The combination of the Lorentz-Lorenz formula with the Lorentz model of dielectric dispersion results in a decrease in the effective resonance frequency of the material when the number density of Lorentz oscillators is large. An equivalence relation is derived that equates the frequency dispersion of the Lorentz model alone with that modified by the Lorentz-Lorenz formula. Negligible differences between the computed ultrashort pulse dynamics are obtained for these equivalent models.
©2003 Optical Society of America
The classical Lorentz model  of dielectric dispersion due to resonance polarization is of fundamental importance in optics as it provides a physically appealing, accurate description of both normal and anomalous dispersion phenomena in the extended optical region of the electromagnetic spectrum from the far infrared up to the near ultraviolet. Of equal importance is the Lorentz-Lorenz formula [2,3] which, as stated in Born and Wolf , “connects Maxwell’s phenomenological theory with the atomistic theory of matter.” It is typically assumed [4,5] that the number density of molecules is sufficiently small so that the Lorentz-Lorenz formula can be simplified to a simple linear relationship between the mean molecular polarizability and the dielectric permittivity. Although the influence of the Lorentz-Lorenz formula on the resulting frequency dispersion can be striking when the number density becomes sufficiently large, the fundamental frequency structure is not altered from that described by the Lorentz model alone; a frequency band of anomalous dispersion with high absorption surrounded by lower and higher frequency regions exhibiting normal dispersion with small absorption.
The fact that the Lorentz model is a causal model  of temporal dispersion has cast it in a central role in both the classical [7–9] and modern [10–11] asymptotic theories of linear dispersive pulse propagation. Although the asymptotic theory is independent of the particular material parameter values chosen for the Lorentz model dielectric considered, the material parameters originally chosen by Brillouin [8,9] and employed in much of the modern asymptotic theory [10–11] correspond to a highly absorptive material for which the Lorentz-Lorenz formula must be applied without approximation. The purpose of this paper is to establish an approximate equivalence relation that equates the frequency dispersion of the Lorentz model alone with that modified by the Lorentz-Lorenz formula. This result then extends the domain of applicability of the asymptotic theory to include the optically dense material originally considered by Brillouin [8,9].
2. The Lorentz-Lorenz formula and the Lorentz model of dielectric dispersion
The Lorentz force acting on a bound electron in a material depends upon the local or effective electromagnetic field present at that molecular site. The effective electric field Eeff(r,t) acting on a molecule at space-time position (r,t) in a polarizable medium with polarization P(r,t) is given by 
where E(r,t) is the external, applied electric field. In a locally linear, homogeneous, isotropic material the electric dipole moment for each molecule is linearly related to the effective electric field through the causal relation
with Fourier transform p̃(r,ω)=α(ω)Ẽeff(r,ω), where α(ω) is the mean polarizability at angular frequency ω. If N denotes the number density of molecules in the material, then the spectrum of the induced polarization in Eq. (1) is given by P̃(r,ω)=N p̃(r,ω). With substitution from the Fourier transform of Eq. (1) one then obtains the expression
which is also referred to as the Clausius-Mossotti relation . It is typically assumed that ε(ω) is sufficiently close to unity that ε(ω)+2≈3 in which case the Lorentz-Lorenz formula simplifies to ε(ω)≈1+4πNα(ω), which is equivalent to the approximation that Eeff(r,t)≈E(r,t).
The Lorentz model  of resonance polarization in dielectrics is based upon the damped harmonic oscillator equation
for the position vector r(t) relative to the nucleus of a bound electron of mass me and charge magnitude qe with (undamped) resonance frequency ω 0 and phenomenological damping constant δ under the action of the local Lorentz force Floc(t)=-qeEeff(t) due to the electric field alone, the magnetic field contribution being assumed negligible by comparison. The solution to this o.d.e. is obtained in the Fourier frequency domain as
The local induced dipole moment is then given by p̃(ω)=-qer̃(ω) which then results in the expression
for the molecular polarizability. Substitution of this expression into the Lorentz-Lorenz formula then yields the final expression
for the complex, relative dielectric permittivity. The complex index of refraction is then given by the branch of the square root of the expression in Eq. (9) that yields a positive imaginary part (attenuation) along the positive real frequency axis. . When the inequality b 2/(6δω 0)≪1 is satisfied, the denominator in Eq. (9) may be approximated by the first two terms in its power series expansion so that
As an example, consider the Lorentz model material parameters chosen by Brillouin [8,9], viz. ω 0=4×1016 r/s, δ=0.28×1016 r/s, b=√20×1016 r/s, which correspond to a highly absorptive dielectric. The angular frequency dispersion of the complex index of refraction for the Lorentz model alone [as given by the square root of the final approximation in Eq. (10)] is illustrated by the solid blue curve in Figure 1. Part (a) of the figure describes the frequency dispersion of the real index of refraction while part (b) describes that for imaginary part . The corresponding solid green curves in Fig. 1 describe the resultant frequency dispersion for this Lorentz model dielectric when the Lorentz-Lorenz relation is used [cf. Eq. (9)]. As can be seen, the Lorentz-Lorenz modified frequency dispersion primarily shifts the resonance frequency to a lower frequency value while increasing both the absorption and the below resonance index of refraction. Notice that b 2/(6δω 0)=2.976 for this choice of material parameters. If the plasma frequency is decreased to the value b=√2×1016 r/s so that b 2/(6δω 0)=0.2976, then the modification of the Lorentz model by the Lorentz-Lorenz relation is relatively small, as exhibited by the second set of curves in Fig. 1.
3. An approximate equivalence relation
Since the primary effect of the Lorentz-Lorenz formula on the Lorentz model is to downshift the effective resonance frequency and increase the low frequency refractive index, consider then determining the resonance frequency ω * appearing in the Lorentz-Lorenz formula for a Lorentz model dielectric that will yield the same value for ε(0) as given by the Lorentz model alone with resonance frequency ω 0. From Eqs. (9) and (10) one then has that
Consider then comparison of the frequency dependence of the expression [cf. Eq. (9)]
of the relative dielectric permittivity for the Lorentz-Lorenz modified Lorentz model dielectric with undamped resonance frequency ω * given by the equivalence relation (12), with the expression [cf. Eq. (10)]
of the approximate relative dielectric permittivity for a Lorentz model dielectric with undamped resonance frequency ω 0. The other two material parameters b and δ are the same in these two expressions.
A comparison of the angular frequency dependence described by Eqs. (14) and (13) with ω * given by the equivalence relation (12) is presented in Fig. 2 for Brillouin’s choice of the material parameters (ω 0=4×1016 r/s, δ=0.28×1016 r/s, b=√20×1016 r/s). The rms error between the two sets of data points presented in Fig. 2 is approximately 2.3×10-16 for the real part and 2.0×10-16 for the imaginary part of the complex index of refraction, with a maximum single point rms error of ~2.5×10-16. The corresponding rms error for the relative dielectric permittivity is ~1.1×10-15 for both the real and imaginary parts with a maximum single point rms error of ~1×10-14. Variation of any of the remaining material parameters in the equivalent Lorentz-Lorenz modified Lorentz model dielectric, including the value of the plasma frequency from that specified in Eq. (12), only results in an increase in the rms error. This approximate equivalence relation between the Lorentz-Lorenz formula modified Lorentz model dielectric and the Lorentz model dielectric alone is then seen to provide a “best fit” in the rms sense between the frequency dependence of the two models.
An approximate equivalence relation between the Lorentz-Lorenz formula modified Lorentz model dielectric and the Lorentz model alone for the complex index of refraction of a single resonance dielectric has been presented. Numerical results show that this approximate equivalence relation provides a “best fit” in the rms sense between the frequency dependence of the two models. This result then extends the domain of applicability of the asymptotic theory of dispersive pulse propagation in a Lorentz model dielectric to include the optically dense material originally considered by Brillouin [8,9] and subsequently used as an example in the modern asymptotic theory [10,11]. In fact, the results are indistinguishable when the two equivalent models are used in a numerical determination of the propagated field due to an input rectangular envelope pulse in a single resonance Lorentz model dielectric using Brillouin’s choice of the material parameters, including the leading and trailing edge precursors.
The research presented in this paper was supported, in part, by the United States Air Force Office of Scientific Research under AFOSR Grant # 49620-01-0306.
References and Links
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