Three and four generalized Lorentzian approximations forthe Voigt line shape
Julio Puerta and Pablo Martin
Three and four generalized Lorentzians in two variables have been obtained to approximate the Voigt func-tion by using the asymptotical Pad6 method. The accuracy has been greatly improved with respect to theone and two generalized Lorentzian approximations reported in a previous paper. Furthermore, the fourLorentzian function is always positive for all values of the normalized collision width and line separation.This approximation gives an accuracy for most of the values of better than 0.0001, and in the worst regionthe absolute error is -0.001. In the limits of low and high pressure adequate limit functions are obtained.A four generalized Lorentzian gives a reliable and easily calculable approximation to the Voigt function formost of the experimental needs.
1. Introduction
In a previous paperl the Voigt function2-6 for theabsorption of a laser line has been approximated by oneand two generalized Lorentzians using the asymptoticalPad6 method.7 These approximations are more accu-rate than previous ones,8 ,
9,1
0 and no new parameters areintroduced. However, for some experiments a higherprecision is required; in addition, these approximationsfail and become negative in a small region where thenormalized collision width p is very small (p 0.1), andthe normalized line separation d is -1 or 2. Thesedisadvantages can be removed by extending our methodto approximations involving three and four generalizedLorentzians in two variables. In this way, we improvethe precision, keeping the advantage of not requiringcomputer programs to perform complicated integra-tions; only simple arithmetical operations are necessary.Furthermore, the four Lorentzian approximation re-mains positive for all positive values of p and d. Thusthere is no infeasible region of negative values as in theone, two, and three Lorentzian cases.
In our extension, we follow the general lines of ourprevious paper': first, we find a multipole approxi-mation following the asymptotical Pad6 procedure;next, by grouping the multipole functions we obtain thegeneralized Lorentzians. Details are in Sec. II.
The analysis of the errors is in Sec. III. Our resultsshow that a precision of better than 0.005 is obtained in
The authors are with Universidad Simon Bolivar, DepartamentoFisica, Apartado Postal 80659, Caracas 1081, Venezuela.
all the computations using three Lorentzians. Disre-garding the region near d = 2 and p < 0.1, the accuracyis higher than 0.001. In the region d < 2.5 and p 0.1,this approximation becomes negative. The exact Voigtfunction is very small in this region, but it is alwayspositive. (Negative values have no physicalmeaning.)
The four Lorentzian functions give a higher order ofprecision, better than 0.0015 for all p and d. Disre-garding a small region around d 2 and d 0.1, theaccuracy is 0.0001. This function gives clearly theprecision required in all the experimental conditions weknow, never becomes negative, and it can be used evenin equations where the Voigt function can appear. Inany case, by looking at the error matrix and the graphicsof Sec. III, the error can be estimated. It is importantto point out that in high pressure limits, the three andfour generalized Lorentzians reduce to the characteristicsimple Lorentzian function of this region.
In the low pressure limit our approximations becomea sum of one variable Lorentzian, which approximatesthe Guassian function. In this limit our results arecoincident with those obtained by the Hilbert-Pad6method."
II. Extension of the Theoretical Analysis for Threeand Four Generalized Lorentzians
In extending the analysis of our previous paper,' theVoigt function V(d,p),
V(dp) = P exp(-y 2 ) dpr _p2+ (d -y)2Y> (1)
is approximated by fractional functions V (d,p) of thetype
where AVD is the Doppler width, v is the laser line fre-quency, v0 is the central absorption line frequency in thegas, and ax is the collision width of the line.'- 6 In thispaper we will work out cases n = 3 and n = 4.
Coefficients pi and q of Eq. (2) are determined byequating the corresponding power series and asymp-totical expansions of the exact function V(z) and theapproximated function Vn (z) in a same manner as ex-plained in a previous work.7 As in our first paper, theVoigt function V(d,p) is just the real part of the com-plex Voigt function V(z), which is also related to theplasma dispersion function Z(z) and complex errorfunction w(z) as follows:
V(z) = - Z(iz) = w(iz) = - expz 2 erfcz. (5)
V4(Z) = Cl + + C; + C2Z-a1 Z-a2 Z-a Z-a2
(12)
where
= -1.2150 + 1.2359i; j, = -0.3085 + 0.0210i, (13)
a2 = -1.3509 + 0.3786i; c2 = 0.5906 - 1.1858i.
We use - to distinguish between the poles of V 4 (z) andV3 (z)-
Looking at the values in Eqs. (11) and (13), we can seethat all the poles of V3 (z) and V4(z) are in the secondthe third quadrants of the complex plane, and no polesare in the first quadrant, which is the one with physicalinterest.
Now as indicated in our first papers the real part ofV 3(z) or V4(z) can be expressed as a sum of two variableLorentzians:
F'' p- aj) + bi~ - (d) , (4V (d,p) = -(4
j=1 (p -aj) 2 + (d - oj)2
where n = 3 or 4, and
aj = aj + ij; c = yj + ij. (15)
The parameters for V3 (z) are from Eq. (11):
a = a3 = -1.1317; 01 = -3 = 0.9050,
a2 = -1.2278; 02 = 0,
The power series and asymptotic expansion are, re-spectively,
2 44 z4V(z) = 1 - Z+z + .
I/ 7r 3\/; 2
V(z) ] 7 2z3 * - - -)
(6)
(7)
For the three Lorentzian cases (n = 3), we retain upto the term in z 3 in the power series and i/Z2 in theasymptotical expansion. For case n = 4, the terms upto z 4 and 1/z must be considered. The same result isobtained by a procedure in which we start from thefractional approximations Z4 2 and Z5 2 (Ref. 4) of the Zfunction, and we then make the transformations indi-cated in Eq. (5).
The results are
,yl = 3 = -0.3277; 1 = 63 = -0.4175,
Y2 = 1.2197; 62 = 0.
And similarly for V4(z) and from Eq. (13), we obtain
&1 = &3 = -1.2150; ,1 = /3 = 1.2359,
2 = &4 = -1.3509; (2 = -4 = 0.3786,(17)
1= 7y = -0.3085; 8 i = 0.0210,
72 = 74 = 0.5906; t2 = -4 = -1.1858.
To get the high and low pressure expressions, it is con-venient to use variables b, ax, and D. Thus
n (D = y yi(a - ajb) + 6j(D - j3 b)b b j=i (a - ajb) 2 + (D + Ojb)2
The high pressure limit is obtained when b goes tozero:
If we compare these results with the values of E, B,C2, F, H, and G shown in Table 1 of Ref. 11 (see /42 and153), we have obtained the same values, except for smalldiscrepancies in the last significant digit.
Ill. Numerical and Graphical Analysis
To analyze adequately the error of the approxima-tions, we have constructed Tables I and II, where theabsolute error is listed for values of d from zero to tenand for p from zero to one hundred. The error has beenmultiplied by a scale factor of 104 to show significantdigits.
From Tables I and II we see that the approximationsare very accurate for d and p near zero and for largevalues of said parameters. For three Lorentzians(Table I) the highest error is -0.006, and it is localizedin the region of very small p, and d is about 2. Thisregion is near the poles of V3(z) [see Eq. (11)], but it isnot the closest one to them. Moving away from thisregion the approximation becomes much better, and theerror is <0.001 for most of the values.
Looking at Table II for the approximation with fourLorentzians, we can point out that for almost all esti-mated values the error is <0.0001. Only there is a smallregion (p < 0.1, d - 2) where the error becomes -0.0015.This occurs near the poles of V4(z), but as in the pre-ceding case, it does not correspond to the region closestto the poles. On the other hand, the error is <10-5 formost values in the table.
Fig. 1. Algebraic maximum and minimum absolute error vs p. In-teger numbers on the ordinate axis must be multiplied by 10-3 to getthe absolute error for the two (dotted line) and three (dashed line)Lorentzian approximations and by 10-4 in the four Lorentzian case
(dot-dash line).
If we compute the Voigt function using the approxi-mation V3(d,p), the function becomes negative for p 0.1 and d 2.5. The exact Voigt function is alwayspositive in the first quadrant. (Negative values haveno physical sense.) However, the small region whereV3(d,p) is negative is not usual, and there the exactfunction attains very small values. This anomalousbehavior was also present in the one and two Lorentzianapproximations.' The four Lorentzian case has theadvantage of being always positive, and at least threedecimals are always coincident with the exact func-tion.
Using Tables I and II, we have drawn three graphs toillustrate the absolute and relative error. In the firstgraph for each value of p (abscissa), we plot the highestpositive value (Amax) and lowest negative values(Amin) in the respective line of the matrix table. In Fig.1 we include the errors for two (see Table II, Ref. 1),three (Table I), and four (Table II) generalized Lo-rentzians. To illustrate the relative error, in Fig. 2 weplot together the curves A4 (four Lorentzians) of Fig. 1
Amax 0lO].................... Vmiax 1: \. ................. . . . . .. . .... . V in/l 0,.a ma xx10] '''''''
10
. minIx104 6 8 P
Fig. 2. Algebraic maximum and minimum absolute errors vs p forthe four Lorentzian approximation (dot-dash line). Scale factor is10-4 as in Fig. 1. Corresponding exact values of the Voigt functionare also shown (dotted lines). In this case, the values have been di-vided by ten to use the same scale 10-3 of the ordinate axis. In thisfigure the 1% of relative error is on the intersection point of the lines
for A4 (max) and the corresponding exact function V.
and the corresponding exact Voigt function for the samevalues of p and d. Adequate scale factors have beenused and shown in the graphs. Clearly in Fig. 1 thehighest errors for a given p do not occur for the samevalue of d in each table.
In Fig. 3 we have proceeded in a similar way as in Fig.1, but now using d as an abscissa instead of p. It nowshows the highest positive and lowest negative error ineach column vs the respective value of d.
IV. Conclusions
The Voigt function has been approximated by threeand four generalized Lorentzians via the asymptoticPad6 method and adequate transformation in thecomplex plane. Considering all values of p and d, theapproximations give better accuracy than previous onesfound by us' and other authors.8-' 0 The three Lo-rentzian cases can be used for almost all values of p andd with an accuracy of better than 0.001. The four Lo-rentzians give a fully reliable approximation for allvalues of p and d; and the accuracy is, for most of thevalues, better than 0.0001. Both approximations areeasily calculable, and to obtain any values of the Voigtfunction we only have to carry out simple arithmeticaloperations.
In the tree Lorentzian case there is a small regionwhere the approximation is not adequate for d 2 andp 0.1. In this region the absolute error reaches thevalue 0.006, and furthermore the function becomesnegative. This anomaly does not appear in the fourLorentzian cases, which is always positive, and in theworst region the highest error is -0.0015 for p - 0 andd 2.
In our procedure we first find fractional approxima-tions for the complex Voigt function, and later wetransform the real part of these approximations in twovariable Lorentzians. All the poles of the complex
20
10
lAmax x10 3]
./ '''
An=Vn - V
A23 =- - -63 =-- -
,,--: A --- ----5A ....
10
AminIxi0 -3]
Fig. 3. Algebraic maximum and minimum absolute error vs d. In-teger numbers on the ordinate axis must be multiplied by 10-3 for thetwo (dotted line) and three (dashed line) Lorentzian approximations
and by 10-4 in the four Lorentzian case (dot-dash line).
fractional approximations are located in the second andthird quadrant of the complex plane. The highest er-rors in each approach are in regions near the poles.
Considering the limit conditions for high and lowpressures, the approximations behave as follows:
(1) In the high pressure region, our two approxima-tions reduce exactly to one Lorentzian in one variableas in the exact Voigt function.
(2) In the low pressure limit, our approximationsreduce, respectively, to a sum of three and four onevariable Lorentzians, which is the result obtained withthe approximation for the Gaussian function using theHilbert-Pad6 method.
We finally conclude that the four generalized Lo-rentzian approximation gives sufficient accuracy for allthe experimental works we know where the Voigtfunction is used.
This paper was completed when one of the authors(P.M.) was on leave for a sabbatical year at UCLA. Heis greatly indebted to the Physics Department, U. Cal-ifornia at Los Angeles, and mainly to Burton D. Friedfor support in this period.
We thank German Torres for his review of the man-uscript.
This research was funded in part by VenezuelanAgency CONICIT, project 31-26-51-0606.
References1. P. Martin and J. Puerta, Appl. Opt. 20, 259 (1981).2. A. Mitchell and M. Zemansky, Resonance Radiation and Excited
Atoms (Cambridge U. P., London, 1971), pp. 93-103.3. D. W. Posener, Aust. J. Phys. 12, 184 (1959).4. A. Reichel, J. Quant. Spectrosc. Radiat. Transfer 8, 1601
(1968).5. H. Farach and H. Teitelbaum, Can. J. Phys. 45, 2913 (1967).6. Z. Kucerovsky, E. Brannen, D. G. Rumbold, and W. J. Sarjeant,
Appl. Opt. 12, 226 (1973).7. P. Martin, J. Zamudio-Cristi, and G. Donoso, J. Math. Phys. 21,
280 (1980).8. J. F. Kielkopf, J. Opt. Soc. Am. 63, 987 (1973).9. A. B. Wertheim, M. A. Butler, K. W. West, and D. N. Buchanan,
Rev. Sci. Instrum. 45, 1369 (1974).10. E. E. Whiting, J. Quant. Spectrosc. Radiat. Transfer 8, 1379
(1968).11. P. Martin,J. Zamudio-Cristi, and G. Donoso, J. Math. Phys. 21,
1332 (1980).
Books continued from page 3866
Optics Lab Manual. By JURGEN R. MEYER-ARENDT andDENIS R. HOLMES. Optics Lab Manual, P.O. Box 364, ForestGrove, Ore. 97116, 2nd ed., 1981. 46 pp. $4.00
This small paperback book contains the 20 experiments performedin the author's laboratory during two terms of Physical Optics. Theexperiments retained from the first edition have all been rewritten.Experiments dropped make room for several new ones. Fiber opticsis used in an experiment to measure the radius of curvature of thecornea of the eye. One experiment is about the zoom lens so widelyused on TV today. Another introduces the student to spatial filteringand optical data processing. Other experiments concern geometricaloptics, wave optics, and the measurement of the Planck constant usingthe photoelectric effect.
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