Abstract

Non-degenerate two-wave mixing in inverted laser materials is investigated both theoretically and experimentally. A complete model of two-wave mixing in gain media in the presence of an athermal refractive index grating is presented and application to beam-combining is investigated in pumped Nd3+ and Yb3+ laser materials. Experimental results are presented for a Nd:yttrium aluminum garnet crystal fiber.

© 2010 Optical Society of America

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References

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  1. M. J. Damzen, A. Boyle, and A. Minassian, “Adaptive gain interferometry: a new mechanism for optical metrology with speckle beams,” Opt. Lett. 30, 2230–2232 (2005).
    [CrossRef] [PubMed]
  2. I. McMichael, R. Saxena, T. Y. Chang, Q. Shu, S. Rand, J. Chen, and H. Tuller, “High-gain nondegenerate two-wave mixing in Cr:YAlO3,” Opt. Lett. 19, 1511–1513 (1994).
    [CrossRef] [PubMed]
  3. A. Brignon, G. Feugnet, J.-P. Huignard, and J.-P. Pocholle, “Multipass degenerate four-wave mixing in a diode-pumped Nd:YVO4 saturable amplifier,” J. Opt. Soc. Am. B 12, 1316–1325 (1995).
    [CrossRef]
  4. P. J. Soan, A. D. Case, M. J. Damzen, and M. H. R. Hutchinson, “High-reflectivity four-wave mixing by saturable gain in Rhodamine 6G dye,” Opt. Lett. 17, 781–783 (1992).
    [CrossRef] [PubMed]
  5. P. Sillard, A. Brignon, and J. P. Huignard, “Gain-grating analysis of a self-starting self-pumped phase-conjugate Nd:YAG loop resonator,” IEEE J. Quantum Electron. 34, 465–472 (1998).
    [CrossRef]
  6. P. Sillard, A. Brignon, J.-P. Huignard, and J.-P. Pocholle, “Self-pumped phase-conjugate diode-pumped Nd:YAG loop resonator,” Opt. Lett. 23, 1093–1095 (1998).
    [CrossRef]
  7. M. J. Damzen, R. P. M. Green, and K. S. Syed, “Self-adaptive solid-state laser oscillator formed by dynamic gain-grating holograms,” Opt. Lett. 20, 1704–1706 (1995).
    [CrossRef] [PubMed]
  8. A. Brignon and J. P. Huignard, “Two-wave mixing in Nd:YAG by gain saturation,” Opt. Lett. 18, 1639–1641 (1993).
    [CrossRef] [PubMed]
  9. O. L. Antipov, S. I. Belyaev, A. S. Kuzhelev, and D. V. Chausov, “Resonant two-wave mixing of optical beams by refractive-index and gain gratings in inverted Nd:YAG,” J. Opt. Soc. Am. B 15, 2276–2282 (1998).
    [CrossRef]
  10. R. Soulard, A. Zinoviev, J. L. Doualan, E. Ivakin, O. Antipov, and R. Moncorgé, “Detailed characterization of pump-induced refractive index changes observed in Nd:YVO4, Nd:GdVO4 and Nd:KGW,” Opt. Express 18, 1553–1568 (2010).
    [CrossRef] [PubMed]
  11. O. L. Antipov, A. S. Kuzhelev, D. V. Chausov, and A. P. Zinov’ev, “Dynamics of refractive-index changes in a Nd:YAG laser crystal under excitation of Nd3+ ions,” J. Opt. Soc. Am. B 16, 1072–1079 (1999).
    [CrossRef]
  12. O. L. Antipov, O. N. Eremeykin, A. P. Savikin, V. A. Vorob’ev, D. V. Bredikhin, and M. S. Kuznetsov, “Electronic changes of refractive index in intensively pumped Nd:YAG laser crystals,” IEEE J. Quantum Electron. 39, 910–918 (2003).
    [CrossRef]
  13. O. L. Antipov, D. V. Bredikhin, O. N. Eremeykin, A. P. Savikin, E. V. Ivakin, and A. V. Sukhadolau, “Electronic mechanism for refractive-index changes in intensively pumped Yb:YAG laser crystals,” Opt. Lett. 31, 763–765 (2006).
    [CrossRef] [PubMed]
  14. J. Margerie, R. Moncorgé, and P. Nagtegaele, “Spectroscopic investigation of the variations in refractive index of a Nd:YAG laser crystal: experiments and crystal-field calculations,” Phys. Rev. B 74, 235108 (2006).
    [CrossRef]
  15. R. Moncorgé, O. N. Ereymekin, J. L. Doualan, and O. L. Antipov, “Origin of athermal refractive index changes observed in Yb3+ doped YAG and KGW,” Opt. Commun. 281, 2526–2530 (2008).
    [CrossRef]
  16. C. H. Henry, “Theory of the linewidth of semiconductor lasers,” IEEE J. Quantum Electron. 18, 259–264 (1982).
    [CrossRef]
  17. P. Yeh, “Two-wave mixing in non linear media,” IEEE J. Quantum Electron. 25, 484–519 (1989).
    [CrossRef]
  18. M. Chi, J. P. Huignard, and P. M. Petersen, “A general theory of two-wave mixing in nonlinear media,” J. Opt. Soc. Am. B 26, 1578–1584 (2009).
    [CrossRef]
  19. E. V. Ivakin, A. V. Sukhadolau, O. L. Antipov, and N. V. Kuleshov, “Transient grating measurements of refractive-index changes in intensively pumped Yb-doped laser crystals,” Appl. Phys. B 86, 315–318 (2007).
    [CrossRef]
  20. A. A. Fotiadi, O. L. Antipov, and P. Megret, “Dynamics of pump-induced refractive index changes in single-mode Yb-doped optical fibers,” Opt. Express 16, 12658–12663 (2008).
    [PubMed]
  21. http://www.fibercryst.com/.
  22. J. Didierjean, M. Castaing, F. Balembois, P. Georges, D. Perrodin, J. M. Fourmigué, K. Lebbou, A. Brenier, and O. Tillement, “High-power laser with Nd:YAG single-crystal fiber grown by the micro-pulling-down technique,” Opt. Lett. 31, 3468–3470 (2006).
    [CrossRef] [PubMed]

2010 (1)

2009 (1)

2008 (2)

A. A. Fotiadi, O. L. Antipov, and P. Megret, “Dynamics of pump-induced refractive index changes in single-mode Yb-doped optical fibers,” Opt. Express 16, 12658–12663 (2008).
[PubMed]

R. Moncorgé, O. N. Ereymekin, J. L. Doualan, and O. L. Antipov, “Origin of athermal refractive index changes observed in Yb3+ doped YAG and KGW,” Opt. Commun. 281, 2526–2530 (2008).
[CrossRef]

2007 (1)

E. V. Ivakin, A. V. Sukhadolau, O. L. Antipov, and N. V. Kuleshov, “Transient grating measurements of refractive-index changes in intensively pumped Yb-doped laser crystals,” Appl. Phys. B 86, 315–318 (2007).
[CrossRef]

2006 (3)

2005 (1)

2003 (1)

O. L. Antipov, O. N. Eremeykin, A. P. Savikin, V. A. Vorob’ev, D. V. Bredikhin, and M. S. Kuznetsov, “Electronic changes of refractive index in intensively pumped Nd:YAG laser crystals,” IEEE J. Quantum Electron. 39, 910–918 (2003).
[CrossRef]

1999 (1)

1998 (3)

1995 (2)

1994 (1)

1993 (1)

1992 (1)

1989 (1)

P. Yeh, “Two-wave mixing in non linear media,” IEEE J. Quantum Electron. 25, 484–519 (1989).
[CrossRef]

1982 (1)

C. H. Henry, “Theory of the linewidth of semiconductor lasers,” IEEE J. Quantum Electron. 18, 259–264 (1982).
[CrossRef]

Antipov, O.

Antipov, O. L.

A. A. Fotiadi, O. L. Antipov, and P. Megret, “Dynamics of pump-induced refractive index changes in single-mode Yb-doped optical fibers,” Opt. Express 16, 12658–12663 (2008).
[PubMed]

R. Moncorgé, O. N. Ereymekin, J. L. Doualan, and O. L. Antipov, “Origin of athermal refractive index changes observed in Yb3+ doped YAG and KGW,” Opt. Commun. 281, 2526–2530 (2008).
[CrossRef]

E. V. Ivakin, A. V. Sukhadolau, O. L. Antipov, and N. V. Kuleshov, “Transient grating measurements of refractive-index changes in intensively pumped Yb-doped laser crystals,” Appl. Phys. B 86, 315–318 (2007).
[CrossRef]

O. L. Antipov, D. V. Bredikhin, O. N. Eremeykin, A. P. Savikin, E. V. Ivakin, and A. V. Sukhadolau, “Electronic mechanism for refractive-index changes in intensively pumped Yb:YAG laser crystals,” Opt. Lett. 31, 763–765 (2006).
[CrossRef] [PubMed]

O. L. Antipov, O. N. Eremeykin, A. P. Savikin, V. A. Vorob’ev, D. V. Bredikhin, and M. S. Kuznetsov, “Electronic changes of refractive index in intensively pumped Nd:YAG laser crystals,” IEEE J. Quantum Electron. 39, 910–918 (2003).
[CrossRef]

O. L. Antipov, A. S. Kuzhelev, D. V. Chausov, and A. P. Zinov’ev, “Dynamics of refractive-index changes in a Nd:YAG laser crystal under excitation of Nd3+ ions,” J. Opt. Soc. Am. B 16, 1072–1079 (1999).
[CrossRef]

O. L. Antipov, S. I. Belyaev, A. S. Kuzhelev, and D. V. Chausov, “Resonant two-wave mixing of optical beams by refractive-index and gain gratings in inverted Nd:YAG,” J. Opt. Soc. Am. B 15, 2276–2282 (1998).
[CrossRef]

Balembois, F.

Belyaev, S. I.

Boyle, A.

Bredikhin, D. V.

O. L. Antipov, D. V. Bredikhin, O. N. Eremeykin, A. P. Savikin, E. V. Ivakin, and A. V. Sukhadolau, “Electronic mechanism for refractive-index changes in intensively pumped Yb:YAG laser crystals,” Opt. Lett. 31, 763–765 (2006).
[CrossRef] [PubMed]

O. L. Antipov, O. N. Eremeykin, A. P. Savikin, V. A. Vorob’ev, D. V. Bredikhin, and M. S. Kuznetsov, “Electronic changes of refractive index in intensively pumped Nd:YAG laser crystals,” IEEE J. Quantum Electron. 39, 910–918 (2003).
[CrossRef]

Brenier, A.

Brignon, A.

Case, A. D.

Castaing, M.

Chang, T. Y.

Chausov, D. V.

Chen, J.

Chi, M.

Damzen, M. J.

Didierjean, J.

Doualan, J. L.

R. Soulard, A. Zinoviev, J. L. Doualan, E. Ivakin, O. Antipov, and R. Moncorgé, “Detailed characterization of pump-induced refractive index changes observed in Nd:YVO4, Nd:GdVO4 and Nd:KGW,” Opt. Express 18, 1553–1568 (2010).
[CrossRef] [PubMed]

R. Moncorgé, O. N. Ereymekin, J. L. Doualan, and O. L. Antipov, “Origin of athermal refractive index changes observed in Yb3+ doped YAG and KGW,” Opt. Commun. 281, 2526–2530 (2008).
[CrossRef]

Eremeykin, O. N.

O. L. Antipov, D. V. Bredikhin, O. N. Eremeykin, A. P. Savikin, E. V. Ivakin, and A. V. Sukhadolau, “Electronic mechanism for refractive-index changes in intensively pumped Yb:YAG laser crystals,” Opt. Lett. 31, 763–765 (2006).
[CrossRef] [PubMed]

O. L. Antipov, O. N. Eremeykin, A. P. Savikin, V. A. Vorob’ev, D. V. Bredikhin, and M. S. Kuznetsov, “Electronic changes of refractive index in intensively pumped Nd:YAG laser crystals,” IEEE J. Quantum Electron. 39, 910–918 (2003).
[CrossRef]

Ereymekin, O. N.

R. Moncorgé, O. N. Ereymekin, J. L. Doualan, and O. L. Antipov, “Origin of athermal refractive index changes observed in Yb3+ doped YAG and KGW,” Opt. Commun. 281, 2526–2530 (2008).
[CrossRef]

Feugnet, G.

Fotiadi, A. A.

Fourmigué, J. M.

Georges, P.

Green, R. P. M.

Henry, C. H.

C. H. Henry, “Theory of the linewidth of semiconductor lasers,” IEEE J. Quantum Electron. 18, 259–264 (1982).
[CrossRef]

Huignard, J. P.

Huignard, J. -P.

Hutchinson, M. H. R.

Ivakin, E.

Ivakin, E. V.

E. V. Ivakin, A. V. Sukhadolau, O. L. Antipov, and N. V. Kuleshov, “Transient grating measurements of refractive-index changes in intensively pumped Yb-doped laser crystals,” Appl. Phys. B 86, 315–318 (2007).
[CrossRef]

O. L. Antipov, D. V. Bredikhin, O. N. Eremeykin, A. P. Savikin, E. V. Ivakin, and A. V. Sukhadolau, “Electronic mechanism for refractive-index changes in intensively pumped Yb:YAG laser crystals,” Opt. Lett. 31, 763–765 (2006).
[CrossRef] [PubMed]

Kuleshov, N. V.

E. V. Ivakin, A. V. Sukhadolau, O. L. Antipov, and N. V. Kuleshov, “Transient grating measurements of refractive-index changes in intensively pumped Yb-doped laser crystals,” Appl. Phys. B 86, 315–318 (2007).
[CrossRef]

Kuzhelev, A. S.

Kuznetsov, M. S.

O. L. Antipov, O. N. Eremeykin, A. P. Savikin, V. A. Vorob’ev, D. V. Bredikhin, and M. S. Kuznetsov, “Electronic changes of refractive index in intensively pumped Nd:YAG laser crystals,” IEEE J. Quantum Electron. 39, 910–918 (2003).
[CrossRef]

Lebbou, K.

Margerie, J.

J. Margerie, R. Moncorgé, and P. Nagtegaele, “Spectroscopic investigation of the variations in refractive index of a Nd:YAG laser crystal: experiments and crystal-field calculations,” Phys. Rev. B 74, 235108 (2006).
[CrossRef]

McMichael, I.

Megret, P.

Minassian, A.

Moncorgé, R.

R. Soulard, A. Zinoviev, J. L. Doualan, E. Ivakin, O. Antipov, and R. Moncorgé, “Detailed characterization of pump-induced refractive index changes observed in Nd:YVO4, Nd:GdVO4 and Nd:KGW,” Opt. Express 18, 1553–1568 (2010).
[CrossRef] [PubMed]

R. Moncorgé, O. N. Ereymekin, J. L. Doualan, and O. L. Antipov, “Origin of athermal refractive index changes observed in Yb3+ doped YAG and KGW,” Opt. Commun. 281, 2526–2530 (2008).
[CrossRef]

J. Margerie, R. Moncorgé, and P. Nagtegaele, “Spectroscopic investigation of the variations in refractive index of a Nd:YAG laser crystal: experiments and crystal-field calculations,” Phys. Rev. B 74, 235108 (2006).
[CrossRef]

Nagtegaele, P.

J. Margerie, R. Moncorgé, and P. Nagtegaele, “Spectroscopic investigation of the variations in refractive index of a Nd:YAG laser crystal: experiments and crystal-field calculations,” Phys. Rev. B 74, 235108 (2006).
[CrossRef]

Perrodin, D.

Petersen, P. M.

Pocholle, J. -P.

Rand, S.

Savikin, A. P.

O. L. Antipov, D. V. Bredikhin, O. N. Eremeykin, A. P. Savikin, E. V. Ivakin, and A. V. Sukhadolau, “Electronic mechanism for refractive-index changes in intensively pumped Yb:YAG laser crystals,” Opt. Lett. 31, 763–765 (2006).
[CrossRef] [PubMed]

O. L. Antipov, O. N. Eremeykin, A. P. Savikin, V. A. Vorob’ev, D. V. Bredikhin, and M. S. Kuznetsov, “Electronic changes of refractive index in intensively pumped Nd:YAG laser crystals,” IEEE J. Quantum Electron. 39, 910–918 (2003).
[CrossRef]

Saxena, R.

Shu, Q.

Sillard, P.

P. Sillard, A. Brignon, and J. P. Huignard, “Gain-grating analysis of a self-starting self-pumped phase-conjugate Nd:YAG loop resonator,” IEEE J. Quantum Electron. 34, 465–472 (1998).
[CrossRef]

P. Sillard, A. Brignon, J.-P. Huignard, and J.-P. Pocholle, “Self-pumped phase-conjugate diode-pumped Nd:YAG loop resonator,” Opt. Lett. 23, 1093–1095 (1998).
[CrossRef]

Soan, P. J.

Soulard, R.

Sukhadolau, A. V.

E. V. Ivakin, A. V. Sukhadolau, O. L. Antipov, and N. V. Kuleshov, “Transient grating measurements of refractive-index changes in intensively pumped Yb-doped laser crystals,” Appl. Phys. B 86, 315–318 (2007).
[CrossRef]

O. L. Antipov, D. V. Bredikhin, O. N. Eremeykin, A. P. Savikin, E. V. Ivakin, and A. V. Sukhadolau, “Electronic mechanism for refractive-index changes in intensively pumped Yb:YAG laser crystals,” Opt. Lett. 31, 763–765 (2006).
[CrossRef] [PubMed]

Syed, K. S.

Tillement, O.

Tuller, H.

Vorob’ev, V. A.

O. L. Antipov, O. N. Eremeykin, A. P. Savikin, V. A. Vorob’ev, D. V. Bredikhin, and M. S. Kuznetsov, “Electronic changes of refractive index in intensively pumped Nd:YAG laser crystals,” IEEE J. Quantum Electron. 39, 910–918 (2003).
[CrossRef]

Yeh, P.

P. Yeh, “Two-wave mixing in non linear media,” IEEE J. Quantum Electron. 25, 484–519 (1989).
[CrossRef]

Zinov’ev, A. P.

Zinoviev, A.

Appl. Phys. B (1)

E. V. Ivakin, A. V. Sukhadolau, O. L. Antipov, and N. V. Kuleshov, “Transient grating measurements of refractive-index changes in intensively pumped Yb-doped laser crystals,” Appl. Phys. B 86, 315–318 (2007).
[CrossRef]

IEEE J. Quantum Electron. (4)

P. Sillard, A. Brignon, and J. P. Huignard, “Gain-grating analysis of a self-starting self-pumped phase-conjugate Nd:YAG loop resonator,” IEEE J. Quantum Electron. 34, 465–472 (1998).
[CrossRef]

O. L. Antipov, O. N. Eremeykin, A. P. Savikin, V. A. Vorob’ev, D. V. Bredikhin, and M. S. Kuznetsov, “Electronic changes of refractive index in intensively pumped Nd:YAG laser crystals,” IEEE J. Quantum Electron. 39, 910–918 (2003).
[CrossRef]

C. H. Henry, “Theory of the linewidth of semiconductor lasers,” IEEE J. Quantum Electron. 18, 259–264 (1982).
[CrossRef]

P. Yeh, “Two-wave mixing in non linear media,” IEEE J. Quantum Electron. 25, 484–519 (1989).
[CrossRef]

J. Opt. Soc. Am. B (4)

Opt. Commun. (1)

R. Moncorgé, O. N. Ereymekin, J. L. Doualan, and O. L. Antipov, “Origin of athermal refractive index changes observed in Yb3+ doped YAG and KGW,” Opt. Commun. 281, 2526–2530 (2008).
[CrossRef]

Opt. Express (2)

Opt. Lett. (8)

M. J. Damzen, R. P. M. Green, and K. S. Syed, “Self-adaptive solid-state laser oscillator formed by dynamic gain-grating holograms,” Opt. Lett. 20, 1704–1706 (1995).
[CrossRef] [PubMed]

P. Sillard, A. Brignon, J.-P. Huignard, and J.-P. Pocholle, “Self-pumped phase-conjugate diode-pumped Nd:YAG loop resonator,” Opt. Lett. 23, 1093–1095 (1998).
[CrossRef]

M. J. Damzen, A. Boyle, and A. Minassian, “Adaptive gain interferometry: a new mechanism for optical metrology with speckle beams,” Opt. Lett. 30, 2230–2232 (2005).
[CrossRef] [PubMed]

O. L. Antipov, D. V. Bredikhin, O. N. Eremeykin, A. P. Savikin, E. V. Ivakin, and A. V. Sukhadolau, “Electronic mechanism for refractive-index changes in intensively pumped Yb:YAG laser crystals,” Opt. Lett. 31, 763–765 (2006).
[CrossRef] [PubMed]

J. Didierjean, M. Castaing, F. Balembois, P. Georges, D. Perrodin, J. M. Fourmigué, K. Lebbou, A. Brenier, and O. Tillement, “High-power laser with Nd:YAG single-crystal fiber grown by the micro-pulling-down technique,” Opt. Lett. 31, 3468–3470 (2006).
[CrossRef] [PubMed]

P. J. Soan, A. D. Case, M. J. Damzen, and M. H. R. Hutchinson, “High-reflectivity four-wave mixing by saturable gain in Rhodamine 6G dye,” Opt. Lett. 17, 781–783 (1992).
[CrossRef] [PubMed]

A. Brignon and J. P. Huignard, “Two-wave mixing in Nd:YAG by gain saturation,” Opt. Lett. 18, 1639–1641 (1993).
[CrossRef] [PubMed]

I. McMichael, R. Saxena, T. Y. Chang, Q. Shu, S. Rand, J. Chen, and H. Tuller, “High-gain nondegenerate two-wave mixing in Cr:YAlO3,” Opt. Lett. 19, 1511–1513 (1994).
[CrossRef] [PubMed]

Phys. Rev. B (1)

J. Margerie, R. Moncorgé, and P. Nagtegaele, “Spectroscopic investigation of the variations in refractive index of a Nd:YAG laser crystal: experiments and crystal-field calculations,” Phys. Rev. B 74, 235108 (2006).
[CrossRef]

Other (1)

http://www.fibercryst.com/.

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Figures (9)

Fig. 1
Fig. 1

Theoretical variations of g 0 L and Δ n L near the laser resonance as a function of δ for g 0 L ( δ = 0 ) = 1.5 and β NR = 3 at the resonance wavelength.

Fig. 2
Fig. 2

TWM process showing the pump beam A 1 and the probe beam A 2 interacting in the laser medium.

Fig. 3
Fig. 3

TWM gain as a function of Ω τ for (a) I 2 / I sat ( λ 0 ) = 0.01 and (b) I 2 / I sat ( λ 0 ) = 0.4 and for different values of the β factor. The small intensity gain-length product g 0 L is 3.

Fig. 4
Fig. 4

TWM gain (a) as a function of I 2 / I sat ( λ 0 ) for g 0 L = 3 and for different values of the β factor and (b) as a function of β for I 2 / I sat ( λ 0 ) = 0.4 and for different values of g 0 L .

Fig. 5
Fig. 5

TWM gain as a function of δ for I 2 / I sat ( λ 0 = λ c ) = 2 and for different values of β NR 0 and g 0 ( λ 0 = λ c ) L .

Fig. 6
Fig. 6

Theoretical calculations of the amount of transferred energy E TWM as a function of the small intensity gain-length product g 0 L at resonance and of the non-resonant variation of the optical path Δ n NR L .

Fig. 7
Fig. 7

Schematic of the experimental setup for the TWM in a Nd 3 + : YAG crystal fiber.

Fig. 8
Fig. 8

Measured TWM gain experienced by the signal beam versus the frequency shift Ω τ for (a) δ = 0.7 , (b) δ = 0.25 , and (c) δ = 0.7 . Comparison of experimental and theoretical results is given for a parameter β NR 0 of 0.24.

Fig. 9
Fig. 9

Variation of g 0 L ( δ ) and Δ n L versus δ for g 0 L ( δ = 0 ) = 3 and β NR 0 = 0.24 .

Tables (1)

Tables Icon

Table 1 Experimental Data of Polarizability Variation, Emission Cross Section at Maximum Gain, and β NR 0 Parameter Used in the Theoretical Calculations

Equations (37)

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

Δ χ = 2 n 0 Δ n ,
Δ χ = 2 n 0 n = n 0 g k 0 ,
g = σ e ( λ 0 ) N ex
Δ n ( λ 0 ) = N ex 2 π 2 P 0 Δ σ ( λ ) ( λ λ 0 ) 2 1 d λ ,
Δ σ = σ e + σ gsa σ esa ,
β ( λ 0 ) = 2 π σ e ( λ 0 ) λ 0 P 0 Δ σ ( λ ) ( λ λ 0 ) 2 1 d λ .
β ( λ 0 ) = β R ( λ 0 ) + β NR ( λ 0 ) ,
σ e ( λ 0 ) = σ e max 1 + [ δ ( λ 0 ) ] 2 ,
β R ( λ 0 ) δ ( λ 0 ) 2 1 / λ 0 1 / λ c Δ λ / λ c 2 ,
β NR ( λ 0 ) = 8 π 2 F l 2 Δ α j ( λ 0 ) n 0 λ 0 σ e ( λ 0 ) .
α j ( ν ) = λ 0 n 0 8 π 2 F l 2 i σ j i ν Δ ν j i ( ν j i 2 ν 2 ) ( ν j i 2 ν 2 ) 2 + ( ν Δ ν j i ) 2 ,
Δ α j ( ν ) = λ 0 n 0 8 π 2 F l 2 ( i σ j i ν Δ ν j i ν j i 2 ν 2 i σ j = 0 , i ν Δ ν j = 0 , i ν j = 0 , i 2 ν 2 ) .
Δ α j ( λ 0 ) Δ α j .
β NR ( λ 0 ) = β NR 0 ( 1 + [ δ ( λ 0 ) ] 2 ) ,
β NR 0 = 8 π 2 F l 2 Δ α j n 0 λ 0 σ e max .
Δ E n 0 2 c 2 2 E t 2 = 1 ε 0 c 2 2 P NL t 2 ,
E = A 1 e i ( ω 1 t k 1 r 1 ) + A 2 e i ( ω 2 t k 2 r 2 ) ,
I ( y , t ) = I S + I M   cos ( K y Ω t ) ,
I S = | A 1 | 2 I sat ( λ 0 ) + | A 2 | 2 I sat ( λ 0 ) ,
I M = 2 | A 1 A 2 | I sat ( λ 0 ) ,
I sat ( λ 0 ) = h c λ 0 σ e ( λ 0 ) τ ,
P NL = ε 0 Δ χ E ,
Δ χ = Δ χ + i Δ χ = i Δ χ ( 1 i β ) .
d N ex d t = W p ( N tot N ex ) N ex τ ( I S + I M   cos ( K y Ω t ) ) N ex τ ,
N ex ( y , t ) = N 0 + N gr e i ( Ω t K y ) + N gr e i ( Ω t K y ) + higher   orders ,
N gr = | N gr | e i Φ .
N gr = N gr ( Ω = 0 ) 1 + i Ω τ ,
τ = τ 1 + I S .
cos ( θ 1 ) d A 1 d z = γ 0 ( λ 0 ) A 1 + γ 1 ( λ 0 ) A 2 [ cos ( Φ ) β ( λ 0 ) sin ( Φ ) ] ,
cos ( θ 2 ) d A 2 d z = γ 0 ( λ 0 ) A 1 + γ 1 ( λ 0 ) A 2 [ cos ( Φ ) + β ( λ 0 ) sin ( Φ ) ] ,
Φ = π arctan ( Ω τ ) .
γ 0 ( λ 0 ) = σ e ( λ 0 ) 2 N ex ( I = 0 ) ( 1 + I s ) 2 ( I M 1 + ( Ω τ ) 2 ) 2 ,
γ 1 ( λ 0 ) = γ 0 ( λ 0 ) ( 1 + I s ( 1 + I s ) 2 I M 2 I M ) 1 1 + ( Ω τ ) 2 .
G TWM = I out , I out , .
Δ n NR L = λ 0 4 π β NR 0 g 0 ( λ 0 = λ c ) L .
Δ n R ( δ ) L = λ 0 4 π δ 1 + δ 2 g 0 ( λ 0 = λ c ) L .
E TWM = I out , I out , I out , = G TWM 1.

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