Abstract

We present a theoretical description of composite nonlinear optical materials having the form of a layered structure of two or more components that differ both in their linear and nonlinear optical properties. We assume that the thickness of each layer is much smaller than an optical wavelength. We present explicit predictions for the second-order nonlinear optical susceptibilities describing second-harmonic generation and the Pockels effect and for the third-order susceptibility describing the nonlinear index of refraction. We find that under experimentally realizable conditions the nonlinear susceptibility of such a composite can exceed those of its constituent materials.

© 1994 Optical Society of America

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  1. J. C. Maxwell Garnett, Philos. Trans. R. Soc. London 203, 385 (1904); Ser. A 205, 237 (1906).
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  2. K. C. Rustagi and C. Flytzanis, Opt. Lett. 9, 344 (1984).
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  6. G. S. Agarwal and S. Dutta Gupta, Phys. Rev. A 38, 5678 (1988).
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  7. J. W. Haus, R. Inguva, and C. M. Bowden, Phys. Rev. A 40, 5729 (1989).
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    [CrossRef] [PubMed]
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  12. A. V. Butenko, W. M. Shalaev, and M. I. Stockman, Sov. Phys. JETP 7, 60 (1988).
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    [CrossRef]
  15. Note that in the notation of Ref. 10 we have set ∊n= 1 in writing Eq. (4) of the present paper, that is, in the present paper we do not treat one medium as the host and the other as the inclusion; both are formally treated as inclusions embedded in vacuum.
  16. D. S. Bethune, J. Opt. Soc. Am. B 8, 367 (1991).
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1992 (1)

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

1991 (1)

1990 (2)

N. C. Kothari, Phys. Rev. A 41, 4486 (1990).
[CrossRef] [PubMed]

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).
[CrossRef]

1989 (2)

1988 (3)

G. S. Agarwal and S. Dutta Gupta, Phys. Rev. A 38, 5678 (1988).
[CrossRef] [PubMed]

V. M. Shalaev and M. I. Stockman, Z. Phys. D 10, 71 (1988); A. V. Butenko, V. M. Shalaev, and M. I. Stockman, Z. Phys. D 10, 81 (1988).
[CrossRef]

A. V. Butenko, W. M. Shalaev, and M. I. Stockman, Sov. Phys. JETP 7, 60 (1988).

1987 (1)

1986 (1)

1985 (1)

1984 (1)

1904 (1)

J. C. Maxwell Garnett, Philos. Trans. R. Soc. London 203, 385 (1904); Ser. A 205, 237 (1906).
[CrossRef]

Agarwal, G. S.

G. S. Agarwal and S. Dutta Gupta, Phys. Rev. A 38, 5678 (1988).
[CrossRef] [PubMed]

Bethune, D. S.

Bloemer, M.

Bowden, C. M.

Boyd, R. W.

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

Butenko, A. V.

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).
[CrossRef]

A. V. Butenko, W. M. Shalaev, and M. I. Stockman, Sov. Phys. JETP 7, 60 (1988).

Chubakov, P. A.

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).
[CrossRef]

Danilova, Yu. E.

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).
[CrossRef]

Dutta Gupta, S.

G. S. Agarwal and S. Dutta Gupta, Phys. Rev. A 38, 5678 (1988).
[CrossRef] [PubMed]

Flytzanis, C.

Hache, F.

Haus, J. W.

Inguva, R.

Kalyaniwalla, N.

Karpov, S. V.

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).
[CrossRef]

Kothari, N. C.

N. C. Kothari, Phys. Rev. A 41, 4486 (1990).
[CrossRef] [PubMed]

Lukasik, J.

Maxwell Garnett, J. C.

J. C. Maxwell Garnett, Philos. Trans. R. Soc. London 203, 385 (1904); Ser. A 205, 237 (1906).
[CrossRef]

Popov, A. K.

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).
[CrossRef]

Rautian, S. G.

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).
[CrossRef]

Ricard, D.

Roussignol, Ph.

Rustagi, K. C.

Safonov, V. P.

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).
[CrossRef]

Shalaev, V. M.

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).
[CrossRef]

V. M. Shalaev and M. I. Stockman, Z. Phys. D 10, 71 (1988); A. V. Butenko, V. M. Shalaev, and M. I. Stockman, Z. Phys. D 10, 81 (1988).
[CrossRef]

Shalaev, W. M.

A. V. Butenko, W. M. Shalaev, and M. I. Stockman, Sov. Phys. JETP 7, 60 (1988).

Sipe, J. E.

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

J. van Kranendonk and J. E. Sipe, in Progress in Optics XV, E. Wolf, ed. (North-Holland, New York, 1977), p. 245.
[CrossRef]

Slabko, V. V.

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).
[CrossRef]

Stockman, M. I.

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).
[CrossRef]

A. V. Butenko, W. M. Shalaev, and M. I. Stockman, Sov. Phys. JETP 7, 60 (1988).

V. M. Shalaev and M. I. Stockman, Z. Phys. D 10, 71 (1988); A. V. Butenko, V. M. Shalaev, and M. I. Stockman, Z. Phys. D 10, 81 (1988).
[CrossRef]

van Kranendonk, J.

J. van Kranendonk and J. E. Sipe, in Progress in Optics XV, E. Wolf, ed. (North-Holland, New York, 1977), p. 245.
[CrossRef]

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

Opt. Lett. (2)

Philos. Trans. R. Soc. London (1)

J. C. Maxwell Garnett, Philos. Trans. R. Soc. London 203, 385 (1904); Ser. A 205, 237 (1906).
[CrossRef]

Phys. Rev. A (4)

N. C. Kothari, Phys. Rev. A 41, 4486 (1990).
[CrossRef] [PubMed]

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

G. S. Agarwal and S. Dutta Gupta, Phys. Rev. A 38, 5678 (1988).
[CrossRef] [PubMed]

J. W. Haus, R. Inguva, and C. M. Bowden, Phys. Rev. A 40, 5729 (1989).
[CrossRef] [PubMed]

Sov. Phys. JETP (1)

A. V. Butenko, W. M. Shalaev, and M. I. Stockman, Sov. Phys. JETP 7, 60 (1988).

Z. Phys. D (2)

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).
[CrossRef]

V. M. Shalaev and M. I. Stockman, Z. Phys. D 10, 71 (1988); A. V. Butenko, V. M. Shalaev, and M. I. Stockman, Z. Phys. D 10, 81 (1988).
[CrossRef]

Other (2)

J. van Kranendonk and J. E. Sipe, in Progress in Optics XV, E. Wolf, ed. (North-Holland, New York, 1977), p. 245.
[CrossRef]

Note that in the notation of Ref. 10 we have set ∊n= 1 in writing Eq. (4) of the present paper, that is, in the present paper we do not treat one medium as the host and the other as the inclusion; both are formally treated as inclusions embedded in vacuum.

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Figures (6)

Fig. 1
Fig. 1

Composite optical material with a layered geometry. The thickness of each layer is assumed to be much smaller than an optical wavelength.

Fig. 2
Fig. 2

Calculation of the mesocopic and the macroscopic fields.

Fig. 3
Fig. 3

Dependence of the effective dielectric constant on the fill fraction fb of component b.

Fig. 4
Fig. 4

Effective nonlinear susceptibility for second-harmonic generation plotted versus the fill fraction of component b with the assumption that only component a possesses a second-order nonlinear optical response.

Fig. 5
Fig. 5

Effective nonlinear susceptibility for the Pockels effect plotted versus the fill fraction of component b, with the assumption that only component a possesses a second-order nonlinear optical response.

Fig. 6
Fig. 6

Effective nonlinear susceptibility describing the nonlinear refractive index plotted versus the fill fraction of component b with the assumption that only component a possesses a third-order nonlinear optical response.

Equations (32)

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eff = f a a + f b b ,
χ eff NL = f a χ a NL + f b χ b NL .
1 eff = f a a + f b b ,
E ( r , ω ) = Δ ( r r ) e ( r , ω ) d r ,
e ( r , ω ) = E ( r , ω ) + 4 π 3 P ( r , ω ) + T 0 ( r r ) c ( | r r | ) · p ( r , ω ) d r 4 π 3 p ( r , ω ) ,
T 0 ( r ) = { ( 3 r ˆ r ˆ U ) / r 3 r > η 0 r < η ,
4 π z ˆ z ˆ · p a ( ω ) [ 4 π 3 p a ( ω ) ] ,
4 π 3 P ( ω ) [ 4 π z ˆ z ˆ · P ( ω ) ] .
e a ( ω ) = E ( ω ) + 4 π z ˆ z ˆ · P ( ω ) 4 π z ˆ z ˆ · p a ( ω ) ,
P ( ω ) = f a p a ( ω ) + f b p b ( ω ) .
e a z ( ω ) = E z ( ω ) + F z ( ω ) 4 π p a z ( ω ) ,
e b z ( ω ) = E z ( ω ) + F z ( ω ) 4 π p b z ( ω ) ,
F z ( ω ) = 4 π f a p a z ( ω ) + 4 π f b p b z ( ω ) .
p a z ( ω ) = χ a ( 1 ) ( ω ) e a z ( ω ) , p b z ( ω ) = χ b ( 1 ) ( ω ) e b z ( ω ) .
e a z ( ω ) = 1 a ( ω ) E z ( ω ) [ f a a ( ω ) + f b b ( ω ) ] ,
e b z ( ω ) = 1 b ( ω ) E z ( ω ) [ f a a ( ω ) + f b b ( ω ) ] .
1 eff ( ω ) = f a a ( ω ) + f b b ( ω ) ,
e a z ( ω ) = eff ( ω ) a ( ω ) E z ( ω ) , e b z ( ω ) = eff ( ω ) b ( ω ) E z ( ω ) .
p a z ( 2 ω ) = χ a ( 2 ) e a z 2 ( ω ) + χ a ( 1 ) ( 2 ω ) e a z ( 2 ω ) ,
p a z ( 2 ω ) = χ a ( 2 ) { E z ( ω ) a ( ω ) [ f a a ( ω ) + f b b ( ω ) ] } 2 + χ a ( 1 ) ( 2 ω ) × [ E z ( 2 ω ) + F z ( 2 ω ) 4 π p a z ( 2 ω ) ] .
p a z ( 2 ω ) = χ a ( 2 ) a ( 2 ω ) { E z ( ω ) a ( ω ) [ f a a ( ω ) + f b b ( ω ) ] } 2 + χ a ( 1 ) ( 2 ω ) a ( 2 ω ) [ E z ( 2 ω ) + F z ( 2 ω ) ] .
P z ( 2 ω ) = χ eff ( 2 ) ( 2 ω = ω + ω ) E z 2 ( ω ) + χ eff ( 1 ) ( 2 ω ) E z ( 2 ω ) ,
χ eff ( 2 ) ( 2 ω = ω + ω ) = f a χ a ( 2 ) a ( 2 ω ) a ( ω ) 2 + f b χ b ( 2 ) b ( 2 ω ) b ( ω ) 2 [ f a a ( ω ) + f b b ( ω ) ] 2 [ f a a ( 2 ω ) + f b b ( 2 ω ) ] .
P a z ( ω ) = 2 χ a ( 2 ) e a z ( ω ) e a z ( 0 ) + χ a ( 1 ) ( ω ) e a z ( ω ) = 2 χ a ( 2 ) E z ( ω ) E z ( 0 ) a ( ω ) [ f a a ( ω ) + f b b ( ω ) ] a ( 0 ) [ f a a ( 0 ) + f b b ( 0 ) ] + χ a ( 1 ) ( ω ) [ E z ( ω ) + F z ( ω ) 4 π p a z ( ω ) ] ,
p a z ( ω ) = 2 χ a ( 2 ) a ( ω ) E z ( ω ) [ f a a ( ω ) + f b b ( ω ) ] × E z ( 0 ) a ( 0 ) [ f a a ( 0 ) + f b b ( 0 ) ] + χ a ( 1 ) ( ω ) a ( ω ) [ E z ( ω ) + F z ( ω ) ] .
P z ( ω ) = 2 χ eff ( 2 ) ( ω = ω + 0 ) E z ( ω ) E z ( 0 ) + χ eff ( 1 ) ( ω ) E z ( ω ) ,
χ eff ( 2 ) ( ω = ω + 0 ) = f a χ a ( 2 ) a ( 0 ) a ( ω ) 2 + f b χ b ( 2 ) b ( 0 ) b ( ω ) 2 [ f a a ( ω ) + f b b ( ω ) ] 2 [ f a a ( 0 ) + f b b ( 0 ) ] .
P a z ( ω ) = 3 χ a ( 2 ) × | E z ( ω ) | 2 E z ( ω ) | a ( ω ) [ f a a ( ω ) + f b b ( ω ) ] | 2 { a ( ω ) [ f a a ( ω ) + f b b ( ω ) ] } + χ a ( 1 ) ( ω ) [ E z ( ω ) + F z ( ω ) 4 π p a z ( ω ) ] ,
P a z ( ω ) = 3 χ a ( 2 ) a ( ω ) × | E z ( ω ) | 2 E z ( ω ) | a ( ω ) [ f a a ( ω ) + f b b ( ω ) ] | 2 { a ( ω ) [ f a a ( ω ) + f b b ( ω ) ] } + χ a ( 1 ) ( ω ) a ( ω ) [ E z ( ω ) + F z ( ω ) ] ,
P z ( ω ) = 3 χ eff ( 3 ) ( ω = ω + ω ω ) | E z ( ω ) | 2 E z ( ω ) + χ eff ( 1 ) ( ω ) E z ( ω ) ,
χ eff ( 3 ) ( ω = ω + ω ω ) = f a χ a ( 3 ) | a ( ω ) | 2 a ( ω ) 2 + f b χ b ( 3 ) | b ( ω ) | 2 b ( ω ) 2 [ f a a ( ω ) + f b b ( ω ) ] 2 [ f a a ( ω ) + f b b ( ω ) ] 2 .
n 2 = 12 π 2 n 0 2 c χ ( 3 ) ( ω = ω + ω ω ) .

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