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Engineering LibreTexts

2.4: Dielectric Susceptibility

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If the incident field is monofrequent, i.e.

E(t)(+)=ˆEeiωt,

and assuming that the inversion w of the atom will be well represented by its time average ws, then the dipole moment will oscillate with the same frequency in the stationary state

d=ˆdeiωt,

and the inversion will adjust to a new stationary value ws. With ansatz (???) and (???) in Eqs. (2.3.98) and (2.3.99), we obtain

ˆd=j2ws1/T2+j(ωωeg)MˆE,

ws=w01+T121/T2|MˆE|2(1/T2)2+(ωegω)2.

We introduce the normalized lineshape function, which is in this case a Lorentzian,

L(ω)=(1/T2)2(1/T2)2+(ωegω)2,

and connect the square of the field |ˆE|2 to the intensity I of a propagating plane wave, according to Equation (2.2.27), I=12ZF|ˆE|2,

ws=w01+IIsL(ω).

Thus the stationary inversion depends on the intensity of the incident light, therefore, w0 can be called the unsaturated inversion, ws the saturated inversion and Is, with

Is=[2T1T2ZF2|MˆE|2|ˆE|2]1,

is the saturation intensity. The expectation value of the dipole operator is then given by

<p>=(Md+Md).

Multiplication with the number of atoms per unit volume N relates the dipole moment of the atom to the complex polarization ˆP+ of the medium, and therefore to the susceptibility according to

ˆP(+)=2NMˆd,

ˆP(+)=ϵxχ(ω)ˆE.

From the definitions (???), (???) and Equation (???) we obtain for the linear susceptibility of the medium

χ(ω)=MMTjNϵ0ws1/T2+j(ωωeg).

which is a tensor. In the following we assume that the direction of the atom is random, i.e. the alignment of the atomic dipole moment M and the electric field is random. Therefore, we have to average over the angle enclosed between the electric field of the wave and the atomic dipole moment, which results in

¯(MxMxMxMyMxMzMyMxMyMyMyMzMzMxMzMyMzMz)=(¯M2x000¯M2y000¯M2y)=13|M|21.

Thus, for homogeneous and isotropic media the susceptibility tensor shrinks to a scalar

χ(ω)=13|M|2jNϵ0ws1/T2+j(ωωeg).

Real and imaginary part of the susceptibility

χ(ω)=χ(ω)+jχ(ω)

are then given by

χ(ω)=|M|2NwsT22(ωegω)3ϵ0L(ω),

χ(ω)=|M|2NwsTs3ϵ0L(ω).

If the incident radiation is weak enough, i.e.

T1T2|MˆE|22L(ω)1

we obtain wsw0. Since w0<0, and especially for optical transitions w0=1, real and imaginary part of the susceptibility are shown in Figure 2.4.

截屏2021-03-26 下午7.02.59.png
Figure 2.4: Real and imaginary part of the complex susceptibility.

The susceptibility computed quantum mechanically compares well with the classical susceptibility derived from the harmonic oscillator model close to the transistion frequency for a transition with reasonably high Q=T2ωab. Note, there is an appreciable deviation far away from resonance. Far off resonance the rotating wave approximation should not be used.

The physical meaning of the real and imaginary part of the susceptibility becomes obvious, when the propagation of a plane electro-magnetic wave through this medium is considered,

E(z,t)={ˆEej(ωtkz)},

which is propagating in the positive z-direction. The propagation constant k is related to the susceptibility by

k=ωμ0ϵ0(1+χ(ω))k0(1+12χ(ω)), with k0=ωμ0ϵ0

for |χ|1. Under this assumption we obtain

k=k0(1+χ2)+jk0χ2.

The real part of the susceptibility contributes to the refractive index n=1+χ/2. In case of χ<0, the imaginary part leads to an exponential damping of the wave. For χ>0 amplification takes place. Amplification of the wave is possible for w0>0, i.e. an inverted medium.

The phase relaxation rate 1/T2 of the dipole moment determines the width of the absorption line or the bandwidth of the amplifier.


This page titled 2.4: Dielectric Susceptibility is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Franz X. Kaertner (MIT OpenCourseWare) via source content that was edited to the style and standards of the LibreTexts platform.

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