ABSORPTION

and the correct expression for the upper-state lifetime is t_n¼ 1

A_n¼ 1 P

mA_nm, (3:3:12)

where the summation is over all statesmwith energyEmlower than the energy leveln (see Fig. 3.6). Numerical values of the “Acoefficients”Anmare usually included in tables of oscillator strengths. Radiative lifetimes of excited atomic states are typically on the order of 10 –100 ns.

the real part of our solution forx(t) is the (real) electron displacement. This approach is used frequently in solving linear equations such as (3.4.3). We solve (3.4.4) by writing x(t)¼ae^i(vtkz), (3:4:5) and after inserting this in (3.4.4) we obtain

(v²2ibvþv²₀)a¼1^e

mE₀: (3:4:6) Therefore the assumed solution (3.4.5) satisfies Eq. (3.4.4) if

a¼ ^1(e=m)E₀

v²v²_0þ2ibv^{, (3:4:7)} and the physically relevant solution is therefore

x(t)¼Re 1^(e=m)E0e^i(vtkz) v²₀v²2ibv

: (3:4:8)

Note that (3.4.8) actually gives only the steady-state solution of (3.4.3). Any solution of the homogeneous version of (3.4.3) can be added to (3.4.8), and the sum will still be a solution of (3.4.3). The homogeneous version is

d²x_hom

dt² þ2b^dxhom

dt þv²₀xhom¼0, (3:4:9) and its general solution is

x_hom¼[Acosv⁰₀tþBsinv⁰₀t]e^bt, (3:4:10) where underdamped oscillation (bv₀) is by far the most common occurrence, so

v⁰₀¼(v²₀b²)¹⁼²v_0: (3:4:11) We will usually neglect the homogeneous part of the full solution to (3.4.3). This is obviously an approximation. The approximation is, however, an excellent one whenever

t1

b^{: (3:4:12)}

Under this condition,e^bt 1, and we can safely neglect the homogeneous component (3.4.10) because it makes only a short-lived transient contribution to the solution.

Even though the damping time 1/bis very short, it is not the shortest time in the pro- blem. Typically, the oscillation periodsT0¼2p_/v₀andT¼2p_/vassociated with the natural oscillation frequencyv₀or the forcing frequencyvare very much shorter. In the case of ordinary optically transparent materials such as atomic vapors, glasses, and many crystals and liquids, bothv₀andvare typically in the neighborhood of 10¹⁵s²¹, andb falls in a wide range of much smaller frequencies:

b10⁶–10¹² s¹v₀,v_: (3:4:13)

3.4 ABSORPTION 79

Relations (3.4.12) and (3.4.13), taken together, imply that times of physical interest must be much longer than an optical period:

tb¹v¹₀ ,v¹_: (3:4:14) That is, steady-state solutions of (3.4.3) are valid for times that are many periods of oscil- lator vibration (T0¼2p_/v₀) and forced vibration (T¼2p_/v) removed fromt¼0, but they cannot be used to predict the oscillator’s response within the first few cycles after t¼0. This is, however, no real restriction in optical physics, as it is equivalent to

t10¹⁵ s (¼10³ps¼1 fs): (3:4:15) One femtosecond (fs) is a time span one or two orders of magnitude smaller than can ordinarily be resolved optically.

To calculate the rate at which energy is absorbed from the field, we consider the rate at which work is done by the field on an oscillator at positionzalong the direction of propa- gation of the presumed plane-wave, monochromatic field:

dW

dt ¼F

^v¼F

^dx_dt ^{¼1^eE0cos(vtkz)}

^dx_dt ^{: (3:4:16)}

The (steady-state) velocitydx/dtfor the oscillator follows by differentiation of (3.4.8):

dx

dt ¼ 1^(e=m)E₀

(v²₀v²)²þ4b²v^2[2bv²cos(vtkz)v(v²₀v²) sin(vtkz)]: (3:4:17) We now use this expression in (3.4.16) and average over times large compared to 1/v as in the preceding section. This amounts to the replacement of cos²(vt2kz) by¹₂and sin(vt2kz) cos(vt2kz) by 0, resulting in

dW dt ¼e²

mE²₀1 b

b²v² (v²₀v²)²þ4b²v²

(3:4:18)

for the cycle-averaged rate of work. We have used the fact that^1

^{^1¼}1, i.e., that1^is a unit vector (see also Problem 3.5).

Sincebv,v₀, the dimensionless quantity in brackets in Eq. (3.4.18) will have a very small value unless the field frequencyvis near the oscillator resonance frequency v₀. More precisely, frequency “detunings”jv₀₂vjmuch larger thanbresult in very little absorption. Thus we make the approximationv₀þv 2v₀, or

(v²₀v²)²¼(v₀v)²(v₀þv)²4v²₀(v₀v)² (3:4:19) in (3.4.18), and likewise approximatev³byv³₀in the numerator and 4b²v²by 4b²v²₀in the denominator inside the brackets:

dW dt e²

mE₀² bv²₀

4v²₀(v₀v)²þ4b²v²₀

¼pe²

4mE²₀ (1=p)b (v₀v)²þb²

: (3:4:20)

We can write the rate of absorption of energy by the oscillator,dW/dt, in terms of the circular frequencyn¼v_/2p_¼c/lof the field:

dW dt ¼ e²

8mE₀² (1=p)dn0

(nn₀)²þdn²₀

, (3:4:21)

where

n₀¼ v₀

2p ^(3:4:22)

and

dn₀¼ b

2p^{: (3:4:23)}

It is also convenient to write the absorption rate in terms of the field intensity

I_n¼¹₂ce₀E²₀, (3:4:24) where the subscript indicates thatI_nis the intensity of the assumed monochromatic field of frequencyn. Thus,

dW dt ¼ e²

4mce₀^{InL(n), (3:4:25)} where the “lineshape function”L(n), which determines the dependence of the absorption on the field frequency, is defined by

L(n)¼ dn₀=p

(nn0)²þdn²₀: (3:4:26) This is called the Lorentzian lineshape function, or Lorentzian distribution, and is plotted in Fig. 3.7.

The Lorentzian function is a mathematically idealized lineshape in several respects.

We have already shown that it is the near-resonance approximation to the more compli- cated function appearing in (3.4.18). The function is defined mathematically for negative frequencies, even though they have no physical significance. It is exactly normalized to unity when integrated over all frequencies, as is easily checked:

ð₁

1dnL(n)¼dn₀ p

ð₁

1

dn

(nn0)²þdn²₀¼1, (3:4:27) and the normalization is approximately the same when only the physical, positive fre- quencies are used. The approximation is excellent for dn₀n₀ [recall (3.4.13)]. In other words, the contribution of the unphysical negative frequencies is negligible

3.4 ABSORPTION 81

because the linewidth is negligible compared to the resonance frequency, and in this senseL(n) is physically as well as mathematically normalized to unity.

The maximum value ofL(n) occurs at the resonance frequencyn¼n0: L(n)_max¼L(n₀)¼ 1

pdn0: (3:4:28) Atn¼n0+dn0we have

L(n₀+dn₀)¼ 1 2pdn0

¼1

2L(n)_max: (3:4:29) Because of this property, 2dn0is called the width of the Lorentzian function, or thefull width at half-maximum(FWHM), and dn0 is called thehalf width at half-maximum (HWHM). The Lorentzian function is fully specified by its width (FWHM or HWHM) and the frequencyn0where it peaks. The peak value of the absorption rate is

dW dt

max

¼ e²

4mce₀^InL(n0)¼ 1 4pe₀

e²

mcdn₀I_n, (3:4:30) and it decreases to half this resonance value when the field is “detuned” from resonance by the half widthdn₀of the Lorentzian function.

Our classical theory thus predicts that the absorption is strongest when the frequency of the light equals the oscillation frequency of the electron oscillator. Far out in the wings of the Lorentzian, wherejnn₀j dn₀, there is very little absorption. A knowledge of the widthdn₀is therefore essential to a quantitative interpretation of absorption data.

–3 –2 –1 0

Lorentzian lineshape L(n)

n – n₀ dn₀ 2dn0

0 1 2 3

Figure 3.7 Lorentzian lineshape function.

To determine the numerical value ofdn0in a given situation, we must consider in some detail the physical origin of this absorption width. This we do in Section 3.8.

We shall see that the absorption rate does not always have the Lorentzian form (3.4.25). However, we can in general write

dW dt ¼ e²

4mce₀^{InS(n), (3:4:31)} where the lineshape functionS(n), whatever its form, is normalized to unity:

ð₁

0

dnS(n)¼1: (3:4:32) As in the case of spontaneous emission, two changes to the classical oscillator theory of absorption are required to obtain quantitatively correct formulas. First, we introduce the oscillator strengthf, in this case replacing (3.4.31) with

dW dt ¼ e²f

4mce₀^{InS(n): (3:4:33)} Second, we account for the fact that atoms are found only in states of allowed energy.

That is, the absorption process proceeds from a state in whichN1atoms are in a state of lower energyE₁to a state in whichN₂atoms are in a state of higher energyE₂¼ E₁þhn₀, as indicated in Fig. 3.8. This suggests the replacement of dW/dt of the classical theory byhv₀dN1=dtsince the actual rate of energy absorption should be proportional to the number of atoms in the lower energy state from which the absorption proceeds. Thus, we relate the rate (3.4.31) at which energy is absorbed according to the classical theory to the rate of change of the “population”N1as follows:

dW dt ¼ d

dt(hv₀N₁), (3:4:34) or

dN₁

dt ¼ 1

4pe₀

pe²f mchv₀

N₁InS(n): (3:4:35)

n0

E₁ E₂

Figure 3.8 Atomic absorption transition of frequencyn0¼(E22E1)/h.

3.4 ABSORPTION 83

Similarly, becausedN2/dt¼2dN1/dt, the rate of change of the upper-level population N2due to absorption of light of frequencynand intensityInis

dN2

dt ¼ 1 4pe₀

pe²f mchv₀

N₁I_nS(n): (3:4:36)

It is convenient to use Eq. (3.3.7) to write the rate Eq. (3.4.35) for absorption in terms of the spontaneous emission rateA₂₁for theE₂!E₁transition. Simple algebra yields

dN1

dt ¼ 1 hn

l²A21

8p

N1InS(n)¼ dN2

dt : (3:4:37)

Because we are assuming that the applied field is close to the transition resonance fre- quencyn₀, we have used nin place ofn₀ in this equation, and for later convenience we have also written the quantity in brackets in terms of the wavelength l_¼c/n rather thann.

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