Abstract

The spatial and temporal configuration of a degenerate four-wave mixing (DFWM) experiment have been analyzed. Numerical calculations and experiments show the clear dependence of the experimental signal value on the wave vector geometry. A commutative time symmetry method was applied to analyze data from DFWM measurements. The method permits the direct formulation of symmetry transformation rules that can be used to analyze the temporal shape of the diffracted pulse. This time symmetry approach was used to describe DFWM measurements on two third-order nonlinear materials, CS2 and polyacetylene thin films. The approach was used to analyze the slow decay temporal component region for DFWM experiments.

© 1999 Optical Society of America

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References

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  1. J. L. Bredas, C. Adant, P. Tackx, A. Persoons, and B. M. Pierce, “Third-order nonlinear response in organic materials: theoretical and experimental aspects,” Chem. Rev. 94, 243–278 (1994).
    [CrossRef]
  2. H. Singh Nalwa and S. Miyata, Nonlinear Optics of Organic Molecules and Polymers (CRC Press, Boca Raton, Fla., 1997).
  3. S. Ohtsuka, K. Tsunetomo, T. Koyama, and S. Tanaka, “Embedment of gold and CdTe particles in glass matrix and their nonlinear optical properties,” Nonlinear Opt. 13, 101–108 (1995).
  4. K. Puech, W. Blau, A. Grund, C. Bubeck, and G. Gardenas, “Picosecond degenerate four-wave mixing in colloidal solutions of gold nanoparticles at high repetition rates,” Opt. Lett. 20, 1613–1615 (1995).
    [CrossRef] [PubMed]
  5. W. Schrof, S. Rozouvan, E. van Keuren, D. Horn, J. Schmitt, and G. Decher, “Nonlinear optics of polyelectrolyte thin films containing gold nanoparticles investigated by wavelength dispersive femtosecond degenerate four wave mixing,” Adv. Mater. 10, 338–341 (1998).
    [CrossRef]
  6. T. Hattori and T. Kobayashi, “Femtosecond dephasing in a polydiacetylene film measured by degenerate four-wave mixing with an incoherent nanosecond laser,” Chem. Phys. Lett. 133, 230–234 (1987).
    [CrossRef]
  7. C. Malouin, A. Villeneuve, G. Vitrant, and R. A. Lessard, “Degenerate four-wave mixing geometry in thin-film waveguides for nonlinear material characterization,” Opt. Lett. 21, 21–23 (1996).
    [CrossRef] [PubMed]
  8. D. Narayana Rao, J. Swiatkiewicz, P. Chopra, S. K. Ghoshal, and P. N. Prasad, “Third order nonlinear interactions in thin films of poly-p phenylenebenzobisthiazole,” Appl. Phys. Lett. 48, 1187–1189 (1986).
    [CrossRef]
  9. K. Minoshima, M. Taiji, and T. Kobayashi, “Femtosecond time-resolved interferometry for the determination of complex nonlinear susceptibility,” Opt. Lett. 16, 1683–1685 (1991).
    [CrossRef] [PubMed]
  10. S. A. Jenekhe, W.-C. Chen, S. Lo, and S. R. Flom, “Large third-order optical nonlinearities in organic polymer superlattices,” Appl. Phys. Lett. 57, 126–128 (1990).
    [CrossRef]
  11. G. M. Carter, “Excited-state dynamics and temporally resolved nonresonant nonlinear-optical processes in polydiacetylenes,” J. Opt. Soc. Am. B 4, 1018–1024 (1987).
    [CrossRef]
  12. K. S. Wong, S. G. Han, and Z. V. Vardeny, “Studies of resonant and preresonant femtosecond degenerate four-wave mixing in unoriented conducting polymers,” J. Appl. Phys. 70, 1896–1898 (1991).
    [CrossRef]
  13. B. I. Greene, J. Orenstein, and S. Schmitt-Rink, “Alloptical nonlinearities in organics,” Science 247, 679–687 (1990).
    [CrossRef] [PubMed]
  14. S. G. Rozuvan, K. P. Rozuvan, I. A. Shaykevich, and Y. A. Tikhonov, “New method and the calculation program of life time determination of inhomogeneously broadened transitions,” in Optical Diagnostics of Materials and Devices for Opto-, Micro- and Quantum Electronics, S. V. Svechnikov, and M. Ya. Valakh, eds., Proc. SPIE 2648, 300–303 (1995).
    [CrossRef]
  15. D. A. Fish and A. K. Powell, “Solutions to vector four-wave mixing in cubic photorefractive materials,” J. Opt. Soc. Am. B 14, 2628–2640 (1997).
    [CrossRef]
  16. P. N. Prasad and D. J. Williams, Introduction to Nonlinear Optical Effects in Molecules and Polymers (Wiley, New York, 1991).
  17. A. Jariv, Kvantovaja Eliectronica (Sovietskoje radio, Moskva, 1973) [translated from A. Jariv, Quantum Electronics (Wiley, New York, 1967)].
  18. H. J. Eichler, P. Günter, and D. W. Pohl, Laser-Induced Dynamic Gratings (Springer-Verlag, Berlin, 1986).
  19. D. A. Kleinman, “Nonlinear dielectric polarization in optical media,” Phys. Rev. 126, 1977–1979 (1962).
    [CrossRef]
  20. W. Schrof, S. Rozouvan, T. Hartmann, V. Belov, H. Mohwald, and E. van Keuren, “Nonlinear optical properties of novel low-bandgap polythiophenes,” J. Opt. Soc. Am. B 15, 889–894 (1998).
    [CrossRef]

1998 (2)

W. Schrof, S. Rozouvan, E. van Keuren, D. Horn, J. Schmitt, and G. Decher, “Nonlinear optics of polyelectrolyte thin films containing gold nanoparticles investigated by wavelength dispersive femtosecond degenerate four wave mixing,” Adv. Mater. 10, 338–341 (1998).
[CrossRef]

W. Schrof, S. Rozouvan, T. Hartmann, V. Belov, H. Mohwald, and E. van Keuren, “Nonlinear optical properties of novel low-bandgap polythiophenes,” J. Opt. Soc. Am. B 15, 889–894 (1998).
[CrossRef]

1997 (1)

1996 (1)

1995 (2)

S. Ohtsuka, K. Tsunetomo, T. Koyama, and S. Tanaka, “Embedment of gold and CdTe particles in glass matrix and their nonlinear optical properties,” Nonlinear Opt. 13, 101–108 (1995).

K. Puech, W. Blau, A. Grund, C. Bubeck, and G. Gardenas, “Picosecond degenerate four-wave mixing in colloidal solutions of gold nanoparticles at high repetition rates,” Opt. Lett. 20, 1613–1615 (1995).
[CrossRef] [PubMed]

1994 (1)

J. L. Bredas, C. Adant, P. Tackx, A. Persoons, and B. M. Pierce, “Third-order nonlinear response in organic materials: theoretical and experimental aspects,” Chem. Rev. 94, 243–278 (1994).
[CrossRef]

1991 (2)

K. S. Wong, S. G. Han, and Z. V. Vardeny, “Studies of resonant and preresonant femtosecond degenerate four-wave mixing in unoriented conducting polymers,” J. Appl. Phys. 70, 1896–1898 (1991).
[CrossRef]

K. Minoshima, M. Taiji, and T. Kobayashi, “Femtosecond time-resolved interferometry for the determination of complex nonlinear susceptibility,” Opt. Lett. 16, 1683–1685 (1991).
[CrossRef] [PubMed]

1990 (2)

S. A. Jenekhe, W.-C. Chen, S. Lo, and S. R. Flom, “Large third-order optical nonlinearities in organic polymer superlattices,” Appl. Phys. Lett. 57, 126–128 (1990).
[CrossRef]

B. I. Greene, J. Orenstein, and S. Schmitt-Rink, “Alloptical nonlinearities in organics,” Science 247, 679–687 (1990).
[CrossRef] [PubMed]

1987 (2)

G. M. Carter, “Excited-state dynamics and temporally resolved nonresonant nonlinear-optical processes in polydiacetylenes,” J. Opt. Soc. Am. B 4, 1018–1024 (1987).
[CrossRef]

T. Hattori and T. Kobayashi, “Femtosecond dephasing in a polydiacetylene film measured by degenerate four-wave mixing with an incoherent nanosecond laser,” Chem. Phys. Lett. 133, 230–234 (1987).
[CrossRef]

1986 (1)

D. Narayana Rao, J. Swiatkiewicz, P. Chopra, S. K. Ghoshal, and P. N. Prasad, “Third order nonlinear interactions in thin films of poly-p phenylenebenzobisthiazole,” Appl. Phys. Lett. 48, 1187–1189 (1986).
[CrossRef]

1962 (1)

D. A. Kleinman, “Nonlinear dielectric polarization in optical media,” Phys. Rev. 126, 1977–1979 (1962).
[CrossRef]

Adant, C.

J. L. Bredas, C. Adant, P. Tackx, A. Persoons, and B. M. Pierce, “Third-order nonlinear response in organic materials: theoretical and experimental aspects,” Chem. Rev. 94, 243–278 (1994).
[CrossRef]

Belov, V.

Blau, W.

Bredas, J. L.

J. L. Bredas, C. Adant, P. Tackx, A. Persoons, and B. M. Pierce, “Third-order nonlinear response in organic materials: theoretical and experimental aspects,” Chem. Rev. 94, 243–278 (1994).
[CrossRef]

Bubeck, C.

Carter, G. M.

Chen, W.-C.

S. A. Jenekhe, W.-C. Chen, S. Lo, and S. R. Flom, “Large third-order optical nonlinearities in organic polymer superlattices,” Appl. Phys. Lett. 57, 126–128 (1990).
[CrossRef]

Chopra, P.

D. Narayana Rao, J. Swiatkiewicz, P. Chopra, S. K. Ghoshal, and P. N. Prasad, “Third order nonlinear interactions in thin films of poly-p phenylenebenzobisthiazole,” Appl. Phys. Lett. 48, 1187–1189 (1986).
[CrossRef]

Decher, G.

W. Schrof, S. Rozouvan, E. van Keuren, D. Horn, J. Schmitt, and G. Decher, “Nonlinear optics of polyelectrolyte thin films containing gold nanoparticles investigated by wavelength dispersive femtosecond degenerate four wave mixing,” Adv. Mater. 10, 338–341 (1998).
[CrossRef]

Fish, D. A.

Flom, S. R.

S. A. Jenekhe, W.-C. Chen, S. Lo, and S. R. Flom, “Large third-order optical nonlinearities in organic polymer superlattices,” Appl. Phys. Lett. 57, 126–128 (1990).
[CrossRef]

Gardenas, G.

Ghoshal, S. K.

D. Narayana Rao, J. Swiatkiewicz, P. Chopra, S. K. Ghoshal, and P. N. Prasad, “Third order nonlinear interactions in thin films of poly-p phenylenebenzobisthiazole,” Appl. Phys. Lett. 48, 1187–1189 (1986).
[CrossRef]

Greene, B. I.

B. I. Greene, J. Orenstein, and S. Schmitt-Rink, “Alloptical nonlinearities in organics,” Science 247, 679–687 (1990).
[CrossRef] [PubMed]

Grund, A.

Han, S. G.

K. S. Wong, S. G. Han, and Z. V. Vardeny, “Studies of resonant and preresonant femtosecond degenerate four-wave mixing in unoriented conducting polymers,” J. Appl. Phys. 70, 1896–1898 (1991).
[CrossRef]

Hartmann, T.

Hattori, T.

T. Hattori and T. Kobayashi, “Femtosecond dephasing in a polydiacetylene film measured by degenerate four-wave mixing with an incoherent nanosecond laser,” Chem. Phys. Lett. 133, 230–234 (1987).
[CrossRef]

Horn, D.

W. Schrof, S. Rozouvan, E. van Keuren, D. Horn, J. Schmitt, and G. Decher, “Nonlinear optics of polyelectrolyte thin films containing gold nanoparticles investigated by wavelength dispersive femtosecond degenerate four wave mixing,” Adv. Mater. 10, 338–341 (1998).
[CrossRef]

Jenekhe, S. A.

S. A. Jenekhe, W.-C. Chen, S. Lo, and S. R. Flom, “Large third-order optical nonlinearities in organic polymer superlattices,” Appl. Phys. Lett. 57, 126–128 (1990).
[CrossRef]

Kleinman, D. A.

D. A. Kleinman, “Nonlinear dielectric polarization in optical media,” Phys. Rev. 126, 1977–1979 (1962).
[CrossRef]

Kobayashi, T.

K. Minoshima, M. Taiji, and T. Kobayashi, “Femtosecond time-resolved interferometry for the determination of complex nonlinear susceptibility,” Opt. Lett. 16, 1683–1685 (1991).
[CrossRef] [PubMed]

T. Hattori and T. Kobayashi, “Femtosecond dephasing in a polydiacetylene film measured by degenerate four-wave mixing with an incoherent nanosecond laser,” Chem. Phys. Lett. 133, 230–234 (1987).
[CrossRef]

Koyama, T.

S. Ohtsuka, K. Tsunetomo, T. Koyama, and S. Tanaka, “Embedment of gold and CdTe particles in glass matrix and their nonlinear optical properties,” Nonlinear Opt. 13, 101–108 (1995).

Lessard, R. A.

Lo, S.

S. A. Jenekhe, W.-C. Chen, S. Lo, and S. R. Flom, “Large third-order optical nonlinearities in organic polymer superlattices,” Appl. Phys. Lett. 57, 126–128 (1990).
[CrossRef]

Malouin, C.

Minoshima, K.

Mohwald, H.

Narayana Rao, D.

D. Narayana Rao, J. Swiatkiewicz, P. Chopra, S. K. Ghoshal, and P. N. Prasad, “Third order nonlinear interactions in thin films of poly-p phenylenebenzobisthiazole,” Appl. Phys. Lett. 48, 1187–1189 (1986).
[CrossRef]

Ohtsuka, S.

S. Ohtsuka, K. Tsunetomo, T. Koyama, and S. Tanaka, “Embedment of gold and CdTe particles in glass matrix and their nonlinear optical properties,” Nonlinear Opt. 13, 101–108 (1995).

Orenstein, J.

B. I. Greene, J. Orenstein, and S. Schmitt-Rink, “Alloptical nonlinearities in organics,” Science 247, 679–687 (1990).
[CrossRef] [PubMed]

Persoons, A.

J. L. Bredas, C. Adant, P. Tackx, A. Persoons, and B. M. Pierce, “Third-order nonlinear response in organic materials: theoretical and experimental aspects,” Chem. Rev. 94, 243–278 (1994).
[CrossRef]

Pierce, B. M.

J. L. Bredas, C. Adant, P. Tackx, A. Persoons, and B. M. Pierce, “Third-order nonlinear response in organic materials: theoretical and experimental aspects,” Chem. Rev. 94, 243–278 (1994).
[CrossRef]

Powell, A. K.

Prasad, P. N.

D. Narayana Rao, J. Swiatkiewicz, P. Chopra, S. K. Ghoshal, and P. N. Prasad, “Third order nonlinear interactions in thin films of poly-p phenylenebenzobisthiazole,” Appl. Phys. Lett. 48, 1187–1189 (1986).
[CrossRef]

Puech, K.

Rozouvan, S.

W. Schrof, S. Rozouvan, E. van Keuren, D. Horn, J. Schmitt, and G. Decher, “Nonlinear optics of polyelectrolyte thin films containing gold nanoparticles investigated by wavelength dispersive femtosecond degenerate four wave mixing,” Adv. Mater. 10, 338–341 (1998).
[CrossRef]

W. Schrof, S. Rozouvan, T. Hartmann, V. Belov, H. Mohwald, and E. van Keuren, “Nonlinear optical properties of novel low-bandgap polythiophenes,” J. Opt. Soc. Am. B 15, 889–894 (1998).
[CrossRef]

Schmitt, J.

W. Schrof, S. Rozouvan, E. van Keuren, D. Horn, J. Schmitt, and G. Decher, “Nonlinear optics of polyelectrolyte thin films containing gold nanoparticles investigated by wavelength dispersive femtosecond degenerate four wave mixing,” Adv. Mater. 10, 338–341 (1998).
[CrossRef]

Schmitt-Rink, S.

B. I. Greene, J. Orenstein, and S. Schmitt-Rink, “Alloptical nonlinearities in organics,” Science 247, 679–687 (1990).
[CrossRef] [PubMed]

Schrof, W.

W. Schrof, S. Rozouvan, T. Hartmann, V. Belov, H. Mohwald, and E. van Keuren, “Nonlinear optical properties of novel low-bandgap polythiophenes,” J. Opt. Soc. Am. B 15, 889–894 (1998).
[CrossRef]

W. Schrof, S. Rozouvan, E. van Keuren, D. Horn, J. Schmitt, and G. Decher, “Nonlinear optics of polyelectrolyte thin films containing gold nanoparticles investigated by wavelength dispersive femtosecond degenerate four wave mixing,” Adv. Mater. 10, 338–341 (1998).
[CrossRef]

Swiatkiewicz, J.

D. Narayana Rao, J. Swiatkiewicz, P. Chopra, S. K. Ghoshal, and P. N. Prasad, “Third order nonlinear interactions in thin films of poly-p phenylenebenzobisthiazole,” Appl. Phys. Lett. 48, 1187–1189 (1986).
[CrossRef]

Tackx, P.

J. L. Bredas, C. Adant, P. Tackx, A. Persoons, and B. M. Pierce, “Third-order nonlinear response in organic materials: theoretical and experimental aspects,” Chem. Rev. 94, 243–278 (1994).
[CrossRef]

Taiji, M.

Tanaka, S.

S. Ohtsuka, K. Tsunetomo, T. Koyama, and S. Tanaka, “Embedment of gold and CdTe particles in glass matrix and their nonlinear optical properties,” Nonlinear Opt. 13, 101–108 (1995).

Tsunetomo, K.

S. Ohtsuka, K. Tsunetomo, T. Koyama, and S. Tanaka, “Embedment of gold and CdTe particles in glass matrix and their nonlinear optical properties,” Nonlinear Opt. 13, 101–108 (1995).

van Keuren, E.

W. Schrof, S. Rozouvan, E. van Keuren, D. Horn, J. Schmitt, and G. Decher, “Nonlinear optics of polyelectrolyte thin films containing gold nanoparticles investigated by wavelength dispersive femtosecond degenerate four wave mixing,” Adv. Mater. 10, 338–341 (1998).
[CrossRef]

W. Schrof, S. Rozouvan, T. Hartmann, V. Belov, H. Mohwald, and E. van Keuren, “Nonlinear optical properties of novel low-bandgap polythiophenes,” J. Opt. Soc. Am. B 15, 889–894 (1998).
[CrossRef]

Vardeny, Z. V.

K. S. Wong, S. G. Han, and Z. V. Vardeny, “Studies of resonant and preresonant femtosecond degenerate four-wave mixing in unoriented conducting polymers,” J. Appl. Phys. 70, 1896–1898 (1991).
[CrossRef]

Villeneuve, A.

Vitrant, G.

Wong, K. S.

K. S. Wong, S. G. Han, and Z. V. Vardeny, “Studies of resonant and preresonant femtosecond degenerate four-wave mixing in unoriented conducting polymers,” J. Appl. Phys. 70, 1896–1898 (1991).
[CrossRef]

Adv. Mater. (1)

W. Schrof, S. Rozouvan, E. van Keuren, D. Horn, J. Schmitt, and G. Decher, “Nonlinear optics of polyelectrolyte thin films containing gold nanoparticles investigated by wavelength dispersive femtosecond degenerate four wave mixing,” Adv. Mater. 10, 338–341 (1998).
[CrossRef]

Appl. Phys. Lett. (2)

D. Narayana Rao, J. Swiatkiewicz, P. Chopra, S. K. Ghoshal, and P. N. Prasad, “Third order nonlinear interactions in thin films of poly-p phenylenebenzobisthiazole,” Appl. Phys. Lett. 48, 1187–1189 (1986).
[CrossRef]

S. A. Jenekhe, W.-C. Chen, S. Lo, and S. R. Flom, “Large third-order optical nonlinearities in organic polymer superlattices,” Appl. Phys. Lett. 57, 126–128 (1990).
[CrossRef]

Chem. Phys. Lett. (1)

T. Hattori and T. Kobayashi, “Femtosecond dephasing in a polydiacetylene film measured by degenerate four-wave mixing with an incoherent nanosecond laser,” Chem. Phys. Lett. 133, 230–234 (1987).
[CrossRef]

Chem. Rev. (1)

J. L. Bredas, C. Adant, P. Tackx, A. Persoons, and B. M. Pierce, “Third-order nonlinear response in organic materials: theoretical and experimental aspects,” Chem. Rev. 94, 243–278 (1994).
[CrossRef]

J. Appl. Phys. (1)

K. S. Wong, S. G. Han, and Z. V. Vardeny, “Studies of resonant and preresonant femtosecond degenerate four-wave mixing in unoriented conducting polymers,” J. Appl. Phys. 70, 1896–1898 (1991).
[CrossRef]

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

Nonlinear Opt. (1)

S. Ohtsuka, K. Tsunetomo, T. Koyama, and S. Tanaka, “Embedment of gold and CdTe particles in glass matrix and their nonlinear optical properties,” Nonlinear Opt. 13, 101–108 (1995).

Opt. Lett. (3)

Phys. Rev. (1)

D. A. Kleinman, “Nonlinear dielectric polarization in optical media,” Phys. Rev. 126, 1977–1979 (1962).
[CrossRef]

Science (1)

B. I. Greene, J. Orenstein, and S. Schmitt-Rink, “Alloptical nonlinearities in organics,” Science 247, 679–687 (1990).
[CrossRef] [PubMed]

Other (5)

S. G. Rozuvan, K. P. Rozuvan, I. A. Shaykevich, and Y. A. Tikhonov, “New method and the calculation program of life time determination of inhomogeneously broadened transitions,” in Optical Diagnostics of Materials and Devices for Opto-, Micro- and Quantum Electronics, S. V. Svechnikov, and M. Ya. Valakh, eds., Proc. SPIE 2648, 300–303 (1995).
[CrossRef]

P. N. Prasad and D. J. Williams, Introduction to Nonlinear Optical Effects in Molecules and Polymers (Wiley, New York, 1991).

A. Jariv, Kvantovaja Eliectronica (Sovietskoje radio, Moskva, 1973) [translated from A. Jariv, Quantum Electronics (Wiley, New York, 1967)].

H. J. Eichler, P. Günter, and D. W. Pohl, Laser-Induced Dynamic Gratings (Springer-Verlag, Berlin, 1986).

H. Singh Nalwa and S. Miyata, Nonlinear Optics of Organic Molecules and Polymers (CRC Press, Boca Raton, Fla., 1997).

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Figures (5)

Fig. 1
Fig. 1

k1, k2, and k3 wave vectors in spherical coordinate system.

Fig. 2
Fig. 2

Coherence length versus variations in the direction of k3 in Eqs. (1), (2), and (3). The k1 and k2 vectors are also presented schematically.

Fig. 3
Fig. 3

Experimental setup for DFWM measurements. 1a, 1b, Ar+ (12 W cw, 514 nm) and Ti:sapphire (1 W, 700–950 nm, 90 fs) lasers; 2, pulse picker; 3a, 3b, delay stages; 4a, 4b, chopper controller and chopper; 5a, 5b, lenses; 6, sample; 7, aperture; 8, Si photodiode; 9, low-noise preamplifier; 10, lock-in amplifier; 11, personal computer with IEEE 488 interface card; 12, step-motor controller.

Fig. 4
Fig. 4

Measured Ik4 signal as a function of delay times between k1, k2, and k3 for (a) CS2 and (b) polyacetylene. The curves connect points with equal value of the Ik4 signal.

Fig. 5
Fig. 5

Measured Ik4 signals from Fig. 4(b), with various values of the second delay time.

Equations (22)

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

k1+k2=k3+k4.
k1+k3˙=k2+k4,
k3+k2=k1+k4.
|k|=|ki|=const.
k1=|k|-χ/20=cos(-χ/2)sin(-χ/2)0|k|,
k2=|k|χ/20=cos(χ/2)sin(χ/2)0|k|,
k3=|k|φθ=sin θ cos φsin θ sin φcos θ|k|.
lc=π/Δk=π/(|k4+k3-k1-k2|)
lc=λ2(1-{1+4 cos(χ/2)[cos(χ/2)-sin θ cos φ]}1/2),
lc=λ2(1-{1+4 sin(χ/2)[sin(χ/2)+sin θ sin φ]}1/2),
lc=λ2(1-{1+4 sin(χ/2)[sin(χ/2)-sin θ sin φ]}1/2).
dE4dzχ(3)E1E2E3 exp(iΔkz).
Ik4(χ(3))2I3[sin(Δkdgrat)/Δk]2,
χ(3)(ω;-ω, ω, -ω)=χ(3)(ω; ω, -ω, -ω)=χ(3)(ω;-ω, -ω, ω),
(2, 1, 3),t1,2(1)=t1,2(1),t1,3(1)=t1,3(1);(3, 1, 2),t1,3(2)=t1,2(1),t1,2(2)=t1,3(1);(1, 2, 3),t1,2(3)=-t1,2(1),t1,3(3)=t1,3(1)-t1,2(1);(1, 3, 2),t1,3(4)=-t1,2(1),t1,2(4)=t1,3(1)-t1,2(1);(3, 2, 1),t1,3(5)=t1,2(1)-t1,3(1),t1,2(5)=-t1,3(1);(2, 3, 1),t1,3(6)=-t1,3(1),t1,2(6)=t1,2(1)-t1,3(1).
I(t12*, t13*)=I(t13*, t12*)=I(-t1,2*, t1,3*-t1,2*)=I(t1,3*-t1,2*, -t1,2*)=I(-t1,3*, t1,2*-t1,3*)=I(t1,2*-t1,3*, -t1,3*),
S01,t13=0;
S02,t12=0;
S12,t23=0.
S01,(k1-k3)+k2=k4;
S02,(k1+k2)-k3=k4;
S12,(k2-k3)+k1=k4.

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