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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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, �??�?ber die Beziehungzwischen der Fortpflanzungsgeschwindigkeit des Lichtes der Körperdichte,�?? Ann. Phys. 9, 641-665 (1880).
  3. L. Lorenz, �??�?ber 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, �??�?ber die Fortpflanzung des Lichtes in Disperdierenden Medien,�?? Ann. Phys. 44, 177-202 (1914).
    [CrossRef]
  8. L. Brillouin, �??�?ber 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.

Ann. Phys. (4)

H. A. Lorentz, �??�?ber die Beziehungzwischen der Fortpflanzungsgeschwindigkeit des Lichtes der Körperdichte,�?? Ann. Phys. 9, 641-665 (1880).

L. Lorenz, �??�?ber die Refractionsconstante,�?? Ann. Phys. 11, 70-103 (1880).
[CrossRef]

A. Sommerfeld, �??�?ber die Fortpflanzung des Lichtes in Disperdierenden Medien,�?? Ann. Phys. 44, 177-202 (1914).
[CrossRef]

L. Brillouin, �??�?ber die Fortpflanzung des Licht in Disperdierenden Medien,�?? Ann. Phys. 44, 203-240 (1914).
[CrossRef]

J. Opt. Soc. Am. B (1)

Other (7)

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).

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

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.

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.

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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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