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Applied Optics

Applied Optics

APPLICATIONS-CENTERED RESEARCH IN OPTICS

  • Editor: Joseph N. Mait
  • Vol. 48, Iss. 30 — Oct. 20, 2009
  • pp: 5655–5663
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Light scattering calculations exploring sensitivity of depolarization ratio to shape changes. II. Single rod-shaped vegetative bacteria in air

Burt V. Bronk and Stephen D. Druger  »View Author Affiliations


Applied Optics, Vol. 48, Issue 30, pp. 5655-5663 (2009)
http://dx.doi.org/10.1364/AO.48.005655


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Abstract

In article I of this series, calculations and graphs of the depolarization ratio, D ( Θ , λ ) = 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 S i j in these papers denote the appropriate Mueller matrix elements. We compare graphs for different size parameters for both D ( Θ , λ ) and the ratio R 34 ( Θ , λ ) = < S 34 > / < S 11 > . The ratio R 34 ( Θ , λ ) 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 ( Θ , λ ) is not very sensitive to these changes, but D ( Θ , λ ) 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 ( Θ , λ ) and R 34 ( Θ , λ ) . 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 ( Θ , λ ) but not for R 34 ( Θ , λ ) .

© 2009 Optical Society of America

1. Introduction

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

5. B. V. Bronk, S. D. Druger, J. Czégé, and W. P. Van De Merwe, “Measuring diameters of rod-shaped bacteria in vivo with polarized light scattering,” Biophys. J. 69, 1170 (1995). [CrossRef] [PubMed]

] for graphs of experimentally measured values of R34(Θ,λ) for those bacteria using microscopically measured values [5

5. B. V. Bronk, S. D. Druger, J. Czégé, and W. P. Van De Merwe, “Measuring diameters of rod-shaped bacteria in vivo with polarized light scattering,” Biophys. J. 69, 1170 (1995). [CrossRef] [PubMed]

, 6

6. B. V. Bronk, W. P. Van De Merwe, and M. Stanley, “An in-vivo measure of average bacterial cell size from a polarized light scattering function,” Cytometry 13, 155–162 (1992). [CrossRef] [PubMed]

] 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

5. B. V. Bronk, S. D. Druger, J. Czégé, and W. P. Van De Merwe, “Measuring diameters of rod-shaped bacteria in vivo with polarized light scattering,” Biophys. J. 69, 1170 (1995). [CrossRef] [PubMed]

, 6

6. B. V. Bronk, W. P. Van De Merwe, and M. Stanley, “An in-vivo measure of average bacterial cell size from a polarized light scattering function,” Cytometry 13, 155–162 (1992). [CrossRef] [PubMed]

]. 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
Y=nka<12
(1)
is fulfilled, where n is the refractive index, k is 2π/λ 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

8. Y. You, G. W. Kattawar, C. Li, and P. Yang, “Internal dipole radiation as a tool for particle identification,” Appl. Opt. 45, 9115–9124 (2006). [CrossRef] [PubMed]

, 9

9. B. T. Draine and P. J. Flatau, “Discrete dipole approximation for scattering calculations,” J. Opt. Soc. Am. A 11, 1491–1499 (1994). [CrossRef]

, 10

10. M. A. Yurkin, V. P. Maltsev, and A. G. Hoekstra, “ I. Theoretical effects,” J. Opt. Soc. Am. A 23, 2578–2591 (2006). [CrossRef]

, 11

11. M. A. Yurkin, V. P. Maltsev, and A. G. Hoekstra, “Convergence of the discrete dipole approximation. II. An extrapolation technique to increase the accuracy,” J. Opt. Soc. Am. A 23, 2592–2601 (2006). [CrossRef]

]. 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,
V=πr2(L2r/3),
(2)
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

5. B. V. Bronk, S. D. Druger, J. Czégé, and W. P. Van De Merwe, “Measuring diameters of rod-shaped bacteria in vivo with polarized light scattering,” Biophys. J. 69, 1170 (1995). [CrossRef] [PubMed]

, 6

6. B. V. Bronk, W. P. Van De Merwe, and M. Stanley, “An in-vivo measure of average bacterial cell size from a polarized light scattering function,” Cytometry 13, 155–162 (1992). [CrossRef] [PubMed]

, 7

7. W. P. Van De Merwe, J. Czege, M. E. Milham, and B. V. Bronk, “Rapid optically based measurements of diameter and length for spherical or rod-shaped bacteria in vivo,” Appl. Opt. 43, 5295–5302 (2004). [CrossRef] [PubMed]

]. 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

5. B. V. Bronk, S. D. Druger, J. Czégé, and W. P. Van De Merwe, “Measuring diameters of rod-shaped bacteria in vivo with polarized light scattering,” Biophys. J. 69, 1170 (1995). [CrossRef] [PubMed]

] 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

12. G. M. Hale and M. R. Querry, “Optical constants of water in the 200nm to 200 micrometer wavelength region,” Appl. Opt. 12, 555–563 (1973). [CrossRef] [PubMed]

, 13

13. E. Arakawa, P. S. Tuminello, B. N. Khare, and M. E. Milham, “Optical properties of ovalbumin in 0.1302.50μm spectral region,” Biopolymers 62, 122–128 (2001). [CrossRef] [PubMed]

] for wavelengths between 0.690 and 1.5μm. 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

14. E. T. Arakawa, P. S. Tuminello, B. N. Khare, and M. E. Milham, “Optical properties of Erwinia herbicola bacteria at 0.190–2.50 micron,” Biopolymers 72, 391–398 (2003). [CrossRef] [PubMed]

]. 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

5. B. V. Bronk, S. D. Druger, J. Czégé, and W. P. Van De Merwe, “Measuring diameters of rod-shaped bacteria in vivo with polarized light scattering,” Biophys. J. 69, 1170 (1995). [CrossRef] [PubMed]

, 6

6. B. V. Bronk, W. P. Van De Merwe, and M. Stanley, “An in-vivo measure of average bacterial cell size from a polarized light scattering function,” Cytometry 13, 155–162 (1992). [CrossRef] [PubMed]

, 7

7. W. P. Van De Merwe, J. Czege, M. E. Milham, and B. V. Bronk, “Rapid optically based measurements of diameter and length for spherical or rod-shaped bacteria in vivo,” Appl. Opt. 43, 5295–5302 (2004). [CrossRef] [PubMed]

] 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

6. B. V. Bronk, W. P. Van De Merwe, and M. Stanley, “An in-vivo measure of average bacterial cell size from a polarized light scattering function,” Cytometry 13, 155–162 (1992). [CrossRef] [PubMed]

, 7

7. W. P. Van De Merwe, J. Czege, M. E. Milham, and B. V. Bronk, “Rapid optically based measurements of diameter and length for spherical or rod-shaped bacteria in vivo,” Appl. Opt. 43, 5295–5302 (2004). [CrossRef] [PubMed]

].)

3. Theoretical Considerations

As in the first paper of this series, the coordinate system used will be that of [1

1. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983).

, 3

3. S. D. Druger, J. Czege, Z. Z. Li, and B. V. Bronk, “Calculations of light scattering measurements predicting sensitivity of depolarization to shape changes of spores and bacteria,” Tech. Rep. ECBC-TR-607 (Edgewood Chemical Biological Center, 2008).

], 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

15. E. M. Purcell and C. R. Pennypacker, “Scattering and absorption of light by nonspherical dielectric grains,” Astrophys. J. 186, 705–714 (1973). [CrossRef]

] with the polarizability of each dipole defined by the Clausius–Mossotti formula, e.g., [3

3. S. D. Druger, J. Czege, Z. Z. Li, and B. V. Bronk, “Calculations of light scattering measurements predicting sensitivity of depolarization to shape changes of spores and bacteria,” Tech. Rep. ECBC-TR-607 (Edgewood Chemical Biological Center, 2008).

, 16

16. S. D. Druger and B. V. Bronk, “Internal and scattered electric fields in the discrete dipole approximation,” J. Opt. Soc. Am. B 16, 2239–2246 (1999). [CrossRef]

], using the numerical value of the refractive index given above. Details of the methods of calculation are given elsewhere, e.g., [3

3. S. D. Druger, J. Czege, Z. Z. Li, and B. V. Bronk, “Calculations of light scattering measurements predicting sensitivity of depolarization to shape changes of spores and bacteria,” Tech. Rep. ECBC-TR-607 (Edgewood Chemical Biological Center, 2008).

, 9

9. B. T. Draine and P. J. Flatau, “Discrete dipole approximation for scattering calculations,” J. Opt. Soc. Am. A 11, 1491–1499 (1994). [CrossRef]

, 1

1. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983).

].

4. Results

The results of the DDA approximation were previously shown to adequately represent the differential scattering function (i.e., Mueller matrix element S11) for a sphere by comparison with calculations using the exact Mie solution in [9

9. B. T. Draine and P. J. Flatau, “Discrete dipole approximation for scattering calculations,” J. Opt. Soc. Am. A 11, 1491–1499 (1994). [CrossRef]

]. In the present study we concentrate on the special cases of the functions D(Θ,λ) and R34(Θ,λ), 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 S11 identical to S22, which in turn makes D(Θ,λ) 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

3. S. D. Druger, J. Czege, Z. Z. Li, and B. V. Bronk, “Calculations of light scattering measurements predicting sensitivity of depolarization to shape changes of spores and bacteria,” Tech. Rep. ECBC-TR-607 (Edgewood Chemical Biological Center, 2008).

] (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 1.2μm, refractive index 1.373, and the scattering wavelength is λ=1.328μm. Taking the ratio of the peaks near 150°, we see that the calculation for 1021 dipoles gives values for D(Θ,λ) a factor of about 5.17 larger than the same calculation with 8025 dipoles. If one multiplies the values of D(Θ,λ) 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(Θ,λ) 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.

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 1.5μm, while its diameter is 1.2μm. 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

2. S. D. Druger, J. Czege, Z. Z. Li, and B. V. Bronk, “Light scattering calculations exploring sensitivity of depolarization to shape changes for: I. Single spores in air,” Appl. Opt. 48, 716–724 (2009). Note: the values used for the indices of refraction for Bacillus cereus spores were mistakenly attributed by one of us (BVB) to M. Querry and M. Milham. The experimental data were actually due to P. S. Tuminello, M. E. Milham, B. N. Khare, and E. T. Arakawa. [CrossRef] [PubMed]

, 3

3. S. D. Druger, J. Czege, Z. Z. Li, and B. V. Bronk, “Calculations of light scattering measurements predicting sensitivity of depolarization to shape changes of spores and bacteria,” Tech. Rep. ECBC-TR-607 (Edgewood Chemical Biological Center, 2008).

] 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(Θ,λ) 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(Θ,λ) for the short stubby capped cylinder is already small. We later show (Fig. 6) that the values of D(Θ,λ) 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(Θ,λ) 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(Θ,λ) 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(Θ,λ) 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(Θ,λ), the R34(Θ,λ) 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 R34(Θ,λ) 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 R34(Θ,λ) and the approach to the mathematically exact zero value for D(Θ,λ) 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(Θ,λ) and R34(Θ,λ) if all other parameters are kept the same. In Fig. 5 the graphs of D(Θ,λ) 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 0.8μm.

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°, and additionally, a smaller hump develops near 160° as the capped cylinder becomes long. The large peak near 90° only moves slightly (except for the shortest cells) to smaller angles as the length increases.

The graphs for D(Θ,λ) 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 0.8μm. 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(Θ,λ) 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(Θ,λ) to its maximum near 90° is often larger for the spores.

When the length increases beyond those shown in the graph, the maximum value of D(Θ,λ) 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(Θ,λ) 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 R34(Θ,λ) for the four sets of Table 1 with averages over length and orientation are presented for a vegetative cell with diameter 0.8μm. 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 R34(Θ,λ) 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, R34(Θ,λ) 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(Θ,λ) versus Θ in Fig. 8a and R34(Θ,λ) versus Θ in Fig. 8b are shown with the diameter of the capped cylinder varied for the constant length, L=5.5μm. As previously observed [2

2. S. D. Druger, J. Czege, Z. Z. Li, and B. V. Bronk, “Light scattering calculations exploring sensitivity of depolarization to shape changes for: I. Single spores in air,” Appl. Opt. 48, 716–724 (2009). Note: the values used for the indices of refraction for Bacillus cereus spores were mistakenly attributed by one of us (BVB) to M. Querry and M. Milham. The experimental data were actually due to P. S. Tuminello, M. E. Milham, B. N. Khare, and E. T. Arakawa. [CrossRef] [PubMed]

, 3

3. S. D. Druger, J. Czege, Z. Z. Li, and B. V. Bronk, “Calculations of light scattering measurements predicting sensitivity of depolarization to shape changes of spores and bacteria,” Tech. Rep. ECBC-TR-607 (Edgewood Chemical Biological Center, 2008).

], 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(Θ,λ) moves to larger angles. In Fig. 9 the graphs for D(Θ,λ), for each of the short lengths of set A, are shown for a single di ameter of 0.65μm.

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

5. B. V. Bronk, S. D. Druger, J. Czégé, and W. P. Van De Merwe, “Measuring diameters of rod-shaped bacteria in vivo with polarized light scattering,” Biophys. J. 69, 1170 (1995). [CrossRef] [PubMed]

]), 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 R34(Θ,λ) 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(Θ,λ) 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 R34(Θ,λ), in contrast, are qualitatively similar in appearance for the long and the short length distributions. There is some additional bumpiness in the graph of R34(Θ,λ) for the long distribution, D, between 60° and 90°, but otherwise the two graphs appear similar.

5. Conclusions

In this paper, we concentrated on determining how the graphs of depolarization ratio, D(Θ,λ), and the Mueller matrix ratio, R34(Θ,λ) 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(Θ,λ) for a perfect homogeneous sphere is identically zero. If the number of dipoles used is not large enough, the graph of D(Θ,λ) depends on the orientation of the pseudosphere modeled with dipoles. In contrast, the value of R34(Θ,λ) 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(Θ,λ) 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(Θ,λ) becomes small, approaching zero as L approaches d in magnitude.

Calculations were made to determine how the graphs of D(Θ,λ) and R34(Θ,λ) 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(Θ,λ) somewhat larger as the major change for aerosolized particles. The graphs of R34(Θ,λ) 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(Θ,λ) and for R34(Θ,λ). As previously noted for R34(Θ,λ), an increasing value of d causes the shape to move to smaller angles. Now we note that this is the case also for D(Θ,λ).

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(Θ,λ) for a long versus a short distribution had a substantial change in shape, whereas the graphs for R34(Θ,λ) had a qualitative resemblance for its major features between the long and short distributions.

Table 1. Length Distribution for Log-Phase E. coli B/r Grown in a Minimal Medium [5, 6, 7] and Similar Length Distributions Used in Modeling

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Table 2. Diameter Distribution for Log-Phase E. coli B/r Grown in a Minimal Medium [5, 6] and Similar One for Models Used in Present Paper

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Table 3. Weightings for Uncorrelated Averages over the Lengths of Model Sets A and D, Table 1, Together with the Model Diameters of Table 2 a

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Fig. 1 Length distribution for log-phase E. coli B/r.
Fig. 2 Graphs of the depolarization ratio versus angle for scattering wavelength 1.328μm for a “dipole sphere” of diameter 1.2μm, refractive index 1.373, at orientation angles ϕ=57°, θ=17° modeled with either 1021 dipoles or 8025 dipoles as indicated. The values of the parameter Y in Eq. (1) are 0.628 for 1021 dipoles and 0.314 for 8025. The values of D(Θ,λ) calculated for orientation angles Θ=ϕ=0 coincide with the horizontal axis on the scale of this graph because of the symmetrical dipole placement. For the other orientation, D(Θ,λ) approaches zero as predicted for increasing dipole number.
Fig. 3 Results on a (a) linear scale and (b) logarithmic scale for D(Θ,λ) averaged over orientations for both the capped cylinder with a shape close to spherical (L=1.5μm; d=1.2μm) and for the pseudosphere for n=1.373 with λ=1.328μm in both cases. The calculations used 8829 dipoles for the sphere and 10,719 dipoles for the capped cylinder, giving a Y value in Eq. (1) of about 0.3 in both cases. The values for the sphere are close to the correct value of zero showing that Eq. (1) is adequate on the scale of the linear graph. For a short capped cylinder, the calculated values of D become small but nonzero as they should in approaching the limiting zero value.
Fig. 4 Graph of the ratio R34(Θ,λ) versus Θ for a sphere of di ameter 1.2μm and n=1.373 and λ=1.328μm, from calculations using the DDA model with 8025 dipoles at two different orientations. Either orientation of the pseudosphere of dipoles is seen to produce a result close to the graph resulting from the exact Mie solution.
Fig. 5 D(Θ,λ) versus Θ for models A, B, C, and D from Table 1 with fixed diameter=0.8μm and averages over the length distribution for each set as well as an orientation average, with λ=1.266μm. Average lengths for each distribution are indicated in the figure.
Fig. 6 D(Θ,λ) versus Θ is shown for vegetative cells having individual lengths from sets A and B with fixed diameter of 0.8μm for (a) n=1.37 (vegetative bacteria) and (b) n=1.505 (spores). The averaging was over orientation only. The length for each calculation is shown in the figure. The scattering wavelength for both figures is 1.266μm.
Fig. 7 (a) R34(Θ,λ) for n=1.37 and λ=1.266μm averaged over orientation and length for the four model sets of Table 1 with fixed diameter of 0.8μm. The average length for each set is indicated in the graph. (b) Graphs are shown for same function and parameters used for the single lengths indicated (with orientation averaging only).
Fig. 8 Orientation averaged (a) D(Θ,λ) and (b) R34(Θ,λ), both calculated for n=1.374 with λ=1.266μm for a single fixed length, L=5.5μm and varying diameters. The radius, R, of the cylinder is indicated for each calculation.
Fig. 9 Depolarization ratio, orientation averaged for capped cylinders of diameter 0.65μm for several single short lengths as shown using n=1.374 and λ=1.266μm.
Fig. 10 Uncorrelated averages of D(Θ,λ) over all lengths and diameters for length sets A (short) are denoted by a dashed line and length set D (long) are denoted by a solid line with weightings as indicated in Table 3. The averages were also over orientations. The index and wavelength were n=1.374 and λ=1.266μm. The same diameter distribution is assumed for both length sets.
Fig. 11 Uncorrelated averages over all lengths and diameters for R34(Θ,λ) with weightings as indicated in Table 3. Averages are over orientations also. The short bacteria are set A denoted by the dashed line. The long bacteria are set D denoted by the solid line. The same diameter distribution is assumed for both sets.
1.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983).

2.

S. D. Druger, J. Czege, Z. Z. Li, and B. V. Bronk, “Light scattering calculations exploring sensitivity of depolarization to shape changes for: I. Single spores in air,” Appl. Opt. 48, 716–724 (2009). Note: the values used for the indices of refraction for Bacillus cereus spores were mistakenly attributed by one of us (BVB) to M. Querry and M. Milham. The experimental data were actually due to P. S. Tuminello, M. E. Milham, B. N. Khare, and E. T. Arakawa. [CrossRef] [PubMed]

3.

S. D. Druger, J. Czege, Z. Z. Li, and B. V. Bronk, “Calculations of light scattering measurements predicting sensitivity of depolarization to shape changes of spores and bacteria,” Tech. Rep. ECBC-TR-607 (Edgewood Chemical Biological Center, 2008).

4.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983), 383, pp. 65–67.

5.

B. V. Bronk, S. D. Druger, J. Czégé, and W. P. Van De Merwe, “Measuring diameters of rod-shaped bacteria in vivo with polarized light scattering,” Biophys. J. 69, 1170 (1995). [CrossRef] [PubMed]

6.

B. V. Bronk, W. P. Van De Merwe, and M. Stanley, “An in-vivo measure of average bacterial cell size from a polarized light scattering function,” Cytometry 13, 155–162 (1992). [CrossRef] [PubMed]

7.

W. P. Van De Merwe, J. Czege, M. E. Milham, and B. V. Bronk, “Rapid optically based measurements of diameter and length for spherical or rod-shaped bacteria in vivo,” Appl. Opt. 43, 5295–5302 (2004). [CrossRef] [PubMed]

8.

Y. You, G. W. Kattawar, C. Li, and P. Yang, “Internal dipole radiation as a tool for particle identification,” Appl. Opt. 45, 9115–9124 (2006). [CrossRef] [PubMed]

9.

B. T. Draine and P. J. Flatau, “Discrete dipole approximation for scattering calculations,” J. Opt. Soc. Am. A 11, 1491–1499 (1994). [CrossRef]

10.

M. A. Yurkin, V. P. Maltsev, and A. G. Hoekstra, “ I. Theoretical effects,” J. Opt. Soc. Am. A 23, 2578–2591 (2006). [CrossRef]

11.

M. A. Yurkin, V. P. Maltsev, and A. G. Hoekstra, “Convergence of the discrete dipole approximation. II. An extrapolation technique to increase the accuracy,” J. Opt. Soc. Am. A 23, 2592–2601 (2006). [CrossRef]

12.

G. M. Hale and M. R. Querry, “Optical constants of water in the 200nm to 200 micrometer wavelength region,” Appl. Opt. 12, 555–563 (1973). [CrossRef] [PubMed]

13.

E. Arakawa, P. S. Tuminello, B. N. Khare, and M. E. Milham, “Optical properties of ovalbumin in 0.1302.50μm spectral region,” Biopolymers 62, 122–128 (2001). [CrossRef] [PubMed]

14.

E. T. Arakawa, P. S. Tuminello, B. N. Khare, and M. E. Milham, “Optical properties of Erwinia herbicola bacteria at 0.190–2.50 micron,” Biopolymers 72, 391–398 (2003). [CrossRef] [PubMed]

15.

E. M. Purcell and C. R. Pennypacker, “Scattering and absorption of light by nonspherical dielectric grains,” Astrophys. J. 186, 705–714 (1973). [CrossRef]

16.

S. D. Druger and B. V. Bronk, “Internal and scattered electric fields in the discrete dipole approximation,” J. Opt. Soc. Am. B 16, 2239–2246 (1999). [CrossRef]

OCIS Codes
(280.1100) Remote sensing and sensors : Aerosol detection
(280.3640) Remote sensing and sensors : Lidar
(290.1090) Scattering : Aerosol and cloud effects
(290.5850) Scattering : Scattering, particles
(280.1415) Remote sensing and sensors : Biological sensing and sensors
(290.5855) Scattering : Scattering, polarization

ToC Category:
Scattering

History
Original Manuscript: June 16, 2009
Revised Manuscript: September 2, 2009
Manuscript Accepted: September 9, 2009
Published: October 12, 2009

Virtual Issues
Vol. 4, Iss. 12 Virtual Journal for Biomedical Optics

Citation
Burt V. Bronk and Stephen D. Druger, "Light scattering calculations exploring sensitivity of depolarization ratio to shape changes. II. Single rod-shaped vegetative bacteria in air," Appl. Opt. 48, 5655-5663 (2009)
http://www.opticsinfobase.org/ao/abstract.cfm?URI=ao-48-30-5655


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References

  1. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983).
  2. S. D. Druger, J. Czege, Z. Z. Li, and B. V. Bronk, “Light scattering calculations exploring sensitivity of depolarization to shape changes for: I. Single spores in air,” Appl. Opt. 48, 716-724 (2009). Note: the values used for the indices of refraction for Bacillus cereus spores were mistakenly attributed by one of us (BVB) to M. Querry and M. Milham. The experimental data were actually due to P. S. Tuminello, M. E. Milham, B. N. Khare, and E. T. Arakawa. [CrossRef] [PubMed]
  3. S. D. Druger, J. Czege, Z. Z. Li, and B. V. Bronk, “Calculations of light scattering measurements predicting sensitivity of depolarization to shape changes of spores and bacteria,” Tech. Rep. ECBC-TR-607 (Edgewood Chemical Biological Center, 2008).
  4. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983), 383, pp. 65-67.
  5. B. V. Bronk, S. D. Druger, J. Czégé, and W. P. Van De Merwe, “Measuring diameters of rod-shaped bacteria in vivo with polarized light scattering,” Biophys. J. 69, 1170 (1995). [CrossRef] [PubMed]
  6. B. V. Bronk, W. P. Van De Merwe, and M. Stanley, “An in-vivo measure of average bacterial cell size from a polarized light scattering function,” Cytometry 13, 155-162(1992). [CrossRef] [PubMed]
  7. W. P. Van De Merwe, J. Czege, M. E. Milham, and B. V. Bronk, “Rapid optically based measurements of diameter and length for spherical or rod-shaped bacteria in vivo,” Appl. Opt. 43, 5295-5302 (2004). [CrossRef] [PubMed]
  8. Y. You, G. W. Kattawar, C. Li, and P. Yang, “Internal dipole radiation as a tool for particle identification,” Appl. Opt. 45, 9115-9124 (2006). [CrossRef] [PubMed]
  9. B. T. Draine and P. J. Flatau, “Discrete dipole approximation for scattering calculations,” J. Opt. Soc. Am. A 11, 1491-1499(1994). [CrossRef]
  10. M. A. Yurkin, V. P. Maltsev, and A. G. Hoekstra, “ I. Theoretical effects,” J. Opt. Soc. Am. A 23, 2578-2591 (2006). [CrossRef]
  11. M. A. Yurkin, V. P. Maltsev, and A. G. Hoekstra, “Convergence of the discrete dipole approximation. II. An extrapolation technique to increase the accuracy,” J. Opt. Soc. Am. A 23, 2592-2601 (2006). [CrossRef]
  12. G. M. Hale and M. R. Querry, “Optical constants of water in the 200 nm to 200 micrometer wavelength region,” Appl. Opt. 12, 555-563 (1973). [CrossRef] [PubMed]
  13. E. Arakawa, P. S. Tuminello, B. N. Khare, and M. E. Milham, “Optical properties of ovalbumin in 0.130-2.50 μm spectral region,” Biopolymers 62, 122-128 (2001). [CrossRef] [PubMed]
  14. E. T. Arakawa, P. S. Tuminello, B. N. Khare, and M. E. Milham, “Optical properties of Erwinia herbicola bacteria at 0.190-2.50 micron,” Biopolymers 72, 391-398(2003). [CrossRef] [PubMed]
  15. E. M. Purcell and C. R. Pennypacker, “Scattering and absorption of light by nonspherical dielectric grains,” Astrophys. J. 186, 705-714 (1973). [CrossRef]
  16. S. D. Druger and B. V. Bronk, “Internal and scattered electric fields in the discrete dipole approximation,” J. Opt. Soc. Am. B 16, 2239-2246 (1999). [CrossRef]

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