Abstract

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

1. Introduction

The classical Lorentz model [1] 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 [4], “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 [6] of temporal dispersion has cast it in a central role in both the classical [79] and modern [1011] 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 [1011] 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 [3]

Eeff(r,t)=E(r,t)+4π3P(r,t),

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

p(r,t)=tα̂(tt)Eeff(r,t)dt

with Fourier transform (r)=α(ω)eff(r), 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 (r)=N (r). With substitution from the Fourier transform of Eq. (1) one then obtains the expression

P˜(r,ω)=χe(ω)E˜(r,ω),

where

χe(ω)=Nα(ω)1(4π3)Nα(ω)

is the electric susceptibility. With the expression ε(ω)=1+4πχe(ω) for the relative dielectric permittivity, one then obtains the Lorentz-Lorenz formula [2,3]

α(ω)=34πNε(ω)1ε(ω)+2,

which is also referred to as the Clausius-Mossotti relation [12]. 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 [1] of resonance polarization in dielectrics is based upon the damped harmonic oscillator equation

me(r¨(t)+2δr˙(t)+ω02r(t))=qeEeff(t)

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

r˜(ω)=qemeω2ω02+2iδωE˜eff(ω).

The local induced dipole moment is then given by (ω)=-qe(ω) which then results in the expression

α(ω)=qe2meω2ω02+2iδω

for the molecular polarizability. Substitution of this expression into the Lorentz-Lorenz formula then yields the final expression

ε(ω)=1(2b23)(ω2ω02+2iδω)1+(b23)(ω2ω02+2iδω)

for the complex, relative dielectric permittivity. The complex index of refraction n(ω)=ε(ω) 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. [12]. 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

ε(ω)(12b23ω2ω02+2iδω)(1b23ω2ω02+2iδω)1b2ω2ω02+2iδω,

which is the usual expression [1,411] for the frequency dispersion of a single resonance Lorentz model dielectric.

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 nr(ω)={ε(ω)} while part (b) describes that for imaginary part nr(ω)={ε(ω)} . 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.

 

Fig. 1. Angular frequency dependence of the real (a) and imaginary (b) parts of the complex index of refraction for a Lorentz model dielectric with (green curves) and without (blue curves) the Lorentz-Lorenz formula for two different values of the material plasma frequency.

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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

1+b2ω02=1+(23)(b2ω*2)1(13)(b2ω*2),

with solution

ω*=ω02+b23.

Consider then comparison of the frequency dependence of the expression [cf. Eq. (9)]

ε(ω)=1(2b23)(ω2ω*2+2iδω)1+(b23)(ω2ω*2+2iδω)

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)]

εapp(ω)=1b2ω2ω02+2iδω

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.

 

Fig. 2. Comparison of the angular frequency dependence of the real (a) and imaginary (b) parts of the complex index of refraction for a single resonance Lorentz model dielectric alone (solid blue curves) and for the equivalent Lorentz-Lorenz formula modified Lorentz model (green circles).

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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.

4. Conclusions

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.

Acknowledgement

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

1. H. A. Lorentz, Versuch einer Theorie der Electrischen und Optischen Erscheinungen in Bewegten Körpern (Teubner, 1906); see also H. A. Lorentz The Theory of Electrons (Dover, 1952).

2. H. A. Lorentz, “Über die Beziehungzwischen der Fortpflanzungsgeschwindigkeit des Lichtes der Körperdichte,” Ann. Phys. 9, 64–665 (1880).

3. L. Lorenz, “Über die Refractionsconstante,” Ann. Phys. 11, 70–103 (1880). [CrossRef]  

4. M. Born and E. Wolf, Principles of Optics, 7th (expanded) edition (Cambridge U. Press, 1999) Ch. 2.

5. J. M. Stone, Radiation and Optics (McGraw-Hill, 1963) Ch. 15.

6. H. M. Nussenzveig, Causality and Dispersion Relations (Academic Press, 1972) Ch. 1.

7. A. Sommerfeld, “Über die Fortpflanzung des Lichtes in Disperdierenden Medien,” Ann. Phys. 44, 177–202 (1914). [CrossRef]  

8. L. Brillouin, “Über die Fortpflanzung des Licht in Disperdierenden Medien,” Ann. Phys. 44, 203–240 (1914). [CrossRef]  

9. L. Brillouin, Wave Propagation and Group Velocity (Academic Press, 1960).

10. K. E. Oughstun and G. C. Sherman. “Propagation of electromagnetic pulses in a linear dispersive medium with absorption (the Lorentz medium),” J. Opt. Soc. Am. B 5, 817–849 (1988).

11. K. E. Oughstun and G. C. Sherman, Electromagnetic Pulse Propagation in Causal Dielectrics (Springer-Verlag, 1994). [CrossRef]  

12. B. K. P. Scaife, Principles of Dielectrics (Oxford, 1989) Ch. 7.

References

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  1. H. A. Lorentz, Versuch einer Theorie der Electrischen und Optischen Erscheinungen in Bewegten Körpern (Teubner, 1906); see also H. A. Lorentz The Theory of Electrons (Dover, 1952).
  2. H. A. Lorentz, “Über die Beziehungzwischen der Fortpflanzungsgeschwindigkeit des Lichtes der Körperdichte,” Ann. Phys. 9, 64–665 (1880).
  3. L. Lorenz, “Über die Refractionsconstante,” Ann. Phys. 11, 70–103 (1880).
    [Crossref]
  4. M. Born and E. Wolf, Principles of Optics, 7th (expanded) edition (Cambridge U. Press, 1999) Ch. 2.
  5. J. M. Stone, Radiation and Optics (McGraw-Hill, 1963) Ch. 15.
  6. H. M. Nussenzveig, Causality and Dispersion Relations (Academic Press, 1972) Ch. 1.
  7. A. Sommerfeld, “Über die Fortpflanzung des Lichtes in Disperdierenden Medien,” Ann. Phys. 44, 177–202 (1914).
    [Crossref]
  8. L. Brillouin, “Über die Fortpflanzung des Licht in Disperdierenden Medien,” Ann. Phys. 44, 203–240 (1914).
    [Crossref]
  9. L. Brillouin, Wave Propagation and Group Velocity (Academic Press, 1960).
  10. K. E. Oughstun and G. C. Sherman. “Propagation of electromagnetic pulses in a linear dispersive medium with absorption (the Lorentz medium),” J. Opt. Soc. Am. B 5, 817–849 (1988).
  11. K. E. Oughstun and G. C. Sherman, Electromagnetic Pulse Propagation in Causal Dielectrics (Springer-Verlag, 1994).
    [Crossref]
  12. B. K. P. Scaife, Principles of Dielectrics (Oxford, 1989) Ch. 7.

1988 (1)

K. E. Oughstun and G. C. Sherman. “Propagation of electromagnetic pulses in a linear dispersive medium with absorption (the Lorentz medium),” J. Opt. Soc. Am. B 5, 817–849 (1988).

1914 (2)

A. Sommerfeld, “Über die Fortpflanzung des Lichtes in Disperdierenden Medien,” Ann. Phys. 44, 177–202 (1914).
[Crossref]

L. Brillouin, “Über die Fortpflanzung des Licht in Disperdierenden Medien,” Ann. Phys. 44, 203–240 (1914).
[Crossref]

1880 (2)

H. A. Lorentz, “Über die Beziehungzwischen der Fortpflanzungsgeschwindigkeit des Lichtes der Körperdichte,” Ann. Phys. 9, 64–665 (1880).

L. Lorenz, “Über die Refractionsconstante,” Ann. Phys. 11, 70–103 (1880).
[Crossref]

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th (expanded) edition (Cambridge U. Press, 1999) Ch. 2.

Brillouin, L.

L. Brillouin, “Über die Fortpflanzung des Licht in Disperdierenden Medien,” Ann. Phys. 44, 203–240 (1914).
[Crossref]

L. Brillouin, Wave Propagation and Group Velocity (Academic Press, 1960).

Lorentz, H. A.

H. A. Lorentz, “Über die Beziehungzwischen der Fortpflanzungsgeschwindigkeit des Lichtes der Körperdichte,” Ann. Phys. 9, 64–665 (1880).

H. A. Lorentz, Versuch einer Theorie der Electrischen und Optischen Erscheinungen in Bewegten Körpern (Teubner, 1906); see also H. A. Lorentz The Theory of Electrons (Dover, 1952).

Lorenz, L.

L. Lorenz, “Über die Refractionsconstante,” Ann. Phys. 11, 70–103 (1880).
[Crossref]

Nussenzveig, H. M.

H. M. Nussenzveig, Causality and Dispersion Relations (Academic Press, 1972) Ch. 1.

Oughstun, K. E.

K. E. Oughstun and G. C. Sherman. “Propagation of electromagnetic pulses in a linear dispersive medium with absorption (the Lorentz medium),” J. Opt. Soc. Am. B 5, 817–849 (1988).

K. E. Oughstun and G. C. Sherman, Electromagnetic Pulse Propagation in Causal Dielectrics (Springer-Verlag, 1994).
[Crossref]

Scaife, B. K. P.

B. K. P. Scaife, Principles of Dielectrics (Oxford, 1989) Ch. 7.

Sherman, G. C.

K. E. Oughstun and G. C. Sherman. “Propagation of electromagnetic pulses in a linear dispersive medium with absorption (the Lorentz medium),” J. Opt. Soc. Am. B 5, 817–849 (1988).

K. E. Oughstun and G. C. Sherman, Electromagnetic Pulse Propagation in Causal Dielectrics (Springer-Verlag, 1994).
[Crossref]

Sommerfeld, A.

A. Sommerfeld, “Über die Fortpflanzung des Lichtes in Disperdierenden Medien,” Ann. Phys. 44, 177–202 (1914).
[Crossref]

Stone, J. M.

J. M. Stone, Radiation and Optics (McGraw-Hill, 1963) Ch. 15.

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 7th (expanded) edition (Cambridge U. Press, 1999) Ch. 2.

Ann. Phys. (4)

H. A. Lorentz, “Über die Beziehungzwischen der Fortpflanzungsgeschwindigkeit des Lichtes der Körperdichte,” Ann. Phys. 9, 64–665 (1880).

L. Lorenz, “Über die Refractionsconstante,” Ann. Phys. 11, 70–103 (1880).
[Crossref]

A. Sommerfeld, “Über die Fortpflanzung des Lichtes in Disperdierenden Medien,” Ann. Phys. 44, 177–202 (1914).
[Crossref]

L. Brillouin, “Über die Fortpflanzung des Licht in Disperdierenden Medien,” Ann. Phys. 44, 203–240 (1914).
[Crossref]

J. Opt. Soc. Am. (1)

K. E. Oughstun and G. C. Sherman. “Propagation of electromagnetic pulses in a linear dispersive medium with absorption (the Lorentz medium),” J. Opt. Soc. Am. B 5, 817–849 (1988).

Other (7)

K. E. Oughstun and G. C. Sherman, Electromagnetic Pulse Propagation in Causal Dielectrics (Springer-Verlag, 1994).
[Crossref]

B. K. P. Scaife, Principles of Dielectrics (Oxford, 1989) Ch. 7.

L. Brillouin, Wave Propagation and Group Velocity (Academic Press, 1960).

M. Born and E. Wolf, Principles of Optics, 7th (expanded) edition (Cambridge U. Press, 1999) Ch. 2.

J. M. Stone, Radiation and Optics (McGraw-Hill, 1963) Ch. 15.

H. M. Nussenzveig, Causality and Dispersion Relations (Academic Press, 1972) Ch. 1.

H. A. Lorentz, Versuch einer Theorie der Electrischen und Optischen Erscheinungen in Bewegten Körpern (Teubner, 1906); see also H. A. Lorentz The Theory of Electrons (Dover, 1952).

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Figures (2)

Fig. 1.
Fig. 1.

Angular frequency dependence of the real (a) and imaginary (b) parts of the complex index of refraction for a Lorentz model dielectric with (green curves) and without (blue curves) the Lorentz-Lorenz formula for two different values of the material plasma frequency.

Fig. 2.
Fig. 2.

Comparison of the angular frequency dependence of the real (a) and imaginary (b) parts of the complex index of refraction for a single resonance Lorentz model dielectric alone (solid blue curves) and for the equivalent Lorentz-Lorenz formula modified Lorentz model (green circles).

Equations (14)

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E eff ( r , t ) = E ( r , t ) + 4 π 3 P ( r , t ) ,
p ( r , t ) = t α ̂ ( t t ) E eff ( r , t ) d t
P ˜ ( r , ω ) = χ e ( ω ) E ˜ ( r , ω ) ,
χ e ( ω ) = N α ( ω ) 1 ( 4 π 3 ) N α ( ω )
α ( ω ) = 3 4 π N ε ( ω ) 1 ε ( ω ) + 2 ,
m e ( r ¨ ( t ) + 2 δ r ˙ ( t ) + ω 0 2 r ( t ) ) = q e E eff ( t )
r ˜ ( ω ) = q e m e ω 2 ω 0 2 + 2 i δ ω E ˜ eff ( ω ) .
α ( ω ) = q e 2 m e ω 2 ω 0 2 + 2 i δ ω
ε ( ω ) = 1 ( 2 b 2 3 ) ( ω 2 ω 0 2 + 2 i δ ω ) 1 + ( b 2 3 ) ( ω 2 ω 0 2 + 2 i δ ω )
ε ( ω ) ( 1 2 b 2 3 ω 2 ω 0 2 + 2 i δ ω ) ( 1 b 2 3 ω 2 ω 0 2 + 2 i δ ω ) 1 b 2 ω 2 ω 0 2 + 2 i δ ω ,
1 + b 2 ω 0 2 = 1 + ( 2 3 ) ( b 2 ω * 2 ) 1 ( 1 3 ) ( b 2 ω * 2 ) ,
ω * = ω 0 2 + b 2 3 .
ε ( ω ) = 1 ( 2 b 2 3 ) ( ω 2 ω * 2 + 2 i δ ω ) 1 + ( b 2 3 ) ( ω 2 ω * 2 + 2 i δ ω )
ε app ( ω ) = 1 b 2 ω 2 ω 0 2 + 2 i δ ω

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