Abstract

In this paper, we develop for the first time to our knowledge an analytical theory of second harmonic generation (SHG) in a generic nonuniform χ(2) medium. It is shown that by varying the properties of the medium gradually enough, the system can enter an autoresonant state in which the phases of the fundamental pump and of the generated second harmonic wave are locked. The effect of autoresonance allows efficient transfer of energy between the waves and, due to the continuous phase-locking in the system, all the energy of the pump could be converted to the second harmonic. Simple closed-form expressions for the waves amplitudes as a function of the longitudinal coordinate are derived, and an explicit criterion for the stability of the autoresonant state is obtained. Our analytical theory is compared to the numerical solution of the coupled mode equations, which are found to be in excellent agreement with each other. The analytical closed-form expressions that we derive could be very useful for practical design of SHG devices with increased performances, such as highly efficient, wideband frequency converters.

© 2013 Optical Society of America

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  1. R. B. Boyd, Nonlinear Optics, 3rd ed. (Academic, 2008), pp. 96–104.
  2. P. A. Franken, A. E. Hill, C. W. Peters, and G. Weinreich, “Generation of optical harmonics,” Phys. Rev. Lett. 7, 118–119 (1961).
    [CrossRef]
  3. J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
    [CrossRef]
  4. T. Suhara and H. Nishihara, “Theoretical analysis of waveguide second-harmonic generation phase matched with uniform and chirped gratings,” IEEE J. Quantum Electron. 26, 1265–1276 (1990).
    [CrossRef]
  5. K. Mizuuchi, K. Yamamoto, M. Kato, and H. Sato, “Broadening of the phase-matching bandwidth in quasi-phase-matched second-harmonic generation,” IEEE J. Quantum Electron. 30, 1596–1604 (1994).
    [CrossRef]
  6. R. A. Haas, “Influence of a constant temperature gradient on the spectral-bandwidth of second-harmonic generation in nonlinear crystals,” Opt. Commun. 113, 523–529 (1995).
    [CrossRef]
  7. S. Richard, “Second-harmonic generation in tapered optical fibers,” J. Opt. Soc. Am. B 27, 1504–1512 (2010).
    [CrossRef]
  8. K. Regelskis, J. Žius, N. Gavrilin, and G. Račiukaitis, “Efficient second-harmonic generation of a broadband radiation by control of the temperature distribution along a nonlinear crystal,” Opt. Express 20, 28544–28556 (2012).
    [CrossRef]
  9. S. Longhi, “Third-harmonic generation in quasi-phase-matched χ(2) media with missing second harmonic,” Opt. Lett. 32, 1791–1793 (2007).
    [CrossRef]
  10. M. Charbonneau-Lefort, B. Afeyan, and M. M. Fejer, “Competing collinear and noncollinear interactions in chirped quasi-phase-matched optical parametric amplifiers,” J. Opt. Soc. Am. B 25, 1402–1413 (2008).
    [CrossRef]
  11. H. Suchowski, D. Oron, A. Arie, and Y. Silberberg, “Geometrical representation of sum frequency generation and adiabatic frequency conversion,” Phys. Rev. A 78, 063821 (2008).
    [CrossRef]
  12. H. Suchowski, V. Prabhudesai, D. Oron, A. Arie, and Y. Silberberg, “Robust adiabatic sum frequency conversion,” Opt. Express 17, 12731–12740 (2009).
    [CrossRef]
  13. C. R. Phillips and M. M. Fejer, “Efficiency and phase of optical parametric amplification in chirped quasi-phase-matched gratings,” Opt. Lett. 35, 3093–3095 (2010).
    [CrossRef]
  14. O. Yaakobi, L. Caspani, M. Clerici, F. Vidal, and R. Morandotti, “Complete energy conversion by autoresonant three-wave mixing in nonuniform media,” Opt. Express 21, 1623–1632 (2013).
    [CrossRef]
  15. O. Yaakobi and L. Friedland, “Autoresonant four-wave mixing in optical fibers,” Phys. Rev. A 82, 023820 (2010).
    [CrossRef]
  16. A. Barak, Y. Lamhot, L. Friedland, and M. Segev, “Autoresonant dynamics of optical guided waves,” Phys. Rev. Lett. 103, 123901 (2009).
    [CrossRef]
  17. S. Trendafilov, V. Khudik, M. Tokman, and G. Shvets, “Hamiltonian description of non-reciprocal light propagation in nonlinear chiral fibers,” Physica B 405, 3003–3006 (2010).
    [CrossRef]
  18. M. Deutsch, B. Meerson, and J. Golub, “Strong plasma wave excitation by a chirped” laser beat wave,” Phys. Fluids B 3, 1773–1780 (1991).
    [CrossRef]
  19. I. Y. Dodin, G. M. Fraiman, V. M. Malkin, and N. J. Fisch, “Amplification of short laser pulses by Raman backscattering in capillary plasmas,” JETP 95, 625–638 (2002).
    [CrossRef]
  20. R. R. Lindberg, A. E. Charman, J. S. Wurtele, and L. Friedland, “Robust autoresonant excitation in the plasma beat-wave accelerator,” Phys. Rev. Lett. 93, 055001 (2004).
    [CrossRef]
  21. O. Yaakobi, L. Friedland, R. R. Lindberg, A. E. Charman, G. Penn, and J. S. Wurtele, “Spatially autoresonant stimulated Raman scattering in nonuniform plasmas,” Phys. Plasmas 15, 032105 (2008).
    [CrossRef]
  22. O. Yaakobi and L. Friedland, “Multidimensional, autoresonant three-wave interactions,” Phys. Plasmas 15, 102104 (2008).
    [CrossRef]
  23. T. Chapman, S. Huller, P. E. Masson-Laborde, W. Rozmus, and D. Pesme, “Spatially autoresonant stimulated Raman scattering in inhomogeneous plasmas in the kinetic regime,” Phys. Plasmas 17, 122317 (2010).
    [CrossRef]
  24. G. Valiulis, V. Jukna, O. Jedrkiewicz, M. Clerici, E. Rubino, and P. DiTrapani, “Propagation dynamics and X-pulse formation in phase-mismatched second-harmonic generation,” Phys. Rev. A 83, 043834 (2011).
    [CrossRef]
  25. G. Porat and A. Arie, “Efficient, broadband and robust frequency conversion by fully nonlinear adiabatic three-wave mixing,” J. Opt. Soc. Am. B 30, 1342–1351 (2013).
    [CrossRef]

2013 (2)

2012 (1)

2011 (1)

G. Valiulis, V. Jukna, O. Jedrkiewicz, M. Clerici, E. Rubino, and P. DiTrapani, “Propagation dynamics and X-pulse formation in phase-mismatched second-harmonic generation,” Phys. Rev. A 83, 043834 (2011).
[CrossRef]

2010 (5)

T. Chapman, S. Huller, P. E. Masson-Laborde, W. Rozmus, and D. Pesme, “Spatially autoresonant stimulated Raman scattering in inhomogeneous plasmas in the kinetic regime,” Phys. Plasmas 17, 122317 (2010).
[CrossRef]

S. Richard, “Second-harmonic generation in tapered optical fibers,” J. Opt. Soc. Am. B 27, 1504–1512 (2010).
[CrossRef]

O. Yaakobi and L. Friedland, “Autoresonant four-wave mixing in optical fibers,” Phys. Rev. A 82, 023820 (2010).
[CrossRef]

S. Trendafilov, V. Khudik, M. Tokman, and G. Shvets, “Hamiltonian description of non-reciprocal light propagation in nonlinear chiral fibers,” Physica B 405, 3003–3006 (2010).
[CrossRef]

C. R. Phillips and M. M. Fejer, “Efficiency and phase of optical parametric amplification in chirped quasi-phase-matched gratings,” Opt. Lett. 35, 3093–3095 (2010).
[CrossRef]

2009 (2)

H. Suchowski, V. Prabhudesai, D. Oron, A. Arie, and Y. Silberberg, “Robust adiabatic sum frequency conversion,” Opt. Express 17, 12731–12740 (2009).
[CrossRef]

A. Barak, Y. Lamhot, L. Friedland, and M. Segev, “Autoresonant dynamics of optical guided waves,” Phys. Rev. Lett. 103, 123901 (2009).
[CrossRef]

2008 (4)

M. Charbonneau-Lefort, B. Afeyan, and M. M. Fejer, “Competing collinear and noncollinear interactions in chirped quasi-phase-matched optical parametric amplifiers,” J. Opt. Soc. Am. B 25, 1402–1413 (2008).
[CrossRef]

H. Suchowski, D. Oron, A. Arie, and Y. Silberberg, “Geometrical representation of sum frequency generation and adiabatic frequency conversion,” Phys. Rev. A 78, 063821 (2008).
[CrossRef]

O. Yaakobi, L. Friedland, R. R. Lindberg, A. E. Charman, G. Penn, and J. S. Wurtele, “Spatially autoresonant stimulated Raman scattering in nonuniform plasmas,” Phys. Plasmas 15, 032105 (2008).
[CrossRef]

O. Yaakobi and L. Friedland, “Multidimensional, autoresonant three-wave interactions,” Phys. Plasmas 15, 102104 (2008).
[CrossRef]

2007 (1)

2004 (1)

R. R. Lindberg, A. E. Charman, J. S. Wurtele, and L. Friedland, “Robust autoresonant excitation in the plasma beat-wave accelerator,” Phys. Rev. Lett. 93, 055001 (2004).
[CrossRef]

2002 (1)

I. Y. Dodin, G. M. Fraiman, V. M. Malkin, and N. J. Fisch, “Amplification of short laser pulses by Raman backscattering in capillary plasmas,” JETP 95, 625–638 (2002).
[CrossRef]

1995 (1)

R. A. Haas, “Influence of a constant temperature gradient on the spectral-bandwidth of second-harmonic generation in nonlinear crystals,” Opt. Commun. 113, 523–529 (1995).
[CrossRef]

1994 (1)

K. Mizuuchi, K. Yamamoto, M. Kato, and H. Sato, “Broadening of the phase-matching bandwidth in quasi-phase-matched second-harmonic generation,” IEEE J. Quantum Electron. 30, 1596–1604 (1994).
[CrossRef]

1991 (1)

M. Deutsch, B. Meerson, and J. Golub, “Strong plasma wave excitation by a chirped” laser beat wave,” Phys. Fluids B 3, 1773–1780 (1991).
[CrossRef]

1990 (1)

T. Suhara and H. Nishihara, “Theoretical analysis of waveguide second-harmonic generation phase matched with uniform and chirped gratings,” IEEE J. Quantum Electron. 26, 1265–1276 (1990).
[CrossRef]

1962 (1)

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
[CrossRef]

1961 (1)

P. A. Franken, A. E. Hill, C. W. Peters, and G. Weinreich, “Generation of optical harmonics,” Phys. Rev. Lett. 7, 118–119 (1961).
[CrossRef]

Afeyan, B.

Arie, A.

Armstrong, J. A.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
[CrossRef]

Barak, A.

A. Barak, Y. Lamhot, L. Friedland, and M. Segev, “Autoresonant dynamics of optical guided waves,” Phys. Rev. Lett. 103, 123901 (2009).
[CrossRef]

Bloembergen, N.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
[CrossRef]

Boyd, R. B.

R. B. Boyd, Nonlinear Optics, 3rd ed. (Academic, 2008), pp. 96–104.

Caspani, L.

Chapman, T.

T. Chapman, S. Huller, P. E. Masson-Laborde, W. Rozmus, and D. Pesme, “Spatially autoresonant stimulated Raman scattering in inhomogeneous plasmas in the kinetic regime,” Phys. Plasmas 17, 122317 (2010).
[CrossRef]

Charbonneau-Lefort, M.

Charman, A. E.

O. Yaakobi, L. Friedland, R. R. Lindberg, A. E. Charman, G. Penn, and J. S. Wurtele, “Spatially autoresonant stimulated Raman scattering in nonuniform plasmas,” Phys. Plasmas 15, 032105 (2008).
[CrossRef]

R. R. Lindberg, A. E. Charman, J. S. Wurtele, and L. Friedland, “Robust autoresonant excitation in the plasma beat-wave accelerator,” Phys. Rev. Lett. 93, 055001 (2004).
[CrossRef]

Clerici, M.

O. Yaakobi, L. Caspani, M. Clerici, F. Vidal, and R. Morandotti, “Complete energy conversion by autoresonant three-wave mixing in nonuniform media,” Opt. Express 21, 1623–1632 (2013).
[CrossRef]

G. Valiulis, V. Jukna, O. Jedrkiewicz, M. Clerici, E. Rubino, and P. DiTrapani, “Propagation dynamics and X-pulse formation in phase-mismatched second-harmonic generation,” Phys. Rev. A 83, 043834 (2011).
[CrossRef]

Deutsch, M.

M. Deutsch, B. Meerson, and J. Golub, “Strong plasma wave excitation by a chirped” laser beat wave,” Phys. Fluids B 3, 1773–1780 (1991).
[CrossRef]

DiTrapani, P.

G. Valiulis, V. Jukna, O. Jedrkiewicz, M. Clerici, E. Rubino, and P. DiTrapani, “Propagation dynamics and X-pulse formation in phase-mismatched second-harmonic generation,” Phys. Rev. A 83, 043834 (2011).
[CrossRef]

Dodin, I. Y.

I. Y. Dodin, G. M. Fraiman, V. M. Malkin, and N. J. Fisch, “Amplification of short laser pulses by Raman backscattering in capillary plasmas,” JETP 95, 625–638 (2002).
[CrossRef]

Ducuing, J.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
[CrossRef]

Fejer, M. M.

Fisch, N. J.

I. Y. Dodin, G. M. Fraiman, V. M. Malkin, and N. J. Fisch, “Amplification of short laser pulses by Raman backscattering in capillary plasmas,” JETP 95, 625–638 (2002).
[CrossRef]

Fraiman, G. M.

I. Y. Dodin, G. M. Fraiman, V. M. Malkin, and N. J. Fisch, “Amplification of short laser pulses by Raman backscattering in capillary plasmas,” JETP 95, 625–638 (2002).
[CrossRef]

Franken, P. A.

P. A. Franken, A. E. Hill, C. W. Peters, and G. Weinreich, “Generation of optical harmonics,” Phys. Rev. Lett. 7, 118–119 (1961).
[CrossRef]

Friedland, L.

O. Yaakobi and L. Friedland, “Autoresonant four-wave mixing in optical fibers,” Phys. Rev. A 82, 023820 (2010).
[CrossRef]

A. Barak, Y. Lamhot, L. Friedland, and M. Segev, “Autoresonant dynamics of optical guided waves,” Phys. Rev. Lett. 103, 123901 (2009).
[CrossRef]

O. Yaakobi and L. Friedland, “Multidimensional, autoresonant three-wave interactions,” Phys. Plasmas 15, 102104 (2008).
[CrossRef]

O. Yaakobi, L. Friedland, R. R. Lindberg, A. E. Charman, G. Penn, and J. S. Wurtele, “Spatially autoresonant stimulated Raman scattering in nonuniform plasmas,” Phys. Plasmas 15, 032105 (2008).
[CrossRef]

R. R. Lindberg, A. E. Charman, J. S. Wurtele, and L. Friedland, “Robust autoresonant excitation in the plasma beat-wave accelerator,” Phys. Rev. Lett. 93, 055001 (2004).
[CrossRef]

Gavrilin, N.

Golub, J.

M. Deutsch, B. Meerson, and J. Golub, “Strong plasma wave excitation by a chirped” laser beat wave,” Phys. Fluids B 3, 1773–1780 (1991).
[CrossRef]

Haas, R. A.

R. A. Haas, “Influence of a constant temperature gradient on the spectral-bandwidth of second-harmonic generation in nonlinear crystals,” Opt. Commun. 113, 523–529 (1995).
[CrossRef]

Hill, A. E.

P. A. Franken, A. E. Hill, C. W. Peters, and G. Weinreich, “Generation of optical harmonics,” Phys. Rev. Lett. 7, 118–119 (1961).
[CrossRef]

Huller, S.

T. Chapman, S. Huller, P. E. Masson-Laborde, W. Rozmus, and D. Pesme, “Spatially autoresonant stimulated Raman scattering in inhomogeneous plasmas in the kinetic regime,” Phys. Plasmas 17, 122317 (2010).
[CrossRef]

Jedrkiewicz, O.

G. Valiulis, V. Jukna, O. Jedrkiewicz, M. Clerici, E. Rubino, and P. DiTrapani, “Propagation dynamics and X-pulse formation in phase-mismatched second-harmonic generation,” Phys. Rev. A 83, 043834 (2011).
[CrossRef]

Jukna, V.

G. Valiulis, V. Jukna, O. Jedrkiewicz, M. Clerici, E. Rubino, and P. DiTrapani, “Propagation dynamics and X-pulse formation in phase-mismatched second-harmonic generation,” Phys. Rev. A 83, 043834 (2011).
[CrossRef]

Kato, M.

K. Mizuuchi, K. Yamamoto, M. Kato, and H. Sato, “Broadening of the phase-matching bandwidth in quasi-phase-matched second-harmonic generation,” IEEE J. Quantum Electron. 30, 1596–1604 (1994).
[CrossRef]

Khudik, V.

S. Trendafilov, V. Khudik, M. Tokman, and G. Shvets, “Hamiltonian description of non-reciprocal light propagation in nonlinear chiral fibers,” Physica B 405, 3003–3006 (2010).
[CrossRef]

Lamhot, Y.

A. Barak, Y. Lamhot, L. Friedland, and M. Segev, “Autoresonant dynamics of optical guided waves,” Phys. Rev. Lett. 103, 123901 (2009).
[CrossRef]

Lindberg, R. R.

O. Yaakobi, L. Friedland, R. R. Lindberg, A. E. Charman, G. Penn, and J. S. Wurtele, “Spatially autoresonant stimulated Raman scattering in nonuniform plasmas,” Phys. Plasmas 15, 032105 (2008).
[CrossRef]

R. R. Lindberg, A. E. Charman, J. S. Wurtele, and L. Friedland, “Robust autoresonant excitation in the plasma beat-wave accelerator,” Phys. Rev. Lett. 93, 055001 (2004).
[CrossRef]

Longhi, S.

Malkin, V. M.

I. Y. Dodin, G. M. Fraiman, V. M. Malkin, and N. J. Fisch, “Amplification of short laser pulses by Raman backscattering in capillary plasmas,” JETP 95, 625–638 (2002).
[CrossRef]

Masson-Laborde, P. E.

T. Chapman, S. Huller, P. E. Masson-Laborde, W. Rozmus, and D. Pesme, “Spatially autoresonant stimulated Raman scattering in inhomogeneous plasmas in the kinetic regime,” Phys. Plasmas 17, 122317 (2010).
[CrossRef]

Meerson, B.

M. Deutsch, B. Meerson, and J. Golub, “Strong plasma wave excitation by a chirped” laser beat wave,” Phys. Fluids B 3, 1773–1780 (1991).
[CrossRef]

Mizuuchi, K.

K. Mizuuchi, K. Yamamoto, M. Kato, and H. Sato, “Broadening of the phase-matching bandwidth in quasi-phase-matched second-harmonic generation,” IEEE J. Quantum Electron. 30, 1596–1604 (1994).
[CrossRef]

Morandotti, R.

Nishihara, H.

T. Suhara and H. Nishihara, “Theoretical analysis of waveguide second-harmonic generation phase matched with uniform and chirped gratings,” IEEE J. Quantum Electron. 26, 1265–1276 (1990).
[CrossRef]

Oron, D.

H. Suchowski, V. Prabhudesai, D. Oron, A. Arie, and Y. Silberberg, “Robust adiabatic sum frequency conversion,” Opt. Express 17, 12731–12740 (2009).
[CrossRef]

H. Suchowski, D. Oron, A. Arie, and Y. Silberberg, “Geometrical representation of sum frequency generation and adiabatic frequency conversion,” Phys. Rev. A 78, 063821 (2008).
[CrossRef]

Penn, G.

O. Yaakobi, L. Friedland, R. R. Lindberg, A. E. Charman, G. Penn, and J. S. Wurtele, “Spatially autoresonant stimulated Raman scattering in nonuniform plasmas,” Phys. Plasmas 15, 032105 (2008).
[CrossRef]

Pershan, P. S.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
[CrossRef]

Pesme, D.

T. Chapman, S. Huller, P. E. Masson-Laborde, W. Rozmus, and D. Pesme, “Spatially autoresonant stimulated Raman scattering in inhomogeneous plasmas in the kinetic regime,” Phys. Plasmas 17, 122317 (2010).
[CrossRef]

Peters, C. W.

P. A. Franken, A. E. Hill, C. W. Peters, and G. Weinreich, “Generation of optical harmonics,” Phys. Rev. Lett. 7, 118–119 (1961).
[CrossRef]

Phillips, C. R.

Porat, G.

Prabhudesai, V.

Raciukaitis, G.

Regelskis, K.

Richard, S.

Rozmus, W.

T. Chapman, S. Huller, P. E. Masson-Laborde, W. Rozmus, and D. Pesme, “Spatially autoresonant stimulated Raman scattering in inhomogeneous plasmas in the kinetic regime,” Phys. Plasmas 17, 122317 (2010).
[CrossRef]

Rubino, E.

G. Valiulis, V. Jukna, O. Jedrkiewicz, M. Clerici, E. Rubino, and P. DiTrapani, “Propagation dynamics and X-pulse formation in phase-mismatched second-harmonic generation,” Phys. Rev. A 83, 043834 (2011).
[CrossRef]

Sato, H.

K. Mizuuchi, K. Yamamoto, M. Kato, and H. Sato, “Broadening of the phase-matching bandwidth in quasi-phase-matched second-harmonic generation,” IEEE J. Quantum Electron. 30, 1596–1604 (1994).
[CrossRef]

Segev, M.

A. Barak, Y. Lamhot, L. Friedland, and M. Segev, “Autoresonant dynamics of optical guided waves,” Phys. Rev. Lett. 103, 123901 (2009).
[CrossRef]

Shvets, G.

S. Trendafilov, V. Khudik, M. Tokman, and G. Shvets, “Hamiltonian description of non-reciprocal light propagation in nonlinear chiral fibers,” Physica B 405, 3003–3006 (2010).
[CrossRef]

Silberberg, Y.

H. Suchowski, V. Prabhudesai, D. Oron, A. Arie, and Y. Silberberg, “Robust adiabatic sum frequency conversion,” Opt. Express 17, 12731–12740 (2009).
[CrossRef]

H. Suchowski, D. Oron, A. Arie, and Y. Silberberg, “Geometrical representation of sum frequency generation and adiabatic frequency conversion,” Phys. Rev. A 78, 063821 (2008).
[CrossRef]

Suchowski, H.

H. Suchowski, V. Prabhudesai, D. Oron, A. Arie, and Y. Silberberg, “Robust adiabatic sum frequency conversion,” Opt. Express 17, 12731–12740 (2009).
[CrossRef]

H. Suchowski, D. Oron, A. Arie, and Y. Silberberg, “Geometrical representation of sum frequency generation and adiabatic frequency conversion,” Phys. Rev. A 78, 063821 (2008).
[CrossRef]

Suhara, T.

T. Suhara and H. Nishihara, “Theoretical analysis of waveguide second-harmonic generation phase matched with uniform and chirped gratings,” IEEE J. Quantum Electron. 26, 1265–1276 (1990).
[CrossRef]

Tokman, M.

S. Trendafilov, V. Khudik, M. Tokman, and G. Shvets, “Hamiltonian description of non-reciprocal light propagation in nonlinear chiral fibers,” Physica B 405, 3003–3006 (2010).
[CrossRef]

Trendafilov, S.

S. Trendafilov, V. Khudik, M. Tokman, and G. Shvets, “Hamiltonian description of non-reciprocal light propagation in nonlinear chiral fibers,” Physica B 405, 3003–3006 (2010).
[CrossRef]

Valiulis, G.

G. Valiulis, V. Jukna, O. Jedrkiewicz, M. Clerici, E. Rubino, and P. DiTrapani, “Propagation dynamics and X-pulse formation in phase-mismatched second-harmonic generation,” Phys. Rev. A 83, 043834 (2011).
[CrossRef]

Vidal, F.

Weinreich, G.

P. A. Franken, A. E. Hill, C. W. Peters, and G. Weinreich, “Generation of optical harmonics,” Phys. Rev. Lett. 7, 118–119 (1961).
[CrossRef]

Wurtele, J. S.

O. Yaakobi, L. Friedland, R. R. Lindberg, A. E. Charman, G. Penn, and J. S. Wurtele, “Spatially autoresonant stimulated Raman scattering in nonuniform plasmas,” Phys. Plasmas 15, 032105 (2008).
[CrossRef]

R. R. Lindberg, A. E. Charman, J. S. Wurtele, and L. Friedland, “Robust autoresonant excitation in the plasma beat-wave accelerator,” Phys. Rev. Lett. 93, 055001 (2004).
[CrossRef]

Yaakobi, O.

O. Yaakobi, L. Caspani, M. Clerici, F. Vidal, and R. Morandotti, “Complete energy conversion by autoresonant three-wave mixing in nonuniform media,” Opt. Express 21, 1623–1632 (2013).
[CrossRef]

O. Yaakobi and L. Friedland, “Autoresonant four-wave mixing in optical fibers,” Phys. Rev. A 82, 023820 (2010).
[CrossRef]

O. Yaakobi, L. Friedland, R. R. Lindberg, A. E. Charman, G. Penn, and J. S. Wurtele, “Spatially autoresonant stimulated Raman scattering in nonuniform plasmas,” Phys. Plasmas 15, 032105 (2008).
[CrossRef]

O. Yaakobi and L. Friedland, “Multidimensional, autoresonant three-wave interactions,” Phys. Plasmas 15, 102104 (2008).
[CrossRef]

Yamamoto, K.

K. Mizuuchi, K. Yamamoto, M. Kato, and H. Sato, “Broadening of the phase-matching bandwidth in quasi-phase-matched second-harmonic generation,” IEEE J. Quantum Electron. 30, 1596–1604 (1994).
[CrossRef]

Žius, J.

IEEE J. Quantum Electron. (2)

T. Suhara and H. Nishihara, “Theoretical analysis of waveguide second-harmonic generation phase matched with uniform and chirped gratings,” IEEE J. Quantum Electron. 26, 1265–1276 (1990).
[CrossRef]

K. Mizuuchi, K. Yamamoto, M. Kato, and H. Sato, “Broadening of the phase-matching bandwidth in quasi-phase-matched second-harmonic generation,” IEEE J. Quantum Electron. 30, 1596–1604 (1994).
[CrossRef]

J. Opt. Soc. Am. B (3)

JETP (1)

I. Y. Dodin, G. M. Fraiman, V. M. Malkin, and N. J. Fisch, “Amplification of short laser pulses by Raman backscattering in capillary plasmas,” JETP 95, 625–638 (2002).
[CrossRef]

Opt. Commun. (1)

R. A. Haas, “Influence of a constant temperature gradient on the spectral-bandwidth of second-harmonic generation in nonlinear crystals,” Opt. Commun. 113, 523–529 (1995).
[CrossRef]

Opt. Express (3)

Opt. Lett. (2)

Phys. Fluids B (1)

M. Deutsch, B. Meerson, and J. Golub, “Strong plasma wave excitation by a chirped” laser beat wave,” Phys. Fluids B 3, 1773–1780 (1991).
[CrossRef]

Phys. Plasmas (3)

O. Yaakobi, L. Friedland, R. R. Lindberg, A. E. Charman, G. Penn, and J. S. Wurtele, “Spatially autoresonant stimulated Raman scattering in nonuniform plasmas,” Phys. Plasmas 15, 032105 (2008).
[CrossRef]

O. Yaakobi and L. Friedland, “Multidimensional, autoresonant three-wave interactions,” Phys. Plasmas 15, 102104 (2008).
[CrossRef]

T. Chapman, S. Huller, P. E. Masson-Laborde, W. Rozmus, and D. Pesme, “Spatially autoresonant stimulated Raman scattering in inhomogeneous plasmas in the kinetic regime,” Phys. Plasmas 17, 122317 (2010).
[CrossRef]

Phys. Rev. (1)

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[CrossRef]

Phys. Rev. A (3)

O. Yaakobi and L. Friedland, “Autoresonant four-wave mixing in optical fibers,” Phys. Rev. A 82, 023820 (2010).
[CrossRef]

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[CrossRef]

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[CrossRef]

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[CrossRef]

Other (1)

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Figures (3)

Fig. 1.
Fig. 1.

Autoresonant evolution of (a) the wave envelopes B j 2 and (b) the phase mismatch Φ versus the normalized distance | α | ξ for a spatial nonuniformity rate α = 0.1 . The numerical solution of Eqs. (9) and (10) is plotted by colored curves. In (a), the fundamental wave B 1 and the second harmonic wave B 2 are plotted by red and blue, respectively, and the black curves represent the value of the analytical expressions Eqs. (19) and (20). The vertical dashed line is located at | α | ξ = 2 where complete pump depletion is obtained and phase-locking is lost. Note that there is an excellent agreement between the numerical solution and the analytical autoresonant solution, and the corresponding curves are overlapping until | α | ξ = 2 .

Fig. 2.
Fig. 2.

Evolution of (a) wave envelopes B j 2 and (b) phase mismatch Φ for large value of the nonuniformity rate α = 10 . Also in this case, there is a very good agreement between the numerical solution (colored lines) and the analytical autoresonant solution (black line). Note that the corresponding curves are overlapping until | α | ξ 5 , where phase-locking is lost. Because of the large value of | α | , the stability condition for autoresonance, given by Eq. (24), is not satisfied after | α | ξ 5 and consequently the final conversion efficiency is relatively low.

Fig. 3.
Fig. 3.

Stability function Ω 2 versus | α | ξ given by the analytical expression in Eq. (25). Satisfying the criterion | α | < Ω 2 continuously is required for the stability of phase-locking.

Equations (34)

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d A 1 d z = i η 1 A 1 * A 2 e i 0 z Δ k ( z ) d z ,
d A 2 d z = i η 2 2 A 1 2 e + i 0 z Δ k ( z ) d z .
d a 1 d ζ = i a 1 * a 2 e i l 0 ζ Δ k ( ζ ) d ζ ,
d a 2 d ζ = i 1 2 a 1 2 e + i l 0 ζ Δ k ( ζ ) d ζ .
d B 1 d ζ = B 1 B 2 sin Φ ,
d B 2 d ζ = 1 2 B 1 2 sin Φ ,
d Φ d ζ = l Δ k + Q cos Φ ,
B 1 2 + 2 B 2 2 = B 1 , 0 2 + 2 B 2 , 0 2 = 1 + 2 B 2 , 0 2 .
d B 2 d ξ = 1 2 B 1 2 sin Φ ,
d Φ d ξ = α ξ + Q cos Φ .
d B 2 d ξ = 1 2 sin Φ ,
d Φ d ξ = α ξ 1 2 B 2 cos Φ ,
Z = 1 2 α ξ ini [ e i α ( ξ 2 ξ ini 2 ) / 2 ξ ini ξ ] ,
α ξ s Q ^ = 0 , i.e. Q ^ = | α | ξ ,
sin Φ ^ = 2 B ^ 1 2 d B ^ 2 d ξ .
B 1 2 + 2 B 2 2 = 1 .
Q ^ = 2 3 B ^ 1 2 2 ( 1 B ^ 1 2 ) .
2 3 B ^ 1 2 2 ( 1 B ^ 1 2 ) = | α | ξ .
B ^ 1 2 = 6 ( | α | ξ ) 2 | α | ξ 6 + ( | α | ξ ) 2 9 , | α | ξ 2 .
B ^ 2 2 = 3 + ( | α | ξ ) 2 + | α | ξ 6 + ( | α | ξ ) 2 18 , | α | ξ 2 .
d Φ d ξ = s ( | α | ξ Q ^ ) s Q ^ δ B 2 ,
d ( δ Φ ) d ξ = s Q ^ δ B 2 .
d ( δ B 2 ) d ξ = s B ^ 1 2 2 sin δ Φ | α | Q ^ ,
H ( δ B 2 , δ Φ , ξ ) = s Q ^ 2 ( δ B 2 ) 2 + V eff ( δ Φ , ξ ) ,
| α | < Ω 2 ,
Ω 2 = ( | α | ξ ) 2 3 2 | α | ξ ( | α | ξ ) 2 + 6 3 + 2 , | α | ξ < 2 .
| Δ k fin Δ k ini | L < 0.23 l 2 ,
l = [ ( c ε 0 8 π 2 ) · n 1 , avg 2 n 2 , avg λ 1 λ 2 d eff 2 I 1 , 0 ] 1 / 2 .
l = [ ( c ε 0 8 π 2 ) · n 1 , a v g n 2 , a v g n 1 , 0 λ 1 λ 2 d e f f 2 I 1 , 0 ] 1 / 2 [ ( c ε 0 8 π 2 ) · n 1 , avg 2 n 2 , avg λ 1 λ 2 d eff 2 I 1 , 0 ] 1 / 2 ,
α = l 2 L ( Δ k fin Δ k ini ) ,
ξ = l Δ k α = L l · Δ k ( Δ k fin Δ k ini ) ,
| α | ξ = l · Δ k · sign ( Δ k fin Δ k ini ) .
| Δ k ini | > 1 2 l ; | Δ k fin | > 2 l .
| Δ k fin Δ k ini | L < Ω 2 l 2 ,

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