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

Optics Express, Vol. 11, Issue 13, pp. 1541-1546 (2003)

http://dx.doi.org/10.1364/OE.11.001541

Acrobat PDF (302 KB)

### 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

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

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

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

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

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

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

## 2. The Lorentz-Lorenz formula and the Lorentz model of dielectric dispersion

*(*

**E**_{eff}*) acting on a molecule at space-time position (*

**r**,t*) in a polarizable medium with polarization*

**r**,t**(**

*P**) is given by [3*

**r**,t3. L. Lorenz, “Über die Refractionsconstante,” Ann. Phys. **11**, 70–103 (1880). [CrossRef]

**(**

*p̃**)=*

**r**,ω*α*(

*ω*)

*(*

**Ẽ**_{eff}*), where*

**r**,ω*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**). With substitution from the Fourier transform of Eq. (1) one then obtains the expression*

**r**,ω*ε*(

*ω*)=1+4

*πχ*(

_{e}*ω*) for the relative dielectric permittivity, one then obtains the Lorentz-Lorenz formula [2,3

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

*ε*(

*ω*) is sufficiently close to unity that

*ε*(

*ω*)+2≈3 in which case the Lorentz-Lorenz formula simplifies to

*ε*(

*ω*)≈1+4

*πNα*(

*ω*), which is equivalent to the approximation that

*(*

**E**_{eff}*)≈*

**r**,t**(**

*E**).*

**r**,t**(**

*r**t*) relative to the nucleus of a bound electron of mass

*m*and charge magnitude

_{e}*q*with (undamped) resonance frequency

_{e}*ω*

_{0}and phenomenological damping constant

*δ*under the action of the local Lorentz force

*(*

**F**loc*t*)=-

*q*(

_{e}**E**_{eff}*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

**(**

*p̃**ω*)=-

*q*(

_{e}**r̃***ω*) which then results in the expression

*b*

^{2}/(6

*δω*

_{0})≪1 is satisfied, the denominator in Eq. (9) may be approximated by the first two terms in its power series expansion so that

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

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

*ω*

_{0}=4×10

^{16}

*r*/

*s*,

*δ*=0.28×10

^{16}

*r*/

*s*,

*b*=√20×10

^{16}

*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

*b*

^{2}/(6

*δω*

_{0})=2.976 for this choice of material parameters. If the plasma frequency is decreased to the value

*b*=√2×10

^{16}

*r*/

*s*so that

*b*

^{2}/(6

*δω*

_{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

*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

*ω*

_{*}given by the equivalence relation (12), with the expression [cf. Eq. (10)]

*ω*

_{0}. The other two material parameters

*b*and

*δ*are the same in these two expressions.

*ω*

_{*}given by the equivalence relation (12) is presented in Fig. 2 for Brillouin’s choice of the material parameters (

*ω*

_{0}=4×10

^{16}

*r*/

*s*,

*δ*=0.28×10

^{16}

*r*/

*s*,

*b*=√20×10

^{16}

*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

**44**, 203–240 (1914). [CrossRef]

*Electromagnetic Pulse Propagation in Causal Dielectrics* (Springer-Verlag, 1994). [CrossRef]

## Acknowledgement

## References and Links

1. | H. A. Lorentz, |

2. | H. A. Lorentz, “Über die Beziehungzwischen der Fortpflanzungsgeschwindigkeit des Lichtes der Körperdichte,” Ann. Phys. |

3. | L. Lorenz, “Über die Refractionsconstante,” Ann. Phys. |

4. | M. Born and E. Wolf, |

5. | J. M. Stone, |

6. | H. M. Nussenzveig, |

7. | A. Sommerfeld, “Über die Fortpflanzung des Lichtes in Disperdierenden Medien,” Ann. Phys. |

8. | L. Brillouin, “Über die Fortpflanzung des Licht in Disperdierenden Medien,” Ann. Phys. |

9. | L. Brillouin, |

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. |

11. | K. E. Oughstun and G. C. Sherman, |

12. | B. K. P. Scaife, |

**OCIS Codes**

(260.2030) Physical optics : Dispersion

(320.5550) Ultrafast optics : Pulses

**ToC Category:**

Research Papers

**History**

Original Manuscript: June 11, 2003

Revised Manuscript: June 20, 2003

Published: June 30, 2003

**Citation**

Kurt Oughstun and Natalie Cartwright, "On the Lorentz-Lorenz formula and the Lorentz model of dielectric dispersion," Opt. Express **11**, 1541-1546 (2003)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-11-13-1541

Sort: Journal | Reset

### References

- 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).
- 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]
- 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.
- 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]
- L. Brillouin, Wave Propagation and Group Velocity (Academic Press, 1960).
- 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]
- B. K. P. Scaife, Principles of Dielectrics (Oxford, 1989) Ch. 7.

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article | Next Article »

OSA is a member of CrossRef.