1. Introduction

The volume scattering function (VSF) is $\beta (\lambda ,\theta ,\phi )$. For natural waters, the scattering is assumed azimuthally symmetric about the incident direction and the VSF is written $\beta (\lambda ,\theta )$. The integrated backscattering coefficient $b_b(\lambda )$ is the integral of $\beta (\lambda ,\theta )$ over the solid angle ($d\mathrm{\Omega }=\sin \theta \mathrm{d}\theta \mathrm{d}\phi$), for the backwards hemisphere,

The backscattering coefficient is one of the inherent optical properties of natural waters, i.e., it is independent of the ambient light field in the water. As such, it has a central role in many problems of optical oceanography, e.g., visibility and water clarity, remote sensing and diffuse reflectance, hydrosol composition and distributions, and even for estimates of bulk refractive index. In general, it is an important component in the characterization of natural waters [1, 2, 3, 4, 5]. It can play an important role in the modeling of ocean waters, e.g., the Fournier– Forand phase function [4, 6]. Sources contributing to the backscattering include the water molecules themselves, suspended organic and inorganic particulates, and air bubbles.

Although a measurement of the VSF can generally be a slow, difficult, and time-consuming process, a rigorous, straightforward determination of the backscattering coefficient involves measuring the VSF and then using Eq. (1) to calculate $b_b$ [7, 8, 9, 10]. Oishi et al. described an instrument to measure the entire VSF using a CCD camera [11], but simpler, single-measurement instrumentation for $b_b$ is desirable. Measurements of various optical properties of the water, together with application of an inversion algorithm [1, 12, 13, 14, 15], have also been used to obtain $b_b$. Other approaches include an integrating cavity that collects backscattered light [2, 16] and the use of cone reflectors to select backscattered light [17, 18, 19].

But, by far the approach that has received the most widespread interest and application is measurement at a single fixed optimum angle. The optimum angle is chosen so that in a variety of typical measurements, there appears to be the tightest correlation between the scattering at that angle and the backscattering coefficient [3, 20, 21, 22, 23, 24, 25]. The optimum angle meter is, in fact, the basis of the only commercially available instruments. But again, it does not really measure $b_b$. It measures backscattering at some specific angle that is anticipated to give a result proportional to the correct $b_b$; this has been extensively investigated and is, in fact, expected to be generally fairly accurate to $\sim 10\%$ [23, 26] or it “…should have a maximum uncertainty of only a few percent”[20]. But even Oishi, a major proponent of the approach, questions “…if it can hold for plankton bloom and coastal water”[19]. Situations can (and will) arise that will distort the VSF outside acceptable conditions for measuring $b_b$ with a fixed-angle meter, for example, the presence of monodisperse distributions of particles.

Instrumentation is described in the following that is as simple to employ (although the theoretical anal ysis is more complicated) as a conventional fixed- angle meter; it is, in fact, a fixed-angle meter, but one that truly measures $b_b$. Both fixed-angle meters involve sending a beam of light into the water and using a single detector to measure backscattered light. But the instrumentation described here collects backscattered light in such a way as to measure the actual $b_b$ for arbitrary $\beta (\lambda ,\theta )$; it does not rely on assumptions about scattering particle types or distributions. The insight that makes this instrumentation possible is to note that if a laser beam propagates along the z axis and a detector is mounted with the normal to its detector aperture parallel to the z axis, one will measure the integral over the detection solid angle of $\beta (\lambda ,\theta )\cos \theta d\theta$. In order to convert $\cos \theta$ to the $\sin \theta$ required by Eq. (1), one must have a detector aperture whose normal is perpendicular to the z axis. And that is exactly what this instrumentation does—it simply sends a well- collimated laser beam into the water and collects backscattered light that passes through an aperture whose normal is perpendicular to the z axis. It also scrambles the backscattered light in an integrating cavity to eliminate all memory of the original scattering direction before it reaches the detector. In some sense, it is a form of the Beutell and Brewer [27] instrument to measure scattering, but in reverse; it introduces the $\sin \theta$ factor in the same way as the Beutell and Brewer instrument. For typical instrument parameters and due to the presence of $\sin \theta$ in the integrand defining $b_b$, most of the contributions to the observed signal come from light that has been backscattered at distances of less than a few centimeters.

The VSF—and therefore the backscattering coefficient as well—is inherently dependent on the wavelength of the scattered light. However, to simplify the readability of equations, explicit use of λ will be omitted in the following; the wavelength dependence will be considered to be implicit.

2. Instrument Theory and Design

Consider a laser beam entering a medium at $z=0$; see Fig. 1. The aperture that defines the directions of the scattered light to be detected is an opening in the form of a cylindrical ring of radius R extending from $z=-Z_0$ to $z=+Z_0$. The laser beam passes through a quartz rod surrounded by a small opaque tube of radius r up to the point $z=0$ where it enters the scattering medium. The laser beam of cross- sectional area A and irradiance $E_0$ propagates along the z axis.

In Fig. 1, consider a scattering volume of length $dz$ and cross-sectional area A at position z. The power $dP$ scattered at an angle θ into the detector aperture in the solid angle defined by ${\theta }_1\le \theta \le {\theta }_2$ is then given by

The theoretical analysis will proceed by first considering the case of small attenuation, i.e. $e^{-cx}\approx 1$; the analysis will then be extended to cases in which the effects of attenuation become significant. The advantage of this approach is that the small attenuation case allows an in-depth understanding of the theory of the instrument without the need for any further approximations. By then including the effects of attenuation, the efficacy of the instrument can be demonstrated at even relatively high values of attenuation.

2A. Small Attenuation

From Fig. 1, ${\theta }_1$ is given in terms of z. Specifically, $\tan (\pi -{\theta }_1)=R/(z-Z_0)$, solving for ${\theta }_1$ gives

For a fixed value of z, the detector only observes light that is scattered at angles θ defined by ${\theta }_1(z)\le \theta \le {\theta }_2(z)$, where ${\theta }_1(z)$ and ${\theta }_2(z)$ are given by Eqs. (4, 5), respectively. On the other hand, for a fixed value of θ, scattered light is only observed from points z between $z_1(\theta )$ and $z_2(\theta )$. These two functions are obtained by inversion of Eqs. (4, 5), respectively; or by examination of Fig. 1,

Physically, the domain of integration of integral I (i.e., the exact $b_b$) covers the combination of the domains named A and B in Fig. 3, whereas the range of integration for the actual detected power, Eq. (3), covers the combination of domains A and C in Fig. 3. The accuracy of the instrument is given by the sum of integrals II, III, and IV; basically, it is determined by the extent to which the power detected over domain C is equal to the power not detected over domain B.

The accuracy is dominated by the choice of $Z_0$ or, more precisely, by the choice of $R/Z_0$. To the extent that $Z_0$ has a small value, integrals II and III in Eq. (15) will make a relatively small contribution compared to integral I; they also have an opposite sign and will tend to cancel. Integral IV is small because $\cos \theta$ is small for angles near $\pi /2$; it will also be small [even in the presence of asymmetries in $\beta (\theta )$ around $\pi /2$] because $\cos \theta$ is an odd function over the interval of integration. Finally, not only will integral IV be small, but the symmetries are such that any contribution from integral IV will always cancel part of any remainder from the sum of integrals II and III.

To demonstrate the high accuracy possible for measurements of $b_b$, the three ratios (${\rho }_2$, ${\rho }_3$, and ${\rho }_4$) in Eq. (16) and their sum (which gives the accuracy of a measurement of $b_b$) were numerically evaluated for 11 examples covering a broad distribution of water types as well as for four examples of monodisperse suspensions of microspheres. The results are shown in Table 1. For these numerical evaluations, the radius is $R=0.01\mathrm{m}$, and the three values considered for the aperture parameter are $Z_0=0.001$, 0.002, and $0.005\mathrm{m}$. For the cases of natural water examples, the values of $b_b$ are shown; they provide an indication of the wide variability in water types being considered (for the sphere suspensions, $b_b$ depends on the number density). Table 1 also shows the conversion factor $\chi ({\theta }_{\text{optimum}})=b_b/2\pi \beta ({\theta }_{\text{optimum}})$ used in the discussions of fixed- angle meters. For the spheres, χ is irrelevant; e.g., for $4\mathrm{\mu m}$ spheres, χ varies from 0.56 to 1.11 over the range $115°\le {\theta }_{\text{optimum}}\le 120°$, and it is 1.43 at $140°$. Of course, conventional fixed-angle meters were never intended to work for particle suspensions of this type; but, by contrast, our instrument can provide an accurate measurement of $b_b$ for any particle suspension.

Clearly, the cases with $Z_0=0.001\mathrm{m}$ (and $R=0.01\mathrm{m}$) provide exceptional accuracy, typically $<0.5\%$ (shown in boldface); these are parameters one might use in an operational instrument. The ratio $R/Z_0$ is 10, and the relevant integration limits are ${\theta }_{10}=84.3°$ and ${\theta }_{20}=95.7°$. It should be emphasized that even when $Z_0=0.001\mathrm{m}$, the aperture to the detector is still large and collects an ap preciable amount of light. Specifically, the open area of the aperture is $A_a=(2Z_0)(2\pi R)=4\pi R{Z}_0$; for $R=0.01\mathrm{m}$ and $Z_0=0.001\mathrm{m}$, it is $A_a=0.0001257{\mathrm{m}}^2\approx 1.26{\mathrm{cm}}^2$.

If a phase function is symmetric about a scattering angle of $\pi /2$, integrals II and III in Eq. (14), or ratios ${\rho }_2$ and ${\rho }_3$ in Eq. (15), will exactly cancel and integral IV (or ratio ${\rho }_4$) will be zero. But most phase functions decrease with increasing angle in the vicinity of $\pi /2$; in such cases, integral II ($\theta \le \pi /2$) has a larger magnitude than integral III ($\theta \ge \pi /2$) and they do not quite cancel, i.e., ${\rho }_2>{\rho }_3$. Because of the negative sign in front of integral IV and the fact that $\cos \theta$ is an odd function around $\pi /2$, its contribution always provides further cancellation of the combination of integrals II and III (${\rho }_2$ and ${\rho }_3$), e.g., see the examples in Table 1. For an uncommon phase function that increases with increasing angle in the vicinity of $\pi /2$, the integral II would have a smaller magnitude than integral III (${\rho }_2<{\rho }_3$), but the sign of integral IV (or ${\rho }_4$) would also then be changed and will still provide further cancellation. Such an uncommon phase function is demonstrated by the example for monodisperse $596\mathrm{nm}$ spheres in Table 1, an example specifically included to demonstrate this case.

The bottom line is that for an infinitesimal detector aperture, this instrument provides an exact measure of $b_b$ regardless of the asymmetries in $\beta (\theta )$ at $90°$; the question then turns to how a finite detector aperture affects measurement accuracy. Although a detailed discussion is provided regarding the balancing of the contributions due to a finite detector aperture, there is nothing unusual about this measurement process; it is typical. To illustrate with a simple and familiar example, consider a measurement of $\beta (\theta )$ at some angle θ. For an infinitesimal detector angular aperture, a measurement of the scattering at angle θ gives exactly the correct result. With a finite angular aperture, the change in the amount of light collected due to scattering at angles less than θ typically tends to balance the change in the amount of light collected at angles greater than θ, giving a relatively accurate result for scattering at the desired angle θ. If the shape of the phase function $\beta (\theta )$ in the vicinity of θ is such that these deviations do not cancel [e.g., $\beta (\theta )$ has a peak at θ], the deviations can still be reduced by making the angular aperture smaller. In the case of our $b_b$ instrument, the magnitude of the net deviation between the measurement and the correct $b_b$ depends on the change in $\beta (\theta )$ as θ increases from below $90°$ to above $90°$. For a $\beta (\theta )$ symmetric around $90°$ between the angles ${\theta }_{10}$ and ${\theta }_{20}$ (even a symmetric peak at $90°$), the deviation is zero. The maximum deviation would occur for an unphysical step function change in $\beta (\theta )$ at $90°$, but even in this case, the deviations from the correct $b_b$ could still be reduced to any desired accuracy by decreasing the width of the detector aperture.

To summarize, the power P that enters the detector corresponds with relatively high accuracy to the backscattering coefficient $b_b$. The proportional quantity $P^{\prime }$, Eqs. (11, 15), shows the errors due to a finite detector aperture in the form of integrals II, III, and IV. As the concentration of scattering particulates increases, these integrals increase in the same proportion that $P^{\prime }$ and $b_b$ increase; thus, their ratios, Eq. (16), to $b_b$ (i.e., ${\rho }_2$, ${\rho }_3$, and ${\rho }_4$) do not change with the particle concentration. Furthermore, these ratios can be made negligible to the level of accuracy desired by the choice of R and $Z_0$ (e.g., Table 1). The detector provides a signal S proportional to the detected power P, or equivalently to $P^{\prime }$, i.e., $S=K_0{P}^{\prime }$ where $K_0$ is the instrument calibration constant that converts detected power to a signal voltage. Thus, Eq. (15) gives

Further insight into the physics of the instrumentation can be obtained by examining the detection solid angle as a function of the distance z at which backscattering occurs. From Fig. 1, the solid angle subtended by the detector aperture at any point z is

Figure 4 is a plot of Eq. (19) for the case $R=0.01\mathrm{m}$ and $Z_0=0.001\mathrm{m}$; it shows that the detection solid angle decreases so quickly that there is very little contribution to the integral from backscattering at large distances. In fact, more than 90% of the total detection solid angle comes from distances z less than $0.021\mathrm{m}$, and more than 99% at distances less than $0.071\mathrm{m}$. This is, of course, to be expected, because the $\sin \theta$ factor in the integrand of the definition of $b_b$ reduces the contributions to $b_b$ as θ approaches $180°$. In fact, $b_b$ is completely independent of the value of the phase function at $180°$. Because relevant distances are short, one would expect good performance even in the presence of attenuation. This sets the stage for examining attenuation effects.

2B. Effects of Attenuation

To evaluate the effects of attenuation, return to Eq. (3), in which c is the attenuation coefficient [the sum of the scattering and absorption coefficient ($c=a+b$)] and x is the total distance that the light travels through the medium. From the geometry in Fig. 1, the latter is

Note, there are no approximations in this expression for the measured power $P^{\prime }(c)$. The first term, label it $P_0^{\prime }(c)$, is the detected power representing $b_b$; it is exactly proportional to $b_b$ when $c=0$. The other three terms, label this combination $P_1^{\prime }(c)$, represent additional (but typically very small) quantities that are part of the observed power; thus, the measured power is proportional to $P^{\prime }(c)=P_0^{\prime }(c)+P_1^{\prime }(c)$. Although in the final analysis a correction factor will be introduced to correct the measured $P^{\prime }(c)$ for attenuation, useful insights can be gained through a further analysis of Eq. (25), specifically, by expanding the exponential functions in the power series.

Consider $P_0^{\prime }(c)$, the first integral in Eq. (25). Expanding the exponential factors, $e^{c{Z}_0}$, $e^{-c{Z}_0}$ in $P_0^{\prime }(c)$ for small ${cZ}_0$ gives

Now consider the three integrals comprising $P_1^{\prime }(c)$. Typically, R and $Z_0$ are small; for the next-generation instrument, they are expected to be $R=0.01\mathrm{m}$, $Z_0=0.001\mathrm{m}$. Consequently, even for $c=\sim 10{\mathrm{m}}^{-1}$, $cR$ and $c{Z}_0$ are small ($<0.1$). Also, throughout the integration range of all three integrals, θ is close to $\pi /2$; therefore, $\sin \theta \approx 1$. Consequently, all the exponentials in the three terms of $P_1(c)$ can be expanded in the power series giving

In order to demonstrate the relatively small effect of c in Eq. (27), as well as the negligible effects of the correction terms given by Eq. (28), all these terms have been numerically evaluated using the MVSM data from October 2009 in Chesapeake Bay [28, 29]. The instrument parameters used in these numerical evaluations are $c=1{\mathrm{m}}^{-1}$, $R=0.01\mathrm{m}$, and $Z_0=0.001\mathrm{m}$. The results are presented in Table 2 and are shown as a percentage of $P_0^{\prime }(c=0)=b_b$; they are calculated from the MVSM data. The sums of the correction terms are indeed negligible.

Clearly, the major impact of attenuation in the measurements is on $P_0^{\prime }(c)$, the first term in Eq. (25), or equivalently, Eq. (27) (the second entry in Table 2). As c increases, $P_0^{\prime }(c)$ decreases, and the change is considerably more than the change produced by all other terms combined (the latter terms were already small corrections). Further insight is possible by an examination of the effects of c in Eq. (27). Unfortunately, the exponential function in the integrand cannot be simply expanded for small $cR$ because, for $cR\ne 0$ the exponent is not always small; i.e., it is $-\infty$ at $\theta =\pi$. Nevertheless, for small $cR$, the product of $\sin \theta$ with the exponential in the integrand is still approximately $\sin \theta$ for all θ; this is clearly demonstrated by the plots in Fig. 5. For example, on this scale, the curve for $cR=0.001$ (e.g., $R=.01\mathrm{m}$, $c=0.1{\mathrm{m}}^{-1}$) cannot be distinguished from the data points for $cR=0$, the latter being just a plot of $\sin \theta$. Even for an order of magnitude larger c (i.e., $1{\mathrm{m}}^{-1}$), the resulting curve for $cR=0.01$ is barely distinguishable from the $\sin \theta$ data points.

The dependence on c for the expected experimentally measured value of $P^{\prime }(c)$, Eq. (25), was compared with the exact value of $b_b=P^{\prime }(0)$, for the wide variety of natural waters considered in Table 1 as well as for a Rayleigh scattering function. Instrument pa rameters for the calculation were $R=0.01\mathrm{m}$ and $Z_0=0.001\mathrm{m}$. The calculated percent error, $[P^{\prime }(c)-b_b]/b_b\times 100$ is plotted as a function of the attenuation c in Fig. 6. These results demonstrate the effectiveness of this $b_b$ meter design, even in the presence of significant attenuation. For example, from Table 2, the measured power $P^{\prime }(c)$ gives $b_b$ to an accuracy of 97.67% when $c=1{\mathrm{m}}^{-1}$; i.e., the attenuation error is only 2.33%. This is consistent with (actually, less than) the attenuation correction for commercial fixed-angle meters—e.g., the user’s manual for the WET Labs ECO BB meter states, “…attenuation error is typically small, about 4% at $a=1{\mathrm{m}}^{-1}$” [30]. Figure 6 shows that at an attenuation of $c=1{\mathrm{m}}^{-1}$, the direct measurement error with our instrumentation is less than 3% for the entire broad distribution of natural water samples considered in Table 1. It is interesting to note that the effects of attenuation on Petzold clear ocean, Petzold coastal ocean, and Rayleigh are nearly identical in Fig. 6, while all the others tend to cluster in a band at slightly smaller percent deviations for any attenuation c.

It is apparent from Fig. 6 that the percent error is almost linear in attenuation. Although the slopes are slightly different for the various phase functions, one should clearly be able to introduce an average attenuation-dependent linear factor that would provide a correction to the measurements observed with this $b_b$ meter; such a factor would improve the accuracy at all typical values of attenuation. The following form provides an attenuation corrected $P_A^{\prime }(c)$:

This attenuation-dependent correction factor was used on all the examples plotted in Fig. 6. The attenuation dependence of the percent difference between $b_b$ and the attenuation corrected $P_A^{\prime }(c)$ is shown in Fig. 7. Of course, this correction requires an additional measurement of c, but as a result, all these data demonstrate uncertainties of less than 3% for attenuations up to $10{\mathrm{m}}^{-1}$; for c less than $1{\mathrm{m}}^{-1}$, measurement uncertainties are a few tenths of $1\mathrm{\%}$.

Now consider the effect that an error $\delta c$ in the perceived value of the attenuation coefficient c will have on the correction algorithm. Explicitly including $\delta c$ in Eq. (29) gives

From Eq. (30) and setting $R=0.01\mathrm{m}$, the fractional uncertainty Δ in $P_A^{\prime }(c)$ due to the uncertainty $\delta c$ can be written

For small c, the denominator of the coefficient in brackets is large and Δ is a small fraction of the uncertainty in c. For very large c, Δ is equal to the uncertainty in c. As an intermediate case, suppose c is $4.0{\mathrm{m}}^{-1}$ and $\delta c$ is $1{\mathrm{m}}^{-1}$ (25% error in c), then the fractional uncertainty Δ in $P_A^{\prime }(c)$ is 2.4%. Or, for another example, suppose $c=1{\mathrm{m}}^{-1}$ with an uncertainty of 10% ($\delta c/c=0.10$); this propagates into an uncertainty in $P_A^{\prime }(c)$ of 0.26%. Clearly, for c less than $\approx 10{\mathrm{m}}^{-1}$, the accuracy in the determination of c does not have much impact on the correction of $b_b$ for attenuation.

3. Instrument Implementation

Measurements were performed in a water tank of approximately $1.80\mathrm{m}\times 0.6\mathrm{m}\times 0.6\mathrm{m}$ that holds up to 600 liters. It was filled with 120 liters of water that had been highly purified using a Millipore water purification system with a $0.2\mathrm{\mu m}$ particle filter as its final stage.

Figure 8 shows a cross section of the first implementation of the $b_b$ meter; the parameters for this prototype instrument were $R=0.01\mathrm{m}$ and $Z_0=0.005\mathrm{m}$. The instrument is $\approx 0.3\mathrm{m}$ in length and $\approx 0.15\mathrm{m}$ in diameter; the weight is below $1\mathrm{kg}$, excluding the detector and power supply for the laser. The green laser ($532\mathrm{nm}$) is a $20\mathrm{mW}$, TTL-controlled, off-the-shelf system from World Star Tech; it was operated at a fixed modulation frequency of $1\mathrm{kHz}$, and it was built directly into the instrument. The power supply for the laser ($3.3\mathrm{V}\mathrm{dc}$, $<0.5\mathrm{A}$) and the electronics to record the signals were left outside the instrument.

The window is a section of a quartz cone. The angle of the cone is $45°$ so that light backscattered at angles between $90°$ and $180°$ will be incident on the window at angles less than $45°$. Under these conditions, the reflectance at the water/quartz interface is typically 0.18%, rising to 0.21% at $35°$ and 0.32% at $45°$. But for a field instrument, there could be problems with fouling in the space between the aperture and the window. Some discussion of other window designs for a next-generation instrument is provided in Appendix A.

The detectors are two Hamamatsu 1P21 photomultiplier tubes (PMTs). Instead of mounting one of them directly behind the aperture, the backscattered light is diffusely reflected through an air-filled integrating cavity and is partially collected by six $0.001\mathrm{m}$ diameter optical fibers that carry it outside the instrument to one PMT that provides the scattering signal. The integrating cavity employs highly reflective, compressed quartz powder with a Lambertian reflecting profile [31]. With this approach, all directionality of the backscattered light is lost and there can be no detection bias based on the backscattering angle.

However, the fibers provided an incredibly poor light transfer efficiency. Basically, the diffuse reflecting walls have essentially 100% reflection efficiency, so all light entering the cavity must leave the cavity either through the six $0.001\mathrm{m}$ diameter signal fibers or back through the entrance aperture. For $Z_0=0.005\mathrm{m}$, the entrance aperture has area $A_{\text{aperture}}=4\pi R{Z}_0=6.3{\mathrm{cm}}^2$. Thus, the fraction of light leav ing through the signal fibers is $A_{\text{fibers}}/(A_{\text{fibers}}+A_{\text{aperture}})\approx 0.74\%$.

In this regard, a comment about the next- generation instrument is in order. Specifically, decreasing $Z_0$ to, for example, $Z_0=0.001\mathrm{m}$ decreases the amount of light entering the aperture by a factor of 5. But, because the aperture is smaller, less light can escape through it and the fraction of light leaving the cavity through the signal fibers will increase to 3.6% from 0.74% or nearly a factor of five (almost no decrease in the total number of detected photons!). Not only would the next-generation instrument have $Z_0=0.001\mathrm{m}$, but instead of fibers, detectors (PMTs or p-i-n diodes) would be mounted directly in the wall of the cavity and would have an active detection area of $\approx 3{\mathrm{cm}}^2$. This does not affect the important aspect of erasing any memory of the directionality of the photons entering through the detector aperture—that memory loss occurs at the first reflection from the diffusely reflecting walls. With detectors mounted directly in the cavity wall, the fraction of the light entering the cavity that would be detected increases to $\approx 70\%$, or a factor of $\approx 20$ more signal.

Some stray light is produced by scattering and reflections from the laser beam inside the instrument before the light is guided through the glass rod into the water. A normalization fiber collects some of this stray laser light and carries it to the second PMT whose output provides the normalization signal.

The scattering and normalization signals produced by the PMTs are fed through separate lock-in amplifiers that only process signals at the $1\mathrm{kHz}$ modulation frequency, thereby blocking any signals produced by sources other than the laser, e.g., ambient light. The signals are sent to an oscilloscope where they are read out to a PC and are simultaneously monitored in real time. The computer samples and averages the dc outputs of the lock-in amplifiers over a $4\mathrm{s}$ time interval and records the average value for that time interval. This is repeated 75 times; these 75 values are then averaged to ob tain a final value for each data point (total of $5\min$ of averaging). The 75 successive measurements improved statistics, but more importantly, they provided a backup check for drifts or systematic variations. None were observed, and the average magnitude of the deviation of these 75 measurements from their final average value was typically less than 0.5%; the maximum deviation of any of the 75 data points from their average value was typically less than 1.5%. The signals with this initial prototype are already so strong that, in practice, one of these 75 data points would generally be sufficient unless the goal is a few tenths of a percent accuracy; with the next-generation instrument and direct-mounted detectors, the signals will be even stronger. The normalized signal is the output of the first lock-in amplifier divided by the output of the second lock-in amplifier.

4. Results

4A. Calibration

To obtain a calibration, the signal (volts) was measured as a function $b_b$ for three sizes of polystyrene spheres at various concentrations. Specifically, the tank was filled with a known amount of highly purified water that had been filtered through a $0.2\mathrm{\mu m}$ filter. The signal from this water was recorded. Then, predetermined amounts of NIST-traceable polymer microspheres [polystyrene latex (PSL)] [32] were added to successively increment the particle concentration and hence $b_b$. The corresponding signal at each particle concentration was measured.

Because the mean particle diameter and the width of the size distribution is known, Mie calculations can be used to determine the necessary amount of the PSL suspension to add to the tank to obtain any desired backscattering coefficient $b_b$. Rather than trying to pipette small amounts of the original suspension into the tank at each successive step, a predilution approach was used. Specifically, a set of identical suspensions were prepared, each with a desired, predetermined amount of PSL spheres corresponding to the desired increase in $b_b$ at each successive measurement. The procedure was as follows:

First, Mie calculations were used to determine the amount of the original PSL suspension needed to produce some total $b_b$ for measurements in the tank.

Second, after sonification of the PSL suspension, the total required amount of PSL suspension was pipetted from the original container into a thoroughly cleaned beaker, using a pipette of appropriate size (Eppendorf pipettes, 0.01, 0.10, and $1.0\mathrm{ml}$ with single-use tips).

Third, $\sim 100\mathrm{ml}$ of pure ${\mathrm{H}}_2\mathrm{O}$ and a few drops of Tween 20 surfactant were added to the beaker, and the diluted PSL suspension was thoroughly mixed by sonification.

Fourth, this suspension was distributed in equal amounts into each of a number N ($\approx 5$) of sample tubes as follows.

Fifth, equal amounts ($1\mathrm{ml}$) of the diluted solution were pipetted into each of the N sample tubes. This was repeated until there was not enough solution to repeat for all N sample tubes.

Sixth, this remaining solution was diluted again by adding $\sim 100\mathrm{ml}$ of pure ${\mathrm{H}}_2\mathrm{O}$; it was then pipetted into the N sample tubes as before. The remaining solution was again diluted and the procedure repeated several times.

Finally, there will still be a residual amount of very highly diluted solution; it is distributed as equally as possible among the N sample tubes, but because it is so highly diluted, negligible error is introduced. This procedure was repeated with larger amounts of the initial PSL solution when larger steps in $b_b$, or higher values of $b_b$ were desired.

The major source of error is the initial pipetting of the concentrated suspension from the original container. The estimated potential error in the concentration of PSL spheres in the tank using this procedure is approximately 3%. To keep the beads in suspension, an aquarium pump was kept turned on in the tank to keep the water circulating. Prior to each calibration run, this pump was run with pure water to ensure that it was clear of any contaminants.

In a preliminary step, a “placebo” experiment was performed; i.e., no particles were introduced, but all steps, from the filling of the tank, dilution of the samples, and the adding and mixing of those samples into the tank were done. Because no particles are added to increase the backscattering coefficient, any change in signal must be caused by unwanted contaminants. The results of this measurement are shown in Fig. 9 and are consistent with the expectation that there will be a small increase in the backscattering in the tank due to contamination from the individual samples. Using the calibration measurement (described in the next paragraph), the average increase as each sample is added corresponds to an increase in the backscattering coefficient of $b_b=0.00005{\mathrm{m}}^{-1}$, or about 6% of the backscattering coefficient of pure water. These data also illustrate the capability to measure $b_b$ for pure water.

The results of measurements with 203, 596, and $3005\mathrm{nm}$ diameter polystyrene spheres are shown in Fig. 10. Linear fits to these data were made, and the fitted equations are also shown in Fig. 10; their slopes are

Because the $Z_0$ for these data is relatively large ($Z_0=0.005\mathrm{m}$), the true calibration constant $K_0$ (the instrumental ratio that gives the conversion from optical power to signal voltage) must be obtained by adjusting for the known systematic shifts in each of the experimentally determined slopes. This is achieved using Eq. (18) together with the three ratios ${\rho }_2$, ${\rho }_3$, and ${\rho }_4$ (listed in Table 1 for each diameter of polystyrene spheres). The results, derived from the fitted K values listed in Eq. (32), are $K_0=295$, 313, and 299 for spheres with diameters of 203, 596, and $3005\mathrm{nm}$, respectively. The average of these three values is taken as the $K_0$ for the instrument:

4B. Absorption Effects

To investigate the effects of absorption on the signal, measurements were taken while keeping $b_b$ constant and adding dye (McCormick black food color, which has no particle scattering and no fluorescence) to the tank to increase the absorption. Because small errors in attenuation are relatively insignificant, a simple transmission measurement was used to measure the attenuation in the tank after each addition of dye. A small detector measures a signal proportional to the transmitted laser power $P(c)$ at four to six distances where the detector package is inserted into the water. A plot of $\ln [P(c)/P(0)]$ versus distance for the four to six data points is fitted to a straight line whose slope is the attenuation coefficient. A frosted glass plate whose area is approximately $5{\mathrm{cm}}^2$ covers the aperture to the detector. This large size was chosen to ensure that the full transmitted beam will always be detected (neither the $b_b$ instrument nor the detector are rigidly mounted in the tank). Of course, measurements taken with this detector will always slightly underestimate the total attenuation in the tank because very small angle forward scattered light will be detected and counted as transmitted light.

The results of the experiment to study attenuation are shown in Fig. 11. At the beginning of this experiment, a reference signal $P(0)$ is measured with only pure water and PSL spheres in the tank at a concentration so as to give $b_b\sim 0.01{\mathrm{m}}^{-1}$. This level for $b_b$ was chosen so as to ensure that the small amount of contaminants invariably introduced into the water sample when adding dye would have a negligible effect on the backscattered signal (e.g., as in the placebo experiment, Fig. 9). The relative signal is then the measured signal $P(c)$ at different attenuations divided by the reference signal $P(0)$. Thus, the relative signal is by definition 100% at the relative attenuation data point labeled $0{\mathrm{m}}^{-1}$.

The theory shown in Fig. 11 is not a fit to the data; it is a calculation of $P(c)$ using Eq. (25) with the phase function $\beta (\theta )$ for $4.3\mathrm{\mu m}$ PSL spheres; the curve plotted is $P(c)$ divided by $P(0)$, i.e., the calculation with $c=0$.

5. Summary and Conclusions

In conclusion, after decades of effort by many investigators to develop and characterize instruments to measure $b_b$, an instrument is described that, for the first time to our knowledge, provides a direct measurement of $b_b$ rather than some other quantity that will hopefully approximate $b_b$. The instrumentation can be constructed (by appropriate choice of instrument parameters R and $Z_0$) to provide measurements of $b_b$ to arbitrarily high accuracy, i.e., 5%, 1%, 0.1%,…, etc. It is instrumentation that for the first time provides a direct and accurate measurement of $b_b$ regardless of the shape of the phase function; it works for any suspension of particles and does not rely on the hope that the sample being measured is consistent with some characteristic natural water type.

Finally, although this instrument concept might be viewed as a version of a fixed-angle meter, it is in some ways actually simpler than the conventional fixed-angle meters that only measure a proxy for $b_b$. Specifically, it has the detector and light source colocated on the z axis rather than having a detector set off to the side at an appreciable angle. That separation of the light source and detector in conventional fixed-angle meters leads to a very complicated scattering volume with undesirable different sensitivities to positions and scattering directions in the observation volume. On the other hand, our instrument might appear more complicated because the detector aperture is unconventional—i.e., the normal to the detector aperture is perpendicular to the z axis rather than looking directly at the scattering volume, but it is still just an aperture to the detector. Overall, the instrument is extremely simple and robust; it is basically a small package that sends a laser beam out into the water and collects backscattered light in an appropriate way. Many improvements in the initial prototype version of the instrument are possible. Two of the most important improvements are (i) for high accuracy and precision, a practical instrument should have parameters $Z_0=0.001\mathrm{m}$ and $R=0.01\mathrm{m}$ and (ii) to dramatically increase the signal strength, the detectors should be mounted directly in the integrating cavity rather than using optical fibers to collect and transmit the light.

A calibration using particles with different VSFs confirms that the theory presented correctly explains the operation of the instrument. The effects of attenuation were investigated, and a simple algorithm was presented that will provide some correction for the effects of attenuation in the measurements. For attenuations less than $1{\mathrm{m}}^{-1}$, the corrected measurements will have errors of less than a few tenths of $1\mathrm{\%}$. At higher attenuations (up to $10{\mathrm{m}}^{-1}$), errors can be a few percent.

Appendix A: Window Considerations

The aperture to the detector must be sealed with a transparent window. At the same time, it is crucial that the window not introduce a significant bias into the angular detection sensitivity. Figure 12 shows the front portion of the instrument with three window possibilities. Figure 12a is reproduced from Fig. 8 and is the implementation used in our prototype instrument.

Figure 12b shows a window in the form of a cylindrical quartz ring whose cross section is a right triangle with the inner acute angle designated ψ. The window surface in contact with the water is the inner face of the cylindrical ring. Light backscattered at angles from $90°$ to $180°$ will be incident on this surface at angles from $0°$ to $90°$, respectively. For unpolarized light, the reflectivity at this water/quartz interface increases from 0.18% for light incident at $0°$, to 0.32% at $45°$, to 100% at $90°$. To evaluate the seriousness of these reflections, first recall Fig. 4. It shows the observation solid angle as a function of the distance z at which backscattering occurs. Now, the angle of incidence on the window is given by $\theta -90°$ where θ, the backscattering angle, is related to z by $z=R/\tan \theta$. Hence, the fraction of light lost due to reflection at the input window can be included as an effective reduction in the observation solid angle as a function of the distance z; this reduction as a function of z is shown in Fig. 13.

The maximum reduction in observation solid angle due to reflection loss at the window actually occurs at normal incidence (when $z=0$). At normal incidence, the decrease is $0.0023\mathrm{sr}$, which is only a 0.18% decrease. As the angle of incidence increases, the percent decrease will monotonically increase, but the actual net decrease in the observation solid angle never exceeds the decrease at $0°$. At angles of incidence greater than $70°$ ($z=0.027\mathrm{m}$), the actual decrease in the solid angle goes into a steady decline to zero when the angle of incidence is $90°$ (where reflection is 100%). Basically, the inherent decrease in the observation solid angle dominates the reflection losses. Variations in the reflection losses at the second surface of the quartz ring in Fig. 12b can be minimized ($\approx 2\%$) by appropriate choice of the angle ψ. However, perhaps the best way to eliminate angle-of-incidence dependencies is by frosting this second surface. Frosting reduces transmission, but there is plenty of signal and the resulting diffuse transmitted light is perfectly acceptable for input to the integrating cavity.

Finally, the window example shown in Fig. 12c is perhaps optimum. Although light transmission will be less, the Teflon diffuser at the input surface eliminates angle-of-incidence dependencies. The use of Teflon for such purposes has been tested [33]. Again, the back surface of the quartz plate can be frosted.

We gratefully acknowledge support from the Robert A. Welch Foundation under grant A-1218, the U.S. Army/REDCOM Edgewood Chemical and Biological Center Aberdeen Proving Ground under contract W911SR-08-C-0019, and the George P. Mitchell Chair in Experimental Physics. We also thank George Kattawar, Deric Gray, and Yu You for many helpful discussions and comments. Finally, we thank Jim Sullivan for some very valuable suggestions on the final manuscript.

Table 1. Deviations from $b_b$ That Would Be Observed for Measurements Made with This Instrumentation in Natural Waters or Suspensions of Microspheres^a

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Table 2. Numerical Evaluations of Eq. (25, 27), and the Various Terms in Eq. (28) Using MVSM Data for $\mathit{\beta }(\mathit{\theta })$ Obtained during October 2009 in Chesapeake Bay [28, 29]^a

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Fig. 1 Conceptual sketch of instrument (cylindrical symmetry around z axis).

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Fig. 2 The functions ${\theta }_1(z)$, ${\theta }_2(z)$ and the functions $z_1(\theta )$, $z_2(\theta )$ are the same curves, respectively, i.e., for the latter, z is read off the horizontal axis as a function of θ on the vertical axis. The light gray and dark gray areas correspond to the domains of integration of the first and second integral in Eq. (9).

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Fig. 3 Domains of integration for the integrals in Eq. (14).

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Fig. 4 Detection solid angle $\mathrm{\Omega }(z)$ with $R=0.01\mathrm{m}$ and $Z_0=0.001\mathrm{m}$.

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Fig. 5 Coefficient of $\beta (\theta )$ in the integrand of Eq. (27).

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Fig. 6 Dependence on attenuation c of the percent deviation from $b_b$ that would be observed in a measurement of $P^{\prime }(c)$, Eq. (25).

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Fig. 7 Percent deviation from $b_b$ of a measurement of $P^{\prime }(c)$ after multiplication by the attenuation correction factor ($1+2.7cR$) to obtain $P_A^{\prime }(c)$.

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Fig. 8 Instrument implementation—cylindrical symmetry around laser beam (except for fibers).

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Fig. 9 “Placebo” experiment showing a slight signal increase as samples are added.

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Fig. 10 Measured data for aqueous suspensions of polymer microspheres with diameters $d=203$, 596, and $3005\mathrm{nm}$. The least-squares fits of the data to the straight lines are also shown.

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Fig. 11 Attenuation effects on the measured signal.

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Fig. 12 Three possibilities for the window at the aperture to the detector.

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Fig. 13 Effective reduction in the observation solid angle as a function of z due to reflection at the quartz window in the aperture of Fig. 12b.

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