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A modified equivalent-width technique for diode-laser line strength measurements
J. Quant. Spectrosc. Radiat. Transfer Vol.42, No. 6, pp. 52t-527, 1989
Printed in Great Britain.All rightsreserved
0022-4073/89 $3.00+ 0.00 Copyright© 1989PergamonPressplc
A MODIFIED EQUIVALENT-WIDTH TECHNIQUE FOR DIODE-LASER LINE STRENGTH MEASUREMENTS L. LARRABEESTROW and LIu ZHENG Physics Department, University of Maryland, Baltimore County, Baltimore, MD 21228, U.S.A. (Recewed 10 March 1989)
Abstract--A modified equivalent-width method is introduced that improves the accuracy of
spectral line strength measurements using diode lasers. This method differs from the standard method in that we do not attempt to calculate the true equivalent width in order to determine the line strength. Instead, we derive the line strength directly from a measurement of the area under the line close to the line center and ignore the normal correction terms. Tests with actual spectra and simulations indicate that this method is accurate to better than 1% with lines broadened by up to 50% by the laser instrument function.
1. I N T R O D U C T I O N Tunable diode lasers are widely used to measure absorption line strengths. Many early measurements assumed that the distortion of a spectral line was small enough to allow the strength to be directly determined from the line-center absorption. J'2 This method did not require either an accurate or smooth frequency scale and was very easy to implement. It was soon recognized that laser jitter and/or the finite bandwidth of the laser can, in some cases, significantly distort the spectral lineY The effect of this distortion on line strengths derived from line-center absorption can be accounted for if the laser-emission profile can be adequately characterized. Normally, some measure o f the laser width is derived from the observed increase in the Doppler width of a line due to the finite bandwidth of the laser. If the functional form of the laser instrument function is known, correction curves can be generated to determine the correct line strength from the observed line-center absorption. Alternatively, the direct method can be employed with distorted lines by increasing the Doppler width used to determine the strength from the line center absorption. The increased Doppler width approximates the laser distortion of the spectral line. 4-7 Unfortunately, it has proven difficult to characterize the laser profile accurately in practice because the line-center absorption is distorted more severely than the line width, which is used to deduce the width of the instrument profile. The average value of the distortion can be determined by measuring the widths o f a large number of Doppler-shaped lines. 3 However, the laser profile or jitter may change with time, even within one continuous mode. This fact prevents the use o f non-linear least-squares fitting for line parameters when the instrument function is fixed, as is normally done in the analysis o f high resolution FTS spectra) If the signal-to-noise ratio is high enough, a Doppler-limited line can be deconvolved and an instrument function recovered. 9 The sensitivity of this process to noise and the care that must be taken to perform the deconvolution properly often make this technique impractical for day to day use. The method o f equivalent widths has the advantage that one does not need to know the instrument profile in order to determine the line strength. I° However, a smooth and accurate frequency scale is required, as well as an accurate 100% transmission level. Early diode-laser spectrometers used lock-in detection for recording the spectra, which produced inaccurate frequency scales since the time scales for variations o f the laser frequency were similar to the time scales for measurements. With the introduction o f the sweep-integration method for the recording o f diode-laser spectra, I much smoother frequency scales could be obtained via signal-averaging. In addition, since spectra can be recorded much more quickly with sweep-integration, empty cell background scans are taken very close in time to the absorption scans before the instrument intensity scale can drift. 521
522
L. LARRABEESTROWand LIU ZHENG
We present here a modified equivalent-width method for the measurement of line strengths using diode-laser spectra. This method differs from the standard method in that we do not attempt to calculate the true equivalent width and from that determine the line strength. Instead, we derive the line strength directly from a measurement of the area under the line close to the line center and ignore the normal correction terms. This technique is suitable for accurate measurements of isolated lines. Tests with actual spectra and simulations indicate that our method is accurate to at least 1% with lines broadened by up to 50% by the laser instrument function. 2. T H E M O D I F I E D
EQUIVALENT-WIDTH
TECHNIQUE
The equivalent width of a spectral line is defined as ~° W=
[1 - T(v - Vo)] d(v - Vo),
(1)
where T(v - V o ) is the observed transmittance, v the frequency, and Vo the line center frequency. W is independent of the width of the spectral line if the time average of the laser-frequency profile is symmetric about the center frequency. An accurate frequency scale is essential for the determination of W. Diode-laser spectra collected using the sweep-integration method generally have very smooth frequency scales that can be accurately calibrated by using known frequency standards and simultaneously recorded etalon fringes. We use cubic splines to interpolate the frequency scale between etalon fringes. The spline interpolation can correct for any non-linear tuning of the laser over an interval that contains several fringes. Diode lasers may also tune non-linearly over very narrow spectral intervals on the order of a Doppler-limited linewidth or smaller. However, these non-linearities generally vary randomly in time. Averaging several hundred spectra reduces any fine-scale non-linear tuning to acceptable levels given the apparent accuracy of the results presented here. However, these random variations of the laser tuning do broaden the observed spectra. In addition, the transmission scale must be well known, especially the 100% transmission level. We determine the 100% transmission scale by obtaining the ratios of the absorption spectra with respect to empty cell scans. Intensity offsets in either the background or absorption scans can introduce transmission errors that must be minimized when determining W. These small errors are magnified when performing a simple equivalent-width measurement since regions of high transmission contribute more to the equivalent width than regions of low transmission. The techniques described here provide a very simple way to minimize these errors and also eliminate the need to determine correction terms to the equivalent width due to the use of finite integration limits in Eq. (1). Following Fig. l, the measured equivalent width WM is related to the true equivalent width W by W = WM+ Ww+ WR+ Ws,
(2)
where Ww is the area in the line wings missed by integrating Eq. (1) over a finite frequency range, WR is the rectangular area missed when an incorrect 100% transmission level is used, and Ws is a correction to WM caused by the use of an incorrect vertical scale." Investigators previously determined the line strength by first calculating these correction terms to obtain W. An iterative theoretical calculation of W using an estimated line strength then gives the true line strength when the calculated and observed equivalent widths agree. All of the equivalent-width corrections depend on the lineshape. Our equivalent-width technique is more direct and less error prone than the traditional approach described above. It is also very easy to use in an interactive computer algorithm. We do not directly calculate the correction terms discussed above but instead compare the experimentally determined WM to a calculated W~ ~c. A Voigt routine generates a line profile that our algorithm treats exactly like the experimental spectrum. The frequency points used in the calculated spectrum are identical to those in the data. This minimizes any small errors due to undersampling of the spectrum, although in general this is not a problem in our data. An iterative
Modified equivalent-width technique I/2Ww
Wr
l~Ww I
1
523
,~----.~//////////~///////~ ,~, , ,I. ,~._~
0.9
I
I
I
I
",\,
.o
I
I
//"
I
I
B
0.9--
.2 0.8 [-
0.8,]
0.7 ~0.5
,
Av ,
,
Frequency Fig. 1. The correct equivalent width is expressed as W = WM+ Ww+ WR+ Ws where Ww is the area in the line wings missed by integrating Eq. (1) over a finite frequency range, WR is the rectangular area missed when an incorrect 100% transmission level is used, and Ws is a correction to Wu caused by the use of an incorrect vertical scale.
0.7
Frequency Fig. 2. The solid line is the distorted profile of the dashed line. See the text for definitions of points A, A', B, and B'.
procedure adjusts the line strength used to calculate ,rl't/'calCMuntil WM and wcalc ,, M agree to a pre-determined accuracy. Figure 2 shows a Voigt absorption line and the same line broadened by a Gaussian instrument function. The amount of instrumental distortion shown in this figure is somewhat larger than observed in this work. We now consider the distorted line to be our experimental spectrum. First, a somewhat subjective decision is made on the integration limits for the calculation of WM. These integration limits, denoted A and B, are forced to be equidistant from the line center, and must be chosen far enough away from the line center that instrument distortion effects are negligible. This condition is easily satisfied for Doppler shaped lines that lose intensity quickly in the wings. For higher pressure lines this condition is again easy to satisfy since Lorentz line wings vary slowly with frequency and are unaffected by instrument distortion. (Points A and B are placed in regions of instrument distortion in Fig. 2 solely for the sake of clarity.) The observed transmission at point A is not used for the integration limit, instead the average transmission in a small region centered about point A is used in order to reduce the effects of noise. The absorption spectrum is then ratioed to a straight line drawn between points A and B. We generally average a 50 point region to determine A and B. A signal-to-noise ratio of at least 300:1 in addition to this 50 point average is sufficient to keep measurement errors below the 1% level due to the uncertainty in the estimate of the average transmission in this region. However, this error estimate depends on the placement of A and B relative to the line center and on the optical depth. If A and B are placed too close to the line center, instrument distortion will introduce errors into the measurement. However, A and B should not be located too far away from the line center because errors in A and B will be magnified the more distant they are from the line center due to the increasing contribution of high transmission regions to the equivalent width. This ratio to a line between A and B desensitizes the equivalent-width measurement to potential transmission errors. It completely eliminates errors due to any frequency independent offsets in either the empty cell background or the absorption scan. For any frequency dependent offset errors this ratio provides a first-order correction. In the normal equivalent width method this ratio introduces a scale error that the Ws term in Eq. (2) corrects. However, we ratio the calculated data to an equivalent straight line between points A' and B' in Fig. 2 that are offset from the line center frequency by the same amount as points A and B. We thus avoid the need for an explicit correction term. W M and U.,-ca~c M are then obtained from a numerical integration of the experimental and calculated spectra. The line strength used in the calculation of W~ ~¢ is varied until the two equivalent widths agree. ,,
524
L. LARRABEESTROW and Ltu ZHENG
If the spectrum is not ratioed to the straight line between points A and B large errors can occur. We performed tests on a line with 30% distortion (percent increase in width due to instrumental broadening), an observed line center transmission of 0.5, and a 100% transmission level that is in error by 0.5%. If this ratio is not performed the line strength determined from the equivalent width was found to be in error by 5-10%. Again, this error depends on the exact selection for A and B. Previous investigators account for this error with the combination of the correction terms Ws and WR- Tests of our algorithm show that the same 0.5% error in the 100% transmission level introduces errors several orders of magnitude smaller than unavoidable systematic errors in the pressure and pathlength. 3. T E S T OF T H E T E C H N I Q U E
WITH EXPERIMENTAL
SPECTRA
We present here tests of our method using diode-laser spectra of several lines in the vI + v2 - v~ band of CO2 centered at 792 cm J. This band will be used by the CLAES (Cryogenic Limb Array Etalon Spectrometer) instrument on UARS (Upper Atmosphere Research Satellite) 12to determine stratospheric temperatures. Models of the temperature retrieval algorithm indicate that relative line strengths accurate to 1% are needed with absolute accuracies of approx. 3%. This band provides a good test of our algorithm because it occurs at a relatively low frequency where the Doppler widths are small (0.0008 cm-l) and consequently the laser distortion is more evident. On the other hand, this band is less than ideal for demonstrating our method because it has a large ground state energy (approx. 1300 cm -m) making the line strengths very temperature sensitive. For example, for low-J lines a 1 K error in temperature will result in a 2% error in the line strength. There are no well established line strength standards at long wavelengths especially in the region where our laser operated. Consequently we could only test our algorithm by (i) comparing line strengths derived from a single line recorded over a wide pressure range and (ii) measuring the strengths of several lines in the same band and comparing their values, assuming that the relative line strengths follow rigid-rotor values. Because of the high ground state energy of these lines intercomparisons of their strengths may be dominated by errors in the measurement of the gas temperature. Figure 3 shows several spectra of the R (6) transition taken between 0.5 and 20 torr. These spectra were recorded using the technique of sweep-integration with a diode-laser spectrometer similar to the one described in Ref. 13. We first ratio the absorption scans to the empty cell background scan and apply our equivalent-width algorithm to the resulting transmission spectra. For the higher pressure spectra a value for the CO2 self-broadening coefficient is required, although even at 20 torr an uncertainty in the broadening coefficient of 5% only produces an error of 2.5% in the line strength. At 1 torr, an error of 5% in the broadening coefficient results in a line strength error of <0.3%. The high pressure scans are included in this test primarily to demonstrate the robustness of our algorithm. Normally we would expect to use this method at pressures of several torr or lower to avoid line overlap and the need for accurate pressure-broadening coefficients. I
I
I
I
I m
Table 1. Line strength measurements of R(6). The uncorrected strengths were determined without a ratio of the spectra to the line defined by A and B. All strengths in both tables are in units of cm-~/(molec-cm -2) x 10 -23 at 296 K. Pressure
Distortion(%)
(tort)
~ ' ~ Zero Level I
I
I
i
i
Frequency Fig. 3. Spectra of the R(6) line at 0.5, 1, 5, I0, and 20 torr. A zero absorption background scan is also shown.
Measured
Uncorrected
Strength
Strength
0.492
18
1.032
0.837
0.974
17
1.018
0.981
4.925
12
1.035
1.072
10.022
7
1.032
1.058
19.990
3
1.041
1.034
Average
1.032
0.996
St. Dev.
0.0084
0.096
% St. Dev.
0.8
9.6
Modified equivalent-width technique
525
A pressure-broadening coefficient of 0.113 cm-~/atm was used for R(6)) 4 If high-pressure lines must be measured, the half-widths may either be known well enough for an equivalent-width determination of the line strength (i.e., many diatomic and triatomic atmospheric gases), or can be determined using a direct measurement of the line width at high pressure. The measured line strengths of R(6) are listed in Table 1. Although the pressure ranges from 0.5 to 20 torr, the standard deviation of the strengths is only 0.8%. The width of the lowest pressure line (0.5 torr) was broadened 18% by the laser. The highest pressure line (20torr) had almost negligible distortion. The close agreement of the line strength values gives us confidence in our algorithm, especially considering the large variations in lineshape and instrument distortion that occur over the wide range of pressures. Of course, the absolute accuracy of the measurements will be lower because of systematic errors in pressure, pathlength, temperature, etc. We were unable to completely saturate any of the transitions measured due to the low strength of this band and consequently we could not test the integrity of the laser mode. However, the observable laser modes were very widely separated at the operating conditions used for this work. In addition, any significant errors in the 0% transmission level would be evident in our line strength values since they were determined over such a wide range of optical depths (see Fig. 3). We have presently measured the strengths of seven other lines in this band (four other R-branch lines and three Q-branch lines). If we assume that the relative strengths follow rigid-rotor values, we find that all of the measurements agree with a standard deviation of 1.5%. This uncertainty probably contains contributions from temperature measurement errors that are magnified by the large ground state energies. We also recorded a series of scans of R(8) and R(10) between 0.5 and approx. 20 torr. The line strengths determined at each pressure agree to within 1% or better. These two lines were distorted more than R(6) [the 0.5 torr R(8) line was 40% wider than calculated, and the 0.5 torr R(10) line 26% wider], providing a further test of our method. Table 2 shows our line strength results for R(8), the most distorted line we measured. Our preliminary results indicate that the AFGL Atmospheric Absorption Line Parameters Compilation ~4value for this band strength is approx. 17% too high. A full analysis of this band will be presented in a future paper. Line strengths obtained using the same algorithm except omitting the ratio of the spectra to a straight line between points A and B are also listed in Table 1. This omission causes the standard deviation of the strengths to increase dramatically, to around 10%. Clearly there are small errors in the experimental transmission scale that are minimized by performing this ratio. Rescaling the observed spectra when doing an equivalent-width measurements is not new. However, in our algorithm we pay little attention to the exact choice for the rescaling level. The insensitivity of our algorithm to the selection of the points A and B is illustrated in Fig. 4. Three very distinct choices for the locations of A and B are used to determine the strength of this 10-torr spectrum of R(8), the most distorted line we measured. The line strengths obtained from the top two choices for A and B in Fig. 4 agree to within 0.2% and demonstrate that if A and B lie reasonably far from the line center identical line strengths are obtained. The bottom choice for
Table 2. Line strength measurements of R(8). Also included are errors that result when our method is applied to noise-free simulated spectra of
R(8). Pressure
Distortion (%)
(tort)
Measured
Deviation from
Simulation
Strength
Average (%)
Error (%)
0.26
0.00
0.466
39
1.223
0.999
24
1.225
0.38
0.00
5,396
14
1.207
-I.00
-0.01
10.358
11
1.221
0.08
-0.03
24.140
0
1.225
0.41
-0.03
Average
1.220
St. Dev.
0.0066
% St. De~:
0.5
526
L. LARRABEESTROWand LIu ZHENG i
I
I
I
i
I
I
/~B
r
i
I
I
"
i
Frequency Fig. 4. Three choices for the equivalent-width integration limits A and B for the R(8) line at 10 torr. Choices (a) and (b) give line strengths that agree to within 0.2%. The strength from choice (c) differs from (a) and (b) by 4.5%. Points A and B are taken to the average transmission between the x markers.
A and B in Fig. 4 is an extreme case that would not be used in practice. However, even this extreme case gives a strength that differs from the others by only 4.5%, demonstrating the robustness of our algorithm. 4. T E S T OF T H E T E C H N I Q U E
WITH SIMULATED
DATA
A further test of our method was performed using simulated spectra broadened with a Gaussian instrument function. These tests were done using a Gaussian halfwidth of 0.75 x 10 -3 c m - l , which causes approximately the same amount of distortion present in R(8), the most distorted line measured. The errors in the line strengths recovered from these noise-free simulated spectra are listed in the last column of Table 2. Clearly, our algorithm reproduces the line strengths to an accuracy well below the systematic error level. It should be noted that even in these tests there is some randomness in the selection of the points A and B for each line strength determination. Simulations performed with a Gaussian instrument function approximately twice as wide as the one used for Table 2 gave similar results. We believe that the main sources of error in our method come from residual non-linearities in the frequency scale and errors in the transmission level at the scaling points A and B due to noise and any remaining instrument distortion. However, these errors can normally be kept below the level of other systematic errors in the determination of line strengths. Because the effective Doppler width method for the measurement of diode-laser line strengths is popular we performed some simulations to test its accuracy. 4-7 (We know of no published tests of the accuracy of this method using simulated data.) The accuracy of this method is sensitive to the minimum transmission of the line since it assumes that k (v) is convolved with the instrument profile instead of T = exp[-k(v)x]. For laser distortion similar to that found in our data, this method gives line strength errors that range from 1 to 11% for pressures in the 1-10 torr region and line center transmissions between 0.1 and 0.9. Clearly, great care must be exercised using the effective Doppler technique with spectra that show instrument distortion. I f the spectral lines are well separated, and the data are recorded using sweep-integration, much higher accuracies are expected using the equivalent-width method reported here. 5. C O N C L U S I O N S We have developed a modification of the normal equivalent-width method for the measurement of line strengths using diode-laser spectra. This method eliminates the explicit calculation of correction terms to the measured equivalent width before determining a line strength. By treating the measured spectrum and a calculated spectrum of the line identically, our algorithm minimizes the introduction of subtle numerical errors into the line strength measurement. The algorithm is quick enough to be performed interactively, the time between selecting the integration limits and obtaining a line strength is only several seconds o n a low-cost computer workstation.
Modified equivalent-width technique
527
As is normally the case for equivalent-width measurements, isolated lines are needed, although small interferences in the wings of a line can be minimized by choosing integration limits close to the line center. However, this technique is geared to the highest accuracy measurements, which are often only required for isolated lines. In addition, a smooth and accurate frequency scale is required, especially for narrow Doppler lines. Although diode-laser spectra recorded using lock-in amplifiers and slow scans may not meet this requirement, we have found that diode-laser spectra recorded using sweep integration have sufficiently accurate frequency scales.
Acknowledgement--This research was supported by the National Aeronautics and Space Administration.
1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14.
REFERENCES D. A. Jennings, Appl. Opt. 19, 2695 (1980). G. Restelli and F. Cappellani, JQSRT 33, 459 (1985). L. L. Strow, JQSRT 29, 395 (1983). B. Fridovich, V. Malathy Devi, and P. P. Das, J. Molec. Spectrose. 81, 264 (1980). V. Malathy Devi, B. Fridovich, G. D. Jones, D. J. S. Snyder, and A. Neuendorffer, Appl. Opt. 21, 1537 (1982). V. Malathy Devi, P. P. Das, A. Bano, K. Narahari Rao, J. M. Flaud, C. Camy-Peyret, and J. P. Chevillard, J. Molec. Spectrosc. 88, 251 (1981). T. Huet, N. Lacome, and A. Levy, J. Molec. Spectrosc. 128, 206 (1988). L. R. Brown, J. S. Margolis, R. H. Norton, and B. D. Stedry, Appl. Spectrosc. 37, 287 (1983). R. D. May, JQSRT 39, 247 (1988). W. S. Benedict, R. Herman, G. E. Moore, and S. Silverman, Can. J. Phys. 34, 850 (1956). M. A. H. Smith, C. P. Rinsland, B. Fridovich, and K. Narahari Rao, in Molecular Spectroscopy, Modern Research, Vol. III, Chap. 3, pp. 111-248, Academic Press, New York, NY (1985). C. A. Reber, "Upper Atmosphere Research Satellite (UARS) Mission," Technical Report 430-1003-001, NASA Goddard Space Flight Center (May 1985). L. L. Strow and B. M. Gentry, J. Chem. Phys. 84, !149 (1986). L. S. Rothman, R. R. Gamache, A. Barb, A. Goldman, J. R. Gillis, L. R. Brown, R. A. Toth, J. M. Flaud, and C. Camy-Peyret, Appl. Opt. 22, 2247 (1983).
Printed in Great Britain.All rightsreserved
0022-4073/89 $3.00+ 0.00 Copyright© 1989PergamonPressplc
A MODIFIED EQUIVALENT-WIDTH TECHNIQUE FOR DIODE-LASER LINE STRENGTH MEASUREMENTS L. LARRABEESTROW and LIu ZHENG Physics Department, University of Maryland, Baltimore County, Baltimore, MD 21228, U.S.A. (Recewed 10 March 1989)
Abstract--A modified equivalent-width method is introduced that improves the accuracy of
spectral line strength measurements using diode lasers. This method differs from the standard method in that we do not attempt to calculate the true equivalent width in order to determine the line strength. Instead, we derive the line strength directly from a measurement of the area under the line close to the line center and ignore the normal correction terms. Tests with actual spectra and simulations indicate that this method is accurate to better than 1% with lines broadened by up to 50% by the laser instrument function.
1. I N T R O D U C T I O N Tunable diode lasers are widely used to measure absorption line strengths. Many early measurements assumed that the distortion of a spectral line was small enough to allow the strength to be directly determined from the line-center absorption. J'2 This method did not require either an accurate or smooth frequency scale and was very easy to implement. It was soon recognized that laser jitter and/or the finite bandwidth of the laser can, in some cases, significantly distort the spectral lineY The effect of this distortion on line strengths derived from line-center absorption can be accounted for if the laser-emission profile can be adequately characterized. Normally, some measure o f the laser width is derived from the observed increase in the Doppler width of a line due to the finite bandwidth of the laser. If the functional form of the laser instrument function is known, correction curves can be generated to determine the correct line strength from the observed line-center absorption. Alternatively, the direct method can be employed with distorted lines by increasing the Doppler width used to determine the strength from the line center absorption. The increased Doppler width approximates the laser distortion of the spectral line. 4-7 Unfortunately, it has proven difficult to characterize the laser profile accurately in practice because the line-center absorption is distorted more severely than the line width, which is used to deduce the width of the instrument profile. The average value of the distortion can be determined by measuring the widths o f a large number of Doppler-shaped lines. 3 However, the laser profile or jitter may change with time, even within one continuous mode. This fact prevents the use o f non-linear least-squares fitting for line parameters when the instrument function is fixed, as is normally done in the analysis o f high resolution FTS spectra) If the signal-to-noise ratio is high enough, a Doppler-limited line can be deconvolved and an instrument function recovered. 9 The sensitivity of this process to noise and the care that must be taken to perform the deconvolution properly often make this technique impractical for day to day use. The method o f equivalent widths has the advantage that one does not need to know the instrument profile in order to determine the line strength. I° However, a smooth and accurate frequency scale is required, as well as an accurate 100% transmission level. Early diode-laser spectrometers used lock-in detection for recording the spectra, which produced inaccurate frequency scales since the time scales for variations o f the laser frequency were similar to the time scales for measurements. With the introduction o f the sweep-integration method for the recording o f diode-laser spectra, I much smoother frequency scales could be obtained via signal-averaging. In addition, since spectra can be recorded much more quickly with sweep-integration, empty cell background scans are taken very close in time to the absorption scans before the instrument intensity scale can drift. 521
522
L. LARRABEESTROWand LIU ZHENG
We present here a modified equivalent-width method for the measurement of line strengths using diode-laser spectra. This method differs from the standard method in that we do not attempt to calculate the true equivalent width and from that determine the line strength. Instead, we derive the line strength directly from a measurement of the area under the line close to the line center and ignore the normal correction terms. This technique is suitable for accurate measurements of isolated lines. Tests with actual spectra and simulations indicate that our method is accurate to at least 1% with lines broadened by up to 50% by the laser instrument function. 2. T H E M O D I F I E D
EQUIVALENT-WIDTH
TECHNIQUE
The equivalent width of a spectral line is defined as ~° W=
[1 - T(v - Vo)] d(v - Vo),
(1)
where T(v - V o ) is the observed transmittance, v the frequency, and Vo the line center frequency. W is independent of the width of the spectral line if the time average of the laser-frequency profile is symmetric about the center frequency. An accurate frequency scale is essential for the determination of W. Diode-laser spectra collected using the sweep-integration method generally have very smooth frequency scales that can be accurately calibrated by using known frequency standards and simultaneously recorded etalon fringes. We use cubic splines to interpolate the frequency scale between etalon fringes. The spline interpolation can correct for any non-linear tuning of the laser over an interval that contains several fringes. Diode lasers may also tune non-linearly over very narrow spectral intervals on the order of a Doppler-limited linewidth or smaller. However, these non-linearities generally vary randomly in time. Averaging several hundred spectra reduces any fine-scale non-linear tuning to acceptable levels given the apparent accuracy of the results presented here. However, these random variations of the laser tuning do broaden the observed spectra. In addition, the transmission scale must be well known, especially the 100% transmission level. We determine the 100% transmission scale by obtaining the ratios of the absorption spectra with respect to empty cell scans. Intensity offsets in either the background or absorption scans can introduce transmission errors that must be minimized when determining W. These small errors are magnified when performing a simple equivalent-width measurement since regions of high transmission contribute more to the equivalent width than regions of low transmission. The techniques described here provide a very simple way to minimize these errors and also eliminate the need to determine correction terms to the equivalent width due to the use of finite integration limits in Eq. (1). Following Fig. l, the measured equivalent width WM is related to the true equivalent width W by W = WM+ Ww+ WR+ Ws,
(2)
where Ww is the area in the line wings missed by integrating Eq. (1) over a finite frequency range, WR is the rectangular area missed when an incorrect 100% transmission level is used, and Ws is a correction to WM caused by the use of an incorrect vertical scale." Investigators previously determined the line strength by first calculating these correction terms to obtain W. An iterative theoretical calculation of W using an estimated line strength then gives the true line strength when the calculated and observed equivalent widths agree. All of the equivalent-width corrections depend on the lineshape. Our equivalent-width technique is more direct and less error prone than the traditional approach described above. It is also very easy to use in an interactive computer algorithm. We do not directly calculate the correction terms discussed above but instead compare the experimentally determined WM to a calculated W~ ~c. A Voigt routine generates a line profile that our algorithm treats exactly like the experimental spectrum. The frequency points used in the calculated spectrum are identical to those in the data. This minimizes any small errors due to undersampling of the spectrum, although in general this is not a problem in our data. An iterative
Modified equivalent-width technique I/2Ww
Wr
l~Ww I
1
523
,~----.~//////////~///////~ ,~, , ,I. ,~._~
0.9
I
I
I
I
",\,
.o
I
I
//"
I
I
B
0.9--
.2 0.8 [-
0.8,]
0.7 ~0.5
,
Av ,
,
Frequency Fig. 1. The correct equivalent width is expressed as W = WM+ Ww+ WR+ Ws where Ww is the area in the line wings missed by integrating Eq. (1) over a finite frequency range, WR is the rectangular area missed when an incorrect 100% transmission level is used, and Ws is a correction to Wu caused by the use of an incorrect vertical scale.
0.7
Frequency Fig. 2. The solid line is the distorted profile of the dashed line. See the text for definitions of points A, A', B, and B'.
procedure adjusts the line strength used to calculate ,rl't/'calCMuntil WM and wcalc ,, M agree to a pre-determined accuracy. Figure 2 shows a Voigt absorption line and the same line broadened by a Gaussian instrument function. The amount of instrumental distortion shown in this figure is somewhat larger than observed in this work. We now consider the distorted line to be our experimental spectrum. First, a somewhat subjective decision is made on the integration limits for the calculation of WM. These integration limits, denoted A and B, are forced to be equidistant from the line center, and must be chosen far enough away from the line center that instrument distortion effects are negligible. This condition is easily satisfied for Doppler shaped lines that lose intensity quickly in the wings. For higher pressure lines this condition is again easy to satisfy since Lorentz line wings vary slowly with frequency and are unaffected by instrument distortion. (Points A and B are placed in regions of instrument distortion in Fig. 2 solely for the sake of clarity.) The observed transmission at point A is not used for the integration limit, instead the average transmission in a small region centered about point A is used in order to reduce the effects of noise. The absorption spectrum is then ratioed to a straight line drawn between points A and B. We generally average a 50 point region to determine A and B. A signal-to-noise ratio of at least 300:1 in addition to this 50 point average is sufficient to keep measurement errors below the 1% level due to the uncertainty in the estimate of the average transmission in this region. However, this error estimate depends on the placement of A and B relative to the line center and on the optical depth. If A and B are placed too close to the line center, instrument distortion will introduce errors into the measurement. However, A and B should not be located too far away from the line center because errors in A and B will be magnified the more distant they are from the line center due to the increasing contribution of high transmission regions to the equivalent width. This ratio to a line between A and B desensitizes the equivalent-width measurement to potential transmission errors. It completely eliminates errors due to any frequency independent offsets in either the empty cell background or the absorption scan. For any frequency dependent offset errors this ratio provides a first-order correction. In the normal equivalent width method this ratio introduces a scale error that the Ws term in Eq. (2) corrects. However, we ratio the calculated data to an equivalent straight line between points A' and B' in Fig. 2 that are offset from the line center frequency by the same amount as points A and B. We thus avoid the need for an explicit correction term. W M and U.,-ca~c M are then obtained from a numerical integration of the experimental and calculated spectra. The line strength used in the calculation of W~ ~¢ is varied until the two equivalent widths agree. ,,
524
L. LARRABEESTROW and Ltu ZHENG
If the spectrum is not ratioed to the straight line between points A and B large errors can occur. We performed tests on a line with 30% distortion (percent increase in width due to instrumental broadening), an observed line center transmission of 0.5, and a 100% transmission level that is in error by 0.5%. If this ratio is not performed the line strength determined from the equivalent width was found to be in error by 5-10%. Again, this error depends on the exact selection for A and B. Previous investigators account for this error with the combination of the correction terms Ws and WR- Tests of our algorithm show that the same 0.5% error in the 100% transmission level introduces errors several orders of magnitude smaller than unavoidable systematic errors in the pressure and pathlength. 3. T E S T OF T H E T E C H N I Q U E
WITH EXPERIMENTAL
SPECTRA
We present here tests of our method using diode-laser spectra of several lines in the vI + v2 - v~ band of CO2 centered at 792 cm J. This band will be used by the CLAES (Cryogenic Limb Array Etalon Spectrometer) instrument on UARS (Upper Atmosphere Research Satellite) 12to determine stratospheric temperatures. Models of the temperature retrieval algorithm indicate that relative line strengths accurate to 1% are needed with absolute accuracies of approx. 3%. This band provides a good test of our algorithm because it occurs at a relatively low frequency where the Doppler widths are small (0.0008 cm-l) and consequently the laser distortion is more evident. On the other hand, this band is less than ideal for demonstrating our method because it has a large ground state energy (approx. 1300 cm -m) making the line strengths very temperature sensitive. For example, for low-J lines a 1 K error in temperature will result in a 2% error in the line strength. There are no well established line strength standards at long wavelengths especially in the region where our laser operated. Consequently we could only test our algorithm by (i) comparing line strengths derived from a single line recorded over a wide pressure range and (ii) measuring the strengths of several lines in the same band and comparing their values, assuming that the relative line strengths follow rigid-rotor values. Because of the high ground state energy of these lines intercomparisons of their strengths may be dominated by errors in the measurement of the gas temperature. Figure 3 shows several spectra of the R (6) transition taken between 0.5 and 20 torr. These spectra were recorded using the technique of sweep-integration with a diode-laser spectrometer similar to the one described in Ref. 13. We first ratio the absorption scans to the empty cell background scan and apply our equivalent-width algorithm to the resulting transmission spectra. For the higher pressure spectra a value for the CO2 self-broadening coefficient is required, although even at 20 torr an uncertainty in the broadening coefficient of 5% only produces an error of 2.5% in the line strength. At 1 torr, an error of 5% in the broadening coefficient results in a line strength error of <0.3%. The high pressure scans are included in this test primarily to demonstrate the robustness of our algorithm. Normally we would expect to use this method at pressures of several torr or lower to avoid line overlap and the need for accurate pressure-broadening coefficients. I
I
I
I
I m
Table 1. Line strength measurements of R(6). The uncorrected strengths were determined without a ratio of the spectra to the line defined by A and B. All strengths in both tables are in units of cm-~/(molec-cm -2) x 10 -23 at 296 K. Pressure
Distortion(%)
(tort)
~ ' ~ Zero Level I
I
I
i
i
Frequency Fig. 3. Spectra of the R(6) line at 0.5, 1, 5, I0, and 20 torr. A zero absorption background scan is also shown.
Measured
Uncorrected
Strength
Strength
0.492
18
1.032
0.837
0.974
17
1.018
0.981
4.925
12
1.035
1.072
10.022
7
1.032
1.058
19.990
3
1.041
1.034
Average
1.032
0.996
St. Dev.
0.0084
0.096
% St. Dev.
0.8
9.6
Modified equivalent-width technique
525
A pressure-broadening coefficient of 0.113 cm-~/atm was used for R(6)) 4 If high-pressure lines must be measured, the half-widths may either be known well enough for an equivalent-width determination of the line strength (i.e., many diatomic and triatomic atmospheric gases), or can be determined using a direct measurement of the line width at high pressure. The measured line strengths of R(6) are listed in Table 1. Although the pressure ranges from 0.5 to 20 torr, the standard deviation of the strengths is only 0.8%. The width of the lowest pressure line (0.5 torr) was broadened 18% by the laser. The highest pressure line (20torr) had almost negligible distortion. The close agreement of the line strength values gives us confidence in our algorithm, especially considering the large variations in lineshape and instrument distortion that occur over the wide range of pressures. Of course, the absolute accuracy of the measurements will be lower because of systematic errors in pressure, pathlength, temperature, etc. We were unable to completely saturate any of the transitions measured due to the low strength of this band and consequently we could not test the integrity of the laser mode. However, the observable laser modes were very widely separated at the operating conditions used for this work. In addition, any significant errors in the 0% transmission level would be evident in our line strength values since they were determined over such a wide range of optical depths (see Fig. 3). We have presently measured the strengths of seven other lines in this band (four other R-branch lines and three Q-branch lines). If we assume that the relative strengths follow rigid-rotor values, we find that all of the measurements agree with a standard deviation of 1.5%. This uncertainty probably contains contributions from temperature measurement errors that are magnified by the large ground state energies. We also recorded a series of scans of R(8) and R(10) between 0.5 and approx. 20 torr. The line strengths determined at each pressure agree to within 1% or better. These two lines were distorted more than R(6) [the 0.5 torr R(8) line was 40% wider than calculated, and the 0.5 torr R(10) line 26% wider], providing a further test of our method. Table 2 shows our line strength results for R(8), the most distorted line we measured. Our preliminary results indicate that the AFGL Atmospheric Absorption Line Parameters Compilation ~4value for this band strength is approx. 17% too high. A full analysis of this band will be presented in a future paper. Line strengths obtained using the same algorithm except omitting the ratio of the spectra to a straight line between points A and B are also listed in Table 1. This omission causes the standard deviation of the strengths to increase dramatically, to around 10%. Clearly there are small errors in the experimental transmission scale that are minimized by performing this ratio. Rescaling the observed spectra when doing an equivalent-width measurements is not new. However, in our algorithm we pay little attention to the exact choice for the rescaling level. The insensitivity of our algorithm to the selection of the points A and B is illustrated in Fig. 4. Three very distinct choices for the locations of A and B are used to determine the strength of this 10-torr spectrum of R(8), the most distorted line we measured. The line strengths obtained from the top two choices for A and B in Fig. 4 agree to within 0.2% and demonstrate that if A and B lie reasonably far from the line center identical line strengths are obtained. The bottom choice for
Table 2. Line strength measurements of R(8). Also included are errors that result when our method is applied to noise-free simulated spectra of
R(8). Pressure
Distortion (%)
(tort)
Measured
Deviation from
Simulation
Strength
Average (%)
Error (%)
0.26
0.00
0.466
39
1.223
0.999
24
1.225
0.38
0.00
5,396
14
1.207
-I.00
-0.01
10.358
11
1.221
0.08
-0.03
24.140
0
1.225
0.41
-0.03
Average
1.220
St. Dev.
0.0066
% St. De~:
0.5
526
L. LARRABEESTROWand LIu ZHENG i
I
I
I
i
I
I
/~B
r
i
I
I
"
i
Frequency Fig. 4. Three choices for the equivalent-width integration limits A and B for the R(8) line at 10 torr. Choices (a) and (b) give line strengths that agree to within 0.2%. The strength from choice (c) differs from (a) and (b) by 4.5%. Points A and B are taken to the average transmission between the x markers.
A and B in Fig. 4 is an extreme case that would not be used in practice. However, even this extreme case gives a strength that differs from the others by only 4.5%, demonstrating the robustness of our algorithm. 4. T E S T OF T H E T E C H N I Q U E
WITH SIMULATED
DATA
A further test of our method was performed using simulated spectra broadened with a Gaussian instrument function. These tests were done using a Gaussian halfwidth of 0.75 x 10 -3 c m - l , which causes approximately the same amount of distortion present in R(8), the most distorted line measured. The errors in the line strengths recovered from these noise-free simulated spectra are listed in the last column of Table 2. Clearly, our algorithm reproduces the line strengths to an accuracy well below the systematic error level. It should be noted that even in these tests there is some randomness in the selection of the points A and B for each line strength determination. Simulations performed with a Gaussian instrument function approximately twice as wide as the one used for Table 2 gave similar results. We believe that the main sources of error in our method come from residual non-linearities in the frequency scale and errors in the transmission level at the scaling points A and B due to noise and any remaining instrument distortion. However, these errors can normally be kept below the level of other systematic errors in the determination of line strengths. Because the effective Doppler width method for the measurement of diode-laser line strengths is popular we performed some simulations to test its accuracy. 4-7 (We know of no published tests of the accuracy of this method using simulated data.) The accuracy of this method is sensitive to the minimum transmission of the line since it assumes that k (v) is convolved with the instrument profile instead of T = exp[-k(v)x]. For laser distortion similar to that found in our data, this method gives line strength errors that range from 1 to 11% for pressures in the 1-10 torr region and line center transmissions between 0.1 and 0.9. Clearly, great care must be exercised using the effective Doppler technique with spectra that show instrument distortion. I f the spectral lines are well separated, and the data are recorded using sweep-integration, much higher accuracies are expected using the equivalent-width method reported here. 5. C O N C L U S I O N S We have developed a modification of the normal equivalent-width method for the measurement of line strengths using diode-laser spectra. This method eliminates the explicit calculation of correction terms to the measured equivalent width before determining a line strength. By treating the measured spectrum and a calculated spectrum of the line identically, our algorithm minimizes the introduction of subtle numerical errors into the line strength measurement. The algorithm is quick enough to be performed interactively, the time between selecting the integration limits and obtaining a line strength is only several seconds o n a low-cost computer workstation.
Modified equivalent-width technique
527
As is normally the case for equivalent-width measurements, isolated lines are needed, although small interferences in the wings of a line can be minimized by choosing integration limits close to the line center. However, this technique is geared to the highest accuracy measurements, which are often only required for isolated lines. In addition, a smooth and accurate frequency scale is required, especially for narrow Doppler lines. Although diode-laser spectra recorded using lock-in amplifiers and slow scans may not meet this requirement, we have found that diode-laser spectra recorded using sweep integration have sufficiently accurate frequency scales.
Acknowledgement--This research was supported by the National Aeronautics and Space Administration.
1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14.
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