Abstract

Here we report the first observation of hyper-Rayleigh light scattering from bacteriorhodopsin in the form of an aqueous suspension of unoriented purple membranes. A typical purple membrane suspension used in our experiments contains approximately 108 randomly oriented purple membranes. Each purple membrane contains approximately 105 bacteriorhodopsin molecules in a two-dimensional crystalline array. Hyper-Rayleigh light scattering is observed when the purple membrane suspension is illuminated with light that has a wavelength of 1064 nm. We propose that the 532-nm scattered light from each of the bacteriorhodopsin molecules in a single purple membrane is coherent, and that the scattered light from different purple membranes is incoherent. This proposal is supported by the following experimental observations: (a) the 532-nm light intensity is proportional to the square of the incident power, (b) the intensity of the 532-nm signal is linearly proportional to the concentration of purple membrane in solution, (c) the scattered 532-nm light is incoherent, (d) the scattered 532-nm light intensity decreases if the size of the purple membranes is reduced while the bacteriorhodopsin concentration is kept constant, and (e) the 532-nm light is due to the retinal chromophore of the bacteriorhodopsin molecule. The ratio of horizontal polarized hyper-Rayleigh scattered light to vertically polarized hyper-Rayleigh scattered light gives the angle (23 ± 4°) of the retinal axis with respect to the plane of the purple membrane. The hyperpolarizability of the bacteriorhodopsin molecule is found to be 5 ± 0.4 × 10−27 esu.

© 1994 Optical Society of America

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References

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  1. R. R. Birge, “Photophysics and molecular electronic applications of the rhodopsins,” Annu. Rev. Phys. Chem. 41, 683–733 (1990).
    [Crossref] [PubMed]
  2. D. Oesterhelt, C. Bräuchle, N. Hampp, “Bacteriorhodopsin: a biological material for information processing,” Q. Rev. Biophys. 24, 425–478 (1991).
    [Crossref] [PubMed]
  3. R. R. Birge, “Protein-based optical computing and memories,” Computer 25, 56–67 (1992).
    [Crossref]
  4. W. Stoeckenius, R. A. Bogomolni, “Bacteriorhodopsin and related pigments of halobacteria,” Annu. Rev. Biochem. 52, 587–616 (1982).
    [Crossref]
  5. R. Henderson, J. M. Baldwin, T. A. Ceska, F. Zemlin, E. Beckmann, K. H. Downing, “Model for the structure of bacteriorhodopsin based on high-resolution electron cryomicroscopy,” J. Mol. Biol. 213, 899–929 (1990).
    [Crossref] [PubMed]
  6. D. J. Williams, “Large optical nonlinearities,” Angew. Chem. Int. Ed. Engl. 23, 690–703 (1984).
    [Crossref]
  7. Z. Chen, A. Lewis, H. Takei, I. Nebenzahl, “Bacteriorhodopsin oriented in polyvinyl alcohol films as an erasable optical storage medium,” Appl. Opt. 30, 5188–5196 (1991).
    [Crossref] [PubMed]
  8. R. W. Terhune, P. D. Maker, C. M. Savage, “Measurements of nonlinear light scattering,” Phys. Rev. Lett. 14, 681–684(1965).
    [Crossref]
  9. K. Clays, A. Persoon, “Hyper-Rayleigh scattering in solution,” Phys. Rev. Lett. 66, 2980–2983 (1991).
    [Crossref] [PubMed]
  10. J. Y. Huang, Z. Chen, A. Lewis, “Second-harmonic generation in purple membrane-poly(vinyl alcohol) films: probing the dipolar characteristics of the bacteriorhodopsin chromophore in bR570 and M412,” J. Phys. Chem. 93, 3314–3320 (1989).
    [Crossref]
  11. R. R. Birge, C.-F. Zhang, “Two-photon double resonance spectroscopy of bacteriorhodopsin. Assignment of the electronic and dipolar properties of the low-lying 1Ag*-like and 1Bu*-like π, π* states,” J. Chem. Phys. 92, 7178–7195 (1990).
    [Crossref]
  12. B. M. Beecher, J. Y. Cassim, “Improved isolation procedures for the purple-membrane of Halobacterium halbium,” Prep. Biochem. 5, 161–178 (1975).
    [Crossref]
  13. J. Jerphaganon, S. K. Kurtz, “Maker fringes: a detailed comparison of theory and experiment for isotropic and uniaxial crystals,” J. Appl. Phys. 41, 1667–1681 (1970).
    [Crossref]
  14. H. Goldstein, Classical Mechanics, 2nd ed. (Addison-Wesley, Reading, Mass.1980), Chap. 4, p. 147.
  15. S. J. Cyvin, J. E. Rauch, J. C. Decius, “Theory of hyper-Raman effects (nonlinear inelastic light scattering); selection rules and depolarization ratios for the second-order polarizability,” J. Chem. Phys. 43, 4083–4095 (1965).
    [Crossref]
  16. M. P. Heyn, R. J. Cherry, U. Muller, “Transient and linear dichroism studies on bacteriorhodopsin: determination of the orientation of the 568 nm All-trans retinal chromophore,” J. Mol. Biol. 117, 607–620 (1977).
    [Crossref] [PubMed]
  17. Y. R. Shen, The Principles of Nonlinear Optics (Wiley, New York, 1984), Chap. 7, p. 101.

1992 (1)

R. R. Birge, “Protein-based optical computing and memories,” Computer 25, 56–67 (1992).
[Crossref]

1991 (3)

D. Oesterhelt, C. Bräuchle, N. Hampp, “Bacteriorhodopsin: a biological material for information processing,” Q. Rev. Biophys. 24, 425–478 (1991).
[Crossref] [PubMed]

Z. Chen, A. Lewis, H. Takei, I. Nebenzahl, “Bacteriorhodopsin oriented in polyvinyl alcohol films as an erasable optical storage medium,” Appl. Opt. 30, 5188–5196 (1991).
[Crossref] [PubMed]

K. Clays, A. Persoon, “Hyper-Rayleigh scattering in solution,” Phys. Rev. Lett. 66, 2980–2983 (1991).
[Crossref] [PubMed]

1990 (3)

R. R. Birge, C.-F. Zhang, “Two-photon double resonance spectroscopy of bacteriorhodopsin. Assignment of the electronic and dipolar properties of the low-lying 1Ag*-like and 1Bu*-like π, π* states,” J. Chem. Phys. 92, 7178–7195 (1990).
[Crossref]

R. Henderson, J. M. Baldwin, T. A. Ceska, F. Zemlin, E. Beckmann, K. H. Downing, “Model for the structure of bacteriorhodopsin based on high-resolution electron cryomicroscopy,” J. Mol. Biol. 213, 899–929 (1990).
[Crossref] [PubMed]

R. R. Birge, “Photophysics and molecular electronic applications of the rhodopsins,” Annu. Rev. Phys. Chem. 41, 683–733 (1990).
[Crossref] [PubMed]

1989 (1)

J. Y. Huang, Z. Chen, A. Lewis, “Second-harmonic generation in purple membrane-poly(vinyl alcohol) films: probing the dipolar characteristics of the bacteriorhodopsin chromophore in bR570 and M412,” J. Phys. Chem. 93, 3314–3320 (1989).
[Crossref]

1984 (1)

D. J. Williams, “Large optical nonlinearities,” Angew. Chem. Int. Ed. Engl. 23, 690–703 (1984).
[Crossref]

1982 (1)

W. Stoeckenius, R. A. Bogomolni, “Bacteriorhodopsin and related pigments of halobacteria,” Annu. Rev. Biochem. 52, 587–616 (1982).
[Crossref]

1977 (1)

M. P. Heyn, R. J. Cherry, U. Muller, “Transient and linear dichroism studies on bacteriorhodopsin: determination of the orientation of the 568 nm All-trans retinal chromophore,” J. Mol. Biol. 117, 607–620 (1977).
[Crossref] [PubMed]

1975 (1)

B. M. Beecher, J. Y. Cassim, “Improved isolation procedures for the purple-membrane of Halobacterium halbium,” Prep. Biochem. 5, 161–178 (1975).
[Crossref]

1970 (1)

J. Jerphaganon, S. K. Kurtz, “Maker fringes: a detailed comparison of theory and experiment for isotropic and uniaxial crystals,” J. Appl. Phys. 41, 1667–1681 (1970).
[Crossref]

1965 (2)

S. J. Cyvin, J. E. Rauch, J. C. Decius, “Theory of hyper-Raman effects (nonlinear inelastic light scattering); selection rules and depolarization ratios for the second-order polarizability,” J. Chem. Phys. 43, 4083–4095 (1965).
[Crossref]

R. W. Terhune, P. D. Maker, C. M. Savage, “Measurements of nonlinear light scattering,” Phys. Rev. Lett. 14, 681–684(1965).
[Crossref]

Baldwin, J. M.

R. Henderson, J. M. Baldwin, T. A. Ceska, F. Zemlin, E. Beckmann, K. H. Downing, “Model for the structure of bacteriorhodopsin based on high-resolution electron cryomicroscopy,” J. Mol. Biol. 213, 899–929 (1990).
[Crossref] [PubMed]

Beckmann, E.

R. Henderson, J. M. Baldwin, T. A. Ceska, F. Zemlin, E. Beckmann, K. H. Downing, “Model for the structure of bacteriorhodopsin based on high-resolution electron cryomicroscopy,” J. Mol. Biol. 213, 899–929 (1990).
[Crossref] [PubMed]

Beecher, B. M.

B. M. Beecher, J. Y. Cassim, “Improved isolation procedures for the purple-membrane of Halobacterium halbium,” Prep. Biochem. 5, 161–178 (1975).
[Crossref]

Birge, R. R.

R. R. Birge, “Protein-based optical computing and memories,” Computer 25, 56–67 (1992).
[Crossref]

R. R. Birge, “Photophysics and molecular electronic applications of the rhodopsins,” Annu. Rev. Phys. Chem. 41, 683–733 (1990).
[Crossref] [PubMed]

R. R. Birge, C.-F. Zhang, “Two-photon double resonance spectroscopy of bacteriorhodopsin. Assignment of the electronic and dipolar properties of the low-lying 1Ag*-like and 1Bu*-like π, π* states,” J. Chem. Phys. 92, 7178–7195 (1990).
[Crossref]

Bogomolni, R. A.

W. Stoeckenius, R. A. Bogomolni, “Bacteriorhodopsin and related pigments of halobacteria,” Annu. Rev. Biochem. 52, 587–616 (1982).
[Crossref]

Bräuchle, C.

D. Oesterhelt, C. Bräuchle, N. Hampp, “Bacteriorhodopsin: a biological material for information processing,” Q. Rev. Biophys. 24, 425–478 (1991).
[Crossref] [PubMed]

Cassim, J. Y.

B. M. Beecher, J. Y. Cassim, “Improved isolation procedures for the purple-membrane of Halobacterium halbium,” Prep. Biochem. 5, 161–178 (1975).
[Crossref]

Ceska, T. A.

R. Henderson, J. M. Baldwin, T. A. Ceska, F. Zemlin, E. Beckmann, K. H. Downing, “Model for the structure of bacteriorhodopsin based on high-resolution electron cryomicroscopy,” J. Mol. Biol. 213, 899–929 (1990).
[Crossref] [PubMed]

Chen, Z.

Z. Chen, A. Lewis, H. Takei, I. Nebenzahl, “Bacteriorhodopsin oriented in polyvinyl alcohol films as an erasable optical storage medium,” Appl. Opt. 30, 5188–5196 (1991).
[Crossref] [PubMed]

J. Y. Huang, Z. Chen, A. Lewis, “Second-harmonic generation in purple membrane-poly(vinyl alcohol) films: probing the dipolar characteristics of the bacteriorhodopsin chromophore in bR570 and M412,” J. Phys. Chem. 93, 3314–3320 (1989).
[Crossref]

Cherry, R. J.

M. P. Heyn, R. J. Cherry, U. Muller, “Transient and linear dichroism studies on bacteriorhodopsin: determination of the orientation of the 568 nm All-trans retinal chromophore,” J. Mol. Biol. 117, 607–620 (1977).
[Crossref] [PubMed]

Clays, K.

K. Clays, A. Persoon, “Hyper-Rayleigh scattering in solution,” Phys. Rev. Lett. 66, 2980–2983 (1991).
[Crossref] [PubMed]

Cyvin, S. J.

S. J. Cyvin, J. E. Rauch, J. C. Decius, “Theory of hyper-Raman effects (nonlinear inelastic light scattering); selection rules and depolarization ratios for the second-order polarizability,” J. Chem. Phys. 43, 4083–4095 (1965).
[Crossref]

Decius, J. C.

S. J. Cyvin, J. E. Rauch, J. C. Decius, “Theory of hyper-Raman effects (nonlinear inelastic light scattering); selection rules and depolarization ratios for the second-order polarizability,” J. Chem. Phys. 43, 4083–4095 (1965).
[Crossref]

Downing, K. H.

R. Henderson, J. M. Baldwin, T. A. Ceska, F. Zemlin, E. Beckmann, K. H. Downing, “Model for the structure of bacteriorhodopsin based on high-resolution electron cryomicroscopy,” J. Mol. Biol. 213, 899–929 (1990).
[Crossref] [PubMed]

Goldstein, H.

H. Goldstein, Classical Mechanics, 2nd ed. (Addison-Wesley, Reading, Mass.1980), Chap. 4, p. 147.

Hampp, N.

D. Oesterhelt, C. Bräuchle, N. Hampp, “Bacteriorhodopsin: a biological material for information processing,” Q. Rev. Biophys. 24, 425–478 (1991).
[Crossref] [PubMed]

Henderson, R.

R. Henderson, J. M. Baldwin, T. A. Ceska, F. Zemlin, E. Beckmann, K. H. Downing, “Model for the structure of bacteriorhodopsin based on high-resolution electron cryomicroscopy,” J. Mol. Biol. 213, 899–929 (1990).
[Crossref] [PubMed]

Heyn, M. P.

M. P. Heyn, R. J. Cherry, U. Muller, “Transient and linear dichroism studies on bacteriorhodopsin: determination of the orientation of the 568 nm All-trans retinal chromophore,” J. Mol. Biol. 117, 607–620 (1977).
[Crossref] [PubMed]

Huang, J. Y.

J. Y. Huang, Z. Chen, A. Lewis, “Second-harmonic generation in purple membrane-poly(vinyl alcohol) films: probing the dipolar characteristics of the bacteriorhodopsin chromophore in bR570 and M412,” J. Phys. Chem. 93, 3314–3320 (1989).
[Crossref]

Jerphaganon, J.

J. Jerphaganon, S. K. Kurtz, “Maker fringes: a detailed comparison of theory and experiment for isotropic and uniaxial crystals,” J. Appl. Phys. 41, 1667–1681 (1970).
[Crossref]

Kurtz, S. K.

J. Jerphaganon, S. K. Kurtz, “Maker fringes: a detailed comparison of theory and experiment for isotropic and uniaxial crystals,” J. Appl. Phys. 41, 1667–1681 (1970).
[Crossref]

Lewis, A.

Z. Chen, A. Lewis, H. Takei, I. Nebenzahl, “Bacteriorhodopsin oriented in polyvinyl alcohol films as an erasable optical storage medium,” Appl. Opt. 30, 5188–5196 (1991).
[Crossref] [PubMed]

J. Y. Huang, Z. Chen, A. Lewis, “Second-harmonic generation in purple membrane-poly(vinyl alcohol) films: probing the dipolar characteristics of the bacteriorhodopsin chromophore in bR570 and M412,” J. Phys. Chem. 93, 3314–3320 (1989).
[Crossref]

Maker, P. D.

R. W. Terhune, P. D. Maker, C. M. Savage, “Measurements of nonlinear light scattering,” Phys. Rev. Lett. 14, 681–684(1965).
[Crossref]

Muller, U.

M. P. Heyn, R. J. Cherry, U. Muller, “Transient and linear dichroism studies on bacteriorhodopsin: determination of the orientation of the 568 nm All-trans retinal chromophore,” J. Mol. Biol. 117, 607–620 (1977).
[Crossref] [PubMed]

Nebenzahl, I.

Z. Chen, A. Lewis, H. Takei, I. Nebenzahl, “Bacteriorhodopsin oriented in polyvinyl alcohol films as an erasable optical storage medium,” Appl. Opt. 30, 5188–5196 (1991).
[Crossref] [PubMed]

Oesterhelt, D.

D. Oesterhelt, C. Bräuchle, N. Hampp, “Bacteriorhodopsin: a biological material for information processing,” Q. Rev. Biophys. 24, 425–478 (1991).
[Crossref] [PubMed]

Persoon, A.

K. Clays, A. Persoon, “Hyper-Rayleigh scattering in solution,” Phys. Rev. Lett. 66, 2980–2983 (1991).
[Crossref] [PubMed]

Rauch, J. E.

S. J. Cyvin, J. E. Rauch, J. C. Decius, “Theory of hyper-Raman effects (nonlinear inelastic light scattering); selection rules and depolarization ratios for the second-order polarizability,” J. Chem. Phys. 43, 4083–4095 (1965).
[Crossref]

Savage, C. M.

R. W. Terhune, P. D. Maker, C. M. Savage, “Measurements of nonlinear light scattering,” Phys. Rev. Lett. 14, 681–684(1965).
[Crossref]

Shen, Y. R.

Y. R. Shen, The Principles of Nonlinear Optics (Wiley, New York, 1984), Chap. 7, p. 101.

Stoeckenius, W.

W. Stoeckenius, R. A. Bogomolni, “Bacteriorhodopsin and related pigments of halobacteria,” Annu. Rev. Biochem. 52, 587–616 (1982).
[Crossref]

Takei, H.

Z. Chen, A. Lewis, H. Takei, I. Nebenzahl, “Bacteriorhodopsin oriented in polyvinyl alcohol films as an erasable optical storage medium,” Appl. Opt. 30, 5188–5196 (1991).
[Crossref] [PubMed]

Terhune, R. W.

R. W. Terhune, P. D. Maker, C. M. Savage, “Measurements of nonlinear light scattering,” Phys. Rev. Lett. 14, 681–684(1965).
[Crossref]

Williams, D. J.

D. J. Williams, “Large optical nonlinearities,” Angew. Chem. Int. Ed. Engl. 23, 690–703 (1984).
[Crossref]

Zemlin, F.

R. Henderson, J. M. Baldwin, T. A. Ceska, F. Zemlin, E. Beckmann, K. H. Downing, “Model for the structure of bacteriorhodopsin based on high-resolution electron cryomicroscopy,” J. Mol. Biol. 213, 899–929 (1990).
[Crossref] [PubMed]

Zhang, C.-F.

R. R. Birge, C.-F. Zhang, “Two-photon double resonance spectroscopy of bacteriorhodopsin. Assignment of the electronic and dipolar properties of the low-lying 1Ag*-like and 1Bu*-like π, π* states,” J. Chem. Phys. 92, 7178–7195 (1990).
[Crossref]

Angew. Chem. Int. Ed. Engl. (1)

D. J. Williams, “Large optical nonlinearities,” Angew. Chem. Int. Ed. Engl. 23, 690–703 (1984).
[Crossref]

Annu. Rev. Biochem. (1)

W. Stoeckenius, R. A. Bogomolni, “Bacteriorhodopsin and related pigments of halobacteria,” Annu. Rev. Biochem. 52, 587–616 (1982).
[Crossref]

Annu. Rev. Phys. Chem. (1)

R. R. Birge, “Photophysics and molecular electronic applications of the rhodopsins,” Annu. Rev. Phys. Chem. 41, 683–733 (1990).
[Crossref] [PubMed]

Appl. Opt. (1)

Z. Chen, A. Lewis, H. Takei, I. Nebenzahl, “Bacteriorhodopsin oriented in polyvinyl alcohol films as an erasable optical storage medium,” Appl. Opt. 30, 5188–5196 (1991).
[Crossref] [PubMed]

Computer (1)

R. R. Birge, “Protein-based optical computing and memories,” Computer 25, 56–67 (1992).
[Crossref]

J. Mol. Biol. (1)

M. P. Heyn, R. J. Cherry, U. Muller, “Transient and linear dichroism studies on bacteriorhodopsin: determination of the orientation of the 568 nm All-trans retinal chromophore,” J. Mol. Biol. 117, 607–620 (1977).
[Crossref] [PubMed]

J. Appl. Phys. (1)

J. Jerphaganon, S. K. Kurtz, “Maker fringes: a detailed comparison of theory and experiment for isotropic and uniaxial crystals,” J. Appl. Phys. 41, 1667–1681 (1970).
[Crossref]

J. Chem. Phys. (2)

R. R. Birge, C.-F. Zhang, “Two-photon double resonance spectroscopy of bacteriorhodopsin. Assignment of the electronic and dipolar properties of the low-lying 1Ag*-like and 1Bu*-like π, π* states,” J. Chem. Phys. 92, 7178–7195 (1990).
[Crossref]

S. J. Cyvin, J. E. Rauch, J. C. Decius, “Theory of hyper-Raman effects (nonlinear inelastic light scattering); selection rules and depolarization ratios for the second-order polarizability,” J. Chem. Phys. 43, 4083–4095 (1965).
[Crossref]

J. Mol. Biol. (1)

R. Henderson, J. M. Baldwin, T. A. Ceska, F. Zemlin, E. Beckmann, K. H. Downing, “Model for the structure of bacteriorhodopsin based on high-resolution electron cryomicroscopy,” J. Mol. Biol. 213, 899–929 (1990).
[Crossref] [PubMed]

J. Phys. Chem. (1)

J. Y. Huang, Z. Chen, A. Lewis, “Second-harmonic generation in purple membrane-poly(vinyl alcohol) films: probing the dipolar characteristics of the bacteriorhodopsin chromophore in bR570 and M412,” J. Phys. Chem. 93, 3314–3320 (1989).
[Crossref]

Phys. Rev. Lett. (2)

R. W. Terhune, P. D. Maker, C. M. Savage, “Measurements of nonlinear light scattering,” Phys. Rev. Lett. 14, 681–684(1965).
[Crossref]

K. Clays, A. Persoon, “Hyper-Rayleigh scattering in solution,” Phys. Rev. Lett. 66, 2980–2983 (1991).
[Crossref] [PubMed]

Prep. Biochem. (1)

B. M. Beecher, J. Y. Cassim, “Improved isolation procedures for the purple-membrane of Halobacterium halbium,” Prep. Biochem. 5, 161–178 (1975).
[Crossref]

Q. Rev. Biophys. (1)

D. Oesterhelt, C. Bräuchle, N. Hampp, “Bacteriorhodopsin: a biological material for information processing,” Q. Rev. Biophys. 24, 425–478 (1991).
[Crossref] [PubMed]

Other (2)

H. Goldstein, Classical Mechanics, 2nd ed. (Addison-Wesley, Reading, Mass.1980), Chap. 4, p. 147.

Y. R. Shen, The Principles of Nonlinear Optics (Wiley, New York, 1984), Chap. 7, p. 101.

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Figures (6)

Fig. 1
Fig. 1

Experimental arrangement for making the SHG measurements. Suitable filters isolate the frequency-doubled light from the incidence light, and the SHG signal is detected by the PMT and is signal averaged (see text).

Fig. 2
Fig. 2

Power spectrum of the scattered light. The interference filter directly in front of the PMT was replaced by a high-throughput monochromator for this measurement.

Fig. 3
Fig. 3

Intensity of the frequency-doubled light plotted versus the square of the incident (1064-nm) light intensity. The straight line demonstrates that the scattered light is due to a second-order effect.

Fig. 4
Fig. 4

Angular dependence of the scattered light intensity. Rayleight light scattering is due to dipole radiation and should have a cos2 θ behavior. The deviation is due to interference from different BR molecules in a particular PM. This is evidence for the coherent light scattering from individual BR molecules in the PM.

Fig. 5
Fig. 5

Intensity of the scattered light is proportional to the PM concentration at constant incident light intensity. This shows that the light scattered by each PM is incoherent. The light intensity (both scattered 532-nm light and incident 1064-nm light) is corrected for absorption by the PM suspension.

Figure 6
Figure 6

Calculated ratio of p-polarized hyper-Rayleigh scattered light to s-polarized scattered light versus the angle of the retinal with respect to the plane of the PM. The measured ratio is 3.5 ± 0.5, leading to an angle of 27° for the inclination of the retinal axis. This is in good agreement with other measurements (see text).

Equations (6)

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

μ i = α i j E j + β i j k E j E k .
c 112 = c 211 = c 121 = - 0.75 β cos 3 γ , c 113 = c 131 = c 311 = 1.5 β sin γ cos 2 γ , c 222 = 0.75 β cos 3 γ , c 223 = c 232 = c 322 = 1.5 β sin γ cos 2 γ , c 333 = 3 β sin 3 γ .
E = ( n 3 ) exp ( i k r ) r [ ( u × k ) × k ] ,
I i 2 ω = A b T PM c 8 π ( E ) 2 = A b T PM 8 π c ( n 3 ) 2 k 4 r 2 ( β i 22 ) 2 ( I ω ) 2 ,
P Q 2 ω = 512 d 2 A ( I ω ) 2 π 3 F ( n ω , n 2 ω ) / c ,
P Q 2 ω / P PM 2 ω = 36 A d 2 λ 4 r 2 F ( n ω , n 2 ω ) / A b T PM A PMT π 4 n 2 N β 222 2 .

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