Abstract

We present a new two-beam technique, the X scan, for measuring both absorptive and refractive components of optical nonlinearities in bulk semiconductors. The sample is excited by a high-intensity, short (picosecond) pump pulse and is probed with weak degenerate or nondegenerate pulses whose relative arrival time and transverse alignment with the pump are varied. X scans at various delays enable separation of free-carrier mediated nonlinearities and instantaneous χ(3) nonlinearities. Varying the relative pump–probe polarizations yields components of the χ(3) tensor. We report new values for degenerate (532 nm) and nondegenerate (532; 683 nm) cross-phase-modulation and two-photon-absorption coefficients for two polarization states and free-carrier absorption and refractive cross sections at 532 and 683 nm.

© 1997 Optical Society of America

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References

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  1. M. Sheik-Bahae, J. Wang, R. DeSalvo, D. J. Hagan, and E. W. Van Stryland, “Measurement of nondegenerate nonlinearities using a two-color Z-scan,” Opt. Lett. 17, 260 (1992).
    [CrossRef]
  2. M. Sheik-Bahae, A. A. Said, T. H. Wei, D. J. Hagan, and E. W. Van Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26, 760 (1990).
    [CrossRef]
  3. J. Wang, M. Sheik-Bahae, A. A. Said, D. J. Hagan, and E. W. Van Stryland, “Time-resolved Z-scan measurements of optical nonlinearities,” J. Opt. Soc. Am. B 11, 1009 (1994).
    [CrossRef]
  4. D. Fournier, A. C. Boccara, W. Jackson, and N. M. Amer, “Sensitive photothermal deflection technique for measuring absorption in optically thin media,” Opt. Lett. 5, 377 (1980).
    [CrossRef] [PubMed]
  5. W. B. Jackson, N. M. Amer, A. C. Boccara, and D. Fournier, “Photothermal deflection spectroscopy and detection,” Appl. Opt. 20, 1333 (1981).
    [CrossRef] [PubMed]
  6. M. Bertolotti, A. Ferrari, C. Sibilia, G. Suber, D. Apostol, and P. Jani, “Photothermal deflection technique for measuring thermal nonlinearities in semiconductor glasses,” Appl. Opt. 27, 1811 (1988).
    [CrossRef] [PubMed]
  7. K. H. Lee, W. R. Cho, J. H. Park, J. S. Kim, S. H. Park, and U. Kim, “Measurement of free-carrier nonlinearities in ZnSe based on the Z-scan technique with a nanosecond laser,” Opt. Lett. 19, 1116 (1994).
    [CrossRef] [PubMed]
  8. J. R. Milward, J. A. K. Kar, C. R. Pidgeon, and B. S. Wherrett, “Photogenerated carrier recombination time in bulk ZnSe,” J. Appl. Phys. 69, 2708 (1991).
    [CrossRef]
  9. V. L. Vinetskii, N. V. Kukhtarev, and M. S. Soskin, “Transformation of intensities and phases of light beams by a transient ‘undisplaced’ holographic grating,” Sov. J. Quantum Electron. 7, 230 (1977).
    [CrossRef]
  10. V. L. Vinetskii, N. V. Kukhtarev, E. N. Sal’kova, and L. G. Sukhoverkhova, “Mechanisms of dynamic conversion of coherent optical beams in CdS,” Sov. J. Quantum Electron. 10, 684 (1980).
    [CrossRef]
  11. G. C. Valley, and A. L. Smirl, “Theory of transient energy transfer in gallium arsenide,” IEEE J. Quantum Electron. 24, 304 (1988).
    [CrossRef]
  12. A. L. Smirl, J. Dubard, A. G. Cui, T. F. Boggess, and G. C. Valley, “Polarization-rotation switch using picosecond pulses in GaAs,” Opt. Lett. 14, 242 (1989).
    [CrossRef] [PubMed]
  13. H. E. Ruda, ed., Widegap II–VI Compounds for Opto-electronic Applications (Chapman and Hall, New York, 1992).
  14. K. Seeger, Semiconductor Physics, An Introduction (Springer-Verlag, New York, 1991), p. 356ff.
  15. D. H. Auston, S. McAfee, C. V. Shank, E. P. Ippen, and O. Teschke, “Picosecond Spectroscopy of Semiconductors,” Solid State Electron. 21, 147 (1978).
    [CrossRef]
  16. R. K. Jain and M. B. Klein, “Degenerate four-wave mixing in semiconductors,” in Optical Phase Conjugation, R. A. Fisher, ed. (Academic, New York, 1983), p. 335.
  17. D. C. Hutchings and E. W. Van Stryland, “Nondegenerate two-photon absorption in zinc blende semiconductors,” J. Opt. Soc. Am. B 9, 2065 (1992).
    [CrossRef]
  18. A. J. Stentz, M. Kauranen, J. J. Maki, G. P. Agrawal, and R. W. Boyd, “Induced focusing and spatial wave breaking from cross-phase modulation in a self-defocusing medium,” Opt. Lett. 17, 19 (1992).
    [CrossRef] [PubMed]
  19. R. DeSalvo, M. Sheik-Bahae, A. A. Said, D. J. Hagan, and E. W. Van Stryland, “Z-scan measurements of the anisotropy of nonlinear refraction and absorption in crystals,” Opt. Lett. 18, 194 (1993).
    [CrossRef] [PubMed]
  20. Y. R. Shen, Principles of Nonlinear Optics (Wiley-Interscience, New York, 1984), p. 51.
  21. R. W. Boyd, Nonlinear Optics (Academic, San Diego, 1992).
  22. J. Feinberg, D. Heiman, A. R. Tanguay, Jr., and R. W. Hellwarth, “Photorefractive effects and light-induced charge migration in barium titanate,” J. Appl. Phys. 51, 1297 (1980).
    [CrossRef]

1994 (2)

1993 (1)

1992 (3)

1991 (1)

J. R. Milward, J. A. K. Kar, C. R. Pidgeon, and B. S. Wherrett, “Photogenerated carrier recombination time in bulk ZnSe,” J. Appl. Phys. 69, 2708 (1991).
[CrossRef]

1990 (1)

M. Sheik-Bahae, A. A. Said, T. H. Wei, D. J. Hagan, and E. W. Van Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26, 760 (1990).
[CrossRef]

1989 (1)

1988 (2)

1981 (1)

1980 (3)

D. Fournier, A. C. Boccara, W. Jackson, and N. M. Amer, “Sensitive photothermal deflection technique for measuring absorption in optically thin media,” Opt. Lett. 5, 377 (1980).
[CrossRef] [PubMed]

V. L. Vinetskii, N. V. Kukhtarev, E. N. Sal’kova, and L. G. Sukhoverkhova, “Mechanisms of dynamic conversion of coherent optical beams in CdS,” Sov. J. Quantum Electron. 10, 684 (1980).
[CrossRef]

J. Feinberg, D. Heiman, A. R. Tanguay, Jr., and R. W. Hellwarth, “Photorefractive effects and light-induced charge migration in barium titanate,” J. Appl. Phys. 51, 1297 (1980).
[CrossRef]

1978 (1)

D. H. Auston, S. McAfee, C. V. Shank, E. P. Ippen, and O. Teschke, “Picosecond Spectroscopy of Semiconductors,” Solid State Electron. 21, 147 (1978).
[CrossRef]

1977 (1)

V. L. Vinetskii, N. V. Kukhtarev, and M. S. Soskin, “Transformation of intensities and phases of light beams by a transient ‘undisplaced’ holographic grating,” Sov. J. Quantum Electron. 7, 230 (1977).
[CrossRef]

Agrawal, G. P.

Amer, N. M.

Apostol, D.

Auston, D. H.

D. H. Auston, S. McAfee, C. V. Shank, E. P. Ippen, and O. Teschke, “Picosecond Spectroscopy of Semiconductors,” Solid State Electron. 21, 147 (1978).
[CrossRef]

Bertolotti, M.

Boccara, A. C.

Boggess, T. F.

Boyd, R. W.

Cho, W. R.

Cui, A. G.

DeSalvo, R.

R. DeSalvo, M. Sheik-Bahae, A. A. Said, D. J. Hagan, and E. W. Van Stryland, “Z-scan measurements of the anisotropy of nonlinear refraction and absorption in crystals,” Opt. Lett. 18, 194 (1993).
[CrossRef] [PubMed]

M. Sheik-Bahae, J. Wang, R. DeSalvo, D. J. Hagan, and E. W. Van Stryland, “Measurement of nondegenerate nonlinearities using a two-color Z-scan,” Opt. Lett. 17, 260 (1992).
[CrossRef]

Dubard, J.

Feinberg, J.

J. Feinberg, D. Heiman, A. R. Tanguay, Jr., and R. W. Hellwarth, “Photorefractive effects and light-induced charge migration in barium titanate,” J. Appl. Phys. 51, 1297 (1980).
[CrossRef]

Ferrari, A.

Fournier, D.

Hagan, D. J.

J. Wang, M. Sheik-Bahae, A. A. Said, D. J. Hagan, and E. W. Van Stryland, “Time-resolved Z-scan measurements of optical nonlinearities,” J. Opt. Soc. Am. B 11, 1009 (1994).
[CrossRef]

R. DeSalvo, M. Sheik-Bahae, A. A. Said, D. J. Hagan, and E. W. Van Stryland, “Z-scan measurements of the anisotropy of nonlinear refraction and absorption in crystals,” Opt. Lett. 18, 194 (1993).
[CrossRef] [PubMed]

M. Sheik-Bahae, J. Wang, R. DeSalvo, D. J. Hagan, and E. W. Van Stryland, “Measurement of nondegenerate nonlinearities using a two-color Z-scan,” Opt. Lett. 17, 260 (1992).
[CrossRef]

M. Sheik-Bahae, A. A. Said, T. H. Wei, D. J. Hagan, and E. W. Van Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26, 760 (1990).
[CrossRef]

Heiman, D.

J. Feinberg, D. Heiman, A. R. Tanguay, Jr., and R. W. Hellwarth, “Photorefractive effects and light-induced charge migration in barium titanate,” J. Appl. Phys. 51, 1297 (1980).
[CrossRef]

Hellwarth, R. W.

J. Feinberg, D. Heiman, A. R. Tanguay, Jr., and R. W. Hellwarth, “Photorefractive effects and light-induced charge migration in barium titanate,” J. Appl. Phys. 51, 1297 (1980).
[CrossRef]

Hutchings, D. C.

Ippen, E. P.

D. H. Auston, S. McAfee, C. V. Shank, E. P. Ippen, and O. Teschke, “Picosecond Spectroscopy of Semiconductors,” Solid State Electron. 21, 147 (1978).
[CrossRef]

Jackson, W.

Jackson, W. B.

Jani, P.

Kar, J. A. K.

J. R. Milward, J. A. K. Kar, C. R. Pidgeon, and B. S. Wherrett, “Photogenerated carrier recombination time in bulk ZnSe,” J. Appl. Phys. 69, 2708 (1991).
[CrossRef]

Kauranen, M.

Kim, J. S.

Kim, U.

Kukhtarev, N. V.

V. L. Vinetskii, N. V. Kukhtarev, E. N. Sal’kova, and L. G. Sukhoverkhova, “Mechanisms of dynamic conversion of coherent optical beams in CdS,” Sov. J. Quantum Electron. 10, 684 (1980).
[CrossRef]

V. L. Vinetskii, N. V. Kukhtarev, and M. S. Soskin, “Transformation of intensities and phases of light beams by a transient ‘undisplaced’ holographic grating,” Sov. J. Quantum Electron. 7, 230 (1977).
[CrossRef]

Lee, K. H.

Maki, J. J.

McAfee, S.

D. H. Auston, S. McAfee, C. V. Shank, E. P. Ippen, and O. Teschke, “Picosecond Spectroscopy of Semiconductors,” Solid State Electron. 21, 147 (1978).
[CrossRef]

Milward, J. R.

J. R. Milward, J. A. K. Kar, C. R. Pidgeon, and B. S. Wherrett, “Photogenerated carrier recombination time in bulk ZnSe,” J. Appl. Phys. 69, 2708 (1991).
[CrossRef]

Park, J. H.

Park, S. H.

Pidgeon, C. R.

J. R. Milward, J. A. K. Kar, C. R. Pidgeon, and B. S. Wherrett, “Photogenerated carrier recombination time in bulk ZnSe,” J. Appl. Phys. 69, 2708 (1991).
[CrossRef]

Said, A. A.

Sal’kova, E. N.

V. L. Vinetskii, N. V. Kukhtarev, E. N. Sal’kova, and L. G. Sukhoverkhova, “Mechanisms of dynamic conversion of coherent optical beams in CdS,” Sov. J. Quantum Electron. 10, 684 (1980).
[CrossRef]

Shank, C. V.

D. H. Auston, S. McAfee, C. V. Shank, E. P. Ippen, and O. Teschke, “Picosecond Spectroscopy of Semiconductors,” Solid State Electron. 21, 147 (1978).
[CrossRef]

Sheik-Bahae, M.

J. Wang, M. Sheik-Bahae, A. A. Said, D. J. Hagan, and E. W. Van Stryland, “Time-resolved Z-scan measurements of optical nonlinearities,” J. Opt. Soc. Am. B 11, 1009 (1994).
[CrossRef]

R. DeSalvo, M. Sheik-Bahae, A. A. Said, D. J. Hagan, and E. W. Van Stryland, “Z-scan measurements of the anisotropy of nonlinear refraction and absorption in crystals,” Opt. Lett. 18, 194 (1993).
[CrossRef] [PubMed]

M. Sheik-Bahae, J. Wang, R. DeSalvo, D. J. Hagan, and E. W. Van Stryland, “Measurement of nondegenerate nonlinearities using a two-color Z-scan,” Opt. Lett. 17, 260 (1992).
[CrossRef]

M. Sheik-Bahae, A. A. Said, T. H. Wei, D. J. Hagan, and E. W. Van Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26, 760 (1990).
[CrossRef]

Sibilia, C.

Smirl, A. L.

A. L. Smirl, J. Dubard, A. G. Cui, T. F. Boggess, and G. C. Valley, “Polarization-rotation switch using picosecond pulses in GaAs,” Opt. Lett. 14, 242 (1989).
[CrossRef] [PubMed]

G. C. Valley, and A. L. Smirl, “Theory of transient energy transfer in gallium arsenide,” IEEE J. Quantum Electron. 24, 304 (1988).
[CrossRef]

Soskin, M. S.

V. L. Vinetskii, N. V. Kukhtarev, and M. S. Soskin, “Transformation of intensities and phases of light beams by a transient ‘undisplaced’ holographic grating,” Sov. J. Quantum Electron. 7, 230 (1977).
[CrossRef]

Stentz, A. J.

Suber, G.

Sukhoverkhova, L. G.

V. L. Vinetskii, N. V. Kukhtarev, E. N. Sal’kova, and L. G. Sukhoverkhova, “Mechanisms of dynamic conversion of coherent optical beams in CdS,” Sov. J. Quantum Electron. 10, 684 (1980).
[CrossRef]

Tanguay , Jr., A. R.

J. Feinberg, D. Heiman, A. R. Tanguay, Jr., and R. W. Hellwarth, “Photorefractive effects and light-induced charge migration in barium titanate,” J. Appl. Phys. 51, 1297 (1980).
[CrossRef]

Teschke, O.

D. H. Auston, S. McAfee, C. V. Shank, E. P. Ippen, and O. Teschke, “Picosecond Spectroscopy of Semiconductors,” Solid State Electron. 21, 147 (1978).
[CrossRef]

Valley, G. C.

A. L. Smirl, J. Dubard, A. G. Cui, T. F. Boggess, and G. C. Valley, “Polarization-rotation switch using picosecond pulses in GaAs,” Opt. Lett. 14, 242 (1989).
[CrossRef] [PubMed]

G. C. Valley, and A. L. Smirl, “Theory of transient energy transfer in gallium arsenide,” IEEE J. Quantum Electron. 24, 304 (1988).
[CrossRef]

Van Stryland, E. W.

Vinetskii, V. L.

V. L. Vinetskii, N. V. Kukhtarev, E. N. Sal’kova, and L. G. Sukhoverkhova, “Mechanisms of dynamic conversion of coherent optical beams in CdS,” Sov. J. Quantum Electron. 10, 684 (1980).
[CrossRef]

V. L. Vinetskii, N. V. Kukhtarev, and M. S. Soskin, “Transformation of intensities and phases of light beams by a transient ‘undisplaced’ holographic grating,” Sov. J. Quantum Electron. 7, 230 (1977).
[CrossRef]

Wang, J.

J. Wang, M. Sheik-Bahae, A. A. Said, D. J. Hagan, and E. W. Van Stryland, “Time-resolved Z-scan measurements of optical nonlinearities,” J. Opt. Soc. Am. B 11, 1009 (1994).
[CrossRef]

M. Sheik-Bahae, J. Wang, R. DeSalvo, D. J. Hagan, and E. W. Van Stryland, “Measurement of nondegenerate nonlinearities using a two-color Z-scan,” Opt. Lett. 17, 260 (1992).
[CrossRef]

Wei, T. H.

M. Sheik-Bahae, A. A. Said, T. H. Wei, D. J. Hagan, and E. W. Van Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26, 760 (1990).
[CrossRef]

Wherrett, B. S.

J. R. Milward, J. A. K. Kar, C. R. Pidgeon, and B. S. Wherrett, “Photogenerated carrier recombination time in bulk ZnSe,” J. Appl. Phys. 69, 2708 (1991).
[CrossRef]

Appl. Opt. (2)

IEEE J. Quantum Electron. (2)

M. Sheik-Bahae, A. A. Said, T. H. Wei, D. J. Hagan, and E. W. Van Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26, 760 (1990).
[CrossRef]

G. C. Valley, and A. L. Smirl, “Theory of transient energy transfer in gallium arsenide,” IEEE J. Quantum Electron. 24, 304 (1988).
[CrossRef]

J. Appl. Phys. (2)

J. Feinberg, D. Heiman, A. R. Tanguay, Jr., and R. W. Hellwarth, “Photorefractive effects and light-induced charge migration in barium titanate,” J. Appl. Phys. 51, 1297 (1980).
[CrossRef]

J. R. Milward, J. A. K. Kar, C. R. Pidgeon, and B. S. Wherrett, “Photogenerated carrier recombination time in bulk ZnSe,” J. Appl. Phys. 69, 2708 (1991).
[CrossRef]

J. Opt. Soc. Am. B (2)

Opt. Lett. (6)

Solid State Electron. (1)

D. H. Auston, S. McAfee, C. V. Shank, E. P. Ippen, and O. Teschke, “Picosecond Spectroscopy of Semiconductors,” Solid State Electron. 21, 147 (1978).
[CrossRef]

Sov. J. Quantum Electron. (2)

V. L. Vinetskii, N. V. Kukhtarev, and M. S. Soskin, “Transformation of intensities and phases of light beams by a transient ‘undisplaced’ holographic grating,” Sov. J. Quantum Electron. 7, 230 (1977).
[CrossRef]

V. L. Vinetskii, N. V. Kukhtarev, E. N. Sal’kova, and L. G. Sukhoverkhova, “Mechanisms of dynamic conversion of coherent optical beams in CdS,” Sov. J. Quantum Electron. 10, 684 (1980).
[CrossRef]

Other (5)

R. K. Jain and M. B. Klein, “Degenerate four-wave mixing in semiconductors,” in Optical Phase Conjugation, R. A. Fisher, ed. (Academic, New York, 1983), p. 335.

Y. R. Shen, Principles of Nonlinear Optics (Wiley-Interscience, New York, 1984), p. 51.

R. W. Boyd, Nonlinear Optics (Academic, San Diego, 1992).

H. E. Ruda, ed., Widegap II–VI Compounds for Opto-electronic Applications (Chapman and Hall, New York, 1992).

K. Seeger, Semiconductor Physics, An Introduction (Springer-Verlag, New York, 1991), p. 356ff.

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Figures (7)

Fig. 1
Fig. 1

Experimental setup. Pulses from a frequency doubled, Nd:YAG laser are split into probe, pump, and sample beams. The sample beam is used to select pump pulses in a narrow (±3%) energy range and to normalize probe energy on a shot-to-shot basis. For nondegenerate studies the probe (1.4 mJ) is directed into a hydrogen-filled Raman cell for generation of an 11-ps Stokes pulse (683 nm), separated from the second Stokes, Rayleigh, and anti-Stokes lines by interference and absorption filters. In the nondegenerate case an additional sampling detector is used to normalize probe energies. The probe is directed through a delay line, a half-wave plate, a polarizer, and a focusing lens into the sample. The pump is directed through a half-wave plate, a polarizer (for energy and polarization control), and a slow-focusing lens into the sample. Pump and probe cross with an angle 2θ=1315° at the sample, which may be translated axially for transverse (X) scanning. After the sample the probe is split to measure energy and beam position. P, pyroelectric energy detector; S, silicon detector; C, CCD camera; F, absorption and interference filters.

Fig. 2
Fig. 2

Probe absorption and deflection for the degenerate case in the cross-polarized configuration. Curves are numerical solutions to Eqs. (1)–(3) for parameters shown in Tables 2 and 3. Each data point represents the average of about 20 shots, and error bars are 1–σ points. (a) Probe absorption (-ΔT/T) as a function of probe delay relative to the pump at the ZnSe sample with pump and probe near optimal transverse alignment for absorption. Zero delay is established independently using a streak camera. (b) Probe absorption and deflection at 380-ps delay as a function of transverse misalignment (normalized to pump spot size wp). Solid circles refer to absorption and open circles to deflection of probe centroid. Zero misalignment (x=0) is established by the numerical fit. (c) Same as (b) but at zero delay. The dashed curve is a numerical solution with cross-phase modulation (β1pr) set to zero.

Fig. 3
Fig. 3

Probe gain, absorption, and deflection for the degenerate case in the parallel-polarized configuration. (a) Probe gain (ΔT/T) as a function of probe delay. Solid curve is a numerical solution to the TET Eqs. (B1)–(B3) for the parameters in Tables 2 and 3. Zero delay is established independently with a streak camera. A slight pump–probe transverse misalignment is accounted for in calculation. (b) Probe absorption and deflection at long delay. Solid curves are solutions to Eqs. (1)–(3) (special case of TET equations). (c) Probe gain at delay of -7.4 ps. Solid curve is a solution to the TET equations. No deflection curve is shown for reasons explained in the text.

Fig. 4
Fig. 4

Same as Fig. 3 (cross-polarized) but for the nondegenerate case. Parameters used for modeling are in Tables 2 and 4.

Fig. 5
Fig. 5

Same as Fig. 4 but for the cross-polarized configuration. In (c) the dashed curve is a numerical solution with cross-phase modulation (β1pr) set to zero.

Fig. 6
Fig. 6

Calculation of probe deflection for the case represented in Fig. 4(c) but 10 ps earlier in probe arrival time, showing the contributions of β1pr (dotted curve) and γD (dashed curve) to the total deflection (solid curve).

Fig. 7
Fig. 7

Histograms for three positions in the X scan producing the absorption curve in Fig. 3(b). The center distribution represents data for the pump and the probe, which are transversely aligned for maximum probe absorption; top and bottom distributions correspond to misalignments of ±0.12wp.

Tables (4)

Tables Icon

Table 1 Third-Order Susceptibility Components of Single and Two-Beam Coupling Constants in Polycrystalline ZnSe

Tables Icon

Table 2 Nominal ZnSe Parameters Used in Modeling

Tables Icon

Table 3 Measured χ(3) and Free-Carrier Coefficients for the Degenerate Case a

Tables Icon

Table 4 Measured χ(3) and Free-Carrier Coefficients for the Nondegenerate Case a

Equations (26)

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

E˜p=Ep exp(iϕp)exp(ikp·r-iωpt),
E˜r=Er exp(iϕr)exp(ikr·r-iωrt),
cos(θ) dEpdz=-12(αp+β2pIp+2β2prIr+σdpN0)Ep,
dϕpdz=kp+β1pIp+2β1prIr+γDpN0,
cos(θ) dErdz=-12(αr+β2rIr+2β2prIp+σarN0)Er,
dϕrdz=kr+β1rIr+2β1rpIp+γDrN0.
N0t=β2pIp22ωp+β2rIr22ωr+4β2prIpIr(ωp+ωr)-N0τr-KN02,
z+1νgtE(r, t)exp[i(k·r-ωt)]
=-2πω2ikc2PNL(r, t),
Pi(3)(r, t)=Pi(3)(k, ω)=χijkl(3)(ω=ωj+ωk+ωl)×Ej(ωj)Ek(ωk)El(ωl)×exp[i(kj+kk+kl)·r-i(ωj+ωk+ωl)t],
Pi(ωr)=6χijkl(ωr=ωr+ωp-ωp)Ej(ωr)×Ek(ωp)El*(ωp)+3χijkl(ωr=ωr+ωr-ωr)×Ej(ωr)Ek(ωr)El*(ωr).
χijkl=(χ1122δijδkl+χ1212δikδjl+χ1221δilδjk)×(1-δijδjkδkl)+χ1111δijδjkδkl,
Picross(ωr)=6[χ1122Ei(ωr)E(ωp)E*(ωp)+χ1212Ei(ωp)E(ωr)E*(ωp)+χ1221Ei*(ωp)E(ωr)E(ωp)]-6(χ1122+χ1212+χ1221-χ1111)×Ei(ωr)Ei(ωp)Ei*(ωp).
Picross(ωr)=6[χ1122E(ωr)E(ωp)·E*(ωp)i]γri+6[χ1212E(ωp)E(ωr)·E*(ωp)+χ1221E*(ωp)E(ωr)·E(ωp)]γpi-6(χ1122+χ1212+χ1221-χ1111)×E(ωr)E(ωp)E*(ωp)γri(γpi)2.
Pcross(ωr)=6χ1122E(ωr)E(ωp)·E*(ωp)(γr1xˆ+γr2yˆ+γr3zˆ)+6[χ1212E(ωp)E(ωr)E*(ωp)+χ1221E*(ωp)E(ωr)·E(ωp)](γp1xˆ+γp2yˆ+γp3zˆ)-6(χ1122+χ1212+χ1221-χ1111)×E(ωr)E(ωp)E*(ωp)[γr1(γp1)2xˆ+γr2(γp2)2yˆ+γr3(γp3)2zˆ].
Pycross(ωr)=6(χ1122+χ1212+χ1221)|E(ωp)|2Ey(ωr),
Pycross(ωr)=6χ1122|E(ωp)|2Ey(ωr).
Pyself(ωr)=3(2χ1122+χ1221)|E(ωr)|2Ey(ωr),
χ1122(ωr=ωr+ωr-ωr)=χ1212(ωr=ωr+ωr-ωr).
N=N0+2N2 cos2π xΛ+v,
cos θ dEpdz=-12(αp+β2pIp+2β2prIr+σaN0)Ep-γDN2Er sin(Δϕ-v)-12σaN2Er cos(Δϕ-v),
cos θ dErdz=-12(αr+β2rIr+2β2rpIp+σaN0)Er+γDN2Ep sin(Δϕ-v)-12σaN2Ep cos(Δϕ-v),
cos θ dΔϕdz=-(β1rIr-β1pIp)+γDN2EpEr-ErEp×cos(Δϕ-v)+σa2N2EpEr+ErEp×sin(Δϕ-v).
N0t=β2p2ωp[(Ip+Ir)2+2IpIr]-N0τr-K(N02+2N22),
N2t=β2pωp(Ip+Ir)IpIr cos(Δϕ-v)-N2τr-2KN0N2,
vt=β2pN2ωp(Ip+Ir)IpIr sin(Δϕ-v).

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