Abstract

The theory of grating oscillations in photorefractive materials is developed in the linear and nonlinear approximations for the drift mechanism of holographic recording. The resonance dependence of diffraction efficiency on the phase-modulation frequency is predicted for the linear regime in which holograms are recorded by two beams, one of which is phase modulated. For long drift lengths (KgL01) the resonance frequency is shown to be Ωr(τ1KgL0)-1. A more pronounced resonance peak is expected for the non-Bragg diffraction orders. In the nonlinear regime of recording, additional resonance maxima at Ωr/p (p is an integer) are found. Grating oscillations are experimentally studied in thin holograms of Bi12TiO20. A sharp resonance for the non-Bragg order in the interval 100–2000 Hz is detected. The position of the resonance is shown to depend on the experimental conditions. The experiment is in excellent agreement with the theory. At a high contrast ratio and in a high external electric field, distortions in the resonance dependence at Ω<Ωr and even a chaotic frequency dependence are found, which points to the nonlinear character of grating oscillations.

[Optical Society of America ]

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References

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  1. S. I. Stepanov , Applications of photorefractive crystals , Rep. Progr. Phys. RPPHAG 57 , 39 ( 1994
    [CrossRef]
  2. T. J. Hall , M. A. Fiddy , and M. S. Ner , Detector for an optical-fiber accoustic sensor using dynamic holographic interferometry , Opt. Lett. OPLEDP 5 , 485 ( 1980
    [CrossRef] [PubMed]
  3. V. M. Petrov and M. P. Petrov , Two- and three-wave mixing in a PRIZ space light modulator , Tech. Phys. Lett. TPLEED 21 , 403 ( 1995
  4. M. P. Petrov , V. M. Petrov , I. S. Zouboulis , and L. P. Xu , Two-wave and induced three-wave mixing on a thin Bi 12 TiO 20 hologram , Opt. Commun. OPCOB8 134 , 599 ( 1997
    [CrossRef]
  5. S. Breugnot , M. Defour , and J. P. Huignard , Photorefractive two-wave mixing: complex amplitudes solutions in the case of a weak signal beam , Opt. Commun. OPCOB8 134 , 599 ( 1997
    [CrossRef]
  6. S. Bian and J. Frejlich , Photorefractive response time measurements in GaAs crystals by phase modulation in two wave mixing , Opt. Lett. OPLEDP 19 , 1702 ( 1994
    [CrossRef] [PubMed]
  7. M. P. Petrov , V. M. Petrov , V. V. Bryksin , I. Zouboulis , A. Gerwens , and E. Kra tzig , Grating oscillations in photorefractive crystals , Opt. Lett. OPLEDP 22 , 1083 ( 1997
    [CrossRef] [PubMed]
  8. W. R. Klein , Theoretical Efficiency of Bragg Devices , Proc. IEEE IEEPAD 54 , 803 ( 1966
    [CrossRef]
  9. 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. JAPIAU 68 , 2216 ( 1990
    [CrossRef]
  10. N. V. Kukhtarev , V. B. Markov , S. G. Odoulov , M. S. Soskin , and V. L. Vinetskii , Holographic storage in electrooptic crystals , Ferroelectrics FEROA8 22 , 949 ( 1979
    [CrossRef]
  11. M. G. Moharam , T. K. Gaylord , R. Magnuson , and L. Young , Holographic grating formation in photorefractive crystals with arbitrary electron transport length , J. Appl. Phys. JAPIAU 50 , 5642 ( 1979
    [CrossRef]
  12. S. I. Stepanov , V. V. Kulikov , and M. P. Petrov , Running holograms in photorefractive Bi 12 SiO 20 crystals , Opt. Commun. OPCOB8 44 , 19 ( 1982
    [CrossRef]
  13. P. Refregier , L. Solymar , K. Rajbenbach , and J. P. Huignard , Two-beam coupling in photorefractive Bi 12 SiO 20 crystals with moving grating: theory and experiments , J. Appl. Phys. JAPIAU 58 , 45 ( 1985
    [CrossRef]
  14. T. E. McClelland , D. J. Webb , B. I. Sturman , M. Mann , and K. N. Ringhofer , Low frequency peculiarities of the photorefractive response in sillenites , Opt. Commun. OPCOB8 113 , 371 ( 1995
    [CrossRef]
  15. M. Vasnetsov , P. Buchhave , and S. Lyuksyutov , Phase modulation spectroscopy of space-charge wave resonances in Bi 12 SiO 20 , Opt. Commun. OPCOB8 137 , 181 ( 1997
    [CrossRef]
  16. N. G. Zhdanova , M. S. Kagan , R. A. Suris , and B. I. Fuks , Trap charge exchange waves in compensated germanium , Pis'ma Zh. Eksp. Teor. Fiz. PZETAB 74 , 346 ( 1978 ), in Russian
  17. R. Orlowski and E. Kra tzig , Holographic method for the determination of photo-induced electron and hole transport in electro-optic crystals , Solid State Commun. SSCOA4 27 , 1351 ( 1978
    [CrossRef]
  18. F. Rickermann , S. Riehemann , K. Buse , D. Dirksen , and G. von Bally , Diffraction efficiency enhancement of holographic gratings in Bi 12 Ti 0.76 V 0.24 O 20 crystals after recording , J. Opt. Soc. Am. B JOBPDE 13 , 2299 ( 1996
    [CrossRef]

Bryksin, V. V

Dirksen, D

Fuks, B. I

N. G. Zhdanova , M. S. Kagan , R. A. Suris , and B. I. Fuks , Trap charge exchange waves in compensated germanium , Pis'ma Zh. Eksp. Teor. Fiz. PZETAB 74 , 346 ( 1978 ), in Russian

Kagan, M. S

N. G. Zhdanova , M. S. Kagan , R. A. Suris , and B. I. Fuks , Trap charge exchange waves in compensated germanium , Pis'ma Zh. Eksp. Teor. Fiz. PZETAB 74 , 346 ( 1978 ), in Russian

Magnuson, R

M. G. Moharam , T. K. Gaylord , R. Magnuson , and L. Young , Holographic grating formation in photorefractive crystals with arbitrary electron transport length , J. Appl. Phys. JAPIAU 50 , 5642 ( 1979
[CrossRef]

Ner, M. S

Petrov, V. M

V. M. Petrov and M. P. Petrov , Two- and three-wave mixing in a PRIZ space light modulator , Tech. Phys. Lett. TPLEED 21 , 403 ( 1995

Rajbenbach, K

P. Refregier , L. Solymar , K. Rajbenbach , and J. P. Huignard , Two-beam coupling in photorefractive Bi 12 SiO 20 crystals with moving grating: theory and experiments , J. Appl. Phys. JAPIAU 58 , 45 ( 1985
[CrossRef]

Rickermann, F

Riehemann, S

Ringhofer, K. N

T. E. McClelland , D. J. Webb , B. I. Sturman , M. Mann , and K. N. Ringhofer , Low frequency peculiarities of the photorefractive response in sillenites , Opt. Commun. OPCOB8 113 , 371 ( 1995
[CrossRef]

Suris, R. A

N. G. Zhdanova , M. S. Kagan , R. A. Suris , and B. I. Fuks , Trap charge exchange waves in compensated germanium , Pis'ma Zh. Eksp. Teor. Fiz. PZETAB 74 , 346 ( 1978 ), in Russian

Xu, L. P

M. P. Petrov , V. M. Petrov , I. S. Zouboulis , and L. P. Xu , Two-wave and induced three-wave mixing on a thin Bi 12 TiO 20 hologram , Opt. Commun. OPCOB8 134 , 599 ( 1997
[CrossRef]

Zhdanova, N. G

N. G. Zhdanova , M. S. Kagan , R. A. Suris , and B. I. Fuks , Trap charge exchange waves in compensated germanium , Pis'ma Zh. Eksp. Teor. Fiz. PZETAB 74 , 346 ( 1978 ), in Russian

Zouboulis, I

Zouboulis, I. S

M. P. Petrov , V. M. Petrov , I. S. Zouboulis , and L. P. Xu , Two-wave and induced three-wave mixing on a thin Bi 12 TiO 20 hologram , Opt. Commun. OPCOB8 134 , 599 ( 1997
[CrossRef]

Other (18)

W. R. Klein , Theoretical Efficiency of Bragg Devices , Proc. IEEE IEEPAD 54 , 803 ( 1966
[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. JAPIAU 68 , 2216 ( 1990
[CrossRef]

N. V. Kukhtarev , V. B. Markov , S. G. Odoulov , M. S. Soskin , and V. L. Vinetskii , Holographic storage in electrooptic crystals , Ferroelectrics FEROA8 22 , 949 ( 1979
[CrossRef]

M. G. Moharam , T. K. Gaylord , R. Magnuson , and L. Young , Holographic grating formation in photorefractive crystals with arbitrary electron transport length , J. Appl. Phys. JAPIAU 50 , 5642 ( 1979
[CrossRef]

S. I. Stepanov , V. V. Kulikov , and M. P. Petrov , Running holograms in photorefractive Bi 12 SiO 20 crystals , Opt. Commun. OPCOB8 44 , 19 ( 1982
[CrossRef]

P. Refregier , L. Solymar , K. Rajbenbach , and J. P. Huignard , Two-beam coupling in photorefractive Bi 12 SiO 20 crystals with moving grating: theory and experiments , J. Appl. Phys. JAPIAU 58 , 45 ( 1985
[CrossRef]

T. E. McClelland , D. J. Webb , B. I. Sturman , M. Mann , and K. N. Ringhofer , Low frequency peculiarities of the photorefractive response in sillenites , Opt. Commun. OPCOB8 113 , 371 ( 1995
[CrossRef]

M. Vasnetsov , P. Buchhave , and S. Lyuksyutov , Phase modulation spectroscopy of space-charge wave resonances in Bi 12 SiO 20 , Opt. Commun. OPCOB8 137 , 181 ( 1997
[CrossRef]

N. G. Zhdanova , M. S. Kagan , R. A. Suris , and B. I. Fuks , Trap charge exchange waves in compensated germanium , Pis'ma Zh. Eksp. Teor. Fiz. PZETAB 74 , 346 ( 1978 ), in Russian

R. Orlowski and E. Kra tzig , Holographic method for the determination of photo-induced electron and hole transport in electro-optic crystals , Solid State Commun. SSCOA4 27 , 1351 ( 1978
[CrossRef]

V. M. Petrov and M. P. Petrov , Two- and three-wave mixing in a PRIZ space light modulator , Tech. Phys. Lett. TPLEED 21 , 403 ( 1995

M. P. Petrov , V. M. Petrov , I. S. Zouboulis , and L. P. Xu , Two-wave and induced three-wave mixing on a thin Bi 12 TiO 20 hologram , Opt. Commun. OPCOB8 134 , 599 ( 1997
[CrossRef]

S. Breugnot , M. Defour , and J. P. Huignard , Photorefractive two-wave mixing: complex amplitudes solutions in the case of a weak signal beam , Opt. Commun. OPCOB8 134 , 599 ( 1997
[CrossRef]

S. I. Stepanov , Applications of photorefractive crystals , Rep. Progr. Phys. RPPHAG 57 , 39 ( 1994
[CrossRef]

T. J. Hall , M. A. Fiddy , and M. S. Ner , Detector for an optical-fiber accoustic sensor using dynamic holographic interferometry , Opt. Lett. OPLEDP 5 , 485 ( 1980
[CrossRef] [PubMed]

S. Bian and J. Frejlich , Photorefractive response time measurements in GaAs crystals by phase modulation in two wave mixing , Opt. Lett. OPLEDP 19 , 1702 ( 1994
[CrossRef] [PubMed]

F. Rickermann , S. Riehemann , K. Buse , D. Dirksen , and G. von Bally , Diffraction efficiency enhancement of holographic gratings in Bi 12 Ti 0.76 V 0.24 O 20 crystals after recording , J. Opt. Soc. Am. B JOBPDE 13 , 2299 ( 1996
[CrossRef]

M. P. Petrov , V. M. Petrov , V. V. Bryksin , I. Zouboulis , A. Gerwens , and E. Kra tzig , Grating oscillations in photorefractive crystals , Opt. Lett. OPLEDP 22 , 1083 ( 1997
[CrossRef] [PubMed]

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Figures (8)

Fig. 1
Fig. 1

Diagram of the direction of propagation for different diffraction orders: The incident beams R and S are writing a thin grating into the crystal and are simultaneously diffracted from it into the first orders (●)1 and ()-1, the second orders (●)2 and ()-2, and the third orders (●)3 and ()-3.

Fig. 2
Fig. 2

(a) Experimental setup showing the incident beams R and S and the beams Bragg (Pos N1) and non-Bragg (Pos N2) diffracted from a thin hologram: 1, argon-ion laser; 2, mirror; 3, electro-optic modulator; 4, polarizer; 5, BTO crystal; 6, analyzer of polarization; 7, photoreceiver; 8, lock-in detector; b, beamsplitter. (b) Magnified view of the BTO crystal interaction volume: 1, electrodes; 2, BTO crystal; 3, holographic grating. Kg, grating wavevector; E0, external electric field.

Fig. 3
Fig. 3

Two-wave mixing signal I2W as a function of Ω/2π for three different externally applied electric fields E0. The total light intensity is I0=IS+IR=600 W m-2, the contrast of the interference pattern is m=0.24, the wave number of the grating is Kg=5.54×104 m-1, and the amplitude of modulation is Θ=0.58 rad. Symbols, measured data; solid curves, fits according to Eq. (29).

Fig. 4
Fig. 4

Non-Bragg diffraction signal INB as a function of Ω/2π for three different externally applied electric fields E0 (I0=600 W m-2, m=0.24, Kg=5.54×104 m-1, Θ=0.58 rad). Symbols, measured data; solid curve, a fit according to Eq. (26).

Fig. 5
Fig. 5

The dependence of the maximum value of INB on the externally applied electric field is shown. Symbols, measured values; solid curve, fit of the relation INB,maxE03 to the experimental data. The experimental parameters are m=0.22, Kg=12.6×104 m-1, and Θ=0.58 rad.

Fig. 6
Fig. 6

Dependence of the non-Bragg diffraction signal INB on frequency Ω/2π for an externally applied of E0=6.25 kV cm-1. The contrast of the interference pattern is m=0.22, the wave number of the grating is Kg=89.7×104 m-1, and the amplitude of modulation is Θ=0.58 rad. Symbols, measured data; solid curve, fit according to Eq. (26).

Fig. 7
Fig. 7

Non-Bragg diffraction signal INB versus frequency Ω 2π for an externally applied field of E0=6.25 kV cm-1. The total light intensity is I0=2000 W m-2, and the contrast of the interference pattern is m1. The wave number of the grating is Kg=28.9×104 m-1, and the amplitude of modulation is Θ=0.56 rad. Symbols, measured data; solid curve, fit according to Eq. (26).

Fig. 8
Fig. 8

Dependence of non-Bragg diffraction signal INB on frequency Ω/2π for maximum possible externally applied electric field E0=11.25 kV cm-1 and low light intensity I0=5 W m-2. The contrast of the interference pattern is m1. The wave number of the grating is Kg=89.7×104 m-1, and the amplitude of modulation is Θ=0.58 rad.

Equations (46)

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(λh/nΛ2)1/2<1.
AS=AS0 exp(-iKSx),
AR=AR0 exp{-i[KRx+Θ cos(Ωt)]},
I(x, t)=I0{1+m cos[Kgx+Θ cos(Ωt)]},
dE(t)dt=-mE0 exp[iΘ cos(Ωt)]+E(t)τ1(1-id),
Esc(x, t)=½E(t)exp(iKgx)+c.c.
E(t)=-mE0+δE(Ω)exp(iΩt)+δE(-Ω)exp(-iΩt),
δE(Ω)=-imE0Θ2(1+ig+gd),
δE(-Ω)=-imE0Θ2(1-ig-gd),g=Ωτ1.
E(t)=-mE0(1+F+iF),
F=Θgd cos(Ωt+γ)[1+2g2(1-d2)+g4(1+d2)2]1/2,
γ=arctang2(1+d2)-12g,
F=Θ[1+g2(3-2d2)+g4(3+d4)+g6(1+d3)2]1/2 cos(Ωt+β)1+2g2(1-d2)+g4(1+d2)2,
β=arctang[1+g2(1+d2)]1+g2+(1-d2).
Ωr=1τ1(d2+1)1/2.
Esc(x)=E0(1-m2)1/21+m cos(Kgx)-1.
Ep0=-m1+(1-m2)1/2|p|E0.
G(x, t)=G0{1+m cos[Kgx+Θ cos(Ωt)]}.
Esc(x, t)=Esc(x)+Re[δE(x)exp(iΩt)],
gdKg-1(1-m2)1/2δE(x)+i[1+m cos(Kgx)]
×[1+m cos(Kgx)+ig]δE(x)
=imΘE0(1-m2)1/2{sin(Kgx)
+A[1+m cos(Kgx)]}.
δE(x)=pδEp(Ω)exp(ipKgx)
δE2(Ω)=i4 m2ΘE0 2+ig(1+ig+gd)(1+ig+2gd),
δE2(-Ω)=i4 m2ΘE0 2-ig(1-ig-gd)(1-ig-2gd),
δE2(t)=δE2(Ω)exp(iΩt)+δE2(-Ω)exp(-iΩt).
Esc(x, t)=Re[E1(t)exp(iKgx)]+Re[E2(t)exp(i2Kgx)].
T(x, t)=exp[iΔϕ(x, t)].
Δϕ(x, t)=ρEsc(x, t),
T(x, t)=uiuJu[Δϕ(t)]exp[iu(Kgx+F)],
INB=i2 AS0ρE*(t)2=14 ISρ2(mE0)2(1+2F)=Iconst+2ISηNBΘgd cos(Ωt+γ)[1+2g2(1-d2)+g4(1+d2)2]1/2,
ηNB=14 (mE0ρ)2
I2W= |AS0 cos(α)+iAR0 exp[iΘ cos(Ωt)]½Δϕ(t)×sin(α)exp(-iF)|2=IconstB+2(sin2 α)IRηBF-sin(2α)×(IRISηB)1/2[Θ cos(Ωt)-F].
I2WIconstB-sin(2α)(ISIRηB)1/2Θg [1+g2(3+d4)+g4(1+d2)2(3-2d2)+g6(1+d2)4]1/21+2g2(1-d2)+g4(1+d2)2 cos(Ωt+ϕ),
ϕ=arctan1+g2(1+d2)g[1-d2+g2(1+d2)2].
T(x, t)=expiρp Re[Ep(t)exp(ipKgx)],
I2NB= |(i/2)AS0ρE2*(t)|2=Iconst2NB-2ISη2NB3ΘgdW(g, d)cos(Ωt+χ),
η2NB=14 ρ2 m4E02[1+(1-m2)1/2]4.
W(g, d)2(1-g2d2)(1-4g2d2)
χ=arctan2-10g2d2+8g4d4g(1-5g2d2+4g4d4).
IΣ=i ρ2 AS0E1*(t)+iAR0 exp[iΘ cos(Ωt)]E2*(t)2=IconstΣ+2(ISηNB)1/2[F(ISηNB)1/2-Φ(IRη2NB)1/2]×1-IRη2NBISηNB1/2,
m=m=2(ISIR)1/2IS+IR,ISIR1/2=1±(1-m2)1/2m.
IΣ=IconstΣ+2(ISηNB)1/2[F(ISηNB)1/2-Φ(IRη2NB)1/2]×1-m2[1±(1-m2)1/2][1+(1-m2)1/2].
Erw(x, t)=Eg exp[i(Ωt-Kgx)],
Ω1/τ1KgL0.

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