Abstract

Formulas for the evaluation of third-order optical susceptibility of nonlinear composites are derived in the mean-field approximation for four different microstructures corresponding to those described by the Maxwell–Garnett theory, the Bruggeman self-consistent theory, the Sheng theory, and the differential effective medium theory. A commonly encountered error in the literature is pointed out, with the correct formulation given. Anomalous dispersion, i.e., surface plasmon resonance of coated spheres, is identified as the source of large optical nonlinear susceptibility enhancement. Examples are given that demonstrate this microstructural enhancement effect. Comparison with experimental data on AuSiO2 granular films shows that large enhancement in the third-order Kerr-type nonlinear susceptibility can indeed be realized at compositions below the percolation threshold, with prediction of even larger enhancement possible.

© 1998 Optical Society of America

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  1. A. V. Butenko, P. A. Chubakov, Yu. E. Danilova, S. V. Karpov, A. K. Popov, S. G. Rautian, V. P. Safonov, V. V. Slabko, V. M. Shalaev, and M. I. Stockman, Z. Phys. D 17, 283 (1990); M. I. Stockman, L. N. Pandey, L. S. Muratov, and T. F. George, Phys. Rev. Lett 72, 2486 (1994).
    [CrossRef] [PubMed]
  2. D. Stroud and P. M. Hui, Phys. Rev. B 37, 8719 (1988).
    [CrossRef]
  3. D. J. Bergman and D. Stroud, in Solid State Physics, H. Ehrenreich and D. Turnbull, eds. (Academic, Boston, Mass., 1992), Vol. 46, p. 147; G. W. Milton, Appl. Phys. A 26, 1207 (1981); J. Appl. Phys. 52, 5286 (1980).
    [CrossRef]
  4. O. Levy and D. J. Bergman, Physica A 207, 157 (1994).
    [CrossRef]
  5. P. Sheng, Introduction to Wave Scattering, Localization, and Mesoscopic Phenomena (Academic, Boston, Mass., 1995), Chap. 3.
  6. R. W. Cohen, G. D. Cody, M. D. Coutts, and B. Abeles, Phys. Rev. B 8, 3689 (1973).
    [CrossRef]
  7. J. C. M. Garnett, Philos. Trans. R. Soc. London 203, 385 (1904); 205, 237 (1906).
  8. J. E. Sipe and R. W. Boyd, Phys. Rev. A 46, 1614 (1992).
    [CrossRef] [PubMed]
  9. D. A. G. Bruggeman, Ann. Phys. (Leipzig) 24, 636 (1935).
    [CrossRef]
  10. P. Sheng, Phys. Rev. Lett. 45, 60 (1980).
    [CrossRef]
  11. P. Sheng, Phys. Rev. B 41, 4507 (1990).
    [CrossRef]
  12. For a generalized account see A. N. Norris, A. J. Callegari, and P. Sheng, J. Mech. Phys. Solids 33, 525 (1985).
    [CrossRef]
  13. J. H. Weaver and H. P. R. Frederikse, in CRC Handbook of Chemistry and Physics, 74th ed., by D. R. Lide, ed. (CRC, Boca Raton, Fla., 1993/1994) Sec. 12-109.
  14. H. B. Liao, R. F. Xiao, J. S. Fu, P. Yu, G. K. L. Wong, and P. Sheng, Appl. Phys. Lett. 70, 1 (1997).
    [CrossRef]
  15. L. L. Chase and E. W. Van Stryland, in CRC Handbook of Laser Science and Technology, M. J. Weber, ed. (CRC, Boca Raton, Fla., 1995), suppl. 2, p. 269.

1997 (1)

H. B. Liao, R. F. Xiao, J. S. Fu, P. Yu, G. K. L. Wong, and P. Sheng, Appl. Phys. Lett. 70, 1 (1997).
[CrossRef]

1994 (1)

O. Levy and D. J. Bergman, Physica A 207, 157 (1994).
[CrossRef]

1992 (1)

J. E. Sipe and R. W. Boyd, Phys. Rev. A 46, 1614 (1992).
[CrossRef] [PubMed]

1990 (1)

P. Sheng, Phys. Rev. B 41, 4507 (1990).
[CrossRef]

1988 (1)

D. Stroud and P. M. Hui, Phys. Rev. B 37, 8719 (1988).
[CrossRef]

1985 (1)

For a generalized account see A. N. Norris, A. J. Callegari, and P. Sheng, J. Mech. Phys. Solids 33, 525 (1985).
[CrossRef]

1980 (1)

P. Sheng, Phys. Rev. Lett. 45, 60 (1980).
[CrossRef]

1973 (1)

R. W. Cohen, G. D. Cody, M. D. Coutts, and B. Abeles, Phys. Rev. B 8, 3689 (1973).
[CrossRef]

1935 (1)

D. A. G. Bruggeman, Ann. Phys. (Leipzig) 24, 636 (1935).
[CrossRef]

Abeles, B.

R. W. Cohen, G. D. Cody, M. D. Coutts, and B. Abeles, Phys. Rev. B 8, 3689 (1973).
[CrossRef]

Bergman, D. J.

O. Levy and D. J. Bergman, Physica A 207, 157 (1994).
[CrossRef]

Boyd, R. W.

J. E. Sipe and R. W. Boyd, Phys. Rev. A 46, 1614 (1992).
[CrossRef] [PubMed]

Bruggeman, D. A. G.

D. A. G. Bruggeman, Ann. Phys. (Leipzig) 24, 636 (1935).
[CrossRef]

Callegari, A. J.

For a generalized account see A. N. Norris, A. J. Callegari, and P. Sheng, J. Mech. Phys. Solids 33, 525 (1985).
[CrossRef]

Cody, G. D.

R. W. Cohen, G. D. Cody, M. D. Coutts, and B. Abeles, Phys. Rev. B 8, 3689 (1973).
[CrossRef]

Cohen, R. W.

R. W. Cohen, G. D. Cody, M. D. Coutts, and B. Abeles, Phys. Rev. B 8, 3689 (1973).
[CrossRef]

Coutts, M. D.

R. W. Cohen, G. D. Cody, M. D. Coutts, and B. Abeles, Phys. Rev. B 8, 3689 (1973).
[CrossRef]

Fu, J. S.

H. B. Liao, R. F. Xiao, J. S. Fu, P. Yu, G. K. L. Wong, and P. Sheng, Appl. Phys. Lett. 70, 1 (1997).
[CrossRef]

Hui, P. M.

D. Stroud and P. M. Hui, Phys. Rev. B 37, 8719 (1988).
[CrossRef]

Levy, O.

O. Levy and D. J. Bergman, Physica A 207, 157 (1994).
[CrossRef]

Liao, H. B.

H. B. Liao, R. F. Xiao, J. S. Fu, P. Yu, G. K. L. Wong, and P. Sheng, Appl. Phys. Lett. 70, 1 (1997).
[CrossRef]

Norris, A. N.

For a generalized account see A. N. Norris, A. J. Callegari, and P. Sheng, J. Mech. Phys. Solids 33, 525 (1985).
[CrossRef]

Sheng, P.

H. B. Liao, R. F. Xiao, J. S. Fu, P. Yu, G. K. L. Wong, and P. Sheng, Appl. Phys. Lett. 70, 1 (1997).
[CrossRef]

P. Sheng, Phys. Rev. B 41, 4507 (1990).
[CrossRef]

For a generalized account see A. N. Norris, A. J. Callegari, and P. Sheng, J. Mech. Phys. Solids 33, 525 (1985).
[CrossRef]

P. Sheng, Phys. Rev. Lett. 45, 60 (1980).
[CrossRef]

Sipe, J. E.

J. E. Sipe and R. W. Boyd, Phys. Rev. A 46, 1614 (1992).
[CrossRef] [PubMed]

Stroud, D.

D. Stroud and P. M. Hui, Phys. Rev. B 37, 8719 (1988).
[CrossRef]

Wong, G. K. L.

H. B. Liao, R. F. Xiao, J. S. Fu, P. Yu, G. K. L. Wong, and P. Sheng, Appl. Phys. Lett. 70, 1 (1997).
[CrossRef]

Xiao, R. F.

H. B. Liao, R. F. Xiao, J. S. Fu, P. Yu, G. K. L. Wong, and P. Sheng, Appl. Phys. Lett. 70, 1 (1997).
[CrossRef]

Yu, P.

H. B. Liao, R. F. Xiao, J. S. Fu, P. Yu, G. K. L. Wong, and P. Sheng, Appl. Phys. Lett. 70, 1 (1997).
[CrossRef]

Ann. Phys. (Leipzig) (1)

D. A. G. Bruggeman, Ann. Phys. (Leipzig) 24, 636 (1935).
[CrossRef]

Appl. Phys. Lett. (1)

H. B. Liao, R. F. Xiao, J. S. Fu, P. Yu, G. K. L. Wong, and P. Sheng, Appl. Phys. Lett. 70, 1 (1997).
[CrossRef]

J. Mech. Phys. Solids (1)

For a generalized account see A. N. Norris, A. J. Callegari, and P. Sheng, J. Mech. Phys. Solids 33, 525 (1985).
[CrossRef]

Phys. Rev. A (1)

J. E. Sipe and R. W. Boyd, Phys. Rev. A 46, 1614 (1992).
[CrossRef] [PubMed]

Phys. Rev. B (3)

R. W. Cohen, G. D. Cody, M. D. Coutts, and B. Abeles, Phys. Rev. B 8, 3689 (1973).
[CrossRef]

D. Stroud and P. M. Hui, Phys. Rev. B 37, 8719 (1988).
[CrossRef]

P. Sheng, Phys. Rev. B 41, 4507 (1990).
[CrossRef]

Phys. Rev. Lett. (1)

P. Sheng, Phys. Rev. Lett. 45, 60 (1980).
[CrossRef]

Physica A (1)

O. Levy and D. J. Bergman, Physica A 207, 157 (1994).
[CrossRef]

Other (6)

P. Sheng, Introduction to Wave Scattering, Localization, and Mesoscopic Phenomena (Academic, Boston, Mass., 1995), Chap. 3.

D. J. Bergman and D. Stroud, in Solid State Physics, H. Ehrenreich and D. Turnbull, eds. (Academic, Boston, Mass., 1992), Vol. 46, p. 147; G. W. Milton, Appl. Phys. A 26, 1207 (1981); J. Appl. Phys. 52, 5286 (1980).
[CrossRef]

J. C. M. Garnett, Philos. Trans. R. Soc. London 203, 385 (1904); 205, 237 (1906).

A. V. Butenko, P. A. Chubakov, Yu. E. Danilova, S. V. Karpov, A. K. Popov, S. G. Rautian, V. P. Safonov, V. V. Slabko, V. M. Shalaev, and M. I. Stockman, Z. Phys. D 17, 283 (1990); M. I. Stockman, L. N. Pandey, L. S. Muratov, and T. F. George, Phys. Rev. Lett 72, 2486 (1994).
[CrossRef] [PubMed]

J. H. Weaver and H. P. R. Frederikse, in CRC Handbook of Chemistry and Physics, 74th ed., by D. R. Lide, ed. (CRC, Boca Raton, Fla., 1993/1994) Sec. 12-109.

L. L. Chase and E. W. Van Stryland, in CRC Handbook of Laser Science and Technology, M. J. Weber, ed. (CRC, Boca Raton, Fla., 1995), suppl. 2, p. 269.

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Figures (5)

Fig. 1
Fig. 1

Solution of Eq. (62) plotted as a function of volume fraction p. The shaded areas correspond to continuous spectra, and the thick lines indicate delta functions.

Fig. 2
Fig. 2

Spectral densities for the four different microstructures at p=0.4.

Fig. 3
Fig. 3

Composite effective dielectric constants for the four different microstructures calculated at λ=620 nm with material parameters given in the text. Note the anomalous dispersions exhibited in the MG and the Sheng theories.

Fig. 4
Fig. 4

Third-order nonlinear enhancement factor |βi| plotted as function of volume fraction p. Left, the enhancement factor in material 1; right, the enhancement factor in material 2. The calculation is for λ=620 nm and material parameter values given in the text.

Fig. 5
Fig. 5

Comparison of the measured third-order nonlinear susceptibility |χ¯(3)| in AuSiO2 composites with theoretical values calculated from the Sheng theory at λ=530 nm. The dashed curve is calculated from the same parameters at λ=620 nm. The compositions where the maximum occurs do not coincide. However, the maximum theory and experimental values of |χ¯(3)| are very close.

Equations (89)

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D=E+A|E|2E+BE2E*,
D=[+χ(3)|E|2]E.
1V dVD=1V dV[E+χ(3)|E|2E]¯E0+χ¯(3)|E0|2E0.
D=0,
×E=0.
Dlin=0,
×Elin=0,
Dlin=Elin.
E=-ϕ,
Elin=-ϕlin.
ϕ|z=0=ϕlin|z=0=0,ϕ|z=L=ϕlin|z=L=-E0L,
δϕ|boundaries=0.
1V dVDE0=-1V dVDϕ0=-1V dV(Dϕ0)+1V dVϕ0D=-1V dSDϕ0=-1V dSDϕ=-1V dVDϕ=1V dVDE,
1V dVDE=¯E02+χ¯(3)|E0|2E02.
1V dVDE*=¯E02+χ¯(3)|E0|2E02
dVDE*=dVDE.
1V dVElin2+2 1V dVElinδE+1V dVχ(3)|Elin|2Elin2
=¯E02+χ¯(3)|E0|2E02.
dVElinδE=-dVElinδϕ=-dV(Elinδϕ)+dVδϕ(Elin).
1V dVElin2+1V dVχ(3)|Elin|2Elin2
=¯E02+χ¯(3)|E0|2E02,
¯E02=1V dVElin2,
χ¯(3)|E0|2E02=1V dVχ(3)|Elin|2Elin2=χ1(3) 1V 1dV|Elin|2Elin2+χ2(3) 1V 2dV|Elin|2Elin2p1χ1(3)|Elin|2Elin21+p2χ2(3)|Elin|2Elin22(β1χ1(3)+β2χ2(3))|E0|2E02,
|Elin|2Elin2i|Elin|2iElin2i,
χ¯(3)=β1χ1(3)+β2χ2(3),
βi=pi |Elin|2iElin2i|E0|2E02.
(ϕlin)=0.
(Δϕ)=-(Δϕlin),
Δ¯E02=1V dV(ΔElin2+2ElinΔE).
Δ¯E02=1V dVΔElin2.
Δ¯E02=Δ1 1V 1dVElin2
¯1 E02=1V 1dVElin2=p1Elin21.
p2Elin22=¯2 E02.
ϕ=-ss-Γ z=-n sn|zs-sn ϕn,
Γ=dVθ(r)G(r-r),
p1|Elin|2=1V 1dV|Elin|2=1V 1dVϕ*ϕ=1V nm |s|2z|nm|z(s*-sn)(s-sm)×1dVϕn* ϕm=n |s|2fn|s-sn|2 E02.
1V dV|Elin|2=¯E02,
1 1V 1dV|Elin|2+2 1V 2dV|Elin|2=¯E02.
p2|Elin|22=¯2-12 p1|Elin|21E02=1-n fns-sn-|s|2fn1-1/s|s-sn|2E02=1-n (|s|2-sn)fn|s-sn|2E02.
¯=21-n fns-sn2[1-F(s)],
p1Elin21=n s2fn(s-sn)2 E02,
p2Elin22=1-n (s2-sn)fn(s-sn)2E02.
p1|Elin|21=dx |s|2μ(x)|s-x|2 E02,
p2|Elin|22=1-dx (|s|2-x)μ(x)|s-x|2E02,
p1Elin21=dx s2μ(x)(s-x)2 E02,
p2Elin22=1-dx (s2-x)μ(x)(s-x)2E02.
F(s)= μ(x)s-x dx.
μ(x)=-1π Im[F(x+i0+)].
¯-2¯+22=p 1-21+22.
F(s)=ps-(1-p)/3.
μ(x)=pδ[x-(1-p)/3].
Elin21=1p ¯1 E02=s2[s-(1-p)/3]2 E02,
Elin22=11-p ¯2 E02=(s-1/3)2+2p/9[s-(1-p)/3]2 E02,
|Elin|21=|s|2|s-(1-p)/3]2 E02,
|Elin|22=11-p 1-(|s|2-(1-p)/3)p|s-(1-p)/3|2 E02,=|s-1/3|2+2p/9|s-(1-p)/3|2 E02.
p 1-¯1+2¯+(1-p) 2-¯2+2¯=0.
F(s)=14s {-1+3p+3s-3[(s-x1)(s-x2)]1/2}.
(1-3p)2-6(1+p)x+9x2=0.
x1=1/3{1+p-2[2p(1-p)]1/2},
x2=1/3{1+p+2[2p(1-p)]1/2}.
μ(x)=(3p-1)2 θ(3p-1)δ(x)+34πx [(x-x1)(x2-x)]1/2x1<x<x20otherwise.
Elin21=1p ¯1 E02=1p -1+3p4+-1+6p-9p2+3s+3ps12[(s-x1)(s-x2)]1/2E02,
Elin22=11-p ¯2E02=11-p 2-3p4+-2-9p+9p2+6s-3ps12[(s-x1)(s-x2)]1/2E02,
|Elin|21=1p  |s|2μ(x)|s-x|2 dxE02,
|Elin|22=11-p 1- (|s|2-x)μ(x)|s-x|2E02.
fD1+(1-f )D2=0.
D1=(¯-2)(1+22)+(2-1)(¯+22)p(2¯+2)(1+22)+2(¯-2)(2-1)p,
D2=(¯-1)(21+2)+(¯+21)(1-2)(1-p)(2¯+1)(21+2)+2(¯-1)(1-2)(1-p),
f=(1-p1/3)3[1-(1-p)1/3]3+(1-p1/3)3,
¯2=-b(s)2a(s)±[b(s)2-4a(s)c(s)]1/22a(s),
a(s)=2s(3s-3+p)(3s-1+p),
b(s)=2(2-3f )(1-p)p+s(-3-5p+2p2+12s+3ps-9s2),
c(s)=(1-s)(2p+4p2-3s-12ps+9s2).
F(s)=1+b(s)2a(s)-27[(s-x1-)(s-x1+)(s-x2-)(s-x2+)(s-x3-)(s-x3+)]1/22a(s).
s6+c5s5+c4s4+c3s3+c2s2+c1s+c0=0,
c0=4(3f-2)2(1-p)2p2729,
c1=4(f-2)(1-p)(3-p)p(1+2p)243,
c2=19+2(31-8f )p81+(12 f-11)p281+4(f-7)p381+4p481,
c3=-89+2(2 f-27)p27+2(7-2 f )p227+4p327,
c4=229+2p-p23,
c5=-83-2p3.
s2(s-1/3)2(s-1)2=0,
s2(s-2/3)2(s-1)2=0,
μ(x)=(2-3f )p3-p θ(2-3f )δ(x)+(3f-1)p2 θ(3f-1)δx-1-p3+(2-3f )(1-p)p2(3-p)×θ(2-3f )δx-1-p3+272|a(x)|π [(x-x1-)×(x1+-x)(x-x2-)(x2+-x)(x-x3-)(x3+-x)]1/2×{[θ(x-x1-)-θ(x-x1+)]+[θ(x-x2-)-θ(x-x2+)]+[θ(x-x3-)-θ(x-x3+)]}.
1-p=¯-12-1 2¯1/3,
μ(x)=3(1-p)(1-x)1/3(2x)4/3π [(1+B)1/3-(1-B)1/3]xl<x<xu0otherwise,
B=1-4(1-p)327(1-x)2x1/2.
xl=23 1-cosπ-ϕ3,
xu=23 1-cosπ+ϕ3,

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