ARMY RESEARCH LABORATORY 

Active Scattering Aerosol 

Spectrometer Probes Made by 

2000050~ 127 : 

Approved for public release; distribution unlimited. 

‘D!i’IC QUALITY IHSPECTED 8 

L_ 
The findings in this report are not to be construed as an 

official Department of the Army position unless so 

designated by other authorized documents. 

Citation of manufacturer’s or trade names does not 

constitute an official endorsement or approval of the use 

thereof. 

Destroy this report when it is no longer needed. Do not 

return it to the originator. 
Army Research Laboratory 

Adelphi, MD 20783-1197 

ARL-TR-1973 

Response Characteristics of 

Active Scattering Aerosol 

Spectrometer Probes Made 

Particle Measuring Systems 

March 2000 

by 

Ronald G. Pinnick, J. David Pendleton, and Gorden Videen 

Information Science and Technology Directorate 

Approved for public release; distribution unlimited. 
Abstract 

Predictions are reported of the size response of various light-scattering 

aerosol counters manufactured by Particle Measuring Systems. Models 

considered are those that exploit the high intensity of light available within 

the cavity of a HeNe gas laser (generically referred to by the manufacturer as “active scattering aerosol spectrometer probes”). The new response 

function properly averages over particle trajectories through nodes, antinodes, and intermediate regions of the intracavity laser beam. Our studies 

address probes having two basic scattering geometries: those that collect 

light scattered over a relatively narrow solid angle (subtending angles between 4” and 22” from the laser beam axis) and those that collect light over 

a rather large solid angle (between 35” and 120”). The new response function predicts smoother dependence on particle size than the previous response function of Pinnick and Auvermann (1979, J. Aerosol Sci. 10: 55-74) 

and is in better agreement with measurement. Response calculations for 

common atmospheric aerosol (water, sulfuric acid, ammonium sulfate, and 

black carbon) reveal the considerable sensitivity of the response to particle 

dielectric properties. Comparison of response calculations with the manufacturer’s calibration reveals conditions for which the manufacturer’s calibration is most appropriate, and the potential for errors (as much as a factor 

of two in sizing) when it is blindly applied. These results should help the 

user of these instruments to more realistically interpret size distribution 

measurements. 
Contents 

1 Introduction 

2 Theoretical Response Function 

3 Response Measurements 

4 Response Predictions for Homogeneous Particles 

5 Response Predictions for Inhomogeneous Particles 

6 Comparison of New Response Function to PAG Theory 

7 Predictions of Response for Multiline Laser Operation 

8 Conclusion 

Appendices 

A 

B 

Theoretical Response Function for Intracavity Laser Probes 27 

A-l Intracavity Scattering ....................... 27 

A-2 Corrected PAG Response Function ............... 32 

A-3 New Response Function ..................... 32 

A-4 Response Function for Multiline Lasing ............ 34 

Normalization of Measured Response to Theory 

Acknowledgments 

1 

5 

7 

11 

15 

19 

23 

25 

27 

37 

39 

. . . 

111 
References 

Distribution 

Report Documentation Page 

iv 

41 

45 

47 
Figures 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

12 

13 

Comparison of theoretical response and measured response 

for PMS narrow-angle scattering aerosol probes . . . . . . . 8 

Same as figure 1, except for PMS wide-angle scattering aerosol 

probes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 

Predicted response for PMS narrow-angle scattering probes 11 

Same as figure 3, except for PMS wide-angle scattering probes 

and also Royce model 236 . . . . . . . . . . . . . . . . . . . . 12 

Relation between PMS-indicated particle radii and predicted 

particle radii for PMS ASASP- and ASASP-300X probes . 13 

Same as figure 5, except for PMS ASASP-X, ASASP-100X, 

LAS-X, LAS-250X, and HS-LAS probes . . . . . . . . . . . . . 13 

Predictions of response for PMS narrow-angle scattering probes 

to inhomogeneous particles . . . . . . . . . . . . . . . . . . . 16 

Same as figure 7, except for PMS wide-angle scattering probes 

and also Royce 236 . . . . . . . . . . . . . . . . . . . . . . . . 18 

Relation between PMS-indicated particle radii and predicted 

particle radii for narrow-angle scattering probes . . . . . . . 20 

Same as figure 9, except for PMS wide-angle scattering probes 21 

Relation between PMS-indicated particle radii and predicted 

particle radii for wide-angle scattering probes. . . . . . . . . 22 

Predictions of response for PMS wide-angle scattering probes 

based on new response function . . . . . . . . . . . . . . . . 22 

Predictions of response for PMS wide-angle scattering probes 

having lasers with sufficiently high gain for lasing in multiple lines. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 
1. Introduction 

Light-scattering particle counters are among the most widely used instruments for aerosol measurement. They have been employed for determining estimates of the tropospheric and stratospheric aerosol burden, for 

monitoring concentrations of particles in clean rooms essential for hightechnology manufacturing, for determining aerosol filter efficiencies, for 

detecting atmospheric aerosol pollutants, for monitoring particulates in 

work place environments, etc. 

These devices work as follows. Air containing particles is drawn through 

an illuminated volume, where light scattered by single particles is sensed 

and converted to an electrical signal whose pulse height is analyzed. The 

pulse height is used to infer particle size. The accumulated measurement of 

many particles results in a size distribution. Particle concentration is determined from total counts. Optical particle counters have the advantage of 

providing rapid, nonintrusive measurement of micrometer-sized aerosol 

particles, and can run continuously with only electrical power required for 

their operation. 

About 25 years ago, Particle Measuring Systems (PMS) in Boulder, CO, began making several laser-based light-scattering particle counters (Schuster 

and Knollenberg, 1972; Knollenberg, 1976) for use by the aerosol community. During the late 1970s and early 198Os, a number of these instruments 

became available. The manufacturer refers to these counters as laser aerosol 

spectrometers, with the implication that they are superior to other lightscattering particle counters. In some of these instruments (the ASASP-300, 

ASASP300X, ASASP-100X, ASASP-X, LAS250X, LAS-X, HS-LAS, a number of aircraft versions of these probes, and various one-of-a-kind models), 

particles are illuminated by the intracavity light from a TEMaa mode HeNe 

laser. The manufacturer generally designates these probes as “active cavity” (probes in which particles are illuminated outside the cavity are designated “passive cavity”). The active-cavity probes are the subject of this 

report. 
There are basically two scattering geometries used in the active-cavity 

probes. In the ASASP- and ASASP300X probes, the scattered light is 

collected by a lens over a relatively small and narrow solid angle subtending angles between 4” and 22” from the laser beam axis. In the more 

sensitive ASASP-X, ASASP-100X, LAS-250X, LAS-X, and HSLAS probes,* 

scattered light is collected by a parabolic mirror, allowing collection over 

a relatively large and wide solid angle subtending angles between 35” and 

120” from the laser beam axis. In both types of probes, the collection solid 

angle has symmetry about the laser beam axis. The “X” in the model designation denotes that the probe is plumbed with a relatively small inlet orifice 

(typically 200 pm in diameter), so that all particles drawn through it intersect the illuminating laser. Because of the high intensity available within 

the cavity (more than 1 W intracavity circulating power for most models 

and up to 25 W for the high-sensitivity HS-LAS model), particles as small 

as 0.1 pm in diameter can be measured. 

However, even with their novel design and considerable sensitivity, these 

instruments are not without limitations: in particular, their size resolution, 

which we address in this report. Their sizing limitations arise from a fundamental fact: the light scattered by a particle depends on the particle’s dielectric properties and shape as well as its size. Thus, unless particles of known 

refractive index are being measured, only an approximate determination of 

particle size can be made from the magnitude of its light-scattering signal. 

Roughly 50 ASASP-300,125 ASASP-X, 400 LAS-X, and 60 HS-LAS probes 

have been sold by PMS to date. Because of their wide use, the performance 

of these instruments has been studied rather extensively (Pinnick and Auverman, 1979; Pinnick and Rosen, 1979; Allan and Ashdown, 1982; Garvey 

and Pinnick, 1983; Solderholm and Salzman, 1984; Chen et al, 1984; Liu et 

aZ, 1985; Szymanski and Liu, 1986; Yamada et al, 1986; Hinds and Kraske, 

1986; Barnard and Harrison, 1988; Knollenberg, 1989; Jeung, 1990; Kim and 

Boatman, 1990; Pueschel et al, 1990; Hering and McMurry, 1991; Liu et al, 

1992; Knollenberg and Veal, 1992; Kim, 1995). 

Many of these studies concentrate on sizing performance where rather 

definitive measurements of response have been carried out on carefully 

prepared monodisperse aerosol test particles of different size, shape, and 

composition. Other studies have carried out theoretical investigations of 

particle response characteristics using the approximate response function 

*Also the Royce 236 counter, for which the optical system was manufactured by PMS 

but sold in the early 1980s by Hiac-Royce, Menlo Park, CA. 
proposed by Pinnick and Auvermann (1979) and Garvey and Pinnick 

(1983) (eq (2) of Pinnick and Auvermann, 1979, hereinafter referred to as 

the PAG response function). 

For the narrow-angle scattering probes (ASASP- and ASASP300X models), these studies reveal (1) a considerable sensitivity of the response to 

particle refractive index; (2) the existence of size regions where, because of 

Mie scattering resonances, particles of different size give the same response; 

and (3) degradation in size resolution for nonspherical particles because 

of the dependence of scattering on particle orientation. The same findings 

generally apply to the more commonly used wide-angle scattering probes 

(ASASP-X, ASASP-100X, LAS-X, LAS-250X, HS-LAS, and Royce 236 models). For both narrow-angle and wide-angle scattering probes, measurements of the response in the cited studies generally agree with the PAG 

response function, although the high-frequency oscillations predicted by 

the PAG response function were not found. 

Although the PAG response function accounts for particle index of refraction, it is only approximate, because it does not account for particles traversing nodes, antinodes, and intermediate regions of the intracavity standing 

wave of the laser, nor does it account for laser mode changes during particle transit. Averaging over particle trajectories and phase smooths the response function and could result in better agreement with measurement. 

For the LAS-X and the high-sensitivity HS-LAS probes, there may be additional smoothing caused by multiline lasing. In these probes, according 

to the manufacturer (Knollenberg, 1989), the laser used has sufficient gain 

to be above threshold for lasing in multiple lines. The PAG response function does not account for multiline lasing. Nor does it treat inhomogeneous 

particles. 

We provide here an improved response function, which correctly averages over particle trajectory through the cavity of a laser and accounts for 

the laser coherence time being shorter than the time of transit of particles 

through the laser beam. This response function may have been suggested 

previously at a conference (Soderholm and Salzman, 1984), but there is insufficient information in the conference proceedings to judge. The new response function was suggested previously by Garvey and Pinnick (1983) 

(their eq (2)) but not recommended by them. 

We compare the new response function, which is a smoother function of 

particle size than the PAG function, to previous response measurements 

of Pinnick and Auvermann (1979), Szymanski and Liu (1986), Garvey and 

3 
Pinnick (1983), and Chen et al (1984). The new response function is in better agreement with measurements of test aerosols compared to the PAG 

function. Thus, the measurements support the superiority of the new more 

general response function over the PAG response function. 

Here we provide response calculations for both narrow-angle and wideangle active-scattering probes, using the new response function, for both 

homogeneous and inhomogeneous particles characteristic of atmospheric 

and pollutant aerosol. For reference, we compare the results to the manufacturer’s calibration, and to the previous PAG response function. 

Finally, we present response calculations for a multiline laser source that 

may be used in the HS-LAS or LAS-X probes. Contrary to the manufacturer’s claim, multiline operation has little effect on the probe response 

characteristics. 

The results should enable users of these probes to better assess their sizing 

limitations. 

4 
2. Theoretical Response Function 

Particles detected and sized in the active-cavity scattering probes are illuminated by the intracavity light from a TEMaa-mode HeNe laser. The 

particle cross section for light scattered into the probe collection aperture 

(defined as the probe response function R) should account for two averaging effects. First, because particles do not pass perpendicularly through the 

beam, they traverse nodes, antinodes, and intermediate regions (fringes) 

of the intracavity standing wave of the laser. Second, the coherence time 

of the laser is generally shorter than the particle transit time, causing the 

fringe positions to change during particle transit. In appendix A, we derive an expression for the response function that accounts for both of these 

effects. For spherical homogeneous particles, the result reduces to 

where k is the wavenumber, 0 is the scattering angle, and S1 (0) and S*(O) 

are the amplitude scattering matrixes (Bohren and Huffman, 1983). 

In appendix A we also derive a corrected expression for the PAG theory, 

which assumes that (1) the particle trajectory is perpendicular to the intracavity standing wave, (2) the particle passes through an antinode, and 

(3) the coherence time of the laser is longer than the particle transit time 

through the beam. The result is 

R-S s e rnax 

siildQ[]Si(B) + Si(7i - @I2 + ]S2(0) - S2(7r - O)l”]. (2) 

@min 

This response function is slightly different from the PAG response function: 

R2 .I Bn’ax sin0 dO[]S,(O) + Si(7r - Q)]* + ]S2(19) + S2(7r - @]*I. 

k” &,in 

(3) 

Finally, we address the issue of multiline lasing. Lasers in some of the 

active-cavity probes have high gain and lase on multiple lines. We show 

5 
in appendix A that for multiwavelength operation, the response function 

is a sum of the response functions at each wavelength weighted by the appropriate irradiance fraction in each wavelength. 

6 
3. Response Measurements 

A definitive test of the validity of the theoretical response function (eq 

(1)) requires measurement of well-characterized monodisperse test particles having a range of sizes and refractive indexes. For active-cavity probes 

there is an abundance of measurements available. Here we concentrate on 

the measurements of Pinnick and Auvermann (1979) for the ASASP-300, 

Garvey and Pinnick (1983) for the ASASP-X, Chen et nl(1984) for the Royce 

236, and Syzmanski and Liu (1986) for the ASASP-300X and LAS-X. 

We first consider the narrow-angle scattering probes (ASASP- and 

ASASP300X). Figure 1 shows the measured response to nonabsorbing 

polystyrene latex and dioctylphthalate (DOT’) particles and to highly absorbing nigrosin dye and carbon black particles, compared to the theoretical response function (eq (1)). For each probe, a single normalization constant relating response voltage to particle cross section is determined from 

a fit of the experimental data for polystyrene latex to the theoretical curve 

for latex. This fitting procedure is described in appendix B. 

The measurements confirm the predicted oscillatory character of the response curve for nonabsorbing latex and DOT’ particles, although the fine 

structure cannot be resolved. The measurements also generally corroborate 

the smooth monotonic response predicted for highly absorbing particles. 

As noted previously by Pinnick and Auvermann (1979), the fall-off in the 

observed response for highly absorbing nigrosin particles for radii greater 

than 1 pm is believed to be due to a reduction of laser power caused by 

the particles. The disagreement between carbon black measurements and 

theory could result from the difficulty in preparing homogeneous spherical 

particles (having refractive index 1.775 - 0.508i as used in the theory) from 

India ink. 

A similar comparison for several models of the wide-angle probes (the 

ASASP-X, LAS-X, and Royce 236) is shown in figure 2. Again, we emphasize that we determined a single normalization factor for each probe by 

doing a weighted fit of the polystyrene latex measurements to the theoretical response. The theoretical response is generally within the error of measurement, or what appear to be errors in measurement, as evidenced by 

7 
Figure 1. Comparison 

of theoretical response 

(eq (1)) and measured 

response for PMS 

narrow-angle scattering 

aerosol probes. Theory 

and measurements are 

shown for 

nonabsorbing 

polystyrene latex and 

dioctylphthalate, and 

highly absorbing 

nigrosin dye and 

carbon black. Data sets 

are offset by 1 to 3 

orders of magnitude on 

vertical scale, as 

indicated. A single 

normalization factor for 

each probe relates 

experimental response 

voltage to 

particle-scattering cross 

section. Predicted broad 

Mie scattering 

resonance features are 

corroborated by 

measurements. 

8 

10-77 

1 O-8, 

10-9, 

3 IO-‘0, 

2 

r 

x 

2 

J2 lo-“= 

3 

0’ 

E 

B 

1 o-‘2, 

1 O-12 1 

1 O-12 I 

7 

4” to 22” collection 

- Theory 

0 

Measurements: 

/ 

/ 

00000 Szymanski & Liu (1986)/ASASP-300X 

l.OV= 1.0x lo-scm2 

/ 

. . . . . Pinnick & Auvermann (1979)/ASASP-300 

1 .O V = 5.8 x IO-9 cm2 

10-Q I / I I , , I I # I 

5 

I”‘, 1 

6 789 10-I 2 3 4 5 6789 2 

1 

Particle radius (pm) 
Figure 2. Same as 

figure 1, except for PMS 

wide-angle scattering 

aerosol probes. 

Predictions agree with 

measurements and 

reveal a generally 

monotonic increase of 

response with particle 

size for both 

nonabsorbing and 

absorbing particles. 

1 o-7 

1 

35” to 

lo-‘* 

1 o-1 &I_. 

v 

Measurements: 

t 

00000 Szymanski & Liu (1986)/LAS-S 

l.OV=6.6xlO-9cm2 

t 

l l l l l Garvey & Pinnick (1983)/ASASP-X 

1 .O V = 5.3 x 10-g cm* 

NUU~ Chen et al (1984)/Royce 236 

1 l.OV=1.51 x10-*cm* 

lo-‘* 

, ( , , , , I I 8 I II,,, I 

5 6 789 2 3 4 5 6789 2 

lo-’ 1 

120” collection 

DOP (1.4845 - Oi) 

(1.67 - 0.26i) 

Particle radius (pm) 

fluctuations in the data points where no error bars are given, for both nonabsorbing and absorbing particles. The theoretical curves and measured response data reveal an almost monotonic increase in response with increasing particle size, although the response increases slowly with increasing 

size for highly absorbing particles with 0.2 pm < T < 0.6 pm. 

Collectively, the comparison of measured response with the theoretical response for both narrow-angle and wide-angle scattering probes reveals 

9 
that (1) the response function (eq (1)) that properly averages over intracavity particle trajectories is in slightly better agreement with measurement 

compared to the previous PAG response function (eq (3)), and (2) the fine 

structure predicted by the response function (eq (1)) is not experimentally 

resolved. 

This latter finding is not surprising, since the amplitude of the fine structure is of the order of the pulse height channel widths in these probes, and 

thus we should not expect them to be resolved. Even the use of a pulse 

height analyzer with greater resolution may not resolve the fine structure 

because of basic limitations of these instruments (laser intensity variations 

for different particle trajectories, etc). 

We conclude that the theoretical response function (eq (1)) can adequately 

predict the response of both narrow-angle and wide-angle scattering probes 

to spherical particles, regardless of their dielectric properties, so long as 

they are of uniform composition. We provide such predictions for some 

common atmospheric aerosols in the next section. 

10 
4. Response Predictions for Homogeneous Particles 

Atmospheric aerosols can be irregular and of mixed composition. However, some occur as spherical and homogeneous particles. We consider four 

types of aerosol particles that are common in the earth’s atmosphere: water droplets, droplets composed of a 75-percent sulfuric acid/25-percent 

water mixture, ammonium sulfate particles, and particles of black carbon. 

In addition, for generality, we consider metal particles of copper. We initially assume all to be spherical and homogeneous. Figures 3 and 4 present 

response calculations for these aerosols for both narrow-angle and wideangle scattering probes. For reference, these calculations are compared to 

the manufacturer’s calibration, which is plotted on the particle-scattering 

cross-section scale after normalization by the factors determined in the previous section (from fitting the polystyrene latex measurements to theory). 

Figure 3. Predicted 

response for PMS 

narrow-angle scattering 

probes (ASASP- and 

ASASP300X). Curves 

for water, 75% sulfuric 

acid in water, 

ammonium sulfate, 

black carbon, and 

copper are shown. 

Response is a sensitive 

function of particle 

refractive index. 

Particles differing by a 

factor of two in radius 

can have same 

response. 

Manufacturer’s 

calibration for a 

ASASP- probe is 

given for reference. 

- l l l l *Manufacturer’s calibration 

l.OV=5.8x10-9cm2 

10-E : 

Theory 

- Water (1.332 - 0;) 

-- - Sulfuric acid (1.428 - 0;) 

- - - - Ammonium sulfate (1.520 -0;) 

------ Blackcarbon(1.950-0.66;) 

.-.---.---.- Metal (0.56 - 3.01;) 

5 6 7 89 2 3 4 5 6 7 8 9 2 

10-l 1 

Particle radius (pm) 

11 
Figure 4. Same as 

figure 3, except for PMS 

wide-angle scattering 

probes (ASASP-X, 

ASASP-100X, LAS-X, 

LAS-250X, HS-LAS 

models) and also Royce 

model 236. As for 

narrow-angle scattering 

probes, particles 

differing by a factor of 

two in radius can have 

same response. 

Manufacturer’s 

calibration is for an 

ASASP-X. 

1 O-7 3 35” to 120” collection 

I l l l l l Manufacturer’s calibration 

l.OV=5.3xlO-9cm* 

10-8: 

Theory 

- Water (1.332 - 0;) 

-_ - Sulfuric acid (1.428 - Oi) 

- - - - Ammonium sulfate (1.520 - Oi) 

- - - - - - Black carbon (1.950 - 0.66;) 

.-----.-.--- Metal (0.56 - 3.01 i) 

10-12 ’ 1 1 1 1 I I I 

5 6 7 8 9 2 3 4 5 6 789 2 

lo-’ 1 

Particle radius (pm) 

These results confirm what is already well known: the response for both 

types of probes depends on particle refractive index as well as size. Measurement of particles having a range of compositions (and refractive indexes) degrades the size resolution of the probes considerably, as particles 

of radii differing by as much as a factor of two can have the same response. 

As is evident from the figures, the manufacturer’s calibration passes 

through the family of curves that represent commonly occurring atmospheric aerosol constituents. Thus, for measurement of unknown aerosol 

that consists of particles of widely different composition (as for example in 

the earth’s tropospheric mixed layer), the manufacturer’s calibration, even 

though it divides size intervals into too many channels to be meaningful, 

might be appropriate for obtaining average size distributions. 

If the manufacturer’s calibration is used to infer particle size, the errors 

resulting from refractive index effects are more easily revealed in figures 5 

and 6, where we plot the PMS (manufacturer’s calibration) indicated radii 

versus the actual (theoretical response function) radii. For both probes, the 

PMS-indicated radius is generally in error by less than 50 percent. Note 

that for the narrow-angle scattering probes, the PMS-indicated radius can 

be multivalued. 

12 
Figure 5. Relation i 

between PMS-indicated 

particle radii (smooth 

curve through 

manufacturer’s 1 

calibration) and 9 

predicted particle radii 5 

8 

(theoretical response g 

; 

function) for PMS a 5 

ASASP- and !! 

$ 4 

ASASP-300X probes. 5 

Solid straight line is a p ” 3 

reference depicting z 

perfect agreement 5 2 

between indicated and 

predicted size. Curves 

for water, 75% sulfuric 

acid in water, 1 o-’ 

ammonium sulfate, 9 

a 

black carbon, and 7 

copper are shown. 6 

Errors in indicated radii 5 

are generally less than 

rt50%,. 

I- 

. . 

4 

5 

40 to 220 collection 

- Water (1.332 - 0;) 

-- - Sulfuric acid (1.428 - 0;) 

--_ - Ammonium sulfate (1.520 - 0;) 

------ Blackcarbon(1.950-0.66i) 

.....-..---- Metal (0.56 - 3.01i) 

8111, 

6 7 8 9 

1 o-’ 

I ,111, I 

2 3 4 5 6 7 6 9 

1 

2 

Particle radius (pm) 

Figure 6. Same as 

figure 5, except for PMS 

ASASP-X, ASASP-100X, 

LAS-X, LAS250X, and 

HS-LAS probes. Errors 

in PMS-indicated radii 

are generally less than 

zt50%,. 

35” to 120” collection 

- Water (1.332 - 0;) 

Sulfuric acid (1.428 - 0;) 

- - - - Ammonium sulfate (1.520 - Oi) 

5 6 7 a 9 2 3 4 5 6769 2 

1 o-1 1 

Particle radius (pm) 
These results give a comprehensive assessment of the response of the PMS 

probes to homogeneous spherical particles, but what about aerosol particles that are inhomogeneous? 

14 
5. Response Predictions for Inhomogeneous Particles 

For many applications, not only does the aerosol of interest consist of particles of different composition, but single particles may be composed of 

mixtures (black carbon in ammonium sulfate, black carbon in quartz, clay 

minerals in quartz, black carbon in water, sulfuric acid with crustal core, 

ammonium sulfate in sulfuric acid, silica shards in black carbon agglomerates, metals in sulfuric acid, etc). Such particles can have very complicated morphology (Gillette and Walker, 1977; Chylek et al, 1981; Pinnick et 

al, 1985; Sheridan and Musselman, 1985; Sheridan, 1989a, 1989b; Sheridan 

et al, 1993; Reitmeijer and Janeczek, 1997). To predict the response of the 

probes to internal mixtures, we modify the response function (eq (1)) by 

choosing the simplest possible model: that of a sphere containing a spherical inclusion (Ngo ef al, 1996). We consider four specific particle types: a 

water droplet containing a black carbon inclusion, a sulfuric acid droplet 

containing quartz, an ammonium sulfate particle containing carbon, and a 

sulfuric acid droplet containing metal. 

We consider the narrow-angle scattering probes first. To calculate the response function for black carbon in a water droplet sulfate host, we neglect the fractal cluster morphology of black carbon (Forrest and Witten, 

1979; Colbeck et al, 1989) and assume a solid black carbon core or inclusion with fixed radius of 0.1 pm enclosed within the host. We calculate the 

response in two ways. First we assume the core is located in the center 

of the host particle and increase the host size. These results are displayed 

by the solid curves in figure 7. Then we relax this assumption and let the 

inclusion be randomly positioned within the particle, and calculate the response for each position. The larger the host particle, the more possible 

positions are considered. The average of these response values is shown by 

the corresponding open circles in figure 7. The standard deviations of the 

response values for different inclusion positions are also calculated but not 

displayed. Results are calculated for only a small number of host particle 

sizes because of the considerable computational requirements. For reference, the manufacturer’s calibration is also shown. 

For simplicity, in calculating the response for other internal mixtures, we fix 

the inclusion size to be the same as for black carbon in water (i.e., inclusion 

15 
Figure 7. Predictions of 

response for PM’S 

narrow-angle scattering 

probes (models 

ASASP- and 

ASASP-300X) to 

inhomogeneous 

particles: black carbon 

inclusions in water, 

quartz in sulfuric acid, 

black carbon in 

ammonium sulfate, and 

metal in sulfuric acid. 

Solid curves are for a 

O.l-pm-radius inclusion 

at center of particle; 

open circles denote 

average response 

values for inclusion 

positioned randomly 

within particle. Data 

sets are offset by 1 to 3 

orders of magnitude on 

vertical scale as 

indicated. Response of 

a composite particle 

rapidly transitions from 

that of inclusion (for 

composite particle 

radius near 0.1 pm) to 

that of host as host 

radius is increased. 

Manufacturer’s 

calibration is shown by 

solid dots. 

1 o-7 4” to 22” collection 

l l l l l Manufacturer’s calibration 

1 .O V = 5.8 x 1 O-9 cm2 

IO-*? 

1 o-91 

5? 

9 lo-‘0: 

r 

x 

% 

= 1()-11-r 

i% 

s 

:: 

; 

1 o-‘2, 

1 O-12 I 

1 O-12 I 

Iconcentric inclusion 

o o o o o average for randomly positioned inclusion 

a carbon (1.95 - 0.66i) in water 1.33 - Oi) 

b silicon (1.54 - Oi) in sulfuric acid (1.428 - Oi) 

c carbon in ammonium sulfate (1.52 - Oi) 

10-12 I , I , , d metal (0.56 - 3.Oi) in sulfuric acid (1.428 - Oi) 

, I I 

5 6789 2 3 4 5 6789 2 

lo-’ 1 

Host particle radius (pm) 
radius T = 0.1 pm). The response calculations for internal mixtures using 

this simple model of a sphere containing a spherical inclusion reveal the 

following: (1) for small host sizes, where the inclusion nearly fills the host, 

the response approximates that of the inclusion, but quickly transitions, as 

the host size increases, to a response approximating that of the host; (2) the 

average response for a randomly positioned inclusion is generally within 

20 percent of the response for a concentric inclusion; (3) the variation in 

response due to changing orientation is small, as revealed by the standard 

deviations of the response for a randomly positioned inclusion generally 

being only about 10 percent of the average response; and (4) as for most 

homogeneous particles, the manufacturer’s calibration is not a very good 

approximation of the predicted response, since the result in some size regions is undersizing, and in others oversizing. 

Figure 8 presents the response of the wide-angle scattering probes to particles containing inclusions. Again, the average response for a randomly 

positioned inclusion is generally within 20 percent of the response for a 

concentric inclusion (detailed comparisons show that for carbon in water, 

the difference is less than 13 percent, for silicon in sulfuric acid 3 percent, 

for carbon in ammonium sulfate 10 percent, and for metal in sulfuric acid 

22 percent). However, unlike for the narrow-angle scattering probes, the 

manufacturer’s calibration is quite a good representation of the probe’s response for some internal mixtures: quartz in sulfuric acid, black carbon in 

ammonium sulfate, and metal in sulfuric acid (so long as the sulfuric acid 

mass fraction dominates). The manufacturer’s calibration is less valid for a 

mixture of black carbon in water. 

17 
Figure 8. Same as 

figure 7, except for PMS 

wide-angle scattering 

probes (ASASP-X, 

ASASP-100X, LAS-X, 

LAS-250X, HS-LAS 

models) and also Royce 

236. 

1 o-7 35” to 120” collection 

l l l l l Manufacturer’s calibration 

l.OV=5.8xlO-scm2 

. 

. 

. 

1 

. f’ 

Inclusion (radius = 0.1 jrm) inside host particle 

. 

.: - concentric inclusion 

. 

oo oo o average for randomly positioned inclusion 

10-12 l ** a carbon (1.95 - 0.66;) in water 1.33 - Oi) 

. b 

-. 

silicon (1.54 - Oi) in sulfuric acid (1.428 - Oi) 

C carbon in ammonium sulfate (1.52 - 0;) 

d 

1 O-12 

metal (0.56 - 3.0;) in sulfuric acid (1.428 - 0;) 

I I I 

5 6789 2 3 4 5 6 789 2 

1 o-1 1 

Host particle radius (pm) 

18 
6. Comparison of New Response Function to PAG Theory 

What are the differences between the results of using the new response 

function (which properly accounts for particle trajectory through nodes, 

antinodes, and intermediate regions of the intracavity laser beam) and the 

results of using the previously published PAG theory approximation (which 

assumes particle trajectories pass through only antinodes of the beam)? 

How do these response functions compare to the manufacturer’s calibration? To address these questions, we have compared the response functions 

by making graphs of the PMS-indicated size (manufacturer’s calibration) 

versus the actual size (predicted). 

Results for particles that are nonabsorbing (~1, = 1.50 - Oi) and absorbing 

(1.95 - 0.66i) are displayed in figure 9 (for narrow-angle scattering probes) 

and figure 10 (wide-angle scattering probes). The difference between the 

new response function (eq (1)) and the PAG response function (eq (3)) results is small, particularly for highly absorbing particles. For nonabsorbing 

particles, the amplitude of the high-frequency oscillations is much reduced 

for the new response function compared to the PAG response function. 

For highly absorbing particles, the response functions for the narrow-angle 

scattering probes are indistinguishable. A similar comparison for the corrected PAG response function (eq (2)) in figure 11 reveals small differences 

from the original PAG response function (eq (3)). 

Figure 12 presents a more direct comparison of the new response function 

and the PAG response function for the wide-angle scattering probes. For 

nonabsorbing particles with refractive index nl = 1.50 - Oi, the new response function is a smoother function of particle size compared to the PAG 

response function. 

19 
Figure 9. Relation a- 4” to 22” collection 

between PMS-indicated 

particle radii (smooth 

curve through 

manufacturer’s icalibration) and 9- 

predicted particle radii _ 

(new response function 2 s_ 

and PAG response 5 5- 

function) for 5 

ZT 4- narrow-angle scattering n 

probes. Solid straight j$ s_ 

line is a reference $ 

depicting perfect 

agreement between 

indicated and predicted 

size. Curves for 

nonabsorbing and - Present theory (1.50 - 0;) 

highly absorbing ------ PAGtheory(1.50-Oi) 

particles are shown. 

Difference between new ‘- ~~~~~~~-~.~~ Present theory (1.95 - 0.66;) 

response function and _-- PAG theory (1.95 - 0.66;) 

PAG response function ; ; 6 1 I I II, 6 

2 3 4 5 6 7 6 9 2 

results is small for both 10-l 1 

absorbing and Particle radius (urn) 

nonabsorbing particles. 

Normalization in factor 

relating manufacturer 

calibration to 

theoretical response is 

1 V = 5.8 x lo-!’ cm’. 

20 
Figure 10. Same as 

figure 9, except for PMS 

wide-angle scattering 

probes. 

Large-amplitude 

osciIlations in PAG 

response function are 

not evident in new 

response function. 

2 1 35” to 120” collection / 

I9- 

6- 

7- 

- Present theory (1.50 - 0;) 

------ PAG theory (1.50-o;) 

..-.---.-... Present theory (1.95 - 0.66;) 

-- - PAG theory (1.95 - 0.66;) 

, , , 1 ,,,, I 

2 3 4 5 6769: 2 

Particle radius (pm) 

21 
Figure 11. Relation 35” to 120” collection 

between PMS-indicated 

particle radii (smooth 

curve through 

manufacturer’s 

calibration) and 

predicted particle radii 

(corrected PAG 

response function, eq 

(2), and PAG function, 

eq (3)) for wide-angle s 

scattering probes. Solid 

straight line is a 

reference depicting .r 

perfect agreement 

between indicated and 

predicted size. Curves 

for nonabsorbing and - Corrected PAG (1.50 - 0;) 

highly absorbing ______ PAG theory (1.50 - 0;) 

particles are shown. 

Normalization factor 

relating manufacturer 

calibration to 

theoretical response is 

1 V = 5.3 x lo-’ cm’. 

5 6769 

1 o-1 

2 3 4 5 6789 2 

Particle radius (pm) 1 

Figure 12. Predictions 

of response for PMS 

wide-angle scattering 

probes based on new 

response function (eq 

(1)) and PAG response 

function (eq (3)), 

compared to 

3 

E 

manufacturer’s r 

g 

calibration. For N‘ 

particles with nr = 1.50 5 

- Oi, manufacturer’s z 

calibration follows PAG g 

theory and undersizes 

particles with radii less 

2 

than 0.2 Llrn. 

22 

10-12 I ’ 1 ’ ’ , I I 

5 6 7 6 9 2 3 4 5 6769 2 

10-l 

Particle radius (pm) 1 
7. Predictions of Response for Multiline Laser Operation 

The manufacturer claims (Knollenberg, 1989) that the HeNe lasers in the 

LAS-X and HS-LAS probes have sufficient gain to operate in multifrequency 

modes, with as many as six lines from yellow to red for the HS-LAS, and 

that the multifrequency operation smooths the response characteristics of 

the instrument. To address this claim, we consider particle scattering in a 

cavity lasing in multiple lines. 

As shown in appendix A, one can calculate the response function for multiline operation by weighing the response function for each line by the intensity of that line (eq (A-27)). According to PMS, the intensity distribution for 

the HS-LAS is typically 80 percent for the 632%nm line, 10 percent for the 

629.4-nm line, and 10 percent for all other lines (private communication, 

John Mitchell, Particle Measuring Systems, 1998). We note that by measuring the light transmitted through the highly reflecting closure mirror of a 

factory-refurbished LAS-X (serial number 9075-0786-252, manufactured in 

July 1986), we found 98-percent intensity in the 632.8-nm line, 2-percent intensity in the 640.1-nm line, and no measurable intensity in the other lines. 

Thus the manufacturer’s claim of multiline operation may be exaggerated. 

Nevertheless, lasers in the HS-LAS probes purportedly have higher gain 

than those in the LAS-X probes. To assess the effect of multiline operation, 

we assume the manufacturer’s claim is correct and calculate the response 

of the HS-LAS (wide-angle scattering probe) for the following distribution 

of intensities: 80 percent at 632.8 nm, 10 percent at 629.4 nm, and 2 percent each at the 593.9-, 604.6-, 611.8-, 635.2-, and 640.1-nm lines. Figure 13 

compares the results for multiline lasing and single-line lasing (all intensities at 632.8 nm). Response calculations are presented for nonabsorbing 

ammonium sulfate and highly absorbing black carbon. In both cases the 

difference in response cannot be distinguished in the figure (even though 

it may not be clearly evident, two curves are displayed for each refractive 

index). Numerical results show that the differences are less than 7 percent 

for ammonium sulfate and less than 2 percent for black carbon. Thus the 

effect of multiline lasing on response is negligible. 

23 
Figure 13. Predictions 

of response for PMS a 

wide-angle scattering I Manufacturer’s calibration 

probes having lasers _ l.OV=5.8xlO-9crn2 b 

with sufficiently high 

gain for lasing in 

multiple lines (LAS-X, 

LAS-250X, HS-LAS 

models). Response 

curves are presented for 

multiline and 

single-line lasing a Theory (1.52 - 0;) 

operation. Curves are 

- Single-line lasing 

presented for 

------ Multiline lasing 

nonabsorbing 

ammonium sulfate b Theory (1.95 - 0.66;) 

(171 = 1.52 - Oi) and -Single-line lasing 

highly absorbing black 

------ Multiline lasing 

carbon (7~7 = 1.95 - 

0.66i) particles (two 

curves are presented for 

each refractive index). 10-121 ” “I I I 

5 6 7 8 9 2 3 4 56 7 89 2 

Multiline operation has 10-l 1 

negligible effect on Particle radius (pm) 

response characteristics. 

24 
8. Conclusion 

We present here an improved response function for light-scattering aerosol 

counters that use intracavity laser illumination. The new response function predicts a smoother size dependence compared to the previous PAG 

response function, particularly for nonabsorbing particles, because it accounts for phase averaging. The response function has been used to investigate the response characteristics of Particle Measuring Systems aerosol 

probes that use intracavity laser illumination. Probes having two scattering geometries have been studied: those that collect light scattered through 

angles ranging from 4” to 22” from the direction of the laser-beam axis (as 

in ASASP- and ASASP-300X style probes), and those that collect light 

scattered 35” to 120” from the laser-beam axis (as in the ASASP-X, ASASP100X, LAS-250X, LAS-X, and HS-LAS probes). The new response function 

is generally in good agreement with measurements on uniform particles 

of polystyrene latex, dioctylphthalate, nigrosin dye, and carbon black. Response calculations for both homogeneous and inhomogeneous particles 

with a variety of refractive indexes are presented and compared to the 

manufacturer’s calibration; this comparison reveals that caution needs to 

be exercised in using the calibration. For the wide-angle scattering probes, 

our study suggests that the manufacturer’s calibration is generally appropriate for sizing sulfate particles, as well as particles containing mixtures of 

black carbon in sulfate and quartz in sulfuric acid. Contrary to the manufacturer’s claim, multiline laser operation has little effect on probe response 

characteristics. 

25 
Appendix A. Theoretical Response Function for Intracavity Laser 

Probes 

A-l Intracavity Scattering 

In active-cavity scattering probes, aerosol particles traverse an intracavity 

laser beam within an optical cell. Scattered light is collected either by a 

lens (for the narrow-angle scattering probes) or by a parabolic mirror (for 

the wide-angle scattering probes). In both instruments, the collection solid 

angle is symmetric about the beam axis, except for small asymmetry caused 

by the aerosol entrance jet and exit port, which protrude into the optical cell 

in the X-probes. These ports reduce in a small but complicated fashion the 

solid angle over which light is collected; this effect is neglected here. 

We consider the intracavity beam to be made up of two counterpropagating beams. The beam propagating towards the collection aperture center is 

labeled beam 1 and the other beam 2. We define an (1:: y: z) Cartesian coordinate system with unit basis vectors (2: y^, 2) so that the z-axis extends 

through the center of the aperture, and E = 0 is defined to be the position where particles cross the z-axis. For our scattering calculations, we neglect the finite spatial extent of the beams and assume an infinite coherence 

length. We model the beams as linearly polarized plane waves with equal 

amplitudes and with parallel polarization in the x-direction. We use the 

exp(-l:&) convention and represent the electrical fields of beam 1 (propagating in the direction of increasing 2) and beam 2 (propagating in the 

direction of decreasing 2) as 

and 

E&r.t: 1) = C!$,,, expfi(kz - wf + @I)]. (A-la) 

Eirlc(r.t: 2) = .?EirZC exI$i(-k,- - ~lt + a%)]. (A-lb) 

The wavelength of the beams in the medium (with index of refraction N) 

surrounding the particle is X. and the wavenumber is k = Nw/c = 27i-/X. 

We explicitly introduce the beam phases GP to facilitate discussion of laser 

coherence. The total electrical field is the sum of the fields of the two beams, 

Ei,,,(r.t) = 1 Ei,,(r,t;p), 

p=1.2 

(A-2) 

27 
and the time-averaged energy distribution of this monochromatic field is 

proportional to 

E&r,t) . ETn,(r$) = JEi,,(r,t; 1) I2 + [E&r& 2) I2 + 2 Re [J&Jr& I) . E$&,G 2)1 

= 2(Ei,c)2[1 + cos(2kz + d&2)] 

= 4(&J2 cos2(l;z + $2): (A-3) 

where Qp12 = @I - ip2. We see that maxima (i.e., antinodes or bright fringes) 

occur in the energy distribution when kz + al2 = nick: n = 0, fl, f2, &3: . . . . 

and the distance between the antinodes of this interference fringe pattern 

is 6 = n/k = x/2. If the coherence time of the beams (T,) is longer than the 

transit time (T,) required for the particle to pass through the beams, then 

a12 is constant and the fringes are fixed in space, but if T, < Tt, then @I2 

becomes a function of time, as the fringes change during transit. 

The two incident electromagnetic waves scatter from the particle crossing 

the beams, and our immediate goal is to obtain the scattered wave solutions 

as functions of the variables and basis vectors of a single coordinate system, 

which we choose to be the (r: 8,$) spherical system (with unit basis vectors 

(F, $, 4) associated with the (5, y, ,z) Cartesian system. To accomplish this 

goal, it is helpful to introduce two other Cartesian systems: (~1, yl,zl) with 

basis vectors (21, QrJ 21) associated with beam 1, and (~2, ~2: 22) with basis 

vectors (22, y^2> 22) associated with beam 2. The origin of these two systems 

is the center of the particle, and 

As the particle crosses the beams, the large spatial dimension of the collection aperture compared to that of the aerosol stream that crosses the 

beam (about 100 LLrn) and particle diameter (a few micrometers) allows us 

to make the approximations 

28 

(A-5) 
The spherical systems associated with these two Cartesian systems are 

(1.; 81. ($1) = (r.. 8: 4), 

(T: 82, &) = (T: ?T - 0: -4). (A-6) 

with unit basis vectors 

(?&ji) = (@:& 

_ h _ _ 

(P. 82.&J) = (?, -8: -4) (A-7) 

We now write the electrical fields of the two incident waves 0, = 1,2) as 

Ei,,(r,t;p) = n^,Ei,, exp[i(k+ - wt + @p)]: (A-8) 

and the far-field scattering solutions as 

(A-9a) 

where 

&,(r;p) = (J%+,& + E4,&) . 

(A-9b) 

The scattered fields can be expressed as functions of amplitude scattering 

matrices (Bohren and Huffman, 1983) with components Sh: (I; = 1 to 4), and 

for incident fields polarized in the ZP direction, 

where ~1~ E cos QP and h-, = Nwp/c. If the particle is a homogeneous sphere, 

then Sa = S’d = 0. 

Recalling equations (A-6) and (A-7), we find that 

Es&-: 1) = EB,; + EdI& 

E,,,(r; 2) = -EQ?? - E@,$. (A-10) 

If the particle does not pass perpendicular to the beam, then it has a small 

velocity component parallel to the beam axis, and the frequencies of the two 

scattered waves are slightly unequal because of small Doppler shifts. An 

appropriate expression given by Jackson (1975, eq (11.8)) can be rewritten 

to obtain the frequency of incident beam p in the particle’s rest frame (PRF) 

as 

(W&RF = W - kv. zp. (A-lla) 

29 
and a second Doppler shift of scattered wave p into the lab system gives 

the frequency of the far-field scattered wave at a point on the aperture (in 

direction r^) as 

WP = (wp)pfw - lix. F> 

= w+awp, (A-llb) 

where 

nwp G -lx * (Fp + F): (A-UC) 

so that the difference between the scattered frequencies at a point on the 

aperture is 

w21 z w2-w1> 

= 2k(v. Z), 

= 2bUZ. (A-12) 

The Doppler shift of each beam is position dependent over the aperture, but 

the difference in frequencies between the two scattered waves (differential 

Doppler shift) is not position dependent. 

The total electric and magnetic fields at point r on the aperture at time t are 

&&,t) = c JL(rJd 

p=1,2 

Hsca(r$) = c Hsca(r$:l-‘). 

p=1,2 

(A-13) 

and the instantaneous Poynting vector is 

S(r.t) = Re[E,,,(r.t)] x Re[H,,.,(r.t)]. (A-14) 

We assume that all radiant power scattered into the aperture solid angle, 

&A(t) = I dRr2r^~ S(r. t). (A-15) 

an 

is delivered to the sensing element, which produces an electronic signal 

proportional to this power. We find that 

r^- S(r,t) = $ Re c c [L&-;p) . EL&-; dl exp[G-+qt + Qpq)l: 

p=1.2 q=1.2 

(A-16) 

30 
where L+,~ z tip - L+ and QPrl s a, - QQ. The time-averaged irradiance of 

each incident beam is 

Iin, = &(E~~~~)2~ (A-17) 

so that 

(A-18a) 

where 

We observe that I/T’& = I&‘,, so that lVl1 and W22 are real. 

In order to obtain a realistic simulation of the response function, we need 

to consider the averaging effect of signal processing by the probe sensor 

electronics. To do this we examine the passage of the particle through the 

beams from entrance at time t = 0 until exit at t = Tt. After the particle 

enters the beams, scattered light begins to reach the sensor, producing a 

signal that peaks as the particle reaches the maximum irradiance at the Eaxis when t _N T;/2. The instrument response to a particle passing through 

the beams is obtained by electronic averaging of the signal generated by 

the sensor over an interval of time T, (a characteristic RC time constant 

of the sensor amplifiers) centered on the occurrence of the peak signal at 

t N Tt/2. We compute the instrument response R by averaging equation 

(A-Ma) over the appropriate interval of time, 

= Re c 1 lfpq(exp[j(tiqpf + aa,,)]) 

jz1.2 p1.2 

= If;11 + TV22 + 2Re[Tl’ir!(exp[i(wnlt + Qi2)])]. (A-19) 

Thus the response function consists of three cross-section components: one 

associated with beam 1, one with beam 2, and an interference term. 

31 
A-2 Corrected PAG Response Function 

We now derive a new expression of the PAG theory, which is applicable 

when the particle’s trajectory is exactly transverse to the beams (i.e., ~21 = 0 

and kl = k2 = k) at the position of an unmoving bright fringe (antinode) 

so that <pr2 = 0. These assumptions reduce equation (A-19) to 

R = w11+ w22 + I4712 + Iv21 ) (A-20) 

where 

Iv11 = ; spmi” & [lSd~)/~ + 1~2(/412 + l&&)12 + IS&)i’] > 

Pmax 

Iv22 = $l;I; & [Isd-~)l~ + lS2(-d12 + lS3(-~)1~ + lS4(-dl”] > 

iv12 = - ; I pmin dp [&(p>S;(-P) - S2(&9;(-P) + s3(~)s;(-P) - S4(,@:(-I-L)] : 

Pmlax 

Iv21 = - ; I 

Wmin dp [S;(/&(-P) - S,*(P)S~(-P) + S3*(&53(-~) - 5%$94(-~)1; 

. pmax 

and where p.min z cos 0min and prnax F cos 8,, We observe that equation 

(A-20) can be rewritten as 

R 2: ;Lzdp [ l&(P) + &(-P)12 + lS2(P) - S2(-P)12 

+ lS3(P) + S3(-P) I2 + lS4(P) - S4(-/I)I”] : (A-21) 

and if the particle is a homogeneous sphere, then the integrands s3(~) = 

s4(p) = S3(-p) = S4(-p) = 0: and we obtain 

Rz.k21;I; dp [ ISI + &(-d12 + IS&) - sT?(-p)12] . (A-22) 

We refer to this result as the corrected PAG equation, because it is a corrected version of the original PAG equation given by equation (3) in the 

main body of the report. 

A-3 New Response Function 

To obtain a more realistic response function, we now consider particles 

crossing the nodes, antinodes, and intermediate regions during transit 

32 
through the intracavity beam. If the beam fringes do not change or move 

during particle transit, then a.12 remains constant, and the interference term 

in equation (A-19) can be simplified, giving 

R = WI1 + I4722 + 2{sin(w~lT,/2)/(wnlT,/2)} Re{lvln exp[i(W2rTt/2 + @12)]}. (A-23) 

Recalling equation (A-12), we find that wzrT,/2 = 27r(z~~T,/X). To estimate 

the magnitude of this term, we assume the diameter of the aerosol stream 

that intersects the laser beam to be 100 pm, and the particle’s velocity transverse to the beam to be 10 m/s, so that the transit time across the beam, Tt, is 

10 ks. Also, we assume that the particle’s velocity is not exactly transverse 

to the r-axis but has a component II, that is, let us say, 2” from the perpendicular direction, so that (v~),~~~ = (10 m/s) tan (2”) = 0.349 prn/ps. Using 

X = 0.6328 pm and Tp = 1.0 ps, we find w2iTe/2 = 27r (0.349)(1.0)/0.6328 

and sill(w*1T,/2)/(w21T,/2) = -0.092. Thus this factor causes a 91-percent 

reduction of the magnitude of the third (interference) term in equation (A23). This reduction is even more for larger values of wzlT,/2 = n(zrZTJS). 

This last expression of w2iTe/2 is equal to 7r times the number of fringes that 

the particle crosses during the time T,, so the interference term of equation 

(A-23) becomes smaller if the particle crosses more fringes. 

Having demonstrated that the interference term of equation (A-23) is small 

for realistic particle trajectories through the beam, we turn our attention to 

an additional effect that may also reduce the magnitude of this term: that 

of finite laser coherence time. We examine the hypothetical situation when 

the particle passes exactly perpendicular through the beam so that ZJ, = 0 

and wpl = 0 (i.e., there is no differential Doppler shift). Even though there 

is no Doppler shift, the fringes are not fixed during particle transit, because 

the laser changes longitudinal modes (a 12 is not constant, as the coherence 

time of the laser, T, 2 0.5 pus, is less than the particle transit time and the 

electronic averaging time). Thus, Qr and <p;? each assume multiple random 

values during particle transit, and we find that 

1 

(exp[i(Wplf + %2)]) = - 

.I 

t=Tt/2+T+ 

T, t=Tt/P-T,/2 

df exp[iQl2(t)] 21 0. (A-24) 

We conclude that the response function that accounts for particles crossing 

fringes of the intracavity beam and the finite coherence time of the laser can 

be approximated by 

33 
R = W11+W22 

=;.r,l:“:” & [ lSl(d12 + lS2W12 + 15’3b)12 + IS~(P)I~ 

+ lsI(-~u>12 + ls2(-d2 + Is3(-/1)12 + Is4(-~#] I (A-25) 

where we have used the simplifying approximation that, since particle trajectory velocities are within a few degrees of being perpendicular to the 

beam, l/(k~)~ 21 l/(k~)~ 21 l/kik2 cx 1/k2. 

If the particle is a homogeneous sphere, then the integrands S3 (P) = S4 (P) = 

S3(-p) = S4(-P) = 0: and we obtain the new response function: 

+ IW-IdI2 + lS2C-P)12] . 

(A-26) 

A-4 Response Function for Multiline Lasing 

Finally, we consider the instrument response for multiline lasing. If the intracavity beam contains multiple frequencies, then the difference between 

these frequencies is approximately a factor of lo6 greater than the differential Doppler frequency considered earlier, and the time-dependent interference terms in the total instantaneous Poynting vector are averaged to zero 

by the sensor. The result is that the time-averaged instantaneous Poynting 

vector is equal to the sum of the time-averaged Poynting vectors of the 

different frequencies, and the instrument’s response is 

where f(x) is the fraction of the total irradiance, 

34 

(A-27a) 

(A-27b) 
Thus the multifrequency response is the weighted sum of the response 

functions (of wavelength A), each weighted by the irradiance fraction at 

A. 

35 
Appendix B. Normalization of Measured Response to Theory 

The measured responses of the PMS probes are fit to the theoretical response as follows. Measurements of the raw response of a particular probe 

(in volts), RTa7L’(~‘n): w h ere T,, is the polystyrene latex particle radius, are 

taken from the appropriate journal reference for a total of N particle sizes. 

We restrict the fit to the polystyrene latex data because these particles have 

well-characterized size and refractive index. The experimental response of 

the instrument (in centimeters squared), Rfp, is then given by 

R ,4”” IZ GRTa7’&,), (B-1) 

where G is the gain (in centimeters squared per volt) to be determined by 

comparison of the experimental response with a theoretical cross section 

(in centimeters squared). The corresponding theoretical response RslJln (rn.) 

is computed with equations (l), (2), or (3) in the main body of the report. 

The approach used here is to determine the gain G by minimizing the absolute value of a function S(G) defined as 

S(G) = 5 ([log,o(RYp>] - [k&&,m(~n)l) . (B-2) 

n=l 

This approach, in which the sum of the difference of the logarithm of response is minimized, weighs all response measurements nearly equally in 

the fitting procedure. 

37 
Acknowledgments 

We thank Steven C. Hill for verifying equation (1) and Dennis Garvey for 

helpful comments. 

39 
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41 
42 

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43 
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I_ TITLE AND SUBTITLE Response Characteristics of Active Scattering Aerosol 5. FUNDING NUMBERS 

Spectrometer Probes Made by Particle Measuring Systems DA PR: B53A 

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i.~~r~ea(a) Ronald G. Einnick, J. David Pendleton, and Gorden 

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13. ABSTRACT (MaxImom 200 wordsj 

Predictions are reported of the size response of various light-scattering aerosol counters manufactured by 

Particle Measuring Systems. Models considered are those that exploit the high intensity of light available 

within the cavity of a HeNe gas laser (generically referred to by the manufacturer as “active scattering 

aerosol spectrometer probes”). The new response function properly averages over particle trajectories 

through nodes, antinodes, and intermediate regions of the intracavity laser beam. Our studies address 

probes having two basic scattering geometries: those that collect light scattered over a relatively narrow 

solid angle (subtending angles between 4” and 22” from the laser beam axis) and those that collect light 

over a rather large solid angle (between 35” and 120”). The new response function predicts smoother 

dependence on particle size than the previous response function of Pinnick and Auvermann (1979, J. 

Aerosol Sci. 10: 55-74) and is in better agreement with measurement. Response calculations for common 

atmospheric aerosol (water, sulfuric acid, ammonium sulfate, and black carbon) reveal the considerable 

sensitivity of the response to particle dielectric properties. Comparison of response calculations with the 

manufacturer’s calibration reveals conditions for which the manufacturer’s calibration is most appropriate, 

and the potential for errors (as much as a factor of two in sizing) when it is blindly applied. These results 

should help the user of these instruments to more realistically interpret size distribution measurements. 

14. SUBJECT TERMS 

Aerosol counter 

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