Abstract

We report a mathematical formulation that successfully describes the holographic photocurrent that is produced, in strongly absorbing photorefractive materials, by the action of a pattern of interference fringes of light vibrating sinusoidally with large amplitude. The large vibrating amplitude produces a sensible enhancement of the photocurrent signal and in this way facilitates measurements. We also show that taking account of the bulk light absorption of the sample is essential for adequately describing the experiment. We measure the first temporal harmonic of the photocurrent, without an externally applied field, as a function of the amplitude and the temporal frequency of the vibrating pattern of fringes and show that these data fit our theoretical model well. From this fit we are able to determine some material parameters for pure and doped photorefractive Bi12TiO20 crystals.

© 2002 Optical Society of America

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References

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  1. M. P. Petrov, I. A. Sokolov, S. I. Stepanov, and G. S. Trofimov, “Non-steady-state photo-electromotive-force induced by dynamic gratings in partially compensated photoconductors,” J. Appl. Phys. 68, 2216–2225 (1990).
    [CrossRef]
  2. S. I. Stepanov, “Photo-electromotive-force effect in semiconductors,” in Semiconductor Devices, H. S. Nalwa, ed., Vol. 2 of Handbook of Advanced Electronic and Photonic Materials and Devices (Academic, San Diego, Calif., 2001), Chap. 6, pp. 205–272.
  3. S. I. Stepanov and G. S. Trofimov, “Transient emf in crystals having ambipolar photoconductivity,” Sov. Phys. Solid State 31, 49–50 (1989).
  4. N. Korneev, D. Mayorga, S. Stepanov, H. Veenhuis, K. Buse, C. Kuper, H. Hesse, and E. Krätzig, “Holographic and non-steady-state photocurrent characterization of photorefractive barium-calcium titanate,” Opt. Commun. 160, 98–102 (1999).
    [CrossRef]
  5. M. A. Bryushinin, G. B. Dubrovsky, and L. A. Sokolov, “Non-steady-state photocurrents in SnS2 crystals,” Appl. Phys. B 68, 871–875 (1999).
    [CrossRef]
  6. R. Bittner, K. Meerholz, and S. Stepanov, “Nonsteady-state photo-EMF effect in photorefractive polymers,” Appl. Phys. Lett. 74, 3723–3725 (1999).
    [CrossRef]
  7. N. A. Korneev and S. Stepanov, “Non-steady-state photoelectromotive force in semiconductor crystals with high light absorption,” J. Appl. Phys. 74, 2736–2741 (1993).
    [CrossRef]
  8. G. S. Trofimov and S. I. Stepanov, “Time-dependent holographic currents in photorefractive crystals,” Sov. Phys. Solid State 28, 1559–1562 (1986).
  9. S. Stepanov and P. Petrov, Photorefractive Materials and Their Applications I, P. Günter and J.-P. Huignard, eds., Vol. 61 of Topics in Applied Physics (Springer-Verlag, Berlin, 1988), Chap. 9, pp. 263–289.
    [CrossRef]
  10. A. A. Freschi, P. M. Garcia, and J. Frejlich, “Charge-carriers diffusion length in photorefractive crystals computed from the initial hologram phase shift,” Appl. Phys. Lett. 71, 2427–2429 (1997).
    [CrossRef]
  11. L. Mosquera, I. de Oliveira, J. Frejlich, A. C. Hernandes, S. Lanfredi, and J. F. Carvalho, “Dark conductivity, photoconductivity and light-induced absorption in photorefractive sillenite crystals,” J. Appl. Phys. 90, 2635–2641 (2001).
    [CrossRef]
  12. I. de Oliveira and J. Frejlich, “Photorefractive running hologram for materials characterization,” J. Opt. Soc. Am. B 18, 291–297 (2001).
    [CrossRef]
  13. M. C. Barbosa, L. Mosquera, and J. Frejlich, “Speed and diffraction efficiency in feedback-controlled running holograms for photorefractive crystal characterization,” Appl. Phys. B 72, 717–721 (2001).
    [CrossRef]
  14. I. de Oliveira and J. Frejlich, “Dielectric relaxation time measurement in absorbing photorefractive materials,” Opt. Commun. 178, 251–255 (2000).
    [CrossRef]
  15. B. C. Grabmaier and R. Oberschmid, “Properties of pure and doped Bi12GeO20 and Bi12SiO20 crystals,” Phys. Status Solidi 96, 199–211 (1986).
    [CrossRef]

2001 (3)

L. Mosquera, I. de Oliveira, J. Frejlich, A. C. Hernandes, S. Lanfredi, and J. F. Carvalho, “Dark conductivity, photoconductivity and light-induced absorption in photorefractive sillenite crystals,” J. Appl. Phys. 90, 2635–2641 (2001).
[CrossRef]

M. C. Barbosa, L. Mosquera, and J. Frejlich, “Speed and diffraction efficiency in feedback-controlled running holograms for photorefractive crystal characterization,” Appl. Phys. B 72, 717–721 (2001).
[CrossRef]

I. de Oliveira and J. Frejlich, “Photorefractive running hologram for materials characterization,” J. Opt. Soc. Am. B 18, 291–297 (2001).
[CrossRef]

2000 (1)

I. de Oliveira and J. Frejlich, “Dielectric relaxation time measurement in absorbing photorefractive materials,” Opt. Commun. 178, 251–255 (2000).
[CrossRef]

1999 (3)

N. Korneev, D. Mayorga, S. Stepanov, H. Veenhuis, K. Buse, C. Kuper, H. Hesse, and E. Krätzig, “Holographic and non-steady-state photocurrent characterization of photorefractive barium-calcium titanate,” Opt. Commun. 160, 98–102 (1999).
[CrossRef]

M. A. Bryushinin, G. B. Dubrovsky, and L. A. Sokolov, “Non-steady-state photocurrents in SnS2 crystals,” Appl. Phys. B 68, 871–875 (1999).
[CrossRef]

R. Bittner, K. Meerholz, and S. Stepanov, “Nonsteady-state photo-EMF effect in photorefractive polymers,” Appl. Phys. Lett. 74, 3723–3725 (1999).
[CrossRef]

1997 (1)

A. A. Freschi, P. M. Garcia, and J. Frejlich, “Charge-carriers diffusion length in photorefractive crystals computed from the initial hologram phase shift,” Appl. Phys. Lett. 71, 2427–2429 (1997).
[CrossRef]

1993 (1)

N. A. Korneev and S. Stepanov, “Non-steady-state photoelectromotive force in semiconductor crystals with high light absorption,” J. Appl. Phys. 74, 2736–2741 (1993).
[CrossRef]

1990 (1)

M. P. Petrov, I. A. Sokolov, S. I. Stepanov, and G. S. Trofimov, “Non-steady-state photo-electromotive-force induced by dynamic gratings in partially compensated photoconductors,” J. Appl. Phys. 68, 2216–2225 (1990).
[CrossRef]

1989 (1)

S. I. Stepanov and G. S. Trofimov, “Transient emf in crystals having ambipolar photoconductivity,” Sov. Phys. Solid State 31, 49–50 (1989).

1986 (2)

G. S. Trofimov and S. I. Stepanov, “Time-dependent holographic currents in photorefractive crystals,” Sov. Phys. Solid State 28, 1559–1562 (1986).

B. C. Grabmaier and R. Oberschmid, “Properties of pure and doped Bi12GeO20 and Bi12SiO20 crystals,” Phys. Status Solidi 96, 199–211 (1986).
[CrossRef]

Barbosa, M. C.

M. C. Barbosa, L. Mosquera, and J. Frejlich, “Speed and diffraction efficiency in feedback-controlled running holograms for photorefractive crystal characterization,” Appl. Phys. B 72, 717–721 (2001).
[CrossRef]

Bittner, R.

R. Bittner, K. Meerholz, and S. Stepanov, “Nonsteady-state photo-EMF effect in photorefractive polymers,” Appl. Phys. Lett. 74, 3723–3725 (1999).
[CrossRef]

Bryushinin, M. A.

M. A. Bryushinin, G. B. Dubrovsky, and L. A. Sokolov, “Non-steady-state photocurrents in SnS2 crystals,” Appl. Phys. B 68, 871–875 (1999).
[CrossRef]

Buse, K.

N. Korneev, D. Mayorga, S. Stepanov, H. Veenhuis, K. Buse, C. Kuper, H. Hesse, and E. Krätzig, “Holographic and non-steady-state photocurrent characterization of photorefractive barium-calcium titanate,” Opt. Commun. 160, 98–102 (1999).
[CrossRef]

Carvalho, J. F.

L. Mosquera, I. de Oliveira, J. Frejlich, A. C. Hernandes, S. Lanfredi, and J. F. Carvalho, “Dark conductivity, photoconductivity and light-induced absorption in photorefractive sillenite crystals,” J. Appl. Phys. 90, 2635–2641 (2001).
[CrossRef]

de Oliveira, I.

L. Mosquera, I. de Oliveira, J. Frejlich, A. C. Hernandes, S. Lanfredi, and J. F. Carvalho, “Dark conductivity, photoconductivity and light-induced absorption in photorefractive sillenite crystals,” J. Appl. Phys. 90, 2635–2641 (2001).
[CrossRef]

I. de Oliveira and J. Frejlich, “Photorefractive running hologram for materials characterization,” J. Opt. Soc. Am. B 18, 291–297 (2001).
[CrossRef]

I. de Oliveira and J. Frejlich, “Dielectric relaxation time measurement in absorbing photorefractive materials,” Opt. Commun. 178, 251–255 (2000).
[CrossRef]

Dubrovsky, G. B.

M. A. Bryushinin, G. B. Dubrovsky, and L. A. Sokolov, “Non-steady-state photocurrents in SnS2 crystals,” Appl. Phys. B 68, 871–875 (1999).
[CrossRef]

Frejlich, J.

L. Mosquera, I. de Oliveira, J. Frejlich, A. C. Hernandes, S. Lanfredi, and J. F. Carvalho, “Dark conductivity, photoconductivity and light-induced absorption in photorefractive sillenite crystals,” J. Appl. Phys. 90, 2635–2641 (2001).
[CrossRef]

I. de Oliveira and J. Frejlich, “Photorefractive running hologram for materials characterization,” J. Opt. Soc. Am. B 18, 291–297 (2001).
[CrossRef]

M. C. Barbosa, L. Mosquera, and J. Frejlich, “Speed and diffraction efficiency in feedback-controlled running holograms for photorefractive crystal characterization,” Appl. Phys. B 72, 717–721 (2001).
[CrossRef]

I. de Oliveira and J. Frejlich, “Dielectric relaxation time measurement in absorbing photorefractive materials,” Opt. Commun. 178, 251–255 (2000).
[CrossRef]

A. A. Freschi, P. M. Garcia, and J. Frejlich, “Charge-carriers diffusion length in photorefractive crystals computed from the initial hologram phase shift,” Appl. Phys. Lett. 71, 2427–2429 (1997).
[CrossRef]

Freschi, A. A.

A. A. Freschi, P. M. Garcia, and J. Frejlich, “Charge-carriers diffusion length in photorefractive crystals computed from the initial hologram phase shift,” Appl. Phys. Lett. 71, 2427–2429 (1997).
[CrossRef]

Garcia, P. M.

A. A. Freschi, P. M. Garcia, and J. Frejlich, “Charge-carriers diffusion length in photorefractive crystals computed from the initial hologram phase shift,” Appl. Phys. Lett. 71, 2427–2429 (1997).
[CrossRef]

Grabmaier, B. C.

B. C. Grabmaier and R. Oberschmid, “Properties of pure and doped Bi12GeO20 and Bi12SiO20 crystals,” Phys. Status Solidi 96, 199–211 (1986).
[CrossRef]

Hernandes, A. C.

L. Mosquera, I. de Oliveira, J. Frejlich, A. C. Hernandes, S. Lanfredi, and J. F. Carvalho, “Dark conductivity, photoconductivity and light-induced absorption in photorefractive sillenite crystals,” J. Appl. Phys. 90, 2635–2641 (2001).
[CrossRef]

Hesse, H.

N. Korneev, D. Mayorga, S. Stepanov, H. Veenhuis, K. Buse, C. Kuper, H. Hesse, and E. Krätzig, “Holographic and non-steady-state photocurrent characterization of photorefractive barium-calcium titanate,” Opt. Commun. 160, 98–102 (1999).
[CrossRef]

Korneev, N.

N. Korneev, D. Mayorga, S. Stepanov, H. Veenhuis, K. Buse, C. Kuper, H. Hesse, and E. Krätzig, “Holographic and non-steady-state photocurrent characterization of photorefractive barium-calcium titanate,” Opt. Commun. 160, 98–102 (1999).
[CrossRef]

Korneev, N. A.

N. A. Korneev and S. Stepanov, “Non-steady-state photoelectromotive force in semiconductor crystals with high light absorption,” J. Appl. Phys. 74, 2736–2741 (1993).
[CrossRef]

Krätzig, E.

N. Korneev, D. Mayorga, S. Stepanov, H. Veenhuis, K. Buse, C. Kuper, H. Hesse, and E. Krätzig, “Holographic and non-steady-state photocurrent characterization of photorefractive barium-calcium titanate,” Opt. Commun. 160, 98–102 (1999).
[CrossRef]

Kuper, C.

N. Korneev, D. Mayorga, S. Stepanov, H. Veenhuis, K. Buse, C. Kuper, H. Hesse, and E. Krätzig, “Holographic and non-steady-state photocurrent characterization of photorefractive barium-calcium titanate,” Opt. Commun. 160, 98–102 (1999).
[CrossRef]

Lanfredi, S.

L. Mosquera, I. de Oliveira, J. Frejlich, A. C. Hernandes, S. Lanfredi, and J. F. Carvalho, “Dark conductivity, photoconductivity and light-induced absorption in photorefractive sillenite crystals,” J. Appl. Phys. 90, 2635–2641 (2001).
[CrossRef]

Mayorga, D.

N. Korneev, D. Mayorga, S. Stepanov, H. Veenhuis, K. Buse, C. Kuper, H. Hesse, and E. Krätzig, “Holographic and non-steady-state photocurrent characterization of photorefractive barium-calcium titanate,” Opt. Commun. 160, 98–102 (1999).
[CrossRef]

Meerholz, K.

R. Bittner, K. Meerholz, and S. Stepanov, “Nonsteady-state photo-EMF effect in photorefractive polymers,” Appl. Phys. Lett. 74, 3723–3725 (1999).
[CrossRef]

Mosquera, L.

L. Mosquera, I. de Oliveira, J. Frejlich, A. C. Hernandes, S. Lanfredi, and J. F. Carvalho, “Dark conductivity, photoconductivity and light-induced absorption in photorefractive sillenite crystals,” J. Appl. Phys. 90, 2635–2641 (2001).
[CrossRef]

M. C. Barbosa, L. Mosquera, and J. Frejlich, “Speed and diffraction efficiency in feedback-controlled running holograms for photorefractive crystal characterization,” Appl. Phys. B 72, 717–721 (2001).
[CrossRef]

Oberschmid, R.

B. C. Grabmaier and R. Oberschmid, “Properties of pure and doped Bi12GeO20 and Bi12SiO20 crystals,” Phys. Status Solidi 96, 199–211 (1986).
[CrossRef]

Petrov, M. P.

M. P. Petrov, I. A. Sokolov, S. I. Stepanov, and G. S. Trofimov, “Non-steady-state photo-electromotive-force induced by dynamic gratings in partially compensated photoconductors,” J. Appl. Phys. 68, 2216–2225 (1990).
[CrossRef]

Sokolov, I. A.

M. P. Petrov, I. A. Sokolov, S. I. Stepanov, and G. S. Trofimov, “Non-steady-state photo-electromotive-force induced by dynamic gratings in partially compensated photoconductors,” J. Appl. Phys. 68, 2216–2225 (1990).
[CrossRef]

Sokolov, L. A.

M. A. Bryushinin, G. B. Dubrovsky, and L. A. Sokolov, “Non-steady-state photocurrents in SnS2 crystals,” Appl. Phys. B 68, 871–875 (1999).
[CrossRef]

Stepanov, S.

N. Korneev, D. Mayorga, S. Stepanov, H. Veenhuis, K. Buse, C. Kuper, H. Hesse, and E. Krätzig, “Holographic and non-steady-state photocurrent characterization of photorefractive barium-calcium titanate,” Opt. Commun. 160, 98–102 (1999).
[CrossRef]

R. Bittner, K. Meerholz, and S. Stepanov, “Nonsteady-state photo-EMF effect in photorefractive polymers,” Appl. Phys. Lett. 74, 3723–3725 (1999).
[CrossRef]

N. A. Korneev and S. Stepanov, “Non-steady-state photoelectromotive force in semiconductor crystals with high light absorption,” J. Appl. Phys. 74, 2736–2741 (1993).
[CrossRef]

Stepanov, S. I.

M. P. Petrov, I. A. Sokolov, S. I. Stepanov, and G. S. Trofimov, “Non-steady-state photo-electromotive-force induced by dynamic gratings in partially compensated photoconductors,” J. Appl. Phys. 68, 2216–2225 (1990).
[CrossRef]

S. I. Stepanov and G. S. Trofimov, “Transient emf in crystals having ambipolar photoconductivity,” Sov. Phys. Solid State 31, 49–50 (1989).

G. S. Trofimov and S. I. Stepanov, “Time-dependent holographic currents in photorefractive crystals,” Sov. Phys. Solid State 28, 1559–1562 (1986).

Trofimov, G. S.

M. P. Petrov, I. A. Sokolov, S. I. Stepanov, and G. S. Trofimov, “Non-steady-state photo-electromotive-force induced by dynamic gratings in partially compensated photoconductors,” J. Appl. Phys. 68, 2216–2225 (1990).
[CrossRef]

S. I. Stepanov and G. S. Trofimov, “Transient emf in crystals having ambipolar photoconductivity,” Sov. Phys. Solid State 31, 49–50 (1989).

G. S. Trofimov and S. I. Stepanov, “Time-dependent holographic currents in photorefractive crystals,” Sov. Phys. Solid State 28, 1559–1562 (1986).

Veenhuis, H.

N. Korneev, D. Mayorga, S. Stepanov, H. Veenhuis, K. Buse, C. Kuper, H. Hesse, and E. Krätzig, “Holographic and non-steady-state photocurrent characterization of photorefractive barium-calcium titanate,” Opt. Commun. 160, 98–102 (1999).
[CrossRef]

Appl. Phys. B (2)

M. A. Bryushinin, G. B. Dubrovsky, and L. A. Sokolov, “Non-steady-state photocurrents in SnS2 crystals,” Appl. Phys. B 68, 871–875 (1999).
[CrossRef]

M. C. Barbosa, L. Mosquera, and J. Frejlich, “Speed and diffraction efficiency in feedback-controlled running holograms for photorefractive crystal characterization,” Appl. Phys. B 72, 717–721 (2001).
[CrossRef]

Appl. Phys. Lett. (2)

A. A. Freschi, P. M. Garcia, and J. Frejlich, “Charge-carriers diffusion length in photorefractive crystals computed from the initial hologram phase shift,” Appl. Phys. Lett. 71, 2427–2429 (1997).
[CrossRef]

R. Bittner, K. Meerholz, and S. Stepanov, “Nonsteady-state photo-EMF effect in photorefractive polymers,” Appl. Phys. Lett. 74, 3723–3725 (1999).
[CrossRef]

J. Appl. Phys. (3)

N. A. Korneev and S. Stepanov, “Non-steady-state photoelectromotive force in semiconductor crystals with high light absorption,” J. Appl. Phys. 74, 2736–2741 (1993).
[CrossRef]

M. P. Petrov, I. A. Sokolov, S. I. Stepanov, and G. S. Trofimov, “Non-steady-state photo-electromotive-force induced by dynamic gratings in partially compensated photoconductors,” J. Appl. Phys. 68, 2216–2225 (1990).
[CrossRef]

L. Mosquera, I. de Oliveira, J. Frejlich, A. C. Hernandes, S. Lanfredi, and J. F. Carvalho, “Dark conductivity, photoconductivity and light-induced absorption in photorefractive sillenite crystals,” J. Appl. Phys. 90, 2635–2641 (2001).
[CrossRef]

J. Opt. Soc. Am. B (1)

Opt. Commun. (2)

I. de Oliveira and J. Frejlich, “Dielectric relaxation time measurement in absorbing photorefractive materials,” Opt. Commun. 178, 251–255 (2000).
[CrossRef]

N. Korneev, D. Mayorga, S. Stepanov, H. Veenhuis, K. Buse, C. Kuper, H. Hesse, and E. Krätzig, “Holographic and non-steady-state photocurrent characterization of photorefractive barium-calcium titanate,” Opt. Commun. 160, 98–102 (1999).
[CrossRef]

Phys. Status Solidi (1)

B. C. Grabmaier and R. Oberschmid, “Properties of pure and doped Bi12GeO20 and Bi12SiO20 crystals,” Phys. Status Solidi 96, 199–211 (1986).
[CrossRef]

Sov. Phys. Solid State (2)

S. I. Stepanov and G. S. Trofimov, “Transient emf in crystals having ambipolar photoconductivity,” Sov. Phys. Solid State 31, 49–50 (1989).

G. S. Trofimov and S. I. Stepanov, “Time-dependent holographic currents in photorefractive crystals,” Sov. Phys. Solid State 28, 1559–1562 (1986).

Other (2)

S. Stepanov and P. Petrov, Photorefractive Materials and Their Applications I, P. Günter and J.-P. Huignard, eds., Vol. 61 of Topics in Applied Physics (Springer-Verlag, Berlin, 1988), Chap. 9, pp. 263–289.
[CrossRef]

S. I. Stepanov, “Photo-electromotive-force effect in semiconductors,” in Semiconductor Devices, H. S. Nalwa, ed., Vol. 2 of Handbook of Advanced Electronic and Photonic Materials and Devices (Academic, San Diego, Calif., 2001), Chap. 6, pp. 205–272.

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Figures (7)

Fig. 1
Fig. 1

Experimental setup: A laser beam of 514.5-nm wavelength is divided into two, filtered, expanded, collimated, and made to interfere over the BTO sample. A PZT in one of the beams vibrates with angular frequency Ω. A lock-in amplifier that measures current and is schematically represented by the operational amplifier (OA) with feedback is tuned to Ω to measure first-harmonic component iΩ of the photocurrent along the K direction in the sample’s volume.

Fig. 2
Fig. 2

First-harmonic component of the holographic current |iΩ| data as a function of amplitude vd of the driving voltage applied to the PZT for I0=IRo+ISo=455 W/m2 and several fixed frequencies Ω/2π. The connecting curves are merely guides for the eye.

Fig. 3
Fig. 3

First-harmonic component of the holographic current |iΩ| data as a function of amplitude vd of the driving voltage applied to the PZT for I0=IRo+ISo=177 W/m2 and several fixed frequencies Ω/2π. The connecting curves are merely guides to the eye.

Fig. 4
Fig. 4

Frequency response of the PZT mirror. The dashed curve is a guide for the eye only.

Fig. 5
Fig. 5

Data from Fig. 2 with vd converted into its corresponding value of KΔ. The continuous curves are the best fit to theory for several values of Ω/2π Hz. Data for 980, 546, and 349 Hz are omitted because they are close to those for 152 Hz.

Fig. 6
Fig. 6

Data from Fig. 3 with vd converted into its corresponding value of KΔ for several frequencies. All data fit the same curve, which is not shown so the data distribution will be more obvious.

Fig. 7
Fig. 7

|iΩ| data (filled circles) plotted as a function of Ω/2π for KΔ=1.1 rad: for BTO from Fig. 5 (curve A) and from Fig. 6 (curve B); for BTO:Ce (curve C); and for BTO:Pb (curve D). The best fits are represented by the continuous curves. The continuous and the dashed curves in the inset are a detailed picture of curve D and the best fits (the corresponding parameters are listed in Table 2), with and without bulk absorption considered, for BTO:Pb.

Tables (2)

Tables Icon

Table 2 Best-Fit Parameters

Equations (42)

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

I=I0+(I0/2)[m(t)exp(iKx)+m(t)¯exp(-iKx)],
m(t)=|m|exp(iϕ)exp(iKΔ sin Ωt),
exp(ikΔ sin Ωt)=l=-+ Jl(KΔ)exp(ilΩt),
m(t)=|m| exp(iϕ)l=-+ Jl(KΔ)exp(ilΩt).
N(x, t)=N0+N02[a(t)exp(iKx)+a(t)¯exp(-iKx)],
ND+(x, t)=ND++ND+2[A(t)exp(iKx)+A(t)¯ exp(-iKx)],
E(x, t)=E0+½[Esc(t)exp(iKx)+Esc(t)¯ exp(-iKx)].
ε0·E(x, t)=q[ND+(x, t)-NA-N(x, t)]
τsc Esc(t)t+Esc(t)=-m(t)Eeff,
τsc=ε0σ0 1+K2LD2-iKLE1+K2ls2-iKlE,
Eeff=E0+iED1+K2ls2-iKlE,
a(t)=Esc(t)iKμτ+m(t)sI0/(sI0+β)-A(t)ND/(ND-ND+)1-iKτμE0+K2Dτ,
-iKε0Esc(t)qND+A(t),
a(t)=i K2LD2-K2ls21+K2LD2 Esc(t)ED+|m|exp(iϕ)1+K2LD2 t=-+Jl(KΔ)exp(ilΩt),
Esc(t)=-|m|Eeffl=-+ Jl(KΔ)exp(ilΩt)1+ilΩτsc,
a(t)=|m|exp(iϕ)1+K2LD2 l=-+Jl(KΔ)exp(ilΩt)-i K2LD2-K2ls21+K2LD2|m| EeffED×l=-+Jl(KΔ)exp(ilΩt)1+ilΩτsc.
j(t)=1L 0LeμN(x, t)Esc(x, t)dx,
j(t)=eμ N02 a(t) Esc(t)¯2+a(t)¯ Esc(t)2.
a(t)=l=-+alΩ exp(ilΩt),
Esc(t)=l=-+EsclΩ exp(ilΩt).
j(t)=j02+jΩ2 exp(iΩt)+j2Ω2 exp(i2Ωt)++c.c.,
j0=σ02(a0Esc0¯+a0¯Esc0+aΩEscΩ¯+aΩ¯EscΩ+a2ΩEsc2Ω¯+a2Ω¯Esc2Ω+),
jΩ=σ02(a0Esc-Ω¯+a0¯EscΩ+aΩEsc0¯+a-Ω¯Esc0+a2ΩEscΩ¯+a-ΩEsc-2Ω¯+aΩ¯Esc2Ω+a-2Ω¯Esc-Ω+),
j2Ω=σ02(a0Esc-2Ω¯+a0¯Esc2Ω+aΩEsc-Ω¯+a-Ω¯EscΩ+a-2Ω¯(Esc0)+a2ΩEsc0¯+),
j0=-σ0|m|2 J0(KΔ)2(1+K2LD2)(1+K2ls2)ED sin ϕ-2σ0|m|2 J1(KΔ)2(1+K2LD2)(1+K2ls2) ED1+Ω2τsc2×sin ϕ-2σ0|m|2 J2(KΔ)2(1+K2LD2)(1+K2ls2)×ED1+4Ω2τsc2 sin ϕ;
jΩ=-σ0|m|2 J0(KΔ)J1(KΔ)1+K2LD2 Ωτsc1+iΩτsc ED1+K2ls2 cos ϕ-σ0|m|2 J1(KΔ)J2(KΔ)1+K2LD2 3Ωτsc(1-iΩτsc)(1+i2Ωτsc)×ED1+K2ls2 cos ϕ;
j2Ω=-σ0|m|2 J0(KΔ)J2(KΔ)1+K2LD2 ED1+K2ls2 sin ϕ-σ0|m|2 J0(KΔ)J2(KΔ)1+K2LD2 ED1+K2ls2 11+i2Ωτsc×sin ϕ-σ0|m|2 J1(KΔ)21+K2LD2 ED1+K2ls2×11+iΩτsc sin ϕ-2σ0|m|2J0(KΔ)J2(KΔ)×K2LD2-K2ls21+K2LD2 ED1+K2ls2 11+i2Ωτsc sin ϕ+σ0|m|2J1(KΔ)2 K2LD2-K2ls21+K2LD2×ED1+K2ls2 1(1+iΩτsc)2 sin ϕ.
jΩ=jRΩ+ijIΩ,
jRΩ=ΩτscJ1(KΔ) J0(KΔ)+3J2(KΔ)+2Ω2τsc2[2J0(KΔ)+3J2(KΔ)](1+Ω2τsc2)(1+4Ω2τsc2) σ0|m|2ED cos ϕ(1+K2LD2)(1+K2ls2),
jIΩ=-Ω2τsc2J1(KΔ) J0(KΔ)+3J2(KΔ)+4Ω2τsc2J0(KΔ)(1+Ω2τsc2)(1+4Ω2τsc2) σ0|m|2ED cos ϕ(1+K2LD2)(1+K2ls2).
Aσ0τsc(1+K2LD2)(1+K2ls2)=ε0(1+K2ls2)2.
jΩ2 exp(iΩt)+jΩ¯2 exp(-iΩt)
=jΩ+jΩ¯2 cos Ωt+i jΩ-jΩ¯2 sin Ωt=jRΩ cos Ωt-jIΩ sin Ωt
=[(jRΩ)2+(jIΩ)2]1/2 cos(Ωt+φΩ).
|jΩ|=σ0|m|2ED cos ϕ(1+K2LD2)(1+K2ls2)J1(KΔ)J0(KΔ)=Aτsc|m|2ED cos ϕJ1(KΔ)J0(KΔ),
|j0Ω|=Ωε0 |m|2ED cos ϕ(1+K2ls2)2J1(KΔ)[J0(KΔ)+3J2(KΔ)]=ΩA|m|2ED cos ϕJ1(KΔ)[J0(KΔ)+3J2(KΔ)],
σ0(z)=σ0(0)exp(-αz),
τsc(z)=τsc(0)exp(αz),
|iΩ|=H0djΩ(z)dz,
v(t)=vd sin Ωt,
KΔ=KPZTΩvd,
LD=kbTqμτ1/2

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