Abstract

We present a comprehensive model to describe the propagation of single-cycle and broadband optical pulses in anisotropic, dispersive, and nonlinear materials. Two nonlinear coupled wave equations describe the dynamics and interactions of optical pulses in uniaxial second-order nonlinear materials. The equations are first order in the propagation coordinate and are valid for arbitrarily wide pulse bandwidth, providing an accurate modeling of the evolution of ultrabroadband pulses also when the separation into different coupled frequency components is not possible or not profitable. We exploit this model to simulate recently observed femtosecond single-cycle multiterahertz transients in gallium selenide and to predict harmonic generation and spectral broadening in the visible and mid-infrared in lithium niobate.

© 2011 Optical Society of America

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  1. R. Huber, F. Tauser, A. Brodschelm, M. Bichler, G. Abstreiter, and A. Leitenstorfer, “How many-particle interactions develop after ultrafast excitation of an electron–hole plasma,” Nature 414, 286–289 (2001).
    [CrossRef] [PubMed]
  2. G. Gunter, A. A. Anappara, J. Hees, A. Sell, G. Biasiol, L. Sorba, S. De Liberato, C. Ciuti, A. Tredicucci, A. Leitenstorfer, and R. Huber, “Sub-cycle switch-on of ultrastrong light–matter interaction,” Nature 458, 178–181 (2009).
    [CrossRef] [PubMed]
  3. T. Brabec and F. Krausz, “Intense few-cycle laser fields: frontiers of nonlinear optics,” Rev. Mod. Phys. 72, 545–591(2000).
    [CrossRef]
  4. C. Chudoba, E. Nibbering, and T. Elsaesser, “Site-specific excited-state solute-solvent interactions probed by femtosecond vibrational spectroscopy,” Phys. Rev. Lett. 81, 3010–3013(1998).
    [CrossRef]
  5. E. J. Heilweil, “Ultrafast glimpses at water and ice,” Science 283, 1467–1468 (1999).
    [CrossRef]
  6. S. Woutersen, U. Emmerichs, and H. J. Bakker, “Femtosecond mid-IR pump probe spectroscopy of liquid water: evidence for a two-component structure,” Science 278, 658–660 (1997).
    [CrossRef]
  7. G. Cerullo and S. De Silvestri, “Ultrafast optical parametric amplifiers,” Rev. Sci. Instrum. 74, 1–18 (2003).
    [CrossRef]
  8. D. Polli, L. Luer, and G. Cerullo, “High-time-resolution pump-probe system with broadband detection for the study of time-domain vibrational dynamics,” Rev. Sci. Instrum. 78, 103108(2007).
    [CrossRef] [PubMed]
  9. C. Manzoni, G. Cerullo, and S. De Silvestri, “Ultrabroadband self-phase-stabilized pulses by difference-frequency generation,” Opt. Lett. 29, 2668–2670 (2004).
    [CrossRef] [PubMed]
  10. F. Junginger, A. Sell, O. Schubert, B. Mayer, D. Brida, M. Marangoni, G. Cerullo, A. Leitenstorfer, and R. Huber, “Single-cycle multiterahertz transients with peak fields above 10 MV/cm,” Opt. Lett. 35, 2645–2647 (2010).
    [CrossRef] [PubMed]
  11. D. Brida, C. Manzoni, G. Cirmi, M. Marangoni, S. Bonora, P. Villoresi, S. De Silvestri, and G. Cerullo, “Few-optical-cycle pulses tunable from the mid-infrared by optical parametric amplifiers,” J. Opt. 12, 013001 (2010).
    [CrossRef]
  12. R. W. Boyd, Nonlinear Optics, 2nd ed. (Academic, 2003).
  13. P. Kinsler and G. H. C. New, “Few-cycle pulse propagation,” Phys. Rev. A 67, 023813 (2003).
    [CrossRef]
  14. J. Moses and F. W. Wise, “Controllable self-steepening of ultrashort pulses in quadratic nonlinear media,” Phys. Rev. Lett. 97, 073903 (2006).
    [CrossRef] [PubMed]
  15. M. Geissler, G. Tempea, A. Scrinzi, M. Schnurer, F. Krausz, and T. Brabec, “Light propagation in field-ionizing media: extreme nonlinear optics,” Phys. Rev. Lett. 83, 2930–2933 (1999).
    [CrossRef]
  16. A. V. Housakou and J. Herrmann, “Supercontinuum generation of higher-order solitons by fission in photonic crystal fibers,” Phys. Rev. Lett. 87, 203901 (2001).
    [CrossRef]
  17. M. Kolesik, J. V. Moloney, and M. Mlejnek, “Unidirectional optical pulse propagation equation,” Phys. Rev. Lett. 89, 283902 (2002).
    [CrossRef]
  18. M. Kolesik and J. V. Moloney, “Nonlinear optical pulse propagation simulation: from Maxwell’s to unidirectional equations,” Phys. Rev. E 70, 036604 (2004).
    [CrossRef]
  19. 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] [PubMed]
  20. P. Kinsler, S. B. P. Radnor, and G. H. C. New, “Theory of directional pulse propagation,” Phys. Rev. A 72, 063807 (2005).
    [CrossRef]
  21. P. Kinsler, “Optical pulse propagation with minimal approximations,” Phys. Rev. A 81, 013819 (2010).
    [CrossRef]
  22. A. Kumar, “Ultrashort pulse propagation in a cubic medium including the Raman effect,” Phys. Rev. A 81, 013807 (2010).
    [CrossRef]
  23. 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]
  24. M. Conforti, F. Baronio, and C. De Angelis, “Ultra-broadband optical phenomena in quadratic nonlinear media,” IEEE Photonics J. 2, 600–610 (2010).
    [CrossRef]
  25. L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon, 1984).
  26. 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]
  27. D. N. Nikogosyan, Nonlinear Optical Crystals: A Complete Survey (Springer, 2005).
  28. M. Conforti, F. Baronio, A. Degasperis, and S. Wabnitz, “Parametric frequency conversion of short optical pulses controlled by a CW background,” Opt. Express 15, 12246–12251 (2007).
    [CrossRef] [PubMed]
  29. 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]
  30. N. E. Yu, J. H. Ro, M. Cha, S. Kurimura, and T. Taira, “Broadband quasi-phase-matched second-harmonic generation in MgO-doped periodically poled LiNbO3 at the communications band,” Opt. Lett. 27, 1046–1048 (2002).
    [CrossRef]
  31. N. E. Yu, S. Kurimura, K. Kitamura, J. H. Ro, M. Cha, S. Ashihara, T. Shimura, K. Kuroda, and T. Taira, “Efficient frequency doubling of a femtosecond pulse with simultaneous group-velocity matching and quasi phase matching in periodically poled, MgO-doped lithium niobate,” Appl. Phys. Lett. 82, 3388–3390(2003).
    [CrossRef]

2010 (6)

D. Brida, C. Manzoni, G. Cirmi, M. Marangoni, S. Bonora, P. Villoresi, S. De Silvestri, and G. Cerullo, “Few-optical-cycle pulses tunable from the mid-infrared by optical parametric amplifiers,” J. Opt. 12, 013001 (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]

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 Photonics J. 2, 600–610 (2010).
[CrossRef]

F. Junginger, A. Sell, O. Schubert, B. Mayer, D. Brida, M. Marangoni, G. Cerullo, A. Leitenstorfer, and R. Huber, “Single-cycle multiterahertz transients with peak fields above 10 MV/cm,” Opt. Lett. 35, 2645–2647 (2010).
[CrossRef] [PubMed]

2009 (1)

G. Gunter, A. A. Anappara, J. Hees, A. Sell, G. Biasiol, L. Sorba, S. De Liberato, C. Ciuti, A. Tredicucci, A. Leitenstorfer, and R. Huber, “Sub-cycle switch-on of ultrastrong light–matter interaction,” Nature 458, 178–181 (2009).
[CrossRef] [PubMed]

2008 (1)

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 (3)

2006 (1)

J. Moses and F. W. Wise, “Controllable self-steepening of ultrashort pulses in quadratic nonlinear media,” Phys. Rev. Lett. 97, 073903 (2006).
[CrossRef] [PubMed]

2005 (1)

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

2004 (2)

C. Manzoni, G. Cerullo, and S. De Silvestri, “Ultrabroadband self-phase-stabilized pulses by difference-frequency generation,” Opt. Lett. 29, 2668–2670 (2004).
[CrossRef] [PubMed]

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 (3)

N. E. Yu, S. Kurimura, K. Kitamura, J. H. Ro, M. Cha, S. Ashihara, T. Shimura, K. Kuroda, and T. Taira, “Efficient frequency doubling of a femtosecond pulse with simultaneous group-velocity matching and quasi phase matching in periodically poled, MgO-doped lithium niobate,” Appl. Phys. Lett. 82, 3388–3390(2003).
[CrossRef]

P. Kinsler and G. H. C. New, “Few-cycle pulse propagation,” Phys. Rev. A 67, 023813 (2003).
[CrossRef]

G. Cerullo and S. De Silvestri, “Ultrafast optical parametric amplifiers,” Rev. Sci. Instrum. 74, 1–18 (2003).
[CrossRef]

2002 (2)

2001 (2)

R. Huber, F. Tauser, A. Brodschelm, M. Bichler, G. Abstreiter, and A. Leitenstorfer, “How many-particle interactions develop after ultrafast excitation of an electron–hole plasma,” Nature 414, 286–289 (2001).
[CrossRef] [PubMed]

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

2000 (1)

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

1999 (2)

E. J. Heilweil, “Ultrafast glimpses at water and ice,” Science 283, 1467–1468 (1999).
[CrossRef]

M. Geissler, G. Tempea, A. Scrinzi, M. Schnurer, F. Krausz, and T. Brabec, “Light propagation in field-ionizing media: extreme nonlinear optics,” Phys. Rev. Lett. 83, 2930–2933 (1999).
[CrossRef]

1998 (1)

C. Chudoba, E. Nibbering, and T. Elsaesser, “Site-specific excited-state solute-solvent interactions probed by femtosecond vibrational spectroscopy,” Phys. Rev. Lett. 81, 3010–3013(1998).
[CrossRef]

1997 (1)

S. Woutersen, U. Emmerichs, and H. J. Bakker, “Femtosecond mid-IR pump probe spectroscopy of liquid water: evidence for a two-component structure,” Science 278, 658–660 (1997).
[CrossRef]

1965 (1)

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]

Abstreiter, G.

R. Huber, F. Tauser, A. Brodschelm, M. Bichler, G. Abstreiter, and A. Leitenstorfer, “How many-particle interactions develop after ultrafast excitation of an electron–hole plasma,” Nature 414, 286–289 (2001).
[CrossRef] [PubMed]

Anappara, A. A.

G. Gunter, A. A. Anappara, J. Hees, A. Sell, G. Biasiol, L. Sorba, S. De Liberato, C. Ciuti, A. Tredicucci, A. Leitenstorfer, and R. Huber, “Sub-cycle switch-on of ultrastrong light–matter interaction,” Nature 458, 178–181 (2009).
[CrossRef] [PubMed]

Ashihara, S.

N. E. Yu, S. Kurimura, K. Kitamura, J. H. Ro, M. Cha, S. Ashihara, T. Shimura, K. Kuroda, and T. Taira, “Efficient frequency doubling of a femtosecond pulse with simultaneous group-velocity matching and quasi phase matching in periodically poled, MgO-doped lithium niobate,” Appl. Phys. Lett. 82, 3388–3390(2003).
[CrossRef]

Bakker, H. J.

S. Woutersen, U. Emmerichs, and H. J. Bakker, “Femtosecond mid-IR pump probe spectroscopy of liquid water: evidence for a two-component structure,” Science 278, 658–660 (1997).
[CrossRef]

Baronio, F.

M. Conforti, F. Baronio, and C. De Angelis, “Ultra-broadband optical phenomena in quadratic nonlinear media,” IEEE Photonics 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]

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]

M. Conforti, F. Baronio, A. Degasperis, and S. Wabnitz, “Parametric frequency conversion of short optical pulses controlled by a CW background,” Opt. Express 15, 12246–12251 (2007).
[CrossRef] [PubMed]

Biasiol, G.

G. Gunter, A. A. Anappara, J. Hees, A. Sell, G. Biasiol, L. Sorba, S. De Liberato, C. Ciuti, A. Tredicucci, A. Leitenstorfer, and R. Huber, “Sub-cycle switch-on of ultrastrong light–matter interaction,” Nature 458, 178–181 (2009).
[CrossRef] [PubMed]

Bichler, M.

R. Huber, F. Tauser, A. Brodschelm, M. Bichler, G. Abstreiter, and A. Leitenstorfer, “How many-particle interactions develop after ultrafast excitation of an electron–hole plasma,” Nature 414, 286–289 (2001).
[CrossRef] [PubMed]

Bonora, S.

D. Brida, C. Manzoni, G. Cirmi, M. Marangoni, S. Bonora, P. Villoresi, S. De Silvestri, and G. Cerullo, “Few-optical-cycle pulses tunable from the mid-infrared by optical parametric amplifiers,” J. Opt. 12, 013001 (2010).
[CrossRef]

Boyd, R. W.

R. W. Boyd, Nonlinear Optics, 2nd ed. (Academic, 2003).

Brabec, T.

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

M. Geissler, G. Tempea, A. Scrinzi, M. Schnurer, F. Krausz, and T. Brabec, “Light propagation in field-ionizing media: extreme nonlinear optics,” Phys. Rev. Lett. 83, 2930–2933 (1999).
[CrossRef]

Brida, D.

F. Junginger, A. Sell, O. Schubert, B. Mayer, D. Brida, M. Marangoni, G. Cerullo, A. Leitenstorfer, and R. Huber, “Single-cycle multiterahertz transients with peak fields above 10 MV/cm,” Opt. Lett. 35, 2645–2647 (2010).
[CrossRef] [PubMed]

D. Brida, C. Manzoni, G. Cirmi, M. Marangoni, S. Bonora, P. Villoresi, S. De Silvestri, and G. Cerullo, “Few-optical-cycle pulses tunable from the mid-infrared by optical parametric amplifiers,” J. Opt. 12, 013001 (2010).
[CrossRef]

Brodschelm, A.

R. Huber, F. Tauser, A. Brodschelm, M. Bichler, G. Abstreiter, and A. Leitenstorfer, “How many-particle interactions develop after ultrafast excitation of an electron–hole plasma,” Nature 414, 286–289 (2001).
[CrossRef] [PubMed]

Cerullo, G.

D. Brida, C. Manzoni, G. Cirmi, M. Marangoni, S. Bonora, P. Villoresi, S. De Silvestri, and G. Cerullo, “Few-optical-cycle pulses tunable from the mid-infrared by optical parametric amplifiers,” J. Opt. 12, 013001 (2010).
[CrossRef]

F. Junginger, A. Sell, O. Schubert, B. Mayer, D. Brida, M. Marangoni, G. Cerullo, A. Leitenstorfer, and R. Huber, “Single-cycle multiterahertz transients with peak fields above 10 MV/cm,” Opt. Lett. 35, 2645–2647 (2010).
[CrossRef] [PubMed]

D. Polli, L. Luer, and G. Cerullo, “High-time-resolution pump-probe system with broadband detection for the study of time-domain vibrational dynamics,” Rev. Sci. Instrum. 78, 103108(2007).
[CrossRef] [PubMed]

C. Manzoni, G. Cerullo, and S. De Silvestri, “Ultrabroadband self-phase-stabilized pulses by difference-frequency generation,” Opt. Lett. 29, 2668–2670 (2004).
[CrossRef] [PubMed]

G. Cerullo and S. De Silvestri, “Ultrafast optical parametric amplifiers,” Rev. Sci. Instrum. 74, 1–18 (2003).
[CrossRef]

Cha, M.

N. E. Yu, S. Kurimura, K. Kitamura, J. H. Ro, M. Cha, S. Ashihara, T. Shimura, K. Kuroda, and T. Taira, “Efficient frequency doubling of a femtosecond pulse with simultaneous group-velocity matching and quasi phase matching in periodically poled, MgO-doped lithium niobate,” Appl. Phys. Lett. 82, 3388–3390(2003).
[CrossRef]

N. E. Yu, J. H. Ro, M. Cha, S. Kurimura, and T. Taira, “Broadband quasi-phase-matched second-harmonic generation in MgO-doped periodically poled LiNbO3 at the communications band,” Opt. Lett. 27, 1046–1048 (2002).
[CrossRef]

Chudoba, C.

C. Chudoba, E. Nibbering, and T. Elsaesser, “Site-specific excited-state solute-solvent interactions probed by femtosecond vibrational spectroscopy,” Phys. Rev. Lett. 81, 3010–3013(1998).
[CrossRef]

Cirmi, G.

D. Brida, C. Manzoni, G. Cirmi, M. Marangoni, S. Bonora, P. Villoresi, S. De Silvestri, and G. Cerullo, “Few-optical-cycle pulses tunable from the mid-infrared by optical parametric amplifiers,” J. Opt. 12, 013001 (2010).
[CrossRef]

Ciuti, C.

G. Gunter, A. A. Anappara, J. Hees, A. Sell, G. Biasiol, L. Sorba, S. De Liberato, C. Ciuti, A. Tredicucci, A. Leitenstorfer, and R. Huber, “Sub-cycle switch-on of ultrastrong light–matter interaction,” Nature 458, 178–181 (2009).
[CrossRef] [PubMed]

Conforti, M.

M. Conforti, F. Baronio, and C. De Angelis, “Ultra-broadband optical phenomena in quadratic nonlinear media,” IEEE Photonics 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]

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]

M. Conforti, F. Baronio, A. Degasperis, and S. Wabnitz, “Parametric frequency conversion of short optical pulses controlled by a CW background,” Opt. Express 15, 12246–12251 (2007).
[CrossRef] [PubMed]

De Angelis, C.

M. Conforti, F. Baronio, and C. De Angelis, “Ultra-broadband optical phenomena in quadratic nonlinear media,” IEEE Photonics 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]

De Liberato, S.

G. Gunter, A. A. Anappara, J. Hees, A. Sell, G. Biasiol, L. Sorba, S. De Liberato, C. Ciuti, A. Tredicucci, A. Leitenstorfer, and R. Huber, “Sub-cycle switch-on of ultrastrong light–matter interaction,” Nature 458, 178–181 (2009).
[CrossRef] [PubMed]

De Silvestri, S.

D. Brida, C. Manzoni, G. Cirmi, M. Marangoni, S. Bonora, P. Villoresi, S. De Silvestri, and G. Cerullo, “Few-optical-cycle pulses tunable from the mid-infrared by optical parametric amplifiers,” J. Opt. 12, 013001 (2010).
[CrossRef]

C. Manzoni, G. Cerullo, and S. De Silvestri, “Ultrabroadband self-phase-stabilized pulses by difference-frequency generation,” Opt. Lett. 29, 2668–2670 (2004).
[CrossRef] [PubMed]

G. Cerullo and S. De Silvestri, “Ultrafast optical parametric amplifiers,” Rev. Sci. Instrum. 74, 1–18 (2003).
[CrossRef]

Degasperis, A.

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]

M. Conforti, F. Baronio, A. Degasperis, and S. Wabnitz, “Parametric frequency conversion of short optical pulses controlled by a CW background,” Opt. Express 15, 12246–12251 (2007).
[CrossRef] [PubMed]

Dudley, J. M.

Elsaesser, T.

C. Chudoba, E. Nibbering, and T. Elsaesser, “Site-specific excited-state solute-solvent interactions probed by femtosecond vibrational spectroscopy,” Phys. Rev. Lett. 81, 3010–3013(1998).
[CrossRef]

Emmerichs, U.

S. Woutersen, U. Emmerichs, and H. J. Bakker, “Femtosecond mid-IR pump probe spectroscopy of liquid water: evidence for a two-component structure,” Science 278, 658–660 (1997).
[CrossRef]

Geissler, M.

M. Geissler, G. Tempea, A. Scrinzi, M. Schnurer, F. Krausz, and T. Brabec, “Light propagation in field-ionizing media: extreme nonlinear optics,” Phys. Rev. Lett. 83, 2930–2933 (1999).
[CrossRef]

Genty, G.

Gunter, G.

G. Gunter, A. A. Anappara, J. Hees, A. Sell, G. Biasiol, L. Sorba, S. De Liberato, C. Ciuti, A. Tredicucci, A. Leitenstorfer, and R. Huber, “Sub-cycle switch-on of ultrastrong light–matter interaction,” Nature 458, 178–181 (2009).
[CrossRef] [PubMed]

Hees, J.

G. Gunter, A. A. Anappara, J. Hees, A. Sell, G. Biasiol, L. Sorba, S. De Liberato, C. Ciuti, A. Tredicucci, A. Leitenstorfer, and R. Huber, “Sub-cycle switch-on of ultrastrong light–matter interaction,” Nature 458, 178–181 (2009).
[CrossRef] [PubMed]

Heilweil, E. J.

E. J. Heilweil, “Ultrafast glimpses at water and ice,” Science 283, 1467–1468 (1999).
[CrossRef]

Herrmann, J.

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

Housakou, A. V.

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

Huber, R.

F. Junginger, A. Sell, O. Schubert, B. Mayer, D. Brida, M. Marangoni, G. Cerullo, A. Leitenstorfer, and R. Huber, “Single-cycle multiterahertz transients with peak fields above 10 MV/cm,” Opt. Lett. 35, 2645–2647 (2010).
[CrossRef] [PubMed]

G. Gunter, A. A. Anappara, J. Hees, A. Sell, G. Biasiol, L. Sorba, S. De Liberato, C. Ciuti, A. Tredicucci, A. Leitenstorfer, and R. Huber, “Sub-cycle switch-on of ultrastrong light–matter interaction,” Nature 458, 178–181 (2009).
[CrossRef] [PubMed]

R. Huber, F. Tauser, A. Brodschelm, M. Bichler, G. Abstreiter, and A. Leitenstorfer, “How many-particle interactions develop after ultrafast excitation of an electron–hole plasma,” Nature 414, 286–289 (2001).
[CrossRef] [PubMed]

Junginger, F.

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] [PubMed]

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 and G. H. C. New, “Few-cycle pulse propagation,” Phys. Rev. A 67, 023813 (2003).
[CrossRef]

Kitamura, K.

N. E. Yu, S. Kurimura, K. Kitamura, J. H. Ro, M. Cha, S. Ashihara, T. Shimura, K. Kuroda, and T. Taira, “Efficient frequency doubling of a femtosecond pulse with simultaneous group-velocity matching and quasi phase matching in periodically poled, MgO-doped lithium niobate,” Appl. Phys. Lett. 82, 3388–3390(2003).
[CrossRef]

Kolesik, M.

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.

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

M. Geissler, G. Tempea, A. Scrinzi, M. Schnurer, F. Krausz, and T. Brabec, “Light propagation in field-ionizing media: extreme nonlinear optics,” Phys. Rev. Lett. 83, 2930–2933 (1999).
[CrossRef]

Kumar, A.

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

Kurimura, S.

N. E. Yu, S. Kurimura, K. Kitamura, J. H. Ro, M. Cha, S. Ashihara, T. Shimura, K. Kuroda, and T. Taira, “Efficient frequency doubling of a femtosecond pulse with simultaneous group-velocity matching and quasi phase matching in periodically poled, MgO-doped lithium niobate,” Appl. Phys. Lett. 82, 3388–3390(2003).
[CrossRef]

N. E. Yu, J. H. Ro, M. Cha, S. Kurimura, and T. Taira, “Broadband quasi-phase-matched second-harmonic generation in MgO-doped periodically poled LiNbO3 at the communications band,” Opt. Lett. 27, 1046–1048 (2002).
[CrossRef]

Kuroda, K.

N. E. Yu, S. Kurimura, K. Kitamura, J. H. Ro, M. Cha, S. Ashihara, T. Shimura, K. Kuroda, and T. Taira, “Efficient frequency doubling of a femtosecond pulse with simultaneous group-velocity matching and quasi phase matching in periodically poled, MgO-doped lithium niobate,” Appl. Phys. Lett. 82, 3388–3390(2003).
[CrossRef]

Landau, L. D.

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

Leitenstorfer, A.

F. Junginger, A. Sell, O. Schubert, B. Mayer, D. Brida, M. Marangoni, G. Cerullo, A. Leitenstorfer, and R. Huber, “Single-cycle multiterahertz transients with peak fields above 10 MV/cm,” Opt. Lett. 35, 2645–2647 (2010).
[CrossRef] [PubMed]

G. Gunter, A. A. Anappara, J. Hees, A. Sell, G. Biasiol, L. Sorba, S. De Liberato, C. Ciuti, A. Tredicucci, A. Leitenstorfer, and R. Huber, “Sub-cycle switch-on of ultrastrong light–matter interaction,” Nature 458, 178–181 (2009).
[CrossRef] [PubMed]

R. Huber, F. Tauser, A. Brodschelm, M. Bichler, G. Abstreiter, and A. Leitenstorfer, “How many-particle interactions develop after ultrafast excitation of an electron–hole plasma,” Nature 414, 286–289 (2001).
[CrossRef] [PubMed]

Lifshitz, E. M.

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

Luer, L.

D. Polli, L. Luer, and G. Cerullo, “High-time-resolution pump-probe system with broadband detection for the study of time-domain vibrational dynamics,” Rev. Sci. Instrum. 78, 103108(2007).
[CrossRef] [PubMed]

Manzoni, C.

D. Brida, C. Manzoni, G. Cirmi, M. Marangoni, S. Bonora, P. Villoresi, S. De Silvestri, and G. Cerullo, “Few-optical-cycle pulses tunable from the mid-infrared by optical parametric amplifiers,” J. Opt. 12, 013001 (2010).
[CrossRef]

C. Manzoni, G. Cerullo, and S. De Silvestri, “Ultrabroadband self-phase-stabilized pulses by difference-frequency generation,” Opt. Lett. 29, 2668–2670 (2004).
[CrossRef] [PubMed]

Marangoni, M.

D. Brida, C. Manzoni, G. Cirmi, M. Marangoni, S. Bonora, P. Villoresi, S. De Silvestri, and G. Cerullo, “Few-optical-cycle pulses tunable from the mid-infrared by optical parametric amplifiers,” J. Opt. 12, 013001 (2010).
[CrossRef]

F. Junginger, A. Sell, O. Schubert, B. Mayer, D. Brida, M. Marangoni, G. Cerullo, A. Leitenstorfer, and R. Huber, “Single-cycle multiterahertz transients with peak fields above 10 MV/cm,” Opt. Lett. 35, 2645–2647 (2010).
[CrossRef] [PubMed]

Mayer, B.

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]

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 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]

Moses, J.

J. Moses and F. W. Wise, “Controllable self-steepening of ultrashort pulses in quadratic nonlinear media,” Phys. Rev. Lett. 97, 073903 (2006).
[CrossRef] [PubMed]

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]

P. Kinsler and G. H. C. New, “Few-cycle pulse propagation,” Phys. Rev. A 67, 023813 (2003).
[CrossRef]

Nibbering, E.

C. Chudoba, E. Nibbering, and T. Elsaesser, “Site-specific excited-state solute-solvent interactions probed by femtosecond vibrational spectroscopy,” Phys. Rev. Lett. 81, 3010–3013(1998).
[CrossRef]

Nikogosyan, D. N.

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

Polli, D.

D. Polli, L. Luer, and G. Cerullo, “High-time-resolution pump-probe system with broadband detection for the study of time-domain vibrational dynamics,” Rev. Sci. Instrum. 78, 103108(2007).
[CrossRef] [PubMed]

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]

Ro, J. H.

N. E. Yu, S. Kurimura, K. Kitamura, J. H. Ro, M. Cha, S. Ashihara, T. Shimura, K. Kuroda, and T. Taira, “Efficient frequency doubling of a femtosecond pulse with simultaneous group-velocity matching and quasi phase matching in periodically poled, MgO-doped lithium niobate,” Appl. Phys. Lett. 82, 3388–3390(2003).
[CrossRef]

N. E. Yu, J. H. Ro, M. Cha, S. Kurimura, and T. Taira, “Broadband quasi-phase-matched second-harmonic generation in MgO-doped periodically poled LiNbO3 at the communications band,” Opt. Lett. 27, 1046–1048 (2002).
[CrossRef]

Schnurer, M.

M. Geissler, G. Tempea, A. Scrinzi, M. Schnurer, F. Krausz, and T. Brabec, “Light propagation in field-ionizing media: extreme nonlinear optics,” Phys. Rev. Lett. 83, 2930–2933 (1999).
[CrossRef]

Schubert, O.

Scrinzi, A.

M. Geissler, G. Tempea, A. Scrinzi, M. Schnurer, F. Krausz, and T. Brabec, “Light propagation in field-ionizing media: extreme nonlinear optics,” Phys. Rev. Lett. 83, 2930–2933 (1999).
[CrossRef]

Sell, A.

F. Junginger, A. Sell, O. Schubert, B. Mayer, D. Brida, M. Marangoni, G. Cerullo, A. Leitenstorfer, and R. Huber, “Single-cycle multiterahertz transients with peak fields above 10 MV/cm,” Opt. Lett. 35, 2645–2647 (2010).
[CrossRef] [PubMed]

G. Gunter, A. A. Anappara, J. Hees, A. Sell, G. Biasiol, L. Sorba, S. De Liberato, C. Ciuti, A. Tredicucci, A. Leitenstorfer, and R. Huber, “Sub-cycle switch-on of ultrastrong light–matter interaction,” Nature 458, 178–181 (2009).
[CrossRef] [PubMed]

Shimura, T.

N. E. Yu, S. Kurimura, K. Kitamura, J. H. Ro, M. Cha, S. Ashihara, T. Shimura, K. Kuroda, and T. Taira, “Efficient frequency doubling of a femtosecond pulse with simultaneous group-velocity matching and quasi phase matching in periodically poled, MgO-doped lithium niobate,” Appl. Phys. Lett. 82, 3388–3390(2003).
[CrossRef]

Sorba, L.

G. Gunter, A. A. Anappara, J. Hees, A. Sell, G. Biasiol, L. Sorba, S. De Liberato, C. Ciuti, A. Tredicucci, A. Leitenstorfer, and R. Huber, “Sub-cycle switch-on of ultrastrong light–matter interaction,” Nature 458, 178–181 (2009).
[CrossRef] [PubMed]

Taira, T.

N. E. Yu, S. Kurimura, K. Kitamura, J. H. Ro, M. Cha, S. Ashihara, T. Shimura, K. Kuroda, and T. Taira, “Efficient frequency doubling of a femtosecond pulse with simultaneous group-velocity matching and quasi phase matching in periodically poled, MgO-doped lithium niobate,” Appl. Phys. Lett. 82, 3388–3390(2003).
[CrossRef]

N. E. Yu, J. H. Ro, M. Cha, S. Kurimura, and T. Taira, “Broadband quasi-phase-matched second-harmonic generation in MgO-doped periodically poled LiNbO3 at the communications band,” Opt. Lett. 27, 1046–1048 (2002).
[CrossRef]

Tauser, F.

R. Huber, F. Tauser, A. Brodschelm, M. Bichler, G. Abstreiter, and A. Leitenstorfer, “How many-particle interactions develop after ultrafast excitation of an electron–hole plasma,” Nature 414, 286–289 (2001).
[CrossRef] [PubMed]

Tempea, G.

M. Geissler, G. Tempea, A. Scrinzi, M. Schnurer, F. Krausz, and T. Brabec, “Light propagation in field-ionizing media: extreme nonlinear optics,” Phys. Rev. Lett. 83, 2930–2933 (1999).
[CrossRef]

Tredicucci, A.

G. Gunter, A. A. Anappara, J. Hees, A. Sell, G. Biasiol, L. Sorba, S. De Liberato, C. Ciuti, A. Tredicucci, A. Leitenstorfer, and R. Huber, “Sub-cycle switch-on of ultrastrong light–matter interaction,” Nature 458, 178–181 (2009).
[CrossRef] [PubMed]

Villoresi, P.

D. Brida, C. Manzoni, G. Cirmi, M. Marangoni, S. Bonora, P. Villoresi, S. De Silvestri, and G. Cerullo, “Few-optical-cycle pulses tunable from the mid-infrared by optical parametric amplifiers,” J. Opt. 12, 013001 (2010).
[CrossRef]

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]

M. Conforti, F. Baronio, A. Degasperis, and S. Wabnitz, “Parametric frequency conversion of short optical pulses controlled by a CW background,” Opt. Express 15, 12246–12251 (2007).
[CrossRef] [PubMed]

Warner, J.

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]

Wise, F. W.

J. Moses and F. W. Wise, “Controllable self-steepening of ultrashort pulses in quadratic nonlinear media,” Phys. Rev. Lett. 97, 073903 (2006).
[CrossRef] [PubMed]

Woutersen, S.

S. Woutersen, U. Emmerichs, and H. J. Bakker, “Femtosecond mid-IR pump probe spectroscopy of liquid water: evidence for a two-component structure,” Science 278, 658–660 (1997).
[CrossRef]

Yu, N. E.

N. E. Yu, S. Kurimura, K. Kitamura, J. H. Ro, M. Cha, S. Ashihara, T. Shimura, K. Kuroda, and T. Taira, “Efficient frequency doubling of a femtosecond pulse with simultaneous group-velocity matching and quasi phase matching in periodically poled, MgO-doped lithium niobate,” Appl. Phys. Lett. 82, 3388–3390(2003).
[CrossRef]

N. E. Yu, J. H. Ro, M. Cha, S. Kurimura, and T. Taira, “Broadband quasi-phase-matched second-harmonic generation in MgO-doped periodically poled LiNbO3 at the communications band,” Opt. Lett. 27, 1046–1048 (2002).
[CrossRef]

Appl. Phys. Lett. (1)

N. E. Yu, S. Kurimura, K. Kitamura, J. H. Ro, M. Cha, S. Ashihara, T. Shimura, K. Kuroda, and T. Taira, “Efficient frequency doubling of a femtosecond pulse with simultaneous group-velocity matching and quasi phase matching in periodically poled, MgO-doped lithium niobate,” Appl. Phys. Lett. 82, 3388–3390(2003).
[CrossRef]

Br. J. Appl. Phys. (1)

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]

IEEE J. Quantum Electron. (1)

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 Photonics J. (1)

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

J. Opt. (1)

D. Brida, C. Manzoni, G. Cirmi, M. Marangoni, S. Bonora, P. Villoresi, S. De Silvestri, and G. Cerullo, “Few-optical-cycle pulses tunable from the mid-infrared by optical parametric amplifiers,” J. Opt. 12, 013001 (2010).
[CrossRef]

Nature (2)

R. Huber, F. Tauser, A. Brodschelm, M. Bichler, G. Abstreiter, and A. Leitenstorfer, “How many-particle interactions develop after ultrafast excitation of an electron–hole plasma,” Nature 414, 286–289 (2001).
[CrossRef] [PubMed]

G. Gunter, A. A. Anappara, J. Hees, A. Sell, G. Biasiol, L. Sorba, S. De Liberato, C. Ciuti, A. Tredicucci, A. Leitenstorfer, and R. Huber, “Sub-cycle switch-on of ultrastrong light–matter interaction,” Nature 458, 178–181 (2009).
[CrossRef] [PubMed]

Opt. Express (2)

Opt. Lett. (3)

Phys. Rev. A (5)

P. Kinsler and G. H. C. New, “Few-cycle pulse propagation,” Phys. Rev. A 67, 023813 (2003).
[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]

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]

Phys. Rev. E (1)

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. (5)

J. Moses and F. W. Wise, “Controllable self-steepening of ultrashort pulses in quadratic nonlinear media,” Phys. Rev. Lett. 97, 073903 (2006).
[CrossRef] [PubMed]

M. Geissler, G. Tempea, A. Scrinzi, M. Schnurer, F. Krausz, and T. Brabec, “Light propagation in field-ionizing media: extreme nonlinear optics,” Phys. Rev. Lett. 83, 2930–2933 (1999).
[CrossRef]

A. V. Housakou and J. Herrmann, “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]

C. Chudoba, E. Nibbering, and T. Elsaesser, “Site-specific excited-state solute-solvent interactions probed by femtosecond vibrational spectroscopy,” Phys. Rev. Lett. 81, 3010–3013(1998).
[CrossRef]

Rev. Mod. Phys. (1)

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

Rev. Sci. Instrum. (2)

G. Cerullo and S. De Silvestri, “Ultrafast optical parametric amplifiers,” Rev. Sci. Instrum. 74, 1–18 (2003).
[CrossRef]

D. Polli, L. Luer, and G. Cerullo, “High-time-resolution pump-probe system with broadband detection for the study of time-domain vibrational dynamics,” Rev. Sci. Instrum. 78, 103108(2007).
[CrossRef] [PubMed]

Science (2)

E. J. Heilweil, “Ultrafast glimpses at water and ice,” Science 283, 1467–1468 (1999).
[CrossRef]

S. Woutersen, U. Emmerichs, and H. J. Bakker, “Femtosecond mid-IR pump probe spectroscopy of liquid water: evidence for a two-component structure,” Science 278, 658–660 (1997).
[CrossRef]

Other (3)

R. W. Boyd, Nonlinear Optics, 2nd ed. (Academic, 2003).

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 (4)

Fig. 1
Fig. 1

Generation of single-cycle idler pulse by parametric mixing of pump and signal waves centered at 1.18 and 1.28 μm , respectively. I p = 25 GW / cm 2 , I s = 40 GW / cm 2 , L = 140 μm . (a) Input (dashed curves) and output (solid curves) power spectrum of extraordinary wave E x (red curves) and ordinary wave E y (blue curves). (b) Power spectrum of ordinary output terahertz component. (c) Single-cycle terahertz transients obtained by filtering E y in the [ 1 100 ] THz range. Dashed curve, in-phase pump-signal input; solid curve, phase difference 3 / 4 π .

Fig. 2
Fig. 2

Evolution of the total power spectrum (decibels). The initial pulse has a Gaussian shape and the parameters are T = 100 fs , λ = 1550 nm , I = 60 GW / cm 2 , Λ QPM = 19.55 μm , and θ = ϕ = π / 2 .

Fig. 3
Fig. 3

Input (dashed curves) and output (solid curves) power spectrum after propagation in a PPMgOLN crystal. Red and blue curves represent extraordinary and ordinary polarized spectra, respectively. Power spectrum is shown both as a function of (a) frequency and of (b) wavelength.

Fig. 4
Fig. 4

(a) Phase-matching curve for Type I SH generation interaction in a PPMg0LN crystal. (b) Level curves of sum frequency wavelength ( λ 3 = λ 1 + λ 2 ) and of mismatch Δ k = 0 (thick blue curves) for V-order (superimposed to λ = 540 nm ) and VII-order QPM (superimposed to λ = 470 nm ).

Tables (1)

Tables Icon

Table 1 Effective Nonlinear Coefficients

Equations (26)

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

× E = B t ,
× H = D t ,
B = μ 0 H ,
D = ε 0 D L + P N L ,
D L , j = ε j k ( t t ) E k ( t ) d t .
A = [ cos ϕ cos θ sin ϕ cos θ sin θ sin ϕ cos ϕ 0 sin θ cos ϕ sin ϕ sin θ cos θ ] .
ε = A ε A T = [ ε o cos 2 θ + ε e sin 2 θ 0 ( ε o ε e ) cos θ sin θ 0 ε o 0 ( ε o ε e ) cos θ sin θ 0 ε o sin 2 θ + ε e cos 2 θ ] .
E = ε 0 1 ε 1 D = ε 0 1 [ ( cos 2 θ ε o + sin 2 θ ε e ) D x ε o 1 D y ε e ε o ε e ε o cos θ sin θ D x ] .
× × E 1 c 2 2 D L t 2 = 1 ε 0 c 2 2 P N L t 2
2 E x z 2 1 c 2 2 D L , x t 2 = 1 ε 0 c 2 2 P N L , x t 2 ,
2 E y z 2 1 c 2 2 D L , y t 2 = 1 ε 0 c 2 2 P N L , y t 2 ,
0 = 1 ε 0 c 2 2 P N L , z t 2 .
2 E m ( z , t ) z 2 1 c 2 2 t 2 + E m ( z , t ) ε m ( t t ) d t = 1 ε 0 c 2 2 t 2 P N L , m ( z , t ) , m = x , y ,
ε x = ( cos 2 θ ε o + sin 2 θ ε e ) 1 ,
ε y = ε o .
2 E ^ m ( z , ω ) z 2 + ω 2 c 2 ε ^ m ( ω ) E ^ m ( z , ω ) = ω 2 ε 0 c 2 P ^ N L , m ( z , ω ) ,
2 U ^ m ( z , ω ) z 2 2 i k m ( ω ) U ^ m ( z , ω ) z = ω 2 ε 0 c 2 P ^ N L , m ( z , ω ) e i k m ( ω ) z .
U ^ m ( z , ω ) z = i ω 2 2 ε 0 c 2 k m ( ω ) P ^ N L , m ( z , ω ) e i k m ( ω ) z ,
E ^ m ( z , ω ) z + i k m ( ω ) E ^ m ( z , ω ) = i ω 2 ε 0 c n m ( ω ) P ^ N L , m ( z , ω ) .
P N L , j = 2 ε 0 d j k l E k E l ,
P NL ( E ) = A P N L ( A T E ) .
E ^ x z + i k x ( ω ) E ^ x = i ω c n x ( ω ) F [ 2 d 1 E x E y + d 2 E y 2 ] , E ^ y z + i k y ( ω ) E ^ y = i ω c n y ( ω ) F [ d 1 E x 2 + 2 d 2 E x E y ] ,
I = [ n x ( ω ) | E ^ x ( ω ) | 2 + n y ( ω ) | E ^ y ( ω ) | 2 ] d ω ,
d I d z = m = x , y n m ( ω ) [ E ^ m z E ^ m * + E ^ m * z E ^ m ] d ω = i ω c [ 2 d 1 ω E ^ x ( ω ) E ^ y ( ω ω ) E ^ x * ( ω ) d ω d ω + d 2 ω E ^ y ( ω ) E ^ y ( ω ω ) E ^ x * ( ω ) d ω d ω + d 1 ω E ^ x ( ω ) E ^ x ( ω ω ) E ^ y * ( ω ) d ω d ω + 2 d 2 ω E ^ x ( ω ) E ^ y ( ω ω ) E ^ y * ( ω ) d ω d ω c.c. ] = i ω c [ J 1 J 2 ] .
d I d z = 2 c { 2 d 1 [ i ω E ^ x ( ω ) ] * [ E ^ x ( ω ) E ^ y ( ω ω ) d ω ] d ω + d 2 [ i ω E ^ x ( ω ) ] * [ E ^ y ( ω ) E ^ y ( ω ω ) d ω ] d ω + d 1 [ i ω E ^ y ( ω ) ] * [ E ^ x ( ω ) E ^ x ( ω ω ) d ω ] d ω + 2 d 2 [ i ω E ^ y ( ω ) ] * [ E ^ x ( ω ) E ^ y ( ω ω ) d ω ] d ω } .
d I d z = 2 c [ 2 d 1 E x t E x E y d t + d 2 E x t E y 2 d t + d 1 E y t E x 2 d t + 2 d 2 E y t E x E y d t ] = 0 ,

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