Abstract

The approximate equivalence relation equating the frequency dispersion of the Lorentz model alone with that modified by the Lorentz-Lorenz formula is shown to also equate the branch points appearing in each of these two descriptions.

© 2003 Optical Society of America

The primary effect [1] of the Lorentz-Lorenz formula, rewritten here to express the relative dielectric permittivity ε(ω) in terms of the mean molecular polarizability α(ω) as

ε(ω)=1+(8π3)Nα(ω)1(4π3)Nα(ω),

with the single resonance Lorentz model of the molecular polarizability

α(ω)=qe2meω2ω02+2iδ0ω

is to downshift the effective resonance frequency and increase the strength of the low frequency behavior from that described by the Lorentz model approximation

εapp(ω)=1b2ω2ω02+2iδω.

Here ω 0 is the undamped resonance frequency of the harmonically bound electron of charge magnitude qe and mass me with number density N, phenomenological damping constant δ and plasma frequency b=4πNqe2me . The approximate expression given in Eq. (3) is obtained from Eq. (1) with Eq. (2) when the inequality b 2/(6δω 0)≪1 is satisfied. The Lorentz-Lorenz relation (1) with the Lorentz model (2) of the molecular polarizability gives the relative dielectric permittivity as

ε(ω)=ω2ω*2+2iδω2b23ω2ω*2+2iδω+b23,

where the undamped resonance frequency is denoted in this expression by ω *. The value of this resonance frequency ω * that will yield the same value for ε(0) as given by Eq. (3) is given by [1]

ω*=ω02+b23.

This approximate equivalence relation provides a “best fit” in the rms sense between the frequency dependence of the Lorentz-Lorenz modified Lorentz model dielectric and the Lorentz model alone [1] for both the dielectric permittivity and the complex index of refraction n(ω)=ε(ω) along the positive real frequency axis.

The branch points of the complex index of refraction for the single resonance Lorentz model dielectric with dielectric permittivity described by Eq. (3) are given by

ωp±=iδ±ω02δ2,
ωz±=iδ±ω02+b2δ2,

and the branch points of the complex index of refraction for the Lorentz-Lorenz modified Lorentz model dielectric with dielectric permittivity described by Eq. (4) are given by

ωp±=iδ±ω*2b23δ2,
ωz±=iδ±ω*2+2b23δ2.

If ω *=ω 0, then the branch points of n(ω) for the Lorentz-Lorenz modified Lorentz model are shifted inward toward the imaginary axis from the branch point locations for the Lorentz model alone provided that the inequality ω*2-b 2/3-δ 2≥0 is satisfied. If the opposite inequality ω*2-b 2/3-δ 2<0 is satisfied, then the branch points ωp± are located along the imaginary axis.

If ω * is given by the equivalence relation (5), then the locations of the branch points of the complex index of refraction n(ω) for the Lorentz-Lorenz modified Lorentz model and the Lorentz model alone are exactly the same. The branch cuts for these two models are then also the same (or can be chosen so). It then follows that the analyticity properties for these two causal models of the dielectric permittivity are the same. This important result is central to the asymptotic description of both ultrawideband signal and ultrashort pulse propagation in Lorentz model dielectrics, particularly when the number density of molecules is large.

References and links

1. K. E. Oughstun and N. A. Cartwright, “On the Lorentz-Lorenz formula and the Lorentz model of dielectric dispersion,” Opt. Express 11, 1541–1546 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-13-1541. [CrossRef]   [PubMed]  

References

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  1. K. E. Oughstun and N. A. Cartwright, "On the Lorentz-Lorenz formula and the Lorentz model of dielectric dispersion," Opt. Express 11, 1541-1546 (2003), <a href= "http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-13-1541">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-13-1541</a>.
    [CrossRef] [PubMed]

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Equations (9)

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ε ( ω ) = 1 + ( 8 π 3 ) N α ( ω ) 1 ( 4 π 3 ) N α ( ω ) ,
α ( ω ) = q e 2 m e ω 2 ω 0 2 + 2 i δ 0 ω
ε app ( ω ) = 1 b 2 ω 2 ω 0 2 + 2 i δ ω .
ε ( ω ) = ω 2 ω * 2 + 2 i δ ω 2 b 2 3 ω 2 ω * 2 + 2 i δ ω + b 2 3 ,
ω * = ω 0 2 + b 2 3 .
ω p ± = i δ ± ω 0 2 δ 2 ,
ω z ± = i δ ± ω 0 2 + b 2 δ 2 ,
ω p ± = i δ ± ω * 2 b 2 3 δ 2 ,
ω z ± = i δ ± ω * 2 + 2 b 2 3 δ 2 .

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