## Abstract

In article I of this series, calculations and graphs of the depolarization ratio, $D(\mathrm{\Theta},\lambda )=1-<{S}_{22}>/<{S}_{11}>$, for light scattered from an ensemble of single-aerosolized *Bacillus* spores using the discrete dipole approximation (DDA) (sometimes also called the coupled dipole approximation) were presented. The $Sij$ in these papers denote the appropriate Mueller matrix elements. We compare graphs for different size parameters for both $D(\mathrm{\Theta},\lambda )$ and the ratio ${R}_{34}(\mathrm{\Theta},\lambda )=<{S}_{34}>/<{S}_{11}>$. The ratio ${R}_{34}(\mathrm{\Theta},\lambda )$ was shown previously to be sensitive to diameters of rod-shaped and spherical bacteria suspended in liquids. The present paper isolates the effect of length changes and shows that ${R}_{34}(\mathrm{\Theta},\lambda )$ is not very sensitive to these changes, but $D(\mathrm{\Theta},\lambda )$ is sensitive to length changes when the aspect ratio becomes small enough. In the present article, we extend our analysis to vegetative bacteria which, because of their high percentage of water, generally have a substantially lower index of refraction than spores. The parameters used for the calculations were chosen to simulate values previously measured for log-phase *Escherichia coli*. Each individual *E. coli* bacterium appears microscopically approximately like a right-circular cylinder, capped smoothly at each end by a hemisphere of the same diameter. With the present model we focus particular attention on determining the effect, if any, of length changes on the graphs of $D(\mathrm{\Theta},\lambda )$ and ${R}_{34}(\mathrm{\Theta},\lambda )$. We study what happens to these two functions when the diameters of the bacteria remain constant and their basic shape remains that of a capped cylinder, but with total length changed by reducing the length of the cylindrical part of each cell. This approach also allows a test of the model, since the limiting case as the length of the cylindrical part approaches zero is exactly a sphere, which is known to give a value identically equal to zero for $D(\mathrm{\Theta},\lambda )$ but not for ${R}_{34}(\mathrm{\Theta},\lambda )$.

© 2009 Optical Society of America

## 1. Introduction

It is well known that a single particle or a cloud of such particles has all its light scattering information described by the 16 elements of the $4\times 4$ Mueller matrix (e.g., [1]). These matrix elements, ${S}_{ij}$, are functions of the scattering angles, the wavelength of the light, and the properties of the scatterers, including their size, shape, index of refraction, and, in the case of single particles or an oriented stream of single particles, each particle’s orientation with respect to the incoming light. In previous papers of this series [2, 3] the behavior of the graphs of two functions of the Mueller matrix elements versus scattering angles for randomly oriented bacterial endospores in air were studied. These two functions were the depolarization ratio, $D(\mathrm{\Theta},\lambda )=1-<{S}_{22}>/<{S}_{11}>$, and the ratio, ${R}_{34}(\mathrm{\Theta},\lambda )=\u3008{S}_{34}\u3009/\u3008{S}_{11}\u3009$, where the brackets indicate that averages were taken over many particles. Averages over orientation angles of the parti cles to simulate random orientation were taken in each case, and size averages were calculated where indicated.

An interpretation of these functions can be based on the definition of the Stokes vector entries [4]. At any scattering angle, for unpolarized input light, ${S}_{11}$ is proportional to the scattered intensity. ${S}_{11}\u2013{S}_{22}$ is a measure of the cross-polarizing elements of the single-particle amplitude matrices, so that $D(\mathrm{\Theta},\lambda )$ is related to the tendency of a scatterer to change the polarization for incoming polarized light. The function, ${R}_{34}$, measures the amount of the third Stokes vector element, *U*, moved to the fourth Stokes element, *V*, and is related to the transformation of linear to circular polarization. This interpretation can be misleading, however. For example, ${R}_{34}$ for an isotropic, nonchiral spherical scatterer has an interesting dynamic behavior as a function of size or wavelength, but the scattering does not give rise to circular polarization because the sphere has no handedness.

In [2, 3], two fairly similar shapes were used in modeling spores, and the results were compared. The shapes used were either a right-circular cylinder capped smoothly on each end with a hemisphere of the same radius, or a prolate spheroid with size pa rameters chosen to make the models comparable. It was found that the two models generally give similar results for corresponding parameters. However, for short stubby spores (i.e., with small aspect ratio), there were sufficient differences between results for the two models that one could expect to distinguish between the two shapes from measured values of $D(\mathrm{\Theta},\lambda )$ and ${R}_{34}(\mathrm{\Theta},\lambda )$ as functions of Θ. These differences in the graphs remained even when averages over a size distribution similar to the experimentally measured one for spores was applied. As expected, shorter spores gave lower values for $D(\mathrm{\Theta},\lambda )$. The difference in the graphs of this parameter for the two similar shapes was shown to become more pronounced as the aspect ratio (length/diameter) became closer to 1. In agreement with previous experimental and calculation results for bacteria suspended in water [5, 6, 7], the results again indicated that the angular locations of the oscillations in the graph of orientation averages of ${R}_{34}(\mathrm{\Theta},\lambda )$ are mainly controlled by the diameter of the particles.

In the present paper, we further explore the same scattering functions with particular attention to determining the effect of length changes. This paper concentrates on randomly oriented aerosols of single vegetative bacteria. We note that vegetative bacteria generally have a much lower index of refraction than spores. We utilize the capped cylinder model, which approximates the apparent shape of *Escherichia coli* well, and which produced good fits [5] for graphs of experimentally measured values of ${R}_{34}(\mathrm{\Theta},\lambda )$ for those bacteria using microscopically measured values [5, 6] for lengths and diameters.

## 2. Parameters Used

The parameters used for the present calculations were chosen to be similar to values measured for log-phase *Escherichia coli* grown in a minimal medi um [5, 6]. Each individual *E. coli* bacterium was modeled as a right-circular cylinder capped smoothly by a hemisphere of the same diameter as the cylinder at each end. This closely resembles the appearance of this bacterial cell *in vivo* in a phase contrast microscope or in electron microscope pictures. The scattering by the particle is modeled by filling the model with dipoles on a simple cubic lattice with the lattice spacing small enough so that

*n*is the refractive index,

*k*is $2\pi /\lambda $ where

*λ*is the scattering wavelength, and

*a*is the spacing between dipoles. This condition has been found to give good convergence for the DDA [8, 9, 10, 11]. An estimate for

*N*, the number of dipoles needed to obtain this convergence, is found by setting $(V/N{)}^{1/3}=a$, where

*V*is the volume of the model microorganism. In the case of the hemispherically capped cylinder model, where

*r*is the cylindrical and hemispherical radius, and

*L*is the overall length including cylinder and the end-cap hemispheres.

For the present calculations, we used parameters similar to those measured for log-phase distributions of the B/r strains of *Escherichia coli* grown in a minimal medium [5, 6, 7]. These and most other vegetative bacteria have refractive indices decidedly different from those of Bacillus spores. The real part of the refractive index for *E. coli* is generally known for the visible spectrum and was measured previously [5] as 1.373. We added an arbitrary and small imaginary index of 0.00097 to that. The real index, just a small increment above that of water, is a good approximate value through the visible and near-IR range. This statement applies also for near-IR because the indices for water and for protein, the major constituents of a vegetative cell, change only in the third decimal place [12, 13] for wavelengths between 0.690 and $1.5\text{\hspace{0.17em}}\mathrm{\mu m}$. We note that a substantially larger real index of about 1.52 was reported for *killed* cells of *Pantoea agglomerans* (formerly called *Erwinia herbicola*) at visible wavelengths [14]. We do not be lieve such a high value would apply to the index for live *Pantoea* cells, which like *E. coli* consist mostly of water.

The length distribution measured for log-phase *Escherichia coli* B/r grown in a minimal medium [5, 6, 7] is shown in Fig. 1 and is tabulated in Table 1. The diameter distribution measured for the same cells is given in Table 2. The diameters were un correlated with length for the same growth, presumably because the cells grow primarily by elongation. (We note that diameters for different media *are* correlated with length for the given medium, i.e., richer media give rise to longer and fatter bacteria for the same species, e.g., [6, 7].)

These experimental distributions were used for our modeling in an earlier version of this research [3]. However, in that publication we often did not satisfy inequality (1), and therefore a number of the older results using longer lengths are only suggestive. We were careful in the present paper to choose parameters and a sufficient number of dipoles such that inequality (1) is always satisfied. In carrying out the calculations to determine the effect of length changes on $D(\mathrm{\Theta},\lambda )$ and ${R}_{34}(\mathrm{\Theta},\lambda )$, we sometimes used reduced (i.e., hypothetical) width and length distributions similar to, but smaller than, the measured ones to reduce computing time and still satisfy condition (1). We are mainly looking for the qualitative effect on the graphs of changes in the size param eters, which can be varied much more readily in computations using the DDA model than experimentally with aerosols. We also sometimes used larger lengths (See Table 1). We note that many species of bacteria having shapes similar to our model have different sizes, some larger, some smaller, than *E. coli*. In previous experiments we measured the stationary phase (i.e., grown to nutrient exhaustion) for several species of bacteria to be much smaller in length and diameter than log phase of the same species grown with the same conditions [6]. For *E. coli* grown in a minimal medium, the experimental volume for stationary phase was $0.39\text{\hspace{0.17em}}{\mathrm{\mu m}}^{3}$, which is smaller than $46\text{\hspace{0.17em}}{\mathrm{\mu m}}^{3}$, the volume of the smallest model (set A, Table 1) used here. We emphasize that we are only trying to study the typical response to size change in the functions studied, not trying to model the particular bacteria *E. coli*. The results can be applied to larger (or smaller) size distributions if the scattering wavelength is increased (or reduced) proportionally along with size, provided there is no large change of absorption between the two wavelengths.

## 3. Theoretical Considerations

As in the first paper of this series, the coordinate system used will be that of [1, 3], with Θ giving the angle between the scattering direction and the direction of the incoming light. The calculations were carried out using the discrete dipole approximation (DDA, also known as coupled dipole approximation) of Purcell and Pennypacker [15] with the polarizability of each dipole defined by the Clausius–Mossotti formula, e.g., [3, 16], using the numerical value of the refractive index given above. Details of the methods of calculation are given elsewhere, e.g., [3, 9, 1].

## 4. Results

The results of the DDA approximation were previously shown to adequately represent the differential scattering function (i.e., Mueller matrix element ${S}_{11}$) for a sphere by comparison with calculations using the exact Mie solution in [9]. In the present study we concentrate on the special cases of the functions $D(\mathrm{\Theta},\lambda )$ and ${R}_{34}(\mathrm{\Theta},\lambda )$, and the effect of *Y* satisfying Eq. (1)].

#### 4A. Convergence to the Identically Zero Depolarization Ratio for a Pseudosphere

For a true sphere, the off-diagonal elements of the amplitude matrix are identically zero by symmetry, making ${S}_{11}$ identical to ${S}_{22}$, which in turn makes $D(\mathrm{\Theta},\lambda )$ equal to zero. Because the sphere modeled with dipoles only approximates a true sphere, we must either take note of its orientation angles or else average over all orientations when the number of dipoles is fairly small. The orientation angles are defined with respect to the laser direction as are the scattering angles shown in Fig. 1 of [3] (i.e., interpret the scattering direction as the orientation direction of the symmetry axis, instead of scattering direction). Then *θ* and *ϕ* become orientation angles.

The graphs for a particular orientation of the “pseudosphere” modeled with two different dipole numbers are shown in Fig. 2. The sphere modeled has diameter $1.2\text{\hspace{0.17em}}\mathrm{\mu m}$, refractive index 1.373, and the scattering wavelength is $\lambda =1.328\text{\hspace{0.17em}}\mathrm{\mu m}$. Taking the ratio of the peaks near $150\xb0$, we see that the calculation for 1021 dipoles gives values for $D(\mathrm{\Theta},\lambda )$ a factor of about 5.17 larger than the same calculation with 8025 dipoles. If one multiplies the values of $D(\mathrm{\Theta},\lambda )$ for the smaller valued graph by this ratio, one obtains a graph that closely approximates the larger graph for the sphere modeled with 8025 dipoles. This is consistent with the calculated value of $D(\mathrm{\Theta},\lambda )$ approaching zero for an arbitrary angle as the dipole number increases and *Y* goes from exceeding to satisfying Eq. (1). As noted in the figure caption, although the graph approaches zero as the di pole number increases, it does have some orientation dependence.

We next discuss the interesting limiting case in which the length, *L*, of the cylindrical portion of the capped cylinder approaches zero. In this case, the exact solution for $D(\mathrm{\Theta},\lambda )$ must approach zero. This is because, as the cylinder's length approaches zero, the model becomes a sphere whose diameter is that of the former cylinder.

This approach to zero is illustrated in Fig. 3, where graphs are shown for a hemispherically capped cylinder whose cylindrical part is short enough that its shape is close to that of a sphere without yet reaching that limit. The total length of the capped cylinder is $1.5\text{\hspace{0.17em}}\mathrm{\mu m}$, while its diameter is $1.2\text{\hspace{0.17em}}\mathrm{\mu m}$. The graphs for both the capped cylinder and for the dipole sphere are shown. Both these calculations are averaged for random orientation over a 9 by 15 grid in *θ* and *ϕ* as are the other calculations where random orientations are used in this paper. This number of orientations was tested in [2, 3] and found to give results indistinguishable from those from a 13 by 21 grid on the scale of these graphs.

The graphs of Fig. 3a have magnitudes of $D(\mathrm{\Theta},\lambda )$ plotted with a linear scale, so that the much smaller values for the dipole pseudosphere show up only as heavy overlays on the horizontal axis. The value of $D(\mathrm{\Theta},\lambda )$ for the short stubby capped cylinder is already small. We later show (Fig. 6) that the values of $D(\mathrm{\Theta},\lambda )$ become monotonically smaller as the length decreases for constant diameter when the aspect ratio is less than three. This is consistent with the magnitude of $D(\mathrm{\Theta},\lambda )$ approaching zero as the capped cylinder shape approaches the (dipole- approximated) shape of a true sphere. The graphs of Fig. 3b have magnitudes plotted on a logarithmic scale so that the values of $D(\mathrm{\Theta},\lambda )$ for the pseudosphere are shown to scale but can be seen to be 100 times smaller than those for the short capped cylinder. These calculations, together with those of Fig. 6, show that the value of $D(\mathrm{\Theta},\lambda )$ becomes small and approaches zero for either a capped cylinder shape as it approaches the shape of a sphere, or for a pseudosphere as it is modeled with more and more dipoles.

In contrast to the above graphs for $D(\mathrm{\Theta},\lambda )$, the ${R}_{34}(\mathrm{\Theta},\lambda )$ graph for the pseudosphere having the same parameters as the pseudosphere used to generate Fig. 3 has a characteristic nonzero oscillation that remains much the same as the dipole number becomes large. In Fig. 4, we see that graph of ${R}_{34}(\mathrm{\Theta},\lambda )$ obtained from the DDA model for scattering from a sphere using 8025 dipoles for either of two *arbitrarily selected particular orientations* gives a graph for both orientations that is qualitatively similar to the result of a calculation using the exact Mie solution for a sphere. Although the difference is a few percent of the Mie value at some locations, the general shape of the DDA graphs and the location of the extrema are quite similar for the two types of calculation. These results for ${R}_{34}(\mathrm{\Theta},\lambda )$ and the approach to the mathematically exact zero value for $D(\mathrm{\Theta},\lambda )$ as the capped cylinder shape approaches that of a sphere gives us additional confidence that the condition of Eq. (1) is adequate to give reasonably accurate graphs.

#### 4B. Rod-Shaped Bacteria

The four sets, A, B, C, and D, of model lengths listed in Table 1 are all approximately proportional to the experimentally measured lengths listed on the left-hand side of Table 1. That distribution was obtained for log-phase *E. coli* B/r grown in a minimal medium. We look first at how changes in length affect $D(\mathrm{\Theta},\lambda )$ and ${R}_{34}(\mathrm{\Theta},\lambda )$ if all other parameters are kept the same. In Fig. 5 the graphs of $D(\mathrm{\Theta},\lambda )$ are plotted for distributions of cells averaged over orientations as well as averaged over each of the four sets of lengths, with each graph calculated for cells of the same diameter of $0.8\text{\hspace{0.17em}}\mathrm{\mu m}$.

We see that the magnitude and shape of the graph of *D* stays about the same until the case of a short capped cylinder with aspect ratio $<2$. Increasing length does have a noticeable effect on the maxi mum near $90\xb0$, and additionally, a smaller hump develops near $160\xb0$ as the capped cylinder becomes long. The large peak near $90\xb0$ only moves slightly (except for the shortest cells) to smaller angles as the length increases.

The graphs for $D(\mathrm{\Theta},\lambda )$ for each of the individual lengths in the short sets A and B of Table 1 are shown in Fig. 6. The calculations are for a fixed diameter of $0.8\text{\hspace{0.17em}}\mathrm{\mu m}$. Figure 6a is for the vegetative cells while Fig. 6b is for the capped cylinder model of the spores. The only parameter difference is that the index of refraction in air is somewhat larger for the spores at $n=1.505$ than at $n=1.37$ for the vegetative cells. In both cases, $D(\mathrm{\Theta},\lambda )$ continues to decrease in magnitude, and the peak moves to slightly larger angle as the length of the cylindrical part decreases toward zero.

The shapes of the graphs are quite similar for spores and for vegetative cells of the same size in air. However, the depolarization is generally larger for the spores, particularly in the back-scattering direction. We also note that the ratio of the direct back-scattering value of $D(\mathrm{\Theta},\lambda )$ to its maximum near $90\xb0$ is often larger for the spores.

When the length increases beyond those shown in the graph, the maximum value of $D(\mathrm{\Theta},\lambda )$ does not change much although there continue to be moderate shape changes with the large peak appearing to move toward the forward scattering direction.

The fact that $D(\mathrm{\Theta},\lambda )$ for a sphere is identically zero requires that the magnitude of this function must decrease for all angles as the length of the cylindrical part of the capped cylinder approaches zero. However, from Figs. 5, 6, and other results it is observed that this decrease in magnitude does not begin to occur until the length is less than about three times the diameter of the end caps, at least for the present wavelength.

In Fig. 7a, graphs of ${R}_{34}(\mathrm{\Theta},\lambda )$ for the four sets of Table 1 with averages over length and orientation are presented for a vegetative cell with diameter $0.8\text{\hspace{0.17em}}\mathrm{\mu m}$. These show the small effect of length change on the shape of the graphs of this function. Examination of these graphs indicates that there are only small changes in appearance for ${R}_{34}(\mathrm{\Theta},\lambda )$ with length changes over this broad range of average lengths. This change includes a slight movement of the graphs toward smaller angles as the length increases. This may be due to longer paths being available to the photons, which tend to loop the surface before being re-emitted as scattered light. One may also note some decrease in magnitude and smoothness as the length increases.

In Fig. 7b, ${R}_{34}(\mathrm{\Theta},\lambda )$ is graphed for the same pa rameters but with single short lengths, and averaging over orientation only. The effect of larger amplitude oscillations for shorter length is more pronounced in this case. This effect is probably due to the fact that the paths a photon can take before scattering for different orientations are more similar in length for shorter cells.

Next, graphs of $D(\mathrm{\Theta},\lambda )$ versus Θ in Fig. 8a and ${R}_{34}(\mathrm{\Theta},\lambda )$ versus Θ in Fig. 8b are shown with the diameter of the capped cylinder varied for the constant length, $L=5.5\text{\hspace{0.17em}}\mathrm{\mu m}$. As previously observed [2, 3], the major features of the graph move to the left toward smaller angles for both functions as the di ameter increases.

As we consider relatively narrow cells, the larger valued part of the graph of $D(\mathrm{\Theta},\lambda )$ moves to larger angles. In Fig. 9 the graphs for $D(\mathrm{\Theta},\lambda )$, for each of the short lengths of set A, are shown for a single di ameter of $0.65\text{\hspace{0.17em}}\mathrm{\mu m}$.

Finally, we consider results based on full averages over lengths and widths as well as orientation using distributions similar to those observed experimentally for vegetative cells. In Table 3 the weightings are calculated by averaging over lengths and diam eters that are uncorrelated, i.e., separately combining all of the lengths of either Model Set, A or D, in Table 1, with all the diameters of the Model Set of Table 2 with the appropriate weights. In view of the fact that it has been observed that the diam eters and lengths for a single log-phase growth of *E. coli* are uncorrelated experimentally (e.g., [5]), the weighting for a given pair *L* and *d* is just the product of the individual weights. The resulting overall distribution should reasonably simulate a real distribution of bacteria. The results of this averaging over all lengths and diameters may be seen for the depolarization ratio in Fig. 10.

The values of ${R}_{34}(\mathrm{\Theta},\lambda )$ for the uncorrelated averages of diameters with lengths of set A and D are plotted together in Fig. 11. Figure 10 shows substantial changes in $D(\mathrm{\Theta},\lambda )$ versus Θ for the shape and magnitude of the graph when the length of the bacterial distribution changes, with the width distribution remaining the same. However the graphs retain some qualitative resemblance in their dependence on scattering angle. The two graphs of ${R}_{34}(\mathrm{\Theta},\lambda )$, in contrast, are qualitatively similar in appearance for the long and the short length distributions. There is some additional bumpiness in the graph of ${R}_{34}(\mathrm{\Theta},\lambda )$ for the long distribution, D, between $60\xb0$ and $90\xb0$, but otherwise the two graphs appear similar.

## 5. Conclusions

In this paper, we concentrated on determining how the graphs of depolarization ratio, $D(\mathrm{\Theta},\lambda )$, and the Mueller matrix ratio, ${R}_{34}(\mathrm{\Theta},\lambda )$ versus Θ are affected by changes in size for an aerosol of rod-shaped vegetative cells. We used size distributions similar to those experimentally measured for log-phase *Escherichia coli* bacteria. Each individual bacterium was modeled as a cylindrical rod smoothly ended with hemispherical end caps.

The value of $D(\mathrm{\Theta},\lambda )$ for a perfect homogeneous sphere is identically zero. If the number of dipoles used is not large enough, the graph of $D(\mathrm{\Theta},\lambda )$ depends on the orientation of the pseudosphere modeled with dipoles. In contrast, the value of ${R}_{34}(\mathrm{\Theta},\lambda )$ for any orientation of the pseudosphere was found for a more modest dipole number to closely resemble that obtained using the exact Mie solution for a sphere.

The calculations graphed in this paper verify that the coupled dipole approximation or DDA, which we used throughout, indeed gives a value approaching zero for $D(\mathrm{\Theta},\lambda )$ for any orientation of the pseudosphere as the number of dipoles increases beyond that needed to satisfy condition (1).

In our capped cylinder model of a bacterium, the shape of the capped cylinder continuously approaches that of a sphere as the length *L* of the model approaches the value, *d*, of the diameter of the cylinder or the hemispherical end caps. We showed that for an orientation average, with a sufficient number of dipoles, the value of $D(\mathrm{\Theta},\lambda )$ becomes small, approaching zero as *L* approaches *d* in magnitude.

Calculations were made to determine how the graphs of $D(\mathrm{\Theta},\lambda )$ and ${R}_{34}(\mathrm{\Theta},\lambda )$ varied as the length changed for fixed diameter of the bacteria. The magnitude of *D* stays roughly constant, as the length, *L*, of the capped cylinder decreases until *L* is less than roughly three times the cylindrical diameter, after which the magnitude of *D* decreases continuously toward zero. If the length becomes quite a bit longer than the diameter, additional features appear in the direct back-scattering direction. A change of index from the vegetative value to that for a *Bacillus* spore makes $D(\mathrm{\Theta},\lambda )$ somewhat larger as the major change for aerosolized particles. The graphs of ${R}_{34}(\mathrm{\Theta},\lambda )$ remain similar to one another as the cell length increases with no change in diameter, except that the magnitude of the oscillations generally is smaller for *longer* cells.

Changes in the diameter, *d*, keeping *L* fixed, in contrast have a noticeable effect on the shape of the graphs both for $D(\mathrm{\Theta},\lambda )$ and for ${R}_{34}(\mathrm{\Theta},\lambda )$. As previously noted for ${R}_{34}(\mathrm{\Theta},\lambda )$, an increasing value of *d* causes the shape to move to smaller angles. Now we note that this is the case also for $D(\mathrm{\Theta},\lambda )$.

When we averaged over *L*
*d* for a complete distribution similar to that found for a log-phase growth of *E. coli*, the graphs of $D(\mathrm{\Theta},\lambda )$ for a long versus a short distribution had a substantial change in shape, whereas the graphs for ${R}_{34}(\mathrm{\Theta},\lambda )$ had a qualitative resemblance for its major features between the long and short distributions.

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