Abstract

An efficient and tunable 176-550 nm source based on the emission of resonant dispersive radiation from ultrafast solitons at 800 nm is demonstrated in a gas-filled hollow-core photonic crystal fiber (PCF). By careful optimization and appropriate choice of gas, informed by detailed numerical simulations, we show that bright, high quality, localized bands of UV light (relative widths of a few percent) can be generated at all wavelengths across this range. Pulse energies of more than 75 nJ in the deep-UV, with relative bandwidths of ~3%, are generated from pump pulses of a few μJ. Excellent agreement is obtained between numerical and experimental results. The effects of positive and negative axial pressure gradients are also experimentally studied, and the coherence of the deep-UV dispersive wave radiation numerically investigated.

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    [CrossRef]
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    [CrossRef] [PubMed]

2012 (3)

2011 (6)

2010 (4)

2009 (2)

2006 (3)

D. R. Austin, C. M. de Sterke, B. J. Eggleton, and T. G. Brown, “Dispersive wave blue-shift in supercontinuum generation,” Opt. Express14(25), 11997–12007 (2006).
[CrossRef] [PubMed]

R. J. Bartula, J. W. Walewski, and S. T. Sanders, “Generation of ultraviolet broadband light in a single-mode fiber,” Appl. Phys. B84(3), 395–400 (2006).
[CrossRef]

J. M. Dudley, G. Genty, and S. Coen, “Supercontinuum generation in photonic crystal fiber,” Rev. Mod. Phys.78(4), 1135–1184 (2006).
[CrossRef]

2005 (2)

A. Suda, M. Hatayama, K. Nagasaka, and K. Midorikawa, “Generation of sub-10-fs, 5-mJ-optical pulses using a hollow fiber with a pressure gradient,” Appl. Phys. Lett.86(11), 111116 (2005).
[CrossRef]

G. Genty, M. Lehtonen, and H. Ludvigsen, “Route to broadband blue-light generation in microstructured fibers,” Opt. Lett.30(7), 756–758 (2005).
[CrossRef] [PubMed]

2004 (2)

I. Cristiani, R. Tediosi, L. Tartara, and V. Degiorgio, “Dispersive wave generation by solitons in microstructured optical fibers,” Opt. Express12(1), 124–135 (2004).
[CrossRef] [PubMed]

M. Kolesik and J. V. Moloney, “Nonlinear optical pulse propagation simulation: From Maxwell’s to unidirectional equations,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.70(3), 036604 (2004).
[CrossRef] [PubMed]

2003 (2)

L. Tartara, I. Cristiani, and V. Degiorgio, “Blue light and infrared continuum generation by soliton fission in a microstructured fiber,” Appl. Phys. B77(2-3), 307–311 (2003).
[CrossRef]

K. M. Hilligsøe, H. N. Paulsen, J. Thøgersen, S. R. Keiding, and J. J. Larsen, “Initial steps of supercontinuum generation in photonic crystal fibers,” J. Opt. Soc. Am. B20(9), 1887–1893 (2003).
[CrossRef]

2002 (4)

2001 (1)

A. V. Husakou and J. Herrmann, “Supercontinuum Generation of Higher-Order Solitons by Fission in Photonic Crystal Fibers,” Phys. Rev. Lett.87(20), 203901 (2001).
[CrossRef] [PubMed]

1995 (3)

G. A. Rines, H. H. Zenzie, R. A. Schwarz, Y. Isyanova, and P. F. Moulton, “Nonlinear conversion of Ti:sapphire laser wavelengths,” IEEE J. Sel. Top. Quantum Electron.1(1), 50–57 (1995).
[CrossRef]

N. Akhmediev and M. Karlsson, “Cherenkov radiation emitted by solitons in optical fibers,” Phys. Rev. A51(3), 2602–2607 (1995).
[CrossRef] [PubMed]

J. N. Elgin, T. Brabec, and S. M. J. Kelly, “A perturbative theory of soliton propagation in the presence of third order dispersion,” Opt. Commun.114(3-4), 321–328 (1995).
[CrossRef]

1993 (1)

V. I. Karpman, “Radiation by solitons due to higher-order dispersion,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics47(3), 2073–2082 (1993).
[CrossRef] [PubMed]

1986 (1)

1964 (1)

E. A. J. Marcatili and Schmeltzer, “Hollow metallic and dielectric waveguides for long distance optical transmission and lasers,” Bell Syst. Tech. J.43, 1783–1809 (1964).

Akhmediev, N.

N. Akhmediev and M. Karlsson, “Cherenkov radiation emitted by solitons in optical fibers,” Phys. Rev. A51(3), 2602–2607 (1995).
[CrossRef] [PubMed]

Austin, D. R.

D. R. Austin, C. M. de Sterke, B. J. Eggleton, and T. G. Brown, “Dispersive wave blue-shift in supercontinuum generation,” Opt. Express14(25), 11997–12007 (2006).
[CrossRef] [PubMed]

D. R. Austin, C. M. de Sterke, T. G. Brown, and B. J. Eggleton, “The nonlinear wavenumber in supercontinuum generation,” in Australian Conference on Optical Fibre Technology/Australian Optical Society, 2006. ACOFT/AOS 2006 (2006), p. 137.
[CrossRef]

Bartula, R. J.

R. J. Bartula, J. W. Walewski, and S. T. Sanders, “Generation of ultraviolet broadband light in a single-mode fiber,” Appl. Phys. B84(3), 395–400 (2006).
[CrossRef]

Biancalana, F.

N. Y. Joly, J. Nold, W. Chang, P. Hölzer, A. Nazarkin, G. K. L. Wong, F. Biancalana, and P. St. J. Russell, “Bright spatially coherent wavelength-tunable deep-UV laser source using an Ar-filled photonic crystal fiber,” Phys. Rev. Lett.106(20), 203901 (2011).
[CrossRef] [PubMed]

M. F. Saleh, W. Chang, P. Hölzer, A. Nazarkin, J. C. Travers, N. Y. Joly, P. St. J. Russell, and F. Biancalana, “Theory of Photoionization-Induced Blueshift of Ultrashort Solitons in Gas-Filled Hollow-Core Photonic Crystal Fibers,” Phys. Rev. Lett.107(20), 203902 (2011).
[CrossRef] [PubMed]

Bielawski, S.

N. Y. Joly, M. Labat, J. C. Travers, K. F. Mak, F. Tani, G. Gatti, T. Tanikawa, B. De Ninno, F. Mahieu, A. Ciocci, V. Petralia, V. Petrillo, J. V. Rau, and S. Bielawski, “Seeding of SPARC-FEL with a tunable fibre-based source,” in (2012).

Boppart, S. A.

Brabec, T.

J. N. Elgin, T. Brabec, and S. M. J. Kelly, “A perturbative theory of soliton propagation in the presence of third order dispersion,” Opt. Commun.114(3-4), 321–328 (1995).
[CrossRef]

Brown, T. G.

D. R. Austin, C. M. de Sterke, B. J. Eggleton, and T. G. Brown, “Dispersive wave blue-shift in supercontinuum generation,” Opt. Express14(25), 11997–12007 (2006).
[CrossRef] [PubMed]

D. R. Austin, C. M. de Sterke, T. G. Brown, and B. J. Eggleton, “The nonlinear wavenumber in supercontinuum generation,” in Australian Conference on Optical Fibre Technology/Australian Optical Society, 2006. ACOFT/AOS 2006 (2006), p. 137.
[CrossRef]

Chang, G.

Chang, W.

N. Y. Joly, J. Nold, W. Chang, P. Hölzer, A. Nazarkin, G. K. L. Wong, F. Biancalana, and P. St. J. Russell, “Bright spatially coherent wavelength-tunable deep-UV laser source using an Ar-filled photonic crystal fiber,” Phys. Rev. Lett.106(20), 203901 (2011).
[CrossRef] [PubMed]

J. C. Travers, W. Chang, J. Nold, N. Y. Joly, and P. St. J. Russell, “Ultrafast nonlinear optics in gas-filled hollow-core photonic crystal fibers,” J. Opt. Soc. Am. B28, A11–A26 (2011).
[CrossRef]

W. Chang, A. Nazarkin, J. C. Travers, J. Nold, P. Hölzer, N. Y. Joly, and P. St. J. Russell, “Influence of ionization on ultrafast gas-based nonlinear fiber optics,” Opt. Express19(21), 21018–21027 (2011).
[CrossRef] [PubMed]

M. F. Saleh, W. Chang, P. Hölzer, A. Nazarkin, J. C. Travers, N. Y. Joly, P. St. J. Russell, and F. Biancalana, “Theory of Photoionization-Induced Blueshift of Ultrashort Solitons in Gas-Filled Hollow-Core Photonic Crystal Fibers,” Phys. Rev. Lett.107(20), 203902 (2011).
[CrossRef] [PubMed]

Chen, C.-M.

Chen, H. H.

Chen, L.-J.

Cheng, J.

Ciocci, A.

N. Y. Joly, M. Labat, J. C. Travers, K. F. Mak, F. Tani, G. Gatti, T. Tanikawa, B. De Ninno, F. Mahieu, A. Ciocci, V. Petralia, V. Petrillo, J. V. Rau, and S. Bielawski, “Seeding of SPARC-FEL with a tunable fibre-based source,” in (2012).

Coen, S.

J. M. Dudley, G. Genty, and S. Coen, “Supercontinuum generation in photonic crystal fiber,” Rev. Mod. Phys.78(4), 1135–1184 (2006).
[CrossRef]

J. M. Dudley and S. Coen, “Coherence properties of supercontinuum spectra generated in photonic crystal and tapered optical fibers,” Opt. Lett.27(13), 1180–1182 (2002).
[CrossRef] [PubMed]

Cristiani, I.

I. Cristiani, R. Tediosi, L. Tartara, and V. Degiorgio, “Dispersive wave generation by solitons in microstructured optical fibers,” Opt. Express12(1), 124–135 (2004).
[CrossRef] [PubMed]

L. Tartara, I. Cristiani, and V. Degiorgio, “Blue light and infrared continuum generation by soliton fission in a microstructured fiber,” Appl. Phys. B77(2-3), 307–311 (2003).
[CrossRef]

De Ninno, B.

N. Y. Joly, M. Labat, J. C. Travers, K. F. Mak, F. Tani, G. Gatti, T. Tanikawa, B. De Ninno, F. Mahieu, A. Ciocci, V. Petralia, V. Petrillo, J. V. Rau, and S. Bielawski, “Seeding of SPARC-FEL with a tunable fibre-based source,” in (2012).

de Sterke, C. M.

D. R. Austin, C. M. de Sterke, B. J. Eggleton, and T. G. Brown, “Dispersive wave blue-shift in supercontinuum generation,” Opt. Express14(25), 11997–12007 (2006).
[CrossRef] [PubMed]

D. R. Austin, C. M. de Sterke, T. G. Brown, and B. J. Eggleton, “The nonlinear wavenumber in supercontinuum generation,” in Australian Conference on Optical Fibre Technology/Australian Optical Society, 2006. ACOFT/AOS 2006 (2006), p. 137.
[CrossRef]

Degiorgio, V.

I. Cristiani, R. Tediosi, L. Tartara, and V. Degiorgio, “Dispersive wave generation by solitons in microstructured optical fibers,” Opt. Express12(1), 124–135 (2004).
[CrossRef] [PubMed]

L. Tartara, I. Cristiani, and V. Degiorgio, “Blue light and infrared continuum generation by soliton fission in a microstructured fiber,” Appl. Phys. B77(2-3), 307–311 (2003).
[CrossRef]

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] [PubMed]

J. M. Dudley, G. Genty, and S. Coen, “Supercontinuum generation in photonic crystal fiber,” Rev. Mod. Phys.78(4), 1135–1184 (2006).
[CrossRef]

J. M. Dudley and S. Coen, “Coherence properties of supercontinuum spectra generated in photonic crystal and tapered optical fibers,” Opt. Lett.27(13), 1180–1182 (2002).
[CrossRef] [PubMed]

Eggleton, B. J.

D. R. Austin, C. M. de Sterke, B. J. Eggleton, and T. G. Brown, “Dispersive wave blue-shift in supercontinuum generation,” Opt. Express14(25), 11997–12007 (2006).
[CrossRef] [PubMed]

D. R. Austin, C. M. de Sterke, T. G. Brown, and B. J. Eggleton, “The nonlinear wavenumber in supercontinuum generation,” in Australian Conference on Optical Fibre Technology/Australian Optical Society, 2006. ACOFT/AOS 2006 (2006), p. 137.
[CrossRef]

Elgin, J. N.

J. N. Elgin, T. Brabec, and S. M. J. Kelly, “A perturbative theory of soliton propagation in the presence of third order dispersion,” Opt. Commun.114(3-4), 321–328 (1995).
[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] [PubMed]

Garmund, M.

Gatti, G.

N. Y. Joly, M. Labat, J. C. Travers, K. F. Mak, F. Tani, G. Gatti, T. Tanikawa, B. De Ninno, F. Mahieu, A. Ciocci, V. Petralia, V. Petrillo, J. V. Rau, and S. Bielawski, “Seeding of SPARC-FEL with a tunable fibre-based source,” in (2012).

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] [PubMed]

J. M. Dudley, G. Genty, and S. Coen, “Supercontinuum generation in photonic crystal fiber,” Rev. Mod. Phys.78(4), 1135–1184 (2006).
[CrossRef]

G. Genty, M. Lehtonen, and H. Ludvigsen, “Route to broadband blue-light generation in microstructured fibers,” Opt. Lett.30(7), 756–758 (2005).
[CrossRef] [PubMed]

Griebner, U.

J. Herrmann, U. Griebner, N. Zhavoronkov, A. Husakou, D. Nickel, J. C. Knight, W. J. Wadsworth, P. St. J. Russell, and G. Korn, “Experimental evidence for supercontinuum generation by fission of higher-order solitons in photonic fibers,” Phys. Rev. Lett.88(17), 173901 (2002).
[CrossRef] [PubMed]

Grüner-Nielsen, L.

Hatayama, M.

A. Suda, M. Hatayama, K. Nagasaka, and K. Midorikawa, “Generation of sub-10-fs, 5-mJ-optical pulses using a hollow fiber with a pressure gradient,” Appl. Phys. Lett.86(11), 111116 (2005).
[CrossRef]

Herrmann, J.

S.-J. Im, A. Husakou, and J. Herrmann, “High-power soliton-induced supercontinuum generation and tunable sub-10-fs VUV pulses from kagome-lattice HC-PCFs,” Opt. Express18(6), 5367–5374 (2010).
[CrossRef] [PubMed]

S.-J. Im, A. Husakou, and J. Herrmann, “Guiding properties and dispersion control of kagome lattice hollow-core photonic crystal fibers,” Opt. Express17(15), 13050–13058 (2009).
[CrossRef] [PubMed]

J. Herrmann, U. Griebner, N. Zhavoronkov, A. Husakou, D. Nickel, J. C. Knight, W. J. Wadsworth, P. St. J. Russell, and G. Korn, “Experimental evidence for supercontinuum generation by fission of higher-order solitons in photonic fibers,” Phys. Rev. Lett.88(17), 173901 (2002).
[CrossRef] [PubMed]

A. V. Husakou and J. Herrmann, “Supercontinuum Generation of Higher-Order Solitons by Fission in Photonic Crystal Fibers,” Phys. Rev. Lett.87(20), 203901 (2001).
[CrossRef] [PubMed]

Hilligsøe, K. M.

Hölzer, P.

N. Y. Joly, J. Nold, W. Chang, P. Hölzer, A. Nazarkin, G. K. L. Wong, F. Biancalana, and P. St. J. Russell, “Bright spatially coherent wavelength-tunable deep-UV laser source using an Ar-filled photonic crystal fiber,” Phys. Rev. Lett.106(20), 203901 (2011).
[CrossRef] [PubMed]

W. Chang, A. Nazarkin, J. C. Travers, J. Nold, P. Hölzer, N. Y. Joly, and P. St. J. Russell, “Influence of ionization on ultrafast gas-based nonlinear fiber optics,” Opt. Express19(21), 21018–21027 (2011).
[CrossRef] [PubMed]

M. F. Saleh, W. Chang, P. Hölzer, A. Nazarkin, J. C. Travers, N. Y. Joly, P. St. J. Russell, and F. Biancalana, “Theory of Photoionization-Induced Blueshift of Ultrashort Solitons in Gas-Filled Hollow-Core Photonic Crystal Fibers,” Phys. Rev. Lett.107(20), 203902 (2011).
[CrossRef] [PubMed]

J. Nold, P. Hölzer, N. Y. Joly, G. K. L. Wong, A. Nazarkin, A. Podlipensky, M. Scharrer, and P. St. J. Russell, “Pressure-controlled phase matching to third harmonic in Ar-filled hollow-core photonic crystal fiber,” Opt. Lett.35(17), 2922–2924 (2010).
[CrossRef] [PubMed]

Homann, C.

Husakou, A.

Husakou, A. V.

A. V. Husakou and J. Herrmann, “Supercontinuum Generation of Higher-Order Solitons by Fission in Photonic Crystal Fibers,” Phys. Rev. Lett.87(20), 203901 (2001).
[CrossRef] [PubMed]

Im, S.-J.

Isyanova, Y.

G. A. Rines, H. H. Zenzie, R. A. Schwarz, Y. Isyanova, and P. F. Moulton, “Nonlinear conversion of Ti:sapphire laser wavelengths,” IEEE J. Sel. Top. Quantum Electron.1(1), 50–57 (1995).
[CrossRef]

Jakobsen, D.

Jespersen, K. G.

Joly, N. Y.

N. Y. Joly, J. Nold, W. Chang, P. Hölzer, A. Nazarkin, G. K. L. Wong, F. Biancalana, and P. St. J. Russell, “Bright spatially coherent wavelength-tunable deep-UV laser source using an Ar-filled photonic crystal fiber,” Phys. Rev. Lett.106(20), 203901 (2011).
[CrossRef] [PubMed]

J. C. Travers, W. Chang, J. Nold, N. Y. Joly, and P. St. J. Russell, “Ultrafast nonlinear optics in gas-filled hollow-core photonic crystal fibers,” J. Opt. Soc. Am. B28, A11–A26 (2011).
[CrossRef]

W. Chang, A. Nazarkin, J. C. Travers, J. Nold, P. Hölzer, N. Y. Joly, and P. St. J. Russell, “Influence of ionization on ultrafast gas-based nonlinear fiber optics,” Opt. Express19(21), 21018–21027 (2011).
[CrossRef] [PubMed]

M. F. Saleh, W. Chang, P. Hölzer, A. Nazarkin, J. C. Travers, N. Y. Joly, P. St. J. Russell, and F. Biancalana, “Theory of Photoionization-Induced Blueshift of Ultrashort Solitons in Gas-Filled Hollow-Core Photonic Crystal Fibers,” Phys. Rev. Lett.107(20), 203902 (2011).
[CrossRef] [PubMed]

J. Nold, P. Hölzer, N. Y. Joly, G. K. L. Wong, A. Nazarkin, A. Podlipensky, M. Scharrer, and P. St. J. Russell, “Pressure-controlled phase matching to third harmonic in Ar-filled hollow-core photonic crystal fiber,” Opt. Lett.35(17), 2922–2924 (2010).
[CrossRef] [PubMed]

N. Y. Joly, M. Labat, J. C. Travers, K. F. Mak, F. Tani, G. Gatti, T. Tanikawa, B. De Ninno, F. Mahieu, A. Ciocci, V. Petralia, V. Petrillo, J. V. Rau, and S. Bielawski, “Seeding of SPARC-FEL with a tunable fibre-based source,” in (2012).

Karlsson, M.

N. Akhmediev and M. Karlsson, “Cherenkov radiation emitted by solitons in optical fibers,” Phys. Rev. A51(3), 2602–2607 (1995).
[CrossRef] [PubMed]

Karpman, V. I.

V. I. Karpman, “Radiation by solitons due to higher-order dispersion,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics47(3), 2073–2082 (1993).
[CrossRef] [PubMed]

Kärtner, F. X.

Keiding, S. R.

Kelley, P. L.

Kelly, S. M. J.

J. N. Elgin, T. Brabec, and S. M. J. Kelly, “A perturbative theory of soliton propagation in the presence of third order dispersion,” Opt. Commun.114(3-4), 321–328 (1995).
[CrossRef]

Kinsler, P.

P. Kinsler, “Optical pulse propagation with minimal approximations,” Phys. Rev. A81(1), 013819 (2010).
[CrossRef]

Knight, J. C.

A. Ortigosa-Blanch, J. C. Knight, and P. St. J. Russell, “Pulse breaking and supercontinuum generation with 200-fs pump pulses in photonic crystal fibers,” J. Opt. Soc. Am. B19(11), 2567–2572 (2002).
[CrossRef]

J. Herrmann, U. Griebner, N. Zhavoronkov, A. Husakou, D. Nickel, J. C. Knight, W. J. Wadsworth, P. St. J. Russell, and G. Korn, “Experimental evidence for supercontinuum generation by fission of higher-order solitons in photonic fibers,” Phys. Rev. Lett.88(17), 173901 (2002).
[CrossRef] [PubMed]

Kolesik, M.

M. Kolesik and J. V. Moloney, “Nonlinear optical pulse propagation simulation: From Maxwell’s to unidirectional equations,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.70(3), 036604 (2004).
[CrossRef] [PubMed]

Korn, G.

J. Herrmann, U. Griebner, N. Zhavoronkov, A. Husakou, D. Nickel, J. C. Knight, W. J. Wadsworth, P. St. J. Russell, and G. Korn, “Experimental evidence for supercontinuum generation by fission of higher-order solitons in photonic fibers,” Phys. Rev. Lett.88(17), 173901 (2002).
[CrossRef] [PubMed]

Labat, M.

N. Y. Joly, M. Labat, J. C. Travers, K. F. Mak, F. Tani, G. Gatti, T. Tanikawa, B. De Ninno, F. Mahieu, A. Ciocci, V. Petralia, V. Petrillo, J. V. Rau, and S. Bielawski, “Seeding of SPARC-FEL with a tunable fibre-based source,” in (2012).

Lang, P.

Larsen, J. J.

Lee, J. H.

Lee, Y. C.

Lehtonen, M.

Ludvigsen, H.

Mahieu, F.

N. Y. Joly, M. Labat, J. C. Travers, K. F. Mak, F. Tani, G. Gatti, T. Tanikawa, B. De Ninno, F. Mahieu, A. Ciocci, V. Petralia, V. Petrillo, J. V. Rau, and S. Bielawski, “Seeding of SPARC-FEL with a tunable fibre-based source,” in (2012).

Mak, K. F.

N. Y. Joly, M. Labat, J. C. Travers, K. F. Mak, F. Tani, G. Gatti, T. Tanikawa, B. De Ninno, F. Mahieu, A. Ciocci, V. Petralia, V. Petrillo, J. V. Rau, and S. Bielawski, “Seeding of SPARC-FEL with a tunable fibre-based source,” in (2012).

Marcatili, E. A. J.

E. A. J. Marcatili and Schmeltzer, “Hollow metallic and dielectric waveguides for long distance optical transmission and lasers,” Bell Syst. Tech. J.43, 1783–1809 (1964).

Menyuk, C. R.

Midorikawa, K.

A. Suda, M. Hatayama, K. Nagasaka, and K. Midorikawa, “Generation of sub-10-fs, 5-mJ-optical pulses using a hollow fiber with a pressure gradient,” Appl. Phys. Lett.86(11), 111116 (2005).
[CrossRef]

Moloney, J. V.

M. Kolesik and J. V. Moloney, “Nonlinear optical pulse propagation simulation: From Maxwell’s to unidirectional equations,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.70(3), 036604 (2004).
[CrossRef] [PubMed]

Moulton, P. F.

G. A. Rines, H. H. Zenzie, R. A. Schwarz, Y. Isyanova, and P. F. Moulton, “Nonlinear conversion of Ti:sapphire laser wavelengths,” IEEE J. Sel. Top. Quantum Electron.1(1), 50–57 (1995).
[CrossRef]

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] [PubMed]

Nagasaka, K.

A. Suda, M. Hatayama, K. Nagasaka, and K. Midorikawa, “Generation of sub-10-fs, 5-mJ-optical pulses using a hollow fiber with a pressure gradient,” Appl. Phys. Lett.86(11), 111116 (2005).
[CrossRef]

Nazarkin, A.

W. Chang, A. Nazarkin, J. C. Travers, J. Nold, P. Hölzer, N. Y. Joly, and P. St. J. Russell, “Influence of ionization on ultrafast gas-based nonlinear fiber optics,” Opt. Express19(21), 21018–21027 (2011).
[CrossRef] [PubMed]

N. Y. Joly, J. Nold, W. Chang, P. Hölzer, A. Nazarkin, G. K. L. Wong, F. Biancalana, and P. St. J. Russell, “Bright spatially coherent wavelength-tunable deep-UV laser source using an Ar-filled photonic crystal fiber,” Phys. Rev. Lett.106(20), 203901 (2011).
[CrossRef] [PubMed]

M. F. Saleh, W. Chang, P. Hölzer, A. Nazarkin, J. C. Travers, N. Y. Joly, P. St. J. Russell, and F. Biancalana, “Theory of Photoionization-Induced Blueshift of Ultrashort Solitons in Gas-Filled Hollow-Core Photonic Crystal Fibers,” Phys. Rev. Lett.107(20), 203902 (2011).
[CrossRef] [PubMed]

J. Nold, P. Hölzer, N. Y. Joly, G. K. L. Wong, A. Nazarkin, A. Podlipensky, M. Scharrer, and P. St. J. Russell, “Pressure-controlled phase matching to third harmonic in Ar-filled hollow-core photonic crystal fiber,” Opt. Lett.35(17), 2922–2924 (2010).
[CrossRef] [PubMed]

Nickel, D.

J. Herrmann, U. Griebner, N. Zhavoronkov, A. Husakou, D. Nickel, J. C. Knight, W. J. Wadsworth, P. St. J. Russell, and G. Korn, “Experimental evidence for supercontinuum generation by fission of higher-order solitons in photonic fibers,” Phys. Rev. Lett.88(17), 173901 (2002).
[CrossRef] [PubMed]

Nold, J.

Ortigosa-Blanch, A.

Paulsen, H. N.

Petralia, V.

N. Y. Joly, M. Labat, J. C. Travers, K. F. Mak, F. Tani, G. Gatti, T. Tanikawa, B. De Ninno, F. Mahieu, A. Ciocci, V. Petralia, V. Petrillo, J. V. Rau, and S. Bielawski, “Seeding of SPARC-FEL with a tunable fibre-based source,” in (2012).

Petrillo, V.

N. Y. Joly, M. Labat, J. C. Travers, K. F. Mak, F. Tani, G. Gatti, T. Tanikawa, B. De Ninno, F. Mahieu, A. Ciocci, V. Petralia, V. Petrillo, J. V. Rau, and S. Bielawski, “Seeding of SPARC-FEL with a tunable fibre-based source,” in (2012).

Podlipensky, A.

Rau, J. V.

N. Y. Joly, M. Labat, J. C. Travers, K. F. Mak, F. Tani, G. Gatti, T. Tanikawa, B. De Ninno, F. Mahieu, A. Ciocci, V. Petralia, V. Petrillo, J. V. Rau, and S. Bielawski, “Seeding of SPARC-FEL with a tunable fibre-based source,” in (2012).

Riedle, E.

Rines, G. A.

G. A. Rines, H. H. Zenzie, R. A. Schwarz, Y. Isyanova, and P. F. Moulton, “Nonlinear conversion of Ti:sapphire laser wavelengths,” IEEE J. Sel. Top. Quantum Electron.1(1), 50–57 (1995).
[CrossRef]

Russell, P. St. J.

S. P. Stark, J. C. Travers, and P. St. J. Russell, “Extreme supercontinuum generation to the deep UV,” Opt. Lett.37(5), 770–772 (2012).
[CrossRef] [PubMed]

N. Y. Joly, J. Nold, W. Chang, P. Hölzer, A. Nazarkin, G. K. L. Wong, F. Biancalana, and P. St. J. Russell, “Bright spatially coherent wavelength-tunable deep-UV laser source using an Ar-filled photonic crystal fiber,” Phys. Rev. Lett.106(20), 203901 (2011).
[CrossRef] [PubMed]

W. Chang, A. Nazarkin, J. C. Travers, J. Nold, P. Hölzer, N. Y. Joly, and P. St. J. Russell, “Influence of ionization on ultrafast gas-based nonlinear fiber optics,” Opt. Express19(21), 21018–21027 (2011).
[CrossRef] [PubMed]

J. C. Travers, W. Chang, J. Nold, N. Y. Joly, and P. St. J. Russell, “Ultrafast nonlinear optics in gas-filled hollow-core photonic crystal fibers,” J. Opt. Soc. Am. B28, A11–A26 (2011).
[CrossRef]

M. F. Saleh, W. Chang, P. Hölzer, A. Nazarkin, J. C. Travers, N. Y. Joly, P. St. J. Russell, and F. Biancalana, “Theory of Photoionization-Induced Blueshift of Ultrashort Solitons in Gas-Filled Hollow-Core Photonic Crystal Fibers,” Phys. Rev. Lett.107(20), 203902 (2011).
[CrossRef] [PubMed]

J. Nold, P. Hölzer, N. Y. Joly, G. K. L. Wong, A. Nazarkin, A. Podlipensky, M. Scharrer, and P. St. J. Russell, “Pressure-controlled phase matching to third harmonic in Ar-filled hollow-core photonic crystal fiber,” Opt. Lett.35(17), 2922–2924 (2010).
[CrossRef] [PubMed]

J. Herrmann, U. Griebner, N. Zhavoronkov, A. Husakou, D. Nickel, J. C. Knight, W. J. Wadsworth, P. St. J. Russell, and G. Korn, “Experimental evidence for supercontinuum generation by fission of higher-order solitons in photonic fibers,” Phys. Rev. Lett.88(17), 173901 (2002).
[CrossRef] [PubMed]

A. Ortigosa-Blanch, J. C. Knight, and P. St. J. Russell, “Pulse breaking and supercontinuum generation with 200-fs pump pulses in photonic crystal fibers,” J. Opt. Soc. Am. B19(11), 2567–2572 (2002).
[CrossRef]

F. Tani, J. C. Travers, and P. St. J. Russell, in preparation.

Saleh, M. F.

M. F. Saleh, W. Chang, P. Hölzer, A. Nazarkin, J. C. Travers, N. Y. Joly, P. St. J. Russell, and F. Biancalana, “Theory of Photoionization-Induced Blueshift of Ultrashort Solitons in Gas-Filled Hollow-Core Photonic Crystal Fibers,” Phys. Rev. Lett.107(20), 203902 (2011).
[CrossRef] [PubMed]

Sanders, S. T.

R. J. Bartula, J. W. Walewski, and S. T. Sanders, “Generation of ultraviolet broadband light in a single-mode fiber,” Appl. Phys. B84(3), 395–400 (2006).
[CrossRef]

Scharrer, M.

Schwarz, R. A.

G. A. Rines, H. H. Zenzie, R. A. Schwarz, Y. Isyanova, and P. F. Moulton, “Nonlinear conversion of Ti:sapphire laser wavelengths,” IEEE J. Sel. Top. Quantum Electron.1(1), 50–57 (1995).
[CrossRef]

Stark, S. P.

Suda, A.

A. Suda, M. Hatayama, K. Nagasaka, and K. Midorikawa, “Generation of sub-10-fs, 5-mJ-optical pulses using a hollow fiber with a pressure gradient,” Appl. Phys. Lett.86(11), 111116 (2005).
[CrossRef]

Tani, F.

N. Y. Joly, M. Labat, J. C. Travers, K. F. Mak, F. Tani, G. Gatti, T. Tanikawa, B. De Ninno, F. Mahieu, A. Ciocci, V. Petralia, V. Petrillo, J. V. Rau, and S. Bielawski, “Seeding of SPARC-FEL with a tunable fibre-based source,” in (2012).

F. Tani, J. C. Travers, and P. St. J. Russell, in preparation.

Tanikawa, T.

N. Y. Joly, M. Labat, J. C. Travers, K. F. Mak, F. Tani, G. Gatti, T. Tanikawa, B. De Ninno, F. Mahieu, A. Ciocci, V. Petralia, V. Petrillo, J. V. Rau, and S. Bielawski, “Seeding of SPARC-FEL with a tunable fibre-based source,” in (2012).

Tartara, L.

I. Cristiani, R. Tediosi, L. Tartara, and V. Degiorgio, “Dispersive wave generation by solitons in microstructured optical fibers,” Opt. Express12(1), 124–135 (2004).
[CrossRef] [PubMed]

L. Tartara, I. Cristiani, and V. Degiorgio, “Blue light and infrared continuum generation by soliton fission in a microstructured fiber,” Appl. Phys. B77(2-3), 307–311 (2003).
[CrossRef]

Tediosi, R.

Thøgersen, J.

Travers, J. C.

S. P. Stark, J. C. Travers, and P. St. J. Russell, “Extreme supercontinuum generation to the deep UV,” Opt. Lett.37(5), 770–772 (2012).
[CrossRef] [PubMed]

W. Chang, A. Nazarkin, J. C. Travers, J. Nold, P. Hölzer, N. Y. Joly, and P. St. J. Russell, “Influence of ionization on ultrafast gas-based nonlinear fiber optics,” Opt. Express19(21), 21018–21027 (2011).
[CrossRef] [PubMed]

J. C. Travers, W. Chang, J. Nold, N. Y. Joly, and P. St. J. Russell, “Ultrafast nonlinear optics in gas-filled hollow-core photonic crystal fibers,” J. Opt. Soc. Am. B28, A11–A26 (2011).
[CrossRef]

M. F. Saleh, W. Chang, P. Hölzer, A. Nazarkin, J. C. Travers, N. Y. Joly, P. St. J. Russell, and F. Biancalana, “Theory of Photoionization-Induced Blueshift of Ultrashort Solitons in Gas-Filled Hollow-Core Photonic Crystal Fibers,” Phys. Rev. Lett.107(20), 203902 (2011).
[CrossRef] [PubMed]

F. Tani, J. C. Travers, and P. St. J. Russell, in preparation.

N. Y. Joly, M. Labat, J. C. Travers, K. F. Mak, F. Tani, G. Gatti, T. Tanikawa, B. De Ninno, F. Mahieu, A. Ciocci, V. Petralia, V. Petrillo, J. V. Rau, and S. Bielawski, “Seeding of SPARC-FEL with a tunable fibre-based source,” in (2012).

Tu, H.

Wadsworth, W. J.

J. Herrmann, U. Griebner, N. Zhavoronkov, A. Husakou, D. Nickel, J. C. Knight, W. J. Wadsworth, P. St. J. Russell, and G. Korn, “Experimental evidence for supercontinuum generation by fission of higher-order solitons in photonic fibers,” Phys. Rev. Lett.88(17), 173901 (2002).
[CrossRef] [PubMed]

Wai, P. K. A.

Walewski, J. W.

R. J. Bartula, J. W. Walewski, and S. T. Sanders, “Generation of ultraviolet broadband light in a single-mode fiber,” Appl. Phys. B84(3), 395–400 (2006).
[CrossRef]

Wang, K.

Wong, G. K. L.

N. Y. Joly, J. Nold, W. Chang, P. Hölzer, A. Nazarkin, G. K. L. Wong, F. Biancalana, and P. St. J. Russell, “Bright spatially coherent wavelength-tunable deep-UV laser source using an Ar-filled photonic crystal fiber,” Phys. Rev. Lett.106(20), 203901 (2011).
[CrossRef] [PubMed]

J. Nold, P. Hölzer, N. Y. Joly, G. K. L. Wong, A. Nazarkin, A. Podlipensky, M. Scharrer, and P. St. J. Russell, “Pressure-controlled phase matching to third harmonic in Ar-filled hollow-core photonic crystal fiber,” Opt. Lett.35(17), 2922–2924 (2010).
[CrossRef] [PubMed]

Xu, C.

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] [PubMed]

Zenzie, H. H.

G. A. Rines, H. H. Zenzie, R. A. Schwarz, Y. Isyanova, and P. F. Moulton, “Nonlinear conversion of Ti:sapphire laser wavelengths,” IEEE J. Sel. Top. Quantum Electron.1(1), 50–57 (1995).
[CrossRef]

Zhavoronkov, N.

J. Herrmann, U. Griebner, N. Zhavoronkov, A. Husakou, D. Nickel, J. C. Knight, W. J. Wadsworth, P. St. J. Russell, and G. Korn, “Experimental evidence for supercontinuum generation by fission of higher-order solitons in photonic fibers,” Phys. Rev. Lett.88(17), 173901 (2002).
[CrossRef] [PubMed]

Appl. Phys. B (2)

R. J. Bartula, J. W. Walewski, and S. T. Sanders, “Generation of ultraviolet broadband light in a single-mode fiber,” Appl. Phys. B84(3), 395–400 (2006).
[CrossRef]

L. Tartara, I. Cristiani, and V. Degiorgio, “Blue light and infrared continuum generation by soliton fission in a microstructured fiber,” Appl. Phys. B77(2-3), 307–311 (2003).
[CrossRef]

Appl. Phys. Lett. (1)

A. Suda, M. Hatayama, K. Nagasaka, and K. Midorikawa, “Generation of sub-10-fs, 5-mJ-optical pulses using a hollow fiber with a pressure gradient,” Appl. Phys. Lett.86(11), 111116 (2005).
[CrossRef]

Bell Syst. Tech. J. (1)

E. A. J. Marcatili and Schmeltzer, “Hollow metallic and dielectric waveguides for long distance optical transmission and lasers,” Bell Syst. Tech. J.43, 1783–1809 (1964).

IEEE J. Sel. Top. Quantum Electron. (1)

G. A. Rines, H. H. Zenzie, R. A. Schwarz, Y. Isyanova, and P. F. Moulton, “Nonlinear conversion of Ti:sapphire laser wavelengths,” IEEE J. Sel. Top. Quantum Electron.1(1), 50–57 (1995).
[CrossRef]

J. Opt. Soc. Am. B (5)

Opt. Commun. (1)

J. N. Elgin, T. Brabec, and S. M. J. Kelly, “A perturbative theory of soliton propagation in the presence of third order dispersion,” Opt. Commun.114(3-4), 321–328 (1995).
[CrossRef]

Opt. Express (8)

W. Chang, A. Nazarkin, J. C. Travers, J. Nold, P. Hölzer, N. Y. Joly, and P. St. J. Russell, “Influence of ionization on ultrafast gas-based nonlinear fiber optics,” Opt. Express19(21), 21018–21027 (2011).
[CrossRef] [PubMed]

I. Cristiani, R. Tediosi, L. Tartara, and V. Degiorgio, “Dispersive wave generation by solitons in microstructured optical fibers,” Opt. Express12(1), 124–135 (2004).
[CrossRef] [PubMed]

S.-J. Im, A. Husakou, and J. Herrmann, “Guiding properties and dispersion control of kagome lattice hollow-core photonic crystal fibers,” Opt. Express17(15), 13050–13058 (2009).
[CrossRef] [PubMed]

D. R. Austin, C. M. de Sterke, B. J. Eggleton, and T. G. Brown, “Dispersive wave blue-shift in supercontinuum generation,” Opt. Express14(25), 11997–12007 (2006).
[CrossRef] [PubMed]

H. Tu and S. A. Boppart, “Optical frequency up-conversion by supercontinuum-free widely-tunable fiber-optic Cherenkov radiation,” Opt. Express17(12), 9858–9872 (2009).
[CrossRef] [PubMed]

J. Cheng, J. H. Lee, K. Wang, C. Xu, K. G. Jespersen, M. Garmund, L. Grüner-Nielsen, and D. Jakobsen, “Generation of Cerenkov radiation at 850 nm in higher-order-mode fiber,” Opt. Express19(9), 8774–8780 (2011).
[CrossRef] [PubMed]

G. Chang, L.-J. Chen, and F. X. Kärtner, “Fiber-optic Cherenkov radiation in the few-cycle regime,” Opt. Express19(7), 6635–6647 (2011).
[CrossRef] [PubMed]

S.-J. Im, A. Husakou, and J. Herrmann, “High-power soliton-induced supercontinuum generation and tunable sub-10-fs VUV pulses from kagome-lattice HC-PCFs,” Opt. Express18(6), 5367–5374 (2010).
[CrossRef] [PubMed]

Opt. Lett. (6)

Phys. Rev. A (2)

N. Akhmediev and M. Karlsson, “Cherenkov radiation emitted by solitons in optical fibers,” Phys. Rev. A51(3), 2602–2607 (1995).
[CrossRef] [PubMed]

P. Kinsler, “Optical pulse propagation with minimal approximations,” Phys. Rev. A81(1), 013819 (2010).
[CrossRef]

Phys. Rev. E Stat. Nonlin. Soft Matter Phys. (1)

M. Kolesik and J. V. Moloney, “Nonlinear optical pulse propagation simulation: From Maxwell’s to unidirectional equations,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.70(3), 036604 (2004).
[CrossRef] [PubMed]

Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics (1)

V. I. Karpman, “Radiation by solitons due to higher-order dispersion,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics47(3), 2073–2082 (1993).
[CrossRef] [PubMed]

Phys. Rev. Lett. (5)

N. Y. Joly, J. Nold, W. Chang, P. Hölzer, A. Nazarkin, G. K. L. Wong, F. Biancalana, and P. St. J. Russell, “Bright spatially coherent wavelength-tunable deep-UV laser source using an Ar-filled photonic crystal fiber,” Phys. Rev. Lett.106(20), 203901 (2011).
[CrossRef] [PubMed]

A. V. Husakou and J. Herrmann, “Supercontinuum Generation of Higher-Order Solitons by Fission in Photonic Crystal Fibers,” Phys. Rev. Lett.87(20), 203901 (2001).
[CrossRef] [PubMed]

J. Herrmann, U. Griebner, N. Zhavoronkov, A. Husakou, D. Nickel, J. C. Knight, W. J. Wadsworth, P. St. J. Russell, and G. Korn, “Experimental evidence for supercontinuum generation by fission of higher-order solitons in photonic fibers,” Phys. Rev. Lett.88(17), 173901 (2002).
[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(22), 223904 (2012).
[CrossRef] [PubMed]

M. F. Saleh, W. Chang, P. Hölzer, A. Nazarkin, J. C. Travers, N. Y. Joly, P. St. J. Russell, and F. Biancalana, “Theory of Photoionization-Induced Blueshift of Ultrashort Solitons in Gas-Filled Hollow-Core Photonic Crystal Fibers,” Phys. Rev. Lett.107(20), 203902 (2011).
[CrossRef] [PubMed]

Rev. Mod. Phys. (1)

J. M. Dudley, G. Genty, and S. Coen, “Supercontinuum generation in photonic crystal fiber,” Rev. Mod. Phys.78(4), 1135–1184 (2006).
[CrossRef]

Other (5)

N. Y. Joly, M. Labat, J. C. Travers, K. F. Mak, F. Tani, G. Gatti, T. Tanikawa, B. De Ninno, F. Mahieu, A. Ciocci, V. Petralia, V. Petrillo, J. V. Rau, and S. Bielawski, “Seeding of SPARC-FEL with a tunable fibre-based source,” in (2012).

K. F. Mak, J. C. Travers, P. Hoelzer, W. Chang, F. Tani, F. Vinzent, N. Joly, and P. St. J. Russell, “Interaction between Kerr and Ionization Induced Nonlinear Fiber Optics,” in CLEO: Science and Innovations, OSA Technical Digest (online) (Optical Society of America, 2012), p. CM1J.5.

D. R. Austin, C. M. de Sterke, T. G. Brown, and B. J. Eggleton, “The nonlinear wavenumber in supercontinuum generation,” in Australian Conference on Optical Fibre Technology/Australian Optical Society, 2006. ACOFT/AOS 2006 (2006), p. 137.
[CrossRef]

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

F. Tani, J. C. Travers, and P. St. J. Russell, in preparation.

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Figures (9)

Fig. 1
Fig. 1

(a) Measured loss (solid curve) and calculated dispersion characteristics (dashed curves) of a kagomé fiber with 27 µm core diameter and filled with Ar at three different pressures. (b) Calculated wavelength of dispersive waves for different gases and at different filling pressure.

Fig. 2
Fig. 2

Schematic of the experimental set-up. Chirped mirrors were employed at the input to compensate for the material dispersion due to the lens and glass window.

Fig. 3
Fig. 3

Measured output spectra of the kagomé HC-PCF (core diameter 27 μm, 9 bar Ar) at various input energies, showing the range of different spectral shapes obtainable in the UV.

Fig. 4
Fig. 4

Experimental (top row) and simulated (bottom row) output spectra of a 10 cm Ar-filled kagomé HC-PCF at various input energies and pressures of (a,e) 5 bar, (b,f) 6.9 bar, (c,g) 9 bar and (d,h) 13.5 bar. The zero-dispersion wavelength is indicated by the dashed black-white line. An additional signal from a HOM dispersive wave can be seen in (d).

Fig. 5
Fig. 5

Experimental generation of coherent ultrashort pulses through resonant dispersive wave emission in gas-filled kagomé HC-PCF. Each peak is the individual normalized spectrum for a specific gas, pressure and pump energy. All the tuning was carried out in an identical length of kagomé HC-PCF (27 µm core diameter), except in the case of neon, where the core diameter was 37 µm.

Fig. 6
Fig. 6

Calculated pressures (green dotted line), ZDWs (blue solid line) and wavelengths of the dispersive wave (red dashed line) in a Kr-filled kagomé PCF with core diameter 27 µm with a pressure gradient between input and output.

Fig. 7
Fig. 7

(a) Experimental output spectra with Kr at constant pressure of 12 bar in 90 cm of kagomé fiber with 27 μm core diameter pumped at 800 nm with a pulse FWHM of ~50 fs; (b) positive pressure gradient of 0 to 12 bar; (c) negative pressure gradient of 12 to 0 bar.

Fig. 8
Fig. 8

(a) Evolution of the squared electric field (green line) and field envelope (blue line) for 9 bar Ar and 1.4 µJ input energy, plotted on a normalized linear scale. The red line represents the high frequency component (200 to 300 nm) of the electric field, with its intensity doubled for better contrast. (b) Simulated spectrograms of the pulse at different stages for the same parameters in (a), obtained with a 10 fs Gaussian gating-pulse, with the ZDW indicated by the red dashed line.

Fig. 9
Fig. 9

(a,b) Simulated pulse evolution and coherence at different point of propagation for 1.5 µJ and 3 µJ of input energy, in a fiber filled with 13.5 bar of Ar. (c, d) Normalized coherence parameter gpq along propagation for the two different energies, with a value of 1 indicating perfect coherence.

Tables (1)

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Table 1 Experimental Parameters for Optimized Dispersive Wave Generation at Various Wavelengths

Equations (5)

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β(ω) β NL (ω)=0,
β sol =β( ω 0 ) ω β | ω 0 [ω ω 0 ] γ P 0 2 ,
β(ω)β( ω 0 ) ω β | ω 0 [ω ω 0 ] γ P c 2 =0.
p(z)= p 0 2 + z L ( p 1 2 p 0 2 ) ,
| g pq (1) (ω) |=| E p * (ω) E q (ω) | E p (ω) | 2 |

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