THE VOIGT PROFILE

wherel₀is expressed in nanometers,MXin grams, andTin kelvins. In these same units the formula

dnD

n₀ 7:1610⁷ T M_{X 1=2}

(3:9:12) for the ratio of the Doppler width to the resonance frequency is also useful.

The Doppler width depends only on the transition frequency, the gas temperature, and the molecular weight of the absorbing species. It is, therefore, much simpler to calculate than the collision-broadened width, which involves the collision cross section. As an example, consider the 632.8-nm line of Ne in the He – Ne laser. SinceMNe¼20.18 g, we obtain from (3.9.11) the Doppler width

dn_D1500 MHz (3:9:13)

forT¼400K. For the 10.6-mm line of CO2and the same temperature, however, we find a much smaller Doppler width:

dnD61 MHz: (3:9:14)

We will now derive the absorption lineshape when both collision broadening and Doppler broadening must be taken into account.

Equation (3.4.26) gives the collision-broadened lineshape for each atom in the gas. If an atom has a velocity componentvmoving away from the source of light of frequency nn0, its absorption curve is Doppler shifted to

S(n,v)¼ (1=p)dn0

(n₀nþvn=c)²þdn²₀: (3:10:1) In other words, the peak absorption for this atom will occur at the field frequencynsuch that (3.9.3) is satisfied:

nn₀þn₀v

c : (3:10:2)

The lineshape function for the gas is obtained by integrating over the velocity distri- bution (3.9.2):

S(n)¼ ð₁

1dvS(n,v) M_X 2pRT ₁₌₂

e^MXv2=2RT

¼ M_X 2pRT ₁₌₂

dn₀ p

ð₁

1

dve^MXv2=2RT (n₀nþn₀v=c)²þdn²₀

¼ 1 p³⁼²

b² dn₀

ð₁

1

dy e^y2

(yþx)²þb², (3:10:3)

where we have made the change of variables

x¼(4 ln 2)¹⁼² n0n dn_D

, (3:10:4)

and we have defined

b¼(4 ln 2)¹⁼²dn₀

dn_D: (3:10:5) The lineshape function (3.10.3) is called theVoigt profile.

In the case when the applied field is tuned exactly to the resonance frequencyn0, we havex¼0 and therefore

S(n₀)¼ b² p³⁼²dn₀

ð₁

1

dy e^y2

y²þb²: (3:10:6) The integral defines a known function:

ð₁

1

dy e^y2 y²þb²¼p

be^b2erfc(b), (3:10:7)

3.10 THE VOIGT PROFILE 109

where

erfc(b)¼ 2 p¹⁼²

ð₁

b

du e^u2 (3:10:8) is thecomplementary error function. From (3.10.6) and (3.10.7), therefore, the lineshape function for the resonance frequencyn¼n0has the value

S(n0)¼ b² p³⁼²dn₀

p

be^b2erfc(b)¼ b

p¹⁼²dn₀e^b2erfc(b)

¼ 4 ln 2 p ₁₌₂

1

dn_De^b2erfc(b): (3:10:9) This function is plotted versus the parameterbin Fig. 3.16.

S(n0) depends strongly on the ratio of the linewidths for collision and Doppler broad- ening. When the collision widthdn0is much greater than the Doppler widthdnD, we haveb1. For large values ofb,

e^b2erfc(b) 1

p¹⁼²b (b1): (3:10:10) In this “collision-broadened limit,” therefore, we have from (3.10.9) the result

S(n₀) 1 pdn0

(b1), (3:10:11)

which is exactly (3.4.28) for the case of pure collision broadening. In the limit in which the Doppler width is much greater than the collision-broadened width, on the other hand, we haveb1, in which case the function

e^b2erfc(b)1 (b1): (3:10:12)

1.0 0.8 0.6 0.4 0.2

0 0.5 1.0 1.5

b e^b2 erfc(b) vs. b

2.5 2.0

Figure 3.16 The functione^b2erfc(b).

Then from (3.10.9)

S(n₀) 1 dnD

4 ln 2 p ₁₌₂

(b1), (3:10:13) which is the result (3.9.9) for pure Doppler broadening and n¼n0. The limits dn0dnD and dn0dnD thus reproduce the results for pure collision broadening and pure Doppler broadening, respectively. In general, for arbitrary values ofb,S(n0), given by (3.10.9), must be evaluated using tables of erfc(b).

For the general case of arbitrary values of both the parameter band the detuning parameterx, the lineshape functionS(n) given by Eq. (3.10.3) must be evaluated from tabulated values of the more complicated function

ð₁

1

dy e^y2

(yþx)²þb²¼p bRe i

p ð₁

1

dy e^y2 xþyþib

!

¼p

bRe[w(xþib)], (3:10:14) wherewis the “error function of complex argument.” Numerical values are tabulated in various mathematical handbooks.

In Table 3.1 we summarize our results for collision broadening and Doppler broad- ening, as well as the more general case of the Voigt profile.

TABLE 3.1 Collision, Doppler, and Voigt Lineshape Functions Collision-Broadening Lineshape

S(n)¼ (1=p)dn₀ (nn0)²þdn²₀ dn0¼collision rate

2p Doppler-Broadening Lineshape

S(n)¼0:939 dnD

e^{2:77(nn0)2=dn2D}

dnD¼2:1510⁵ 1 l₀

T M

1=2

" # MHz

T¼gas temperature (K)

M¼molecular weight (g) of absorber l0¼wavelength (nm) of absorption line Voigt Lineshape

S(n)¼0:939 dnD

Rew(xþib) x¼1:67n0n

dnD

b¼1:67dn0

dn_D

w¼error function of complex argument

3.10 THE VOIGT PROFILE 111

† Without going to numerical tables, and even without a study of the asymptotic properties of w(xþib), it is possible to evaluate the Voigt integral (3.10.3) in several limits because both fac- tors in the integrand are normalized lineshapes themselves. There are three limits of interest, as shown in Fig. 3.17.

Collisional Limit (dn0dnD) In this case S(n, v) is very broad and slowly varying compared to the narrow Gaussian velocity distribution (Fig. 3.17a). Since the Gaussian is normalized to unity, it acts like the delta function d(v), and the Voigt integral reduces to S(n)¼S(n, v¼0), which is just the original collisional Lorentzian lineshape given in (3.4.26).

Doppler Limit(dn_Ddn₀) In this case the reverse is true (Fig. 3.17b), and the collisional functionS(n,v) acts like the delta functiond(n₀2nþvn/c). Thus, the Voigt integral gives back the Gaussian function (3.9.9). Except at high pressures or in cases where the Doppler distribution is altered by atomic beam collimation it is usually valid to assume that the inequalitydnDdn0

is accurate and the Doppler limit applies.

Far-Wing Limit(jnn0j dnD,dn0) This case refers to the spectral region far from line center, far outside the half widths of either the collisional or Doppler factors in the Voigt inte- grand. Thus, the integrand is the product of two peaked functions. Each peak falls in the remote wing of the other function (see Fig. 3.17c). Here the qualitative difference between Gaussian and Lorentzian functions is significant. The Gaussian is much more compact. It falls to zero much more rapidly than the Lorentzian. Because the Lorentzian’s wings are falling rela- tively slowly, as 1/n²for largen, it still has nonzero value at the position of the Gaussian peak.

However, the value of the Gaussian function is effectively zero by comparison near the Lorentzian’s peak. Thus, the contribution of the Gaussian function in the Lorentzian wing is much greater than that of the Lorentzian function in the Gaussian wing, and the Voigt integral can be replaced by (3.4.26) in its far wing:

S(n)! dn0=p

(nn0)²: (3:10:15)

This result, which can be derived more formally from the asymptotic behavior of the complementary error function, is anomalous in the sense that the lineshape behaves like a Lorentzian in the far wing even if the broadening is principally Doppler, not

collisional (dnDdn0). †

In document Laser Physics by Milonni P.W., Eberly J.H. .pdf (Page 127-131)