Signal-to-noise ratio analysis in laser absorptionspectrometers using optical multipass cells
Peter Werle and Franz Slemr
In high resolution absorption spectrometers with conventional light sources, the signal-to-noise ratio (SNR) isusually limited by the thermal noise level of the detector-preamplifier combination, which is independent ofthe light source power. However, the noise in many laser absorption spectrometers is dominated by theexcess or shot noise which is dependent on the transmitted laser power, and which in turn is dependent on thenumber of reflections in a multipass cell. The optimum absorption path length for a high frequencymodulated (FM) and a conventional wavelength modulated (WM) diode laser absorption spectrometer isinvestigated in this paper. The major result is that, due to the power attenuation by the multipass cell, thebest SNR of a shot noise limited FM spectrometer is achieved at substantially shorter absorption paths, whencompared with the excess noise limited WM spectrometer. This finding implies that the implementation ofthe FM technique in absorption spectrometers with multipass cells can improve the SNR only by 1 order ofmagnitude. Although desirable, this is substantially less than the improvement of 2 orders of magnitudeexpected in quantum limited conditions with a single pass cell.
1. IntroductionA large number of atmospheric trace gases can be
monitored by absorption spectroscopy. 1-3 The ab-sorptions that have to be detected are usually small,and to achieve sensitivity adequate for environmentalmonitoring, absorption spectrometers require long op-tical paths.' In instruments with a limited size, longabsorption path lengths up to several hundreds ofmeters2 have usually been provided by multireflectionoptical systems of which the most well known are thesystems invented by White4 5 and by Herriott et a. 6 7
In all these systems the sensitivity gained by lengthen-ing the absorption path is offset by the increased atten-uation of the radiation power throughput, because ofthe imperfect reflectivity of the mirrors. Consequent-ly, to achieve the highest SNR each absorption spec-trometer has to be operated with an optimal number ofreflections in multireflection absorption systems.
The increased sensitivity of absorption spectro-meters leads to a greater probability of interference by
The authors are with Fraunhofer Institute for Atmospheric Envi-ronment Research, Postfach 1343, D-8100 Garmisch-Partenkirchen,Federal Republic of Germany.
other atmospheric trace gases. Such interference isusually substantially reduced by operating the spec-trometer at low pressures, where the pressure broaden-ing is suppressed. Therefore, high resolution spec-troscopy is required to achieve specificity in themeasurement of atmospheric trace gases.1 Unfortu-nately, conventional light sources have low spectralpower density. The detectable optical density of highresolution absorption spectrometers with convention-al light sources, such as a Nernst glower, is then usuallylimited by the thermal noise of the detector-preampli-fier combination, which is independent of the incidentlight power. In these conditions the best SNR can beachieved by adjusting the absorption signal to a maxi-mum, i.e., by adjusting the number of reflections to anoptimal value given by the reflectivity of the mirrors.Stephens8 derived an equation to show that the opti-mal number of reflections is reached when the reflec-tion losses reduce the light power to l/e of its initialvalue.
To overcome the limits set by the low spectral powerdensity of conventional light sources, lasers are in-creasingly used in high resolution absorption spectros-copy for atmospheric trace gas monitoring.39 Thedetectable optical density of such laser absorptionspectrometers is usually limited by amplitude andphase fluctuations of the laser itself. Most of thesespectrometers were operated at low modulation fre-quencies and in these conditions the noise is usuallydominated by the /f laser noise.
Fig. 1. Noise frequency spectrum of a lead-salt diode laser.
It has been shown that 1/f laser noise can be substan-tially reduced by using high frequency modulationtechniques. With a ring dye laser, Hall et al.10 foundthat the noise level at 1 MHz is lower by 80 dB than at10 kHz. For lead-salt diode lasers, which are used formonitoring atmospheric trace gases and for high reso-lution IR spectroscopy, similar measurements showedthat the noise level decreased by 40 dB between 2 and200 MHz." In terms of sensitivity, modulation tech-niques operated at 200 MHz should improve the SNRby 2 orders of magnitude. As schematically illustratedin Fig. 1, the noise spectrum with sufficient laser powerbecomes frequency independent above a certain fre-quency WWN. The frequency, CWWN, denotes the transi-tion from the frequency dependent 1/f or pink noise tofrequency independent white noise governed by ther-mal noise and quantum noise (shot noise). Based onthese observations, a high frequency modulation (FM)techniquel2"13 was recently developed to avoid the limi-tations of 1/f noise in conventional derivative tech-niques and to achieve shot noise limited performance.The basic idea of the high frequency modulation (FM)technique is to move, in modulation and detectionfrequency space, from the currently used low modula-tion frequencies (derivative technique using, lock-inamplifiers = WM technique) to frequencies beyondthe above mentioned WWN frequency.
Both 1/f noise and shot noise are strongly dependenton the laser power incident on the detector. Conse-quently, the optimal path length of an absorption spec-trometer limited by detector thermal noise8 will notapply to the FM and WM techniques. In this paper,therefore, we investigate the SNR of 1/f noise and shotnoise limited laser absorption spectrometers as a func-tion of the laser power, the mirror reflectivity, and thenumber of passes in a multipass absorption cell. Theresults show that the optimal number of reflections hasto be calculated individually for the available powerand the noise spectrum of each laser. Due to thedifferent power dependences of the SNR in both cases,less reflections are optimum for shot noise limited FMspectrometers than for wavelength modulated ones.The practical consequences of the analysis are dis-cussed.
II. Formal Description of the SNR in Derivative and FMSpectrometers Using Multipass Cells
The absorption of light by atmospheric trace gases isusually small, and then where absorption occurs signalS is proportional to the optical path length. In amultipass cell optical path length L = n b, where n isthe number of passes and b is the base length of themultipass cell. The signal is also proportional to radi-ation power PD incident on the detector after passingthrough the multipass cell. The number density andmodulated absorption coefficient for the species understudy are assumed to be constant for the present analy-sis. Neglecting all other losses, this power decreaseswith an increasing number of passes, and increaseswith the increasing reflectivity of mirrors R. The sig-nal of a laser absorption spectrometer with fixed basislength is then proportional to
S nPD = nRn-lpo, (1)
where Po is the laser power incident on the detectorwith a single pass of the cell.
The noise generated by a laser absorption spectrom-eter consists mainly of three components: thermalnoise, shot noise, and laser excess noise. The laserexcess noise is referred to as 1/f noise because it isfrequency dependent. Each of these components hasa different dependence on the laser power incident onthe detector:
(2)NTN = constant,
NSN = aPD,
NEX = MPD,
(3)
(4)
where NTN, NSN, and NEX are the rms detector noisecurrents due to thermal, shot, and excess noise, respec-tively, and a and $(X) are proportionality coeffi-cients." The original analysis by Stephens8 consid-ered only the thermal noise in detecting system NTN.
The SNR of a laser absorption spectrometer can bewritten as
SNR nPD/(NTN + NSN + NE01/2
nPD/(UTN + 'PD + 0(')PD (5)
where, in the denominator, shot noise increases withincreasing power, and thermal noise is power indepen-dent. Power level Pmin can then be defined for whichshot noise equals thermal noise:
(6)NSN(Pmin)2
= NTN = Pmin-
Using the variables y(PD) and (w), where
'Y(PD) = 1 + PD/Pmin,
a(W) = M(w)/NN,
Eq. (5) can be rewritten in a more convenient form:
SNR - nPD/NTN['y(PD) + b(O)PM]' 2 . (9)
The meaning of the variables becomes clear from thisexpression. Variable y(PD) describes how much high-er the sum of thermal and shot noise is above thermal
noise; at y(PD) > 2, shot noise dominates, whereas aty(PD) 1, thermal noise dominates and shot noise canbe neglected. y(PD) is frequency independent sinceboth thermal noise and shot noise are also frequencyindependent. Variable 6(w) takes into account thefrequency dependence of the laser excess noise and is adecreasing function of modulation frequency Wnmod Itcan be calculated from the ratio of laser excess noise tothe total white noise level using
NEX/(NTN + NN) = 6PD/Y(PD)-
1000 40 80 120 160 200 240 280
n Reflections
Fig. 2. Light power incident on detector PD and variable -y as afunction of number of passes in a multipass cell. Variable y de-scribes how much higher the white noise, consisting of thermal noise
and shot noise, is in comparison with the detector thermal noise.
10
9
8
7
6
5
200 240 280
102
Fa
101
(10)
As shown in Fig. 1 the laser excess noise decreases withincreasing modulation frequency, and at frequencyWWN it becomes equal to the white noise:
(WWOPD= Y(PD)- (11)
At a modulation frequency of Wmod = WWN, the totalnoise level is 3 dB above the white noise level. Vari-able 6(w) is thus used to distinguish the shot noiselimited FM case from the laser excess noise limitedWM case.
As PD = Rn-'Po, expressions (9) and (7) can berewritten as
(12)
and y(n) = 1 + Rn-lPo/Pmi.Depending on the modulation frequency, two differ-
ent cases can be investigated: low frequency modula-tion (WM) and high frequency modulation (FM). Inthe WM case, w << WWN and 6(w) i. The SNR is thendictated by the laser excess noise:
3
2(13)
In the FM case, w > WWN and 6(w) << 1, consequently,6(w)R2n-2Po << y(n) and the SNR is dominated bywhite noise:
SNR(n) nRn-P0/NTNy(n)l12 . (i4)
With an increasing number of passes the power inci-dent on the detector will decrease until the quantumnoise level becomes smaller than the thermal noise ofthe detector-preamplifier combination, i.e., lime(n) =1 for n - -. Therefore, both the FM and WM casesswitch to the thermal noise-limited regime with anincreasing number of passes and
lim SNR = nRn'PoINTN. (15)n Ad
This is the expression used by Stephens8 for estima-tion of the optimal number of passes.
Ill. Results and DiscussionEquation (12) is used to calculate the SNR as a
function of number of passes for conditions usuallyencountered in tunable diode laser absorption spectro-meters (TDLASs). The values of variable 6(w) arederived from the measured wideband noise character-istics of a lead-salt diode laser.11 In this work, a 40-dBnoise reduction was observed when the modulationfrequency was changed from -10 kHz, where the laserexcess noise dominates, to -200 MHz. The laser pow-
0
0 25 50 75 100 125 150 175
White-Cell Pathlength [m]
Fig. 3. Signal-to-noise ratio of a frequency modulated laser absorp-tion spectrometer as a function of number of passes in a multipasscell for different laser output power. The calculation is based ondetector noise equivalent power of 67 uW, mirror reflectivity of97.5%, cell base length of 62.5 cm, and variable (w) (see definition in
text) equal to 3 X 10-5.
er incident on detector PD was 521 ,W and minimumpower Pmin calculated from the effective detector ther-mal noise was 67 ,.W. From these results values of b(w)can be estimated for two limiting cases:
WM: ( << IWN) = 0.3,uW-2,(16)
FM: (W >> WN) = 3 X 1O-5,W-2 ,
respectively. Using these values for 6 the SNR can becalculated as a function of the number of reflections nand mirror reflectivity R.
A central point of this analysis is the laser powerlevel in the detector. Figure 2 shows the power inci-dent on detector PD and variable y(n) as a function ofthe number of passes n. The dependence is calculated
Fig. 4. Signal-to-noise ratio of a wavelength modulated laser ab-sorption spectrometer as a function of number of passes in a multi-pass cell for different laser output power. The input parameters arethe same as in Fig. 3 with the exception of variable 5(w) which is 0.3 in
this case.
for a laser power P0 of 521 AW and a measured reflec-tivity of 97.5% for the mirrors in the White cell. Thelatter value is lower than the specification which is-98.6%; the difference may be caused by some dust onthe mirror surface. As already mentioned, Pmin corre-sponds to 67,gW, and this power level is reached after80 passes. Therefore, shot noise limited performanceof the FM spectrometer can only be expected with lessthan 80 passes; a higher number of passes will lead to adecrease in the SNR.
In Fig. 3, the SNR calculated from Eq. (14) for FMabsorption spectroscopy is plotted as a function of thenumber of passes at various laser powers. Pmin isassumed to be 67 ALW, the path length is calculatedusing the cell base length of 62.5 cm, and b(w) is 3 X10-5 as derived in the last section. The calculatedoptimal number of passes varies between 40 and 70,corresponding to a path length of -25-45 m, depend-ing on the laser power. With increasing laser power,the optimum number of passes increases only slowly-an increase of less than a factor of 2 for a 32 timesincrease in laser power.
The equivalent calculation for SNR from Eq. (13)for WM absorption spectroscopy is shown in Fig. 4.With the exception of (W), which is set to 0.3, all theother parameters are the same as before. The opti-mum number of passes again increases with increasedlaser power. In comparison with the previous calcula-tion, the number of optimal passes is much larger andincreases more steeply with increasing power than inthe case of FM spectroscopy. The number of optimalpasses between 120 and 200 corresponds to the numberof passes used in commercial WM-TDLAS instru-ments.
1 4
In Fig. 5 the FM and WM limiting cases are com-
210
110
10 1 I I I l I I I l I I l
0 40 80 120 160 200 240 280
n Reflections
Fig. 5. Signal-to-noise ratio of frequency modulated (FM) andwavelength modulated (WM) laser absorption spectrometers andtheir ratio as a function of number of passes in a multipass cell forlaser power of 521 /AW. The input parameters are the same as in
Figs. 3 and 4.
pared for a laser with a power of 521,gW. The ratio ofSNRFM and SNRWM is also plotted as a function of thenumber of passes. The ratio shows the highest value,-100, at very short path lengths. The ratio corre-sponds to the calculated improvement in sensitivity byapplication of the FM technique.s Given a practicalnumber of passes, however, the sensitivity improve-ment is substantially smaller. At an optimal numberof passes for the FM technique of -60, the SNRFM isonly about 40 times better than SNRWM; and when theFM and WM techniques are each operated at an opti-mal number of passes, introduction of the FM tech-nique will only improve the SNR by about a factor of14. The optimal number of passes for the FM tech-nique is only about one-third of the optimal number ofpasses for the WM technique. In practical terms, cellswith a lower number of passes can be built more cheap-ly and adjusted more easily than cells designed for ahigh number of passes.
Another interesting aspect can be derived from Fig.5. If a conventional derivative multipath spectrome-ter is equipped with FM modulation and detectioncircuitry and the adjustment of the absorption cell isnot changed, no substantial sensitivity gain will befound. To achieve better sensitivity with the FMtechnique the number of reflections must be reduced.
The SNR for WM and FM techniques are shown inFigs. 6 and 7, respectively, as a function of the reflectiv-ity of the mirrors. For this calculation the same pa-rameters were used as for Fig. 5. In both cases theoptimal number of passes and the achievable SNRincrease rapidly with the improvement of mirror re-flectivity. Improving the reflectivity from 96.5 to98.5% improves the achievable SNR in both cases byalmost a factor of 2.5, and it increases the optimalnumber of passes by a factor of -2.5. This resultstresses the need for high reflectivity mirrors in multi-pass systems for trace gas analysis.
_ The smaller number of passes required for optimalW SNR in FM spectrometers has other practical conse-w quences. Cells with a smaller number of passes are
cheaper to construct and to adjust. Since they can bedesigned with smaller volume, faster measurementscan be made. This property is especially importantbecause only in cells with a small volume can the po-tential high speed of FM measurements be utilized.
If, on the other hand, ultrasensitive measurementsrequire long absorption path length, high reflectivitymirrors and high power lasers are a prerequisite to getthe full potential of FM techniques.
This work was carried out with support from theGerman Ministry of Research & Technology and it is
600 part of EUROTRAC, an environmental program ofEUREKA.
Fig. 6. Siabsorptior
103 10
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Fig. 7. Siabsorption
IV. ConThe si
used foried usinjderivatiiconditio:mal nurnto be mufor WMboth tec]of passerprove thmultipa,Althoug]lower thprovemEwithout
g11a1-tU-HULe ra11o 01 a wavelength mQUuaL11Q VY IVIl) laser Referencesnspectrometer as a function of number of passes at differ-
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