1008 J. Opt. Soc. Am., Vol. 68, No. 7, July 1978 . 0030-3941/78/6807-1008$00.50 © 1978 Optical Society of America 1008

Calculations of line absorption for the Voigt profile* Isamu Kato, Akira Iwasaki, Akihiro Adachi, and Tsukasa Shimizu

School of Science and Engineering, Waseda University, 4-170, Nishi-Ohkubo, Shinjuku-ku, Tokyo 160, Japan ( Received 3 February 1978)

line-absorption calculations of the 388.9 nm triplet and the 501.6 nm singlet lines of helium have been performed for the Voigt profile. These results can be used to measure the absolute concentration of He 2s 3S and 2s 1S metastable atoms for the various discharge conditions by the reabosorption method.

The reabsorption method has been widely used to measure the concentration of metastable atoms in a plasma. In most reabsorption measurements, however, the mechanism of the line broadening has been assumed to be the Doppler effect only. In this paper, the influence on the absorption coefficient of collisipnal and natural broadening in addition to Doppler broadening is taken into account and the line absorptions are calculated for the resulting Voigt profile. The calculations are carried out for the 388.9 nm triplet and 501.6 nm singlet lines of helium. From this analysis, the difference between the line absorption when Doppler broadening only is present and that when natural, collisional, and Doppler broadening are present is shown.

For a multiplet line, the absorption coefficient kv at fre­quency v can be written as follows:

where

and

In the above, k0 is the summation of the absorption coeffi­cients at the peak of the purely Doppler profile of each com­ponent of the multiplet line, the normalized coefficient Ci is the relative intensity of each component with ∑Ci = 1, and the dimensionless frequency ∆wi is the frequency shift of the ith component from the central frequency of the least-intensity component v1. The parameter a is the ratio of the natural and collisional breadths (Δ V N ,Δ V C ) to the Doppler breadth (ΔvD)

and is expressed by

According to Ref. 1, the parameter a can be calculated as follows:

For the 388.9 nm line,

and for the 501.6 nm line,

where T is the gas temperature in degrees Kelvin and P is the He gas pressure in Torr and is reduced to a temperature of 293 K. These relations are shown in Fig. 1.

FIG. 1. Values of the parameter a for various discharge conditions. Solid lines are for the 388.9 nm line and dashed lines are for the 501.6 nm line.

FIG. 2. Dependence of the line absorption AL on k0l for the 388.9 nm line. 1: a = 0 (purely Doppler profile); 2: a = 0.0125; 3: a = 0.05; 4: a = 0.1; 5: a = 0.5.

The line absorption AL is written theoretically as

where l is the length of the plasma. The dependence of AL on k0l for the 388.9 and 501.6 nm

lines, which are calculated for the absorption coefficients represented by Eq. (1), are shown in Figs. 2 and 3, respectively. The dashed lines in these figures are calculated for the case in which only the Doppler effect is present. As shown in Figs. 2 and 3, under some discharge conditions the calculations of the line absorptions must be carried out for the Voigt profile where the natural, collisional, and Doppler broadenings are taken into account.

The line absorption AL can be determined experimentally. In the reabsorption method, the light emerging from one side of the tube is reflected back into the tube by a plane mirror whose reflectivity at the wavelength of investigated line is known. The absorption ratio AL is determined by comparing the signal due to the light entering the photomultiplier di­rectly when the mirror is covered, Iwm, with the signal in­volving Iwm and the light reflected back through the tube when

FIG. 3. Dependence of the line absorption AL on k0l for the 501.6 nm line. 1: a = 0 (purely Doppler profile); 2: a = 0.0105; 3: a = 0.05; 4: a = 0.1; 5: a = 0.5.

the mirror is uncovered, Im, namely,

where r is the effective reflectivity of the mirror. Then, using these calculations and the well-known relation2

the absolute concentration of helium 2s 3S and 2s 1S meta-stable atoms N can be obtained for the various discharge conditions.

Equation (10), however, is applicable to the range where the population density of the upper level of these lines is much larger than that of the lower level. Therefore these transitions should be used for the measurements under conditions of low excitation or in afterglow studies in which the states above the metastable levels have decayed.

*Work partly supported by the Kawakami Foundation. 1I. Ya. Fugol, D. A. Myshkis, and O. N. Grigorashchenko, "Absorption

methods for measuring the absolute concentration of metastable helium atoms in a plasma," Opt. Spectrosc. 31, 282-285 (1971).

2A. C. G. Mitchell and M. W. Zemansky, Resonance Radiation and Excited Atoms, 2nd ed. (Cambridge U.P., London, 1971), Chap. III.

1009 J. Opt. Soc. Am., Vol. 68, No. 7, July 1978 0030-3941/78/6807-1009$00.50 © 1978 Optical Society of America 1009