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when an intense electric field is applied to a suspension containing highly polarized particled embedded in a host fluid , the particles will form chains due to the induced dipole moments inside the particles , and hence complex anisotropic structures occur . also , the apparent viscosity of the suspension will be enhanced greatly . these structures can have anisotropic physical properties , such as the effective conductivity and permittivity @xcite . in this case , optical nonlinearity enhancement was found as well @xcite . a convenient method of probing the nonlinear characteristics of the field - induced structures is to measure the harmonics of the nonlinear polarization under the application of a sinusoidal ( ac ) electric field @xcite . therefore it is possible to perform a real - time monitoring of the field - induced aggregation process in such a suspension by measuring the optical nonlinear ac responses both parallel and perpendicular to the uniaxial anisotropic axis . recently , an anisotropic maxwell - garnett theory @xcite was applied to study the nonlinear polarization of these anisotropic structures @xcite . anisotropic optical nonlinear ac response was indeed found .    when a nonlinear composite with nonlinear dielectric particles embedded in a host medium , or with a nonlinear host medium is subjected to a sinusoidal field , the electrical response in the composite will in general be a superposition of many sinusoidal functions @xcite . it is natural to investigate the effects of a nonlinear characteristics on the optical nonlinear ac response in a colloidal suspension which can be regarded as a nonlinear composite medium @xcite . the strength of the nonlinear polarization should be reflected in the magnitude of the harmonics . for these field - induced anisotropic structures , we found that as more and more chains are formed the harmonics of local field and induced dipole moment increase ( decrease ) for longitudinal ( transverse ) field case @xcite .    on the other hand , the effect of nonlinear characteristics on the inter - particle force has been analyzed in a colloidal suspension of nonlinear particles @xcite and further extended to a nonlinear host medium @xcite . actually , the intrinsic dielectric dispersion often occurs due to the surface conductivity or the inhomogeneous structure ( e.g. , coating effect @xcite ) which often exist in colloidal suspensions . hence it is of particular importance to discuss the effect of an intrinsic dielectric dispersion on the nonlinear polarization . in this regard , we will use a perturbation expansion method to calculate the nonlinear ac response of a nonlinear composite , i.e. , harmonics of local field as well as induced dipole moment . let us consider the system in which the suspended particles of radius @xmath0 with a nonlinear dielectric constant are suspended in a host medium of linear dielectric constant @xmath1 . the nonlinear characteristic gives rise to a field - dependent dielectric coefficient @xcite . in this case , the electric displacement - electric field relation inside the spheres is given by : @xmath2with @xmath3 , where @xmath4 and @xmath5 are the linear coefficient and the weak nonlinear coefficient of the suspended particles , respectively . here @xmath6 is expressed as a debye expression , @xmath7 is the dielectric constant at high frequency , @xmath8 is the intrinsic dispersion of the particle , @xmath9 is the intrinsic characteristic frequency , with @xmath10 being the intrinsic relaxation time , and @xmath11 is the angular frequency of the external field . in eq.([nl - coefficient ] ) , we have adopted an approximation : the local field inside the particles is assumed to be uniform . this assumption is called the decoupling approximation @xcite . it has been shown that such an approximation yields a lower bound for the accurate result for the local field @xcite . we further assumed that @xmath5 is independent of frequency , which is a valid assumption for low - frequency processes in colloidal suspensions . as a result , the induced dipole moment under an applied field @xmath12 along @xmath13axis , with @xmath14 being time , is given by @xcite : @xmath15where @xmath16 is the effective dielectric constant of the suspension , and @xmath17 is the field - dependent dipole factor : @xmath18 in order to obtain the effective dielectric constant @xmath16 , we invoke the maxwell - garnett approximation ( mga ) for anisotropic composites @xcite . for the longitudinal field case when the ac field is applied along the uniaxial anisotropic axis , the mga has the form = f . [ mga ] with @xmath19 being the volume fraction of particles , whereas for an transverse field case when the ac field is applied perpendicular to the uniaxial anisotropic axis , the mga expression is obtained by replacing @xmath20 with @xmath21 @xcite . here @xmath22 and @xmath23 denote the local field factors parallel and perpendicular to the uniaxial anisotropic axis . these local field factors are defined as the ratio of the local field in the particles to the lorentz cavity field @xcite . for isotropic composites , @xmath24 , while both @xmath20 and @xmath25 will deviate from unity for an anisotropic distribution of particles in composites . these @xmath26 factors have been evaluated in a tetragonal lattice of dipole moments @xcite and in various field - structured composites @xcite . we resort to the spectral representation approach @xcite to represent the local field inside the particle @xcite : |*e*_1|^2 = _ 0 ^ 1x , [ ee-1 ] with the material parameter @xmath27 as well as the spectral density function @xmath28 $ ] , which satisfies the sum rule @xmath29 . hence eq.([ee-1 ] ) has the form |*e*_1|^2 = ^2(t ) . [ average1 ] actually , this is a self - consistent equation for @xmath30 , which can be solved at least numerically @xcite . in what follows , we will use a perturbation method instead to obtain analytic expressions for harmonics of the local electric field and the induced dipole moment .    on the other hand , due to the uniform local field in each particle , eq.([average1 ] ) can also be directly derived from @xcite :     if we apply a sinusoidal electric field , i.e. @xmath31 , the induced dipole moment @xmath32 will depend on time sinusoidally , too . by virtue of the inversion symmetry , @xmath32 is a superposition of odd - order harmonics such that = p_t+p_33t+ . also , the local electric field contains similar harmonics = e_t+e_33t+ . these harmonic coefficients can be extracted from the time dependence of the solution of @xmath32 and @xmath33 . in this section , we will apply the perturbation expansion @xcite to extract the harmonics of the local electric field and the induced dipole moment . it is known that the perturbation expansion method is applicable to weak nonlinearity only , limited by the convergence of the series expansion . we expand @xmath32 and @xmath34 into a taylor expansion , taking @xmath35 as a perturbative quantity : = _ n=0^ _ _ 1=_1(_1|*e*_1|^2)^n , = _ m=0^ _ _ 1=_1(_1|*e*_1|^2)^m . in view of weak nonlinearity , we can rewrite eqs.(10 ) and ( 11 ) , keeping the lowest orders of @xmath35 and @xmath36 : @xmath37 where @xmath38,\nonumber\\ j_1&=&\frac{3|\ep_2|}{\rho},\ \   j_3=\frac{27}{2}|\ep_2|^3\frac{f\beta-1}{\rho|\rho|}(\frac{1}{|\rho|^2}+\frac{1}{(\rho^{2})^*}),\nonumber\end{aligned}\ ] ] with @xmath39 .    in the case of a sinusoidal field , we can expand @xmath40 in terms of the first and the third harmonics . the comparison with eq.(8 ) yields the harmonics of the induced dipole moment : p _ e+h_3_1 e^3 ,  p_3 = -h_3 _ 1 e^3 . [ p3w ] similarly , we find the harmonics of the local electric field _ 1 ^ 1/2e _ = j_1_1 ^ 1/2e+j_3(_1 ^ 1/2e)^3 ,  _ 1 ^ 1/2e_3 = -j_3(_1 ^ 1/2e)^3 . [ e3w ]    in the above analysis , we have used the identity @xmath41 to obtain the first and the third harmonics . similar analysis can be used to extract the higher - order harmonics , by retaining more terms in the series expansion . for instance , to obtain fifth harmonics , we may take into account the identity @xmath42 , and then obtain the harmonics of induced dipole moment and local electric field , respectively : @xmath43 where @xmath44 + \frac{3\ep_2 ^ 2}{\rho(\ep_1 + 2\ep_2)^2}\nonumber\\ & & [ \frac{(1 + 3f - f\beta)\rho-(1-f\beta)(\rho+3f\ep_1 - 3f\ep_2)}{\rho}- \frac{2\rho+6f(\ep_1-\ep_2)}{\ep_1 + 2\ep_2 } ] \},\nonumber\end{aligned}\ ] ] and @xmath45 where @xmath46 because of the weak nonlinearity under consideration , the effect of high order harmonics is quite small . thus , without loss of generality , we only consider the taylor expansion up to second term . in this regard , we will use eqs.([p3w ] ) and ( [ e3w ] ) to investigate the nonlinear ac response in the numerical calculations in next section . we are now in a position to perform numerical calculations to investigate the dispersion effect on the harmonics of local electric field as well as induced dipole moment . we let @xmath47 , @xmath48 , @xmath49 , where @xmath50 is the dielectric constant of free space . additionally , set @xmath51 to normalize the induced dipole moment , where @xmath52 is given by the anisotropic mga ( eq.([mga ] ) ) without nonlinear characteristic . obviously , the expressions for the normalized local electric field and the normalized induced dipole moment are functions of a single variable @xmath53 @xcite . in view of the weak nonlinearity , we set @xmath54 for our calculation .    in fig.1(a)@xmath55(c ) , we investigate the harmonics of local field and induced dipole moment versus frequency for different intrinsic dispersion strengths @xmath8 . it is obvious that increasing the dispersion strength reduces the harmonics of both the local field and the induced dipole moment . we find that , for increasing frequency , the harmonics of local field always increases . however , the harmonics of induced dipole moment exhibits a more complicated behavior . for small @xmath8 , the ratio of the third to first harmonics of induced dipole moment decreases as the frequency increases , which is qualitatively in good agreement with the experimental data @xcite . however , for large @xmath8 , its harmonic ratio also increases as the frequency increases . we expect this behavior can be demonstrated by experiment in the future .    in fig.2 , we investigate the harmonics of local field and induced dipole moment versus frequency for different intrinsic relaxation times @xmath10 . a longitudinal field case ( @xmath56 ) is discussed in fig.2(a ) , while a transverse field case ( @xmath57 ) in fig.2(b ) . it is shown that , in either case , increasing relaxation time @xmath10 may enhance the harmonics of both local field and induced dipole moment . in addition , the transverse field case may predict stronger harmonics than the longitudinal field case within high - frequency region , which is actually valid for different dispersion strengths as well ( no figures shown herein ) .    on the other hand , klingenberg claimed that increasing external electric field leads to increasing harmonics of electric current @xcite . in this sense , we have also calculated the effect of external electric field on the nonlinear ac response of the induced dipole moment ( no figures shown herein ) . obviously , our calculations qualitatively demonstrate the experimental results @xcite as well . here a few comments on our results are in order . in the present study , we have examined the case of nonlinear particles suspending in a linear host . our considerations may be extended to a nonlinear host medium @xcite . we have studied the effect of an intrinsic dispersion on the nonlinear ac response of a colloidal suspension . actually , our formalism can be readily generalized to discuss two or more intrinsic dielectric dispersions of particles ( or host fluid ) . in view of the fact that the intrinsic dielectric dispersion often occurs due to the surface conductivity or the inhomogeneous structure ( e.g. , coating effect @xcite ) , we believe such effect also plays an important role in the dielectric behavior of biological cells , such as the electrorotation , dielectrophoresis and electro - orientation . the perturbation method is used in the present work , but we believe similar results may also be predicted by a self - consistent theory @xcite .    in summary , a perturbation method has been employed to compute the local electric field and the induced dipole moment for suspensions in which the suspended particles have a nonlinear characteristic and an intrinsic dispersion , in an attempt to investigate the frequency effect on the nonlinear ac response . the results showed , for weak intrinsic dispersion strength , the ratio of the third to first harmonics of the induced dipole moment decreases as the frequency increases , which is qualitatively in agreement with experimental result . however , for a strong dispersion strength , the harmonics ratio increases as the frequency increases . moreover , an increase in the intrinsic relaxation time may increase the strength of harmonics . this work was supported by the research grants council of the hong kong sar government under project numbers cuhk 4284/00p and cuhk 4245/01p . k.w.y . thanks prof . a. r. day for his interest in our work and his suggestion to examine the effects of inhomogeneities on the dielectric response of biological cell suspensions . j. e. martin , r. a. anderson and c. p. tigges , in _ nanostructured materials : clusters , composites , and thin films _ , edited by v. m. shalaev and m. moskovitz ( acs symposium series * 679 * , 1996 ) , chap . 6 . j. e. martin , r. a. anderson and c. p. tigges , j. chem . phys . * 108 * , 3765 ( 1998 ) ; * 108 * , 7887 ( 1998 ) . k. p. yuen , m. f. law , k. w. yu and ping sheng , phys . e * 56 * , r1322 ( 1997 ) . d. j. klingenberg , mrs bulletin * 23 * , 30 ( 1998 ) .      o. levy , d. j. bergman and d. stroud , phys . e * 52 * , 3184 ( 1995 ) . g. q. gu , p. m. hui and k. w. yu , physica b * 279 * , 62 ( 2000 ) . jones t. k. wan , g. q. gu and k. w. yu , phys . e * 63 * , 052501 ( 2001 ) . jones t. k. wan , g. q. gu and k. w. yu , int . b * 14 * , 603 ( 2000 ) . l. gao , jones t. k. wan , k. w. yu and z. y. li , phys . e * 61 * , 6011 ( 2000 ) . l. gao and z. y. li , j. appl . * 87 * , 1620 ( 2000 ) ; see also l. gao , jones t. k. wan , k. w. yu and z. y. li , j. appl . phys . * 88 * , 1893 ( 2000 ) . k. w. yu , p. m. hui and d. stroud , phys . b * 47 * , 14150 ( 1993 ) . k. w. yu , p. m. hui and h. c. lee , phys . a * 210 * , 115 ( 1996 ) ; see also w. m. v. wan , h. c. lee , p. m. hui and k. w. yu , phys . rev . b * 54 * , 3946 ( 1996 ) . j. e. martin and r. a. anderson , j. chem 111 * , 4273 ( 1999 ) . d. j. bergman , phys . rep . * 43 * , 379 ( 1978 ) .
when a sinusoidal ( ac ) field is applied to a suspension containing nonlinear dielectric particles , the electrical response will generally consist of ac fields at frequencies of the higher - order harmonics . 
 the situation is further complicated by an intrinsic dielectric dispersion which often occurs due to the surface conductivity or inhomogeneous structure of the particles . 
 we perform a perturbation method to investigate the effect of intrinsic dielectric dispersion on the harmonics of local field as well as induced dipole moment . 
 the results showed , for weak intrinsic dispersion strength , the ratio of the third to first harmonics of the induced dipole moment decreases as the frequency increases , which is qualitatively in agreement with experimental result . 
 however , for a strong dispersion strength , the harmonics ratio increases as the frequency increases . 
 moreover , an increase in the intrinsic relaxation time may increase the strength of harmonics .