Abstract

Rigorous coupled-wave reflection grating analysis and Kukhtarev’s equations have been solved in the time domain to determine the optical intensity, electron density, and dielectric modulation in BaTiO3. A novel Green’s-function approach has been developed to analyze Kukhtarev’s material equations. The Green’s-function approach allowed Kukhtarev’s equations to be reduced to a matrix form, from which the electron density could be obtained. A temporal state variable matrix equation was also developed from which full time-dependent solutions of Kukhtarev’s equations could be determined. An electron balance equation was developed from which the different terms in Kukhtarev’s equation could be studied and compared. Numerical simulations were carried out that showed the growth of a photorefractive BaTiO3 reflection grating. The simulation showed that an asymmetric, blazelike pattern resulted for the dielectric modulation. The blazelike pattern was shown to arise from spatial differentiation of a cusplike shape that the electron-density function assumed for its solution.

[Optical Society of America ]

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References

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  1. F. Zhao , K. Sayano , H. Miller , J. Corwin , and N. Werner , Improved UV photorefractive LiNbO 3 crystals and their applications in Ca K-line narrow-bandwidth imaging filters , in Photorefractive Fiber and Crystal Devices: Materials, Optical Properties, and Applications III , F. T. Yu and S. Yin, eds., Proc. SPIE PSISDG 3137 , 246 253 ( 1997
    [CrossRef]
  2. T. Honda , Hexagonal pattern formation due to counterpropagation in KNbO 3 , Opt. Lett. OPLEDP 18 , 598 600 ( 1993
    [CrossRef]
  3. X. Yi , S. H. Lin , P. Yeh , and K. Y. Hsu , Contradirectional 2-wave mixing with partially coherent waves in photorefractive crystals , Opt. Lett. OPLEDP 21 , 1123 1125 ( 1996
    [CrossRef] [PubMed]
  4. P. P. Banerjee and J. M. Jarem , Transient wave mixing and recording kinetics in photorefractive barium titanate: a nonlinear coupled-mode approach , Opt. Eng. OPEGAR 34 , 2254 2260 ( 1995
    [CrossRef]
  5. J. Jarem and P. P. Banerjee , A nonlinear transient analysis of two- and multi-wave mixing in a photorefractive material using rigorous coupled mode diffraction theory , Opt. Commun. OPCOB8 132 , 825 842 ( 1996
    [CrossRef]
  6. J. M. Jarem and P. P. Banerjee , Exact, dynamical analysis of the Kukhtarev equations in photorefractive barium titanate using rigorous coupled-wave diffraction theory , J. Opt. Soc. Am. A JOAOD6 13 , 819 831 ( 1996
    [CrossRef]
  7. P. P. Banerjee and J. M. Jarem , Effect of induced longitudinal electrostatic field components during two-wave and multiwave mixing in photorefractive materials , J. Opt. Soc. Am. B JOBPDE 13 , 2610 2614 ( 1996
    [CrossRef]
  8. M. G. Moharam and T. K. Gaylord , Coupled-wave analysis of reflection gratings , Appl. Opt. APOPAI 20 , 240 244 ( 1981
    [CrossRef] [PubMed]
  9. M. G. Moharam and T. K. Gaylord , Chain-matrix analysis of arbitrary-thickness dielectric reflection gratings , J. Opt. Soc. Am. JOSAAH 72 , 187 190 ( 1982
    [CrossRef]
  10. Z. Zylberberg and E. Marom , Rigorous coupled-wave analysis of pure reflection gratings , J. Opt. Soc. Am. JOSAAH 73 , 392 401 ( 1983
    [CrossRef]
  11. E. N. Glytsis and T. K. Gaylord , Rigorous three-dimensional coupled-wave diffraction analysis of single and cascaded anisotropic gratings , J. Opt. Soc. Am. A JOAOD6 4 , 2061 2080 ( 1987
    [CrossRef]
  12. J. Jarem and P. P. Banerjee , Time domain state variable analysis of induced reflection gratings in photorefractive materials , in Holographic Materials IV , T. Trout, ed., Proc. SPIE PSISDG 3294 , 161 170 ( 1998
    [CrossRef]

Corwin, J

F. Zhao , K. Sayano , H. Miller , J. Corwin , and N. Werner , Improved UV photorefractive LiNbO 3 crystals and their applications in Ca K-line narrow-bandwidth imaging filters , in Photorefractive Fiber and Crystal Devices: Materials, Optical Properties, and Applications III , F. T. Yu and S. Yin, eds., Proc. SPIE PSISDG 3137 , 246 253 ( 1997
[CrossRef]

Jarem, J

J. Jarem and P. P. Banerjee , A nonlinear transient analysis of two- and multi-wave mixing in a photorefractive material using rigorous coupled mode diffraction theory , Opt. Commun. OPCOB8 132 , 825 842 ( 1996
[CrossRef]

Werner, N

F. Zhao , K. Sayano , H. Miller , J. Corwin , and N. Werner , Improved UV photorefractive LiNbO 3 crystals and their applications in Ca K-line narrow-bandwidth imaging filters , in Photorefractive Fiber and Crystal Devices: Materials, Optical Properties, and Applications III , F. T. Yu and S. Yin, eds., Proc. SPIE PSISDG 3137 , 246 253 ( 1997
[CrossRef]

Other (12)

P. P. Banerjee and J. M. Jarem , Transient wave mixing and recording kinetics in photorefractive barium titanate: a nonlinear coupled-mode approach , Opt. Eng. OPEGAR 34 , 2254 2260 ( 1995
[CrossRef]

J. Jarem and P. P. Banerjee , A nonlinear transient analysis of two- and multi-wave mixing in a photorefractive material using rigorous coupled mode diffraction theory , Opt. Commun. OPCOB8 132 , 825 842 ( 1996
[CrossRef]

F. Zhao , K. Sayano , H. Miller , J. Corwin , and N. Werner , Improved UV photorefractive LiNbO 3 crystals and their applications in Ca K-line narrow-bandwidth imaging filters , in Photorefractive Fiber and Crystal Devices: Materials, Optical Properties, and Applications III , F. T. Yu and S. Yin, eds., Proc. SPIE PSISDG 3137 , 246 253 ( 1997
[CrossRef]

J. Jarem and P. P. Banerjee , Time domain state variable analysis of induced reflection gratings in photorefractive materials , in Holographic Materials IV , T. Trout, ed., Proc. SPIE PSISDG 3294 , 161 170 ( 1998
[CrossRef]

E. N. Glytsis and T. K. Gaylord , Rigorous three-dimensional coupled-wave diffraction analysis of single and cascaded anisotropic gratings , J. Opt. Soc. Am. A JOAOD6 4 , 2061 2080 ( 1987
[CrossRef]

T. Honda , Hexagonal pattern formation due to counterpropagation in KNbO 3 , Opt. Lett. OPLEDP 18 , 598 600 ( 1993
[CrossRef]

J. M. Jarem and P. P. Banerjee , Exact, dynamical analysis of the Kukhtarev equations in photorefractive barium titanate using rigorous coupled-wave diffraction theory , J. Opt. Soc. Am. A JOAOD6 13 , 819 831 ( 1996
[CrossRef]

M. G. Moharam and T. K. Gaylord , Coupled-wave analysis of reflection gratings , Appl. Opt. APOPAI 20 , 240 244 ( 1981
[CrossRef] [PubMed]

P. P. Banerjee and J. M. Jarem , Effect of induced longitudinal electrostatic field components during two-wave and multiwave mixing in photorefractive materials , J. Opt. Soc. Am. B JOBPDE 13 , 2610 2614 ( 1996
[CrossRef]

X. Yi , S. H. Lin , P. Yeh , and K. Y. Hsu , Contradirectional 2-wave mixing with partially coherent waves in photorefractive crystals , Opt. Lett. OPLEDP 21 , 1123 1125 ( 1996
[CrossRef] [PubMed]

M. G. Moharam and T. K. Gaylord , Chain-matrix analysis of arbitrary-thickness dielectric reflection gratings , J. Opt. Soc. Am. JOSAAH 72 , 187 190 ( 1982
[CrossRef]

Z. Zylberberg and E. Marom , Rigorous coupled-wave analysis of pure reflection gratings , J. Opt. Soc. Am. JOSAAH 73 , 392 401 ( 1983
[CrossRef]

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Figures (6)

Fig. 1
Fig. 1

Geometry of the BaTiO3 photorefractive (PR) crystal.

Fig. 2
Fig. 2

Evolution of the dielectric modulation function Δε shown from its initial value (dashed curve). INC, incident.

Fig. 3
Fig. 3

Reflected diffraction efficiency shown as a function of time.

Fig. 4
Fig. 4

Different terms of Eq. (33), which has been named the electron-density balance equation, shown at the time 30.1 ms. A, B, and C refer, respectively, to the expressions i[Gi(Γ3/Γ2) ×(ND/NA-1)(δi,0+Ii/C)]exp(jiκy) [see RHS of Eq. (33)], i(-Gi[(Γ1-Γ4)/Γ4]{i[(dΔεi-i/dy)+j(i-i)κΔεi-i]ni}) ×exp(jiκy) [see LHS of Eq. (33)], and n=ini exp(jiκy) [see LHS of Eq. (33)]. These terms are the most significant terms in Eq. (33).

Fig. 5
Fig. 5

Time derivative of the dielectric modulation function. This term is calculated in Eq. (30). yloc=-y-123Λ. The time derivative of the dielectric modulation function to a very good approximation satisfies dΔε/dt=-Γ2(dn/dy).

Fig. 6
Fig. 6

A, Maximum; B, dc level; and C, minimum optical intensity envelopes shown as a function of time and of distance across the crystal.

Equations (46)

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S=i(Sxia^x+Syia^y)exp(-jkxix+jξiy),
U=i(Uzia^z)exp(-jkxix+jξiy),
×S=-jU,×U=jε=S,ε==εxxεxy0εyxεyy000εzz.
V_y=a11̲̲a12̲̲a21̲̲a22̲̲V_,
a11̲̲=j(K̲̲εyy̲̲-1εyx̲̲-ξ=),a12̲̲=j(-K̲̲εyy̲̲-1K̲̲+I̲̲),
a21̲̲=j(εxx̲̲-εxy̲̲εyy̲̲-1εyx̲̲),a22̲̲=j(-ξ=+εxy̲̲εyy̲̲-1K̲̲),
K̲̲=[kxiδi,i],εαβ̲̲=(εαβi-i],(α, β)=(x, y),
ξ==[ξiδi,i],δi,i=1,i=i0,ii.
Sxl(2)=i=-MTi=MTn=1NTCnlSxilexp(jξi+qnl)yexp(-jkxix),
Uzl(2)=i=-MTi=MTn=1NTCnlUzilexp(jξi+qnl)yexp(-jkxix).
Hz(1)=i=-MTi=MT[H0δi,0 exp(jky10y)+ri exp(-jky1iy)]exp(-jkxix),
Hz(3)=i=-MTi=MT[ti exp(-jky3iy)]exp(-jkxix),
I(y)ExlExl*+EylEyl*.
Γ2 2ny2+Γ1Δε ny+{-Γ3[1+I(y)/C]+(Γ1-Γ4) Δεy-Γ3Γ4(1+n)}n
=Γ3nt-[1+I(y)/C]NDNA-1
+[1+I(y)/C] Δεy,
Δεt=-Γ1nΔε-Γ2 ny.
I(y)=i=-Mci=McIi(y)exp(jiκy),
n(y)=i=-Mci=Mcni(y)exp(jiκy),
Δε(y)=i=-Mci=McΔεi(y)exp(jiκy),
i2niy2+2jiκ niy-[α2+(iκ)2]niexp(jiκy)
=iHi exp(jiκy),
α=Γ3Γ2 (1+Γ4)1/2,
Hi=iTi-ini+Fi
Ti=-jiκ Γ1Γ2 Δεi+Γ3Γ2 (Ii/C)-Γ1-Γ4Γ2 Δεiy+jiκΔεi+Γ3Γ4Γ2 ni,
Fi=-Γ1Γ2 iΔεi-i niy+Γ3Γ2 nit-(δi,0+Ii/C)NDNA-1+1Γ2 iΔεi-iy+j(i-i)κΔεi-i×(δi,0+Ii/C).
2ni(y)y2+2jiκ ni(y)y-[α2+(iκ)2]ni(y)=Hi(y),
i=-MC,,MC.
ni(0)=ni(-L)=0,i=-MC,,MC.
gi(y|y)=gi-(y|y),-Ly<ygi+(y|y),y<y0,
gi-(y|y)={exp[s1(y+L)]-exp[s2(y+L)]}{exp[-s2y]-exp[-s1y]}{exp[s1L]-exp[s2L]}(s1-s2),
gi+(y|y)={exp[s1y]-exp[s2y]}{exp[s1L-s2y]-exp[s2L-s1y]}{exp[s1L]-exp[s2L]}(s1-s2),
ni(y)=-L0gi(y|y)Hi(y)dy.
gi(y|y)=-1s1-s2 exp[s1(y-y)],-Ly<yexp[s2(y-y)],y<y0.
ni(y)=-GiHi(y),
Gi=1α2+(iκ)2.
i[δi,i+GiTi-i]ni=iLi,ini=-GiFi
ni+i[GiTi-i]ni=-GiFi.
ni(y)=iLi,i-1(-GiFi),
Δεit=-Γ1i(ni-iΔεi)-Γ2niy+jiκni.
Δεit+i[Ai,iΔεi]=fi,
Ai,i=Γ1ni-i+(Γ1jiκ)iLi,i-1Gi ni-iy+(iiκ2)iLi,i-1Gi[δi-i,0+Li-i/C],
fi=-Γ2 niy-(Γ2jiκ)iLi,i-1Gi×-Γ3Γ2 nit+Γ3Γ2 [δi,0+Ii/C]NDNA-1-1Γ2 i Δεi-iy [δi,0+Ii/C].
n(y)+ii(GiTi-i)niexp(jiκy)
=i[-GiFi]exp(jiκy).
Δε(y, tF)̲=n=12MC+1Δεn(y, tI)exp(-λnδt)+fn1-exp(-λnδt)λnVn,

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