Abstract

A fully analytical toolbox for supercontinuum generation relying on scenarios without pulse splitting is presented. Furthermore, starting from the new insights provided by this formalism about the physical nature of direct and cascaded dispersive wave emission, a unified description of this radiation in both normal and anomalous dispersion regimes is derived. Previously unidentified physics of broadband spectra reported in earlier works is successfully explained on this basis. Finally, a foundry-compatible few-millimeters-long silicon waveguide allowing octave-spanning supercontinuum generation pumped at telecom wavelengths in the normal dispersion regime is designed, hence showcasing the potential of this new analytical approach.

© 2016 Optical Society of America

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References

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2016 (2)

N. Vermeulen, J. Cheng, J. E. Sipe, and H. Thienpont, “Opportunities for wideband wavelength conversion in foundry-compatible silicon waveguides covered with graphene,” IEEE J. Sel. Top. Quantum Electron. 22, 8100113 (2016).
[Crossref]

G. Xu, A. Mussot, A. Kudlinski, S. Trillo, F. Copie, and M. Conforti, “Shock wave generation triggered by a weak background in optical fibers,” Opt. Lett. 41, 2656–2659 (2016).
[Crossref] [PubMed]

2015 (3)

2014 (5)

2013 (5)

2012 (3)

2011 (2)

2010 (3)

2007 (3)

A. V. Gorbach and D. V. Skryabin, “Light trapping in gravity-like potentials and expansion of supercontinuum spectra in photonic-crystal fibres,” Nature Photon. 1, 653–657 (2007).
[Crossref]

A. D. Bristow, N. Rotenberg, and H. M. van Driel, “Two-photon absorption and Kerr coefficients of silicon for 850–2200 nm,” Appl. Phys. Lett. 90, 191104 (2007).
[Crossref]

L. Yin, Q. Lin, and G. P. Agrawal, “Soliton fission and supercontinuum generation in silicon waveguides,” Opt. Lett. 32, 391–393 (2007).
[Crossref] [PubMed]

2006 (3)

2005 (2)

D. V. Skryabin and A. V. Yulin, “Theory of generation of new frequencies by mixing of solitons and dispersive waves in optical fibers,” Phys. Rev. E 72, 016619 (2005).
[Crossref]

A. Ferrando, M. Zacarés, P. Fernández de Córdoba, D. Binosi, and Á. Montero, “Forward-backward equations for nonlinear propagation in axially invariant optical systems,” Phys. Rev. E 71, 016601 (2005).
[Crossref]

2004 (2)

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]

I. Cristiani, R. Tediosi, L. Tartara, and V. Degiorgio, “Dispersive wave generation by solitons in microstructured optical fibers,” Opt. Express 12, 124–135 (2004).
[Crossref] [PubMed]

2003 (1)

X. Fang, N. Karasawa, R. Morita, R. S. Windeler, and M. Yamashita, “Nonlinear propagation of a-few-optical-cycle pulses in a photonic crystal fiber — Experimental and theoretical studies beyond the slowly varying-envelope approximation,” IEEE Photon. Technol. Lett. 15, 233–235 (2003).
[Crossref]

2002 (2)

2001 (2)

1999 (1)

1997 (1)

1995 (1)

N. Akhmediev and M. Karlsson, “Cherenkov radiation emitted by solitons in optical fibers,” Phys. Rev. A 51, 2602–2607 (1995).
[Crossref] [PubMed]

1992 (1)

1989 (2)

J. E. Rothenberg and D. Grischkowsky, “Observation of the formation of an optical intensity shock and wave breaking in the nonlinear propagation of pulses in optical fibers,” Phys. Rev. Lett. 62, 531–534 (1989).
[Crossref] [PubMed]

J. E. Rothenberg, “Femtosecond optical shocks and wave breaking in fiber propagation,” J. Opt. Soc. Am. B 6, 2392–2401 (1989).
[Crossref]

1987 (1)

Y. Kodama and A. Hasegawa, “Nonlinear pulse propagation in a monomode dielectric guide,” IEEE J. Quantum Electron. 23, 510–524 (1987).
[Crossref]

1986 (1)

1985 (1)

1972 (1)

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. Phys. JETP 34, 62–70 (1972).

Agrawal, G. P.

Akhmediev, N.

N. Akhmediev and M. Karlsson, “Cherenkov radiation emitted by solitons in optical fibers,” Phys. Rev. A 51, 2602–2607 (1995).
[Crossref] [PubMed]

Amiranashvili, Sh.

Sh. Amiranashvili and A. Demircan, “Ultrashort optical pulse propagation in terms of analytic signal,” Adv. Opt. Technol. 2011, 989515 (2011).
[Crossref]

Anderson, D.

Andrés, P.

Austin, D. R.

Baets, R.

Baronio, F.

M. Conforti, F. Baronio, and S. Trillo, “Resonant radiation shed by dispersive shock waves,” Phys. Rev. A 89, 013807 (2014).
[Crossref]

Barry, L. P.

Beausoleil, R. G.

Binosi, D.

A. Ferrando, M. Zacarés, P. Fernández de Córdoba, D. Binosi, and Á. Montero, “Forward-backward equations for nonlinear propagation in axially invariant optical systems,” Phys. Rev. E 71, 016601 (2005).
[Crossref]

Blanco-Redondo, A.

Bollond, P. G.

Boppart, S. A.

Bristow, A. D.

A. D. Bristow, N. Rotenberg, and H. M. van Driel, “Two-photon absorption and Kerr coefficients of silicon for 850–2200 nm,” Appl. Phys. Lett. 90, 191104 (2007).
[Crossref]

Broderick, N. G. R.

K. E. Webb, M. Erkintalo, Y. Xu, N. G. R. Broderick, J. M. Dudley, G. Genty, and S. G. Murdoch, “Nonlinear optics of fibre event horizons,” Nat. Commun. 5, 4969 (2014).
[PubMed]

Broeng, J.

Brown, T. G.

Castelló-Lurbe, D.

Chang, G.

Chau, A. H. L.

Chen, C.

Chen, H. H.

Chen, L.

Cheng, J.

N. Vermeulen, J. Cheng, J. E. Sipe, and H. Thienpont, “Opportunities for wideband wavelength conversion in foundry-compatible silicon waveguides covered with graphene,” IEEE J. Sel. Top. Quantum Electron. 22, 8100113 (2016).
[Crossref]

Coen, S.

Conforti, M.

Copie, F.

Cristiani, I.

Cui, Y.

Daniel, B. A.

Dave, U.

de Sterke, C. M.

Degiorgio, V.

Demircan, A.

Sh. Amiranashvili and A. Demircan, “Ultrashort optical pulse propagation in terms of analytic signal,” Adv. Opt. Technol. 2011, 989515 (2011).
[Crossref]

Desaix, M.

Drummond, P. D.

Dudley, J. M.

K. E. Webb, M. Erkintalo, Y. Xu, N. G. R. Broderick, J. M. Dudley, G. Genty, and S. G. Murdoch, “Nonlinear optics of fibre event horizons,” Nat. Commun. 5, 4969 (2014).
[PubMed]

M. Erkintalo, Y. Q. Xu, S. G. Murdoch, J. M. Dudley, and G. Genty, “Cascaded phase matching and nonlinear symmetry breaking in fiber frequency combs,” Phys. Rev. Lett. 109, 223904 (2012).
[Crossref]

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

J. M. Dudley, L. P. Barry, P. G. Bollond, J. D. Harvey, R. Leonhardt, and P. D. Drummond, “Direct measurement of pulse distortion near the zero-dispersion wavelength in an optical fiber by frequency-resolved optical gating,” Opt. Lett. 22, 457–459 (1997).
[Crossref] [PubMed]

Eggleton, B. J.

Erkintalo, M.

K. E. Webb, M. Erkintalo, Y. Xu, N. G. R. Broderick, J. M. Dudley, G. Genty, and S. G. Murdoch, “Nonlinear optics of fibre event horizons,” Nat. Commun. 5, 4969 (2014).
[PubMed]

K. E. Webb, Y. Q. Xu, M. Erkintalo, and S. G. Murdoch, “Generalized dispersive wave emission in nonlinear fibers,” Opt. Lett. 38, 151–153 (2013).
[Crossref] [PubMed]

Y. Q. Xu, M. Erkintalo, G. Genty, and S. G. Murdoch, “Cascaded Bragg scattering in fiber optics,” Opt. Lett. 38, 142–144 (2013).
[Crossref] [PubMed]

M. Erkintalo, Y. Q. Xu, S. G. Murdoch, J. M. Dudley, and G. Genty, “Cascaded phase matching and nonlinear symmetry breaking in fiber frequency combs,” Phys. Rev. Lett. 109, 223904 (2012).
[Crossref]

Fang, X.

X. Fang, N. Karasawa, R. Morita, R. S. Windeler, and M. Yamashita, “Nonlinear propagation of a-few-optical-cycle pulses in a photonic crystal fiber — Experimental and theoretical studies beyond the slowly varying-envelope approximation,” IEEE Photon. Technol. Lett. 15, 233–235 (2003).
[Crossref]

Fernández de Córdoba, P.

A. Ferrando, M. Zacarés, P. Fernández de Córdoba, D. Binosi, and Á. Montero, “Forward-backward equations for nonlinear propagation in axially invariant optical systems,” Phys. Rev. E 71, 016601 (2005).
[Crossref]

Ferrando, A.

A. Ferrando, M. Zacarés, P. Fernández de Córdoba, D. Binosi, and Á. Montero, “Forward-backward equations for nonlinear propagation in axially invariant optical systems,” Phys. Rev. E 71, 016601 (2005).
[Crossref]

Gaeta, A. L.

Genty, G.

K. E. Webb, M. Erkintalo, Y. Xu, N. G. R. Broderick, J. M. Dudley, G. Genty, and S. G. Murdoch, “Nonlinear optics of fibre event horizons,” Nat. Commun. 5, 4969 (2014).
[PubMed]

Y. Q. Xu, M. Erkintalo, G. Genty, and S. G. Murdoch, “Cascaded Bragg scattering in fiber optics,” Opt. Lett. 38, 142–144 (2013).
[Crossref] [PubMed]

M. Erkintalo, Y. Q. Xu, S. G. Murdoch, J. M. Dudley, and G. Genty, “Cascaded phase matching and nonlinear symmetry breaking in fiber frequency combs,” Phys. Rev. Lett. 109, 223904 (2012).
[Crossref]

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

G. Genty, M. Lehtonen, H. Ludvigsen, J. Broeng, and M. Kaivola, “Spectral broadening of femtosecond pulses into continuum radiation in microstructured fibers,” Opt. Express 10, 1083–1098 (2002).
[Crossref] [PubMed]

Gorbach, A. V.

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, “Light trapping in gravity-like potentials and expansion of supercontinuum spectra in photonic-crystal fibres,” Nature Photon. 1, 653–657 (2007).
[Crossref]

A. V. Gorbach, D. V. Skryabin, J. M. Stone, and J. C. Knight, “Four-wave mixing of solitons with radiation and quasi-nondispersive wave packets at the short-wavelength edge of a supercontinuum,” Opt. Express 14, 9854–9863 (2006).
[Crossref] [PubMed]

Gorza, S.

Gosciniak, J.

Green, W. M. J.

Griffith, A. G.

Grischkowsky, D.

J. E. Rothenberg and D. Grischkowsky, “Observation of the formation of an optical intensity shock and wave breaking in the nonlinear propagation of pulses in optical fibers,” Phys. Rev. Lett. 62, 531–534 (1989).
[Crossref] [PubMed]

Harvey, J. D.

Hasegawa, A.

Y. Kodama and A. Hasegawa, “Nonlinear pulse propagation in a monomode dielectric guide,” IEEE J. Quantum Electron. 23, 510–524 (1987).
[Crossref]

Herrmann, J.

A. V. Husakou and J. Herrmann, “Supercontinuum generation of higher-order solitons by fission in photonic crystal fibers,” Phys. Rev. Lett. 27, 203901 (2001).
[Crossref]

Husakou, A. V.

A. V. Husakou and J. Herrmann, “Supercontinuum generation of higher-order solitons by fission in photonic crystal fibers,” Phys. Rev. Lett. 27, 203901 (2001).
[Crossref]

Husko, C.

Islam, M. N.

Ji, W.

Johnson, A. M.

Johnson, A. R.

Joshi, C.

Kaivola, M.

Karasawa, N.

X. Fang, N. Karasawa, R. Morita, R. S. Windeler, and M. Yamashita, “Nonlinear propagation of a-few-optical-cycle pulses in a photonic crystal fiber — Experimental and theoretical studies beyond the slowly varying-envelope approximation,” IEEE Photon. Technol. Lett. 15, 233–235 (2003).
[Crossref]

Karlsson, M.

N. Akhmediev and M. Karlsson, “Cherenkov radiation emitted by solitons in optical fibers,” Phys. Rev. A 51, 2602–2607 (1995).
[Crossref] [PubMed]

Kärtner, F. X.

Keller, U.

Kelley, P. L.

Kim, J.

Klenner, A.

Knight, J. C.

Kockaert, P.

Kodama, Y.

Y. Kodama and A. Hasegawa, “Nonlinear pulse propagation in a monomode dielectric guide,” IEEE J. Quantum Electron. 23, 510–524 (1987).
[Crossref]

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.

Kuyken, B.

Læsgaard, J.

Lamb, E. S.

Lamont, M. R. E.

Lau, R. K. W.

Lee, Y. C.

Lefrancois, S.

Lehtonen, M.

Leo, F.

Leonhardt, R.

Lin, Q.

Lipson, M.

Lisak, M.

Liu, X.

Ludvigsen, H.

Luke, K.

Mayer, A. S.

Menyuk, C. R.

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]

Montero, Á.

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Other (3)

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

Y. R. Shen, The Principles of Nonlinear Optics (John Wiley & Sons, 2003).

See ePIXfab at http://www.epixfab.eu and Institute of Microelectronics (IME) at https://www.a-star.edu.sg/ .

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Figures (5)

Fig. 1
Fig. 1 (a)–(b): Evolution of the generalized lengths for N = 10 in (a) the normal dispersion and (b) the anomalous dispersion regimes. (c)–(d): Comparison of the analytically calculated (normalized) generalized wave-breaking distance (red squares) with the numerical results (blue circles) in (c) the normal dispersion and (d) the anomalous dispersion regimes. The parameters considered in theses cases are included in the insets of (c) and (d). See details of the formula for ξGWB in Table 1.
Fig. 2
Fig. 2 Illustration of the frequency overlapping and the resulting FWM processes in (a) the normal dispersion regime, s2 = 1, and (b) the anomalous dispersion regime, s2 = −1. The schematic plots include the absolute instantaneous frequency, ω(t) = ω0+δω(t) (continuous lines), and instantaneous power, P(t) (dashed lines). Thick lines highlight the frequencies that can overlap. The time shifting induced by dispersion can induce the FWM processes that are represented.
Fig. 3
Fig. 3 Direct DW emission in (a) the normal regime and (b) the anomalous regime. Cascaded DW emission in (c) the normal regime and (d) the anomalous regime. HOD has been calculated according to the analytical results in Table 2 using δωSPM = δωol and |β2| = 1 ps2m−1, γ0 = 4 W−1m−1, T0 = 0.5 ps and ν0 corresponding to 1550 nm. The position of the resonances are successfully predicted in our framework in all these cases. The propagations have been stopped at the distances ξ, where the resonances achieve their maximum power levels while additional processes not included in our model have not impacted the dynamics yet. Yellow dashed arrows represent the spectral broadening relying on SPM and green solid arrows indicate the FWM processes that have been considered. The dispersion profiles, including arrows to indicate the pumping frequency, have been added as insets.
Fig. 4
Fig. 4 (a) Output spectra simulated through Eq. (6) including higher order effects (blue solid curve) and without them (green dashed line). (b) Evolution of the generalized lengths [cf. Fig. 1(a)]. (c) Plot of the relative β1, i.e., the inverse of the group velocity. The green window includes the frequencies that can overlap in the leading pulse edge. A and N indicate anomalous and normal dispersion, respectively. (d) Linear phase mismatch (only negative values are represented). The green solid curve points out processes with group-velocity matching (see details in the text).
Fig. 5
Fig. 5 (a) Output spectra at several distances between zGWB and the total length of the waveguide, zWG, according to [21]. (b) Evolution of the generalized lengths [cf. Fig. 1(b)]. (c) Plot of the relative β1, i.e., the inverse of the group velocity. The green window includes the frequencies that can overlap. A and N indicate anomalous and normal dispersion, respectively. (d) Linear phase mismatch. The green solid curve points out the processes with group-velocity matching (see details in the text).

Tables (2)

Tables Icon

Table 1 Characteristic lengths, ξGWB and ξol, and the maximum chirps generated by SPM at such lengths. Here a can be considered an auxiliary variable, σ2, and ϒ are input-pulse form factors. In particular, they equal to 16/35 and 4 / 27 , respectively, for a sech pulse and 8 / 27 and 2 / e for a Gaussian pulse. σ ˜ 2 is an auxiliary parameter equals to 1 / 2 (see details in Appendix). Remember that N2 = LD/LNL.

Tables Icon

Table 2 Four different scenarios for SC relying on direct and cascaded DW. The optimal HOD parameters that induces group-velocity matching and phase matching for those FWM and the spectral broadening produced through each mechanism (ωDW or ωCDW) are also included. δωSPM can be estimated by δωol in Table 1.

Equations (12)

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z A ˜ ( z , ω ω 0 ) = i β 2 2 ( ω ω 0 ) 2 A ˜ ( z , ω ω 0 ) + i γ 0 ω 0 1 [ | A ( z , t ) | 2 A ( z , t ) ] ,
D 1 ( z ) = β 2 2 d ω ( ω ω 0 ) 2 | A ˜ ( z , ω ω 0 ) | 2 d ω | A ˜ ( z , ω ω 0 ) | 2 ,
NL 1 ( z ) = γ 0 2 d t | A ( z , t ) | 4 d t | A ( z , t ) | 2 .
d d ξ [ NL 1 ( ξ ) NL 1 ( 0 ) ] = 2 s 2 N [ NL 1 ( ξ ) NL 1 ( 0 ) ] 2 ( s 2 σ 2 [ 1 NL 1 ( ξ ) NL 1 ( 0 ) ] ) 1 / 2 ,
ξ GWB = s 2 2 N σ 2 1 / 2 1 s 2 / 2 1 d a a 2 [ s 2 ( 1 a ) ] 1 / 2 .
z A ˜ ( z , ω ω 0 ) = i β p ( ω ) A ˜ ( z , ω ω 0 ) + i γ 0 ω 0 1 [ | A ( z , t ) | 2 A ( z , t ) ] ,
NL 1 ( z ) + D 1 ( z ) = NL 1 ( 0 ) + D 1 ( 0 ) ,
d d z NL 1 ( z ) = β 2 γ 0 d t t φ ( z , t ) t | A ( z , t ) | 2 | A ( z , t ) | 2 d t | A ( z , t ) | 2 ,
σ 2 a 2 ( γ 0 P 0 b ) 2 = s 2 N 2 ( 1 a ) ,
d a d ξ = 2 s 2 σ 2 ( γ 0 P 0 b ) a 3 ,
d a d ξ = 2 s 2 N a 2 ( s 2 σ 2 [ 1 a ] ) 1 / 2 ,
δ ω max ( ξ ) = σ 2 1 / 2 N ( s 2 [ 1 a ( ξ ) ] ) 1 / 2 ϒ T 0 1

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