nonlinearity: nonlinear coefficients Γ entering the necessary condition for MI [see Eq. (26)] calculated for AlGaAs waveguide at (a) λp=1.65μm and (b) SOI waveguide at λp=2.2μm. Dashed curves show their factorized approximations, see Eq. (9).

Fig. 5.
Fig. 5.

MI in AlGaAs waveguide: (a) group velocity dispersion, conventional MI is possible in the range of anomalous GVD, β2<0. Shaded area indicates the region of unconventional MI due to the dispersion of nonlinearity. (b) Condition for γ<0, see Eq. (28). (c) γ calculated for λp=1.665μm from Eq. (27) by using Taylor expansion coefficients of γ˜(ω), solid curve, and γ(ω), dashed curve. Red/gray circles correspond to γ calculated from Eq. (26).

Fig. 6.
Fig. 6.

Dispersion of nonlineriaty induced MI in AlGaAs waveguide: δβ (a) and calculated gain (b), for the pump wavelength λp=1.665μm and pump power |Ap|2=150W.

Fig. 7.
Fig. 7.

MI development with 100 ps pulse excitation: (a) output spectra after z=0.4mm propagation distance. Thin solid curve corresponds to wavelength-independent nonlinearity, dotted red/gray curve to self-steepening only, and thick solid curve to full dispersion of nonlinearity. Pump peak power is 150 W. (b) Conversion efficiency as a function of the pump peak power. Full/open circles correspond to the full/self-steepening only dispersion of nonlinearity. Full squares correspond to the same as full circles, but for 10 ps pulse excitation. Vertical dashed line indicates the threshold power for the case of three interacting waves [13]. The signal peak power is fixed to 0.1 mW in all simulations.

Fig. 8.
Fig. 8.

MI development with 10 ps pulse excitation: (a) evolution of the spectrum with propagation distance and (b) output signal in time domain after the propagation distance of z=0.8mm. The inset zooms in the central region of the pulse, where formation of a periodic sequence of ultrashort pulses is clearly visible. The pump/seed peak power is 150W/0.1mW; other parameters are the same as in Fig. 7.

Fig. 9.
Fig. 9.

TPA rate in bulk silicon: experimental data adapted from [23] (circles) and analytical fit used in our calculations (solid curve).

Fig. 10.
Fig. 10.

(a) Calculated GVD and (b) nonlinear coefficient (real part), for the quasi-TE mode of SOI waveguide shown in Fig. 1(b). Solid/dashed curve in (b) corresponds to the modified/conventional (γ˜/γ) nonlinear coefficient. In numerical simulations, we used polynomial fits of orders ND=6 and NG=4 to reproduce β2(ω) and γ˜(ω) dependencies, respectively.

Fig. 11.
Fig. 11.

Spectral broadening in SOI waveguide pumped at λ0=2.2μm by a 100 fs pulse with 100 W peak power. The result is obtained for the case of wavelength-independent nonlinear coefficient, γ˜=const. TPA is neglected.

Fig. 12.
Fig. 12.

Spectral broadening in SOI waveguide: geometrical dispersion of nonlinearity. Output spectra at z=1mm calculated for the cases of wavelength-independent nonlinearity γ˜=Const. (thin solid curve), self-steepening nonlinearity γ˜=ωΓ0 (dashed red/gray curve), and fully dispersive nonlinearity γ˜=ωΓ0Γ (thick solid curve). Input parameters are the same as in Fig. 11.

Fig. 13.
Fig. 13.

Spectral broadening in SOI waveguide: dispersion of TPA. Output spectra at z=1mm calculated for the cases of no TPA (thin solid curve), wavelength-independent TPA αTPA0.03 (dashed red/gray curve), and dispersive TPA (thick solid curve) as in Fig. 9. Full geometrical dispersion of nonlinearity is taken into account. Input parameters are the same as in Fig. 11.

Equations (29)

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E⃗(r⃗,t)=12πE(r⃗,ω)eiωtdω+c.c.,
E(r⃗,ω)=Iω1/2Aω(z)eω(x,y),
Iω=+(eω×hω*+eω*×hω)e^zdxdy.
D(ω)=ϵ0[ϵ(ω)E(ω)+χ(3)(ω;ω1,ω2,ω3)E(ω1)E*(ω2)E(ω3)],
izAω=βωAωω2πΓωω1ω2ω3Aω1Aω2*Aω3dω1dω2,
ω3=ωω1+ω2,
Γnklm=ϵ0InIkIlIm+χnklmζnklmdxdy,
ζnklm=ρ[(en*em)(ekel*)+(en*ek)(emel*)+(en*el*)(ekem)]+3(1ρ)i=x,y,zeni*ekieli*emi.
Γnklmgngkglgm,gn=Γnnnn1/4Γn1/4.
An=(g0/gn)Ψn,
izΨω=βωΨωγ˜ω2πΨω1Ψω2*Ψω3dω1dω2,
γ˜ω=ωΓ0Γω.
P=ω|Ψω|2γ˜ωdω=const.,
N=|Ψω|2γ˜ωdω=const.
ψ(z,t)=12πΨω(z)ei(ω0+δ)tdδ,
iza=D^(iτ)aG^(iτ)(|a|2a),
D^(iτ)=n=2NDβnn!(iτ)n,
G^(iτ)=n=0NGγ˜nn!(iτ)n.
γ˜0=γ0,
γ˜1=12(γ0ω0+γ1),
γ˜2=γ22(γ1γ0/ω0)24γ0.
izAp=βpApωpΓp|Ap|2Ap,
izAs=βsAs2ωsΓsp|Ap|2AsωsΓ4Ap2Ai*,
izAi=βiAi2ωiΓip|Ap|2AiωiΓ4Ap2As*.
4γ|Ap|2<δβ<4γ+|Ap|2,
γ±=(ωsΓsp+ωiΓipωpΓp±ωsωiΓ4)/2,
2γ=γ˜0+γ˜2δ2(γ˜02+γ˜22δ2)2γ˜12δ2.
G=γ˜0γ˜2γ˜12<1.
χ(3)=χ(1+iαTPA).

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