Abstract

We derive unidirectional pulse propagation equations to describe extreme high-intensity and ultrabroadband optical interactions in uniaxial crystals, showing both second- and third-order nonlinear optical susceptivities. We focus our attention on the anisotropic nature of the quadratic and cubic nonlinear response of beta-barium-borate (β-BaB2O4, BBO) crystals. Two nonlinearly coupled first-order (in the propagation coordinate) equations describe the dynamics and interactions of the ordinary and extraordinary field polarizations, and are valid for arbitrarily wide pulse bandwidth. We exploit this model to predict harmonic and supercontinuum generation in BBO crystals under the strong and competing influence of quadratic and cubic susceptivities.

© 2013 Optical Society of America

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  31. Note that the reported dm coefficients have reversed sign with respect to the ones usually reported in literature [16]. This is because the reference frames have been traditionally selected such that, for θ=0, ϕ=0, Ee ad Eo are directed in −x and −y direction, respectively [28]. In our reference frame we have Ee,o=Ex,y, i.e., simply the x and y electric field components. This change of sign is insignificant owing to the scaling properties of Eq. (23): E→−E, dm→−dm.
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    [CrossRef]
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2012

2011

2010

M. Conforti, F. Baronio, and C. De Angelis, “Ultra-broadband optical phenomena in quadratic nonlinear media,” IEEE Photon. J. 2, 600–610 (2010).
[CrossRef]

J. Sung, S. Lee, T. Yu, T. Jeong, and J. Lee, “0.1 Hz 1.0 PW Ti:sapphire laser,” Opt. Lett. 35, 3021–3023 (2010).
[CrossRef]

M. Conforti, F. Baronio, and C. De Angelis, “Nonlinear envelope equation for broadband optical pulses in quadratic media,” Phys. Rev. A 81, 053841 (2010).
[CrossRef]

M. Bache, O. Bang, B. Zhou, J. Moses, and F. Wise, “Optical Cherenkov radiation in ultrafast cascaded second-harmonic generation,” Phys. Rev. A 82, 063806 (2010).
[CrossRef]

F. Baronio, M. Conforti, C. De Angelis, A. Degasperis, M. Andreana, V. Couderc, and A. Barthelemy, “Velocity-locked solitary waves in quadratic media,” Phys. Rev. Lett. 104, 113902 (2010).
[CrossRef]

P. Kinsler, “Optical pulse propagation with minimal approximations,” Phys. Rev. A 81, 013819 (2010).
[CrossRef]

A. Kumar, “Ultrashort pulse propagation in a cubic medium including the Raman effect,” Phys. Rev. A 81, 013807 (2010).
[CrossRef]

2009

2008

M. Bache, O. Bang, W. Krolikowski, J. Moses, and F. Wise, “Limits to compression with cascaded quadratic soliton compressors,” Opt. Express 16, 3273–3287 (2008).
[CrossRef]

F. Baronio, M. Conforti, A. Degasperis, and S. Wabnitz, “Three-wave trapponic solitons for tunable high-repetition rate pulse train generation,” IEEE J. Quantum Electron. 44, 542–546 (2008).
[CrossRef]

2007

G. Genty, P. Kinsler, B. Kibler, and J. M. Dudley, “Nonlinear envelope equation modeling of sub-cycle dynamics and harmonic generation in nonlinear waveguides,” Opt. Express 15, 5382–5387 (2007).
[CrossRef]

V. V. Lozhkarev, G. I. Freidman, V. N. Ginzburg, E. V. Katin, E. A. Khazanov, A. V. Kirsanov, G. A. Luchinin, A. N. Malshakov, M. A. Martyanov, O. V. Palashov, A. K. Poteomkin, A. M. Sergeev, A. A. Shaykin, and I. V. Yakovlev, “Compact 0.56 petawatt laser system based on optical parametric chirped pulse amplification in KDP crystals,” Laser Phys. Lett. 4, 421–427 (2007).
[CrossRef]

J. Price, T. Monro, H. Ebendorff-Heidepriem, F. Poletti, P. Horak, V. Finazzi, J. Leong, P. Petropoulos, J. Flanagan, G. Brambilla, X. Feng, and D. Richardson, “Mid-IR supercontinuum generation from nonsilica microstructured optical fibers,” IEEE J. Sel. Top. Quantum Electron. 13, 738–749 (2007).
[CrossRef]

2006

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

2005

P. Kinsler, S. B. P. Radnor, and G. H. C. New, “Theory of directional pulse propagation,” Phys. Rev. A 72, 063807 (2005).
[CrossRef]

2004

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

2003

2002

M. Kolesik, J. V. Moloney, and M. Mlejnek, “Unidirectional optical pulse propagation equation,” Phys. Rev. Lett. 89, 283902 (2002).
[CrossRef]

P. S. Banks, M. D. Feit, and M. D. Perry, “High intensity third-harmonic generation,” J. Opt. Soc. Am. B 19, 102–118 (2002).
[CrossRef]

2001

A. V. Housakou and J. Herrmnann, “Supercontinuum generation of higher-order solitons by fission in photonic crystal fibers,” Phys. Rev. Lett. 87, 203901 (2001).
[CrossRef]

2000

T. Brabec and F. Krausz, “Intense few-cycle laser fields: frontiers of nonlinear optics,” Rev. Mod. Phys. 72, 545–591 (2000).
[CrossRef]

1987

1965

J. E. Midwinter and J. Warner, “The effects of phase matching method and of uniaxial crystal symmetry on the polar distribution of second-order non-linear optical polarization,” Br. J. Appl. Phys. 16, 1135–1142 (1965).
[CrossRef]

J. E. Midwinter and J. Warner, “The effects of phase matching method and of crystal symmetry on the polar dependence of third-order non-linear optical polarization,” Br. J. Appl. Phys. 16, 1667 (1965).
[CrossRef]

Akahane, Y.

Andreana, M.

F. Baronio, M. Conforti, C. De Angelis, A. Degasperis, M. Andreana, V. Couderc, and A. Barthelemy, “Velocity-locked solitary waves in quadratic media,” Phys. Rev. Lett. 104, 113902 (2010).
[CrossRef]

Aoyama, M.

Bache, M.

B. B. Zhou, A. Chong, F. W. Wise, and M. Bache, “Ultrafast and octave-spanning optical nonlinearities from strongly phase-mismatched quadratic interactions,” Phys. Rev. Lett. 109, 043902 (2012).
[CrossRef]

M. Bache, O. Bang, B. Zhou, J. Moses, and F. Wise, “Optical Cherenkov radiation by cascaded nonlinear interaction: an efficient source of few-cycle energetic near- to mid-IR pulses,” Opt. Express 19, 22557–22562 (2011).
[CrossRef]

M. Bache, O. Bang, B. Zhou, J. Moses, and F. Wise, “Optical Cherenkov radiation in ultrafast cascaded second-harmonic generation,” Phys. Rev. A 82, 063806 (2010).
[CrossRef]

M. Bache, O. Bang, W. Krolikowski, J. Moses, and F. Wise, “Limits to compression with cascaded quadratic soliton compressors,” Opt. Express 16, 3273–3287 (2008).
[CrossRef]

M. Bache, H. Guo, B. Zhou, and X. Zheng, “The anisotropic Kerr nonlinear refractive index of β-BaB2O4,” arXiv: 1209.3158v1 (2012).

Bang, O.

Banks, P. S.

Baronio, F.

M. Levenius, M. Conforti, F. Baronio, V. Pasiskevicius, F. Laurell, C. De Angelis, and K. Gallo, “Multistep quadratic cascading in broadband optical parametric generation,” Opt. Lett. 37, 1727–1729 (2012).
[CrossRef]

M. Conforti, F. Baronio, and C. De Angelis, “Modeling of ultrabroadband and single-cycle phenomena in anisotropic quadratic crystals,” J. Opt. Soc. Am. B 28, 1231–1237 (2011).
[CrossRef]

F. Baronio, M. Conforti, C. De Angelis, A. Degasperis, M. Andreana, V. Couderc, and A. Barthelemy, “Velocity-locked solitary waves in quadratic media,” Phys. Rev. Lett. 104, 113902 (2010).
[CrossRef]

M. Conforti, F. Baronio, and C. De Angelis, “Nonlinear envelope equation for broadband optical pulses in quadratic media,” Phys. Rev. A 81, 053841 (2010).
[CrossRef]

M. Conforti, F. Baronio, and C. De Angelis, “Ultra-broadband optical phenomena in quadratic nonlinear media,” IEEE Photon. J. 2, 600–610 (2010).
[CrossRef]

F. Baronio, M. Conforti, A. Degasperis, and S. Wabnitz, “Three-wave trapponic solitons for tunable high-repetition rate pulse train generation,” IEEE J. Quantum Electron. 44, 542–546 (2008).
[CrossRef]

Barthelemy, A.

F. Baronio, M. Conforti, C. De Angelis, A. Degasperis, M. Andreana, V. Couderc, and A. Barthelemy, “Velocity-locked solitary waves in quadratic media,” Phys. Rev. Lett. 104, 113902 (2010).
[CrossRef]

Brabec, T.

T. Brabec and F. Krausz, “Intense few-cycle laser fields: frontiers of nonlinear optics,” Rev. Mod. Phys. 72, 545–591 (2000).
[CrossRef]

Brambilla, G.

J. Price, T. Monro, H. Ebendorff-Heidepriem, F. Poletti, P. Horak, V. Finazzi, J. Leong, P. Petropoulos, J. Flanagan, G. Brambilla, X. Feng, and D. Richardson, “Mid-IR supercontinuum generation from nonsilica microstructured optical fibers,” IEEE J. Sel. Top. Quantum Electron. 13, 738–749 (2007).
[CrossRef]

Chai, L.

Chen, H.

Chong, A.

B. B. Zhou, A. Chong, F. W. Wise, and M. Bache, “Ultrafast and octave-spanning optical nonlinearities from strongly phase-mismatched quadratic interactions,” Phys. Rev. Lett. 109, 043902 (2012).
[CrossRef]

Coen, S.

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

Conforti, M.

M. Levenius, M. Conforti, F. Baronio, V. Pasiskevicius, F. Laurell, C. De Angelis, and K. Gallo, “Multistep quadratic cascading in broadband optical parametric generation,” Opt. Lett. 37, 1727–1729 (2012).
[CrossRef]

M. Conforti, F. Baronio, and C. De Angelis, “Modeling of ultrabroadband and single-cycle phenomena in anisotropic quadratic crystals,” J. Opt. Soc. Am. B 28, 1231–1237 (2011).
[CrossRef]

F. Baronio, M. Conforti, C. De Angelis, A. Degasperis, M. Andreana, V. Couderc, and A. Barthelemy, “Velocity-locked solitary waves in quadratic media,” Phys. Rev. Lett. 104, 113902 (2010).
[CrossRef]

M. Conforti, F. Baronio, and C. De Angelis, “Nonlinear envelope equation for broadband optical pulses in quadratic media,” Phys. Rev. A 81, 053841 (2010).
[CrossRef]

M. Conforti, F. Baronio, and C. De Angelis, “Ultra-broadband optical phenomena in quadratic nonlinear media,” IEEE Photon. J. 2, 600–610 (2010).
[CrossRef]

F. Baronio, M. Conforti, A. Degasperis, and S. Wabnitz, “Three-wave trapponic solitons for tunable high-repetition rate pulse train generation,” IEEE J. Quantum Electron. 44, 542–546 (2008).
[CrossRef]

Couderc, V.

F. Baronio, M. Conforti, C. De Angelis, A. Degasperis, M. Andreana, V. Couderc, and A. Barthelemy, “Velocity-locked solitary waves in quadratic media,” Phys. Rev. Lett. 104, 113902 (2010).
[CrossRef]

Dai, N.

De Angelis, C.

M. Levenius, M. Conforti, F. Baronio, V. Pasiskevicius, F. Laurell, C. De Angelis, and K. Gallo, “Multistep quadratic cascading in broadband optical parametric generation,” Opt. Lett. 37, 1727–1729 (2012).
[CrossRef]

M. Conforti, F. Baronio, and C. De Angelis, “Modeling of ultrabroadband and single-cycle phenomena in anisotropic quadratic crystals,” J. Opt. Soc. Am. B 28, 1231–1237 (2011).
[CrossRef]

F. Baronio, M. Conforti, C. De Angelis, A. Degasperis, M. Andreana, V. Couderc, and A. Barthelemy, “Velocity-locked solitary waves in quadratic media,” Phys. Rev. Lett. 104, 113902 (2010).
[CrossRef]

M. Conforti, F. Baronio, and C. De Angelis, “Ultra-broadband optical phenomena in quadratic nonlinear media,” IEEE Photon. J. 2, 600–610 (2010).
[CrossRef]

M. Conforti, F. Baronio, and C. De Angelis, “Nonlinear envelope equation for broadband optical pulses in quadratic media,” Phys. Rev. A 81, 053841 (2010).
[CrossRef]

Degasperis, A.

F. Baronio, M. Conforti, C. De Angelis, A. Degasperis, M. Andreana, V. Couderc, and A. Barthelemy, “Velocity-locked solitary waves in quadratic media,” Phys. Rev. Lett. 104, 113902 (2010).
[CrossRef]

F. Baronio, M. Conforti, A. Degasperis, and S. Wabnitz, “Three-wave trapponic solitons for tunable high-repetition rate pulse train generation,” IEEE J. Quantum Electron. 44, 542–546 (2008).
[CrossRef]

Dudley, J. M.

Ebendorff-Heidepriem, H.

J. Price, T. Monro, H. Ebendorff-Heidepriem, F. Poletti, P. Horak, V. Finazzi, J. Leong, P. Petropoulos, J. Flanagan, G. Brambilla, X. Feng, and D. Richardson, “Mid-IR supercontinuum generation from nonsilica microstructured optical fibers,” IEEE J. Sel. Top. Quantum Electron. 13, 738–749 (2007).
[CrossRef]

Fang, X.

Farrell, C.

Feit, M. D.

Fejer, M. M.

Feng, X.

J. Price, T. Monro, H. Ebendorff-Heidepriem, F. Poletti, P. Horak, V. Finazzi, J. Leong, P. Petropoulos, J. Flanagan, G. Brambilla, X. Feng, and D. Richardson, “Mid-IR supercontinuum generation from nonsilica microstructured optical fibers,” IEEE J. Sel. Top. Quantum Electron. 13, 738–749 (2007).
[CrossRef]

Fermann, M. E.

Finazzi, V.

J. Price, T. Monro, H. Ebendorff-Heidepriem, F. Poletti, P. Horak, V. Finazzi, J. Leong, P. Petropoulos, J. Flanagan, G. Brambilla, X. Feng, and D. Richardson, “Mid-IR supercontinuum generation from nonsilica microstructured optical fibers,” IEEE J. Sel. Top. Quantum Electron. 13, 738–749 (2007).
[CrossRef]

Flanagan, J.

J. Price, T. Monro, H. Ebendorff-Heidepriem, F. Poletti, P. Horak, V. Finazzi, J. Leong, P. Petropoulos, J. Flanagan, G. Brambilla, X. Feng, and D. Richardson, “Mid-IR supercontinuum generation from nonsilica microstructured optical fibers,” IEEE J. Sel. Top. Quantum Electron. 13, 738–749 (2007).
[CrossRef]

Freidman, G. I.

V. V. Lozhkarev, G. I. Freidman, V. N. Ginzburg, E. V. Katin, E. A. Khazanov, A. V. Kirsanov, G. A. Luchinin, A. N. Malshakov, M. A. Martyanov, O. V. Palashov, A. K. Poteomkin, A. M. Sergeev, A. A. Shaykin, and I. V. Yakovlev, “Compact 0.56 petawatt laser system based on optical parametric chirped pulse amplification in KDP crystals,” Laser Phys. Lett. 4, 421–427 (2007).
[CrossRef]

Gallo, K.

Genty, G.

Ginzburg, V.

Ginzburg, V. N.

V. V. Lozhkarev, G. I. Freidman, V. N. Ginzburg, E. V. Katin, E. A. Khazanov, A. V. Kirsanov, G. A. Luchinin, A. N. Malshakov, M. A. Martyanov, O. V. Palashov, A. K. Poteomkin, A. M. Sergeev, A. A. Shaykin, and I. V. Yakovlev, “Compact 0.56 petawatt laser system based on optical parametric chirped pulse amplification in KDP crystals,” Laser Phys. Lett. 4, 421–427 (2007).
[CrossRef]

Guo, H.

M. Bache, H. Guo, B. Zhou, and X. Zheng, “The anisotropic Kerr nonlinear refractive index of β-BaB2O4,” arXiv: 1209.3158v1 (2012).

Hartl, I.

Herrmnann, J.

A. V. Housakou and J. Herrmnann, “Supercontinuum generation of higher-order solitons by fission in photonic crystal fibers,” Phys. Rev. Lett. 87, 203901 (2001).
[CrossRef]

Horak, P.

J. Price, T. Monro, H. Ebendorff-Heidepriem, F. Poletti, P. Horak, V. Finazzi, J. Leong, P. Petropoulos, J. Flanagan, G. Brambilla, X. Feng, and D. Richardson, “Mid-IR supercontinuum generation from nonsilica microstructured optical fibers,” IEEE J. Sel. Top. Quantum Electron. 13, 738–749 (2007).
[CrossRef]

Housakou, A. V.

A. V. Housakou and J. Herrmnann, “Supercontinuum generation of higher-order solitons by fission in photonic crystal fibers,” Phys. Rev. Lett. 87, 203901 (2001).
[CrossRef]

Hu, M.

Huang, L.

Inoue, N.

Ivanov, M.

F. Krausz and M. Ivanov, “Attosecond physics,” Rev. Mod. Phys. 81, 163–234 (2009).
[CrossRef]

Jeong, T.

Jiang, J.

Katin, E. V.

V. V. Lozhkarev, G. I. Freidman, V. N. Ginzburg, E. V. Katin, E. A. Khazanov, A. V. Kirsanov, G. A. Luchinin, A. N. Malshakov, M. A. Martyanov, O. V. Palashov, A. K. Poteomkin, A. M. Sergeev, A. A. Shaykin, and I. V. Yakovlev, “Compact 0.56 petawatt laser system based on optical parametric chirped pulse amplification in KDP crystals,” Laser Phys. Lett. 4, 421–427 (2007).
[CrossRef]

Khazanov, E.

Khazanov, E. A.

V. V. Lozhkarev, G. I. Freidman, V. N. Ginzburg, E. V. Katin, E. A. Khazanov, A. V. Kirsanov, G. A. Luchinin, A. N. Malshakov, M. A. Martyanov, O. V. Palashov, A. K. Poteomkin, A. M. Sergeev, A. A. Shaykin, and I. V. Yakovlev, “Compact 0.56 petawatt laser system based on optical parametric chirped pulse amplification in KDP crystals,” Laser Phys. Lett. 4, 421–427 (2007).
[CrossRef]

Kibler, B.

Kinsler, P.

P. Kinsler, “Optical pulse propagation with minimal approximations,” Phys. Rev. A 81, 013819 (2010).
[CrossRef]

G. Genty, P. Kinsler, B. Kibler, and J. M. Dudley, “Nonlinear envelope equation modeling of sub-cycle dynamics and harmonic generation in nonlinear waveguides,” Opt. Express 15, 5382–5387 (2007).
[CrossRef]

P. Kinsler, S. B. P. Radnor, and G. H. C. New, “Theory of directional pulse propagation,” Phys. Rev. A 72, 063807 (2005).
[CrossRef]

Kiriyama, H.

Kirsanov, A. V.

V. V. Lozhkarev, G. I. Freidman, V. N. Ginzburg, E. V. Katin, E. A. Khazanov, A. V. Kirsanov, G. A. Luchinin, A. N. Malshakov, M. A. Martyanov, O. V. Palashov, A. K. Poteomkin, A. M. Sergeev, A. A. Shaykin, and I. V. Yakovlev, “Compact 0.56 petawatt laser system based on optical parametric chirped pulse amplification in KDP crystals,” Laser Phys. Lett. 4, 421–427 (2007).
[CrossRef]

Kolesik, M.

M. Kolesik, P. T. Whalen, and J. V. Moloney, “Theory and simulation of ultrafast intense pulse propagation in extended media,” IEEE J. Sel. Top. Quantum Electron. 18, 494–506(2012).
[CrossRef]

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

M. Kolesik, J. V. Moloney, and M. Mlejnek, “Unidirectional optical pulse propagation equation,” Phys. Rev. Lett. 89, 283902 (2002).
[CrossRef]

Krausz, F.

F. Krausz and M. Ivanov, “Attosecond physics,” Rev. Mod. Phys. 81, 163–234 (2009).
[CrossRef]

T. Brabec and F. Krausz, “Intense few-cycle laser fields: frontiers of nonlinear optics,” Rev. Mod. Phys. 72, 545–591 (2000).
[CrossRef]

Krolikowski, W.

Kumar, A.

A. Kumar, “Ultrashort pulse propagation in a cubic medium including the Raman effect,” Phys. Rev. A 81, 013807 (2010).
[CrossRef]

Landau, L. D.

L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon, 1984).

Langrock, C.

Laurell, F.

Lee, J.

Lee, S.

Lee, Y.

Leong, J.

J. Price, T. Monro, H. Ebendorff-Heidepriem, F. Poletti, P. Horak, V. Finazzi, J. Leong, P. Petropoulos, J. Flanagan, G. Brambilla, X. Feng, and D. Richardson, “Mid-IR supercontinuum generation from nonsilica microstructured optical fibers,” IEEE J. Sel. Top. Quantum Electron. 13, 738–749 (2007).
[CrossRef]

Levenius, M.

Li, J.

Lifshitz, E. M.

L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon, 1984).

Lozhkarev, V.

Lozhkarev, V. V.

V. V. Lozhkarev, G. I. Freidman, V. N. Ginzburg, E. V. Katin, E. A. Khazanov, A. V. Kirsanov, G. A. Luchinin, A. N. Malshakov, M. A. Martyanov, O. V. Palashov, A. K. Poteomkin, A. M. Sergeev, A. A. Shaykin, and I. V. Yakovlev, “Compact 0.56 petawatt laser system based on optical parametric chirped pulse amplification in KDP crystals,” Laser Phys. Lett. 4, 421–427 (2007).
[CrossRef]

Luchinin, G. A.

V. V. Lozhkarev, G. I. Freidman, V. N. Ginzburg, E. V. Katin, E. A. Khazanov, A. V. Kirsanov, G. A. Luchinin, A. N. Malshakov, M. A. Martyanov, O. V. Palashov, A. K. Poteomkin, A. M. Sergeev, A. A. Shaykin, and I. V. Yakovlev, “Compact 0.56 petawatt laser system based on optical parametric chirped pulse amplification in KDP crystals,” Laser Phys. Lett. 4, 421–427 (2007).
[CrossRef]

Lundquist, T. R.

Ma, J.

Malshakov, A. N.

V. V. Lozhkarev, G. I. Freidman, V. N. Ginzburg, E. V. Katin, E. A. Khazanov, A. V. Kirsanov, G. A. Luchinin, A. N. Malshakov, M. A. Martyanov, O. V. Palashov, A. K. Poteomkin, A. M. Sergeev, A. A. Shaykin, and I. V. Yakovlev, “Compact 0.56 petawatt laser system based on optical parametric chirped pulse amplification in KDP crystals,” Laser Phys. Lett. 4, 421–427 (2007).
[CrossRef]

Martyanov, M. A.

V. V. Lozhkarev, G. I. Freidman, V. N. Ginzburg, E. V. Katin, E. A. Khazanov, A. V. Kirsanov, G. A. Luchinin, A. N. Malshakov, M. A. Martyanov, O. V. Palashov, A. K. Poteomkin, A. M. Sergeev, A. A. Shaykin, and I. V. Yakovlev, “Compact 0.56 petawatt laser system based on optical parametric chirped pulse amplification in KDP crystals,” Laser Phys. Lett. 4, 421–427 (2007).
[CrossRef]

Menyuk, C.

Midwinter, J. E.

J. E. Midwinter and J. Warner, “The effects of phase matching method and of uniaxial crystal symmetry on the polar distribution of second-order non-linear optical polarization,” Br. J. Appl. Phys. 16, 1135–1142 (1965).
[CrossRef]

J. E. Midwinter and J. Warner, “The effects of phase matching method and of crystal symmetry on the polar dependence of third-order non-linear optical polarization,” Br. J. Appl. Phys. 16, 1667 (1965).
[CrossRef]

Mironov, S.

Mlejnek, M.

M. Kolesik, J. V. Moloney, and M. Mlejnek, “Unidirectional optical pulse propagation equation,” Phys. Rev. Lett. 89, 283902 (2002).
[CrossRef]

Moloney, J. V.

M. Kolesik, P. T. Whalen, and J. V. Moloney, “Theory and simulation of ultrafast intense pulse propagation in extended media,” IEEE J. Sel. Top. Quantum Electron. 18, 494–506(2012).
[CrossRef]

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

M. Kolesik, J. V. Moloney, and M. Mlejnek, “Unidirectional optical pulse propagation equation,” Phys. Rev. Lett. 89, 283902 (2002).
[CrossRef]

Monro, T.

J. Price, T. Monro, H. Ebendorff-Heidepriem, F. Poletti, P. Horak, V. Finazzi, J. Leong, P. Petropoulos, J. Flanagan, G. Brambilla, X. Feng, and D. Richardson, “Mid-IR supercontinuum generation from nonsilica microstructured optical fibers,” IEEE J. Sel. Top. Quantum Electron. 13, 738–749 (2007).
[CrossRef]

Moses, J.

New, G. H. C.

P. Kinsler, S. B. P. Radnor, and G. H. C. New, “Theory of directional pulse propagation,” Phys. Rev. A 72, 063807 (2005).
[CrossRef]

Nikogosyan, D. N.

D. N. Nikogosyan, Nonlinear Optical Crystals: A Complete Survey (Springer, 2005).

Palashov, O. V.

V. V. Lozhkarev, G. I. Freidman, V. N. Ginzburg, E. V. Katin, E. A. Khazanov, A. V. Kirsanov, G. A. Luchinin, A. N. Malshakov, M. A. Martyanov, O. V. Palashov, A. K. Poteomkin, A. M. Sergeev, A. A. Shaykin, and I. V. Yakovlev, “Compact 0.56 petawatt laser system based on optical parametric chirped pulse amplification in KDP crystals,” Laser Phys. Lett. 4, 421–427 (2007).
[CrossRef]

Pasiskevicius, V.

Pelc, J. S.

Perry, M. D.

Petropoulos, P.

J. Price, T. Monro, H. Ebendorff-Heidepriem, F. Poletti, P. Horak, V. Finazzi, J. Leong, P. Petropoulos, J. Flanagan, G. Brambilla, X. Feng, and D. Richardson, “Mid-IR supercontinuum generation from nonsilica microstructured optical fibers,” IEEE J. Sel. Top. Quantum Electron. 13, 738–749 (2007).
[CrossRef]

Phillips, C. R.

Poletti, F.

J. Price, T. Monro, H. Ebendorff-Heidepriem, F. Poletti, P. Horak, V. Finazzi, J. Leong, P. Petropoulos, J. Flanagan, G. Brambilla, X. Feng, and D. Richardson, “Mid-IR supercontinuum generation from nonsilica microstructured optical fibers,” IEEE J. Sel. Top. Quantum Electron. 13, 738–749 (2007).
[CrossRef]

Poteomkin, A. K.

V. V. Lozhkarev, G. I. Freidman, V. N. Ginzburg, E. V. Katin, E. A. Khazanov, A. V. Kirsanov, G. A. Luchinin, A. N. Malshakov, M. A. Martyanov, O. V. Palashov, A. K. Poteomkin, A. M. Sergeev, A. A. Shaykin, and I. V. Yakovlev, “Compact 0.56 petawatt laser system based on optical parametric chirped pulse amplification in KDP crystals,” Laser Phys. Lett. 4, 421–427 (2007).
[CrossRef]

Price, J.

J. Price, T. Monro, H. Ebendorff-Heidepriem, F. Poletti, P. Horak, V. Finazzi, J. Leong, P. Petropoulos, J. Flanagan, G. Brambilla, X. Feng, and D. Richardson, “Mid-IR supercontinuum generation from nonsilica microstructured optical fibers,” IEEE J. Sel. Top. Quantum Electron. 13, 738–749 (2007).
[CrossRef]

Radnor, S. B. P.

P. Kinsler, S. B. P. Radnor, and G. H. C. New, “Theory of directional pulse propagation,” Phys. Rev. A 72, 063807 (2005).
[CrossRef]

Reid, D. T.

Richardson, D.

J. Price, T. Monro, H. Ebendorff-Heidepriem, F. Poletti, P. Horak, V. Finazzi, J. Leong, P. Petropoulos, J. Flanagan, G. Brambilla, X. Feng, and D. Richardson, “Mid-IR supercontinuum generation from nonsilica microstructured optical fibers,” IEEE J. Sel. Top. Quantum Electron. 13, 738–749 (2007).
[CrossRef]

Sergeev, A. M.

V. V. Lozhkarev, G. I. Freidman, V. N. Ginzburg, E. V. Katin, E. A. Khazanov, A. V. Kirsanov, G. A. Luchinin, A. N. Malshakov, M. A. Martyanov, O. V. Palashov, A. K. Poteomkin, A. M. Sergeev, A. A. Shaykin, and I. V. Yakovlev, “Compact 0.56 petawatt laser system based on optical parametric chirped pulse amplification in KDP crystals,” Laser Phys. Lett. 4, 421–427 (2007).
[CrossRef]

Serrels, K. A.

Shaykin, A. A.

V. V. Lozhkarev, G. I. Freidman, V. N. Ginzburg, E. V. Katin, E. A. Khazanov, A. V. Kirsanov, G. A. Luchinin, A. N. Malshakov, M. A. Martyanov, O. V. Palashov, A. K. Poteomkin, A. M. Sergeev, A. A. Shaykin, and I. V. Yakovlev, “Compact 0.56 petawatt laser system based on optical parametric chirped pulse amplification in KDP crystals,” Laser Phys. Lett. 4, 421–427 (2007).
[CrossRef]

Sung, J.

Tashchilina, A.

Ueda, H.

Vedagarbha, P.

Wabnitz, S.

F. Baronio, M. Conforti, A. Degasperis, and S. Wabnitz, “Three-wave trapponic solitons for tunable high-repetition rate pulse train generation,” IEEE J. Quantum Electron. 44, 542–546 (2008).
[CrossRef]

Wai, P.

Wang, C.

Warner, J.

J. E. Midwinter and J. Warner, “The effects of phase matching method and of crystal symmetry on the polar dependence of third-order non-linear optical polarization,” Br. J. Appl. Phys. 16, 1667 (1965).
[CrossRef]

J. E. Midwinter and J. Warner, “The effects of phase matching method and of uniaxial crystal symmetry on the polar distribution of second-order non-linear optical polarization,” Br. J. Appl. Phys. 16, 1135–1142 (1965).
[CrossRef]

Whalen, P. T.

M. Kolesik, P. T. Whalen, and J. V. Moloney, “Theory and simulation of ultrafast intense pulse propagation in extended media,” IEEE J. Sel. Top. Quantum Electron. 18, 494–506(2012).
[CrossRef]

Wise, F.

Wise, F. W.

B. B. Zhou, A. Chong, F. W. Wise, and M. Bache, “Ultrafast and octave-spanning optical nonlinearities from strongly phase-mismatched quadratic interactions,” Phys. Rev. Lett. 109, 043902 (2012).
[CrossRef]

Yakovlev, I. V.

V. V. Lozhkarev, G. I. Freidman, V. N. Ginzburg, E. V. Katin, E. A. Khazanov, A. V. Kirsanov, G. A. Luchinin, A. N. Malshakov, M. A. Martyanov, O. V. Palashov, A. K. Poteomkin, A. M. Sergeev, A. A. Shaykin, and I. V. Yakovlev, “Compact 0.56 petawatt laser system based on optical parametric chirped pulse amplification in KDP crystals,” Laser Phys. Lett. 4, 421–427 (2007).
[CrossRef]

Yamakawa, K.

Yu, T.

Zheltikov, A. M.

Zheng, X.

M. Bache, H. Guo, B. Zhou, and X. Zheng, “The anisotropic Kerr nonlinear refractive index of β-BaB2O4,” arXiv: 1209.3158v1 (2012).

Zhou, B.

M. Bache, O. Bang, B. Zhou, J. Moses, and F. Wise, “Optical Cherenkov radiation by cascaded nonlinear interaction: an efficient source of few-cycle energetic near- to mid-IR pulses,” Opt. Express 19, 22557–22562 (2011).
[CrossRef]

M. Bache, O. Bang, B. Zhou, J. Moses, and F. Wise, “Optical Cherenkov radiation in ultrafast cascaded second-harmonic generation,” Phys. Rev. A 82, 063806 (2010).
[CrossRef]

M. Bache, H. Guo, B. Zhou, and X. Zheng, “The anisotropic Kerr nonlinear refractive index of β-BaB2O4,” arXiv: 1209.3158v1 (2012).

Zhou, B. B.

B. B. Zhou, A. Chong, F. W. Wise, and M. Bache, “Ultrafast and octave-spanning optical nonlinearities from strongly phase-mismatched quadratic interactions,” Phys. Rev. Lett. 109, 043902 (2012).
[CrossRef]

Appl. Opt.

Br. J. Appl. Phys.

J. E. Midwinter and J. Warner, “The effects of phase matching method and of uniaxial crystal symmetry on the polar distribution of second-order non-linear optical polarization,” Br. J. Appl. Phys. 16, 1135–1142 (1965).
[CrossRef]

J. E. Midwinter and J. Warner, “The effects of phase matching method and of crystal symmetry on the polar dependence of third-order non-linear optical polarization,” Br. J. Appl. Phys. 16, 1667 (1965).
[CrossRef]

IEEE J. Quantum Electron.

F. Baronio, M. Conforti, A. Degasperis, and S. Wabnitz, “Three-wave trapponic solitons for tunable high-repetition rate pulse train generation,” IEEE J. Quantum Electron. 44, 542–546 (2008).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron.

J. Price, T. Monro, H. Ebendorff-Heidepriem, F. Poletti, P. Horak, V. Finazzi, J. Leong, P. Petropoulos, J. Flanagan, G. Brambilla, X. Feng, and D. Richardson, “Mid-IR supercontinuum generation from nonsilica microstructured optical fibers,” IEEE J. Sel. Top. Quantum Electron. 13, 738–749 (2007).
[CrossRef]

M. Kolesik, P. T. Whalen, and J. V. Moloney, “Theory and simulation of ultrafast intense pulse propagation in extended media,” IEEE J. Sel. Top. Quantum Electron. 18, 494–506(2012).
[CrossRef]

IEEE Photon. J.

M. Conforti, F. Baronio, and C. De Angelis, “Ultra-broadband optical phenomena in quadratic nonlinear media,” IEEE Photon. J. 2, 600–610 (2010).
[CrossRef]

J. Opt. Soc. Am. B

Laser Phys. Lett.

V. V. Lozhkarev, G. I. Freidman, V. N. Ginzburg, E. V. Katin, E. A. Khazanov, A. V. Kirsanov, G. A. Luchinin, A. N. Malshakov, M. A. Martyanov, O. V. Palashov, A. K. Poteomkin, A. M. Sergeev, A. A. Shaykin, and I. V. Yakovlev, “Compact 0.56 petawatt laser system based on optical parametric chirped pulse amplification in KDP crystals,” Laser Phys. Lett. 4, 421–427 (2007).
[CrossRef]

Opt. Express

Opt. Lett.

Phys. Rev. A

M. Conforti, F. Baronio, and C. De Angelis, “Nonlinear envelope equation for broadband optical pulses in quadratic media,” Phys. Rev. A 81, 053841 (2010).
[CrossRef]

M. Bache, O. Bang, B. Zhou, J. Moses, and F. Wise, “Optical Cherenkov radiation in ultrafast cascaded second-harmonic generation,” Phys. Rev. A 82, 063806 (2010).
[CrossRef]

P. Kinsler, S. B. P. Radnor, and G. H. C. New, “Theory of directional pulse propagation,” Phys. Rev. A 72, 063807 (2005).
[CrossRef]

P. Kinsler, “Optical pulse propagation with minimal approximations,” Phys. Rev. A 81, 013819 (2010).
[CrossRef]

A. Kumar, “Ultrashort pulse propagation in a cubic medium including the Raman effect,” Phys. Rev. A 81, 013807 (2010).
[CrossRef]

Phys. Rev. E

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

Phys. Rev. Lett.

F. Baronio, M. Conforti, C. De Angelis, A. Degasperis, M. Andreana, V. Couderc, and A. Barthelemy, “Velocity-locked solitary waves in quadratic media,” Phys. Rev. Lett. 104, 113902 (2010).
[CrossRef]

B. B. Zhou, A. Chong, F. W. Wise, and M. Bache, “Ultrafast and octave-spanning optical nonlinearities from strongly phase-mismatched quadratic interactions,” Phys. Rev. Lett. 109, 043902 (2012).
[CrossRef]

A. V. Housakou and J. Herrmnann, “Supercontinuum generation of higher-order solitons by fission in photonic crystal fibers,” Phys. Rev. Lett. 87, 203901 (2001).
[CrossRef]

M. Kolesik, J. V. Moloney, and M. Mlejnek, “Unidirectional optical pulse propagation equation,” Phys. Rev. Lett. 89, 283902 (2002).
[CrossRef]

Rev. Mod. Phys.

F. Krausz and M. Ivanov, “Attosecond physics,” Rev. Mod. Phys. 81, 163–234 (2009).
[CrossRef]

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

T. Brabec and F. Krausz, “Intense few-cycle laser fields: frontiers of nonlinear optics,” Rev. Mod. Phys. 72, 545–591 (2000).
[CrossRef]

Other

Note that the reported dm coefficients have reversed sign with respect to the ones usually reported in literature [16]. This is because the reference frames have been traditionally selected such that, for θ=0, ϕ=0, Ee ad Eo are directed in −x and −y direction, respectively [28]. In our reference frame we have Ee,o=Ex,y, i.e., simply the x and y electric field components. This change of sign is insignificant owing to the scaling properties of Eq. (23): E→−E, dm→−dm.

M. Bache, H. Guo, B. Zhou, and X. Zheng, “The anisotropic Kerr nonlinear refractive index of β-BaB2O4,” arXiv: 1209.3158v1 (2012).

L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon, 1984).

D. N. Nikogosyan, Nonlinear Optical Crystals: A Complete Survey (Springer, 2005).

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Figures (5)

Fig. 1.
Fig. 1.

Dependence of SHG efficiency on the pump peak intensity at the fundamental frequency, obtained without (blue stars) and with (red circles) consideration of the cubic nonlinear effects.

Fig. 2.
Fig. 2.

Evolution of the total power spectrum |E^x|2+|E^y|2 (decibels). The initial pulse has duration T=30fs, wavelength λ0=630nm The peak intensity is (a) 10GW/cm2, and (b) 2TW/cm2. Crystal’s orientation: θ=38° and ϕ=90°.

Fig. 3.
Fig. 3.

(a) Temporal propagation and (b) field spectrum evolution (decibels) of the ordinarily polarized electric field envelope in BBO crystal. The initial pulse has duration T=20fs, wavelength λ0=1200nm, and peak intensity of 120GW/cm2. Crystal’s orientation: θ=19° and ϕ=90°.

Fig. 4.
Fig. 4.

(a) Temporal propagation and (b) field spectrum evolution (decibels) of the ordinarily polarized electric field envelope in BBO crystal. The initial pulse has duration T=20fs, wavelength λ0=1200nm, and peak intensity of 120GW/cm2. Crystal’s orientation: θ=16.2° and ϕ=90°.

Fig. 5.
Fig. 5.

(a) Temporal propagation and (b) field spectrum evolution of the ordinarily polarized electric field envelope in BBO crystal. The initial pulse has duration T=30fs, wavelength λ0=2000nm, and peak intensity of 130GW/cm2. Crystal’s orientation: θ=80° and ϕ=90°.

Tables (2)

Tables Icon

Table 1. Effective Quadratic Nonlinear Coefficientsa

Tables Icon

Table 2. Effective Cubic Nonlinear Coefficientsa

Equations (26)

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

×E=Bt,
×H=Dt,
B=μ0H,
D=DL+PNL,
DL,j=ε0εjk(tt)Ek(t)dt.
A=[cosϕcosθsinϕcosθsinθsinϕcosϕ0sinθcosϕsinϕsinθcosθ].
ε=AεAT=[εocos2θ+εesin2θ0(εoεe)cosθsinθ0εo0(εoεe)cosθsinθ0εosin2θ+εecos2θ].
E=ε01ε1D=ε01[(cos2θεo+sin2θεe)Dxεo1DyεeεoεeεocosθsinθDx].
××E1ε0c22DLt2=1ε0c22PNLt2.
2Exz21ε0c22DL,xt2=1ε0c22PNL,xt2,
2Eyz21ε0c22DL,yt2=1ε0c22PNL,yt2,
0=1ε0c22PNL,zt2.
2Em(z,t)z21c22t2+Em(z,t)εm(tt)dt=1ε0c22t2PNL,m(z,t),m=x,y,
εx=(cos2θεo+sin2θεe)1,
εy=εo.
2E^m(z,ω)z2+ω2c2ε^m(ω)E^m(z,ω)=ω2ε0c2P^NL,m(z,ω),
E^m(z,ω)=F^m(z,ω)eikm(ω)z+B^m(z,ω)eikm(ω)z.
(2F^mz22ikm(ω)F^mz)eikm(ω)z+(2B^mz2+2ikm(ω)B^mz)eikm(ω)z=ω2ε0c2P^NL,m,
z(F^mzeikm(ω)z+B^mzeikm(ω)z)ikm(ω)(F^mzeikm(ω)zB^mzeikm(ω)z)=ω2ε0c2P^NL,m,
F^m(z,ω)z=i2km(ω)ω2ε0c2P^NL,m(z,ω)e+ikm(ω)z,B^m(z,ω)z=+i2km(ω)ω2ε0c2P^NL,m(z,ω)eikm(ω)z.
E^m(z,ω)z+ikm(ω)E^m(z,ω)=iω2ε0cnm(ω)P^NL,m(z,ω).
PNL,j=ε0(χjkl(2)EkEl+χjklm(3)EkElEm),
PNL(E)=APNL(ATE).
E^xz+ikx(ω)E^x=iωcnx(ω)P^x,E^yz+iky(ω)E^y=iωcny(ω)P^y,
Px=d0Ex2+2d1ExEy+d2Ey2+12(c0Ex3+3c1Ex2Ey+3c2Ey2Ex+c3Ey3),
Py=d1Ex2+2d2ExEy+d3Ey2+12(c1Ex3+3c2Ex2Ey+3c3Ey2Ex+c4Ey3).

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