12W. R. Hunter and G. Hass, "Thickness of absorbing films necessaryto measure their optical constants using the reflectance-vs-angle-of-incidence method," J. Opt. Soc. Am. 64, 429-433 (1974).
13 W. R. Hunter, "Errors in using the reflectance vs angle of incidencemethod for measuring optical constants," J. Opt. Soc. Am. 55,1197-1204 (1965).
14 K. Siegbahn, C. Nordling, A. Fahlman, R. Nordberg, K. Hamrin,J. Hedman, G. Johansson, T. Bergmark, S.-E. Karlsson, I. Lindgren,
and B. Lindberg, in ESCA: Atomic, Molecular and Solid StateStructure Studied by Means of Electron Spectroscopy, (Almquist& Wiksell, Stockholm, 1967).
15A. Seignac and S. Robin, "Proprietes Optiques de Couches Mincede Platine dans L'Ultraviolet Lointain," Solid State Commun. 11,217-219 (1972).
16 R. C. Linton, NASA Technical Note TN D-7061, October 1972(unpublished).
Optoacoustic measurement of optical absorption in acetylenesmoke
D. M. Roessler and F. R. FaxvogPhysics Department, General Motors Research Laboratories, Warren, Michigan 48090
(Received 28 June 1979)
Optoacoustic measurements of acetylene smoke in a simple nonresonant cell at wavelengths of0.5145 and 10.6 ,um give specific absorption coefficients A A of 8.340.9 and 0.76±0.1 m2 g-1, re-spectively. In the visible region, about 85% of the total optical attenuation is due to absorption. Thesedata are consistent with a simple model in which the smoke is comprised of agglomerated carbonspheres. The experimentally determined responsivity of the optoacoustic cell is in good agreementwith theory.
INTRODUCTION
Optoacoustic spectroscopy is now widely recognized asa useful tool. Examples of its applications appear in a recentbook edited by Pao.1 The high sensitivity has made it par-ticularly important in monitoring low concentrations of ma-terial.2' 3 Other applications have included the studies of re-laxation times4 of molecules and the measurement of aero-sols.
5 ,6 In the present work, however, we are primarily con-cerned with its use in the absolute measurement of opticalabsorption.
Simple transmission measurements are often adequate todetermine the total optical attenuation or extinction in ma-terials. However, the relative contributions of light absorp-tion and scattering to the total attenuation are usually difficultto establish. The optoacoustic effect is intriguing in that, inprinciple, it depends on only the absorbed fraction of light andcan, therefore, be used to measure the latter. It is simple todemonstrate that if an optoacoustic signal is generated by ablack smoke having some given transmission, then a "white"smoke (such as cigarette smoke or water vapor in the visibleregion) having the same transmission will give a much weakeroptoacoustic signal. The relevant factor is that for carbona-ceous smokes the attenuation is determined primarily byabsorption, while in the "white" smokes the dominant processis scattering. In this paper we describe an optoacoustic de-termination of the absorption in acetylene smoke in the visibleand infrared spectral regions, and discuss the applicability ofMie theory to such smoke.
THEORY
Excellent descriptions of the optoacoustic effect can befound in the literature, for example in the book edited by Pao.'
For the purposes of the work reported here a very simple ac-count is adequate.
Consider a sample chamber containing an optically ab-sorbing material such as black smoke. Absorption of light bythe smoke will generate heat and subsequently produce apressure rise in the sample chamber. Modulation of the in-cident light beam at audio frequencies allows a microphoneto be used to monitor the resultant pressure fluctuations asan acoustic signal.
For the simple case of nonresonant operation (such that themodulation frequency is not near an acoustic resonance of thesample chamber) and where we assume that the pressure riseis uniform throughout the cell, it can be shown (see Appendix)that the optoacoustic signal S can be expressed as
S = R(bA/bE)[1 - exp(-bEL)IW/L, (1)
where W is the optical power incident on the sample containedin a cell of length L, bE is the extinction coefficient of thesample, bA is the absorption coefficient, and R is the respon-sivity of the cell. The cell responsivity is a function of the cellgeometry, the chopping frequency, the microphone sensitivity,and the thermal properties of the gas in the cell (usually airin the case of smoke samples).
For the case of weak attenuation such that bEL << 1, Eq. (1)reduces to
S - RAAMCW, (2)
where Mc is the sample mass concentration and AA the spe-cific absorption given by AA = bA/MC-
In the present work we have considered a wide range of massconcentrations, and from Eq. (1) we see that, once the cellresponsivity R has been determined, the fractional absorption
FIG. 1. Schematic of optoacoustic measurement apparatus (not toscale).
of the sample bA/bE can be obtained by measuring the opto-acoustic signal as a function of sample mass concentration.Further, if the attenuation is weak, then the specific absorp-tion AA itself can be derived via Eq. (2). The response itselfcan be determined by calibrating the cell with a strongly ab-sorbing gas of known extinction coefficient, and then applyingEq. (2). In this case scattering may be neglected and we canset bA/bE z 1.
EXPERIMENTAL
A schematic of the apparatus is given in Fig. 1 (not to scale).The sample cell consists of a brass cylinder of 100-mm lengthand 50-mm diameter through which a hole of 10-mm diameterhas been bored. Small ports are provided to permit aerosolor smoke samples to be pumped through the cell. The endsof the cell are closed by windows of quartz for work in thevisible and of zinc selenide for measurements in the infraredregion. An argon ion laser was used to provide 0.5145-Amwavelength radiation and a cw CO2 laser used at 10.6 ,um. Amechanical chopper provided square-wave modulation of thelaser at 500 Hz. The pressure changes in the cell which arisefrom the periodic laser heating were monitored by a smallhearing aid electret microphone (Knowles BT 1753) which wasmounted flush with the inside wall. The microphone signalwas fed to a lock-in amplifier (P.A.R. HR-8) whose outputvoltage was monitored on a chart recorder.
Smoke was generated in a 0.6 X 0.3 X 0.3-m container bythe fuel-rich combustion of an acetylene/air mixture in aburner comprised of a modified acetylene welding tip attachedto a gas mixing chamber. Steady mass concentrations ofsmoke of up to 250 mg m- 3 were pumped to the optoacousticcell from this smoke generator by controlling the gas mixingratio. In practice, mass concentrations of 100 mgm- 3 or moregradually contaminate cell windows and contribute unwantedbackground signals, but were included in the present work inorder to extend the mass concentration over a wide range.
The smoke was pumped through the cell at rates of 0.1 to1.0 Lmin-' (chosen as a compromise between excessive flownoise and fast time response). A 47-mm diameter Nucleporefilter of 0.2-gm pore size was placed immediately after the cellto collect the smoke particles. The mass concentration Mcwas then determined from the weight of the collected soot andthe volume of air pumped through the cell as monitored by awet test meter. Typically a few tenths of a milligram of sootwas collected for each experiment. The mass concentrationthus determined will be less than that in the optical path of
1700 J. Opt. Soc. Am., Vol. 69, No. 12, December 1979
the cell to the extent that some loss of particles occurs on thecell walls and in the connection to the filter. However, suchlosses were estimated to be less than the uncertainty in themass determination (±5%), by comparing the amounts of sootcollected when the filter was placed first at the entrance andthen at the exit port of the cell.
A power meter was used to measure the laser power trans-mitted by the cell. For the work in the visible region, thespecific extinction of acetylene smoke is sufficiently high7 thatit can be measured directly for the higher mass concentrationsby simply observing the transmission change and applyingBeer's Law, i.e., IT = Io exp(-bEL).
The cell was calibrated in the visible region with a mixtureof NO2 in air as this gas has appreciable absorption at theargon laser wavelengths. Terhune and Anderson 6 found anabsorption coefficient at 0.5145 gm of about 6 X 10-4m- ppm-' for NO2 at atmospheric pressure. The maximumabsorption measured in our optoacoustic cell (L = 0.1 m) wasabout 0.9 m- 1 . An absorption coefficient of 6 X 10-4m-1 ppm-1 thus implies that our maximum gas concentrationwas about 1500 ppm. By varying this concentration, and si-multaneously measuring both the optoacoustic signal S andthe transmission loss in the cell, we generated the data shownin Fig. 2. The optoacoustic signal is therefore calibrated interms of the extinction coefficient bE, and the cell responsecan be determined.
A best fit of Eq. (1) to the data of Fig. 2, with bA bE forNO 2 in air, gives R = 13.6 ± 0.5 mV/(m-' W). For the con-centrations of NO2 used for calibration, the thermal propertiesof the cell medium are essentially unchanged from those of air.The cell response R can then be written (see Appendix forderivation) as
R = 4(y - 1)uML/7rcoV , (3)
where y is the ratio of the specific heats of air (taken to be1.403), (eM is the microphone sensitivity (typical manufac-turer's specifications give aM = 10 mV Pa-' at a frequency of500 Hz), V is the cell volume, and w is the angular chopping
5S
a
0
0
1.2
1.0
0.8
0.6
0.4
0.2
0 0.2 0.4 0.6 0.8 1.0
EXTINCTION COEFFICIENT bE (m1m
FIG. 2. Optoacoustic signal of various concentrations of nitrogen dioxidein air as a function of the extinction coefficient at 0.5145 Aim.
D. M. Roessler and F. R. Faxvog 1700
ZC
C.1)
0
2.5
2.0
1.5
1.0
0.5 Ad A = 0.5145pI. m
00 50 100 150 200 250
MASS CONCENTRATION Mc (mg m 3)
FIG. 3. Optoacoustic signal of acetylene smoke at 0.5145 gm as afunction of smoke mass concentration. The data scatter is associatedprimarily with the mass determination.
frequency. For our cell length L of 0.1 m and diameter 10 mmwe thus have R = 14.7 mV/(m- 1 W). The agreement betweentheory and experiment is thus remarkable, particularly whenone realizes that the volume V is slightly greater than that ofthe cylinder (7rLD2/4) used in the calculation because of thesmall cavities corresponding to the entrance and exit ports andto the microphone. With R determined we are now able todetermine the fractional absorption of acetylene smoke on thebasis of Eq. (1).
ABSORPTION AT 0.5145-lim WAVELENGTH
The optoacoustic signal was measured as a function of theacetylene smoke mass concentration and the results are shownin Fig. 3. The laser power shown (100 mW) is the averagepower incident on the smoke, and was determined via thetransmitted power of the empty cell corrected for the windowlosses. The mass concentration range covered in Fig. 3 is largeenough that the nonlinearity with signal is significant. Thedata scatter primarily reflects the uncertainty in the soot massdetermination (from filter weighing before and after deposi-tion).
The data of Fig. 3 can be matched to Eq. (1) provided onealso knows the value of bE. We separately determined 7 thespecific extinction AE = bE/MC of acetylene smoke to be 9.8+ 0.8 m2 g-1 and checked this at the higher mass concentra-tions by monitoring the transmission through the optoacousticcell. Using Eq. (1) we thus obtain RbA/bE = 11.48 i 0.8mV/(m 1l W), and hence bA/bE = 0.84 + 0.09 (since R = 13.6± 0.5 mV/(m-1 W)). Since AE = 9.8 m2 g-1 we also obtain thespecific absorption AA = 8.3 a 0.9 m 2 g 1-. Thus for acetylene
smoke more than 80% of the total light attenuation at0.5145-Mrm wavelength is due to absorption rather than scat-tering.
It is instructive to compare this experimental value withwhat is predicted by applying Mie theory to a carbonaceoussmoke. We take a very simple model which assumes thesmoke to consist of small unagglomerated carbon spheres
having a log-normal mass distribution m(D), where D is thesphere diameter. If the extinction cross section per unitparticle mass 8 is RE(D) for a given sphere, then the totalspecific extinction AE is simply
AE = (1/MC) f RE(D) m(D) dD. (4)
Corresponding formulas apply for the specific absorptionAA and the specific scattering As. The extinction cross sec-tion per unit particle mass RE is directly related to the ex-tinction cross section CE by
RE = CE/[p(4/3)r(D/2) 3], (5)
where p is the particle density. The computation of CE fromthe Mie theory is straightforward.
In Fig. 4 we present the computed values of the specificextinction, absorption, and scattering for spheres having acomplex refractive index m = 1.75 - 0.5i, density p = 1 g cm- 3 ,
and the size distribution indicated. In our notation, the massdistribution m(D) has its maximum value at D3 and thenumber distribution has its peak at the particle diameter D1 .The geometric mean standard deviation a is defined as thestandard deviation of logjOD and is zero for a singly sizeddistribution. The relationship between the mean particlediameter D1 and the mean mass diameter D3 is given bylogjo(D 3 /D 1) = 6.91 a2.
The choice of refractive index and density in Fig. 4 issomewhat arbitrary, although the value of a is quite typicalof carbon aerosols. The specific extinction AE of variousblack smokes covers the range from about 4 m 2 g-1 (Ref. 9) toabout 10 m2 g 1 or above (Ref. 7 and references therein).Thus for small spheres (D1 < 0.05 gm), this simple model cansimulate the optical properties of black smokes reasonablywell. Note also that for such spheres Fig. 4 indicates that onlya small fraction of the total extinction is due to scatteringprocesses.
Mean Particle Diameter Di (pm)
0.01
E
0
U
Yl
12
8
4
00.01
0.10
1 .000.10
Mean Mass Diameter D3 (pm)
FIG. 4. Theoretical variation of the specific absorption, scattering andtotal extinction of spheres of density p = 1 g cm-3 and refractive index m= 1.75-0.5iat X = 0.5145 Am as a function of their size distribution. Themean diameters D1 and D3, and the geometric mean standard deviation arare defined in the text.
1701 J. Opt. Soc. Am., Vol. 69, No. 12, December 1979 D. M. Roessler and F.R. Faxvog
1.0
ze
0
C
C
0.8
0.6
0.4 L-0.01 0.10
MEAN MASS DIAMETER D3 (jIml
1.0
FIG. 5. Theoretical dependence of the fractional absorption, bA/bE, at0.5145 ,m of carbon smokes as a function of smoke particle size distri-bution. Data are presented for singly-sized particles (a = 0) and verydisperse systems (a = 0.4). The refractive index of m = 2.0-1.0i is ap-propriate for a model of small graphite-like spheres; that of m = 1.56-0.47i(Ref. 12) may be appropriate for agglomerated spheres (see text).,
Figure 5 illustrates this feature more explicitly and removesthe assumptions about particle density by presenting thecomputed ratio bA/bE (or AA/AE). Furthermore, we see thatfor a given size distribution the choice of the refractive indexin the calculations is not critical and that for mean mass di-ameters D i less than about 0. 1,im, more than 80% of the totallight extinction is due to absorption.
We have commented elsewhere 7 on the possible effects ofagglomeration of such spheres on the optical properties. Inparticular, we assume that the unagglomerated spheres havea refractive index close to that of graphite,1 0 say 2.0-1.Oi, butthat when clustered into more loosely packed larger spheresthe effective refractive index is lowered." The value of m= 1.56-0.47i reported by Dalzell and Sarofim12 for acetylenesoot may correspond to such a situation. Experimental de-termination of the size distribution for some carbonaceoussmokes indicate that D:3 is usually about 0.1 Am and invariablyless than 0.15 gm. 1 3,1 4 Thus from Fig.5 we see that whetherone includes agglomeration or not, the Mie theory predictsthat, at a wavelength of 0.5145 Am, the fractional absorptionbA/bEg of carbonaceous smokes is unlikely to be less than 0.7and may be considerably higher. Thus the simple model ofcarbonaceous spheres predicts a fractional absorption con-sistent with that determined experimentally, i.e., 0.84 + 0.09at 0.5145,im.
ABSORPTION AT 10.6-,lm WAVELENGTH
For the measurements at 10.6 gm, the CO2 laser was usedinstead of the argon laser, and ZnSe windows replaced thequartz windows on the cell. Although the responsivity of thecell is not a function of the cell window material, according tothe simple deviation given in the Appendix, we occasionallynoted discrepancies in cell response following removal andreplacement of windows. This was attributed to defectivesealing of the windows, but confirmed the need for separatecalibration of the cell at 10.6 gm to check seal integrity.
Many common organic solvents have high absorption at10.6 gm so that their vapor produces appreciable optical at-tenuation in even a short cell such as that used here. We,therefore, introduced trichloroethylene vapor and simulta-neously monitored the optoacoustic signal and the opticaltransmission. This allows the extinction coefficient bE of thevapor to be determined via Beer's Law. The results are shownin Fig. 6, and applying Eq. (1) as in the case of NO 2, we findR = 13.6 + 0.3 mV/(m-1 W), in excellent agreement with thecalibration at 0.5145 ,m. The nonlinearity of Eq. (1) is moreapparent in Fig. 6 than in Fig. 2 because of the larger value ofbE in the former.
Although very large extinction coefficients were measured,the concentration of the trichloroethylene is still too small toaffect the cell response via a change in the specific-heat ratio-y. The vapor pressure of trichloroethylene is about 70 mmHg at room temperature,1 5 so that the maximum concentra-tion in the cell is less than 10%. If we assume the ratio ofspecific heats for pure trichloroethylene to be 1.250, then amixture of 10% in air would have -y 1.39 which is little dif-ferent from that of air (1.40). Further, at the maximumconcentrations of the vapor in the cell, the extinction coeffi-cient was found to be over 50 m-1 corresponding to opticaltransmission of less than 1%. For the values of bE shown inFig. 6, the concentration of trichloroethylene vapor is thereforewell under 10% and does not affect the cell response. Fromthese data the cell responsivity was found to be 13.6 + 0.3mV/(m'1 W), in excellent agreement with the NO 2 calibration(R = 13.6 mV/(m'1 W).
Acetylene smoke has a smaller specific extinction at 10.6,gm than at 0.5145 Aum so that determination of bE fromtransmission measurements in a short cell is difficult.However, the fact that bEL << 1 allows us to use the relation-ship given in Eq. (2) to describe the dependence of opto-
200
150
5C1
C
C
100
50
U 2 4 6 8
EXTINCTION COEFFICIENT b (m I;
FIG. 6. Calibration curve for the optoacoustic cell at 10.6-,m wavelength,with trichloroethylene vapor as the absorbing sample. The cell responsivityR is 13.6 + 0.3 mV/(m-1 W).
I). M. Roessler and F. R. Faxvog 1702
E
"'4
0
0
0 100 200
MASS CONCENTRATION Mc (mg m 3)
FIG. 7. Optoacoustic signal of acetylene smoke at 10.6 Aim as a functionof smoke particle mass concentration. For the mass concentration rangeshown, the plot is approximately linear.
acoustic signal on smoke mass concentration. The data at10.6 Am are shown in Fig. 7. Using the experimentally de-termined cell response R = 13.6 i 0.3 mV/(m IW) andapplying Eq. (2) we find AA = 0.76 i 0.1 m 2 gel. The con-tribution of any CO2 absorption was checked by placing thefilter at the entrance to the cell and was found to be negligible.There appear to be little data in the literature with which tocompare this directly, but Volz16 has measured the spectraltransmission of a number of dusts and soots. For the latterhe shows AE - 0.5 m2 ga1 near 10 Am. In principle the par-ticles comprising the smoke may be considered so smallcompared to the light wavelength (10.6,4m) that they may beregarded as Rayleigh absorbers and hence their extinctionshould be almost entirely determined by their optical ab-sorption, i.e., AA/AE - 1. Thus our value of AA is some 50%higher than that of Volz. We note that independent mea-surements 7 of AE at 10.6 Am for acetylene smoke in a long-path cell gave a value of 0.94 + 0.1 m 2 g' .
We have used the Mie theory to calculate the specific ab-sorption for a distribution of small carbon spheres as a func-tion of wavelength. The results are shown in Fig. 8 togetherwith the experimental values reported in this work at 0.5145Aim and 10.6 jm. We have chosen a range of refractive indicesm believed to encompass those applicable to acetylene smoke.It will be appreciated, of course, that the refractive index isa function of wavelength (although only weak for carbon inthe visible and infrared) and that, therefore, none of the in-dividual curves in Fig. 8 for fixed values of m can be taken asthe actual variation of AA with wavelength.
As noted previously, the data of 0.5145,4m can be matchedreasonably well by the simple model of either agglomeratedor unagglomerated spheres. However, at 10.6 jm, the valueof 0.76 m2 gl1 cannot be matched for a particle density of 2gcm- 3 (helium pycnometry measurements give p = 2.05g cm 3 for the acetylene smoke particles). The data' 2 ofDalzell and Sarofim on acetylene soot suggest that m is about4.85 - 3.85i at 10.6 /Am, which gives AA = 0.13 m2 g-1 if p = 2g cm- 3. However, if agglomeration occurs the effective den-sity of the particle is lowered (because of the loose packing of
1703 J. Opt. Soc. Am., Vol. 69, No. 12, December 1979
the clustered spheres) and so is the effective refractive index.Either effect increases AA. Further, agglomeration thatproduces nonspherical shapes in the resultant clusters is alsoexpected to increase the value of AA over that from spheres. 7
Thus some form of agglomeration must be invoked to obtaina satisfactory match between theory and experiment for boththe visible and infrared regions.
Neither clustering nor nonsphericity is expected to alter thepredominance of absorption rather than scattering for car-bonaceous smoke at 10.6 gm, so that AE should have aboutthe same value as AA. In the present case we have AA/AE =
0.76/0.94. The uncertainties in the determination of both AA
and AE are large (more than 10%) so that it is not clear if a realdiscrepancy exists. Extinction measurements on diesel ex-haust particulates1 7 at 10.6 gm gave a value of AE = 0.83 + 0.2m 2 g'-, similar to that for acetylene smoke.
CONCLUSIONS
We have been able to use the optoacoustic effect in a smallnonresonant cell to measure the specific absorption AA ofacetylene smoke at 0.5145 and 10.6 Am. In the visible region,about 85% of the total light extinction by the smoke is due toabsorption. At 10.6 jm the uncertainty in the determinationof the absorption and extinction does not permit us to estab-lish whether indeed the smoke particles act as true Rayleighabsorbers, or if there is still non-negligible scattering. Butoverall, the experimental data suggest that agglomeration ofthe individual smoke particles is probably significant in de-termining the optical properties.
ACKNOWLEDGMENTS
It is a pleasure to thank T. VanSteenkiste of the GMRPhysics Department for his expert technical assistancethroughout this work. We also appreciated very helpful in-teractions with C. W. Bruce at the White Sands Missile Range,
20
10
0
0
0.1 L0.1 1.0 10 20
WAVELENGTH (hm)
FIG. 8. Mie theory calculation of the specific absorption as a function ofwavelength for small spheres of mean particle diameter D1 = 0.04 gim anddensity p = 2 gcm- 3. The refractive index m of acetylene smoke isprobably less than 2.0-1.0i in the visible region but may be as high as5.0-4.0i(Ref. 12) at 10 jim. The experimental values of AA found in thiswork are also shown.
D. M. Roessler and F. R. Faxvog 1703
New Mexico, when we first contemplated using the opto-acoustic effect.
APPENDIX: DERIVATION OF OPTOACOUSTICCELL RESPONSE
Neglecting the effects of acoustic losses produced by heatconduction and viscosity, we use the wave equation
I 1 (2p (by -1) (H12 p _ (yl _Hc2 (t 2 U2 b
The linear term in the time dependence simply represents thedc heating of the cell and is ultimately balanced by the cellheat losses which have been neglected in Eq. (Al). The si-nusoidal voltage produced by the microphone will be ampwhere (,, is the microphone sensitivity (mVPa-1) at thechopping frequency f. This signal, fed to a lock-in amplifierset at the fundamental frequency, produces a dc outputvoltage which is the rms value of the input, i.e., armp/V/.This final output voltage S is thus, from Eq. (A7),
(Al)
Here, we assume that the absorbed light is equivalent to a heatsource H(r,t) for the subsequent change in acoustic pressurep. The symbol c is the velocity of sound in the samplechamber medium and -y is the ratio of the specific heats of thismedium (cp/c0 ).
For the cell used in the present work, and at modulationfrequencies well away from the acoustic resonances of thechamber, we further assume that the pressure rise is uniformthroughout the cell volume V. Equation (Al) thus be-comes
at = ('- 1)H.Ci
(A2)
The heat generated in a volume element Adz is given by
dH = bAI(z) Adz, (A3)
where 1(z) = Ioexp(-bsz) is the light beam intensity at adistance z into the sample, bA and bE are the absorption andextinction coefficients, respectively, and A is the cross sec-tional area of the light beam. The average heat generated perunit volume thoughout the cell is SdH/V, or
H = (AIV)(bA/bE)[1 - exp(-bEL)H, (A4)
where L is the cell length. We assume that negligible heat isgenerated by cell wall absorption of any light scattered fromthe main beam.
The chopper produces a square-wave time dependence forthe laser beam intensity and we express the latter as its Fou-rier sum
Here, l0 is half the peak intensity (i.e., the average intensity)and w is the angular chopping frequency 27rf.
Substituting for I(t) in Eq. (A4) gives
H A( -e-bEL) 1 + 4 E sin[(2n - 1)wt] AVbE 7r n=1 (2n-1)
where we have put the laser average incident power W =
A&o
Integrating Eq. (A2) and substituting for H from Eq. (A6)thus gives
PW=(-y - l) W b, (1 , cbEL)p (t ) = -- 1-e~EV bE
X ~t _ 4 cos(2n -i1)ct (A7)
S - 4(- 1)km bA (1 - e-bEL)W.N/-9rwV BE
It is convenient to express this in the form
S = R(bAI/bE)(1 - e-bEL)W/L,
(A8)
(A9)
where R may be termed the responsivity of the cell and hasthe units mV/(m-1 W). The remainder of the expression onthe right-hand side of Eq. (A9) is simply the fraction of inci-dent power absorbed by the sample per unit cell length. Fora given cell geometry, microphone sensitivity, and choppingfrequency, R is a constant (provided the sample examined isnot of such high concentration that it alters the specific heatsratio of the cell medium, usually air).
'Optoacoustic Spectroscopy and Detection, edited by Y.-H. Pao,(Academic, New York, 1977).
2 L. B. Kreuzer, "Ultralow gas concentration infrared absorptionspectroscopy," J. Appl. Phys, 42, 2934-2943 (1971).
:C. K. N. Patel and R. J. Kerl, "A new optoacoustic cell with improvedperformance," Appl. Phys. Lett. 30, 578-579 (1977).
4P. V. Slobodskaya and N. F. Tkachenko, "Study of the RelaxationTime for the Vibrational Energy of the NO Molecule by Means ofthe Spectrophone," Opt. Spectrosc. 29, 138-142 (1970).
SC. W. Bruce and R. G. Pinnick, "In-situ measurements of aerosolabsorption with a resonant cw laser spectrophone," Appl. Opt. 16,1762-1765 (1977).
(R. W. Terhune and J. E. Anderson, "Spectrophone measurement ofthe absorption of visible light by aerosols in the atmosphere," Opt.Lett. 1, 70-72 (1377).
7 D. M. Roessler and F. R. Faxvog, "Optical properties of acetylenesmoke at 0.5145 um and 10.6 pum wavelengths," J. Opt. Soc. Am.(to be published).
8F. R. Faxvog and D. M. Roessler, "Carbon aerosol visibility vs particlesize distribution," Appl Opt. 17, 2612-2616 (1978).
9W. F. Stoecker, "Smoke Density Measurement," Mech. Engr. 72,793-798 (1950).
"'R. Tsu, H. J. Gonzalez, and I. C. Hernandez, "Observation ofSplitting of the E2e Mode and Two-Phonon Spectrum in Graph-ites," Solid State Commun. 27, 507-510 (1978).
l IS. C. Graham, "The Refractive Indices of Isolated and AggregatedSoot Particles," Combustion Sci. Technol. 9, 159-163 (1974).
'2 W. H. Dalzell and A. F. Sarofim, "Optical Constants of Soot andTheir Application to Heat-Flux Calculations," J. Heat Transfer91, 100-104 (1969).
':p. J. Groblicki and C. R. Begeman, "Particle Size Variation in DieselCar Exhaust," SAE Paper 790421, February 1979.
"D. F. Dolan and D. B. Kittelson, "Diesel Exhaust Aerosol ParticleSize Distributions-Comparison of Theory and Experiment," SAEpaper 780110, February 1978, pp. 1-7.
' 5Handbook of Chemistry and Physics, 58th Ed. (CRC, Cleveland,Ohio, 1977) p. D-192.
16 F. E. Volz, "Infrared Absorption by Atmospheric Aerosol Sub-stances," J. Geophys. Res. 77, 1017-1031 (1972).
17F. R. Faxvog and D. M. Roessler, "Optoacoustic Measurement ofDiesel Particulate Emissions," J. Appl. Phys. (to be published).
1704 J. Opt. Soc. Am., Vol. 69, No. 12, December 1979 D. M. Roessler and F. R. Faxvog 1704
