Abstract

The interaction of four waves propagating in a medium with third-order nonlinearities is considered. In general, nonlinear phase shifts detune the interaction before the transfer of energy from the pump to the daughter waves is complete. Fortunately, however, it is possible to compensate for this nonlinear detuning by imposing a linear wave-number mismatch on the interaction. Simple criteria are given that allow the weaker pump wave to be depleted completely and, hence, the energy transfer to be maximized. Some exact solutions of the four-wave equations are used to illustrate the underlying wave physics, and some useful approximate solutions are described. As an application of the general theory, near-collinear four-wave mixing in a Kerr medium is discussed briefly.

© 1993 Optical Society of America

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References

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  1. C. J. McKinstrie and X. D. Cao, “Nonlinear detuning of three-wave interactions,” J. Opt. Soc. Am. B 10, 898–912 (1993), and references therein.
    [CrossRef]
  2. Y. Inoue, “Resonant four-wave interaction in a dispersive medium,” J. Phys. Soc. Jpn. 39, 1092–1099 (1975).
    [CrossRef]
  3. T. J. M. Boyd and J. G. Turner, “Four-wave interactions in plasmas,” Lett. Math. Phys. 1, 477–484 (1977).
    [CrossRef]
  4. C. J. McKinstrie and G. G. Luther, “Solitary-wave solutions of the generalised three-wave and four-wave equations,” Phys. Lett. A 127, 14–18 (1988).
    [CrossRef]
  5. Y. Chen and A. W. Snyder, “Four-photon parametric mixing in optical fibers: effect of pump depletion,” Opt. Lett. 14, 87–89 (1989).
    [CrossRef] [PubMed]
  6. Y. Chen, “Four-wave mixing in optical fibers: exact solution,” J. Opt. Soc. Am. B 6, 1986–1993 (1989).
    [CrossRef]
  7. C. J. McKinstrie, G. G. Luther, and S. H. Batha, “Signal enhancement in collinear four-wave mixing,” J. Opt. Soc. Am. B 7, 340–344 (1990).
    [CrossRef]
  8. X. D. Cao and C. J. McKinstrie, “The optimization of sum-frequency generation,” paper 1P10, presented at the 21st Annual Anomalous Absorption Conference, Banff, Alberta, CanadaApril 15–19, 1991. Although it was concerned primarily with three-wave interactions, this presentation included several results for four-wave interactions.
  9. B. Kryzhanovsky, A. Karapetyan, and B. Glushko, “Theory of energy exchange and conversion via four-wave mixing in a nondissipative χ(3)material,” Phys. Rev. A 44, 6036–6042 (1991).
    [CrossRef] [PubMed]
  10. R. Y. Chiao, P. L. Kelley, and E. Garmire, “Stimulated four-photon interaction and its influence on stimulated Rayleigh-wing scattering,” Phys. Rev. Lett. 17, 1158–1161 (1966).
    [CrossRef]
  11. J. H. Marburger and J. F. Lam, “Effect of nonlinear index changes on degenerate four-wave mixing,” Appl. Phys. Lett. 35, 249–251 (1979).
    [CrossRef]
  12. R. Bingham and C. N. Lashmore-Davies, “On the nonlinear development of the Langmuir modulational instability,” J. Plasma Phys. 21, 51–69 (1979).
    [CrossRef]
  13. R. O. Dendy and D. Ter Haar, “On the integration of a three-wave set of equations,” Phys. Lett. A 97, 129–130 (1983).
    [CrossRef]
  14. R. O. Dendy and D. Ter Haar, “On the nonlinear development of the Langmuir modulational instability,” J. Plasma Phys. 31, 67–79 (1984).
    [CrossRef]
  15. G. Cappelliini and S. Trillo, “Third-order three-wave mixing in single-mode fibers: Exact solutions and spatial instability effects,” J. Opt. Soc. Am. B 8, 824–838 (1991).
    [CrossRef]
  16. 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]
  17. J. M. Manley and H. E. Rowe, “Some general properties of nonlinear elements—Part I. General energy relations,” Proc. IRE 44, 904–913 (1956).
    [CrossRef]
  18. M. T. Weiss, “Quantum derivation of energy relations analogous to those for nonlinear reactances,” Proc. IRE 45, 1012–1013 (1957).
  19. M. Kauranen, J. J. Maki, A. L. Gaeta, and R. W. Boyd, “Two-beam-excited conical emission,” Opt. Lett. 16, 943–945 (1991), and references therein.
    [CrossRef] [PubMed]
  20. M. Kauranen, A. L. Gaeta, and C. J. McKinstrie, “Transverse instabilities of two intersecting laser beams in a nonlinear Kerr medium,” J. Opt. Soc. Am. B (to be published).
  21. G. P. Agrawal, Nonlinear Fiber Optics (Academic, Boston, Mass., 1989), Chap. 10.
  22. C. J. McKinstrie and S. H. Batha, “Energy cascading in the beat-wave accelerator,” in New Developments in Particle Acceleration Techniques, S. Turner, ed. (Centre Européen de la Recherche Nucléaire, Geneva, 1987), pp. 443–457, and references therein.
  23. R. R. E. Salomaa and S. J. Karttunen, “Dephasing effects in beat-wave acceleration,” in New Developments in Particle Acceleration Techniques, S. Turner, ed. (Centre Européen de la Recherche Nucléaire, Geneva, 1987), pp. 458–471.
  24. R. A. Sammut and S. J. Garth, “Power exchange between generated frequencies in single-mode optical fibers,” J. Opt. Soc. Am. B 8, 2097–2101 (1991).
    [CrossRef]
  25. R. Lytel, “Pump-depletion effects in noncollinear degenerate four-wave mixing in Kerr media,” J. Opt. Soc. Am. B 3, 1580–1584 (1986), and references therein.
    [CrossRef]
  26. P. F. Byrd and M. D. Friedman, Handbook of Elliptic Functions for Engineers and Scientists, 2nd ed. (Springer-Verlag, New York, 1971), pp. 18–21, 72, 77, 112, and 133.
  27. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (National Bureau of Standards, Washington, D.C., 1964), p. 591.
  28. R. W. Boyd, Nonlinear Optics (Academic, Boston, Mass., 1992), Chaps. 3 and 5.

1993 (1)

1991 (4)

1990 (1)

1989 (2)

1988 (1)

C. J. McKinstrie and G. G. Luther, “Solitary-wave solutions of the generalised three-wave and four-wave equations,” Phys. Lett. A 127, 14–18 (1988).
[CrossRef]

1986 (1)

1984 (1)

R. O. Dendy and D. Ter Haar, “On the nonlinear development of the Langmuir modulational instability,” J. Plasma Phys. 31, 67–79 (1984).
[CrossRef]

1983 (1)

R. O. Dendy and D. Ter Haar, “On the integration of a three-wave set of equations,” Phys. Lett. A 97, 129–130 (1983).
[CrossRef]

1979 (2)

J. H. Marburger and J. F. Lam, “Effect of nonlinear index changes on degenerate four-wave mixing,” Appl. Phys. Lett. 35, 249–251 (1979).
[CrossRef]

R. Bingham and C. N. Lashmore-Davies, “On the nonlinear development of the Langmuir modulational instability,” J. Plasma Phys. 21, 51–69 (1979).
[CrossRef]

1977 (1)

T. J. M. Boyd and J. G. Turner, “Four-wave interactions in plasmas,” Lett. Math. Phys. 1, 477–484 (1977).
[CrossRef]

1975 (1)

Y. Inoue, “Resonant four-wave interaction in a dispersive medium,” J. Phys. Soc. Jpn. 39, 1092–1099 (1975).
[CrossRef]

1966 (1)

R. Y. Chiao, P. L. Kelley, and E. Garmire, “Stimulated four-photon interaction and its influence on stimulated Rayleigh-wing scattering,” Phys. Rev. Lett. 17, 1158–1161 (1966).
[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]

1957 (1)

M. T. Weiss, “Quantum derivation of energy relations analogous to those for nonlinear reactances,” Proc. IRE 45, 1012–1013 (1957).

1956 (1)

J. M. Manley and H. E. Rowe, “Some general properties of nonlinear elements—Part I. General energy relations,” Proc. IRE 44, 904–913 (1956).
[CrossRef]

Abramowitz, M.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (National Bureau of Standards, Washington, D.C., 1964), p. 591.

Agrawal, G. P.

G. P. Agrawal, Nonlinear Fiber Optics (Academic, Boston, Mass., 1989), Chap. 10.

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]

Batha, S. H.

C. J. McKinstrie, G. G. Luther, and S. H. Batha, “Signal enhancement in collinear four-wave mixing,” J. Opt. Soc. Am. B 7, 340–344 (1990).
[CrossRef]

C. J. McKinstrie and S. H. Batha, “Energy cascading in the beat-wave accelerator,” in New Developments in Particle Acceleration Techniques, S. Turner, ed. (Centre Européen de la Recherche Nucléaire, Geneva, 1987), pp. 443–457, and references therein.

Bingham, R.

R. Bingham and C. N. Lashmore-Davies, “On the nonlinear development of the Langmuir modulational instability,” J. Plasma Phys. 21, 51–69 (1979).
[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. W.

Boyd, T. J. M.

T. J. M. Boyd and J. G. Turner, “Four-wave interactions in plasmas,” Lett. Math. Phys. 1, 477–484 (1977).
[CrossRef]

Byrd, P. F.

P. F. Byrd and M. D. Friedman, Handbook of Elliptic Functions for Engineers and Scientists, 2nd ed. (Springer-Verlag, New York, 1971), pp. 18–21, 72, 77, 112, and 133.

Cao, X. D.

C. J. McKinstrie and X. D. Cao, “Nonlinear detuning of three-wave interactions,” J. Opt. Soc. Am. B 10, 898–912 (1993), and references therein.
[CrossRef]

X. D. Cao and C. J. McKinstrie, “The optimization of sum-frequency generation,” paper 1P10, presented at the 21st Annual Anomalous Absorption Conference, Banff, Alberta, CanadaApril 15–19, 1991. Although it was concerned primarily with three-wave interactions, this presentation included several results for four-wave interactions.

Cappelliini, G.

Chen, Y.

Chiao, R. Y.

R. Y. Chiao, P. L. Kelley, and E. Garmire, “Stimulated four-photon interaction and its influence on stimulated Rayleigh-wing scattering,” Phys. Rev. Lett. 17, 1158–1161 (1966).
[CrossRef]

Dendy, R. O.

R. O. Dendy and D. Ter Haar, “On the nonlinear development of the Langmuir modulational instability,” J. Plasma Phys. 31, 67–79 (1984).
[CrossRef]

R. O. Dendy and D. Ter Haar, “On the integration of a three-wave set of equations,” Phys. Lett. A 97, 129–130 (1983).
[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]

Friedman, M. D.

P. F. Byrd and M. D. Friedman, Handbook of Elliptic Functions for Engineers and Scientists, 2nd ed. (Springer-Verlag, New York, 1971), pp. 18–21, 72, 77, 112, and 133.

Gaeta, A. L.

M. Kauranen, J. J. Maki, A. L. Gaeta, and R. W. Boyd, “Two-beam-excited conical emission,” Opt. Lett. 16, 943–945 (1991), and references therein.
[CrossRef] [PubMed]

M. Kauranen, A. L. Gaeta, and C. J. McKinstrie, “Transverse instabilities of two intersecting laser beams in a nonlinear Kerr medium,” J. Opt. Soc. Am. B (to be published).

Garmire, E.

R. Y. Chiao, P. L. Kelley, and E. Garmire, “Stimulated four-photon interaction and its influence on stimulated Rayleigh-wing scattering,” Phys. Rev. Lett. 17, 1158–1161 (1966).
[CrossRef]

Garth, S. J.

Glushko, B.

B. Kryzhanovsky, A. Karapetyan, and B. Glushko, “Theory of energy exchange and conversion via four-wave mixing in a nondissipative χ(3)material,” Phys. Rev. A 44, 6036–6042 (1991).
[CrossRef] [PubMed]

Inoue, Y.

Y. Inoue, “Resonant four-wave interaction in a dispersive medium,” J. Phys. Soc. Jpn. 39, 1092–1099 (1975).
[CrossRef]

Karapetyan, A.

B. Kryzhanovsky, A. Karapetyan, and B. Glushko, “Theory of energy exchange and conversion via four-wave mixing in a nondissipative χ(3)material,” Phys. Rev. A 44, 6036–6042 (1991).
[CrossRef] [PubMed]

Karttunen, S. J.

R. R. E. Salomaa and S. J. Karttunen, “Dephasing effects in beat-wave acceleration,” in New Developments in Particle Acceleration Techniques, S. Turner, ed. (Centre Européen de la Recherche Nucléaire, Geneva, 1987), pp. 458–471.

Kauranen, M.

M. Kauranen, J. J. Maki, A. L. Gaeta, and R. W. Boyd, “Two-beam-excited conical emission,” Opt. Lett. 16, 943–945 (1991), and references therein.
[CrossRef] [PubMed]

M. Kauranen, A. L. Gaeta, and C. J. McKinstrie, “Transverse instabilities of two intersecting laser beams in a nonlinear Kerr medium,” J. Opt. Soc. Am. B (to be published).

Kelley, P. L.

R. Y. Chiao, P. L. Kelley, and E. Garmire, “Stimulated four-photon interaction and its influence on stimulated Rayleigh-wing scattering,” Phys. Rev. Lett. 17, 1158–1161 (1966).
[CrossRef]

Kryzhanovsky, B.

B. Kryzhanovsky, A. Karapetyan, and B. Glushko, “Theory of energy exchange and conversion via four-wave mixing in a nondissipative χ(3)material,” Phys. Rev. A 44, 6036–6042 (1991).
[CrossRef] [PubMed]

Lam, J. F.

J. H. Marburger and J. F. Lam, “Effect of nonlinear index changes on degenerate four-wave mixing,” Appl. Phys. Lett. 35, 249–251 (1979).
[CrossRef]

Lashmore-Davies, C. N.

R. Bingham and C. N. Lashmore-Davies, “On the nonlinear development of the Langmuir modulational instability,” J. Plasma Phys. 21, 51–69 (1979).
[CrossRef]

Luther, G. G.

C. J. McKinstrie, G. G. Luther, and S. H. Batha, “Signal enhancement in collinear four-wave mixing,” J. Opt. Soc. Am. B 7, 340–344 (1990).
[CrossRef]

C. J. McKinstrie and G. G. Luther, “Solitary-wave solutions of the generalised three-wave and four-wave equations,” Phys. Lett. A 127, 14–18 (1988).
[CrossRef]

Lytel, R.

Maki, J. J.

Manley, J. M.

J. M. Manley and H. E. Rowe, “Some general properties of nonlinear elements—Part I. General energy relations,” Proc. IRE 44, 904–913 (1956).
[CrossRef]

Marburger, J. H.

J. H. Marburger and J. F. Lam, “Effect of nonlinear index changes on degenerate four-wave mixing,” Appl. Phys. Lett. 35, 249–251 (1979).
[CrossRef]

McKinstrie, C. J.

C. J. McKinstrie and X. D. Cao, “Nonlinear detuning of three-wave interactions,” J. Opt. Soc. Am. B 10, 898–912 (1993), and references therein.
[CrossRef]

C. J. McKinstrie, G. G. Luther, and S. H. Batha, “Signal enhancement in collinear four-wave mixing,” J. Opt. Soc. Am. B 7, 340–344 (1990).
[CrossRef]

C. J. McKinstrie and G. G. Luther, “Solitary-wave solutions of the generalised three-wave and four-wave equations,” Phys. Lett. A 127, 14–18 (1988).
[CrossRef]

M. Kauranen, A. L. Gaeta, and C. J. McKinstrie, “Transverse instabilities of two intersecting laser beams in a nonlinear Kerr medium,” J. Opt. Soc. Am. B (to be published).

C. J. McKinstrie and S. H. Batha, “Energy cascading in the beat-wave accelerator,” in New Developments in Particle Acceleration Techniques, S. Turner, ed. (Centre Européen de la Recherche Nucléaire, Geneva, 1987), pp. 443–457, and references therein.

X. D. Cao and C. J. McKinstrie, “The optimization of sum-frequency generation,” paper 1P10, presented at the 21st Annual Anomalous Absorption Conference, Banff, Alberta, CanadaApril 15–19, 1991. Although it was concerned primarily with three-wave interactions, this presentation included several results for four-wave interactions.

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]

Rowe, H. E.

J. M. Manley and H. E. Rowe, “Some general properties of nonlinear elements—Part I. General energy relations,” Proc. IRE 44, 904–913 (1956).
[CrossRef]

Salomaa, R. R. E.

R. R. E. Salomaa and S. J. Karttunen, “Dephasing effects in beat-wave acceleration,” in New Developments in Particle Acceleration Techniques, S. Turner, ed. (Centre Européen de la Recherche Nucléaire, Geneva, 1987), pp. 458–471.

Sammut, R. A.

Snyder, A. W.

Stegun, I. A.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (National Bureau of Standards, Washington, D.C., 1964), p. 591.

Ter Haar, D.

R. O. Dendy and D. Ter Haar, “On the nonlinear development of the Langmuir modulational instability,” J. Plasma Phys. 31, 67–79 (1984).
[CrossRef]

R. O. Dendy and D. Ter Haar, “On the integration of a three-wave set of equations,” Phys. Lett. A 97, 129–130 (1983).
[CrossRef]

Trillo, S.

Turner, J. G.

T. J. M. Boyd and J. G. Turner, “Four-wave interactions in plasmas,” Lett. Math. Phys. 1, 477–484 (1977).
[CrossRef]

Weiss, M. T.

M. T. Weiss, “Quantum derivation of energy relations analogous to those for nonlinear reactances,” Proc. IRE 45, 1012–1013 (1957).

Appl. Phys. Lett. (1)

J. H. Marburger and J. F. Lam, “Effect of nonlinear index changes on degenerate four-wave mixing,” Appl. Phys. Lett. 35, 249–251 (1979).
[CrossRef]

J. Opt. Soc. Am. B (6)

J. Phys. Soc. Jpn. (1)

Y. Inoue, “Resonant four-wave interaction in a dispersive medium,” J. Phys. Soc. Jpn. 39, 1092–1099 (1975).
[CrossRef]

J. Plasma Phys. (2)

R. Bingham and C. N. Lashmore-Davies, “On the nonlinear development of the Langmuir modulational instability,” J. Plasma Phys. 21, 51–69 (1979).
[CrossRef]

R. O. Dendy and D. Ter Haar, “On the nonlinear development of the Langmuir modulational instability,” J. Plasma Phys. 31, 67–79 (1984).
[CrossRef]

Lett. Math. Phys. (1)

T. J. M. Boyd and J. G. Turner, “Four-wave interactions in plasmas,” Lett. Math. Phys. 1, 477–484 (1977).
[CrossRef]

Opt. Lett. (2)

Phys. Lett. A (2)

R. O. Dendy and D. Ter Haar, “On the integration of a three-wave set of equations,” Phys. Lett. A 97, 129–130 (1983).
[CrossRef]

C. J. McKinstrie and G. G. Luther, “Solitary-wave solutions of the generalised three-wave and four-wave equations,” Phys. Lett. A 127, 14–18 (1988).
[CrossRef]

Phys. Rev. (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]

Phys. Rev. A (1)

B. Kryzhanovsky, A. Karapetyan, and B. Glushko, “Theory of energy exchange and conversion via four-wave mixing in a nondissipative χ(3)material,” Phys. Rev. A 44, 6036–6042 (1991).
[CrossRef] [PubMed]

Phys. Rev. Lett. (1)

R. Y. Chiao, P. L. Kelley, and E. Garmire, “Stimulated four-photon interaction and its influence on stimulated Rayleigh-wing scattering,” Phys. Rev. Lett. 17, 1158–1161 (1966).
[CrossRef]

Proc. IRE (2)

J. M. Manley and H. E. Rowe, “Some general properties of nonlinear elements—Part I. General energy relations,” Proc. IRE 44, 904–913 (1956).
[CrossRef]

M. T. Weiss, “Quantum derivation of energy relations analogous to those for nonlinear reactances,” Proc. IRE 45, 1012–1013 (1957).

Other (8)

X. D. Cao and C. J. McKinstrie, “The optimization of sum-frequency generation,” paper 1P10, presented at the 21st Annual Anomalous Absorption Conference, Banff, Alberta, CanadaApril 15–19, 1991. Although it was concerned primarily with three-wave interactions, this presentation included several results for four-wave interactions.

M. Kauranen, A. L. Gaeta, and C. J. McKinstrie, “Transverse instabilities of two intersecting laser beams in a nonlinear Kerr medium,” J. Opt. Soc. Am. B (to be published).

G. P. Agrawal, Nonlinear Fiber Optics (Academic, Boston, Mass., 1989), Chap. 10.

C. J. McKinstrie and S. H. Batha, “Energy cascading in the beat-wave accelerator,” in New Developments in Particle Acceleration Techniques, S. Turner, ed. (Centre Européen de la Recherche Nucléaire, Geneva, 1987), pp. 443–457, and references therein.

R. R. E. Salomaa and S. J. Karttunen, “Dephasing effects in beat-wave acceleration,” in New Developments in Particle Acceleration Techniques, S. Turner, ed. (Centre Européen de la Recherche Nucléaire, Geneva, 1987), pp. 458–471.

P. F. Byrd and M. D. Friedman, Handbook of Elliptic Functions for Engineers and Scientists, 2nd ed. (Springer-Verlag, New York, 1971), pp. 18–21, 72, 77, 112, and 133.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (National Bureau of Standards, Washington, D.C., 1964), p. 591.

R. W. Boyd, Nonlinear Optics (Academic, Boston, Mass., 1992), Chaps. 3 and 5.

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Figures (10)

Fig. 1
Fig. 1

(a) Action flux densities of the stronger pump and probe waves and (b) the relative phase of the four waves plotted as functions of position for the case in which λ = 0.50, ρ = 0.81, and δ = 0.00. For these parameters the first of solutions (3.8) is relevant. The action flux densities are normalized to the initial action flux density of the stronger pump wave, and the relative phase is measured in units of π. Distance is measured in units of the growth length of the linear four-wave instability.

Fig. 2
Fig. 2

Peak action flux density of the signal wave, normalized to the initial action flux density of the stronger pump wave, plotted as a function of the initial wave-number mismatch δ for the case in which ρ = 0.81. (a) λ = 0.50, (b) λ = 2.00. In both cases the first of Eqs. (3.10) is relevant.

Fig. 3
Fig. 3

(a) Normalized action flux densities of the stronger pump and probe waves and (b) the relative phase of the four waves in units of π, plotted as functions of position for the case in which λ = 0.50, ρ = 0.81, and δ = 0.40. For these parameters the first of solutions (3.14) is relevant.

Fig. 4
Fig. 4

(a) Normalized action flux densities of the stronger pump and probe waves and (b) the relative phase of the four waves in units of π, plotted as functions of position for the case in which λ = 0.50, and = 0.01. For these parameters the first of solutions (4.7) is relevant.

Fig. 5
Fig. 5

(a) Normalized action flux densities of the stronger pump and probe waves and (b) the relative phase of the four waves in units of π, plotted as functions of position for the case in which λ = 0.50, ρ = 0.81, and = 0.01. For these parameters solution (4.11) is relevant.

Fig. 6
Fig. 6

Exact solution (4.11) and approximate solution (3.14) superimposed for the case in which λ = 0.50, ρ = 0.81, and = 0.01. (a) Normalized action flux density of the probe wave; (b) relative phase of the four waves in units of π.

Fig. 7
Fig. 7

(a) Normalized action flux densities of the stronger pump and probe waves and (b) the relative phase of the four waves in units of π, plotted as functions of position for the case in which λ = 2.00, ρ = 0.16, and = 0.02. For these parameters solution (4.25) is relevant.

Fig. 8
Fig. 8

(a) Normalized action flux densities of the stronger pump and probe waves and (b) the relative phase of the four waves in units of π, plotted as functions of position for the case in which λ = 2.00, ρ = 0.81, and = 0.75. For these parameters solution (4.25) is relevant.

Fig. 9
Fig. 9

Potential function Q(F) associated with Eq. (4.3). (a) Q(F) displayed for the case in which λ = 2.00, ρ = 0.81, and = 0.75, as in Fig. 8; (b) 100Q(F) displayed for the case in which λ = 2.00, ρ = 0.16, and = 0.02, as in Fig. 7.

Fig. 10
Fig. 10

Interaction geometry for near-collinear four-wave mixing. Notice that the angles of inclination of the daughter waves need not equal those of the pump waves. The angles of inclination are all exaggerated for clarity.

Equations (126)

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

ω 1 = ω 2 + ω 3 , k 1 = k 2 + k 3 .
ω 1 + ω 2 = ω 3 + ω 4 , k 1 + k 2 = k 3 + k 4 .
s 1 d z A 1 = i A 2 * A 3 A 4 + i ( δ 1 + λ 1 β | A β | 2 ) A 1 , s 3 d z A 3 = i A 4 * A 1 A 2 + i ( δ 3 + λ 3 β | A β | 2 ) A 3 ,
A α = F α 1 / 2 exp ( i ϕ α ) .
s 1 d z F 1 = 2 ( F 1 F 2 F 3 F 4 ) 1 / 2 sin ϕ , s 3 d z F 3 = 2 ( F 1 F 2 F 3 F 4 ) 1 / 2 sin ϕ
s 1 d z ϕ 1 = ( F 2 F 3 F 4 / F 1 ) 1 / 2 cos ϕ + ( δ 1 + λ 1 β F β ) , s 3 d z ϕ 3 = ( F 4 F 1 F 2 / F 3 ) 1 / 2 cos ϕ + ( δ 3 + λ 3 β F β ) ,
ϕ = ϕ 1 + ϕ 2 ϕ 3 ϕ 4
H = 2 ( F 1 F 2 F 3 F 4 ) 1 / 2 cos ϕ ( δ α + 1 2 λ α β F β ) F α ,
s α d F α d z = H ϕ α , s α d ϕ α d z = H F α .
( d z F 3 ) 3 = 4 F 1 F 2 F 3 F 4 [ H + ( δ α + 1 2 λ α β F β ) F α ] 2 .
d z ( s 1 F 1 s 2 F 2 ) = 0 , d z ( s 3 F 3 s 4 F 4 ) = 0.
d z ( s 1 F 1 + s 2 F 2 + s 3 F 3 + s 4 F 4 ) = 0.
F 1 ( ) = 1 , F 2 ( ) = ρ , F 3 ( ) = F 4 ( ) = 0 ,
F 3 ( z ) = F 4 ( z ) = F ( z ) , F 1 ( z ) = 1 F ( z ) , F 2 ( z ) = ρ f ( z ) .
δ ¯ = 1 2 ( δ 1 + δ 2 δ 3 δ 4 ) , λ ¯ α = 1 4 ( λ 1 α + λ 2 α λ 3 α λ 4 α ) ,
δ = δ ¯ + 2 ( λ ¯ 1 + λ ¯ 2 ρ ) , λ = λ ¯ 1 + λ ¯ 2 λ ¯ 3 λ ¯ 4 ,
H + ( δ α + 1 2 λ α β F β ) F α = 2 ( δ λ F ) F .
( d z F ) 2 = 4 F 2 [ ( 1 F ) ( ρ F ) ( δ λ F ) 2 ] .
a = 4 ( ρ δ 2 ) , b = 4 [ 2 δ λ ( 1 + ρ ) ] , c = 4 ( 1 λ 2 ) ,
F ( z ) = { 2 a b + ( b 2 4 a c ) 1 / 2 cosh ( a 1 / 2 z ) if a > 0 4 b 4 c + b 2 z 2 if a = 0 , b > 0 , c < 0
cos ϕ = ( δ λ F ) [ ( 1 F ) ( ρ F ) ] 1 / 2 .
F max = { ( 1 + ρ ) 2 δ λ [ 4 ( δ λ ) ( δ λ ρ ) + ( 1 ρ ) 2 ] 1 / 2 2 ( 1 λ 2 ) if λ 2 1 ρ δ 2 1 + ρ 2 δ if λ = ± 1 .
δ = λ ρ .
λ 2 1 / ρ .
δ = ρ 1 / 2 sgn ( λ )
F ( z ) = { ρ ( 1 λ 2 ρ ) ( 1 λ 2 ρ ) + ( 1 ρ ) sin h 2 [ ( ρ λ 2 ρ 2 ) 1 / 2 z ] if λ 2 < 1 / ρ ρ 1 + ρ ( 1 ρ ) z 2 if λ 2 = 1 / ρ .
F ( z ) = 1 1 + exp [ 2 ( 1 λ 2 ) 1 / 2 z ]
F 1 ( 0 ) = 1 , F 2 ( 0 ) = ρ , F 3 ( 0 ) = , F 4 ( 0 ) = 0.
F 4 ( z ) = F ( z ) , F 1 ( z ) = 1 F ( z ) , F 2 ( z ) = ρ F ( z ) , F 3 ( z ) = + F ( z ) .
( d z F ) 2 = 4 F [ ( 1 F ) ( ρ F ) ( + F ) F ( δ λ F ) 2 ] ,
δ = δ ¯ + 2 ( λ ¯ 1 + λ ¯ 2 ρ + λ ¯ 3 )
cos ϕ = ( δ λ F ) F [ ( 1 F ) ( ρ F ) ( + F ) F ] 1 / 2 .
δ = λ ρ .
F ( z ) = { sinh 2 [ ( 1 + λ 2 ) 1 / 2 z ] ( 1 + λ 2 ) + sinh 2 [ ( 1 + λ 2 ) 1 / 2 z if λ 2 1 z 2 1 + z 2 if = λ 2 1 .
cos ϕ = λ [ F / ( + F ) ] 1 / 2 .
F ( z ) = tanh 2 ( 1 / 2 z ) ,
f ± = ( 1 λ 2 ρ ) ± [ ( 1 λ 2 ρ ) 2 + 4 ( 1 λ 2 ) ] 1 / 2 2 ( 1 λ 2 ) .
F ( z ) = ρ f sn 2 ( k z , m ) f + ρ cn 2 ( k z , m ) ,
k 2 = ( 1 λ 2 ) ( ρ f ) f + , m 2 = [ ρ ( f + f ) ] / [ f + ( ρ f ) ] .
k l = K ( m ) ,
l ~ log [ 4 ρ 1 / 2 ( 1 λ 2 ρ ) / ( ρ ) 1 / 2 ] ( ρ λ 2 ρ 2 ) 1 / 2
F ( z ) sinh 2 [ ( ρ λ 2 ρ 2 ) 1 / 2 z ] ( 1 λ 2 ρ ) .
l 1 ~ log [ 4 ( ρ λ 2 ρ 2 ) / ] 2 ( ρ λ 2 ρ 2 ) 1 / 2 .
l 2 ~ log [ 4 ( 1 λ 2 ρ ) / ( 1 ρ ) ] 2 ( ρ λ 2 ρ 2 ) 1 / 2 .
F ( z ) = ρ f sn 2 ( k z , m ) f + ρ cn 2 ( k z , m ) ,
k 2 = ( ρ + ) ( 1 ρ ) , m 2 = [ ρ ( 1 ρ ) ] / [ ( ρ + ) ( 1 ρ ) ] .
F ( l + z ) ρ sec h 2 [ ( ρ ρ 2 ) 1 / 2 z ] .
l ~ log [ 4 ( ρ ρ 2 ) 1 / 2 / 1 / 2 ] ( ρ ρ 2 ) 1 / 2 .
F ( z ) = ρ sn 2 ( k z , m ) ,
k 2 = , m 2 = ρ [ ( 1 ρ ) ] / .
± = [ ( λ 2 1 ) 2 ± | λ | ( 1 ρ ) 1 / 2 ] 2 .
F ( z ) = ρ β [ 1 cn ( k z , m ) ] ( α + β ) + ( α β ) cn ( k z , m ) ,
k 2 = 4 ( λ 2 1 ) α β , m 2 = [ ρ 2 ( α β ) 2 ] / 4 α β
α 2 = ( + ρ ) ( 1 ρ ) / ( λ 2 1 ) , β 2 = / ( λ 2 1 ) .
k l = 2 K ( m ) .
g δ ¯ = k ( ω 0 ) ω 2 / 2 + ( q x 2 + q y 2 k x 2 ) / 2 k ( ω 0 ) ,
g = 24 π 2 k ( ω 0 ) | χ ( 3 ) | I 1 / c .
λ ¯ 1 = λ ¯ 2 = λ ¯ 3 = λ ¯ 4 = σ / 8 ,
2 ( λ ¯ 1 + λ ¯ 2 ρ + λ ¯ 3 ) = σ ( 1 + ρ ) / 4 ,
λ = σ / 2 .
δ ¯ = σ ( 1 + ρ ) / 4 .
δ ¯ = σ ( 1 ρ ) / 4 .
d F [ ( f + F ) ( 1 F ) ( F f ) F ] 1 / 2 = ± 2 ( 1 λ 2 ) 1 / 2 d z
c y d u [ ( a u ) ( b u ) ( u c ) ( u d ) ] 1 / 2 ,
a = f + , b = ρ , c = 0 , d = f .
υ 2 = ( b d ) ( u c ) ( b c ) ( u d ) ,
2 [ ( a c ) ( b d ) ] 1 / 2 0 υ ( y ) d υ [ ( 1 υ 2 ) ( 1 m 2 υ 2 ) ] 1 / 2 ,
m 2 = ( b c ) ( a d ) ( a c ) ( b d ) .
F ( z ) = c ( b d ) d ( b c ) sn 2 ( k z , m ) ( b d ) ( b c ) sn 2 ( k z , m ) ,
k 2 = ( 1 λ 2 ) ( a c ) ( b d ) .
k 2 = 1 2 { ρ ( 1 λ 2 ρ ) + 2 + ρ [ ( 1 λ 2 ρ ) 2 + 4 ( 1 λ 2 ) ] 1 / 2 } ,
m 2 = 2 ρ [ ( 1 λ 2 ρ ) 2 + 4 ( 1 λ 2 ) ] 1 / 2 ρ ( 1 λ 2 ρ ) 2 + 2 + ρ [ ( 1 λ 2 ρ ) 2 + 4 ( 1 λ 2 ) ] 1 / 2 .
sn ( K + x ) = cn ( x ) / dn ( x ) , cn ( K + x ) = ( 1 m 2 ) 1 / 2 sn ( x ) / dn ( x ) ,
dn ( x ) = [ 1 m 2 sn 2 ( x ) ] 1 / 2 ,
F ( l + z ) = b ( a c ) a ( b c ) sn 2 ( k z , m ) ( a c ) ( b c ) sn 2 ( k z , m ) .
k 2 ρ ( 1 λ 2 ρ ) + [ ( 1 λ 2 ρ ) λ 2 ρ ( 1 ρ ) ] / ( 1 λ 2 ρ ) ,
m 2 1 ( 1 ρ ) / [ ρ ( 1 λ 2 ρ ) 2 ] .
K ( m ) log [ 4 / ( 1 m 2 ) 1 / 2 ]
sn ( x ) tanh ( x ) , cn ( x ) sech ( x ) ,
F ( l + z ) ρ ( 1 λ 2 ρ ) ( 1 λ 2 ρ ) + ( 1 ρ ) sin h 2 [ ( ρ λ 2 ρ 2 ) 1 / 2 z ] ,
d F [ ( ρ F ) ( F f ) F ] 1 / 2 = ± 2 ( 1 ρ ) 1 / 2 d z .
b y d u [ ( a u ) ( u b ) ( u c ) ] 1 / 2 ,
a = ρ , b = 0 , c = f .
υ 2 = ( a c ) ( u b ) ( a b ) ( u c ) ,
2 ( a c ) 1 / 2 0 υ ( y ) d υ [ ( 1 υ 2 ) ( 1 m 2 υ 2 ) ] 1 / 2 ,
m 2 = ( a b ) / ( a c ) .
F ( z ) = b ( a c ) c ( a b ) sn 2 ( k z , m ) ( a c ) ( a b ) sn 2 ( k z , m ) ,
k 2 = ( 1 ρ ) ( a c ) .
k 2 = ( ρ + ) ( 1 ρ ) ,
m 2 = ρ ( 1 ρ ) ( ρ + ) ( 1 ρ ) .
F ( l + z ) = a ( a b ) sn 2 ( k z , m ) .
m 2 1 / [ ρ ( 1 ρ ) ] ,
F ( l + z ) ρ sec h 2 [ ( ρ ρ 2 ) 1 / 2 z ] .
d F [ ( f F ) ( ρ F ) F ] 1 / 2 = ± 2 [ ( 1 ρ ) ] 1 / 2 d z .
c y d u [ ( a u ) ( b u ) ( u c ) ] 1 / 2 ,
a = f , b = ρ , c = 0.
υ 2 = ( u c ) / ( b c ) ,
2 ( a c ) 1 / 2 0 υ ( y ) d υ [ ( 1 υ 2 ) ( 1 m 2 υ 2 ) ] 1 / 2 ,
m 2 = ( b c ) / ( a c ) .
F ( z ) = c + ( b c ) sn 2 ( k z , m ) ,
k 2 = [ ( 1 ρ ) ] ( a c ) .
k 2 = ,
m 2 = ρ [ ( 1 ρ ) ] / .
F ( z ) tan h 2 ( 1 / 2 z ) ,
d F [ F ( ρ F ) ( f + F ) ( f F ) ] 1 / 2 = ± 2 ( λ 2 1 ) 1 / 2 d z ,
b y d u { ( a u ) ( u b ) [ ( u c ) 2 + d 2 ] } 1 / 2 ,
a = ρ , b = 0 , c = ( f + + f ) / 2 , d = ( f + f ) / 2 i .
υ = β ( a u ) α ( u b ) β ( a u ) + α ( u b ) ,
α 2 = ( a c ) 2 + d 2 , β 2 = ( b c ) 2 + d 2 ,
1 ( α β ) 1 / 2 υ ( y ) 1 d υ [ ( 1 υ 2 ) ( 1 m 2 + m 2 υ 2 ) ] 1 / 2 ,
m 2 = ( a b ) 2 ( α β ) 2 4 α β .
F ( z ) = ( α b + β a ) + ( α b β a ) cn ( k z , m ) ( α + β ) + ( α β ) cn ( k z , m ) ,
k 2 = 4 ( λ 2 1 ) α β .
k 2 = 4 [ ( + ρ ) ( 1 ρ ) ] 1 / 2 ,
m 2 = ( λ 2 1 ) ρ 2 [ ( + ρ ) 1 / 2 ( 1 ρ ) 1 / 2 1 / 2 ] 2 4 [ ( + ρ ) ( 1 ρ ) ] 1 / 2 .
E ( t , r ) = E ( ω , r ) exp ( i ω t ) d ω ,
( c 2 2 + n 2 ω 0 2 ) E ( ω 0 ) = 4 π ω 0 2 P nl ( ω 0 ) ,
n = [ 1 + 4 π χ ( 1 ) ( ω 0 ) ] 1 / 2
P nl ( ω 0 ) = 3 χ ( 3 ) ( ω 0 ; ω 0 , ω 0 , ω 0 ) | E ( ω 0 ) | 2 E ( ω 0 )
E ( ω 0 ) = α E α exp ( i k α · r ) ,
k 1 = [ k x 0 , k z ( ω 0 , k x , 0 ) ] , k 2 = [ k x , 0 , k z ( ω 0 , k x , 0 ) ] , k 3 = [ q x , q y , q z ( ω 0 , q x , q y ) ] , k 4 = [ q x , q y , q z ( ω 0 , q x , q y ) ] .
s 1 d z E 1 = i κ E 2 * E 3 E 4 exp ( i Δ k z ) + i κ ( 1 2 | E 1 | 2 + | E 2 | 2 + | E 3 | 2 + | E 4 | 2 ) E 1 , s 3 d z E 3 = i κ E 4 * E 1 E 2 exp ( i Δ k z ) + i κ ( | E 1 | 2 + | E 2 | 2 + 1 2 | E 3 | 2 + | E 4 | 2 ) E 1 ,
κ = 12 π ω 0 χ ( 3 ) / n c
Δ k = ( q x 2 + q y 2 k x 2 ) / k ( ω 0 )
I α = n c | E α | 2 / 2 π .
g = 24 π 2 ω 0 | χ ( 3 ) | I 1 / n 2 c 2 .

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