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.

[Optical Society of America ]

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  1. E. Strohkendl , J. Jonathan , and R. W. Hellwarth , Hole electron competition in photorefractive gratings , Opt. Lett. OPLEDP 11 , 312 314 ( 1986
    [CrossRef]
  2. G. Valley , Simultaneous electron/hole transport in photorefractive materials , J. Appl. Phys. JAPIAU 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 JOBPDE 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 JOBPDE 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. APPLAB 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. JAPIAU 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. OPLEDP 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 JOBPDE 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 JOBPDE 9 , 1693 1697 ( 1992
    [CrossRef]
  10. P. Re fre gier , L. Solymar , H. 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 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 FEROA8 22 , 949 960 ( 1979
    [CrossRef]

Other (11)

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

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

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 JOBPDE 11 , 1743 1757 ( 1994
[CrossRef]

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 JOBPDE 9 , 1666 1672 ( 1992
[CrossRef]

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. APPLAB 21 , 752 754 ( 1996

S. Zhivkova and M. Miteva , Holographic recording in photorefractive crystals with simultaneous electron hole transport and two active centers , J. Appl. Phys. JAPIAU 68 , 3099 3103 ( 1990
[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. OPLEDP 21 , 752 754 ( 1996
[CrossRef] [PubMed]

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 JOBPDE 13 , 2352 2360 ( 1996
[CrossRef]

L.-K. Dai , C. Gu , and P. Yeh , Effect of position dependent time constant on photorefractive two-wave mixing , J. Opt. Soc. Am. B JOBPDE 9 , 1693 1697 ( 1992
[CrossRef]

P. Re fre gier , L. Solymar , H. 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 57 ( 1985
[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 , 949 960 ( 1979
[CrossRef]

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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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