Abstract

The photorefractive effect is demonstrated in a single crystal of ferroelectric Ba-modified Ce-doped (SrCa)2NaNb5O15, for the first time to our knowledge. From two-beam coupling the gain and the time response were found. The maximum intensity gain with ordinary light at a 488-nm wavelength was Γ − 10 cm−1, while the photorefractive time response was determined to be 1.85 s at an incident intensity of I = 1W/cm2. The phase shift of the index gratings from the intensity grating was found to vary with angle from ~28° at a 79° crossing angle to 55° at a 13° crossing angle. The dark decay was measured, and the two-beam coupling intensity gain was found to oscillate with time. The oscillation period and the dc component were found to depend on the crossing angle. Beam fanning was observed with a transmitted beam reduction of 1.61 optical density and a time response of τ ~ 5.6 s at I = 230 mW/cm2. Finally, anisotropic conical diffraction was observed in the form of a partial ring pattern about the transmitted (extraordinary-polarized) incident beam in a direction opposite the beam fan. When the dependence of the angle between the ring and the transmitted beam is fitted to the change in the incident angle, it is possible to measure the birefringence, which was neno = −0.051.

© 1994 Optical Society of America

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  1. J.-P. Huignard and A. Marrakchi, Opt. Commun. 38, 249 (1981).
    [Crossref]
  2. J. O. White and A. Yariv, Appl. Phys. Lett. 37, 5 (1980).
    [Crossref]
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    [Crossref] [PubMed]
  4. P. Günter and J.-P. Huignard, eds., Photorefractive Materials and Their Applications I and II, Vols. 61 and 62 of Topics in Applied Physics (Springer-Verlag, Berlin, 1988, 1989).
    [Crossref]
  5. M. Cronin-Golomb and A. Yariv, J. Appl. Phys. 57, 4906 (1985).
    [Crossref]
  6. R. R. Neurgaonkar, W. K. Cory, J. R. Oliver, E. J. Sharp, G. L. Wood, M. J. Miller, and G. J. Salamo, Mat. Res. Bull. 23, 1459 (1988).
    [Crossref]
  7. M. E. Lines and A. M. Glass, Principles and Applications of Ferroelectrics and Related Materials (Clarendon, Oxford, 1977).
  8. S. Ducharme, J. Feinberg, and R. R. Neurgaonkar, IEEE J. Quantum Electron. QE-23, 2116 (1987).
    [Crossref]
  9. F. P. Strohkendl, J. M. C. Jonathan, and R. W. Hellwarth, Opt. Lett. 11, 312 (1986).
    [Crossref]
  10. G. L. Wood, W. W. Clark, M. J. Miller, E. J. Sharp, G. J. Salamo, and R. R. Neurgaonkar, IEEE J. Quantum Electron. QE-23, 2126 (1987).
    [Crossref]
  11. D. Mahgerefteh and J. Feinberg, Phys. Rev. Lett. 64, 2195 (1990).
    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref]

1992 (1)

1991 (4)

D. Fluck, P. Amrhein, and P. Günter, J. Opt. Soc. Am. B 8, 2196 (1991).
[Crossref]

C. Gu, J. Hong, H. Li, D. Psaltis, and P. Yeh, J. Appl. Phys. 69, 1167 (1991).
[Crossref]

J. H. Hong, F. Vachss, S. Campbell, and P. Yeh, J. Appl. Phys. 69, 2835 (1991).
[Crossref]

B. Sturman, J. Opt. Soc. Am. B 8, 1333 (1991).
[Crossref]

1990 (2)

D. Mahgerefteh and J. Feinberg, Phys. Rev. Lett. 64, 2195 (1990).
[Crossref] [PubMed]

W. Krolikowski, M. Cronin-Golomb, and B. S. Chen, Appl. Phys. Lett. 57, 7 (1990).
[Crossref]

1989 (2)

J. Otten, A. Ozols, M. Reinfelde, and K. H. Ringhofer, Opt. Commun. 72, 175 (1989).
[Crossref]

L. B. Au and L. Solymar, Appl. Phys. 49, 339 (1989).
[Crossref]

1988 (2)

R. R. Neurgaonkar, W. K. Cory, J. R. Oliver, E. J. Sharp, G. L. Wood, M. J. Miller, and G. J. Salamo, Mat. Res. Bull. 23, 1459 (1988).
[Crossref]

P. Yeh, A. E. T. Chiou, and J. Hong, Appl. Opt. 27, 2093 (1988).
[Crossref] [PubMed]

1987 (2)

S. Ducharme, J. Feinberg, and R. R. Neurgaonkar, IEEE J. Quantum Electron. QE-23, 2116 (1987).
[Crossref]

G. L. Wood, W. W. Clark, M. J. Miller, E. J. Sharp, G. J. Salamo, and R. R. Neurgaonkar, IEEE J. Quantum Electron. QE-23, 2126 (1987).
[Crossref]

1986 (2)

1985 (1)

M. Cronin-Golomb and A. Yariv, J. Appl. Phys. 57, 4906 (1985).
[Crossref]

1983 (1)

G. C. Valley and M. B. Klein, Opt. Eng. 22, 704 (1983).
[Crossref]

1982 (1)

S. G. Odoulov, JETP Lett. 35, 10 (1982).

1981 (1)

J.-P. Huignard and A. Marrakchi, Opt. Commun. 38, 249 (1981).
[Crossref]

1980 (1)

J. O. White and A. Yariv, Appl. Phys. Lett. 37, 5 (1980).
[Crossref]

1979 (1)

N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, and V. L. Vinetskii, Ferroelectrics,  22, 949, 961 (1979).
[Crossref]

Amrhein, P.

D. Fluck, P. Amrhein, and P. Günter, J. Opt. Soc. Am. B 8, 2196 (1991).
[Crossref]

M. Z. Zha, P. Amrhein, and P. Günter, Digest of Topical Meeting on Photorefractive Materials, Effects, and Devices (Optical Society of America, Washington, D.C., 1990), p. 291.

Au, L. B.

L. B. Au and L. Solymar, Appl. Phys. 49, 339 (1989).
[Crossref]

Bacher, G. D.

Campbell, S.

J. H. Hong, F. Vachss, S. Campbell, and P. Yeh, J. Appl. Phys. 69, 2835 (1991).
[Crossref]

Chen, B. S.

W. Krolikowski, M. Cronin-Golomb, and B. S. Chen, Appl. Phys. Lett. 57, 7 (1990).
[Crossref]

Chiou, A. E. T.

Clark, W. W.

G. L. Wood, W. W. Clark, M. J. Miller, E. J. Sharp, G. J. Salamo, and R. R. Neurgaonkar, IEEE J. Quantum Electron. QE-23, 2126 (1987).
[Crossref]

Cory, W. K.

R. R. Neurgaonkar, W. K. Cory, J. R. Oliver, E. J. Sharp, G. L. Wood, M. J. Miller, and G. J. Salamo, Mat. Res. Bull. 23, 1459 (1988).
[Crossref]

Cronin-Golomb, M.

W. Krolikowski, M. Cronin-Golomb, and B. S. Chen, Appl. Phys. Lett. 57, 7 (1990).
[Crossref]

M. Cronin-Golomb and A. Yariv, J. Appl. Phys. 57, 4906 (1985).
[Crossref]

Cudney, R. S.

Ducharme, S.

S. Ducharme, J. Feinberg, and R. R. Neurgaonkar, IEEE J. Quantum Electron. QE-23, 2116 (1987).
[Crossref]

Ewbank, M. S.

M. S. Ewbank, P. Yeh, and J. Feinberg, Opt. Commun. 59, 423 (1986).
[Crossref]

Feinberg, J.

R. S. Cudney, G. D. Bacher, R. M. Pierce, and J. Feinberg, Opt. Lett. 17, 67 (1992).
[Crossref] [PubMed]

D. Mahgerefteh and J. Feinberg, Phys. Rev. Lett. 64, 2195 (1990).
[Crossref] [PubMed]

S. Ducharme, J. Feinberg, and R. R. Neurgaonkar, IEEE J. Quantum Electron. QE-23, 2116 (1987).
[Crossref]

M. S. Ewbank, P. Yeh, and J. Feinberg, Opt. Commun. 59, 423 (1986).
[Crossref]

Fluck, D.

Glass, A. M.

M. E. Lines and A. M. Glass, Principles and Applications of Ferroelectrics and Related Materials (Clarendon, Oxford, 1977).

Gu, C.

C. Gu, J. Hong, H. Li, D. Psaltis, and P. Yeh, J. Appl. Phys. 69, 1167 (1991).
[Crossref]

Günter, P.

D. Fluck, P. Amrhein, and P. Günter, J. Opt. Soc. Am. B 8, 2196 (1991).
[Crossref]

M. Z. Zha, P. Amrhein, and P. Günter, Digest of Topical Meeting on Photorefractive Materials, Effects, and Devices (Optical Society of America, Washington, D.C., 1990), p. 291.

Hellwarth, R. W.

Hong, J.

C. Gu, J. Hong, H. Li, D. Psaltis, and P. Yeh, J. Appl. Phys. 69, 1167 (1991).
[Crossref]

P. Yeh, A. E. T. Chiou, and J. Hong, Appl. Opt. 27, 2093 (1988).
[Crossref] [PubMed]

Hong, J. H.

J. H. Hong, F. Vachss, S. Campbell, and P. Yeh, J. Appl. Phys. 69, 2835 (1991).
[Crossref]

Huignard, J.-P.

J.-P. Huignard and A. Marrakchi, Opt. Commun. 38, 249 (1981).
[Crossref]

Jonathan, J. M. C.

Klein, M. B.

G. C. Valley and M. B. Klein, Opt. Eng. 22, 704 (1983).
[Crossref]

Krolikowski, W.

W. Krolikowski, M. Cronin-Golomb, and B. S. Chen, Appl. Phys. Lett. 57, 7 (1990).
[Crossref]

Kukhtarev, N. V.

N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, and V. L. Vinetskii, Ferroelectrics,  22, 949, 961 (1979).
[Crossref]

Li, H.

C. Gu, J. Hong, H. Li, D. Psaltis, and P. Yeh, J. Appl. Phys. 69, 1167 (1991).
[Crossref]

Lines, M. E.

M. E. Lines and A. M. Glass, Principles and Applications of Ferroelectrics and Related Materials (Clarendon, Oxford, 1977).

Mahgerefteh, D.

D. Mahgerefteh and J. Feinberg, Phys. Rev. Lett. 64, 2195 (1990).
[Crossref] [PubMed]

Markov, V. B.

N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, and V. L. Vinetskii, Ferroelectrics,  22, 949, 961 (1979).
[Crossref]

Marrakchi, A.

J.-P. Huignard and A. Marrakchi, Opt. Commun. 38, 249 (1981).
[Crossref]

Miller, M. J.

R. R. Neurgaonkar, W. K. Cory, J. R. Oliver, E. J. Sharp, G. L. Wood, M. J. Miller, and G. J. Salamo, Mat. Res. Bull. 23, 1459 (1988).
[Crossref]

G. L. Wood, W. W. Clark, M. J. Miller, E. J. Sharp, G. J. Salamo, and R. R. Neurgaonkar, IEEE J. Quantum Electron. QE-23, 2126 (1987).
[Crossref]

Neurgaonkar, R. R.

R. R. Neurgaonkar, W. K. Cory, J. R. Oliver, E. J. Sharp, G. L. Wood, M. J. Miller, and G. J. Salamo, Mat. Res. Bull. 23, 1459 (1988).
[Crossref]

S. Ducharme, J. Feinberg, and R. R. Neurgaonkar, IEEE J. Quantum Electron. QE-23, 2116 (1987).
[Crossref]

G. L. Wood, W. W. Clark, M. J. Miller, E. J. Sharp, G. J. Salamo, and R. R. Neurgaonkar, IEEE J. Quantum Electron. QE-23, 2126 (1987).
[Crossref]

Odoulov, S. G.

S. G. Odoulov, JETP Lett. 35, 10 (1982).

Odulov, S. G.

N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, and V. L. Vinetskii, Ferroelectrics,  22, 949, 961 (1979).
[Crossref]

Oliver, J. R.

R. R. Neurgaonkar, W. K. Cory, J. R. Oliver, E. J. Sharp, G. L. Wood, M. J. Miller, and G. J. Salamo, Mat. Res. Bull. 23, 1459 (1988).
[Crossref]

Otten, J.

J. Otten, A. Ozols, M. Reinfelde, and K. H. Ringhofer, Opt. Commun. 72, 175 (1989).
[Crossref]

Ozols, A.

J. Otten, A. Ozols, M. Reinfelde, and K. H. Ringhofer, Opt. Commun. 72, 175 (1989).
[Crossref]

Pierce, R. M.

Psaltis, D.

C. Gu, J. Hong, H. Li, D. Psaltis, and P. Yeh, J. Appl. Phys. 69, 1167 (1991).
[Crossref]

Reinfelde, M.

J. Otten, A. Ozols, M. Reinfelde, and K. H. Ringhofer, Opt. Commun. 72, 175 (1989).
[Crossref]

Ringhofer, K. H.

J. Otten, A. Ozols, M. Reinfelde, and K. H. Ringhofer, Opt. Commun. 72, 175 (1989).
[Crossref]

Salamo, G. J.

R. R. Neurgaonkar, W. K. Cory, J. R. Oliver, E. J. Sharp, G. L. Wood, M. J. Miller, and G. J. Salamo, Mat. Res. Bull. 23, 1459 (1988).
[Crossref]

G. L. Wood, W. W. Clark, M. J. Miller, E. J. Sharp, G. J. Salamo, and R. R. Neurgaonkar, IEEE J. Quantum Electron. QE-23, 2126 (1987).
[Crossref]

Sharp, E. J.

R. R. Neurgaonkar, W. K. Cory, J. R. Oliver, E. J. Sharp, G. L. Wood, M. J. Miller, and G. J. Salamo, Mat. Res. Bull. 23, 1459 (1988).
[Crossref]

G. L. Wood, W. W. Clark, M. J. Miller, E. J. Sharp, G. J. Salamo, and R. R. Neurgaonkar, IEEE J. Quantum Electron. QE-23, 2126 (1987).
[Crossref]

Solymar, L.

L. B. Au and L. Solymar, Appl. Phys. 49, 339 (1989).
[Crossref]

Soskin, M. S.

N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, and V. L. Vinetskii, Ferroelectrics,  22, 949, 961 (1979).
[Crossref]

Strohkendl, F. P.

Sturman, B.

Vachss, F.

J. H. Hong, F. Vachss, S. Campbell, and P. Yeh, J. Appl. Phys. 69, 2835 (1991).
[Crossref]

Valley, G. C.

G. C. Valley and M. B. Klein, Opt. Eng. 22, 704 (1983).
[Crossref]

Vinetskii, V. L.

N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, and V. L. Vinetskii, Ferroelectrics,  22, 949, 961 (1979).
[Crossref]

White, J. O.

J. O. White and A. Yariv, Appl. Phys. Lett. 37, 5 (1980).
[Crossref]

Wood, G. L.

R. R. Neurgaonkar, W. K. Cory, J. R. Oliver, E. J. Sharp, G. L. Wood, M. J. Miller, and G. J. Salamo, Mat. Res. Bull. 23, 1459 (1988).
[Crossref]

G. L. Wood, W. W. Clark, M. J. Miller, E. J. Sharp, G. J. Salamo, and R. R. Neurgaonkar, IEEE J. Quantum Electron. QE-23, 2126 (1987).
[Crossref]

Yariv, A.

M. Cronin-Golomb and A. Yariv, J. Appl. Phys. 57, 4906 (1985).
[Crossref]

J. O. White and A. Yariv, Appl. Phys. Lett. 37, 5 (1980).
[Crossref]

Yeh, P.

J. H. Hong, F. Vachss, S. Campbell, and P. Yeh, J. Appl. Phys. 69, 2835 (1991).
[Crossref]

C. Gu, J. Hong, H. Li, D. Psaltis, and P. Yeh, J. Appl. Phys. 69, 1167 (1991).
[Crossref]

P. Yeh, A. E. T. Chiou, and J. Hong, Appl. Opt. 27, 2093 (1988).
[Crossref] [PubMed]

M. S. Ewbank, P. Yeh, and J. Feinberg, Opt. Commun. 59, 423 (1986).
[Crossref]

Zha, M. Z.

M. Z. Zha, P. Amrhein, and P. Günter, Digest of Topical Meeting on Photorefractive Materials, Effects, and Devices (Optical Society of America, Washington, D.C., 1990), p. 291.

Appl. Opt. (1)

Appl. Phys. (1)

L. B. Au and L. Solymar, Appl. Phys. 49, 339 (1989).
[Crossref]

Appl. Phys. Lett. (2)

W. Krolikowski, M. Cronin-Golomb, and B. S. Chen, Appl. Phys. Lett. 57, 7 (1990).
[Crossref]

J. O. White and A. Yariv, Appl. Phys. Lett. 37, 5 (1980).
[Crossref]

Ferroelectrics (1)

N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, and V. L. Vinetskii, Ferroelectrics,  22, 949, 961 (1979).
[Crossref]

IEEE J. Quantum Electron. (2)

S. Ducharme, J. Feinberg, and R. R. Neurgaonkar, IEEE J. Quantum Electron. QE-23, 2116 (1987).
[Crossref]

G. L. Wood, W. W. Clark, M. J. Miller, E. J. Sharp, G. J. Salamo, and R. R. Neurgaonkar, IEEE J. Quantum Electron. QE-23, 2126 (1987).
[Crossref]

J. Appl. Phys. (3)

C. Gu, J. Hong, H. Li, D. Psaltis, and P. Yeh, J. Appl. Phys. 69, 1167 (1991).
[Crossref]

J. H. Hong, F. Vachss, S. Campbell, and P. Yeh, J. Appl. Phys. 69, 2835 (1991).
[Crossref]

M. Cronin-Golomb and A. Yariv, J. Appl. Phys. 57, 4906 (1985).
[Crossref]

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

JETP Lett. (1)

S. G. Odoulov, JETP Lett. 35, 10 (1982).

Mat. Res. Bull. (1)

R. R. Neurgaonkar, W. K. Cory, J. R. Oliver, E. J. Sharp, G. L. Wood, M. J. Miller, and G. J. Salamo, Mat. Res. Bull. 23, 1459 (1988).
[Crossref]

Opt. Commun. (3)

J.-P. Huignard and A. Marrakchi, Opt. Commun. 38, 249 (1981).
[Crossref]

J. Otten, A. Ozols, M. Reinfelde, and K. H. Ringhofer, Opt. Commun. 72, 175 (1989).
[Crossref]

M. S. Ewbank, P. Yeh, and J. Feinberg, Opt. Commun. 59, 423 (1986).
[Crossref]

Opt. Eng. (1)

G. C. Valley and M. B. Klein, Opt. Eng. 22, 704 (1983).
[Crossref]

Opt. Lett. (2)

Phys. Rev. Lett. (1)

D. Mahgerefteh and J. Feinberg, Phys. Rev. Lett. 64, 2195 (1990).
[Crossref] [PubMed]

Other (3)

M. E. Lines and A. M. Glass, Principles and Applications of Ferroelectrics and Related Materials (Clarendon, Oxford, 1977).

P. Günter and J.-P. Huignard, eds., Photorefractive Materials and Their Applications I and II, Vols. 61 and 62 of Topics in Applied Physics (Springer-Verlag, Berlin, 1988, 1989).
[Crossref]

M. Z. Zha, P. Amrhein, and P. Günter, Digest of Topical Meeting on Photorefractive Materials, Effects, and Devices (Optical Society of America, Washington, D.C., 1990), p. 291.

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Figures (13)

Fig. 1
Fig. 1

Transmission of SCNN versus wavelength in nanometers over the visible portion of the spectrum.

Fig. 2
Fig. 2

Experimental arrangement for measuring two-beam coupling. The 488-nm Ar+ -ion line has ordinary polarization, and both beams have the same incident angle, so that the space-charge field is along the c axis. Pol. rot., polarization rotator; E’s, ordinary electric field; M’s, mirrors; D, detector.

Fig. 3
Fig. 3

Gain versus the full external crossing angle (FECA). The filled circles represent the experimental data points, and the solid curve is the theoretical fit with r13ξ = 26 pm/V and Neff = 27 × 1016 cm−3.

Fig. 4
Fig. 4

Gain and tan Φ versus the FECA. The experimental gain data are shown as filled circles, while the experimentally determined tangent phase angle (right y axis) is shown as open triangles. The theoretical fit to the gain with an internal unmodulated dc field is shown as the solid curve, and the dashed curve shows the subsequent tan(Φ) fit.

Fig. 5
Fig. 5

Gain and tan(Φ) versus FECA. The curves show the best theoretical fits with an intensity-modulated photovoltaic current. This is the smallest E0 and Neff that can fit the gain. Larger values of both can still fit the gain but do not change the tan(Φ) fit by much.

Fig. 6
Fig. 6

Gain and tan(Φ) versus FECA. The theoretical curves are derived for a modulated photovoltaic β coefficient. These are the smallest E0 and Neff values that fit both data sets.

Fig. 7
Fig. 7

Gain versus the time the gratings have been in the dark, read out by two weak coupling probe-pulse beams. The oscillations are fitted to a cosine function with the amplitude A, period T, and dc offset DC. These oscillations were measured at four different FECA’s with these results: (a) FECA = 2θ = 13°, A = 2.37, T = 191.7 min, DC = 0.235; (b) FECA = 2θ = 18°, A = 3.85, T = 172.5 min, DC = 0.139; (c) FECA = 2θ = 44.5°, A = 5.0, T = 145 min, DC = 2.5; (d) FECA = 2θ = 79°, A = 3.89, T = 553 min, DC = 7.5.

Fig. 8
Fig. 8

Measured dark periods at the four crossing angles versus the FECA (filled circles). The solid curve is a fit to Eq. (18) for the space-charge period oscillation with a, b, and c as fitting parameters. The constant c is related to the internal electric field.

Fig. 9
Fig. 9

Amplitude of the dark oscillations versus the FECA (filled circles). The solid line serves only to connect the points.

Fig. 10
Fig. 10

Experimental arrangement for measuring the transmitted beam during beam fanning. Symbols are as in Fig. 2; S, shutter; A, aperture.

Fig. 11
Fig. 11

Cross section of the index ellipsoid. The ordinary circle lies outside the extraordinary ellipse. The extraordinary polarized incident beam ki and fan beams kf (in the shaded region) write many gratings with one k that equals the magnitude (and opposite direction) of a grating written by ki and the anisotropic scattered beam k0. The internal incident beam propagates at θi to the normal of the crystal face, and the ring is formed at αi.

Fig. 12
Fig. 12

Wave-vector geometry corresponding to equation 2kikf = k0. The internal cone angle αi and the incident angle θi are shown relative to the crystal axes, and θi is fixed. kf rotates about 2ki and trace out an ellipse. k0 has a fixed magnitude for all angles, and the cone angle is determined for the one point at which k0 touches the kf ellipse.

Fig. 13
Fig. 13

Observed ring angle versus the incident angle in degrees. The filled circles mark experimental data points, and the solid curve represents the theoretical fit, with no and ne given.

Equations (21)

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

I c / I u ~ exp ( Γ L ) ,
Γ = k n o 3 r 13 Im ( E sc ) / [ m cos ( θ i ) ] ,
E sc ( t ) = [ i ξ ( k g ) k B T / e ] [ m / ( 1 + I d d / I ) ] × { k g [ 1 + ( k g / k 0 ) 2 ] - 1 } [ 1 - exp ( - t / τ ) ] ,
τ - 1 = I / A + σ d / 0 ,
1 / A = ( e / ω 0 ) [ 1 + ( k g / k 0 ) 2 ] × { s ( N D - N A ) μ / [ γ N A ( 1 + ( k g / K ) 2 ) ] } ,
η ( t ) = L d ( t ) / I t = ( 1 - R ) exp ( - α L ) sin 2 ξ d exp [ - ( t / τ ) ] } ,
tan Φ = Γ / [ ( 4 ξ d / m L ) 2 - Γ 2 ] 1 / 2 .
Im ( E sc ) = m ξ ( k g ) E q [ E d ( E d + E q ) + E 0 2 ] × [ ( E d + E q ) 2 + E 0 2 ] - 1 ,
Re ( E sc ) = - m ξ ( k g ) E q 2 E 0 [ ( E d + E q ) 2 + E 0 2 ] - 1 ,
tan Φ = Im ( E s c ) / Re ( E sc ) = - { E q [ E d ( E d + E q ) + E 0 2 ] } / E q 2 E 0 ,
j z = σ E 0 + k B T μ ( d n / d z ) - β ( E 1 + E 2 ) ( E 1 + E 2 ) * ,
j = j 0 + j 1 exp ( i k g z ) , n = n 0 + n 1 exp ( i k g z ) , E = E 0 + E 1 exp ( i k g z ) ,
j 0 = e μ n o E 0 - β I 0 = 0 ;
Im ( E sc ) = m ξ ( k g ) E q [ E d ( E q + E d ) ] [ ( E d + E q ) 2 + E 0 2 ] - 1 ,
tan Φ = ( E d + E q ) / E 0 .
Im ( E sc ) = m ξ ( k g ) E q [ E d ( E q + E d ) - E β 1 E 0 ] × [ ( E d + E q ) + E 0 2 ] - 1 , tan Φ = [ E d ( E q + E d ) - E β 1 E 0 ] × [ E 0 E d + E β 1 ( E d + E q ) ] - 1 ,
l * = Φ v Γ - 1 Σ ,
T = 2 π τ di [ ( 1 + τ R / τ D ) 2 + ( τ R / τ E ) 2 ] [ ( τ R / τ E ) ( τ di / τ I - 1 ) ] - 1 ,
T = a { [ 1 + b sin 2 ( Θ / 2 ) ] 2 + c sin 2 ( Θ / 2 ) } sin ( Θ / 2 ) ,
( k f · x ^ ) 2 / k 2 n e 2 + ( k f · z ^ ) 2 / k 2 n o 2 = 1 ,
cos ( α i ) = 2 n 1 ( θ ) / [ n o + n 1 ( θ ) ] ,

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