Lorentz-Lorenz

On the Lorentz-Lorenz formula and the

Lorentz model of dielectric dispersionKurt E. Oughstun and Natalie A. Cartwright

College of Engineering & Mathematics, University of Vermont, Burlington, VT [email protected], [email protected]

Abstract: The combination of the Lorentz-Lorenz formula with the Lorentzmodel of dielectric dispersion results in a decrease in the effectiveresonance frequency of the material when the number density of Lorentzoscillators is large. An equivalence relation is derived that equates thefrequency dispersion of the Lorentz model alone with that modified by theLorentz-Lorenz formula. Negligible differences between the computedultrashort pulse dynamics are obtained for these equivalent models.

©2003 Optical Society of America

OCIS codes: (260.2030) Dispersion; (320.5550) Pulses.

References and Links1. 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, 641-665 (1880).3. L. Lorenz, “Über die Refractionsconstante,” Ann. Phys. 11, 70-103 (1880).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).8. L. Brillouin, “Über die Fortpflanzung des Licht in Disperdierenden Medien,” Ann. Phys. 44, 203-240

(1914).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).12. B. K. P. Scaife, Principles of Dielectrics (Oxford, 1989) Ch. 7.

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1. Introduction

The classical Lorentz model [1] of dielectric dispersion due to resonance polarization is offundamental importance in optics as it provides a physically appealing, accurate description ofboth normal and anomalous dispersion phenomena in the extended optical region of theelectromagnetic spectrum from the far infrared up to the near ultraviolet. Of equal importanceis the Lorentz-Lorenz formula [2,3] which, as stated in Born and Wolf [4], “connectsMaxwell’s phenomenological theory with the atomistic theory of matter.” It is typicallyassumed [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 molecularpolarizability and the dielectric permittivity. Although the influence of the Lorentz-Lorenzformula on the resulting frequency dispersion can be striking when the number densitybecomes sufficiently large, the fundamental frequency structure is not altered from that

(C) 2003 OSA 30 June 2003 / Vol. 11, No. 13 / OPTICS EXPRESS 1541#2595 - $15.00 US Received June 11, 2003; Revised June 20, 2003

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described by the Lorentz model alone; a frequency band of anomalous dispersion with highabsorption surrounded by lower and higher frequency regions exhibiting normal dispersionwith small absorption.

The fact that the Lorentz model is a causal model [6] of temporal dispersion has cast it ina central role in both the classical [7-9] and modern [10-11] asymptotic theories of lineardispersive pulse propagation. Although the asymptotic theory is independent of the particularmaterial parameter values chosen for the Lorentz model dielectric considered, the materialparameters originally chosen by Brillouin [8,9] and employed in much of the modernasymptotic theory [10-11] correspond to a highly absorptive material for which the Lorentz-Lorenz formula must be applied without approximation. The purpose of this paper is toestablish an approximate equivalence relation that equates the frequency dispersion of theLorentz model alone with that modified by the Lorentz-Lorenz formula. This result thenextends the domain of applicability of the asymptotic theory to include the optically densematerial 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 effectiveelectromagnetic field present at that molecular site. The effective electric field E reff t,( )acting on a molecule at space-time position r,t( ) in a polarizable medium with polarization

P r,t( ) is given by [3]

E r E r P reff t t t, , ,( ) = ( ) + ( )43p

, (1)

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 theeffective electric field through the causal relation

p r E r, ˆ ,t t t t dtt

eff( ) = - ¢( ) ¢( ) ¢-•Ú a (2)

with Fourier transform ˜ , ˜ ,p r E rw a w w( ) = ( ) ( )eff , where a w( ) is the mean polarizability

at angular frequency w . If N denotes the number density of molecules in the material, then

the spectrum of the induced polarization in Eq. (1) is given by ˜ , ˜ ,P r p rw w( ) = ( )N . Withsubstitution from the Fourier transform of Eq. (1) one then obtains the expression

˜ , ˜ ,P r E rw c w w( ) = ( ) ( )e , (3)

where

c w a wp a we

N

N( ) = ( )

- ( ) ( )1 4 3 (4)

is the electric susceptibility. With the expression e w pc w( ) = + ( )1 4 e for the relative

dielectric permittivity, one then obtains the Lorentz-Lorenz formula [2,3]

a wp

e we w

( ) = ( ) -( ) +

34

12N

, (5)


which is also referred to as the Clausius-Mossotti relation [12]. It is typically assumed thate w( ) is sufficiently close to unity that e w( ) + ª2 3 in which case the Lorentz-Lorenz

formula simplifies to e w p a w( ) ª + ( )1 4 N , which is equivalent to the approximation that

E r E reff t t, ,( ) ª ( ) .

The Lorentz model [1] of resonance polarization in dielectrics is based upon the dampedharmonic oscillator equation

m t t t q te e eff˙̇ ˙r r r E( ) + ( ) + ( )( ) = - ( )2 02d w (6)

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 w0 and phenomenological damping

constant d under the action of the local Lorentz force F Eloc e efft q 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 Eww w dw

w( ) =- +

( )q m

ie e

eff202 2

. (7)

The local induced dipole moment is then given by ˜ ˜p rw w( ) = - ( )qe which then results in

the expression

a ww w dw

( ) = -- +

q m

ie e2

202 2

(8)

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

e ww w dww w dw

( ) =- ( ) - +( )+ ( ) - +( )

1 2 3 2

1 3 2

2 202

2 202

b i

b i (9)

for the complex, relative dielectric permittivity. The complex index of refraction

n w e w( ) = ( ) 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 b2 06 1dw( ) << is satisfied, the denominator in Eq. (9) may be

approximated by the first two terms in its power series expansion so that

e ww w dw w w dw w w dw

( ) ª -- +

ÊËÁ

ˆ¯̃

-- +

ÊËÁ

ˆ¯̃ª -

- +1

2 32

132

12

2

202

2

202

2

202

b

i

b

i

b

i, (10)

which is the usual expression [1,4-11] for the frequency dispersion of a single resonanceLorentz model dielectric.

As an example, consider the Lorentz model material parameters chosen by Brillouin [8,9],

viz. w0164 10= ¥ r s/ , d = ¥0 28 1016. /r s , b r s= ¥20 1016 / , which correspond

to a highly absorptive dielectric. The angular frequency dispersion of the complex index ofrefraction for the Lorentz model alone [as given by the square root of the final approximationin 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 w e w( ) = ¬ ( ){ } while part (b)

describes that for imaginary part ni w e w( ) = ¡ ( ){ } . The corresponding solid green

curves in Fig. 1 describe the resultant frequency dispersion for this Lorentz model dielectricwhen the Lorentz-Lorenz relation is used [cf. Eq. (9)]. As can be seen, the Lorentz-Lorenzmodified frequency dispersion primarily shifts the resonance frequency to a lower frequencyvalue while increasing both the absorption and the below resonance index of refraction.

Notice that b2 06 2 976dw( ) = . for this choice of material parameters. If the plasma

frequency is decreased to the value b r s= ¥2 1016 / so that b2 06 0 2976dw( ) = . ,

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 complexindex 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.

3. An approximate equivalence relation

Since the primary effect of the Lorentz-Lorenz formula on the Lorentz model is to downshiftthe effective resonance frequency and increase the low frequency refractive index, considerthen determining the resonance frequency w* appearing in the Lorentz-Lorenz formula for a

Lorentz model dielectric that will yield the same value for e 0( ) as given by the Lorentz

model alone with resonance frequency w0 . From Eqs. (9) and (10) one then has that

11 2 3

1 1 32

02

2 2

2 2+ =+ ( )( )- ( )( )

*

*

bb

bw

ww

, (11)

with solution

w w* = +02 2 3b . (12)

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

e ww w dww w dw

( ) =- ( ) - +( )+ ( ) - +( )

*

*

1 2 3 2

1 3 2

2 2 2

2 2 2

b i

b i (13)

0 1 2 3 4 5 6 7 8 9 10

x 1016

0

0.5

1

1.5

2

2.5

3

w - r/s

n r(w)

b2 = 20X1032r/s

b2 = 2X1032r/s

(a)0 1 2 3 4 5 6 7 8 9 10

x 1016

0

0.5

1

1.5

2

2.5

3

w - r/s

n i(w)

(b)

b2 = 20X1032r/s

b2=2X1032r/s


of the relative dielectric permittivity for the Lorentz-Lorenz modified Lorentz model dielectricwith undamped resonance frequency w* given by the equivalence relation (12), with theexpression [cf. Eq. (10)]

e ww w dwapp

b

i( ) = -

- +1

2

2

202 (14)

of the approximate relative dielectric permittivity for a Lorentz model dielectric withundamped resonance frequency w0 . The other two material parameters b and d are the

same in these two expressions.

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

A comparison of the angular frequency dependence described by Eqs. (14) and (13) withw* given by the equivalence relation (12) is presented in Fig. 2 for Brillouin’s choice of the

material parameters (w0164 10= ¥ r s/ , d = ¥0 28 1016. /r s , b r s= ¥20 1016 / ).

The rms error between the two sets of data points presented in Fig. 2 is approximately2.3 10-16 for the real part and 2.0 10-16 for the imaginary part of the complex index ofrefraction, with a maximum single point rms error of ~2.5 10-16. The corresponding rmserror for the relative dielectric permittivity is ~1.1 10-15 for both the real and imaginaryparts with a maximum single point rms error of ~1 10-14. Variation of any of theremaining material parameters in the equivalent Lorentz-Lorenz modified Lorentz modeldielectric, including the value of the plasma frequency from that specified in Eq. (12), onlyresults in an increase in the rms error. This approximate equivalence relation between theLorentz-Lorenz formula modified Lorentz model dielectric and the Lorentz model dielectricalone is then seen to provide a “best fit” in the rms sense between the frequency dependenceof the two models.

4. Conclusions

An approximate equivalence relation between the Lorentz-Lorenz formula modified Lorentzmodel dielectric and the Lorentz model alone for the complex index of refraction of a singleresonance dielectric has been presented. Numerical results show that this approximateequivalence relation provides a “best fit” in the rms sense between the frequency dependenceof the two models. This result then extends the domain of applicability of the asymptotic

0 1 2 3 4 5 6 7 8 9 10

x 1016

0

0.5

1

1.5

2

2.5

3

w - r/s

n r(w

)

(a)0 1 2 3 4 5 6 7 8 9 10

x 1016

0

0.5

1

1.5

2

2.5

w - r/s

n i(w)

(b)


theory of dispersive pulse propagation in a Lorentz model dielectric to include the opticallydense material originally considered by Brillouin [8,9] and subsequently used as an examplein the modern asymptotic theory [10,11]. In fact, the results are indistinguishable when thetwo equivalent models are used in a numerical determination of the propagated field due to aninput rectangular envelope pulse in a single resonance Lorentz model dielectric usingBrillouin’s choice of the material parameters, including the leading and trailing edgeprecursors.

Acknowledgement

The research presented in this paper was supported, in part, by the United States Air ForceOffice of Scientific Research under AFOSR Grant # 49620-01-0306.


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