Abstract

Two-beam intensity coupling is calculated for photorefractive crystals with two types of movable charge carrier in the undepleted-pump approximation. The analytical expressions are derived for the temporal evolution of the space-charge field; for weak coupling they are used for calculation of the transmitted beam intensities. The results of the calculation are compared with the experimental observations in photorefractive tin hypothiodiphosphate (Sn2P2S6). All experimental data are in reasonable quantitative agreement with the calculations.

© 1998 Optical Society of America

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References

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  1. E. Strohkendl, J. Jonathan, and R. W. Hellwarth, “Hole–electron competition in photorefractive gratings,” Opt. Lett. 11, 312–314 (1986).
    [CrossRef]
  2. G. Valley, “Simultaneous electron/hole transport in photorefractive materials,” J. Appl. Phys. 59, 3363–3366 (1986).
    [CrossRef]
  3. M. C. Bashaw, M. Jeganathan, and L. Hesselink, “Theory of two-center transport in photorefractive media for low-intensity, continuous-wave illumination in the quasi-steady-state limit,” J. Opt. Soc. Am. B 11, 1743–1757 (1994).
    [CrossRef]
  4. M. C. Bashaw, T.-P. Ma, and R. C. Barker, “Comparison of single and two-species models of electron–hole transport in photorefractive media,” J. Opt. Soc. Am. B 9, 1666–1672 (1992).
    [CrossRef]
  5. S. G. Odoulov, A. N. Shumelyuk, G. Brost, and K. Magde, “Enhancement of beam coupling in the near infrared for tin hypothiodiphosphate,” Appl. Phys. Lett. 21, 752–754 (1996).
  6. S. Zhivkova and M. Miteva, “Holographic recording in photorefractive crystals with simultaneous electron–hole transport and two active centers,” J. Appl. Phys. 68, 3099–3103 (1990).
    [CrossRef]
  7. S. G. Odoulov, A. N. Shumelyuk, U. Hellwig, R. A. Rupp, and A. A. Grabar, “Photorefractive beam coupling in tin hypothiodiphosphate in the near infrared,” Opt. Lett. 21, 752–754 (1996).
    [CrossRef] [PubMed]
  8. S. G. Odoulov, A. N. Shumelyuk, U. Hellwig, R. A. Rupp, A. A. Grabar, and I. M. Stoyka, “Photorefraction in tin hypothiodiphosphate in the near infrared,” J. Opt. Soc. Am. B 13, 2352–2360 (1996).
    [CrossRef]
  9. L.-K. Dai, C. Gu, and P. Yeh, “Effect of position dependent time constant on photorefractive two-wave mixing,” J. Opt. Soc. Am. B 9, 1693–1697 (1992).
    [CrossRef]
  10. P. Réfrégier, L. Solymar, H. Rajbenbach, and J. P. Huignard, “Two beam coupling in photorefractive Bi12SiO20 crystals with moving grating: theory and experiments,” J. Appl. Phys. 58, 45–57 (1985).
    [CrossRef]
  11. N. V. Kukhtarev, V. B. Markov, S. G. Odoulov, M. S. Soskin, and V. L. Vinetskii, “Holographic storage in electrooptic crystals,” Ferroelectrics 22, 949–960 (1979).
    [CrossRef]

1996 (3)

1994 (1)

1992 (2)

1990 (1)

S. Zhivkova and M. Miteva, “Holographic recording in photorefractive crystals with simultaneous electron–hole transport and two active centers,” J. Appl. Phys. 68, 3099–3103 (1990).
[CrossRef]

1986 (2)

E. Strohkendl, J. Jonathan, and R. W. Hellwarth, “Hole–electron competition in photorefractive gratings,” Opt. Lett. 11, 312–314 (1986).
[CrossRef]

G. Valley, “Simultaneous electron/hole transport in photorefractive materials,” J. Appl. Phys. 59, 3363–3366 (1986).
[CrossRef]

1985 (1)

P. Réfrégier, L. Solymar, H. Rajbenbach, and J. P. Huignard, “Two beam coupling in photorefractive Bi12SiO20 crystals with moving grating: theory and experiments,” J. Appl. Phys. 58, 45–57 (1985).
[CrossRef]

1979 (1)

N. V. Kukhtarev, V. B. Markov, S. G. Odoulov, M. S. Soskin, and V. L. Vinetskii, “Holographic storage in electrooptic crystals,” Ferroelectrics 22, 949–960 (1979).
[CrossRef]

Barker, R. C.

Bashaw, M. C.

Brost, G.

S. G. Odoulov, A. N. Shumelyuk, G. Brost, and K. Magde, “Enhancement of beam coupling in the near infrared for tin hypothiodiphosphate,” Appl. Phys. Lett. 21, 752–754 (1996).

Dai, L.-K.

Grabar, A. A.

Gu, C.

Hellwarth, R. W.

Hellwig, U.

Hesselink, L.

Huignard, J. P.

P. Réfrégier, L. Solymar, H. Rajbenbach, and J. P. Huignard, “Two beam coupling in photorefractive Bi12SiO20 crystals with moving grating: theory and experiments,” J. Appl. Phys. 58, 45–57 (1985).
[CrossRef]

Jeganathan, M.

Jonathan, J.

Kukhtarev, N. V.

N. V. Kukhtarev, V. B. Markov, S. G. Odoulov, M. S. Soskin, and V. L. Vinetskii, “Holographic storage in electrooptic crystals,” Ferroelectrics 22, 949–960 (1979).
[CrossRef]

Ma, T.-P.

Magde, K.

S. G. Odoulov, A. N. Shumelyuk, G. Brost, and K. Magde, “Enhancement of beam coupling in the near infrared for tin hypothiodiphosphate,” Appl. Phys. Lett. 21, 752–754 (1996).

Markov, V. B.

N. V. Kukhtarev, V. B. Markov, S. G. Odoulov, M. S. Soskin, and V. L. Vinetskii, “Holographic storage in electrooptic crystals,” Ferroelectrics 22, 949–960 (1979).
[CrossRef]

Miteva, M.

S. Zhivkova and M. Miteva, “Holographic recording in photorefractive crystals with simultaneous electron–hole transport and two active centers,” J. Appl. Phys. 68, 3099–3103 (1990).
[CrossRef]

Odoulov, S. G.

S. G. Odoulov, A. N. Shumelyuk, G. Brost, and K. Magde, “Enhancement of beam coupling in the near infrared for tin hypothiodiphosphate,” Appl. Phys. Lett. 21, 752–754 (1996).

S. G. Odoulov, A. N. Shumelyuk, U. Hellwig, R. A. Rupp, A. A. Grabar, and I. M. Stoyka, “Photorefraction in tin hypothiodiphosphate in the near infrared,” J. Opt. Soc. Am. B 13, 2352–2360 (1996).
[CrossRef]

S. G. Odoulov, A. N. Shumelyuk, U. Hellwig, R. A. Rupp, and A. A. Grabar, “Photorefractive beam coupling in tin hypothiodiphosphate in the near infrared,” Opt. Lett. 21, 752–754 (1996).
[CrossRef] [PubMed]

N. V. Kukhtarev, V. B. Markov, S. G. Odoulov, M. S. Soskin, and V. L. Vinetskii, “Holographic storage in electrooptic crystals,” Ferroelectrics 22, 949–960 (1979).
[CrossRef]

Rajbenbach, H.

P. Réfrégier, L. Solymar, H. Rajbenbach, and J. P. Huignard, “Two beam coupling in photorefractive Bi12SiO20 crystals with moving grating: theory and experiments,” J. Appl. Phys. 58, 45–57 (1985).
[CrossRef]

Réfrégier, P.

P. Réfrégier, L. Solymar, H. Rajbenbach, and J. P. Huignard, “Two beam coupling in photorefractive Bi12SiO20 crystals with moving grating: theory and experiments,” J. Appl. Phys. 58, 45–57 (1985).
[CrossRef]

Rupp, R. A.

Shumelyuk, A. N.

Solymar, L.

P. Réfrégier, L. Solymar, H. Rajbenbach, and J. P. Huignard, “Two beam coupling in photorefractive Bi12SiO20 crystals with moving grating: theory and experiments,” J. Appl. Phys. 58, 45–57 (1985).
[CrossRef]

Soskin, M. S.

N. V. Kukhtarev, V. B. Markov, S. G. Odoulov, M. S. Soskin, and V. L. Vinetskii, “Holographic storage in electrooptic crystals,” Ferroelectrics 22, 949–960 (1979).
[CrossRef]

Stoyka, I. M.

Strohkendl, E.

Valley, G.

G. Valley, “Simultaneous electron/hole transport in photorefractive materials,” J. Appl. Phys. 59, 3363–3366 (1986).
[CrossRef]

Vinetskii, V. L.

N. V. Kukhtarev, V. B. Markov, S. G. Odoulov, M. S. Soskin, and V. L. Vinetskii, “Holographic storage in electrooptic crystals,” Ferroelectrics 22, 949–960 (1979).
[CrossRef]

Yeh, P.

Zhivkova, S.

S. Zhivkova and M. Miteva, “Holographic recording in photorefractive crystals with simultaneous electron–hole transport and two active centers,” J. Appl. Phys. 68, 3099–3103 (1990).
[CrossRef]

Appl. Phys. Lett. (1)

S. G. Odoulov, A. N. Shumelyuk, G. Brost, and K. Magde, “Enhancement of beam coupling in the near infrared for tin hypothiodiphosphate,” Appl. Phys. Lett. 21, 752–754 (1996).

Ferroelectrics (1)

N. V. Kukhtarev, V. B. Markov, S. G. Odoulov, M. S. Soskin, and V. L. Vinetskii, “Holographic storage in electrooptic crystals,” Ferroelectrics 22, 949–960 (1979).
[CrossRef]

J. Appl. Phys. (3)

S. Zhivkova and M. Miteva, “Holographic recording in photorefractive crystals with simultaneous electron–hole transport and two active centers,” J. Appl. Phys. 68, 3099–3103 (1990).
[CrossRef]

G. Valley, “Simultaneous electron/hole transport in photorefractive materials,” J. Appl. Phys. 59, 3363–3366 (1986).
[CrossRef]

P. Réfrégier, L. Solymar, H. Rajbenbach, and J. P. Huignard, “Two beam coupling in photorefractive Bi12SiO20 crystals with moving grating: theory and experiments,” J. Appl. Phys. 58, 45–57 (1985).
[CrossRef]

J. Opt. Soc. Am. B (4)

Opt. Lett. (2)

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Figures (4)

Fig. 1
Fig. 1

Energy-level diagram. CB, conduction band; VB, valence band.

Fig. 2
Fig. 2

Schematic representation of the experimental setup. M’s, mirrors; P, prism; BS, beam splitter; EOM, electro-optic modulator; ND, neutral-density filter; D, detector; OSC, oscillator.

Fig. 3
Fig. 3

Calculated (left) and measured (right) temporal variations of the transmitted signal wave intensity. The total intensity of the two light waves is 15 W/cm2 at λ=1 μm; the fringe spacing Λ=1 μm.

Fig. 4
Fig. 4

Transient (open squares) and steady-state (filled circles) gain factor versus frequency detuning. The solid curve shows the results calculated with the parameters given in Table 1.

Tables (1)

Tables Icon

Table 1 Parameters of the SPS Sample Used in the Calculation

Equations (75)

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E1=A1 exp(-ik1r+iωt),
E2=A2 exp[-ik2r+i(ω+Ω)t],
I=I0[1+m exp(iKx+iΩt)]
N1it=(βn+snI)(N1-N1i)-γnN1in,
N2it=-(βp+spI)N2i+γp(N2-N2i)p,
nt=N1it+1e jnx,
pt=-N2it-1e jpx,
jn=eμnnE+eDn nx,
jp=eμppE-eDp px,
Ex=e0 (p-n+N1i-N2i-Na),
n(t, x)=n0(t)+n1(t)exp(iKx+iΩt)+c.c.,
p(t, x)=p0(t)+p1(t)exp(iKx+iΩt)+c.c.,
N1i(t, x)=N¯1(t)+M1(t)exp(iKx+iΩt)+c.c.,
N2i(t, x)=N¯2(t)+M2(t)exp(iKx+iΩt)+c.c.,
E(t, x)=E1(t)exp(iKx+iΩt)+c.c.
N¯1=const.,
n0=(βn+snI0)(N1-N¯1)/γnN¯1,
N¯2=const.,
p0=(βp+spI0)N¯2/γp(N2-N¯2)
n1=msnI0(N1-N¯1)-M1(βn+snI0+γnn0+iΩ)-M1t/γnN¯1,
p1=mspI0N¯2-M2(βp+spI0+γpp0+iΩ)-M2t/γp(N2-N¯2).
Na=p0-n0+N¯1-N¯2
E1=(p1-n1+M1-M2)c/(0iK).
M1t+iΩM1+e0 μnn0(p1-n1+M1-M2)
-K2Dnn1=0,
M2t+iΩM2-e0 μpp0(p1-n1+M1-M2)
-K2Dpp1=0.
τmn=0eμnn0
(dielectricrelaxationtimeforelectronsinthedark),
τmp=0eμpp0
(dielectricrelaxationtimeforholesinthedark),
τn=1γnN¯1(free-electronlifetime),
τp=1γp(N2-N¯2)(free-holelifetime),
LDn2=Dnτn(electrondiffusionlength),
LDp2=Dpτp(holediffusionlength),
Eqn=eN¯1(N1-N¯1)0KN1(limitingspace-chargefieldthatcouldbecreatedbytheelectrons),
Eqp=eN¯2(N2-N¯2)0KN2(limitingspace-chargefieldthatcouldbecreatedbytheholes),
M1t 1+τnτmn+K2Dnτn
+M11-τn(βn+snI0+γnn0+iΩ)τmn+K2Dnτn(βn+snI0+γnn0+iΩ)+M2t -τpτmp+M2-1+τp(βp+spI0+γpp0+iΩ)τmp
=-τpτmn mspI0N¯2+τnτmn msnI0(N1-N¯1)+K2DnτnmsnI0(N1-N¯1),
M2t 1+τpτmp+K2Dpτp
+M21-τp(βp+spI0+γpp0+iΩ)τmp+K2Dpτp(βp+spI0+γpp0+iΩ)+M1t -τnτmn+M1-1+τn(βn+snI0+γnn0+iΩ)τmn
=-τnτmp msnI0(N1-N¯1)+τpτmp mspI0N¯2+K2DpτpmspI0N¯2.
M1t (1+K2Dnτn)+M1τmn [1+iΩτmn
+K2Dnτnτmn(βn+snI0+γnn0+iΩ)]+M2-1τmn
=K2DnτnmsnI0(N1-N¯1),
M2t (1+K2Dpτp)+M2τmp [1+iΩτmp
+K2Dpτpτmp(βp+spI0+γpp0+iΩ)]+M1-1τmp
=K2DpτpmspI0N¯2,
A1=-B1[1+iΩτmn(1+lDn2K2)+(ED/Eqn)],
A2=-[τmp(1+LDp2K2)]-1,
B1=-[τmn(1+LDn2K2)]-1,
B2=-A2[1+iΩτmp(1+lDp2K2)+(ED/Eqp)],
C1=-(m0/e)KEDB1 snI0βn+snI0,
C2=-(m0/e)KEDA2 spI0βp+spI0,
M1t+A1M1+B1M2=C1,
M2t+A2M1+B2M2=C2.
M1(t)=-B1C2-B2C1α1α2-α1+B2A2 K1 exp(α1t)-α2+B2A2 K2 exp(α2t),
M2(t)=-A2C1-A1C1α1α2+K1 exp(α1t)+K2 exp(α2t).
Esp=ie0K (M2-M1),
K1=α1C2+A1C2-A2C1α1(α1-α2),
K2=-α1C2+A1C2-A2C1α2(α1-α2);
α1,2=-(A1+B2)±[(A1-B1)2+4A2B1]1/22.
Γ=Re[(2πin3reffEsc)/mλ],
M1[1-exp(-A1t)]C1/A1,
M2-[1-exp(-B2t)]C1A2/A1B2.
Γ(t)ED/(1+lSn2K2)1+(τmnΩ)2(1+lDn2K2)2/(1+lSn2K2)2 ×-1+lSp2K2(1+lSp2K2)2+(τmpΩ)2(1+lDp2K2)2 ×1-(τmpΩ)(1+lDp2K2)1+lSp2K2 sin(Ωt)+cos(Ωt)exp-tτmp 1+lSp2K21+lDp2K2+1-(τmnΩ)(1+lDn2K2)1+lSn2K2 sin(Ωt)+cos(Ωt)exp-tτmn 1+lSn2K21+lDn2K2.
Γ(t)ED1+lSn2K2 -1(1+lSp2K2)×1-exp-tτmp 1+lSp2K21+lDp2K2+1-exp-tτmn 1+lSn2K21+lDn2K2.
Γ(t)EDlSp2K2(1+lSn2K2)(1+lSp2K2).
Γ(t)ED(1+lSn2K2)(1+lSn2K2)2+(τmnΩ)2(1+lDn2K2)2×1-τmnΩ 1+lDn2K21+lSn2K2 sin(Ωt)+cos(Ωt)×exp-tτmn 1+lSn2K21+lDn2K2.
Γ(t)ED(1+lSn2K2)[1+(τmnΩ)2(1+lDn2K2)2/(1+lSn2K2)2].
Γ(t)ED(1+lSn2K2).
1+lSp2K2(1+lSp2K2)2+(τmpΩ)2(1+lDp2K2)2
Γ(t)ED/(1+lSn2K2)1+(τmnΩ)2(1+lDn2K2)2/(1+lSn2K2)2×1-cos(Ωt)exp-tτmn 1+lSn2K21+lDn2K2.
4π2n3reffλ cos θkBTe

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