(6). We define E. Elin. E and lin. , where E . The boundary conditions for and are ..... 0, which is a sixth- order polynomial equation for s: s6 c5s5 c4s4 c3s3 c2s2.
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J. Opt. Soc. Am. B / Vol. 15, No. 3 / March 1998
Ma et al.
Third-order optical nonlinearity enhancement through composite microstructures Hongru Ma,* Rongfu Xiao, and Ping Sheng Department of Physics, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong, China Received September 8, 1997 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 [S0740-3224(98)01503-3] OCIS codes: 190.4710, 230.3990.
1. INTRODUCTION AND FORMULATION Optical nonlinearity of composite materials can be enhanced from those of component materials through local field and resonant scattering effects, both of which are sensitively dependent on the composite microstructure(s). The idea that local fields bring about enhancement of nonlinear susceptibility was put forward by Butenko et al. and Stockman et al.,1 who observed the enhancement in metal fractals and estimated the enhancement factor theoretically. Here we examine the microstructure implications of four effective medium theories, corresponding to different composite microstructures, for the enhancement of third-order Kerr-type optical nonlinear optical coefficients. A commonly encountered error in evaluation of third-order nonlinear optical coefficients is pointed out. Illustrations and comparison of the microstructural enhancement effect are given for the four theories. Comparison with a recent experiment on AuSiO2 films shows that large enhancement of the third-order nonlinear susceptibility is indeed possible. The local constitutive relation of the composite system is given by D 5 e E 1 A u Eu E 1 BE E* , 2
2
(1a)
where e is a (spatially dependent) dielectric constant and A and B are (spatially dependent) third-order Kerr nonlinear susceptibilities. Inasmuch as we show below that the microstructural enhancements of A and B are the same, here we consider only the usual simplified version of the constitutive relation: D 5 @ e 1 x ~ 3 ! u Eu 2 # E.
(1b)
The effective dielectric constant ¯e and the effective thirdorder susceptibility ¯x ( 3 ) are defined by the spatial average of D: 0740-3224/98/031022-08$15.00
1 V
E
dVD 5
1 V
E
dV @ e E 1 x ~ 3 ! u Eu 2 E#
[ ¯e E0 1 ¯x ~ 3 ! u E0 u 2 E0 .
(2)
Here E0 5 (1/V) * dVE is the applied (averaged) electric field, taken to be real. The factor exp(2ivt) is dropped from the expression for simplicity. Below we follow the general approach of Stroud and Hui2 in our derivation of the formulas for ¯e and ¯x ( 3 ) . In the quasi-static limit we have ¹ • D 5 0, ¹ 3 E 5 0.
(3)
The corresponding linear problem is given by ¹ • Dlin 5 0, ¹ 3 Elin 5 0,
(4)
Dlin 5 e Elin .
(5)
where
From Eqs. (3) and (4) we can define E 5 2¹ f , Elin 5 2¹ f lin .
(6)
We define E 5 Elin 1 d E and f 5 f lin 1 d f , where d E 5 2¹ d f . The boundary conditions for f and df are
f u z50 5 f linu z50 5 0,
f u z5L 5 f linu z5L 5 2E 0 L,
(7)
from which it follows that
d f u boundaries 5 0. © 1998 Optical Society of America
(8)
Ma et al.
Vol. 15, No. 3 / March 1998 / J. Opt. Soc. Am. B
E
By multiplying Eq. (2) by E0 5 2¹ f 0 5 ¹(E0 z) and noticing that f 5 f 0 on the boundary, we can write the lefthand side of Eq. (2) as 1 V
E
1 dVD • E0 5 2 V 1 52 V
E E
1 1 V
E E E E
1 52 V
E
dS • Df 0
dVD • ¹ f
1 V
E
E
dV e Elin2 1
1 V
(9)
E
dVD • E* 5 ¯e E0 2 1 ¯x ~ 3 ! u E0 u 2 E0 2
(10)
¯x ~ 3 ! u E0 u 2 E0 2 5
1 V
dVD • E* 5
dV e Elin2 1 2 1 V
E
E E
5 x ~13 !
dV e Elin2 ,
(16)
dV x ~ 3 ! u Elinu 2 Elin2
1 V
1 x ~23 !
E
dV u Elinu 2 Elin2
1
1 V
E
dV u Elinu 2 Elin2
2
[ p 1 x ~13 ! ^ u Elinu 2 Elin2 & 1
(11)
E
[ ~ b 1 x ~13 ! 1 b 2 x ~23 ! ! u E0 u 2 E0 2 , dVD • E.
(12)
Equation (12) is used below in the evaluation of spatially averaged u Eu 2 . From the above energy formulation of the problem it is clear that the enhancement factors for A and B [Eq. (1a)] are the same and are just the enhancement factor for x ( 3 ) . So far the formulation is exact. In general, it is difficult to have an accurate evaluation of an effective nonlinear optical coefficient because of the lack of an analytical solution approach to the nonlinear differential equation that results from nonlinear dielectric constant(s) of the components. However, the magnitude of the nonlinear coefficient is usually small, which means the problem can be treated in the perturbation sense; i.e., one can evaluate the electric field by ignoring the nonlinearity. To first order in the nonlinearity, therefore, we have
1
(15)
1 p 2 x ~23 ! ^ u Elinu 2 Elin2 & 2
E
E
(14)
dV x ~ 3 ! u Elinu 2 Elin2
1 V
or
1 V
E
¯e E0 2 5
Because both f and f * have the same boundary value f 0 , it is easy to see that f in Eq. (9) can be replaced by f * , and we have 1 V
dV d f ¹ • ~ e Elin! .
which implies that
dVD • E,
dVD • E 5 ¯e E0 2 1 ¯x ~ 3 ! u E0 u 2 E0 2 .
dV¹ • ~ e Elind f !
5 ¯e E0 2 1 ¯x ~ 3 ! u E0 u 2 E0 2 ,
so that 1 V
dV e Elin • ¹ d f
The first term on the right-hand side of Eq. (14) can be converted into a surface integral and is therefore zero because d f 5 0 at the boundary. The second term on the right-hand side of Eq. (14) is identically zero from Maxwell’s equation. It follows that
dV f 0 ¹ • D
1 52 V
E E E
1
dV¹ • ~ Df 0 !
dS • Df
1 V
52
dVD • ¹ f 0
1 52 V
5
dV e Elin • d E 5 2
1023
1 V
E
dV e Elin • d E
dV x ~ 3 ! u Elinu 2 Elin2 5 ¯e E0 2 1 ¯x ~ 3 ! u E0 u 2 E0 2 .
(13)
The second term on the left-hand side is zero because
(17)
where ^ ... & i means volume averaging inside material i. x 1(3 ) , b 1 and x 2(3 ) , b 2 are defined as the third-order nonlinear susceptibilities and enhancement factors of material 1 and material 2, respectively. One way to calculate the averages in Eq. (17) is to decouple the averages as
^ u Elinu 2 Elin2 & i ' ^ u Elinu 2 & i ^ Elin2 & i ,
(18a)
which implies that ¯x ~ 3 ! 5 b 1 x ~13 ! 1 b 2 x ~23 ! ,
(18b)
with
bi 5 pi
^ u Elinu 2 & i ^ Elin2 & i u E0 u 2 E0 2
.
(18c)
This approximation is exact when the field inside material i is a constant and becomes poor when the variation of the field inside material i is large. We denote Eq. (18a) the mean-field approximation. It should be noted here that, even though Eq. (12) holds, using * dVD • E* instead of * dVD • E will not result in Eq. (17) because * dV e (Elin • d E* 1 Elin* • d E) is not zero in general when the field is complex [resulting from complex dielectric constant(s)]. Thus the correct quantity to be averaged is u Elinu 2 Elin2 , not u Elinu 4 .
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Ma et al.
Now we relate ^ u Elinu 2 & i and ^ Elin2 & i to the effective dielectric constant and its spectral representation. Consider the linear electrostatic problem ¹ • ~ e ¹ f lin! 5 0.
(19)
If we vary e to e 1 D e , then f lin changes to f lin 1 D f , and Eq. (19) leads to ¹ • ~ e ¹D f ! 5 2¹ • ~ D e ¹ f lin! ,
D¯e E0
1 5 V
E
dV ~ D e Elin 1 2 e Elin • DE! . 2
1 V
E
dVD e Elin2 .
p 1 ^ u Elinu 2 & 5 5
(21)
5
1
1 V
E
5
dVElin2
(23)
(24)
]¯e E 2. ]e2 0
(25)
Here p 1 and p 2 are the volume fractions of components 1 and 2, respectively. Equations (24) and (25) are widely used to calculate the averaged squared electric field. Below we show that the same formulas cannot be used to calculate ^ u Elinu 2 & i when the dielectric constant(s) are complex.
2. SOLUTION IN THE SPECTRAL REPRESENTATION
( n
s^nuz& f , s 2 sn n
(26)
where s n and f n are the nth eigenvalue and eigenfunction of the operator G, defined as
dV u Elinu 2
1
1
1 V
dV¹ f * • ¹ f u s u 2 ^ z u n &^ m u z &
(( n
~ s * 2 s n !~ s 2 s m !
m
E
dV¹ f n * • ¹ f m usu2f n
(
E0 2 .
u s 2 s nu 2
(28)
dV e u Elinu 2 5 ¯e E0 2 ,
which implies that 1 V
E
dV u Elinu 2 1 e 2
1
Alternatively,
S F F
¯e
e2
5 12
5 12
1 V
E
(29)
dV u Elinu 2 5 ¯e E0 2 .
(30)
2
D
e1
2
p 1 ^ u Elinu 2 & 1 E0 2
e2
fn
(
s 2 sn
n
2
u s u 2 fn ~ 1 2 1/s !
~ usu2 2 sn!fn
(
u s 2 s nu 2
n
G
u s 2 s nu 2
G
E0 2 .
E0 2
(31)
To see how this expression differs from ^ Elin2 & , we wish to write ^ Elin2 & also in the spectral representation form. Because the effective dielectric constant can be expressed as3
S
¯e 5 e 2 1 2
To calculate ^ u Elinu 2 & i we go to the spectral representation.3 The spectral representation has been used in analyzing the effective dielectric constant and other effective properties of composites. In a recent publication by Levy and Bergman4 it was also used in the study of nonlinear optical bistability. In the spectral representation the solution of the electrostatic equation can be written in operator form as s z52 f52 s2G
V
E
1 V
p 2 ^ u Elinu 2 & 2 5
The same procedure gives
(27)
From Eq. (11) we have
1
dVElin2 5 p 1 ^ Elin2 & 1 .
p 2 ^ Elin2 & 2 5
1
E E
1
e1
E
V
n
or
]¯e 1 E 25 ]e1 0 V
1
3
(22)
By setting D e 5 D e 1 u (r), where u (r) is the characteristic function of material 1, defined to be 1 in component 1 and 0 otherwise, we have D¯e E0 2 5 D e 1
dV u ~ r8 ! ¹ 8 G~ r 2 r8 ! • ¹ 8 ,
where G(r 2 r8 ) 5 1/4p (r 2 r8 ) denotes the Green’s function for the Laplacian operator. Here s 5 e 2 /( e 2 2 e 1 ) is the only material parameter of the problem. The last equality of Eq. (26) is valid only when r is inside the grains of component 1. Now
The term * dV e Elin • DE is zero, as proved in Eq. (14) [note that the meaning of DE is not the same as that of d E in Eq. (14); here it is comes from any variation of dielectric constant, whereas in Eq. (14) it is from the nonlinearity]. So we have D¯e E0 2 5
E
(20)
where D f 5 0 at the boundaries. The induced change in the effective dielectric constant is given by 2
G5
( n
D
fn [ e 2 @ 1 2 F ~ s !# , s 2 sn
(32)
it follows from Eqs. (24) and (25) that p 1 ^ Elin2 & 1 5
(
F
n
s 2f n ~ s 2 sn!2
p 2 ^ Elin2 & 2 5 1 2
( n
E0 2 ,
~ s2 2 sn!fn ~ s 2 sn!2
(33)
G
E0 2 . (34)
It is clear that when e are real, Eqs. (28) and (31) are the same as Eqs. (33) and (34), as expected. However, u ^ Elin2 & u Þ ^ u Elinu 2 & when e are complex. A common error in the literature is to treat them as the same even when e are complex.
Ma et al.
Vol. 15, No. 3 / March 1998 / J. Opt. Soc. Am. B
When the G operator has a continuous spectrum, Eqs. (28), (31), (33), and (34) can be written in integral form as p 1 ^ u Elinu 2 & 1 5
E
dx
usu2m~ x !
F E
p 2 ^ u Elinu 2 & 2 5 1 2
p 1 ^ Elin2 & 1 5 p 2 ^ Elin2 & 2
E
dx
E0 2 ,
us 2 xu2
dx
us 2 xu2
~s 2 x!
F E
5 12
dx
(35)
~ usu2 2 x !m~ x !
s 2m~ x ! 2
G
~ s 2 x !2
E0 . (38)
Here m (x) is the spectral density of operator G. It is related to F(s) by the relation F~ s ! 5
E
(39)
When we write s as s 1 i0 1, the right-hand side of Eq. (39) becomes P * @ m (x)/(s 2 x) # dx 2 i p m (s), and thus m (x) is given by 1 m ~ x ! 5 2 Im@ F ~ x 1 i0 1!# . p
(40)
The advantage of using the spectral density to evaluate the dielectric constant and the optical nonlinearities lies in the separation of the geometric contribution from the material contribution. In effect, once a given type of microstructure is known and its associated spectral density calculated, all effective dielectric and optical properties can be simply evaluated from the material parameters of the components. Below we give explicit formulas for the four types of microstructure associated with their respective effective medium theories.
3. EVALUATION OF ¯x (3) FOR THE FOUR EFFECTIVE MEDIUM THEORIES A. Dispersion Microstructure (Maxwell–Garnett Theory) The dispersion microstructure pertains to the geometry of the colloidal systems, for example. It has a dispersed component (1) in a matrix component (2). Because the dispersed component can never form an infinite connected network, this particular microstructure precludes the existence of a percolation threshold, except the trivial one at p 5 1. As the dispersion microstructure can be regarded as formed by similar structure unit consisting of a sphere of component 1 coated by a layer of matrix material (component 2),5 optically there can be a so-called anomalous dispersion, i.e., surface plasmon resonance of a coated sphere, if component 1 is metal.6 The Maxwell–Garnett (MG) effective dielectric constant is the solution of the equation5,7 ¯e 2 e 2 e1 2 e2 5p . ¯e 1 2 e 2 e1 1 2e2
(42)
(41)
(43)
Thus we have
^ Elin2 & 1 5
2
m~ x ! dx. s2x
p . s 2 ~ 1 2 p ! /3
m ~ x ! 5 p d @ x 2 ~ 1 2 p ! /3 # .
(36) (37)
G
F~ s ! 5
From Eq. (32), the func-
From Eq. (40),
E0 2 ,
E0 2 ,
~ s2 2 x !m~ x !
Here p 1 5 p and p 2 5 1 2 p. tion F(s) is therefore
1025
5
^ Elin2 & 2 5 5
^ u Elinu 2 & 1 5
^ u Elinu 2 & 2 5 5
1 ]¯e p ]e1
E0 2 s2
E0 2 ,
@ s 2 ~ 1 2 p ! /3# 2
1
]¯e
1 2 p ]e2
E0 2
~ s 2 1/3! 2 1 2p/9 @ s 2 ~ 1 2 p ! /3# 2
usu2 u s 2 ~ 1 2 p ! /3] 2
1 12p
F
(44)
12
E0 2 ,
E0 2 ,
~ u s u 2 2 ~ 1 2 p ! /3! p
u s 2 ~ 1 2 p ! /3u 2
u s 2 1/3u 2 1 2p/9 u s 2 ~ 1 2 p ! /3u 2
(45)
E0 2 .
(46)
G
E0 2 ,
(47)
¯x ( 3 ) can be directly evaluated from Eqs. (18b) and (18c). Sipe and Boyd8 have discussed the enhancement of nonlinear susceptibility of composite optical materials in the MG model. Their calculation took account of the tensorial nature of the nonlinear interaction and did not use the decoupling approximation as we did in Eq. (18a). A comparison of our results with those of Sipe and Boyd reveals that the enhancement factors inside the sphere are identical but the enhancement factor of the host is slightly different. B. Symmetric Microstructure (Bruggeman’s Self-Consistent Theory) The symmetric microstructure is so labeled because it satisfies the symmetry of the interchange between p 1 and p 2 simultaneously with the interchange of the material parameters of the two components. At either p → 0 or p → 1 the symmetric microstructure is somewhat similar to the dispersion microstructure. But at intermediate compositions there can be a matrix inversion and hence a percolation threshold for either component. At p . 1/3, component 1 forms an infinite connected network. At p , 2/3, component 2 forms an infinite connected network. At 1/3 , p , 2/3, both components are connected in the symmetric microstructure. It should be noted that the anomalous dispersion effect is generally absent in the symmetric microstructure, unlike in the dispersion micro-
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Ma et al.
structure. In Bruggeman’s self-consistent theory (EMA) the effective dielectric constant is the solution of 5,9
p
e 1 2 ¯e e 2 2 ¯e 1 ~1 2 p! 5 0. e 1 1 2¯e e 2 1 2¯e
^ Elin2 & 2 5 5
(48)
1 2 p ]e2 1 12p 1
Here, p 1 5 p, p 2 5 1 2 p, and the function F(s) is given by
]¯e
1
H
E0 2
2 2 3p 4
22 2 9p 1 9p 2 1 6s 2 3ps 12@~ s 2 x 1 !~ s 2 x 2 !# 1/2
J
E0 2 , (54)
1 $ 21 1 3p 1 3s 2 3 @~ s 2 x 1 !~ s 2 x 2 !# 1/2% . 4s (49)
^ u Elinu 2 & 1 5
The phase of s 2 x i is restricted within (0,2p), where x 1 and x 2 are given by the solutions of the following equation:
^ u Elinu 2 & 2 5
F~ s ! 5
~ 1 2 3p ! 2 2 6 ~ 1 1 p ! x 1 9x 2 5 0.
p
E
usu2m~ x ! us 2 xu2
1 12p
F E 12
dxE0 2 ,
(55)
~ usu2 2 x !m~ x !
us 2 xu2
G
E0 2 .
(56)
(50)
¯x ( 3 ) can be evaluated from Eqs. (18b) and (18c).
(51)
C. Granular Metal Microstructure (The Sheng Theory) In granular metals, both the anomalous dispersion effect and the percolation threshold in electrical conductivity were observed. Therefore the previous two microstructures are inadequate for the description of granular metals. The granular microstructure as proposed by Sheng10 consists of two types of structure unit: a sphere of component 1 coated by a layer of component 2 and the reverse case. This type of microstructure has been shown to give the best description of the electrical and optical properties of granular metal films.10 For spherically shaped structure units the effective dielectric constant is given by the solution of the equation
The results are x 1 5 1/3$ 1 1 p 2 2 @ 2p ~ 1 2 p !# 1/2% , x 2 5 1/3$ 1 1 p 1 2 @ 2p ~ 1 2 p !# 1/2% .
1
To calculate the spectral density from Eq. (40) we consider s as x 1 i0 1 in Eq. (49), where x 5 Re(s), and extract the imaginary part of the resulting expression. It is clear that the contribution has two parts. The first part comes from the s in the denominator of F(s), which gives a d function to the imaginary part with residue (3p 2 1)/2 when p . 1/3 and is zero when p , 1/3. The other part comes from the expression inside the square root when x 1 , x , x 2 . Thus the spectral density is
fD 1 1 ~ 1 2 f ! D 2 5 0.
(57)
where
m~ x ! 5
~ 3p 2 1 ! u ~ 3p 2 1 ! d ~ x ! 2
1
H
3 @~ x 2 x 1 !~ x 2 2 x !# 1/2 4px
x1 , x , x2
0
otherwise
It follows that
5
~¯e 2 e 2 !~ e 1 1 2 e 2 ! 1 ~ e 2 2 e 1 !~¯e 1 2 e 2 ! p , ~ 2¯e 1 e 2 !~ e 1 1 2 e 2 ! 1 2 ~¯e 2 e 2 !~ e 2 2 e 1 ! p
D2 5
~¯e 2 e 1 !~ 2 e 1 1 e 2 ! 1 ~¯e 1 2 e 1 !~ e 1 2 e 2 !~ 1 2 p ! , ~ 2¯e 1 e 1 !~ 2 e 1 1 e 2 ! 1 2 ~¯e 2 e 1 !~ e 1 2 e 2 !~ 1 2 p !
.
(52)
^ Elin2 & 1 5
D1 5
f5 1 ]¯e p ]e1 1 p
H
1
E0 2
21 1 6p 2 9p 2 1 3s 1 3ps 12@~ s 2 x 1 !~ s 2 x 2 !# 1/2
@ 1 2 ~ 1 2 p ! 1/3# 3 1 ~ 1 2 p 1/3! 3
,
(58)
The solution in the s p 1 5 p, and p 2 5 1 2 p. 5 e 2 /( e 2 2 e 1 ) notation can be written as
21 1 3p 4
~ 1 2 p 1/3! 3
J
¯e 2b ~ s ! @ b ~ s ! 2 2 4a ~ s ! c ~ s !# 1/2 5 6 , e2 2a ~ s ! 2a ~ s !
E0 2 , (53)
where
(59)
Ma et al.
Vol. 15, No. 3 / March 1998 / J. Opt. Soc. Am. B
a ~ s ! 5 2s ~ 3s 2 3 1 p !~ 3s 2 1 1 p ! ,
m~ x ! 5
b ~ s ! 5 2 ~ 2 2 3f !~ 1 2 p ! p 1 s ~ 23 2 5p 1 2p 2 1 12s 1 3ps 2 9s 2 ! ,
~ 2 2 3f ! p u ~ 2 2 3f ! d ~ x ! 32p
12p ~ 3f 2 1 ! p u ~ 3f 2 1 ! d x 2 2 3
1
~ 2 2 3f !~ 1 2 p ! p 2~ 3 2 p !
c ~ s ! 5 ~ 1 2 s !~ 2p 1 4p 2 2 3s 2 12ps 1 9s 2 ! .
(60)
The F(s) function in this formulation is given by the formula
F~ s ! 5 1 1
F S
3 u ~ 2 2 3f ! d x 2 1 2
b~ s ! @~ s 2 x 12!~ s 2 x 11!~ s 2 x 22!~ s 2 x 21!~ s 2 x 32!~ s 2 x 31!# 1/2 2 27 . 2a ~ s ! 2a ~ s !
To choose the correct branch we restrict the phase of s 2 x i6 to within (0,2p). Here x 16 , x 26 , and x 36 are solutions of the equation b 2 2 4ac 5 0, which is a sixthorder polynomial equation for s: s 6 1 c 5 s 5 1 c 4 s 4 1 c 3 s 3 1 c 2 s 2 1 c 1 s 1 c 0 5 0, (62) where
c1 5
4 ~ f 2 2 !~ 1 2 p !~ 3 2 p ! p ~ 1 1 2p ! , 243
c2 5
2 ~ 31 2 8f ! p 1 ~ 12 f 2 11! p 2 1 1 9 81 81 1
D
DG
(61)
27 @~ x 2 x 12!~ x 11 2 x !~ x 2 x 22! 2 u a~ x !u p
3 ~ x 21 2 x !~ x 2 x 32!~ x 31 2 x !# 1/2 3 $ @ u ~ x 2 x 12! 2 u ~ x 2 x 11!# 1 @ u ~ x 2 x 22! 2 u ~ x 2 x 21!# (66)
It should be noted that the percolation threshold for component 1 is given by the solution to the equation 2 2 3f 5 0, i.e., at p l 5 0.45507. Similarly, the solution of 3f 2 1 5 0, given by p u 5 1 2 p l 5 0.54493, denotes the percolation threshold for component 2 in the Sheng theory. Between the two thresholds the microstructure is characterized by biconnectedness for both components. From the known spectral density m (x), ^ Elin2 & 1 , ^ Elin2 & 2 , ^ u Elinu 2 & 1 , and ^ u Elinu 2 & 2 can be calculated from Eqs. (35), (36), (37), and (38), respectively. ¯x ( 3 ) can then be evaluated from Eqs. (18b) and (18c).
4p 4 4~ f 2 7 !p3 1 , 81 81
2 ~ 2 f 2 27! p 8 c3 5 2 1 9 27
D. Hierarchical Microstructure (The Differential Effective Medium Theory) The hierarchical microstructure is best described as a dispersion of component 1 particles in a matrix that is built
4p 3 2~ 7 2 2 f !p2 1 , 1 27 27 p2 22 1 2p 2 , 9 3
2p 8 c5 5 2 2 . 3 3
1
p 3
1 @ u ~ x 2 x 32! 2 u ~ x 2 x 31!# % .
4 ~ 3f 2 2 ! 2 ~ 1 2 p ! 2 p 2 , c0 5 729
c4 5
S
1
1027
(63)
The solution of Eqs. (63) cannot be found analytically. Numerical solution shows that all six roots are real and are in the interval [0,1]. In the limit of p 5 0, Eq. (62) reduces to s 2 ~ s 2 1/3! 2 ~ s 2 1 ! 2 5 0,
(64)
and the six solutions are x 12 5 x 11 5 0, x 22 5 x 21 5 1/3, and x 32 5 x 31 5 1. When p 5 1 we have s 2 ~ s 2 2/3! 2 ~ s 2 1 ! 2 5 0,
(65)
and the six solutions are x 12 5 x 11 5 0, x 22 5 x 21 5 2/3, and x 32 5 x 31 5 1. The numerical solutions of the six roots in the interval 0 , p , 1 are plotted in Fig. 1. The spectral density m (x) is found to be
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.
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J. Opt. Soc. Am. B / Vol. 15, No. 3 / March 1998
Ma et al.
up successively through a homogenization process whereby smaller particles of component 1 are dispersed in a (previously) homogenized medium. The hierarchical microstructure has been shown to give a good description of the electrical and elastic properties of sedimentary rocks.11 The effective dielectric constant of the differential effective medium (DEM) is given by the solution of 9,12 12p5
S D
¯e 2 e 1 e 2 e 2 2 e 1 ¯e
where p 1 5 p and p 2 5 1 2 p.
1/3
,
(67)
The spectral density is
with
m~ x ! 5
H
A3 ~ 1 2 p !~ 1 2 x ! 1/3 ~ 2x ! 4/3p
@~ 1 1 B ! 1/3 2 ~ 1 2 B ! 1/3#
0
Plots of the effective dielectric constant as function of Au volume fraction are shown in Fig. 3. The EMA and the DEM theories are similar. The real parts vary almost monotonically from p 5 0 (SiO2) to p 5 1 (Au). The imaginary parts are relatively flat as a function of p. The MG and the Sheng theories exhibit anomalous dispersions in their effective dielectric constants related physically to the plasmon resonance of coated spheres and manifested here as large undulations in the real and the imaginary parts of the effective dielectric constant. This resonance is identified as the cause of large thirdorder nonlinear susceptibility enhancement, as Fig. 4 shows, where the enhancement factor u b i u is plotted as a
xl , x , xu
,
(68)
otherwise
with
F
B5 12
4~ 1 2 p !3 27~ 1 2 x ! 2 x
G
1/2
.
(69)
x l and x u are determined by the requirement that B be real:
F F
S S
xl 5
p2f 2 1 2 cos 3 3
xu 5
p1f 2 1 2 cos 3 3
DG DG
,
(70)
,
(71)
and f 5 arccos@2(1 2 p)3 2 1#, 0 < f < p . From the spectral density the averaged field intensities and ¯x ( 3 ) can be obtained as in the other cases. Fig. 2. Spectral densities for the four different microstructures at p 5 0.4.
4. ILLUSTRATIONS AND COMPARISON WITH EXPERIMENTS To illustrate the nonlinear enhancement effect we use the derived formulas for different microgeometries to study the AuSiO2 composite system at a wavelength of 620 nm, where the dielectric constant of Au is 29.97 1 0.822i (Ref. 13) and the dielectric constant of SiO2 is taken to be 2.25. Figure 2 shows plots of the spectral densities of various microgeometries at the Au volume fraction of 40%. A vertical line indicates a delta function, with the weight of the delta function given by its height. We see that the EMA and the DEM theories have similar spectral densities but that the EMA has an extra delta function with weight (3p 2 1)/2 when the volume fraction is larger than 1/3. Physically, this reflects the fact that the EMA has a percolation threshold at p 5 1/3, whereas the DEM has none. The MG theory has the simplest spectral density, consisting of a delta function at x 5 p/3 with weight p. The Sheng theory has a spectrum rich in structures; it contains three branches of continuous spectra with a delta function between. There is also a percolation threshold at p 5 0.45507.
Fig. 3. Composite effective dielectric constants for the four different microstructures calculated at l 5 620 nm with material parameters given in the text. Note the anomalous dispersions exhibited in the MG and the Sheng theories.
Ma et al.
Fig. 4. Third-order nonlinear enhancement factor u b i u 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 l 5 620 nm and material parameter values given in the text.
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tive u ¯x ( 3 ) u in this system. In Fig. 5, u ¯x ( 3 ) u calculated from the Sheng theory is plotted as a function of Au volume fraction p together with the measured data. The param(3) eters used are x Au 5 8 3 1028 esu, which we obtained by (3) fitting the data point at p 5 0.67, and x SiO 52 2 (3) 15 212 esu. This value of x Au should itself be re3 10 garded as an effective value because the local fluctuations in the electric field, neglected in our present calculations, could cause an enhancement relative to its true value. It is seen that the agreement between theory and experiment is reasonably good at l 5 530 nm, aside from the composition dependence. With the same parameters, we predict that at l 5 620 nm the effective u ¯x ( 3 ) u can reach the value of ;1023 esu. Experiments are under way to check this prediction. To conclude, we have derived the spectral densities of different microgeometries with related formulas for calculating the effective dielectric constant and the enhancement factors for third-order nonlinearities. We have shown that the largest enhancement source is the anomalous dispersion, i.e., the plasmon resonance of the coated spheres, realized by the MG and the Sheng theories. Comparison with experiment shows the large enhancement to be indeed realizable. Even larger enhancement is predicted to be possible. *Present address, Department of Physics, Shanghai
Jiao Tong University, Shanghai, China.
REFERENCES 1.
2. 3. Fig. 5. Comparison of the measured third-order nonlinear susceptibility u ¯x (3) u in AuSiO2 composites with theoretical values calculated from the Sheng theory at l 5 530 nm. The dashed curve is calculated from the same parameters at l 5 620 nm. The compositions where the maximum occurs do not coincide. However, the maximum theory and experimental values of u ¯x (3) u are very close.
4. 5. 6. 7.
function of p. It is seen that the enhancements are small for the EMA and the DEM but can be quite large in the MG and the Sheng microgeometries. At volume fractions somewhat below that of the percolation threshold the enhancement can be more than 4 orders of magnitude. The fact that the MG and the Sheng microgeometries both exhibit an enhancement peak at ;p 5 0.5 is due to the coincidence of the resonance frequency for the coated spheres at that particular composition, with the driven frequency of 620 nm. A recent experiment14 on the optic third-order nonlinearity in granular AuSiO2 films has shown a large effec-
8. 9. 10. 11. 12. 13. 14. 15.
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