Particle size and refractive index retrieval from the ... - OSA Publishing

Particle size and refractive index retrieval from the backscattering spectrum of white light using the Twomey iterative method: simulation and experiment Jing Zhou, Fred Moshary, Barry Gross, and Samir Ahmed

The Twomey iterative method has been applied to the retrieval of hydrosol microphysical properties. In particular, we focused on the retrieval of single and multimode particle size distributions from both simulated and experimental backscattering spectra in the 400–800 nm wavelength range. Assuming a known refractive index, both single-mode and multimode distributions were successfully retrieved through the introduction of an initial distribution biased toward larger particles. The simulation results were experimentally verified with standard polystyrene particles suspended in water within the diameter range of 0.2–2 ␮m for both narrow and broad monomodal distributions as well as more complicated multimode distributions. Finally, the technique was extended to the retrieval of an unknown refractive index. © 2006 Optical Society of America OCIS codes: 290.1350, 290.3200.

1. Introduction

Light-scattering techniques for the retrieval of particle properties have attracted a great deal of attention over the past 30 years. They have been proven useful for the remote sensing of aerosols and hydrosols in environmental science,1,2 monitoring combustion processes, and characterizing cells for pharmaceutical and medical purposes.3– 6 Light scattering has significant advantages over other sizing methods such as electron microscopy and sieving because it can provide noncontact high-quality in situ measurements in real time. Laser-based optical systems have been used to obtain precise multiangle and兾or multiwavelength scattering profiles that can be analyzed to resolve particle sizes.1,7,8 However, these approaches require complicated geometric configurations, expensive and bulky equipment, and demanding and time-consuming adjustments. Recently, spectral extinction and backscattering measurements using compact fiber probes coupled to white-light sources have generated much interest in

The authors are with the Optical Remote Sensing Laboratory, Department of Electrical Engineering, City College, City University of New York. B. Gross’s e-mail address is [email protected]. Received 16 September 2005; revised 20 January 2006; accepted 10 March 2006; posted 19 April 2006 (Doc. ID 64852). 0003-6935/06/266876-10$15.00/0 © 2006 Optical Society of America 6876

APPLIED OPTICS 兾 Vol. 45, No. 26 兾 10 September 2006

particle sizing applications.3,5,9,10 The compact optical arrangement and inexpensive design are ideal for in situ and online measurement in water. Particle size retrievals from light scattering usually involve the solution of the ill-posed Fredholm equation of the first kind. This is notoriously difficult in that more than one solution can satisfy this equation within a given measurement error. It follows that small changes in the measurement data can result in large variations in the solution that may be completely unphysical though mathematically acceptable. Various numerical schemes have been developed to find optimal physical solutions.11,12 To reduce the instability due to noise, it is often sufficient to restrict the retrieved distribution to a parameterized analytic function such as Gaussian or lognormal distribution and apply nonlinear least-squares fitting to optimize the distribution parameters.3–5 However, in most cases of interest, such shape-constraint methods impose too many assumptions on the actual nature of the size distribution and do not allow for multiple modes. Alternatively, since the backscatter spectrum exhibits oscillations that depend on particle size, a Fourier transformation of the spectra can be also used to retrieve size information, but it is mainly applicable to nearly monodispersed particle analysis.6,8 In a more general approach, constrained linear inversion (regularization) methods have been used to help stabilize the linear retrieval problem13,14 and applied to retrieve atmospheric aerosols from lidar data.1 To obtain an op-

timal solution, a careful balance between the smoothness of the retrieved solution and the degree in which the physical model reproduces the measurement data must occur. This scheme is referred to as regularization, and the balance between smoothness and goodness of fit is quite complex and hardly unique. It becomes even more problematic when multimode distributions are considered. Other approaches include stochastic optimization methods such as the inverse Monte Carlo15,16 (IMC) and genetic algorithms17 that have been applied to determine particle size distribution (PSD) from both angular scattering and wavelength-dependent extinction and backscatter. These methods in general have the advantage of ensuring positivity while smoothness is maintained by appropriate termination conditions. These approaches are nevertheless, computer intensive. To avoid time-consuming calculations, an iterative inversion method such as the modified Chahine algorithm18 can be used to obtain PSD from spectral extinction data. However, the latter requires that the kernel response function has a smooth kernel with a single maximum value that is not appropriate for highly oscillatory backscatter kernels. To eliminate the restrictions of the Chahine method, another iterative approach proposed by Twomey that does not require a highly peaked kernel function was considered. The Twomey iterative method (TIM) had been applied to analysis of data from indirect measurement techniques including multiwavelength extinction coefficient measurements19 and cascade impactor measurements in aerosol science,11 but the application of this approach to retrieve hydrosol microphysics from the backscattering spectrum of white light had not been explored. In this paper the TIM was applied to the retrieval of PSD from the backscatter spectrum over the wavelength range 0.4 ⱕ ␭ ⱕ 0.8 ␮m. The TIM is described in Section 2. In Section 3, we report on the extensive tests of the TIM using computer simulated backscatter spectra generated from both single-mode and multimode distributions. Within this context, the TIM’s sensitivity to measurement noise is examined. In Section 4, experimental backscatter spectra for both monodispersed and polydispersed distributions of polystyrene particles suspended in water were measured and used to validate the numerical procedure for retrieving the hydrosol size distribution. In Section 5, an iterative procedure is introduced to obtain the average particle refractive index for a monodispersed system, and it is applied to both simulated and experimental data. Conclusions are given in Section 6. 2. Twomey Iterative Method

For a system of spherical particles in the single scattering regime, the backscatter spectrum b共␭兲 (per unit length) and the particle size distribution f共r兲 can be related by the following Fredholm integral equation of the first kind20:

b共␭兲 ⫽ N0

冕

⬁

Qb共␭, r, n兲␲r2f共r兲dr,

(1)

0

where Qb is the Mie backscatter efficiency at particular wavelength ␭ for particles with radius r and refractive index n (relative to the surrounding medium). N0 is the total number density of the particles, and f共r兲 is the normalized size distribution function such that 冕0⬁ f共r兲dr ⫽ 1. In practice, b共␭兲 is determined on a set of M discrete wavelengths ␭1, ␭2, . . . , ␭M and the distribution function f共r兲 has finite lower and upper limits. If we divide the finite size range 关rmin, rmax兴 into N intervals with equal spacing ⌬r ⫽ 共rmax ⫺ rmin兲兾N and approximate f共r兲 on the jth interval 关rj, rj⫹1兴 by f共rj ⫹ ⌬r兾2兲, Eq. (1) can be expressed in the following linear discrete form: b共␭i兲 ⫽ bi ⫽ N0

N

兺 Kijfj j⫽1

i ⫽ 1, 2, . . . , M,

(2)

where Kij ⫽

冕

rj⫹1

Qb共␭i, r, n兲␲r2dr

j ⫽ 1, 2, . . . , N,

rj

(3) fj is the number density of particles within the size range 关rj, rj⫹1兴, and rj ⫽ rmin ⫹ 共j ⫺ 1兲⌬r. Unfortunately, using the number density may result in unstable retrievals since even for relatively high number densities at smaller radii particles, the scattering can be dominated by large particulates. To help reduce this problem, a weighting function w共r兲 can be used to modify the inversion process. In particular, it has been shown1 that retrieving the particle volume density, v共r兲 ⫽ w共r兲f共r兲, where w共r兲 ⫽ 共4兾3兲␲r3, is more stable. In the volume representation bi ⫽ N0

N

兺 K៮ ijvj, j⫽1

(4)

where

冕

៮ ij ⫽ K

rj⫹1

rj

Qb共␭i, r兲␲r2 dr, w共r兲

and N0 is the constant total number density. Hereafter, we first solve for v共r兲 and then transform back to particle size distribution via fj ⫽

vj . w共rj ⫹ ⌬r兾2兲

The TIM begins with an initialization of the distribution vj0. In the kth iteration, bîk is the calculated result from vjk as N

៮ ijvj k i ⫽ 1, 2, . . . , M. bî k ⫽ 兺 K

(5)

j⫽1

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normal PSD function: f共r; rm, ␴兲 ⫽

1 r␴冑2␲

冋

exp ⫺

共ln r ⫺ ln rm兲2 2␴2

册

,

(8)

where rm is the mode radius, and the ␴ is the logarithm of the width of the distribution. When ␴ is small, the corresponding standard deviation of the size distribution 共␦兲 is defined as

␦⫽

冑冕

⬁

共r⫺r៮兲2f共r兲dr,

(9)

0

Fig. 1. Flowchart of the TIM.

Next, by comparing bî k with the actual measurement data bi and introducing high-frequency components that are spectrally similar to the kernels into the solution, vjk⫹1 for the next iteration is recalculated as k⫹1

vj

⫽

再冋冉

冊册冎

bi ៮ ij 1⫹ ⫺1 K ˆbi k

M

兿 i⫽1

k

vj .

冑兺冉冊 N

i⫽1

bi ⫺ bîk , bîk 2

␹2 ⫽

冑兺共v M

j⫽1

⫺vjk兲2,

k⫹1

j

(7)

where ␹1 is root-mean-error (rme) that describes the average relative deviation of the fitted data from the measured data, and ␹2 measures the average difference in vj between consecutive iterations. The iteration stops when either ␹1 is less than a specified value chosen to be at approximately the input noise level or ␹2 reaches a very small value (10⫺4 was used here). The iteration flow is shown in Fig. 1. 3. Simulated Retrieval A.

Implementation

The computer-generated spectra were calculated based on Eq. (1) by assuming a two-parameter log6878

b˜共␭兲 ⫽ b共␭兲共1 ⫹ ␥␧兲,

(6)

¯ is always Note that since bi兾bîk ⫺1 ⱖ ⫺1 and K ij less than 1, the adjustment factor is always greater than zero so that vjk can never turn negative with a positive initial vj0. Also, to eliminate calibration issues, only the shape of the backscatter spectrum is considered here so that N0 can be absorbed by normalizing bi and bî. The following two metrics were used to test for convergence: 1 ␹1 ⫽ N

and is approximately ␴rm, where rm is the mean size radius and is nearly equal to the first moment of the distribution r៮ ⫽ 冕0⬁ rf共r兲dr. Unless otherwise stated, as in Section 5, the refractive index of the particles is fixed at np ⫽ 1.589 for all the wavelength values within the spectral window. Considering that the average refractive index of water is 1.33, the relative particle refractive index is n ⫽ 1.1917. To investigate the sensitivity and stability of TIM retrieval for various signal-to-noise ratios (SNRs) different levels of random noise were superposed on the generated data b共␭兲 to simulate real measurement errors:


(10)

where ␥ is the relative noise level to the signal (⫽ 1兾SNR) and ε is a normally distributed random number with zero mean and unit standard deviation. The inversion is repeated with 50 different noise ensembles to determine the statistics in the retrieval process. In this work, the size (radius) range was set to 关0.1–2 ␮m兴. This is reasonable given that the backscattering spectrum provides the most information when the particle sizes are of the same order as the measurement wavelengths. The wavelength range from 0.4 to 0.8 ␮m is sampled at 5 nm resulting in a total of N ⫽ 81 points over the wavelength range. This sampling frequency is sufficient to sample the oscillatory features of the backscatter spectrum for all monodispersed particles in the given size range. To determine the number of size bins over the size range considered, it is important to note that, on one hand, a large number of size bins introduces instability into the inversion, whereas on the other, the ability to resolve a narrow distribution is determined by the interval between the grid points. We have found that the most robust choice in our simulations were to make the number of radius bins equal to the number of wavelength bins so M ⫽ N ⫽ 81. B.

Initialization and Numerical Properties

If the initial guess is chosen so that bi兾bî 0 is close to 1, there will be little correction and the iteration will stop immediately since the correction factor ap-

Fig. 2. Uniform and power-law initialization of PSD f(r).

proaches unity. It is often common to assume a uniform distribution for the initialization. However, through numerical simulations, we observed that this choice of initial distribution tends to produce final results that overestimate the value at small size bins because of their small scattering cross section. In other words, the initial distribution should not be homogeneous over the size window since small radii contribute little to the resultant backscatter spectrum. Instead we use a power distribution rp, where p is an exponent usually chosen to be p ⫽ 6, for the retrieval. However, in the presence of high noise of the order of 30%, it may be necessary to use an even higher exponent. The initialization mentioned here is for f共r兲 instead of volume density distribution v共r兲; therefore the initial v共r兲 used in the simulation is rp⫹3. The initial PSD of uniform and power-law functions were plotted together in Fig. 2. Since absolute particle number density requires absolute calibration, we choose to normalize all PSDs to unit area. In all simulations, the residual ␹1 usually drops sharply within the first 100 iterations and then levels off as the noise level is reached. This threshold is almost always reached before the ␹2 condition is satisfied except for very small noise levels. To see how our algorithm speed compares to the IMC method, the processing time of the TIM algorithm Pentium 4 2.4 GHz machine inversion result including the uncertainty test of the noisy data can be obtained by the TIM in approximately 10 s, whereas 120 s were needed for the statistical methods such as IMC on a SGI ONYX兾R10000 mainframe.15 Considering that our machine is three times faster than the SGI computer in terms of floating point calculation ability, our method should be approximately four times faster than the IMC method. C. Single-Mode Distribution

A set of synthetic backscatter spectra were generated from monomodal lognormal PSDs by varying the modal radius in the range of rm ⫽ 关0.2, 1.8兴 ␮m with

Fig. 3. Ratio between retrieved and modal radius R ⫽ rret兾rm (left) and retrieved width (right) as a function of input modal radius rm. The solid line is the reference line for perfect retrieval and the dashed line (left) gives the maximum error bounds for power-law initialization. Triangles, power-law initialization; stars, uniform initialization. (a), (b) ␦ ⫽ 0.01 ␮m; (c), (d) ␦ ⫽ 0.05 ␮m; and (e), (f) ␦ ⫽ 0.1 ␮m.

a step size of 0.1 ␮m for three different standard deviations 共␦ ⫽ 0.01, 0.05, 0.1␮m兲 to test the TIM retrieval. In these simulations, both uniform and power-law initial distributions were used. The optical backscatter spectra were modified by adding 5% random noise when utilizing power-law initial distribution. Adding 5% noise to the uniform initialization resulted in very large error fluctuations, so for comparison, uniform initial distribution retrievals were performed only for the noise-free spectra. In Figs. 3(a), 3(c), and 3(e), the ratios between the retrieved radius and the modal radius R ⫽ rret兾rm are plotted as functions of rm. The retrieved distribution widths in terms of standard deviation ␦ret versus rm are also displayed in Figs. 3(b), 3(d), and 3(f) for three input ␦ values [Figs. 3(a) and 3(b), ␦ ⫽ 0.01 ␮m; Figs. 3(c) and 3(d), ␦ ⫽ 0.05 ␮m; and Figs. 3(e) and 3(f), ␦ ⫽ 0.1 ␮m). In general, we observe that the power-law initialization is far more accurate in retrieving the mean particle radius than the uniform initialization. The rret obtained from the uniform initial guess is smaller than the input rm except for sufficiently narrow distributions and small radii. This can be explained by realizing that relatively large values in small radii bins contribute little to the resultant backscatter spectrum and therefore limit the sensitivity of the TIM to make corrections in the small size bins. To see this more clearly, we plot in Figs. 4 – 6 the retrieved PSD together with the corresponding input distributions (solid curve) in the left panels and the fitted backscatter spectrum (dots) compared with the synthetic spectrum (solid curve) in the right panels for two selected particle sizes at each ␦ value: Fig. 4, rm ⫽ 0.3 and 1.8 ␮m with ␦ ⫽ 0.01 ␮m; Fig. 5, rm ⫽ 0.9 and 1.5 ␮m with ␦ ⫽ 0.05 ␮m; and Fig. 6, rm ⫽ 0.7 and 1.5 ␮m with ␦ ⫽ 0.1 ␮m. In these fig10 September 2006 兾 Vol. 45, No. 26 兾 APPLIED OPTICS

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Fig. 4. Retrieval results for two selected modal radii with distribution width ␦ ⫽ 0.01 ␮m. (a), (b) rm ⫽ 0.3 ␮m; (c), (d) rm ⫽ 1.8 ␮m. (a), (c) Retrieved PSD functions (triangles), exact distribution (solid curve). (b), (d) Input backscatter signal (solid curve), fitted backscatter spectrum (dots).

ures, the triangle symbols represent the retrieval results for the power-law initial distribution. For the purpose of illustration and to minimize confusion, the retrieved PSD using uniform initialization was shown only in Fig. 5 (the star symbol). As seen in Fig. 5, when uniform initialization is used, the iteration can reach the termination criteria even with spurious spikes located at small size bins. On the other hand, the inversion depends much less on the choice of initialization for distributions with small radii and narrow width (see the ␦ ⫽ 0.01 ␮m cases) whose backscattering spectrum is easier to resolve. To compensate for the overestimation at small size

Fig. 5. Retrieval results for two selected modal radii with distribution width ␦ ⫽ 0.05 ␮m. (a), (b) rm ⫽ 0.9 ␮m; (c), (d) rm ⫽ 1.5 ␮m. (a), (c) Retrieved PSD functions using power-law initialization (triangles) and uniform initialization (stars) as well as exact distribution (solid curve). (b), (d) Input backscatter signal (solid curve), fitted backscatter spectrum (dots). 6880


Fig. 6. Retrieval results with 5% noise for two selected modal radii with distribution width ␦ ⫽ 0.1 ␮m. (a), (b) rm ⫽ 0.7 ␮m; (c), (d) rm ⫽ 1.5 ␮m. (a), (c) Retrieved PSD functions (triangles), exact distribution (solid curve). (b), (d) Input backscatter signal (solid curve), fitted backscatter spectrum (dots).

bins without degrading the inversion results for small particle size distributions, an initial distribution biased toward the large sizes is required. As shown in Figs. 3(a) and 3(c), the power-law initial distribution effectively removes the spurious spikes at the lower end of the size window and retrieves the mean radius rret with much higher accuracy: within 5% for ␦ ⫽ 0.01 ␮m and 15% for ␦ ⫽ 0.05 ␮m. The retrieved width ␦ret for these two ␦ values [Figs. 3(b) and 3(d)] was also vastly improved when a power-law initialization was used. It should be mentioned here that this kind of initialization works for particulates of all sizes. One notable feature is the remarkable ability of the TIM to retrieve the size distributions for such a wide range of particle sizes considering that the size windows are significantly larger than the wavelength ranges of the backscattering spectra. This results from the inherent positivity constraint of the TIM and the compensating effect of the power-law initial distribution. As the width of the PSD function becomes broader, there are fewer and fewer features in the backscatter data within the limited spectral range. A direct result of this is that there will be a broader class of solutions that are indistinguishable from the backscatter spectra in terms of a noise level threshold within the residual ␹1. This is the case for ␦ ⫽ 0.1 ␮m, and the solution depends more strongly on the choice of the initial guess. The accuracy of the retrieved size radius is within 40% in this case. In Figs. 4 and 5, for small widths, we see that there is sufficient spectral structure to accurately retrieve the distributions. In Fig. 6, the TIM solution is far less accurate and only partially retrieves the PSD. However, even in this case, for sufficiently large particles, rm ⱖ 1.3 ␮m, the retrievals are much more accurate. This may be explained by observing that for

Fig. 7. Retrieved average radius rret (left) and standard deviation ␦ret (right) compared with the input values (solid line) as a function of noise for two modal sizes rm ⫽ 0.6 ␮m (triangles) and 1.2 ␮m (circles). ␦ ⫽ 0.01 ␮m.

a given absolute width, the fractional width becomes smaller with increasing mode size, resulting in more spectral features. D. Noise Sensitivity of Twomey Iterative Method Retrieval

Various noise levels were added to the backscatter data to test the sensitivity and stability of the TIM retrieval. Figure 7 shows the retrievals of mean radius rret and the standard deviation ␦ret for two sharp distributions 共␦ ⫽ 0.01 ␮m兲 with radii rm ⫽ 0.6 and 1.2 ␮m for variable noise levels up to 30%. For rm ⫽ 0.6 ␮m, errors in the retrieval are negligible over the noise range considered. However, for rm ⫽ 1.2 ␮m particles, as the noise increases, the retrieval significantly underestimates the mean particle sizes and the error in the retrieved mean radius increases substantially as spurious spikes appear in the smaller particle size bins in the final PSD solution. The comparatively large uncertainties associated with the retrieval with noise level ⬎5% show that these spikes are strongly dependent on the noise. Therefore the power-law initial distribution can also ensure the stability of the solution by effectively reducing the possibility of spurious spikes at the lower end of the size windows. The retrieved distribution widths ␦ret were proportional to the noise since the iteration stops before it reaches the noise level. At the same noise level, the retrieved distribution for 0.6 ␮m particles shows much better accuracy and stability than that of 1.2 ␮m particles. Examining the backscatter spectra of these two particles, the oscillatory features for the 1.2 ␮m particle are sharper and shallower than that of 0.6 ␮m, which makes them more difficult to be resolved in the presence of the same level of noise. Figure 8 shows the retrieved PSD functions for rm ⫽ 0.6 ␮m (left) and 1.2 ␮m (right) with ␦ ⫽ 0.05 ␮m with three different noise levels 共5%, 15%, 30%兲 added to the simulated backscattering spectra. As expected, the retrieval de-

Fig. 8. Retrieved PSD (stars) with three different levels of noise for two distributions ␦ ⫽ 0.05 ␮m; rm ⫽ 0.6 ␮m (left) and 1.2 ␮m (right). (a), (b), 5%; (c), (d), 15%; (e), (f), 30%. Solid curve, exact distribution.

grades at higher noise levels, but the bell shape of the input distributions can still be recovered with noise levels as high as 30%. The typical values of noise encountered in our laboratory experiments (described in Section 4) were approximately 3%. E.

Multimode Distribution

In this subsection, we apply the TIM to the retrieval of a mixture of four single-mode lognormal distributions fi共r, ri, ␴兲 with modal width ␴ ⫽ 0.01: 4

f共r兲 ⫽ 兺 ci fi共r; ri, 0.01兲.

(11)

i⫽1

The relative amplitudes ci were chosen for simplicity to be equal, and c was normalized to unity (i.e., ci ⫽ 0.25). Two different four-mode distributions are considered: (i) r ⫽ 0.5, 0.6, 0.7, 0.8 ␮m, (ii) r ⫽ 1.4, 1.5, 1.6, 1.7 ␮m. Figure 9 is the retrieved PSD from synthetic backscattering spectra with 1% [Figs. 9(a) and 9(b)] and 5% [Figs. 9(c) and 9(d)] noise level, respectively. It can be seen from Fig. 9 that the TIM can distinguish the correct location of all the multiple peaks for both distributions. The accuracy of the retrieved relative amplitudes of each mode is limited by the resolution of the size bins. For higher noise 共5%兲, the four peaks are still distinguishable, but large uncertainties in small size bins occur for the case (ii) distribution. This is again due to the fact that the shallower and sharper oscillatory patterns in backscattering spectra for large particles are more affected by noise. 4. Experimental Retrieval A.

Experimental Setup

Ten aqueous suspensions (w ⫽ 10% solids by weight) of monodispersed polystyrene microspheres (5000 Series Duke Scientific) were used to test the retrieval 10 September 2006 兾 Vol. 45, No. 26 兾 APPLIED OPTICS

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Fig. 9. Retrieved PSD (stars with broken curve) for two fourmode distributions: (a), (c), r ⫽ 0.5, 0.6, 0.7, 0.8 ␮m; (b), (d), r ⫽ 1.4, 1.5, 1.6, 1.7 ␮m, compared with the exact distributions (solid curve).

of narrow monomode distributions. The diameters of the microspheres were 0.2, 0.5, 0.67, 0.74, 0.82, 0.87, 0.93, 1.0, 1.3, and 2.02 ␮m with size fluctuations within 3%–5% of the nominal diameters as stated by the manufacturer. The refractive index of polystyrene is 1.589 at 589 nm and its density is ␳s ⫽ 1.05 g兾cm3. Vs ⫽ 20 ␮l of each sample was diluted in V ⫽ 200 ml of distilled water. This 200 ml diluted solution was placed in a deep, thin black container so that the influence of the container walls on scattering signal was negligible. This was experimentally verified. The number concentration of particles in the solution is estimated by N0 ⬇ Vs␳w兾␳sVv. ␳ is the water density 共1.0 g兾cm3兲 and v is the volume of a single particle 共4兾3兲 ␲r3. To obtain experimental data for the sample with a broad distribution, we mixed seven monodispersed microspheres with various concentrations to approximate the lognormal distribution with rm ⫽ 0.41 ␮m and ␴ ⫽ 0.1 assuming that the distribution for each of the standard samples is a lognormal distribution with ␴ ⫽ 0.03, as shown in Fig. 10. Among these seven samples, six of them were from Duke Scientific (with diameters of 0.67, 0.74, 0.81, 0.87, 0.93, and 1.0 ␮m) and were also used for narrow distribution testing. One was from Polyscience with a mode diameter of dm ⫽ 0.77 ␮m, ␦ ⫽ 0.025 ␮m, and a solid concentration of w ⫽ 2.62%. All the samples (described above) were sufficiently diluted to be in the single scattering regime. This was verified through linearity tests including repeated dilution and superposition of component signals. The extinction coefficients for all cases were less than 0.25 cm⫺1. The experimental setup is shown in Fig. 11. A backscatter fiber probe from Ocean Optics Inc. (FDP14UV200-11-5-Ref) was used for the measurement. The probe tip has one fiber in the center collecting the backscatter signal into the spectrometer (Ocean Optics, S2000) and six fibers around it delivering the white light to the sample. One additional fiber is split 6882


Fig. 10. Broad distribution by superposition of several narrow distributions: dm ⫽ 0.67, 0.74, 0.77, 0.81, 0.87, 0.93, and 1.0 ␮m, with modal width ␴ ⫽ 0.03.

from the illumination fiber bundle for light source monitoring. The diameters for all the fibers were 200 ␮m and the fiber NA was 0.22. The light source was a 150 W dc regulated QTH fiber optic illuminator from Dolan Jenner (DC-950) and was coupled through a focusing lens into the fiber. To avoid the specular reflections, the fiber probe was tilted by 10° and placed just beneath the surface of the sample solutions. The backscatter signal spectrum was collected over the wavelength range of 0.4–0.75 ␮m (limited by the light source) and 71 wavelength points in this range, at intervals of 5 nm, were used for the retrievals. The collected signal Ic was spectrally calibrated against the backscatter spectrum from a

Fig. 11. Experimental arrangement: White light is delivered to the sample container by the six illuminating fibers. The backscattered signal is then collected by one fiber in the center of the probe and directed to the spectrometer. To avoid the specular reflections, the fiber probe is titled by 10° and placed just beneath the surface of the sample solutions. A fraction of the white-light power is delivered directly to the spectrometer by the reference fiber for light source monitoring. All the fibers have a diameter of 200 ␮m and a NA of 0.22.

Table 1. Narrow Distribution Retrieval from Experimental Backscattering Data

Sample diameter (␮m) Retrievals

d (␮m) ␦ret (␮m)

0.2

0.5

0.67

0.74

0.82

0.87

0.93

1.0

1.3

2.02

0.78 0.03

0.51 0.01

0.67 0.03

0.76 0.03

0.84 0.04

0.89 0.04

0.93 0.04

1.03 0.06

1.19 0.07

2.03 0.07

Spectralon plate (Labsphere PT# SRT-99-020) Is after subtracting the dark background Ib so that bexp ⫽ 共Ic ⫺ Ib兲兾共Is ⫺ Ib兲. Again, since only the relative backscatter spectrum matters, bexp is normalized such that the norm of the vector containing 71 measured backscatter points is unity. B.

Results

The retrieved mean diameters and the standard deviations are given in Table. 1 for all the samples studied. Except for the 0.2 ␮m particle, which is at the edge of the retrieval window, the recovered mean sizes are consistent with nominal sizes to within 8% accuracy, which is reasonable considering the claimed 3%–5% size uniformity of these samples. The retrieved PSDs and the fitted backscatter spectra are also plotted in Fig. 12 for three samples: 0.5, 0.93, and 0.02 ␮m. The backscatter spectra calculated from the output PSD function match with the measured data with a fitting residual of less than 10%. The retrieval results for the broad distribution are shown in Fig. 13. For comparison we have also plotted the experimental data and the retrieved spectra on the same graph. As seen from Fig. 10, due to the limited availability of diameter sizes for standard particles, the resulting mixture is not exactly a lognormal distribution but has a dip resulting in a bimodal shape that is retrieved. The retrieved PSD has the mean radius rret ⫽ 0.436 ␮m and ␦ret ⫽ 0.052 ␮m. Finally, we mixed four of the monodispersed par-

Fig. 12. Retrieved PSD (left) from the experimental backscatter signals (right, solid curve with circles) for three samples with nearly monodispersed distribution. (a), (b), dm ⫽ 0.5 ␮m; (c), (d), dm ⫽ 0.93 ␮m; (e), (f), dm ⫽ 2.02 ␮m. For comparison, the fitted backscatter data (dots) are also plotted in (b), (d), and (f).

ticles with different concentrations. The four samples chosen were 0.5, 0.67, 0.82, and 1.0 ␮m in diameter and their relative mixing ratios are proportional to 1兾r3: 8.5:3.8:2:1. We plot in Fig. 14 the retrieved PSD. The TIM can locate the four peaks accurately, while the discrepancy in the relative amplitude among the modes is due mainly to the limited resolutions of the discrete size bins, which is consistent with the simulation results. 5. Refractive Index Retrieval

Up to this point, the refractive index of the particles was assumed to be known np ⫽ 1.589. However, in many actual experimental situations, the refractive index may not be known a priori. Fortunately, the oscillatory features in the backscatter spectral data from particles with narrow distribution are sensitive to the relative refractive index. Taking advantage of this property, several refractive indices can be tried in the inversion and the refractive index with the minimum fitting residual ␹1 would then be the retrieved refractive index. This approach has already been suggested,15 but the speed and simplicity of TIM retrieval compared with the IMC method makes this procedure more efficient. The usefulness of this method is tested on four simulated narrow distributions with rm ⫽ 0.25, 0.6, 1.2, and 1.8 ␮m and with their modal widths ␴ adjusted so that all of them have the same standard deviation ␦ ⫽ 0.01 ␮m. The backscattering spectra were generated by using three different values of relative refractive index: 1.05 (rep-

Fig. 13. Retrieved PSD (left) from the experimental backscatter signals (right, solid curve with circles) for broad distribution: rm ⫽ 0.41 ␮m, ␴ ⫽ 0.1, and rret ⫽ 0.436 ␮m, ␦ret ⫽ 0.052 ␮m. For comparison, the fitted backscatter data (dots) are also plotted. 10 September 2006 兾 Vol. 45, No. 26 兾 APPLIED OPTICS

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Fig. 14. Multimode distribution retrieval from experimental backscatter data.

resentative of algal particles), 1.113, and 1.196; 5% noise was added to the spectra. Independent TIM inversions for 15 evenly spaced values of real relative refractive index of the particles in the range of 1.02 to 1.3 were carried out. The residual ␹1 as a function of refractive index is plotted in Fig. 15. Only when the refractive index used in the retrieval is close to the actual particle refractive index does the residual reach the minimum, except for the smallest particles 共0.25 ␮m兲. The retrieved refractive index was 1.04 for input index n ⫽ 1.05, in the range of 1.1–1.12 for n ⫽ 1.113 and in the range of 1.18 –1.2 for n ⫽ 1.196. The error was less than 10% and was limited by the grid spacing of the refractive index. In the case of the 0.25 ␮m particles, ␹1 approached the noise level at multiple refractive indices. This was due to the limited oscillations found in the backscatter spectra of small particles making the fitting residuals less sensitive to the particle refractive index used in the retrieval. For the experimental verification, two samples, the first comprising polystyrene particles 共d ⫽ 2.02 ␮m, n ⫽ 1.196兲 and the second comprised

Fig. 15. Refractive index retrieval of simulated backscatter spectra for four size distributions with ␦ ⫽ 0.01 ␮m and three different relative refractive indices. Pluses (⫹), rm ⫽ 0.25 ␮m; circles (°), rm ⫽ 0.6 ␮m; crosses (⫻), rm ⫽ 1.2 ␮m, and stars (*), rm ⫽ 1.8 ␮m. The retrieved refractive indices are indicated by the arrows: n ⫽ 1.05 (left), 1.113 (middle), and 1.196 (right). 6884


Fig. 16. Refractive index retrieval of experimental data. Triangles, polystyrene particles with d ⫽ 2.02 ␮m and n ⫽ 1.196 from Duke Scientific; stars, polymethacrylate particles with d ⫽ 1 ␮m and n ⫽ 1.113 from Sigma Aldrich.

of polymethacrylate particles (d ⫽ 1 ␮m and n ⫽ 1.113) from Sigma Aldrich were used to test this refractive index retrieval procedure. It can be seen from Fig. 16 that both the refractive indices and the mean diameters were recovered fairly well with n ⫽ 1.2, d ⫽ 2.01 ␮m and n ⫽ 1.1, d ⫽ 1.1 ␮m for polystyrene and polymethacrylate particles, respectively. 6. Conclusion

The TIM has been applied to the retrieval of various particle size distributions including both single-mode narrow and broad distributions as well as multimode distributions from both simulated and experimental backscatter spectra in the range of 0.4–0.8 ␮m. This method has a built-in positivity constraint, and no a priori information about the size distributions is needed for the retrieval. Fast retrieval computation speed and lack of any requirements for peaked kernels make this technique suitable for the online size analysis of waterborne particles by using broadband optical scattering. With an initial guess distribution that is biased toward large sizes, single-mode PSDs can be recovered when the size ranges lie within the numerical retrieval window 关0.1, 2 ␮m兴 and the standard deviation ␦ of the size distributions is less than 0.1 ␮m. This large dynamic range is much wider than we would expect from the measurement spectral range 共0.4–0.8 ␮m兲. Numerical simulation also shows that the TIM is an effective and stable technique for inverting data containing random noise. For simple monomode hydrosols, the TIM can be used to determine the size distributions with widths up to 0.05 ␮m even when the data is corrupted by 30% noise. Furthermore, mixtures containing up to four single modes have also been successfully retrieved by this method both numerically and experimentally. To find the particle refractive index, independent inversions at many refractive index values were performed, and the optimal index was chosen by minimizing the error residual ␹1. This procedure was successfully applied to both simulated and experimental spectra for narrow distributions, and the re-

fractive index ranging from 1.05 to 1.2 was obtained within an accuracy of 10% except for particles with radii close to 0.1 ␮m. This work was partially supported under the Center for Optical Sensing and Imaging (COSI) under NASA grant NCC-1-03009.

References 1. D. Muller, U. Wandinger, and A. Ansmann, “Microphysical particle parameters from extinction and backscatter lidar data by inversion with regularization: theory,” Appl. Opt. 38, 2346 – 2357 (1999). 2. K. S. Shifrin, “Spectral attenuation and aerosol particle size distribution,” Appl. Opt. 35, 2114 –2124 (1996). 3. L. T. Perelman, V. Backman, M. Wallace, G. Zonios, R. Manoharan, A. Nusrat, S. Shields, M. Seiler, C. Lima, T. Hamano, I. Itzkan, J. Van Dam, J. M. Crawford, and M. S. Feld, “Observation of periodic fine structure in reflectance from biological tissue: a new technique for measuring nuclear size distribution,” Phys. Rev. Lett. 80, 627– 630 (1998). 4. V. Backman, “Polarized light scattering spectroscopy for quantitative measurement of epithelial cellular structure in situ,” IEEE. Sel. Top. Quantum Electron. 5, 1019 –1026 (1999). 5. M. Bartlett, G. Huang, and L. Larcom, “Measurement of particle size distribution in mammalian cells in vitro by use of polarized light spectroscopy,” Appl. Opt. 43, 1296 –1307 (2004). 6. A. Amelink, M. P. L. Bard, S. A. Burgers, and H. Sterenborg, “Single-scattering spectroscopy for the endoscopic analysis of particle size in superficial layers of turbid media,” Appl. Opt. 42, 4095– 4101 (2003). 7. A. P. Nefedov, “Analysis of particle sizes, concentration, and refractive index in measurement of light transmittance in the forward-scatter-angle range,” Appl. Opt. 36, 1357–1366 (1997).

8. K. A. Semyanov, “Single-particle sizing from light scattering by spectral decomposition,” Appl. Opt. 26, 5110 –5115 (2004). 9. Y. Kitagawa, “Fiber-optic particle size monitor based on whitelight scattering,” Appl. Opt. 31, 859 – 865 (1992). 10. M. Canpolat, “Particle size analysis of turbid media with a single optical fiber in contact with the medium to deliver and detect white light,” Appl. Opt. 40, 3792–3799 (2001). 11. M. Kandlikar and G. Ramachandran, “Inverse methods for analyzing aerosol spectrometer measurements: a critical review,” J. Aerosol Sci. 30, 413– 437 (1999). 12. S. Twomey, Introduction to the Mathematics of Inversion in Remote Sensing and Indirect Measurement (Elsevier, 1977). 13. M. D. King, D. M. Byrne, B. M. Herman, and J. A. Reagan, “Aerosol size distributions obtained by inversion of special optical depth measurements,” J. Atmos. Sci. 35, 2153–2167 (1978). 14. P. C. Hansen, Rank-Deficient and Discrete Ill-Posed Problems: Numerical Aspects of Linear Inversion (SIAM, 1998). 15. D. A. Ligon, J. B. Gillespie, and P. Pellegrino, “Aerosol properties from spectral extinction and backscatter estimated by an inverse Monte Carlo method,” Appl. Opt. 39, 4402– 4410 (2000). 16. D. A. Ligon, T. W. Chen, and J. B. Gillespie, “Determination of aerosol parameters from light-scattering data using an inverse Monte Carlo technique,” Appl. Opt. 35, 4297– 4303. 17. M. Ye, S. Wang, Y. Lu, T. Hu, Z. Zhu, and Y. Xu. “Inversion of particle-size distribution from angular light-scattering data with genetic algorithm,” Appl. Opt. 38, 2677–2685 (1999). 18. F. Ferri, “Modified version of the Chahine algorithm to invert spectral extinction data for particle sizing,” Appl. Opt. 34, 5829 –5829 (1995). 19. R. Hitzenberger and R. Rizzi, “Retrieved and measured aerosol mass size distributions: a comparison,” Appl. Opt. 25, 546 –553 (1986). 20. H. C. Van De Hulst, Light Scattering by Small Particles (Dover, 1981).

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