yields by far the most accurate values. All o ther methods yield values tha t are inferior by 1 order of magnitude. The systematic lower value for Cw using first harmonic detection can be explained as follows. In the individual data there is a tendency toward lower values at high pressure. A nonzero base line can be the explanation, since at high pressure a larger scan and modulat ion depth ¢0a are necessary. Therefore we find tha t second harmonic detect ion combined with a complete line fit yields the most accurate parameters , which are generally in good agreement with the literature. 5' 19-21

Since the absorption coefficient of the transit ion used is 1.3 × 10 -5 cm -1, the SNR is of the order of 30 (see Fig. 7) and Wa~'c is approximately 0.06, a detect ion limit of the order of 10 -8 cm -1 can be obtained (time constant 1 s). In Fig. 4 we have shown tha t the maximum signal of the second harmonic detect ion is independent of pressure in the Lorentz region and tha t Doppler and Lorentz regions can be easily distinguished (see also section II, A).

IV. CONCLUSIONS

We have presented a general theory for FM applied to absorption lines with a Voigt profile and have obtained experimental evidence for its validity. We found that the detection limit is of the same order of magnitude for first and second harmonic FM detection. In that case the modulation depth ought to be at least of the order of the full width of the absorption line involved.

We have presented and compared a number of detec- tion schemes for determining line shape parameters and we have shown that under our conditions the second harmonic detection technique combined with a complete curve fit yields the most accurate results. Our theoretical model can be used to determine line shape parameters

for absorption lines with a Voigt profile using FM with arbitrari ly large modulat ion depths.

In addition we have found more accurate values for line center, self-shift, and self-broadening (due to colli- sions) for a part icular rotat ional transit ion of OCS than are found in the li terature.

ACKNOWLEDGMENTS This work is part of the research program of the "Stichting Fundamenteel

Onderzoek der Materie" (FOM) and was made possible by financial support received from the "Stichting voor Zuiver Wetenschappelijk Onderzoek" (ZWO).

One of us (S.C.M.) thanks the FOM for the financial support which made it possible for him to stay in The Netherlands. The authors are grateful to Prof.dr. C. Th. J. Alkemade for his continued interest and they appreciate the suggestions he has made during numerous discussions.

1. W. Snelleman, T. C. Rains, K. W. Yee, H. D. Cook, and O. Menis, Anal. Chem. 42, 394 (1970).

2. C. H. Townes, A. L. Schawlow, Microwave Spectroscopy. (McGraw-Hill, New York, 1955).

3. Y. Uchida and S. Hattori, Oyo Butsuri 44, 852 (1975). 4. W. C. King and W. Gordy, Phys. Rev. 93, 407 (1954). 5. S. C. M. Luijendijk, J. Phys. B 1O, 1741 (1977). 6. M. L. Olson, D. L. Grieble, and P. R. Griffiths, Appl. Spectrosc. 34, 50 (1980). 7. D. L. Grieble, M. L. Olson, J. N. P. Sun, and P. R. Griffiths, Appl. Spectrosc.

34, 56 (1980). 8. P. E. Perkins, Appl. Spectrosc. 34, 617 {1980). 9. M. R. Whitbeck, Appl. Spectrosc. 35, 93 (1981).

10. E. A. Rinehart, R. H. Kleen, and C. C. Lin, J. Mol. Spectrosc. 5, 458 (1960). 11. R. Karplus, Phys. Rev. 73, 1027 (1948). 12. R. Karplus and J. Schwinger, Phys. Rev. 73, 1020 (1948). 13. M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions. (U.S.

Department of Commerce, Washington, DC, 1964}. 14. R. Arndt, J. Appl. Phys. 36, 2522 (1965). 15. R. W. Parsons and J. A. Roberts, J. Mol. Spectrosc. 18, 412 {1965). 16. J. Reid, J. Shewchun, B. K. Garside and E. A. Ballik, Appl. Opt. 17, 300

(1978). 17. I. R. Dagg, J. A. Roberts, and R. W. Parsons, J. Mol. Spectrosc. 63, 241

(1976). 18. J. H. Carpenter, J. D. Cooper, J. B. Simpson, J. G. Smith, and D. H. Whiffen,

J. Phys. E. 7, 678 (1974). 19. Y. Morino and C. Matsumara, Bull. Chem. Soc. Jpn. 40, 1095 (1967). 20. A. Dubrulle, J. Demalson, J. Burie, and D. Boucher, Z. Naturforsch. 35A,

471 (1980). 21. W. R. McGillivray, J. Phys. B. 9, 2511 {1976).

Saturation and Prefilter Distortions of Observed Excitation Spectral Profiles

E. H. PIEPMEIER* and G. J. BEENEN Department of Chemistry, Oregon State University, Corvallis, Oregon 97331

W h e n the w a v e l e n g t h o f a laser is tuned across a spectra l l ine, the spectra l pos i t ions o f the o b s e r v e d p e a k s o f the absorpt ion , f luorescence , or ion izat ion s igna l s m a y be d is torted b y satura- t ion and pref i l ter effects . Theoret i ca l m o d e l s are p r e s e n t e d w h i c h predict the spectra l shifts that m a y occur w h e n there is a n a s y m m e t r y in e i ther the l ine or laser spectra l prof i les . Ex- amples are p r e s e n t e d that s h o w the appearance o f an ex tra p e a k , a n d a d r a m a t i c s h i f t e q u a l i n m a g n i t u d e to the h a l f - w i d t h of a profi le . Spectra l b r o a d e n i n g is predic ted for both s y m m e t r i c and a s y m m e t r i c s p e c t r a l p r o f i l e s . L i n e reversa l is a pred ic ted

Received 16 Ju ly 1981; revision received 2 November 1981. * Au thor to whom correspondence should be addressed.

Volume 36, Number 3, 1982

pref i l ter effect , in the a b se n c e o f saturat ion. Sa turat ion and pref i l ter e f fects are t reated s e p a r a t e l y to e m p h a s i z e their indi- v idua l character i s t ics . Index Headings: H o l l o w c a t h o d e l a m p s ; Lasers , dye; Optogal - van ic spec troscopy; Techniques , spectroscopic ; Spectra l l i n e

profi les; Sa tura t ion broadening; Pref i l ter effects .

INTRODUCTION

The absorption of a laser beam by a population of free atoms or molecules has many applications. One applica- tion in our laboratory is calibrating the wavelength scale

APPLIED SPECTROSCOPY 235

of a tunable laser by scanning the wavelength across well-known spectral lines, z The absorption may be ob- served directly, or by a subsequent fluorescence or ioni- zation (or other impedance change) signal [e.g., Refs. (2- 5)]. Automatic laser wavelength stabilization systems that use such signals in a feedback loop have been reported. 3

Unfortunately the spectral positions of the absorption, fluorescence, and ionization signals may be distorted by saturation effects which occur when the irradiance of the absorbing population is as high as that produced by some typical lasers. Prefilter effects may also produce distor- tions for fluorescence and ionization signals. Experimen- tally observed saturation and prefilter distortions have been recently reported 5 for the case where laser-induced impedance changes in a hollow cathode lamp were used to tune a dye laser. The observed distortions include spectral line broadening, line shifts, and line reversal. These observations led to the following study, which uses mathematical models to help explain the distortions, to predict their magnitudes, and to indicate ways to avoid or compensate for the distortions. Although developed initially to explain observed spectral profile distortions obtained using impedance change signals, the models are generally applicable to experiments that use absorption, fluorescence, or ionization signals to detect the interac- tion of the laser beam with an absorbing population.

First the saturation model will be presented, followed by typical calculated results which show spectral shifts and distortions for both one-component and two-com- ponent spectral lines. Then a separate treatment for the prefflter distortions will be presented.

I. SATURATION MODEL

The model chosen to predict the saturation effects assumes steady-state conditions and homogeneous illu- mination of a homogeneous population of two-energy level particles. 6 The steady-state assumption is a good approximation for pulsed lasers when the pulses are significantly longer than the times between energy level transitions. Homogeneous illumination implies that the laser beam is not significantly attenuated as it passes through the population; i.e., that prefilter effects are negligible. The model also assumes that an excited par- ticle does not change its Doppler velocity before it leaves the excited state. This is a reasonable assumption for low-pressure hollow cathode lamps where collisions that change the Doppler velocity of an atom are 1 order of magnitude less frequent than many optical transitions of interest, and for higher pressure cells with high quench- ing rates, such as some flames. Those flames and electri- cal plasmas that have collisions which change the Dop- pler velocity of excited atoms without quenching them require a more complicated treatment.

This model is adequate to help the reader gain an insight as to the causes of the spectral distortions and their magnitudes. The model also provides the basis upon which more cumbersome models could be developed for complicated systems ff it were desired to fit experimental data to a more accurate model.

When high irradiance produced by a laser beam is used to stimulate the transitions between two energy levels, a large fraction of the total population of atoms or mole-

236 Volume 36, Number 3, 1982

cules may be in the excited state. In such a saturated state, the absorptivity is not constant, but is a function of irradiance; and fluorescence or ionization signals are no longer directly proportional to irradiance (although they remain directly proportional to concentration). To simplify the saturation model we will assume that a small fraction, say less than 5%, of the laser beam is absorbed by the absorbing population. Then the fluorescence and ionization signals are directly proportional to the absorp- tivity or absorption coefficient. When Doppler broaden- ing is absent, the absorption coefficient k(h) for the population illuminated by a steady state monochromatic beam of wavelength ~ is given 6 by

~f(~,~o) k(~) = k(~o) O)

1 + (E(~) /a(~o)sat ) =of(h, ~o)

where k(ho) = peak value of the absorption coefficient at low irradiance; ~.~(h, ho) = the Lorentzian function which describes the shape of the spectral line due to finite lifetimes (Fig. lc); E(~) = irradiance produced by the laser beam (W cm-2); E(~o)sat = saturation irradiance: the irradiance produced by a monochromatic beam at ~o that reduces k()~o) to one-half its value at zero irradi- ance. At irradiances that are low enough to make the denominator essentially unity, Eq. (1) has the Lorentzian shape of ~(~, ho). As the irradiance increases, Ref. 6 shows that Eq. (1) retains its Lorentzian shape, its width increases by the factor [1 + ( n ( ~ ) / n ( k o ) s a t ) ] 1/2 and its height decreases by the factor [1 + (E(k)/E(ko)~at)] -1. For a perfectly monochromatic beam, and an absorbing pop- ulation with a Gaussian Doppler profile, the absorption coefficient would be the Voigt function obtained by con- voluting Eq (1) with the Gaussian Doppler profile. 6 For a polychromatic beam, the influence of each wavelength component of the beam must be considered before con- voluting with the Doppler profile.

When a polychromatic laser beam is used, each spec- tral component at wavelength ~ of the beam, such as point A1 in Fig. lb, interacts with the population to the extent indicated by the magnitude of the Lorentzian function ~(~, )~o) of the population at that wavelength (point B1 in Fig. lc). The total interaction of the fixed

(0) DOPPLER PROFILE X ~ e

(¢) FILE

WAVELENGTH, X

FI6. 1. Typical laser and line spectra] profiles. Labeled points are used in the "Saturation Mode]" section to describe the interactions of the profiles, a, Gaussian profile, g(~o, hD); b, laser profile, E~,~; and c, Lorentzian profile, ~(~, ~o).

wavelength polychromatic beam with the population is then found by integrating the product of the Lorentzian and laser profiles over all wavelengths. To obtain a normalized shape function for this interaction, the result of the integration is divided by the total irradiance pro- duced by the beam (a constant for a spectral scan) to obtain

f 5P(X, Xo)E~,xrdX

L()kr, )~o) -- (2)

f Ex,xrd)~

where Ex.x~ is the irradiance at each X produced by a fixed wavelength polychromatic beam that has a central or characteristic wavelength at Xr (Fig. lb). If the laser beam is scanned across the ~ line, Xr now changes, and at low irradiances the shape of the resulting spectral profile is found by evaluating L(Xr, ~ ) at each Xr, assuming that Doppler broadening is absent. The absorption coefficient for a polychromatic beam (for high or low irradiances) is found by replacing Lf(X, Xo) in Eq. (1) with L()~r, ~o), to obtain

k(ho)L(h~, Xo) kpo~y (X,.) = (3)

1 + (E/E(Xo)~t)L(X~, Xo)

The contribution of Doppler broadening will now be considered since it is significant in hollow cathode lamps, and in most other convenient atomic and molecular cells, even when pressure broadening is negligible. To simplify the model we will consider the case where an excited atom does not change its Doppler velocity before it either emits or is quenched to the ground state. Under these conditions, the total absorption coefficient can be found by first multiplying the absorption coefficient for each Doppler velocity subset, Eq. (3), by the fraction g(Xo, Xo) for the total population that is in that subset, doing this for all Doppler velocity subsets and adding the results together to obtain, in integral form,

/~po,y(X,-) = k(Xo) f g(Xo, XD) J

(4) . [ L(Xr, Xo) ]dho

1 + (E/E(Xo)=t)L(Xr, Xo)

In Eq. (4), ~kD is the peak wavelength of the Doppler distribution function, a Gaussian for this study (Fig. la), and/~mly(Xr) is the observed absorption coefficient for the Doppler broadening population irradiated by a polychro- matie laser beam tuned so that its central or character- istic wavelength is at Xr. As the laser beam is scanned across a spectral line, X~ varies and the shape of the resulting spectral profile is given by the shape of the absorption coefficient calculated from Eq. (4).

Spectral profiles were calculated from Eq. (4) taking advantage of a fast convolution subroutine that had been written in assembly language for our PDP-11 computer. The convolution subroutine numerically integrates the product of two functions over 100 equally spaced points, and does this for each of 100 equally spaced values of the differences between the characteristic peak values of the two functions. The subroutine assumes values of zero for all points outside the finite set of points used for each

function. The selected laser profile was convoluted with the Lorentzian profile to obtain L()kr, )ko) for 100 values of Xo. Each of these points was then used with the ratio E / E()ko)sat to calculate the term in brackets in Eq. (4) for the 100 values of Xo. The result was convoluted with the Gaussian Doppler profile to obtain the final result, the absorption coefficients for 100 values of Xr. Each of the profiles was scaled to the same maximum value so that changes in profile shapes could be readily observed. When two or more overlapping absorption peaks are used, the calculations are done separately for each peak, and the results added together to obtain the final total profile, just before the total profile is scaled. The location of the peak maximum was calculated from the three highest points using the parabolic interpolation suggested by Bevington. 7

H. RESULTS AND DISCUSSION

The cases that were studied include spectral lines with one component and also lines with two components. For the results reported here, the Lorentz broadening of the lines was one tenth the Doppler broadening, typical for the case of a Doppler broadened Na line with Lorentz broadening caused primarily by natural lifetime broad- ening. Therefore, the spectral profile of a single compo- nent was primarily Gaussian in shape. It should be em- phasized, however, that the Lorentz broadening could not be omitted because its strong tailing characteristic makes a significant contribution to the increased tailing that is produced by saturation broadening.

The influences of both symmetric and asymmetric laser profiles were studied. For the symmetric case, the laser was assumed to have a Gaussian spectral profile, which was indicated by the shape of experimental, low irradiance (unsaturated) excitation profile scans obtained with several different low-pressure cells using fluores- cence and impedance change signals. For the case of an arbitrarily chosen asymmetric laser profile, the width parameter for the Gaussian rise of the laser profile was set equal one half the width parameter for the Gaussian fall (Fig. lb). An asymmetric profile would be caused by a laser cavity spectral gain curve that was not flat or symmetrical over the laser spectral profile.

A. Saturation Effects. In all cases studied, high it- radiances that cause saturation were found to produce peak broadening and add Lorentzian tailing to the wings of the absorption profile as predicted by Ref. 6. Such broadening can cause trouble if not taken into consider- ation when only one side of a peak is being used to calibrate or stabilize a laser. Spectral shifts in peak position do not occur at any sample irradiance when both the laser and the line profiles are symmetrical.

Spectral shifts in the observed profiles are produced by increased irradiances when there is an asymmetry in either the laser spectral profile or in the absorption profile (due to two or more hyperfine components). Fig. 2 shows the type of spectral shifts that occur when irradiance (E in Fig. 2) is increased from zero to 10 times the saturation irradiance (Esat, in Fig. 2) for an asymmet- rical laser spectral profile. All of the peak heights have been scaled to the same height in order to emphasize changes in peak shape and position. The small vertical bars just above the top of the peak show the positions of

APPLIED SPECTROSCOPY 237

the maxima of each of the four superimposed spectral profiles. The vertical bars in the middle show the peak positions tha t are obtained by using the midpoint posi- t ion at half height. The bars were calculated at their proper location and then offset vertically for clarity.

To be generally useful, the peak shifts are expressed as a percentage of the width of the peak obtained for low (or zero) irradiance. For Fig. 2a, where the laser width at half height was one half the Doppler width (0.002 nm), the peak maximum and half -maximum positions shifted only 2% for irradiances up to 10 times the saturat ion irradiance. When the laser width was twice the Doppler width (Fig. 2b), the peak maximum shifted 0.4%, but the position at half maximum shifted 22% of the width at half maximum when the irradiance was increased to 10 t imes the saturat ion irradiance.

For the case of a symmetr ical laser, Fig. 3 shows the spectral distortions tha t occur for the asymmetr ic Na(I) 589.0-nm line tha t has six hyperfine components, s Two hyperfine components were used to simulate the results since these six components group themselves into essen- tially two hyperfine components with a height ratio of 3:5 and a splitting or separat ion of 0.002 nm. Th e Doppler width was also 0.002 nm, corresponding to a translational t empera tu re of 520 K. The laser profile is symmetr ical for these curves. In this case, the narrower laser width produced the largest spectral shift, 11%, in the peak maximum (Fig. 3a), whereas the position at half-maxi- mum changed by less than 0.2%. High irradiances for the

(a) ~ (b) . ~

/I//l\ I///1111 ////,//\ .,.., "-"////'.////

////

FIG. 2. Normalized ~'alculated spectral profiles showing the influence of sample irradiance with an asymmetric laser spectral profile. Sample irradiance is indicated by the normalizing ratio E/E~at, which is equal to E(k)/E(ko)~at as defined in Eq. (1). The laser half-width equals one half the Doppler width for (a) and twice the Doppler width for (b). Small vertical dashes show locations of maxima and half-width posi- tions. The innermost rising and falling edges belong to the E/E~t = 0 peak; working outward from the center, the falling edges of the succes- sive peaks are in the same order as the rising edges.

I

~ Laser = I/2 Doppler " ~ LOSo.021 H / I ~ ~ ~ er -2 x Doppler

Fro. 3. Normalized calculated spectral profiles showing the influence of irradiance, E/E~at, on a sample that has an asymmetric line profile. The six-component hyperfine structure of the Na(I) 589.0-nm line is indicated by the vertical bars at the bottom of the profiles.

2 3 8 Volume 36, Number 3, 1982

wider laser profile (Fig. 3b), produced shifts of only 1.8% and 2.3% for the half -maximum position and peak maxi- mum, respectively.

Saturat ion effects can produce larger spectral shifts and changes in peak shape when overlapping lines having significantly different transit ion probabilities are present in a spectral profile. This s tudy was able to simulate the changes in shal~e and the 0.055/k shift observed 5 for the Ne(I) 6678.28 A line by assuming tha t this profile was produced by two overlapping lines (one unidentified) with widely different saturat ion irradiance parameters . Th e results are shown in Fig. 4, where all of the peak heights have been normalized in order to emphasize the changes in peak shape and position tha t occur as the irradiance increases. Th e low irradiance reference peak is shown in each of the four diagrams to make compari- sons easier. At low irradiances, the left line dominates the profile, whereas the r ight-hand line, 20 t imes weaker, contr ibutes a tailing to the r ight-hand side of the peak. It is assumed tha t the saturat ion irradiance for the right- hand line is 106 t imes higher than for the left line, so tha t the left-hand line saturates much more easily. For the irradiances shown, the left-hand line shows the intensi ty and broadening effects of saturation, whereas the inten- sity of the r ight-hand line continues to increase l inearly with irradiance. In the upper r ight-hand diagram in Fig. 4, the intensities of the two lines are comparable, with the r ight-hand line more intense than the left. The inten- sity of the r ight-hand line continues to increase propor- t ionally with irradiance, gaining a larger influence over the profile, whereas the left line is essentially saturated. In the final diagram in the lower r ight-hand corner, the r ight-hand line clearly dominates the profile; the left- hand peak contr ibutes Lorentz ian broadening to the pro- file tails and a left-hand shoulder tha t is barely visible. The result is a significant shift of what appears to be one peak from one position to another, due to a change in irradiance.

i i

~0

~-500

,

~-o

~100

~ 0

WAVELENGTH FIG. 4. Normalized calculated spectral profiles showing the influence of irradiance on a sample that has two overlapping lines with signif- icantly different saturation parameters. Numbers are values of the irradiance ratio E/Est.

In our studies it was also found that saturation can also cause the appearance of an apparent third peak between two peaks that are fairly well resolved at low irradiances. Fig. 5 shows the low and high irradiances profiles for a hypothetical case where the Doppler pro- files for the lines are well separated and the laser profile is wide enough so that it overlaps both lines when it is tuned to a position between the two lines. At low irradi- ance (Fig. 5b), the profile shows two lines that are fairly well separated. At high irradiance, a third peak appears in the center (Fig. 5c). This central peak is the result of the simultaneous excitation of both lines by the wings of the laser spectral profile, which produces a (relatively unsaturated) signal that is more intense because of con- tributions from two fines, than the significantly saturated signal that is produced when the laser is tuned to the center of either one of the lines. This saturation effect should be kept in mind because it could mislead one into believing that there are more lines present than are actually there.

B. P re f i l t e r Effect . A relatively simple model is ad- equate to show the distortions caused by absorption of the laser beam before its reaching the excitation region. The effects of such absorption are known as prefilter effects. For simplicity of calculation, the model assumes that the laser passes through a relatively long homoge- neous region of absorbing atoms where the prefilter distortion occurs, before it reaches a short second region that contains the atoms that contribute to the impedance change or fluorescent signal. The second region is as- sumed to be short enough that no prefilter distortion occurs within it. The atoms in both regions are assumed to have the same spectral profile. Saturation effects are not included in this model in order to emphasize the prefilter effect distortions. Saturation would tend to smooth out spectral features caused by the prefilter effect, and would make additional contributions to shifts when asymmetry is present.

Fig. 6 shows distortions that occur in the observed spectral profiles when the absorbance of the prefilter region is increased, for example by increasing the absorb-

(G) pL=~=~ ~ 2 HYPERFLNE COMPONENTS

E/Es°t = IOj~/~Xx~X ~

/ \

FIG. 5. Calculated spectral profiles of a sample line with two hyperfine components showing how a high irradiance, E/E~t, can distort the profile to make it appear that a third component is present, a, Laser and Doppler line profiles; b, low irradiance, unsaturated profile; c, high irradiance profile.

ing population by increasing the current of a hollow cathode lamp. The prefilter region absorbances, A, shown in Fig. 6 are the peak absorbances of the spectral profiles for monochromatic light; observed absorbances for the polychromatic laser beam would be somewhat less. The peak intensities of the spectral profiles have been nor- malized so that the peak shapes and shifts can be more easily compared.

The two hyperfine components and Doppler width were chosen to simulate the Na(I) 589.0-nm line. The left-hand diagram in Fig. 6 shows the results for a very narrow symmetrical Gaussian laser spectral profile. With no prefilter absorption, A = 0, the weaker hyperfine component appears as a hump on the shoulder of the main peak. As the absorbance of the prefilter region increases, the peak maximum shifts towards this hyper- fine component by 21% (of the width at half-height for A = 0) when A = 1.1. The position at half-maximum shifts only 3%. Reversal of the line profile has begun to show up where the strongest hy-perfine component is now strongly absorbing the center of the laser spectral profile.

When the laser is twice as broad as the Doppler width, the profile appears as one peak for no prefilter absorption, but when A = 1.1 a hump is obvious on the shoulder of the peak on the side of the strongest hyperfine compo- nent. More of the laser beam is being absorbed when the laser is tuned to the weaker hyperfine component. The peak maximum shifted 22% towards the weaker hyperfine component, whereas the width at half-maximum shifted 2%. At higher prefilter absorbances, line reversal becomes apparent. Such reversal was observed by Beenen and Piepmeier s for the Na(I) 589.0-nm line using laser-in- duced impedance change signals for a pulsed hollow cathode lamp.

The distortions predicted by this study emphasize the desirability of using low sample irradiances and sample cells that do not have significant prefilter effects, when- ever an atomic or molecular spectral line is being used to identify the wavelength of a laser. Recording spectral profiles for different irradiances and prefilter conditions helps to detect line shifts and reversals that may not be apparent when only the location of the peak maximum is used as a measure of spectral position. In some cases the position of the peak at half-height may be a better indication of peak position; but in other cases the oppo- site may be true. Even when wavelength accuracy is not required, but the desire is to produce the maximum analytical signal, the wavelength at which the maximum

A = 0 - -~ ~t~L/ A = 0 Lo=er = 0.43 - - ~ / / / / Loser • 0.1 • Doppler 0.43 2 • DotBpl~ 0.65 --~ ~ rT_~,: 065

• o.sT/ I.I -

FIG. 6. Normalized calculated spectral profiles for a two-component line showing the influence of (prefilter) absorption of the laser beam before the beam reaches the region where these signals are generated. A-values are prefilter absorbances.

APPLIED SPECTROSCOPY 239

signal occurs will depend upon the sample irradiance and prefdter effects, which may change significantly if not carefully controlled.

ACKNOWLEDGMENT

This material is based upon work supported by the National Science Foun- dation under Grant CHE-7901759.

1. G. J. Beenen, J. W. Hosch, and E. H. Piepmeier, Appl. Spectrosc. 35, 593 (1981).

2. D. S. King, P. K. Schenck, K. C. Smyth, and J. C. Travis, Appl. Opt. 16, 2617 (1977).

3. R. B. Green, R. A. Keller, G. G. Luther, P. K. Schenck, and J. C. Travis, IEEE J. Quantum Electron. OE-13, 63 (1977).

4. R. A. Keller, R. Engleman, Jr., B. A. Palmer, Appl. Opt. 19, 836 (1980). 5. G. J. Beenen and E. H. Piepmeier, Anal. Chem. 53, 239 (1981). 6. E. H. Piepmeier, in Analytical Laser Spectroscopy. N. Omenetto, Ed. (Wiley,

New York, 1979), Chap. 3. 7. P. R. Bevington, Data Reduction and Error Analysis for the Physical

Sciences. (McGraw Hill, New York, 1969), p. 210. 8. W. Hartig and H. Walther, Appl. Phys. 1, 171 (1973).

Phase Correct ion in FT-IR

D. B. CHASE E. I. du Pont de Nemours & Company, Inc., Central Research & Development, Experimental Station, Wilmington, Delaware 19898

T w o approaches to phase correction in FT-IR spectroscopy are discussed. Using synthetic data, both t h e Mer tz m e t h o d a n d the symmetrization-convolution method are evaluated with respect to the accurate reproduction o f a single-beam spectrum. T h e v a r i a t i o n in photometric accuracy w i t h (1) t he n u m b e r o f points used in the phase calculation, and (2) apodization in the phase calculation is e x a m i n e d .

Index Headings: F ' r - IR , phase correction; I n f r a r e d ; I n t e r f e r o m e - try, IR.

INTRODUCTION

There are three distinct arithmetic operations involved in converting an interferogram to a single-beam spec- trum. These are apodization, Fourier transformation, and phase correction. Both apodization and Fourier transfor- mation have been well studied, and the effect on the spectrum of different apodizing functions and different length transforms are fairly well understood) The proc- ess of phase correction, however, is still unclear to many users, and a systematic variation of the parameters dur- ing phase correction is complicated by the lack of an appropriate method for evaluating the quality and pho- tometric accuracy of the phase-corrected spectrum. In order to determine the effect of various parameters on phase correction, simulated spectral data have been gen- erated with a known phase-error function, and various phase corrections applied. These results can be used to decide the most appropriate method of phase correction for a given experiment.

I. DISCUSSION

There are essentially only two methods of phase cor- rection. The effects of phase-error can be removed either in the interferogram or in the calculated spectrum. The two methods discussed here are the symmetrization-con- volution approach 2 and the Mertz methodJ Both start from a common point, the calculation of the phase-error function. This is done by taking a short region around

Received 26 October 1981; revision received 21 December 1981.

240 Volume 36, Number 3, 1982

the centerburst and doing a full double-sided complex transform on the data. The phase error is then defined a s

0(~) = tan -1 \Re(v)] (1)

The parameters involved in this operation are the number of data points around the centerburst (this de- termines the resolution of the phase spectrum), the num- ber of points transformed, and the apodization function used. After phase calculation the two methods differ in their approach, but any error in the calculation of the phase function will be carried through in both.

A. Mertz Method. After phase calculation the inter- ferogram is multiplied by a linear left-ramp function that goes from 0 to 1 and has a value of 0.5 at the centerburst. This is necessary to avoid counting information around the centerburst in the hill transform twice and thus giving false intensity informationJ The interferogram may be apodized, and a full complex Fourier transform is performed yielding a real and an imaginary spectrum. Phase correction is made by taking the proper linear combination of real and imaginary components of the calculated spectrum, as shown below.

S( r i ) = R ( l , i ) c o s ( 0 ( r i ) ) + I ( p i ) s i n ( 8 ( r i ) ) (2)

where S(pi) = phase corrected spectrum; R(pi) = real component after Fourier transformation (FFT); I(pi) = imaginary component after FFT; 8@i) = phase function.

Since 0(pi) was calculated for a lower resolution, a combination of zero filling in the phase calculation step and linear interpolation is used to define 0 for ~i of the higher resolution spectrum.

One of the major problems in this approach is the multiplication of the interferogram by the left-ramp func- tion. The assumptions being made in this approach are strictly valid only for a totally symmetric interferogram, and then, only the real component of the spectrum is accurately reproduced. ~

APPLIED SPECTROSCOPY