Abstract

Supercontinuum generation is a key process for nonlinear tailored light generation and strongly depends on the dispersion of the underlying waveguide. Here we reveal the nonlinear dynamics of soliton-based supercontinuum generation in case the waveguide includes a strongly dispersive resonance. Assuming a gas-filled hollow core fiber that includes a Lorentzian-type dispersion term, effects such as multi-color dispersive wave emission and cascaded four-wave mixing have been identified to be the origin of the observed spectral broadening, greatly exceeding the bandwidths of corresponding non-resonant fibers. Moreover, we obtain large spectral bandwidth at low soliton numbers, yielding broadband spectra within the coherence limit. Due to the mentioned advantages, we believe the concept of resonance-enhanced supercontinuum generation to be highly relevant for future nonlinear light sources.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]

2019 (5)

2018 (1)

F. Tani, F. Köttig, D. Novoa, R. Keding, and P. Stj Russell, “Effect of anti-crossings with cladding resonances on ultrafast nonlinear dynamics in gas-filled photonic crystal fibers,” Photonics Res. 6(2), 84–88 (2018).
[Crossref]

2017 (3)

F. Meng, B. Liu, S. Wang, J. Liu, Y. Li, C. Wang, A. M. Zheltikov, and M. Hu, “Controllable two-color dispersive wave generation in argon-filled hypocycloid-core kagome fiber,” Opt. Express 25(26), 32972–32984 (2017).
[Crossref]

R. Sollapur, D. Kartashov, M. Zürch, A. Hoffmann, T. Grigorova, G. Sauer, A. Hartung, A. Schwuchow, J. Bierlich, J. Kobelke, M. Chemnitz, M. A. Schmidt, and C. Spielmann, “Resonance-enhanced multi-octave supercontinuum generation in antiresonant hollow-core fibers,” Light: Sci. Appl. 6(12), e17124 (2017).
[Crossref]

S. Pumpe, M. Chemnitz, J. Kobelke, and M. A. Schmidt, “Monolithic optofluidic mode coupler for broadband thermo- and piezo-optical characterization of liquids,” Opt. Express 25(19), 22932–22946 (2017).
[Crossref]

2015 (2)

A. Hartung, J. Kobelke, A. Schwuchow, K. Wondraczek, J. Bierlich, J. Popp, T. Frosch, and M. A. Schmidt, “Origins of modal loss of antiresonant hollow-core optical fibers in the ultraviolet,” Opt. Express 23(3), 2557–2565 (2015).
[Crossref]

X. Jiang, N. Y. Joly, M. A. Finger, F. Babic, G. K. L. Wong, J. C. Travers, and P. S. J. Russell, “Deep-ultraviolet to mid-infrared supercontinuum generated in solid-core ZBLAN photonic crystal fibre,” Nat. Photonics 9(2), 133–139 (2015).
[Crossref]

2014 (1)

P. S. J. Russell, P. Hölzer, W. Chang, A. Abdolvand, and J. C. Travers, “Hollow-core photonic crystal fibres for gas-based nonlinear optics,” Nat. Photonics 8(4), 278–286 (2014).
[Crossref]

2013 (1)

H. Tu and S. A. Boppart, “Coherent fiber supercontinuum for biophotonics,” Laser Photonics Rev. 7(5), 628–645 (2013).
[Crossref]

2012 (1)

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(22), 223904 (2012).
[Crossref]

2011 (1)

2010 (1)

2009 (1)

S. Roy, S. K. Bhadra, and G. P. Agrawal, “Dispersive waves emitted by solitons perturbed by third-order dispersion inside optical fibers,” Phys. Rev. A 79(2), 023824 (2009).
[Crossref]

2008 (2)

A. Börzsönyi, Z. Heiner, M. P. Kalashnikov, A. P. Kovács, and K. Osvay, “Dispersion measurement of inert gases and gas mixtures at 800 nm,” Appl. Opt. 47(27), 4856–4863 (2008).
[Crossref]

C. F. Kaminski, R. S. Watt, A. D. Elder, J. H. Frank, and J. Hult, “Supercontinuum radiation for applications in chemical sensing and microscopy,” Appl. Phys. B 92(3), 367–378 (2008).
[Crossref]

2007 (1)

E. N. Tsoy and C. M. De Sterke, “Theoretical analysis of the self-frequency shift near zero-dispersion points: Soliton spectral tunneling,” Phys. Rev. A 76(4), 043804 (2007).
[Crossref]

2006 (2)

2004 (2)

2003 (1)

P. Russell, “Photonic crystal fibers,” Science 299(5605), 358–362 (2003).
[Crossref]

2002 (1)

T. Udem, R. Holzwarth, and T. W. Hänsch, “Optical frequency metrology,” Nature 416(6877), 233–237 (2002).
[Crossref]

2000 (2)

J. C. Knight, J. Arriaga, T. A. Birks, A. Ortigosa-Blanch, W. J. Wadsworth, and P. S. J. Russell, “Anomalous dispersion in photonic crystal fiber,” IEEE Photonics Technol. Lett. 12(7), 807–809 (2000).
[Crossref]

J. K. Ranka, R. S. Windeler, and A. J. Stentz, “Visible continuum generation in air–silica microstructure optical fibers with anomalous dispersion at 800 nm,” Opt. Lett. 25(1), 25–27 (2000).
[Crossref]

1999 (1)

R. Cregan, B. J. Mangan, T. A. Birks, J. Knight, P. S. J. Russell, D. C. Allan, and P. Roberts, “Single-Mode Photonic Band Gap Guidance of Light in Air.,” Science 285(5433), 1537–1539 (1999).
[Crossref]

1997 (1)

1995 (1)

N. Akhmediev and M. Karlsson, “Cherenkov radiation emitted by solitons in optical fibers,” Phys. Rev. A 51(3), 2602–2607 (1995).
[Crossref]

1964 (1)

E. A. J. Marcatili and R. A. Schmeltzer, “Hollow Metallic and Dielectric Waveguides for Long Distance Optical Transmission and Lasers,” Bell Syst. Tech. J. 43(4), 1783–1809 (1964).
[Crossref]

Abdolvand, A.

P. S. J. Russell, P. Hölzer, W. Chang, A. Abdolvand, and J. C. Travers, “Hollow-core photonic crystal fibres for gas-based nonlinear optics,” Nat. Photonics 8(4), 278–286 (2014).
[Crossref]

Agrawal, G. P.

S. Roy, S. K. Bhadra, and G. P. Agrawal, “Dispersive waves emitted by solitons perturbed by third-order dispersion inside optical fibers,” Phys. Rev. A 79(2), 023824 (2009).
[Crossref]

G. P. Agrawal, Nonlinear Fiber Optics, 5th Ed. (Academic Press, 2015)

Akhmediev, N.

N. Akhmediev and M. Karlsson, “Cherenkov radiation emitted by solitons in optical fibers,” Phys. Rev. A 51(3), 2602–2607 (1995).
[Crossref]

Allan, D. C.

R. Cregan, B. J. Mangan, T. A. Birks, J. Knight, P. S. J. Russell, D. C. Allan, and P. Roberts, “Single-Mode Photonic Band Gap Guidance of Light in Air.,” Science 285(5433), 1537–1539 (1999).
[Crossref]

Ando, R. F.

Arriaga, J.

J. C. Knight, J. Arriaga, T. A. Birks, A. Ortigosa-Blanch, W. J. Wadsworth, and P. S. J. Russell, “Anomalous dispersion in photonic crystal fiber,” IEEE Photonics Technol. Lett. 12(7), 807–809 (2000).
[Crossref]

Babic, F.

X. Jiang, N. Y. Joly, M. A. Finger, F. Babic, G. K. L. Wong, J. C. Travers, and P. S. J. Russell, “Deep-ultraviolet to mid-infrared supercontinuum generated in solid-core ZBLAN photonic crystal fibre,” Nat. Photonics 9(2), 133–139 (2015).
[Crossref]

Bache, M.

Benabid, F.

Bhadra, S. K.

S. Roy, S. K. Bhadra, and G. P. Agrawal, “Dispersive waves emitted by solitons perturbed by third-order dispersion inside optical fibers,” Phys. Rev. A 79(2), 023824 (2009).
[Crossref]

Bierlich, J.

R. Sollapur, D. Kartashov, M. Zürch, A. Hoffmann, T. Grigorova, G. Sauer, A. Hartung, A. Schwuchow, J. Bierlich, J. Kobelke, M. Chemnitz, M. A. Schmidt, and C. Spielmann, “Resonance-enhanced multi-octave supercontinuum generation in antiresonant hollow-core fibers,” Light: Sci. Appl. 6(12), e17124 (2017).
[Crossref]

A. Hartung, J. Kobelke, A. Schwuchow, K. Wondraczek, J. Bierlich, J. Popp, T. Frosch, and M. A. Schmidt, “Origins of modal loss of antiresonant hollow-core optical fibers in the ultraviolet,” Opt. Express 23(3), 2557–2565 (2015).
[Crossref]

Birks, T. A.

G. Humbert, W. J. Wadsworth, S. G. Leon-Saval, J. C. Knight, T. A. Birks, P. St. J. Russell, M. J. Lederer, D. Kopf, K. Wiesauer, E. I. Breuer, and D. Stifter, “Supercontinuum generation system for optical coherence tomography based on tapered photonic crystal fibre,” Opt. Express 14(4), 1596–1603 (2006).
[Crossref]

J. C. Knight, J. Arriaga, T. A. Birks, A. Ortigosa-Blanch, W. J. Wadsworth, and P. S. J. Russell, “Anomalous dispersion in photonic crystal fiber,” IEEE Photonics Technol. Lett. 12(7), 807–809 (2000).
[Crossref]

R. Cregan, B. J. Mangan, T. A. Birks, J. Knight, P. S. J. Russell, D. C. Allan, and P. Roberts, “Single-Mode Photonic Band Gap Guidance of Light in Air.,” Science 285(5433), 1537–1539 (1999).
[Crossref]

T. A. Birks, J. C. Knight, and P. S. J. Russell, “Endlessly single-mode photonic crystal fiber,” Opt. Lett. 22(13), 961–963 (1997).
[Crossref]

Boppart, S. A.

H. Tu and S. A. Boppart, “Coherent fiber supercontinuum for biophotonics,” Laser Photonics Rev. 7(5), 628–645 (2013).
[Crossref]

Börzsönyi, A.

Boyd, R. W.

R. W. Boyd, Nonlinear Optics, 3rd Ed. (Academic Press, 2008).

Braun, A.

C. Jain, A. Braun, J. Gargiulo, B. Jang, G. Li, H. Lehmann, S. A. Maier, and M. A. Schmidt, “Hollow Core Light Cage: Trapping Light behind Bars,” ACS Photonics 6(3), 649–658 (2019).
[Crossref]

Breuer, E. I.

Chafer, M.

Chang, W.

P. S. J. Russell, P. Hölzer, W. Chang, A. Abdolvand, and J. C. Travers, “Hollow-core photonic crystal fibres for gas-based nonlinear optics,” Nat. Photonics 8(4), 278–286 (2014).
[Crossref]

Chemnitz, M.

R. Sollapur, D. Kartashov, M. Zürch, A. Hoffmann, T. Grigorova, G. Sauer, A. Hartung, A. Schwuchow, J. Bierlich, J. Kobelke, M. Chemnitz, M. A. Schmidt, and C. Spielmann, “Resonance-enhanced multi-octave supercontinuum generation in antiresonant hollow-core fibers,” Light: Sci. Appl. 6(12), e17124 (2017).
[Crossref]

S. Pumpe, M. Chemnitz, J. Kobelke, and M. A. Schmidt, “Monolithic optofluidic mode coupler for broadband thermo- and piezo-optical characterization of liquids,” Opt. Express 25(19), 22932–22946 (2017).
[Crossref]

Coen, S.

J. M. Dudley, G. Genty, and S. Coen, “Supercontinuum Generation in Photonic Crystal Fibre,” Rev. Mod. Phys. 78(4), 1135–1184 (2006).
[Crossref]

Cordier, M.

Cregan, R.

R. Cregan, B. J. Mangan, T. A. Birks, J. Knight, P. S. J. Russell, D. C. Allan, and P. Roberts, “Single-Mode Photonic Band Gap Guidance of Light in Air.,” Science 285(5433), 1537–1539 (1999).
[Crossref]

De Sterke, C. M.

E. N. Tsoy and C. M. De Sterke, “Theoretical analysis of the self-frequency shift near zero-dispersion points: Soliton spectral tunneling,” Phys. Rev. A 76(4), 043804 (2007).
[Crossref]

Debord, B.

Delaye, P.

Diamanti, E.

Dudley, J. M.

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(22), 223904 (2012).
[Crossref]

V. Pureur and J. M. Dudley, “Nonlinear spectral broadening of femtosecond pulses in solid-core photonic bandgap fibers,” Opt. Lett. 35(16), 2813–2815 (2010).
[Crossref]

J. M. Dudley, G. Genty, and S. Coen, “Supercontinuum Generation in Photonic Crystal Fibre,” Rev. Mod. Phys. 78(4), 1135–1184 (2006).
[Crossref]

Elder, A. D.

C. F. Kaminski, R. S. Watt, A. D. Elder, J. H. Frank, and J. Hult, “Supercontinuum radiation for applications in chemical sensing and microscopy,” Appl. Phys. B 92(3), 367–378 (2008).
[Crossref]

Erkintalo, M.

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(22), 223904 (2012).
[Crossref]

Finger, M. A.

X. Jiang, N. Y. Joly, M. A. Finger, F. Babic, G. K. L. Wong, J. C. Travers, and P. S. J. Russell, “Deep-ultraviolet to mid-infrared supercontinuum generated in solid-core ZBLAN photonic crystal fibre,” Nat. Photonics 9(2), 133–139 (2015).
[Crossref]

Frank, J. H.

C. F. Kaminski, R. S. Watt, A. D. Elder, J. H. Frank, and J. Hult, “Supercontinuum radiation for applications in chemical sensing and microscopy,” Appl. Phys. B 92(3), 367–378 (2008).
[Crossref]

Frosch, T.

Gargiulo, J.

B. Jang, J. Gargiulo, R. F. Ando, A. Lauri, S. A. Maier, and M. A. Schmidt, “Light guidance in photonic band gap guiding dual-ring light cages implemented by direct laser writing,” Opt. Lett. 44(16), 4016–4019 (2019).
[Crossref]

C. Jain, A. Braun, J. Gargiulo, B. Jang, G. Li, H. Lehmann, S. A. Maier, and M. A. Schmidt, “Hollow Core Light Cage: Trapping Light behind Bars,” ACS Photonics 6(3), 649–658 (2019).
[Crossref]

Genty, G.

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(22), 223904 (2012).
[Crossref]

J. M. Dudley, G. Genty, and S. Coen, “Supercontinuum Generation in Photonic Crystal Fibre,” Rev. Mod. Phys. 78(4), 1135–1184 (2006).
[Crossref]

G. Genty, M. Lehtonen, and H. Ludvigsen, “Effect of cross-phase modulation on supercontinuum generated in microstructured fibers with sub-30 fs pulse,” Opt. Express 12(19), 4614–4624 (2004).
[Crossref]

Gérome, F.

Gorse, A.

Granzow, N.

Grigorova, T.

R. Sollapur, D. Kartashov, M. Zürch, A. Hoffmann, T. Grigorova, G. Sauer, A. Hartung, A. Schwuchow, J. Bierlich, J. Kobelke, M. Chemnitz, M. A. Schmidt, and C. Spielmann, “Resonance-enhanced multi-octave supercontinuum generation in antiresonant hollow-core fibers,” Light: Sci. Appl. 6(12), e17124 (2017).
[Crossref]

Habib, M. S.

Hänsch, T. W.

T. Udem, R. Holzwarth, and T. W. Hänsch, “Optical frequency metrology,” Nature 416(6877), 233–237 (2002).
[Crossref]

Hartung, A.

R. F. Ando, A. Hartung, B. Jang, and M. A. Schmidt, “Approximate model for analyzing band structures of single-ring hollow-core anti-resonant fibers,” Opt. Express 27(7), 10009–10021 (2019).
[Crossref]

R. Sollapur, D. Kartashov, M. Zürch, A. Hoffmann, T. Grigorova, G. Sauer, A. Hartung, A. Schwuchow, J. Bierlich, J. Kobelke, M. Chemnitz, M. A. Schmidt, and C. Spielmann, “Resonance-enhanced multi-octave supercontinuum generation in antiresonant hollow-core fibers,” Light: Sci. Appl. 6(12), e17124 (2017).
[Crossref]

A. Hartung, J. Kobelke, A. Schwuchow, K. Wondraczek, J. Bierlich, J. Popp, T. Frosch, and M. A. Schmidt, “Origins of modal loss of antiresonant hollow-core optical fibers in the ultraviolet,” Opt. Express 23(3), 2557–2565 (2015).
[Crossref]

Heiner, Z.

Hoffmann, A.

R. Sollapur, D. Kartashov, M. Zürch, A. Hoffmann, T. Grigorova, G. Sauer, A. Hartung, A. Schwuchow, J. Bierlich, J. Kobelke, M. Chemnitz, M. A. Schmidt, and C. Spielmann, “Resonance-enhanced multi-octave supercontinuum generation in antiresonant hollow-core fibers,” Light: Sci. Appl. 6(12), e17124 (2017).
[Crossref]

Hölzer, P.

P. S. J. Russell, P. Hölzer, W. Chang, A. Abdolvand, and J. C. Travers, “Hollow-core photonic crystal fibres for gas-based nonlinear optics,” Nat. Photonics 8(4), 278–286 (2014).
[Crossref]

Holzwarth, R.

T. Udem, R. Holzwarth, and T. W. Hänsch, “Optical frequency metrology,” Nature 416(6877), 233–237 (2002).
[Crossref]

Hu, M.

Hult, J.

C. F. Kaminski, R. S. Watt, A. D. Elder, J. H. Frank, and J. Hult, “Supercontinuum radiation for applications in chemical sensing and microscopy,” Appl. Phys. B 92(3), 367–378 (2008).
[Crossref]

Humbert, G.

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics, 3rd Ed. (John Wiley & Sons: New York, 1999).

Jain, C.

C. Jain, A. Braun, J. Gargiulo, B. Jang, G. Li, H. Lehmann, S. A. Maier, and M. A. Schmidt, “Hollow Core Light Cage: Trapping Light behind Bars,” ACS Photonics 6(3), 649–658 (2019).
[Crossref]

Jang, B.

Jiang, X.

X. Jiang, N. Y. Joly, M. A. Finger, F. Babic, G. K. L. Wong, J. C. Travers, and P. S. J. Russell, “Deep-ultraviolet to mid-infrared supercontinuum generated in solid-core ZBLAN photonic crystal fibre,” Nat. Photonics 9(2), 133–139 (2015).
[Crossref]

Joly, N. Y.

X. Jiang, N. Y. Joly, M. A. Finger, F. Babic, G. K. L. Wong, J. C. Travers, and P. S. J. Russell, “Deep-ultraviolet to mid-infrared supercontinuum generated in solid-core ZBLAN photonic crystal fibre,” Nat. Photonics 9(2), 133–139 (2015).
[Crossref]

Kalashnikov, M. P.

Kaminski, C. F.

C. F. Kaminski, R. S. Watt, A. D. Elder, J. H. Frank, and J. Hult, “Supercontinuum radiation for applications in chemical sensing and microscopy,” Appl. Phys. B 92(3), 367–378 (2008).
[Crossref]

Karlsson, M.

N. Akhmediev and M. Karlsson, “Cherenkov radiation emitted by solitons in optical fibers,” Phys. Rev. A 51(3), 2602–2607 (1995).
[Crossref]

Kartashov, D.

R. Sollapur, D. Kartashov, M. Zürch, A. Hoffmann, T. Grigorova, G. Sauer, A. Hartung, A. Schwuchow, J. Bierlich, J. Kobelke, M. Chemnitz, M. A. Schmidt, and C. Spielmann, “Resonance-enhanced multi-octave supercontinuum generation in antiresonant hollow-core fibers,” Light: Sci. Appl. 6(12), e17124 (2017).
[Crossref]

Keding, R.

F. Tani, F. Köttig, D. Novoa, R. Keding, and P. Stj Russell, “Effect of anti-crossings with cladding resonances on ultrafast nonlinear dynamics in gas-filled photonic crystal fibers,” Photonics Res. 6(2), 84–88 (2018).
[Crossref]

Knight, J.

R. Cregan, B. J. Mangan, T. A. Birks, J. Knight, P. S. J. Russell, D. C. Allan, and P. Roberts, “Single-Mode Photonic Band Gap Guidance of Light in Air.,” Science 285(5433), 1537–1539 (1999).
[Crossref]

Knight, J. C.

Kobelke, J.

Kopf, D.

Köttig, F.

F. Tani, F. Köttig, D. Novoa, R. Keding, and P. Stj Russell, “Effect of anti-crossings with cladding resonances on ultrafast nonlinear dynamics in gas-filled photonic crystal fibers,” Photonics Res. 6(2), 84–88 (2018).
[Crossref]

Kovács, A. P.

Lægsgaard, J.

Lauri, A.

Lederer, M. J.

Lehmann, H.

C. Jain, A. Braun, J. Gargiulo, B. Jang, G. Li, H. Lehmann, S. A. Maier, and M. A. Schmidt, “Hollow Core Light Cage: Trapping Light behind Bars,” ACS Photonics 6(3), 649–658 (2019).
[Crossref]

Lehtonen, M.

Leon-Saval, S. G.

Li, G.

C. Jain, A. Braun, J. Gargiulo, B. Jang, G. Li, H. Lehmann, S. A. Maier, and M. A. Schmidt, “Hollow Core Light Cage: Trapping Light behind Bars,” ACS Photonics 6(3), 649–658 (2019).
[Crossref]

Li, Y.

Liu, B.

Liu, J.

Ludvigsen, H.

Maier, S. A.

B. Jang, J. Gargiulo, R. F. Ando, A. Lauri, S. A. Maier, and M. A. Schmidt, “Light guidance in photonic band gap guiding dual-ring light cages implemented by direct laser writing,” Opt. Lett. 44(16), 4016–4019 (2019).
[Crossref]

C. Jain, A. Braun, J. Gargiulo, B. Jang, G. Li, H. Lehmann, S. A. Maier, and M. A. Schmidt, “Hollow Core Light Cage: Trapping Light behind Bars,” ACS Photonics 6(3), 649–658 (2019).
[Crossref]

Mangan, B. J.

R. Cregan, B. J. Mangan, T. A. Birks, J. Knight, P. S. J. Russell, D. C. Allan, and P. Roberts, “Single-Mode Photonic Band Gap Guidance of Light in Air.,” Science 285(5433), 1537–1539 (1999).
[Crossref]

Marcatili, E. A. J.

E. A. J. Marcatili and R. A. Schmeltzer, “Hollow Metallic and Dielectric Waveguides for Long Distance Optical Transmission and Lasers,” Bell Syst. Tech. J. 43(4), 1783–1809 (1964).
[Crossref]

Markos, C.

Meng, F.

Murdoch, S. G.

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(22), 223904 (2012).
[Crossref]

Novoa, D.

F. Tani, F. Köttig, D. Novoa, R. Keding, and P. Stj Russell, “Effect of anti-crossings with cladding resonances on ultrafast nonlinear dynamics in gas-filled photonic crystal fibers,” Photonics Res. 6(2), 84–88 (2018).
[Crossref]

Orieux, A.

Ortigosa-Blanch, A.

J. C. Knight, J. Arriaga, T. A. Birks, A. Ortigosa-Blanch, W. J. Wadsworth, and P. S. J. Russell, “Anomalous dispersion in photonic crystal fiber,” IEEE Photonics Technol. Lett. 12(7), 807–809 (2000).
[Crossref]

Osvay, K.

Popp, J.

Pumpe, S.

Pureur, V.

Ranka, J. K.

Roberts, P.

R. Cregan, B. J. Mangan, T. A. Birks, J. Knight, P. S. J. Russell, D. C. Allan, and P. Roberts, “Single-Mode Photonic Band Gap Guidance of Light in Air.,” Science 285(5433), 1537–1539 (1999).
[Crossref]

Roy, S.

S. Roy, S. K. Bhadra, and G. P. Agrawal, “Dispersive waves emitted by solitons perturbed by third-order dispersion inside optical fibers,” Phys. Rev. A 79(2), 023824 (2009).
[Crossref]

Russell, P.

P. Russell, “Photonic crystal fibers,” Science 299(5605), 358–362 (2003).
[Crossref]

Russell, P. S. J.

X. Jiang, N. Y. Joly, M. A. Finger, F. Babic, G. K. L. Wong, J. C. Travers, and P. S. J. Russell, “Deep-ultraviolet to mid-infrared supercontinuum generated in solid-core ZBLAN photonic crystal fibre,” Nat. Photonics 9(2), 133–139 (2015).
[Crossref]

P. S. J. Russell, P. Hölzer, W. Chang, A. Abdolvand, and J. C. Travers, “Hollow-core photonic crystal fibres for gas-based nonlinear optics,” Nat. Photonics 8(4), 278–286 (2014).
[Crossref]

N. Granzow, M. A. Schmidt, A. S. Tverjanovich, L. Wondraczek, and P. S. J. Russell, “Band-gap guidance in chalcogenide-silica photonic crystal fibers,” Opt. Lett. 36(13), 2432–2434 (2011).
[Crossref]

A. V. Yulin, D. V. Skryabin, and P. S. J. Russell, “Four-wave mixing of linear waves and solitons in fibers with higher-order dispersion,” Opt. Lett. 29(20), 2411–2413 (2004).
[Crossref]

J. C. Knight, J. Arriaga, T. A. Birks, A. Ortigosa-Blanch, W. J. Wadsworth, and P. S. J. Russell, “Anomalous dispersion in photonic crystal fiber,” IEEE Photonics Technol. Lett. 12(7), 807–809 (2000).
[Crossref]

R. Cregan, B. J. Mangan, T. A. Birks, J. Knight, P. S. J. Russell, D. C. Allan, and P. Roberts, “Single-Mode Photonic Band Gap Guidance of Light in Air.,” Science 285(5433), 1537–1539 (1999).
[Crossref]

T. A. Birks, J. C. Knight, and P. S. J. Russell, “Endlessly single-mode photonic crystal fiber,” Opt. Lett. 22(13), 961–963 (1997).
[Crossref]

Russell, P. St. J.

Sauer, G.

R. Sollapur, D. Kartashov, M. Zürch, A. Hoffmann, T. Grigorova, G. Sauer, A. Hartung, A. Schwuchow, J. Bierlich, J. Kobelke, M. Chemnitz, M. A. Schmidt, and C. Spielmann, “Resonance-enhanced multi-octave supercontinuum generation in antiresonant hollow-core fibers,” Light: Sci. Appl. 6(12), e17124 (2017).
[Crossref]

Schmeltzer, R. A.

E. A. J. Marcatili and R. A. Schmeltzer, “Hollow Metallic and Dielectric Waveguides for Long Distance Optical Transmission and Lasers,” Bell Syst. Tech. J. 43(4), 1783–1809 (1964).
[Crossref]

Schmidt, M. A.

C. Jain, A. Braun, J. Gargiulo, B. Jang, G. Li, H. Lehmann, S. A. Maier, and M. A. Schmidt, “Hollow Core Light Cage: Trapping Light behind Bars,” ACS Photonics 6(3), 649–658 (2019).
[Crossref]

B. Jang, J. Gargiulo, R. F. Ando, A. Lauri, S. A. Maier, and M. A. Schmidt, “Light guidance in photonic band gap guiding dual-ring light cages implemented by direct laser writing,” Opt. Lett. 44(16), 4016–4019 (2019).
[Crossref]

R. F. Ando, A. Hartung, B. Jang, and M. A. Schmidt, “Approximate model for analyzing band structures of single-ring hollow-core anti-resonant fibers,” Opt. Express 27(7), 10009–10021 (2019).
[Crossref]

R. Sollapur, D. Kartashov, M. Zürch, A. Hoffmann, T. Grigorova, G. Sauer, A. Hartung, A. Schwuchow, J. Bierlich, J. Kobelke, M. Chemnitz, M. A. Schmidt, and C. Spielmann, “Resonance-enhanced multi-octave supercontinuum generation in antiresonant hollow-core fibers,” Light: Sci. Appl. 6(12), e17124 (2017).
[Crossref]

S. Pumpe, M. Chemnitz, J. Kobelke, and M. A. Schmidt, “Monolithic optofluidic mode coupler for broadband thermo- and piezo-optical characterization of liquids,” Opt. Express 25(19), 22932–22946 (2017).
[Crossref]

A. Hartung, J. Kobelke, A. Schwuchow, K. Wondraczek, J. Bierlich, J. Popp, T. Frosch, and M. A. Schmidt, “Origins of modal loss of antiresonant hollow-core optical fibers in the ultraviolet,” Opt. Express 23(3), 2557–2565 (2015).
[Crossref]

N. Granzow, M. A. Schmidt, A. S. Tverjanovich, L. Wondraczek, and P. S. J. Russell, “Band-gap guidance in chalcogenide-silica photonic crystal fibers,” Opt. Lett. 36(13), 2432–2434 (2011).
[Crossref]

Schwuchow, A.

R. Sollapur, D. Kartashov, M. Zürch, A. Hoffmann, T. Grigorova, G. Sauer, A. Hartung, A. Schwuchow, J. Bierlich, J. Kobelke, M. Chemnitz, M. A. Schmidt, and C. Spielmann, “Resonance-enhanced multi-octave supercontinuum generation in antiresonant hollow-core fibers,” Light: Sci. Appl. 6(12), e17124 (2017).
[Crossref]

A. Hartung, J. Kobelke, A. Schwuchow, K. Wondraczek, J. Bierlich, J. Popp, T. Frosch, and M. A. Schmidt, “Origins of modal loss of antiresonant hollow-core optical fibers in the ultraviolet,” Opt. Express 23(3), 2557–2565 (2015).
[Crossref]

Skryabin, D. V.

Sollapur, R.

R. Sollapur, D. Kartashov, M. Zürch, A. Hoffmann, T. Grigorova, G. Sauer, A. Hartung, A. Schwuchow, J. Bierlich, J. Kobelke, M. Chemnitz, M. A. Schmidt, and C. Spielmann, “Resonance-enhanced multi-octave supercontinuum generation in antiresonant hollow-core fibers,” Light: Sci. Appl. 6(12), e17124 (2017).
[Crossref]

Spielmann, C.

R. Sollapur, D. Kartashov, M. Zürch, A. Hoffmann, T. Grigorova, G. Sauer, A. Hartung, A. Schwuchow, J. Bierlich, J. Kobelke, M. Chemnitz, M. A. Schmidt, and C. Spielmann, “Resonance-enhanced multi-octave supercontinuum generation in antiresonant hollow-core fibers,” Light: Sci. Appl. 6(12), e17124 (2017).
[Crossref]

Stentz, A. J.

Stifter, D.

Stj Russell, P.

F. Tani, F. Köttig, D. Novoa, R. Keding, and P. Stj Russell, “Effect of anti-crossings with cladding resonances on ultrafast nonlinear dynamics in gas-filled photonic crystal fibers,” Photonics Res. 6(2), 84–88 (2018).
[Crossref]

Tani, F.

F. Tani, F. Köttig, D. Novoa, R. Keding, and P. Stj Russell, “Effect of anti-crossings with cladding resonances on ultrafast nonlinear dynamics in gas-filled photonic crystal fibers,” Photonics Res. 6(2), 84–88 (2018).
[Crossref]

Travers, J. C.

X. Jiang, N. Y. Joly, M. A. Finger, F. Babic, G. K. L. Wong, J. C. Travers, and P. S. J. Russell, “Deep-ultraviolet to mid-infrared supercontinuum generated in solid-core ZBLAN photonic crystal fibre,” Nat. Photonics 9(2), 133–139 (2015).
[Crossref]

P. S. J. Russell, P. Hölzer, W. Chang, A. Abdolvand, and J. C. Travers, “Hollow-core photonic crystal fibres for gas-based nonlinear optics,” Nat. Photonics 8(4), 278–286 (2014).
[Crossref]

Tsoy, E. N.

E. N. Tsoy and C. M. De Sterke, “Theoretical analysis of the self-frequency shift near zero-dispersion points: Soliton spectral tunneling,” Phys. Rev. A 76(4), 043804 (2007).
[Crossref]

Tu, H.

H. Tu and S. A. Boppart, “Coherent fiber supercontinuum for biophotonics,” Laser Photonics Rev. 7(5), 628–645 (2013).
[Crossref]

Tverjanovich, A. S.

Udem, T.

T. Udem, R. Holzwarth, and T. W. Hänsch, “Optical frequency metrology,” Nature 416(6877), 233–237 (2002).
[Crossref]

Wadsworth, W. J.

Wang, C.

Wang, S.

Watt, R. S.

C. F. Kaminski, R. S. Watt, A. D. Elder, J. H. Frank, and J. Hult, “Supercontinuum radiation for applications in chemical sensing and microscopy,” Appl. Phys. B 92(3), 367–378 (2008).
[Crossref]

Wiesauer, K.

Windeler, R. S.

Wondraczek, K.

Wondraczek, L.

Wong, G. K. L.

X. Jiang, N. Y. Joly, M. A. Finger, F. Babic, G. K. L. Wong, J. C. Travers, and P. S. J. Russell, “Deep-ultraviolet to mid-infrared supercontinuum generated in solid-core ZBLAN photonic crystal fibre,” Nat. Photonics 9(2), 133–139 (2015).
[Crossref]

Xu, Y. Q.

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(22), 223904 (2012).
[Crossref]

Yulin, A. V.

Zaquine, I.

Zheltikov, A. M.

Zürch, M.

R. Sollapur, D. Kartashov, M. Zürch, A. Hoffmann, T. Grigorova, G. Sauer, A. Hartung, A. Schwuchow, J. Bierlich, J. Kobelke, M. Chemnitz, M. A. Schmidt, and C. Spielmann, “Resonance-enhanced multi-octave supercontinuum generation in antiresonant hollow-core fibers,” Light: Sci. Appl. 6(12), e17124 (2017).
[Crossref]

ACS Photonics (1)

C. Jain, A. Braun, J. Gargiulo, B. Jang, G. Li, H. Lehmann, S. A. Maier, and M. A. Schmidt, “Hollow Core Light Cage: Trapping Light behind Bars,” ACS Photonics 6(3), 649–658 (2019).
[Crossref]

Appl. Opt. (1)

Appl. Phys. B (1)

C. F. Kaminski, R. S. Watt, A. D. Elder, J. H. Frank, and J. Hult, “Supercontinuum radiation for applications in chemical sensing and microscopy,” Appl. Phys. B 92(3), 367–378 (2008).
[Crossref]

Bell Syst. Tech. J. (1)

E. A. J. Marcatili and R. A. Schmeltzer, “Hollow Metallic and Dielectric Waveguides for Long Distance Optical Transmission and Lasers,” Bell Syst. Tech. J. 43(4), 1783–1809 (1964).
[Crossref]

IEEE Photonics Technol. Lett. (1)

J. C. Knight, J. Arriaga, T. A. Birks, A. Ortigosa-Blanch, W. J. Wadsworth, and P. S. J. Russell, “Anomalous dispersion in photonic crystal fiber,” IEEE Photonics Technol. Lett. 12(7), 807–809 (2000).
[Crossref]

J. Opt. Soc. Am. B (1)

Laser Photonics Rev. (1)

H. Tu and S. A. Boppart, “Coherent fiber supercontinuum for biophotonics,” Laser Photonics Rev. 7(5), 628–645 (2013).
[Crossref]

Light: Sci. Appl. (1)

R. Sollapur, D. Kartashov, M. Zürch, A. Hoffmann, T. Grigorova, G. Sauer, A. Hartung, A. Schwuchow, J. Bierlich, J. Kobelke, M. Chemnitz, M. A. Schmidt, and C. Spielmann, “Resonance-enhanced multi-octave supercontinuum generation in antiresonant hollow-core fibers,” Light: Sci. Appl. 6(12), e17124 (2017).
[Crossref]

Nat. Photonics (2)

P. S. J. Russell, P. Hölzer, W. Chang, A. Abdolvand, and J. C. Travers, “Hollow-core photonic crystal fibres for gas-based nonlinear optics,” Nat. Photonics 8(4), 278–286 (2014).
[Crossref]

X. Jiang, N. Y. Joly, M. A. Finger, F. Babic, G. K. L. Wong, J. C. Travers, and P. S. J. Russell, “Deep-ultraviolet to mid-infrared supercontinuum generated in solid-core ZBLAN photonic crystal fibre,” Nat. Photonics 9(2), 133–139 (2015).
[Crossref]

Nature (1)

T. Udem, R. Holzwarth, and T. W. Hänsch, “Optical frequency metrology,” Nature 416(6877), 233–237 (2002).
[Crossref]

Opt. Express (7)

M. Cordier, A. Orieux, B. Debord, F. Gérome, A. Gorse, M. Chafer, E. Diamanti, P. Delaye, F. Benabid, and I. Zaquine, “Active engineering of four-wave mixing spectral correlations in multiband hollow-core fibers,” Opt. Express 27(7), 9803–9814 (2019).
[Crossref]

R. F. Ando, A. Hartung, B. Jang, and M. A. Schmidt, “Approximate model for analyzing band structures of single-ring hollow-core anti-resonant fibers,” Opt. Express 27(7), 10009–10021 (2019).
[Crossref]

A. Hartung, J. Kobelke, A. Schwuchow, K. Wondraczek, J. Bierlich, J. Popp, T. Frosch, and M. A. Schmidt, “Origins of modal loss of antiresonant hollow-core optical fibers in the ultraviolet,” Opt. Express 23(3), 2557–2565 (2015).
[Crossref]

S. Pumpe, M. Chemnitz, J. Kobelke, and M. A. Schmidt, “Monolithic optofluidic mode coupler for broadband thermo- and piezo-optical characterization of liquids,” Opt. Express 25(19), 22932–22946 (2017).
[Crossref]

F. Meng, B. Liu, S. Wang, J. Liu, Y. Li, C. Wang, A. M. Zheltikov, and M. Hu, “Controllable two-color dispersive wave generation in argon-filled hypocycloid-core kagome fiber,” Opt. Express 25(26), 32972–32984 (2017).
[Crossref]

G. Genty, M. Lehtonen, and H. Ludvigsen, “Effect of cross-phase modulation on supercontinuum generated in microstructured fibers with sub-30 fs pulse,” Opt. Express 12(19), 4614–4624 (2004).
[Crossref]

G. Humbert, W. J. Wadsworth, S. G. Leon-Saval, J. C. Knight, T. A. Birks, P. St. J. Russell, M. J. Lederer, D. Kopf, K. Wiesauer, E. I. Breuer, and D. Stifter, “Supercontinuum generation system for optical coherence tomography based on tapered photonic crystal fibre,” Opt. Express 14(4), 1596–1603 (2006).
[Crossref]

Opt. Lett. (6)

Photonics Res. (1)

F. Tani, F. Köttig, D. Novoa, R. Keding, and P. Stj Russell, “Effect of anti-crossings with cladding resonances on ultrafast nonlinear dynamics in gas-filled photonic crystal fibers,” Photonics Res. 6(2), 84–88 (2018).
[Crossref]

Phys. Rev. A (3)

E. N. Tsoy and C. M. De Sterke, “Theoretical analysis of the self-frequency shift near zero-dispersion points: Soliton spectral tunneling,” Phys. Rev. A 76(4), 043804 (2007).
[Crossref]

N. Akhmediev and M. Karlsson, “Cherenkov radiation emitted by solitons in optical fibers,” Phys. Rev. A 51(3), 2602–2607 (1995).
[Crossref]

S. Roy, S. K. Bhadra, and G. P. Agrawal, “Dispersive waves emitted by solitons perturbed by third-order dispersion inside optical fibers,” Phys. Rev. A 79(2), 023824 (2009).
[Crossref]

Phys. Rev. Lett. (1)

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(22), 223904 (2012).
[Crossref]

Rev. Mod. Phys. (1)

J. M. Dudley, G. Genty, and S. Coen, “Supercontinuum Generation in Photonic Crystal Fibre,” Rev. Mod. Phys. 78(4), 1135–1184 (2006).
[Crossref]

Science (2)

R. Cregan, B. J. Mangan, T. A. Birks, J. Knight, P. S. J. Russell, D. C. Allan, and P. Roberts, “Single-Mode Photonic Band Gap Guidance of Light in Air.,” Science 285(5433), 1537–1539 (1999).
[Crossref]

P. Russell, “Photonic crystal fibers,” Science 299(5605), 358–362 (2003).
[Crossref]

Other (3)

R. W. Boyd, Nonlinear Optics, 3rd Ed. (Academic Press, 2008).

J. D. Jackson, Classical Electrodynamics, 3rd Ed. (John Wiley & Sons: New York, 1999).

G. P. Agrawal, Nonlinear Fiber Optics, 5th Ed. (Academic Press, 2015)

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Figures (10)

Fig. 1.
Fig. 1. (a) Sketch of hollow-capillary waveguide. Spectral distribution of (b) phase indices and (c) group velocity dispersion in case the waveguide includes (solid red) or excludes (dashed blue) one resonance (simulation parameters are given in the main text). Colour bar represents different dispersion regimes (gray: normal dispersion; orange: anomalous dispersion). The two green dots in (c) represent the zero-dispersion frequencies of AR-HCF model.
Fig. 2.
Fig. 2. Simulated energy/spectral evolution of SCG process with ((a) and (b)) and without ((c) and (d)) resonance. The left column refers to a pump wavelength of $700\;\textrm{nm}$, the right to $850\; \textrm{nm}$. The black dotted lines represent the zero-dispersion frequencies, the magenta dashed lines the spectral location of the resonance. Corresponding dispersion domains are indicated on the top of each plot. The two plots on the right include the respective soliton numbers as additional $\textrm{y}$-axis scale.
Fig. 3.
Fig. 3. Simulated pulse propagation (bottom row) and phase-mismatching rate (red: DW, black: FWM) for different soliton numbers ((a)${\textrm{N}_\textrm{s}} = 7$, (b) ${\textrm{N}_\textrm{s}} = 10$, (c) ${\textrm{N}_\textrm{s}} = 15$). The 80 fs pump pulse is centered at 800 nm and the resonance at 930 nm. In the bottom plots, the black dotted lines represent the zero-dispersion frequencies, the magenta dashed lines the resonance frequencies.
Fig. 4.
Fig. 4. Dependence of frequencies of the generated DWs as functions of soliton number, determined from the energy/spectral evolutions using the example configurations shown in Fig. 3. The black dotted lines represent the zero-dispersion frequencies, the magenta dashed line the resonance frequency and the red dash-dotted line the pump frequency.
Fig. 5.
Fig. 5. Spectrograms at selected propagation distances inside the resonance-enhanced fiber calculated for the conditions of Fig. 3(c) (black dotted lines: zero-dispersion frequencies, magenta dashed line: resonance). The parabolic black dashed curve shows changes in group index with frequency (known as $\textrm{c}/{\textrm{v}_\textrm{g}}$).
Fig. 6.
Fig. 6. Simulated energy/spectral SC evolution for six different situations of relative spectral distance between pump and resonance wavelength (resonance wavelength: 930 nm, ${\textrm{A}_1} = 2.5\; \textrm{THz}$, ${\textrm{B}_1} = 10\; \textrm{THz}$, input pulse: 80 fs). The right axis shows the initial soliton number associated with the input pulse energy on the respective left axis. The black dotted lines represent the zero-dispersion frequencies, the magenta dashed lines the resonance frequencies.
Fig. 7.
Fig. 7. Spectral SC bandwidth (at −20 dB level, pump energy 12 µJ) and SC onset energy versus spectral distance between the pump and the resonance wavelength (resonance wavelength: 930 nm, ${\textrm{A}_1} = 2.5\; \textrm{THz}$, ${\textrm{B}_1} = 10\; \textrm{THz}$). The input pulse is located at 800 nm (pulse duration: 80 fs).
Fig. 8.
Fig. 8. Simulated energy/spectral SC evolution for the resonance-enhanced fiber situation when pumping within (a) the short-wavelength (${{\lambda }_\textrm{p}} = 800\; \textrm{nm}$, ${\Delta \lambda } > 0$) and (b) the long-wavelength (${{\lambda }_\textrm{p}} = 1430\textrm{nm}$, ${\Delta \lambda } < 0$) anomalous-dispersion region. The resonance is fixed to $930\; \textrm{nm}$ with amplitude ${\textrm{A}_1} = 2.5\; \textrm{THz}$ and damping ${\textrm{B}_1} = 10\; \textrm{THz}$. The input pulse has a FWHM of $80\; \textrm{fs}$. The black dotted lines represent the zero-dispersion frequencies, the magenta dashed lines the resonance frequencies.
Fig. 9.
Fig. 9. Impact of resonance amplitude ${\textrm{A}_1}$ on SCG. Calculated spectral distribution of the group velocity dispersion (a) and simulated energy/spectral SC evolution for three different resonance amplitude cases ((b-d), indicated by the label in the lower right corner of each plot) with the same damping ($\textrm{B}\, = \,10\; \textrm{THz}$)). Inset in (a) shows the ${{\beta }_2}$/${\omega }$ dependence on a larger scale. The black dotted lines in (b)-(d) represent the zero-dispersion frequencies, the magenta dashed lines the resonance frequencies.
Fig. 10.
Fig. 10. (a) Spectral distribution of the DW phase mismatch within the scope of the approximation defined in the main text (green: numerical solution incl. a non-zero damping parameter, dashed blue: analytic approximation using Eq. (5)). The curves refer to the following parameters: ${\textrm{n}_\textrm{g}} = 1$, ${{\lambda }_{\textrm{res}}} = 0.93$ µm, ${{\lambda }_{\textrm{sol}}} = 0.8$ µm, ${\textrm{B}_1} = 10\; \textrm{THz}$. The different brightness of the colour of the curves refer to different values of ${\textrm{A}_1}$ (from bright to dark: $0.5\; \textrm{THz}$, $1\; \textrm{THz}$, $1.5\; \textrm{THz}$, $2\; \textrm{THz}$, $2.5\; \textrm{THz}$, $3\; \textrm{THz}$). The yellow dots indicate the spectral location of the corresponding DW calculated using Eq. (6). (b) Normalized DW frequency as a function of resonance amplitude (using Eq. (6)) for various spectral separations of soliton and resonance ${\Delta }{{\lambda }_{\textrm{rs}}} = {{\lambda }_{\textrm{sol}}} - {{\lambda }_{\textrm{res}}}$ (from blue to red: $- 330\; \textrm{nm}$, $- 280\; \textrm{nm}$, $- 230\; \textrm{nm}$, $- 180\; \textrm{nm}$, $- 130\; \textrm{nm}$, $- 80\; \textrm{nm}$, $- 30\; \textrm{nm}$).

Tables (3)

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Table 1. Parameters and results of the configuration simulated in Fig. 2

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Table 2. Various parameters of the simulations shown in Fig. 6 (resonance wavelength: 930 nm). The third order dispersion length is given by L D = T 0 3 / | β 3 |

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Table 3. Parameters of the three configurations in Fig. 9 with different resonance amplitudes (pump: 800 nm)

Equations (6)

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n e f f n g ( ω , p , T ) 1 2 ( u 11 c R ω ) 2 + q A q 2 ω q 2 ω 2 + i B q ω
A ~ z i [ β ( ω ) β 0 β 1 ( ω ω 0 ) ] A ~ = i γ ( ω ) F 1 ( | A | 2 A )
Δ β D W ( ω ) = β ( ω ) [ β ( ω s o l ) + ( ω ω s o l ) β 1 , s o l + 1 2 γ P s o l ]
Δ β F W M ( ω ) = β ( ω s ) + β ( ω i ) 2 β ( ω 0 ) + 2 γ P 0
Δ β D W ( ω ) = 1 2 ( ω ω s o l ) 2 ( c 2 u 11 2 R 2 ω ω s o l 2 + A 1 2 ( ω 1 ω ) ( ω 1 ω s o l ) 2 )
ω D W n o r m = ( 1 + ( R A 1 ω s o l c 0 u 11 Δ ω r s ) 2 ) 1

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