Light scattering calculations exploring sensitivity ofdepolarization ratio to shape changes. II. Single
rod-shaped vegetative bacteria in air
Burt V. Bronk1,2,* and Stephen D. Druger3
1Air Force Research Lab, 711 HPW, Wright Patterson Air Force Base, Ohio 45433-5707, USA2U.S. Army ECBC, Aberdeen Proving Ground Maryland, 21010-5424, USA
3Department of Physics and Applied Physics, University of Massachusetts—Lowell,Lowell, Massachusetts 01854, USA
It is well known that a single particle or a cloud ofsuch particles has all its light scattering informationdescribed by the 16 elements of the 4 × 4Mueller ma-trix (e.g., [1]). These matrix elements, Sij, are func-tions of the scattering angles, the wavelength of
the light, and the properties of the scatterers, includ-ing their size, shape, index of refraction, and, in thecase of single particles or an oriented stream of singleparticles, each particle’s orientation with respect tothe incoming light. In previous papers of this series[2,3] the behavior of the graphs of two functions ofthe Mueller matrix elements versus scattering an-gles for randomly oriented bacterial endospores inair were studied. These two functions were the depo-larization ratio, DðΘ; λÞ ¼ 1− < S22 > = < S11 >, and
the ratio, R34ðΘ; λÞ ¼ hS34i=hS11i, where the bracketsindicate that averages were taken over many parti-cles. Averages over orientation angles of the parti-cles to simulate random orientation were taken ineach case, and size averages were calculated whereindicated.An interpretation of these functions can be based
on the definition of the Stokes vector entries [4]. Atany scattering angle, for unpolarized input light, S11is proportional to the scattered intensity. S11–S22 is ameasure of the cross-polarizing elements of the sin-gle-particle amplitude matrices, so that DðΘ; λÞ is re-lated to the tendency of a scatterer to change thepolarization for incoming polarized light. The func-tion, R34, measures the amount of the third Stokesvector element, U, moved to the fourth Stokes ele-ment, V , and is related to the transformation of lin-ear to circular polarization. This interpretation canbe misleading, however. For example, R34 for anisotropic, nonchiral spherical scatterer has an inter-esting dynamic behavior as a function of size orwavelength, but the scattering does not give rise tocircular polarization because the sphere has nohandedness.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 cylindercapped smoothly on each end with a hemisphere ofthe same radius, or a prolate spheroid with size pa-rameters chosen to make the models comparable. Itwas found that the two models generally give similarresults for corresponding parameters. However, forshort stubby spores (i.e., with small aspect ratio),there were sufficient differences between results forthe two models that one could expect to distinguishbetween the two shapes from measured values ofDðΘ; λÞ and R34ðΘ; λÞ as functions of Θ. These differ-ences in the graphs remained even when averagesover a size distribution similar to the experimentallymeasured one for spores was applied. As expected,shorter spores gave lower values for DðΘ; λÞ. Thedifference in the graphs of this parameter for thetwo similar shapes was shown to become morepronounced as the aspect ratio (length/diameter)became closer to 1. In agreement with previous ex-perimental and calculation results for bacteria sus-pended in water [5–7], the results again indicatedthat the angular locations of the oscillations in thegraph of orientation averages of R34ðΘ; λÞ are mainlycontrolled by the diameter of the particles.In the present paper, we further explore the same
scattering functions with particular attention to de-termining the effect of length changes. This paperconcentrates on randomly oriented aerosols of singlevegetative bacteria. We note that vegetative bacteriagenerally have a much lower index of refraction thanspores. We utilize the capped cylinder model, whichapproximates the apparent shape of Escherichia coliwell, and which produced good fits [5] for graphs ofexperimentally measured values of R34ðΘ; λÞ for
those bacteria using microscopically measuredvalues [5,6] for lengths and diameters.
2. Parameters Used
The parameters used for the present calculationswere chosen to be similar to values measured forlog-phase Escherichia coli grown in a minimal medi-um [5,6]. Each individual E. coli bacterium wasmodeled as a right-circular cylinder capped smoothlyby a hemisphere of the same diameter as the cylinderat each end. This closely resembles the appearance ofthis bacterial cell in vivo in a phase contrast micro-scope or in electron microscope pictures. The scatter-ing by the particle is modeled by filling the modelwith dipoles on a simple cubic lattice with the latticespacing small enough so that
Y ¼ nka <12
ð1Þ
is fulfilled, where n is the refractive index, k is 2π=λwhere λ is the scattering wavelength, and a is thespacing between dipoles. This condition has beenfound to give good convergence for the DDA [8–11].An estimate for N, the number of dipoles neededto obtain this convergence, is found by settingðV=NÞ1=3 ¼ a, where V is the volume of the modelmicroorganism. In the case of the hemisphericallycapped cylinder model,
V ¼ πr2ðL − 2r=3Þ; ð2Þwhere r is the cylindrical and hemispherical radius,and L is the overall length including cylinder and theend-cap hemispheres.
For the present calculations, we used parameterssimilar to those measured for log-phase distributionsof the B/r strains of Escherichia coli grown in a mini-mal medium [5–7]. These and most other vegetativebacteria have refractive indices decidedly differentfrom those of Bacillus spores. The real part of the re-fractive index for E. coli is generally known for thevisible spectrum and was measured previously [5]as 1.373. We added an arbitrary and small imaginaryindex of 0.00097 to that. The real index, just a smallincrement above that of water, is a good approximatevalue through the visible and near-IR range. Thisstatement applies also for near-IR because the in-dices for water and for protein, the major constitu-ents of a vegetative cell, change only in the thirddecimal place [12,13] for wavelengths between 0.690and 1:5 μm. We note that a substantially larger realindex of about 1.52 was reported for killed cells ofPantoea agglomerans (formerly called Erwiniaherbicola) at visible wavelengths [14]. We do not be-lieve such a high value would apply to the index forlive Pantoea cells, which like E. coli consist mostlyof water.
The length distribution measured for log-phaseEscherichia coli B/r grown in a minimal medium[5–7] is shown in Fig. 1 and is tabulated in Table 1.
The diameter distribution measured for the samecells is given in Table 2. The diameters were un-correlated with length for the same growth, presum-ably because the cells grow primarily by elongation.(We note that diameters for different media are cor-related with length for the given medium, i.e., richermedia give rise to longer and fatter bacteria for thesame 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 sa-tisfy inequality (1), and therefore a number of theolder results using longer lengths are only sugges-tive. We were careful in the present paper to chooseparameters and a sufficient number of dipoles suchthat inequality (1) is always satisfied. In carrying outthe calculations to determine the effect of lengthchanges on DðΘ; λÞ and R34ðΘ; λÞ, we sometimes usedreduced (i.e., hypothetical) width and length distri-butions similar to, but smaller than, the measuredones to reduce computing time and still satisfy con-dition (1). We are mainly looking for the qualitativeeffect on the graphs of changes in the size param-eters, which can be varied much more readily in com-putations using the DDA model than experimentallywith aerosols. We also sometimes used larger lengths(See Table 1). We note that many species of bacteria
having shapes similar to our model have differentsizes, some larger, some smaller, than E. coli. In pre-vious experiments wemeasured the stationary phase(i.e., grown to nutrient exhaustion) for several spe-cies of bacteria to be much smaller in length and dia-meter than log phase of the same species grown withthe same conditions [6]. For E. coli grown in a mini-mal medium, the experimental volume for stationaryphase was 0:39 μm3, which is smaller than 46 μm3,the volume of the smallest model (set A, Table 1) usedhere. We emphasize that we are only trying to studythe typical response to size change in the functionsstudied, not trying to model the particular bacteriaE. coli. The results can be applied to larger (or smal-ler) size distributions if the scattering wavelength isincreased (or reduced) proportionally along with size,provided there is no large change of absorptionbetween the two wavelengths.
3. Theoretical Considerations
As in the first paper of this series, the coordinate sys-tem used will be that of [1,3], withΘ giving the anglebetween the scattering direction and the direction ofthe incoming light. The calculations were carried outusing the discrete dipole approximation (DDA, alsoknown as coupled dipole approximation) of Purcelland Pennypacker [15] with the polarizability of eachdipole defined by the Clausius–Mossotti formula,e.g., [3,16], using the numerical value of the refrac-tive index given above. Details of the methods of cal-culation are given elsewhere, e.g., [3,9,1].
4. Results
The results of the DDA approximation were pre-viously shown to adequately represent the differen-tial scattering function (i.e., Mueller matrix elementS11) for a sphere by comparison with calculationsusing the exact Mie solution in [9]. In the presentstudy we concentrate on the special cases of the func-tions DðΘ; λÞ and R34ðΘ; λÞ, and the effect of Y satis-fying Eq. (1)].
A. Convergence to the Identically Zero DepolarizationRatio for a Pseudosphere
For a true sphere, the off-diagonal elements of theamplitude matrix are identically zero by symmetry,making S11 identical to S22, which in turn makesDðΘ; λÞ equal to zero. Because the sphere modeled
Fig. 1. Length distribution for log-phase E. coli B/r.
Table 1. Length Distribution for Log-Phase E. coli B/r Grown in a Minimal Medium [5–7] and Similar Length Distributions Used in Modeling
with dipoles only approximates a true sphere, wemust either take note of its orientation angles or elseaverage over all orientations when the number of di-poles is fairly small. The orientation angles are de-fined with respect to the laser direction as are thescattering angles shown in Fig. 1 of [3] (i.e., interpretthe scattering direction as the orientation directionof 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 dipolenumbers are shown in Fig. 2. The sphere modeledhas diameter 1:2 μm, refractive index 1.373, and thescattering wavelength is λ ¼ 1:328 μm. Taking the ra-tio of the peaks near 150°, we see that the calculationfor 1021 dipoles gives values for DðΘ; λÞ a factor ofabout 5.17 larger than the same calculation with8025 dipoles. If one multiplies the values of DðΘ; λÞfor the smaller valued graph by this ratio, one ob-tains a graph that closely approximates the largergraph for the sphere modeled with 8025 dipoles. Thisis consistent with the calculated value of DðΘ; λÞ ap-proaching zero for an arbitrary angle as the dipolenumber 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 orientationdependence.
We next discuss the interesting limiting case inwhich the length, L, of the cylindrical portion of thecapped cylinder approaches zero. In this case, theexact solution for DðΘ; λÞ must approach zero. Thisis because, as the cylinder's length approaches zero,the model becomes a sphere whose diameter is thatof the former cylinder.
This approach to zero is illustrated in Fig. 3, wheregraphs are shown for a hemispherically capped cylin-der whose cylindrical part is short enough that itsshape is close to that of a sphere without yet reachingthat limit. The total length of the capped cylinder is1:5 μm, while its diameter is 1:2 μm. The graphs forboth the capped cylinder and for the dipole sphere
Table 2. Diameter Distribution for Log-Phase E. coli B/r Grownin a Minimal Medium [5,6] and Similar One for Models Used
in Present Paper
ExperimentalDiameter
(μm)Experimental
Percent
ModelDiameter
(μm)
0.85 14 0.650.90 36 0.700.95 36 0.751 14 0.80
Fig. 2. Graphs of the depolarization ratio versus angle for scat-tering wavelength 1:328 μm for a “dipole sphere” of diameter1:2 μm, refractive index 1.373, at orientation angles ϕ ¼ 57°, θ ¼17° modeled with either 1021 dipoles or 8025 dipoles as indicated.The values of the parameter Y in Eq. (1) are 0.628 for 1021 dipolesand 0.314 for 8025. The values ofDðΘ; λÞ calculated for orientationangles Θ ¼ ϕ ¼ 0 coincide with the horizontal axis on the scaleof this graph because of the symmetrical dipole placement. Forthe other orientation, DðΘ; λÞ approaches zero as predicted forincreasing dipole number.
Fig. 3. Results on a (a) linear scale and (b) logarithmic scale forDðΘ; λÞ averaged over orientations for both the capped cylinderwith a shape close to spherical (L ¼ 1:5 μm; d ¼ 1:2 μm) and forthe pseudosphere for n ¼ 1:373 with λ ¼ 1:328 μm in both cases.The calculations used 8829 dipoles for the sphere and 10,719 di-poles for the capped cylinder, giving aY value in Eq. (1) of about 0.3in both cases. The values for the sphere are close to the correctvalue of zero showing that Eq. (1) is adequate on the scale ofthe linear graph. For a short capped cylinder, the calculated valuesof D become small but nonzero as they should in approaching thelimiting zero value.
are shown. Both these calculations are averaged forrandom orientation over a 9 by 15 grid in θ and ϕ asare the other calculations where random orienta-tions are used in this paper. This number of orienta-tions was tested in [2,3] and found to give resultsindistinguishable from those from a 13 by 21 gridon the scale of these graphs.The graphs of Fig. 3(a) have magnitudes of DðΘ; λÞ
plotted with a linear scale, so that the much smallervalues for the dipole pseudosphere show up only asheavy overlays on the horizontal axis. The value ofDðΘ; λÞ for the short stubby capped cylinder is al-ready small. We later show (Fig. 6) that the valuesof DðΘ; λÞ become monotonically smaller as thelength decreases for constant diameter when the as-pect ratio is less than three. This is consistent withthe magnitude of DðΘ; λÞ approaching zero as thecapped cylinder shape approaches the (dipole-approximated) shape of a true sphere. The graphsof Fig. 3(b) have magnitudes plotted on a logarithmicscale so that the values of DðΘ; λÞ for the pseudo-sphere are shown to scale but can be seen to be100 times smaller than those for the short capped cy-linder. These calculations, together with those ofFig. 6, show that the value of DðΘ; λÞ becomes smalland approaches zero for either a capped cylindershape as it approaches the shape of a sphere, or fora pseudosphere as it is modeled with more and moredipoles.In contrast to the above graphs for DðΘ; λÞ, the
R34ðΘ; λÞ graph for the pseudosphere having thesame parameters as the pseudosphere used to gener-ate Fig. 3 has a characteristic nonzero oscillationthat remains much the same as the dipole numberbecomes large. In Fig. 4, we see that graph ofR34ðΘ; λÞ obtained from the DDAmodel for scatteringfrom a sphere using 8025 dipoles for either of twoarbitrarily selected particular orientations gives a
graph for both orientations that is qualitatively simi-lar to the result of a calculation using the exact Miesolution for a sphere. Although the difference is a fewpercent of the Mie value at some locations, the gen-eral shape of the DDA graphs and the location of theextrema are quite similar for the two types of calcu-lation. These results for R34ðΘ; λÞ and the approachto the mathematically exact zero value for DðΘ; λÞ asthe capped cylinder shape approaches that of asphere gives us additional confidence that the condi-tion of Eq. (1) is adequate to give reasonably accurategraphs.
B. Rod-Shaped Bacteria
The four sets, A, B, C, and D, of model lengths listedin Table 1 are all approximately proportional to theexperimentally measured lengths listed on the left-hand side of Table 1. That distribution was obtainedfor log-phase E. coli B/r grown in a minimal medium.We look first at how changes in length affect DðΘ; λÞand R34ðΘ; λÞ if all other parameters are kept thesame. In Fig. 5 the graphs of DðΘ; λÞ are plottedfor distributions of cells averaged over orientationsas well as averaged over each of the four sets oflengths, with each graph calculated for cells of thesame diameter of 0:8 μm.
We see that the magnitude and shape of the graphof D stays about the same until the case of a shortcapped cylinder with aspect ratio <2. Increasinglength does have a noticeable effect on the maxi-mum near 90°, and additionally, a smaller hump de-velops near 160° as the capped cylinder becomeslong. The large peak near 90° only moves slightly (ex-cept for the shortest cells) to smaller angles as thelength increases.
The graphs for DðΘ; λÞ for each of the individuallengths in the short sets A and B of Table 1 are shownin Fig. 6. The calculations are for a fixed diameter of0:8 μm. Figure 6(a) is for the vegetative cells whileFig. 6(b) is for the capped cylinder model of the
Fig. 4. Graph of the ratio R34ðΘ; λÞ versus Θ for a sphere of di-ameter 1:2 μm and n ¼ 1:373 and λ ¼ 1:328 μm, from calculationsusing the DDA model with 8025 dipoles at two different orienta-tions. Either orientation of the pseudosphere of dipoles is seen toproduce a result close to the graph resulting from the exact Miesolution.
Fig. 5. DðΘ; λÞ versus Θ for models A, B, C, and D from Table 1with fixed diameter ¼ 0:8 μm and averages over the length distri-bution for each set as well as an orientation average, withλ ¼ 1:266 μm. Average lengths for each distribution are indicatedin the figure.
spores. The only parameter difference is that the in-dex of refraction in air is somewhat larger for thespores at n ¼ 1:505 than at n ¼ 1:37 for the vegeta-tive cells. In both cases,DðΘ; λÞ continues to decreasein magnitude, and the peak moves to slightly largerangle as the length of the cylindrical part decreasestoward zero.The shapes of the graphs are quite similar for
spores and for vegetative cells of the same size inair. However, the depolarization is generally largerfor the spores, particularly in the back-scattering di-rection. We also note that the ratio of the direct back-scattering value of DðΘ; λÞ 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 DðΘ; λÞ does notchange much although there continue to be moderate
shape changes with the large peak appearing tomove toward the forward scattering direction.
The fact thatDðΘ; λÞ for a sphere is identically zerorequires that the magnitude of this function must de-crease for all angles as the length of the cylindricalpart of the capped cylinder approaches zero. How-ever, from Figs. 5 and 6, and other results it is ob-served that this decrease in magnitude does notbegin to occur until the length is less than aboutthree times the diameter of the end caps, at least forthe present wavelength.
In Fig. 7(a), graphs of R34ðΘ; λÞ for the four sets ofTable 1 with averages over length and orientationare presented for a vegetative cell with diameter0:8 μm. These show the small effect of length changeon the shape of the graphs of this function. Examina-tion of these graphs indicates that there are onlysmall changes in appearance for R34ðΘ; λÞ withlength changes over this broad range of average
Fig. 6. DðΘ; λÞ versus Θ is shown for vegetative cells having in-dividual lengths from sets A and B with fixed diameter of 0:8 μmfor (a) n ¼ 1:37 (vegetative bacteria) and (b) n ¼ 1:505 (spores).The averaging was over orientation only. The length for each cal-culation is shown in the figure. The scattering wavelength for bothfigures is 1:266 μm.
Fig. 7. (a) R34ðΘ; λÞ for n ¼ 1:37 and λ ¼ 1:266 μm averaged overorientation and length for the four model sets of Table 1 with fixeddiameter of 0:8 μm. The average length for each set is indicated inthe graph. (b) Graphs are shown for same function and parametersused for the single lengths indicated (with orientation averagingonly).
lengths. This change includes a slight movement ofthe graphs toward smaller angles as the lengthincreases. This may be due to longer paths beingavailable to the photons, which tend to loop the sur-face before being re-emitted as scattered light. Onemay also note some decrease in magnitude andsmoothness as the length increases.In Fig. 7(b), R34ðΘ; λÞ is graphed for the same pa-
rameters but with single short lengths, and aver-aging over orientation only. The effect of largeramplitude oscillations for shorter length is more pro-
nounced in this case. This effect is probably due tothe fact that the paths a photon can take before scat-tering for different orientations are more similar inlength for shorter cells.
Next, graphs of DðΘ; λÞ versus Θ in Fig. 8(a) andR34ðΘ; λÞ versus Θ in Fig. 8(b) are shown with thediameter of the capped cylinder varied for the con-stant length, L ¼ 5:5 μm. As previously observed[2,3], the major features of the graph move to the lefttoward smaller angles for both functions as the di-ameter increases.
As we consider relatively narrow cells, the largervalued part of the graph of DðΘ; λÞ moves to largerangles. In Fig. 9 the graphs for DðΘ; λÞ, for each ofthe short lengths of set A, are shown for a single di-ameter of 0:65 μm.
Finally, we consider results based on full averagesover lengths and widths as well as orientation usingdistributions similar to those observed experimen-tally for vegetative cells. In Table 3 the weightingsare calculated by averaging over lengths and diam-eters that are uncorrelated, i.e., separately combin-ing all of the lengths of either Model Set, A or D,in Table 1, with all the diameters of the Model Setof Table 2 with the appropriate weights. In view ofthe fact that it has been observed that the diam-eters and lengths for a single log-phase growth ofE. coli are uncorrelated experimentally (e.g., [5]),the weighting for a given pair L and d is just the
Fig. 8. Orientation averaged (a) DðΘ; λÞ and (b) R34ðΘ; λÞ, bothcalculated for n ¼ 1:374with λ ¼ 1:266 μm for a single fixed length,L ¼ 5:5 μm and varying diameters. The radius, R, of the cylinder isindicated for each calculation.
Fig. 9. Depolarization ratio, orientation averaged for capped cy-linders of diameter 0:65 μm for several single short lengths asshown using n ¼ 1:374 and λ ¼ 1:266 μm.
Table 3. Weightings ` for Uncorrelated Averages over the Lengths of Model Sets A and D, Table 1, Together with the Model Diameters of Table 2a
Average Length 1.365 Average Length 5.89 Average Diameter 0.725aThe entries in the table are the frequency for the given size with length from either Set A or Set D in the row, and the diameter from the
column. The frequencies are obtained by multiplying the frequency for the particular length by the frequency for the particular diameter.
product of the individual weights. The resulting over-all distribution should reasonably simulate a realdistribution of bacteria. The results of this averagingover all lengths and diameters may be seen for thedepolarization ratio in Fig. 10.The values of R34ðΘ; λÞ for the uncorrelated
averages of diameters with lengths of set A and Dare plotted together in Fig. 11. Figure 10 shows sub-stantial changes inDðΘ; λÞ versusΘ for the shape andmagnitude of the graphwhen the length of the bacter-ial distribution changes, with the width distributionremaining the same.However the graphs retain somequalitative resemblance in their dependence on scat-tering angle. The two graphs of R34ðΘ; λÞ, in contrast,are qualitatively similar in appearance for the longand the short length distributions. There is some ad-ditional bumpiness in the graph of R34ðΘ; λÞ for the
long distribution, D, between 60° and 90°, but other-wise the two graphs appear similar.
5. Conclusions
In this paper, we concentrated on determining howthe graphs of depolarization ratio, DðΘ; λÞ, and theMueller matrix ratio, R34ðΘ; λÞ versus Θ are affectedby changes in size for an aerosol of rod-shapedvegetative cells. We used size distributions similarto those experimentally measured for log-phaseEscherichia coli bacteria. Each individual bacteriumwas modeled as a cylindrical rod smoothly endedwith hemispherical end caps.
The value of DðΘ; λÞ for a perfect homogeneoussphere is identically zero. If the number of dipolesused is not large enough, the graph of DðΘ; λÞdepends on the orientation of the pseudosphere mod-eled with dipoles. In contrast, the value of R34ðΘ; λÞfor any orientation of the pseudosphere was found fora more modest dipole number to closely resemblethat obtained using the exact Mie solution for asphere.
The calculations graphed in this paper verify thatthe coupled dipole approximation or DDA, which weused throughout, indeed gives a value approachingzero for DðΘ; λÞ for any orientation of the pseudo-sphere as the number of dipoles increases beyondthat needed to satisfy condition (1).
In our capped cylinder model of a bacterium,the shape of the capped cylinder continuously ap-proaches that of a sphere as the length L of the modelapproaches the value, d, of the diameter of the cylin-der or the hemispherical end caps. We showed thatfor an orientation average, with a sufficient numberof dipoles, the value of DðΘ; λÞ becomes small, ap-proaching zero as L approaches d in magnitude.
Calculations were made to determine how thegraphs of DðΘ; λÞ and R34ðΘ; λÞ varied as the lengthchanged for fixed diameter of the bacteria. The mag-nitude of D stays roughly constant, as the length, L,of the capped cylinder decreases until L is less thanroughly three times the cylindrical diameter, afterwhich the magnitude of D decreases continuously to-ward zero. If the length becomes quite a bit longerthan the diameter, additional features appear in thedirect back-scattering direction. A change of indexfrom the vegetative value to that for a Bacillus sporemakes DðΘ; λÞ somewhat larger as the major changefor aerosolized particles. The graphs of R34ðΘ; λÞremain similar to one another as the cell length in-creases with no change in diameter, except that themagnitude of the oscillations generally is smaller forlonger cells.
Changes in the diameter, d, keeping L fixed, in con-trast have a noticeable effect on the shape of thegraphs both for DðΘ; λÞ and for R34ðΘ; λÞ. As pre-viously noted for R34ðΘ; λÞ, an increasing value of dcauses the shape to move to smaller angles. Nowwe note that this is the case also for DðΘ; λÞ.
When we averaged over L d for a complete distri-bution similar to that found for a log-phase growth of
Fig. 10. Uncorrelated averages of DðΘ; λÞ over all lengths anddiameters for length sets A (short) are denoted by a dashed lineand length set D (long) are denoted by a solid line with weightingsas indicated in Table 3. The averages were also over orientations.The index and wavelength were n ¼ 1:374 and λ ¼ 1:266 μm. Thesame diameter distribution is assumed for both length sets.
Fig. 11. Uncorrelated averages over all lengths and diameters forR34ðΘ; λÞ with weightings as indicated in Table 3. Averages areover orientations also. The short bacteria are set A denoted bythe dashed line. The long bacteria are set D denoted by the solidline. The same diameter distribution is assumed for both sets.
E. coli, the graphs of DðΘ; λÞ for a long versus a shortdistribution had a substantial change in shape,whereas the graphs for R34ðΘ; λÞ had a qualitativeresemblance for its major features between the longand short distributions.
References and Notes1. 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 scatter-
ing calculations exploring sensitivity of depolarization toshape changes for: I. Single spores in air,” Appl. Opt. 48,716–724 (2009). Note: the values used for the indices of refrac-tion for Bacillus cereus spores were mistakenly attributed byone of us (BVB) to M. Querry and M. Milham. The experimen-tal data were actually due to P. S. Tuminello, M. E. Milham, B.N. Khare, and E. T. Arakawa.
3. S. D. Druger, J. Czege, Z. Z. Li, and B. V. Bronk, “Calculationsof light scattering measurements predicting sensitivity ofdepolarization to shape changes of spores and bacteria,” Tech.Rep. ECBC-TR-607 (Edgewood Chemical Biological Cen-ter, 2008).
4. C. F. Bohren and D. R. Huffman, Absorption and Scattering ofLight 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 withpolarized light scattering,” Biophys. J. 69, 1170 (1995).
6. B. V. Bronk, W. P. Van De Merwe, and M. Stanley, “Anin-vivo measure of average bacterial cell size from apolarized light scattering function,” Cytometry 13, 155–162(1992).
7. W. P. Van De Merwe, J. Czege, M. E. Milham, and B. V. Bronk,“Rapid optically based measurements of diameter and lengthfor spherical or rod-shaped bacteria in vivo,” Appl. Opt. 43,5295–5302 (2004).
8. Y. You, G. W. Kattawar, C. Li, and P. Yang, “Internal dipoleradiation as a tool for particle identification,” Appl. Opt. 45,9115–9124 (2006).
9. B. T. Draine and P. J. Flatau, “Discrete dipole approximationfor scattering calculations,” J. Opt. Soc. Am. A 11, 1491–1499(1994).
10. M. A. Yurkin, V. P. Maltsev, and A. G. Hoekstra, “ I. Theoreticaleffects,” J. Opt. Soc. Am. A 23, 2578–2591 (2006).
11. M. A. Yurkin, V. P. Maltsev, and A. G. Hoekstra, “Convergenceof the discrete dipole approximation. II. An extrapolationtechnique to increase the accuracy,” J. Opt. Soc. Am. A 23,2592–2601 (2006).
12. G. M. Hale and M. R. Querry, “Optical constants of water inthe 200nm to 200 micrometer wavelength region,” Appl. Opt.12, 555–563 (1973).
13. E. Arakawa, P. S. Tuminello, B. N. Khare, and M. E. Milham,“Optical properties of ovalbumin in 0:130–2:50 μm spectralregion,” Biopolymers 62, 122–128 (2001).
14. E. T. Arakawa, P. S. Tuminello, B. N. Khare, andM. E. Milham, “Optical properties of Erwinia herbicolabacteria at 0.190–2.50 micron,” Biopolymers 72, 391–398(2003).
15. E. M. Purcell and C. R. Pennypacker, “Scattering and absorp-tion of light by nonspherical dielectric grains,” Astrophys. J.186, 705–714 (1973).
16. S. D. Druger and B. V. Bronk, “Internal and scattered electricfields in the discrete dipole approximation,” J. Opt. Soc. Am. B16, 2239–2246 (1999).
