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On the Lorentz-Lorenz formula and the Lorentz model of dielectric dispersion

Kurt E. Oughstun and Natalie A. Cartwright

Open Access Open Access

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

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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.

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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).

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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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