Analytic T-matrix solution of light scattering from capsule and bi-sphere particles: Applications to spore detection

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Journal of Quantitative Spectroscopy &

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Radiative Transfer 108 (2007) 81–105

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Analytic T-matrix solution of light scattering from capsule andbi-sphere particles: Applications to spore detection

Dmitry Petrova,1, Gorden Videenb,c,�, Yuriy Shkuratova, Maryna Kaydasha

aAstronomical Institute of Kharkov V.N. Karazin National University, 35 Sumskaya Street, Kharkov 61022, UkrainebArmy Research Laboratory, AMSRD-ARL-CI-EM, 2800 Powder Mill Road, Adelphi, MA 20783, USA

cAstronomical Institute ‘‘Anton Pannekoek,’’ University of Amsterdam, Kruislaan 403, 1098 SJ Amsterdam, The Netherlands

Received 22 January 2007; received in revised form 19 April 2007; accepted 29 April 2007

Abstract

The derivation of the Sh-matrices using the T-matrix method allows the shape-dependent parameters to be separated

from size- and refractive-index-dependent parameters. This separation also allows for the surface integrals to be solved and

for analytic solutions to the light scattering from some irregular morphologies to be found. In this manuscript we derive

solutions for capsule-shaped and bi-sphere particles. We analyze the two-dimensional patterns resulting from these two

different simulants of Bacillus subtilis spores. The large differences between the patterns suggest that simple

approximations to complicated particles may not be adequate.

Published by Elsevier Ltd.

1. Introduction

Elastic scattering has the potential to provide automated real-time information about a particle system in alow-cost detector. Such detectors exploit characteristics in the light-scattering signal to identify particles ofinterest, and many works have been devoted to particle detection and characterization [1–17]. Probably, themost well-known feature is the drop-off in the forward intensity, and minima locations, used to size particles.One goal of modeling with regard to detector design is to search for identifying features in the scatteringpatterns from particles of interest that can be used to differentiate them from other particles that may bepresent in the natural background. In order to do this successfully, it is necessary to have accurate models ofthe particles that the detector is likely to interrogate; otherwise, the potential detector system may be searchingfor identifying features that are visible in the model particles, but are not present in the real particles.

In some cases, the particles of interest have irregular morphology and are heterogeneous in composition. Insome cases, the morphology and composition are not even well characterized. In the case of Bacillus subtilis

spores, this is the case. Fig. 1a shows a photomicrograph of such a spore. The morphology is oblong, capsule

e front matter Published by Elsevier Ltd.

srt.2007.04.010

ing author. Army Research Laboratory, AMSRD-ARL-CI-EM, 2800 Powder Mill Road, Adelphi, MA 20783, USA.

4 1871; fax: +1 301 394 4797.

esses: [email protected] (D. Petrov), [email protected] (G. Videen).

7 707 50 63.

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dx.doi.org/10.1016/j.jqsrt.2007.04.010

mailto:[email protected]

mailto:[email protected]

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Fig. 1. Examples of (a) Bacillus subtillis, (b) a capsule-shaped particle, and (c) a bi-sphere.

D. Petrov et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 108 (2007) 81–10582

shaped, similar to a cylinder having hemispherical endcaps, with the addition of extensive surface structure.Although the average refractive index has been measured, the internal morphology is largely unknown [7].Naturally, this makes modeling such particles a challenge. Some previous modeling studies have focused onthe scattering properties of aggregates of such particles [8–10]. In these studies the particle shape is assumed tobe that of a capsule. Although the modeling was performed using the discrete-dipole approximation (DDA), anumerical technique in which the complex surface morphology and heterogeneities could be considered, themodeling focused on the signal resulting from the agglomeration itself.

Recently, it has been suggested that a model based on a bi-sphere could be used to model the scatteringproperties of B. subtilis spores, since ‘‘from the general scattering theory, the difference between the patternfrom such a cluster and a cylindrical particle is not appreciable [11].’’ The advantage of using the bi-sphereapproximation is that rapid algorithms exist for calculating their scattering. Comparisons with experimental

ARTICLE IN PRESSD. Petrov et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 108 (2007) 81–105 83

data appear to confirm the validity of the model. These results appear at odds with the conclusions drawn byLi et al. [13], who used the finite-difference time-domain (FDTD) algorithm to compare patterns calculatedfrom finite cylinders, ellipsoids, and coated ellipsoids and found significant differences between the resultingscattering patterns. In addition, results of calculations from simulated water droplets shaped like spheroidparticles, stadium particles, and bi-spheres [14,15] differ strongly in fundamental ways; however, the dropletsin these modeling studies are significantly larger than the spores of interest, so it is possible that theseconclusions do not extrapolate to the smaller size regime. The bi-sphere system also has relevance innanotechnology, where surface plasmons can be used to interrogate aggregates of nanoparticles [1–6].

In this manuscript we examine and compare the light scattering resulting from capsule-shaped particles andbi-spheres in the size regime similar to that of B. subtilis spores. While such calculations can be made with anumerical technique like the DDA or FDTD, we exploit a new algorithm based upon the T-matrix technique[16–18]. Recent modifications to this technique have allowed the shape-dependent parameters to be separatedfrom the size- and refractive-index-dependent parameters. This allows for rapid calculations of size orrefractive index polydispersions of similarly shaped particles to be performed using the T-matrix technique.One additional benefit of the Sh-matrices is that the surface integrals that need to be solved to find theelements of the T-matrix are now much simpler and, in some cases, can be solved analytically. Recently, aderivation of an analytical solution for light scattering from Chebyshev particles has been presented [18]. Inthis manuscript we provide derivations of analytical solutions of capsule-shaped and bi-sphere particles. Inthis manuscript we consider the homogeneous particles, although our approach allows us to carry out thecalculations for particles with layered structure also [12]. The primary advantage of the analytical solutions isthat calculations may be performed extremely rapidly, e.g. with speeds comparable to Lorenz–Miecalculations. In addition, they are not subject to the computational errors resulting from inverting largematrices. In what follows we present the derivations of the light-scattering solution of the capsule-shaped andbi-sphere particles. We then present and compare results from these two types of systems.

2. Shape matrices for particles with a symmetry axis

In this section we present the equations necessary for calculating the Sh-matrices from particles having anaxis of symmetry. The complete derivation for arbitrary particles has been given previously [18]. Following themain idea of the T-matrix method, we expand the incident and scattered fields in series over appropriate vectorspherical wave functions:

Einc r; g;fð Þ ¼X1n¼1

Xn

m¼�n

amnRgMmn r; g;fð Þ þ bmnRgNmn r; g;fð Þ½ �, (1)

Esca r; g;fð Þ ¼X1n¼1

Xn

m¼�n

pmnMmn r; g;fð Þ þ qmnNmn r; g;fð Þ� �

, (2)

where RgMmn r; g;fð Þ, RgNmn r; g;fð Þ, Mmn r; g;fð Þ, Nmn r; g;fð Þ and amn, bmn, pmn, qmn are, respectively, thevector spherical wave functions and corresponding coefficients of expansion, r being a distance, g and f are,respectively, the polar and azimuth angles of spherical coordinate system with point of origin in the particlecenter. The functions RgMmn r; g;fð Þ and RgNmn r; g;fð Þ are finite at the origin (the prefix Rg means‘‘regular’’). The expansion coefficients of the scattered field pmn; qmn are related to the incident field expansioncoefficients amn; bmn through the equations:

pmn ¼X1n0¼1

Xn0

m0¼�n0

T11mnm0n0am0n0 þ T12

mnm0n0bm0n0� �

(3)

and

qmn ¼X1n0¼1

Xn0

m0¼�n0

T21mnm0n0am0n0 þ T22

mnm0n0bm0n0� �

, (4)

ARTICLE IN PRESSD. Petrov et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 108 (2007) 81–10584

which are valid due to the linearity of the Maxwell equations. The transition matrix (T-matrix) is

Tmnm0n0 ¼T11

mnm0n0 T12mnm0n0

T21mnm0n0 T22

mnm0n0

!. (5)

It can be shown that the following formulae for calculating the T-matrix may be obtained from the Maxwellequations:

Tmnm0n0 ¼ � RgQmnm0n0ð Þ Qmnm0n0ð Þ�1. (6)

The matrices RgQmnm0n0 and Qmnm0n0 are:

RgQmnm0n0 ¼RgQ11

mnm0n0 RgQ12mnm0n0

RgQ21mnm0n0 RgQ22

mnm0n0

!(7)

and

Qmnm0n0 ¼Q11

mnm0n0 Q12mnm0n0

Q21mnm0n0 Q22

mnm0n0

!, (8)

where

Q11mnm0n0 ¼ �i m0J21

mnm0n0 þ J12mnm0n0

� �, (9)


mnm0n0 þ J22mnm0n0

� �, (10)


mnm0n0 þ J11mnm0n0

� �, (11)


mnm0n0 þ J21mnm0n0

� �, (12)

RgQ11mnm0n0 ¼ �i m0Rg J21

mnm0n0 þRg J12mnm0n0

� �, (13)



� �, (14)



� �, (15)



� �. (16)

For particles having an axis of symmetry,

Rg J11mnm0n0 ¼ �2pdmm0 i �1ð Þ

m0�mAnn0

Z p

0

dy sin y pm0n0 yð Þtmn yð Þ þ pmn yð Þtm0n0 yð Þ½ ��

jn0 m0Rð Þjn Rð ÞR2�, (17)

Rg J12mnm0n0 ¼ 2pdmm0 �1ð Þ

m0�mAnn0

R p0 dy sin y pm0n0 yð Þpmn yð Þ þ tmn yð Þtm0n0 yð Þ½ � jn0 m0Rð Þj0n Rð ÞR2

� �þR p0 dy sin ydmn yð Þtm0n0 yð Þjn0 m0Rð Þjn Rð Þ qR

qy

� �" #

, (18)

Rg J21mnm0n0 ¼ 2pdmm0 �1ð Þ

m0�m�1Ann0

R p0 dy sin y pm0n0 yð Þpmn yð Þ þ tmn yð Þtm0n0 yð Þ½ � j0n0 m0Rð Þjn Rð ÞR2

� �þ 1

m0

R p0 dy sin ydm0n0 yð Þtmn yð Þjn0 m0Rð Þjn Rð Þ qR

qy

� �" #

,

(19)

Rg J22mnm0n0 ¼ �2pdmm0 i �1ð Þ

m0�mAnn0

R p0 dy sin y pm0n0 yð Þtmn yð Þ þ pmn yð Þtm0n0 yð Þ½ � j0n0 m0Rð Þj0n Rð ÞR2

� �þR p0 dy sin ydmn yð Þpm0n0 yð Þj0n0 m0Rð Þjn Rð Þ qR

qy

� �þ 1

m0

R p0dy sin ydm0n0 yð Þpmn yð Þjn0 m0Rð Þj0n Rð Þ qR

qy

� �2664

3775,

(20)


J11mnm0n0 ¼ �2pdmm0 i �1ð Þ

m0�mAnn0

Z p

0

dy sin y pm0n0 yð Þtmn yð Þ þ pmn yð Þtm0n0 yð Þ½ ��

jn0 m0Rð Þhn Rð ÞR2� �

, (21)

J12mnm0n0 ¼ 2pdmm0 �1ð Þ

m0�mAnn0

R p0dy sin y pm0n0 yð Þpmn yð Þ þ tmn yð Þtm0n0 yð Þ½ � jn0 m0Rð Þh0n Rð ÞR2� �

þR p0 dy sin ydmn yð Þtm0n0 yð Þjn0 m0Rð Þhn Rð Þ qR

qy

� �" #

, (22)

J21mnm0n0 ¼ �2pdmm0 �1ð Þ

m0�mAnn0

R p0 dy sin y pm0n0 yð Þpmn yð Þ þ tmn yð Þtm0n0 yð Þ½ � j0n0 m0Rð Þhn Rð ÞR2

� �þ 1

m0

R p0 dy sin ydm0n0 yð Þtmn yð Þjn0 m0Rð Þhn Rð Þ qR

qy

� �" #

, (23)

J22mnm0n0 ¼ �2pdmm0 i �1ð Þ

m0�mAnn0

R p0 dy sin y pm0n0 yð Þtmn yð Þ þ pmn yð Þtm0n0 yð Þ½ � j0n0 m0Rð Þh0n Rð ÞR2

� �þR p0 dy sin ydmn yð Þpm0n0 yð Þj0n0 m0Rð Þhn Rð Þ qR

qy

� �þ 1

m0

R p0 dy sin ydm0n0 yð Þpmn yð Þjn0 m0Rð Þh0n Rð Þ qR

qy

� �2664

3775. (24)

The angular functions in formula (17)–(24) are defined as follows:

pmn yð Þ ¼m

sin ydn0m yð Þ, (25)

tmn yð Þ ¼d

dydn0m yð Þ, (26)

where dn0m yð Þ is the Wigner d-function:

dsmn yð Þ ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffisþmð Þ! s�mð Þ! sþ nð Þ! s� nð Þ!

p Xmin sþm;s�nð Þ

k¼0

�1ð Þkcosðy=2Þ� �2s�2kþm�n

sinðy=2Þ� �2k�mþn

k! sþm� kð Þ! s� n� kð Þ! n�mþ kð Þ!, (27)

Ann0 ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2n0 þ 1ð Þ

4pn0 n0 þ 1ð Þ

s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2nþ 1ð Þ

4pn nþ 1ð Þ

s. (28)

R ¼ R yð Þ defines the shape of the irregular particles in spherical coordinates; y is the polar angle, and jn Rð Þ,hn Rð Þ are the spherical Bessel functions of the first and third kinds, respectively; the derivatives j0nðRÞ and h0nðRÞ

can be found from the recurrence relations of the spherical Bessel functions.The T-matrix depends only on the physical and geometrical characteristics of the scattering particle, such as

the size parameter, shape, relative refractive index, and orientation; and is completely independent of thedistance, propagation directions and polarization states of the incident and scattered fields. This means thatthe T-matrix needs to be computed only once and then can be used in calculations for any direction andpolarization state of the incident or scattered field. We have suggested further simplifications [16–18] thatisolate the influence of particle shape and the parameters X and m0. The simplified expressions forEqs. (17)–(24) are given in Appendix A.

3. Capsule-shaped particles and bi-spheres

As can be seen from the previous section, solving the Sh-matrices requires making a number of integrationsover the azimuthal angle. These integrations can be solved analytically for the capsule-shaped particles, theshape of which is described by the following equation (see examples in Fig. 1a):

R0 yð Þ ¼

cos y; 0pyo p4;

12 sin y ;

p4pyo 3p

4

� cos y; 3p4pyop:

8><>: , (29)

The Sh-elements (Eqs. (A.9)–(A.26)) is expressed in Appendix B.


The T-matrix has been used previously to calculate the light scattered by bi-spheres [19]. Using the Sh-matrix method, we are able to solve the surface integrals analytically. The bi-sphere shape is described by thefollowing equation (see examples in Fig. 1c):

R0 yð Þ ¼ cos yj j. (30)

The bi-sphere Sh-elements are expressed by the same equations, with the only differences occurring inEq. (B.21) for I

ðyÞmnm0n0 zð Þ and (B.34) for I 0

ðyÞmnm0n0 zð Þ. Expressions for these elements are given in Appendix C.

4. Results and discussion

We have coded the equations of the appendices. The bi-sphere results are consistent with those previouslypublished [11,14], and capsule results are consistent with DDA simulations. Here we present the normalizedintensity and degree of linear polarization of capsule-shaped (solid line) and bi-sphere (points) particles havingsizes 1.5� 3.0 mm illuminated by light with wavelength l ¼ 700 nm. The refractive index of the particles ischosen to be m0 ¼ 1.59+0i for different conditions of illumination. Figs. 2–5 correspond to the scatteringintensities and linear polarizations calculated in the plane of incidence with the particles in fixed orientation.There are a number of similarities between the patterns: neglecting the oscillations, the signal levels areapproximately the same, and the amplitude of oscillation, related to the visibility of the minima, is

1e-005

0.0001

0.001

0.01

0.1

1

I

0 60 120 180

ϑ°

-80

-40

0

40

80

capsule

bi-sphere

P,%

1.5 x 3.0 μmλ=700 nm

m0=1.59+0i

fixed orientationsϕ=0°

Fig. 2. Normalized intensity and degree of linear polarization of a capsule (solid line) and a bi-sphere (points) having sizes 1.5� 3.0mmilluminated by light with wavelength l ¼ 700 nm. The refractive index of the particles is m0 ¼ 1.59+0i. Calculations are at fixed

orientation, when the incident light strikes the particles at angle j ¼ 01, parallel to the major particle axis.

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1e-005

0.0001

0.001

0.01

0.1

1

I

-180 -120 -60 0 60 120 180

ϑ°

-80

-40

0

40

80

P,%

capsulebi-sphere

1.5 x 3.0 μmλ=700 nm

m0=1.59+0i



orientation, when the incident light strikes the particles at angle j ¼ 451 to the major particle axis. Note that the range of scattering angle

extends from y ¼ �1801 to 1801, since the mirror symmetry is broken.

D. Petrov et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 108 (2007) 81–105 87

approximately the same also. In the forward-scattering region, the oscillations tend to overlap to some extent.In this region, diffraction dominates. It is also where data is taken to be used in algorithms to size particles.This suggests that such algorithms would yield similar sizes for both types of particles. One primary differenceis the extremely low-frequency oscillations in the scattering from the capsule-shaped particles. This is mostobvious in the backscatter region of Fig. 2 (around y�1801) and the mid-range section (around y�901) ofFig. 3. These are explained by the fact that the capsule-shaped particle, acting like a long antenna, can bettersupport this much lower frequency mode; whereas, the individual bi-spheres cannot.

While the differences are subtle between the phase functions at fixed orientation, at average orientations,shown in Fig. 4, they are not. Here the amplitude of oscillations, and resulting visibility, of the bi-sphereparticles are much greater than for the capsule-shaped particles. This is because there is greater variancebetween the patterns of the capsule-shaped particles at different orientations than for the bi-spheres.This variance tends to average out the oscillations, whereas, for the bi-spheres, the smaller variancepreserves the oscillations in the average. The reason for the smaller variance is that the bi-sphere componentspreserve many of the scattering characteristics of the individual spheres; i.e., the independent scatteringcomponents have a strong role in the scattering [20]. Since these independent-scattering components of thebi-sphere do not depend on orientation, the variance between the patterns with the particles in differentorientations is not great.

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1e-006

1e-005

0.0001

0.001

0.01

0.1

1

I

0 60 120 180

ϑ°

-80

-40

0

40

80

P,%

capsulebi-sphere

1.5 x 3.0 μmλ=700 nm

m0=1.59+0i



orientation when the incident light strikes the particles at angle j ¼ 901, perpendicular to the major particle axis.


Figs. 6 and 7 show maps of the forward-scattering intensities and polarizations for the bi-sphere (left panels)and capsule (right panels) at fixed orientations. Due to the symmetry, the top panels show a series ofconcentric rings for both particle types. As seen in Fig. 2 for different-size particles, the maxima and minimalocations of the different particles coincide near the center, before drifting out of phase at larger-scatteringangles. Note that the range of polarizations is relatively small for both types of particles, compared to those ofthe particles in other orientations.

When the particle systems are oriented at j ¼ 451 to the incident beam, differences between the scatteringfrom the bi-sphere and capsule are very apparent. The shapes of the fringes from both systems are now curved;however, the fringes are very different. For the bi-sphere particle, there are two sets of intersecting fringes;whereas, only one set of fringes is obvious for the capsule. The polarization features for these two particlestilted at 451 (shown in the middle panels of Fig. 7) are quite different, and their shapes are similar as for theintensities.

The shape of the fringes continue to evolve when the particle systems are oriented at j ¼ 901 to the incidentbeam (shown in the bottom panels of Figs. 6 and 7). For the bi-sphere, two sets of fringes are apparent:circular fringes from the spheres and linear fringes due to the interference of light scattered from the twospherical particles, as discussed and quantified by Videen et al. [7] For the capsule, only the one set of closedfringes is apparent, and their shape is eccentric, with minor axes corresponding to the major axis of the capsule

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0.0001

0.001

0.01

0.1

1

I

0 60 120 180

ϑ°

-80

-40

0

40

P,%

capsule

bi-sphere

1.5 x 3.0 μm

λ=700 nm

m0=1.59+0i

averaged overorientations

Fig. 5. Normalized intensity and degree of linear polarization of a capsule (solid line) and a bi-sphere (points) having sizes 1.5� 3.0mmilluminated by light with wavelength l ¼ 700 nm. The refractive index of the particles is m0 ¼ 1.59+0i. Calculations are orientation

averages.


and vice versa. In addition, a bright vertical band is apparent that can be attributed to scattering from thecylindrical portion of the capsule. Such a band was seen for the stadium particles studied by Videen et al. [7],and can be explained by diffraction. The shapes of these fringes are seen in the polarizations in Fig. 7.

Figs. 8 and 9 show maps of the back-scattering intensities and polarizations for the bi-sphere (left panels)and capsule (right panels) at fixed orientations. As for the forward-scattering hemisphere, the top panels showa series of concentric rings for both particle types due to the particle symmetry. Note that the range ofpolarizations is much greater in the back-scattering hemisphere than in the forward-scattering hemispheresshown in Fig. 7. These maxima and minima also appear sharper for the polarization curves than for theintensity curves.

When the particle systems are oriented at j ¼ 451 to the incident beam, differences between the scatteringfrom the bi-sphere and capsule become more apparent. The shapes of the fringes are now curved, but unlikethe forward-scattering hemispheres, multiple sets of fringes are apparent for both particles. Again, the shapesof the fringes that evolve are different. For the bi-sphere particle, the fringes have noticeably higher visibility.One interesting feature of the capsule particle is the specular reflection from the surface of the cylindricalportion of the capsule that provides a maximum scattering intensity at the left edge of the scatteringhemisphere at y ¼ 901. This feature is also visible in the forward-scattering hemisphere on the right edge of thescattering hemisphere, but is not as obvious because the relative difference in intensities is much less in the

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Fig. 6. Maps of the forward-scattering hemisphere (logarithm scale) by a bi-sphere (left panel) and a capsule (right panel) for particles at

several fixed orientations. Different angles of particle rotation j around the axis perpendicular to the major particle axis are used: j ¼ 01

(the incident light is parallel to the major particle axis), j ¼ 451, j ¼ 901, and j ¼ 1351 correspond to the 1–4 rows, respectively. The

refractive index of particles is m0 ¼ 1.59+0i, and the size parameter is X ¼ 10.0.


latter case. The polarization features for these two particles tilted at 451 (shown in the middle panels of Fig. 9)are quite different both in fringe structure and in the distribution of magnitudes. For the capsule particles,large island regions of negative polarization appear along and near the axis of the particle; whereas, for the bi-sphere, the islands of positive and negative polarization appear smaller and their distribution more uniform.

When the particle systems are oriented at j ¼ 901 to the incident beam, similarities between the scatteringfrom the bi-sphere and capsule almost completely disappear. For the bi-sphere, two sets of fringes areapparent: circular fringes from the spheres and linear fringes due to the interference of light scattered from thetwo spherical particles, as discussed for the forward-scattering hemisphere. For the cylindrical particles,neither of these features is dominant. The pattern resembles that of a cross, and the frequencies of the fringes

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Fig. 7. The same as in Fig. 6, but for the linear degree of polarization.


on the major and minor axes are different, due to the different particle dimensions along these axes. Theshapes of these fringes are seen in the polarizations of Fig. 9.

5. Conclusions

Using the Sh-matrix formalism, we have derived analytical T-matrix solutions for two different particletypes, bi-sphere and capsule-shaped particles, making it possible to make rapid calculations. Such particles areof interest as simulants of simple biological spores, most commonly B. subtilis. We find that the resultingscattering patterns from these particles are quite different, especially when the particles are not oriented so thattheir major axes are oriented along the direction of the incident beam. This suggests that great care must be

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Fig. 8. The same as in Fig. 6, but for the backscatter hemisphere.


taken in developing model simulant particles, so that the scattering features in the simulant are actuallypresent in the actual particle system.

While we agree with previous researchers who suggest that the back-scattering hemispheres are very rich ininformation [6], we find their interpretation more problematic. Our studies suggest that the forward-scatteringhemispheres are significantly different even for these similar particles and provide significant information thatcan be used for discrimination. In addition, features in the forward scattering can be interpreted usingdiffraction theory, suggesting that a simple Fourier analysis could provide basic morphological information.While the back-scattering hemisphere appears to be rich in information, few algorithms have yet to bedeveloped to interpret this information and extract particle properties necessary for identification or evendiscrimination. One of the exceptions is the backscattering intensity surge and accompanying polarizationphenomena that have been topics of extensive research in recent years [21,22]; however, these are derived fromthe one-dimensional phase function, so the richness of the full two-dimensional back scattering is not fullyutilized.

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Fig. 9. The same as in Fig. 7, but for the backscatter hemisphere.


Appendix A

In this appendix we provide simplified expressions for Eqs. (17)–(24) [16–18]. In these expressions themorphological information is contained within the shape matrix Sh.

Rg J11mnm0n0 X ;m0ð Þ ¼ X nþn0þ2 m0ð Þ

n0X1k1¼0

Xm0ð Þ2k1

k1!G n0 þ k1 þ ð3=2Þ� �

�X1k2¼0

Xð Þ2k2

k2!G nþ k2 þ ð3=2Þ� �RgSh11

mnm0n0 ;k1þk2, ðA:1Þ



n0X1k1¼0

Xm0ð Þ2k1

k1!G n0 þ k1 þ ð3=2Þ� �X1

k2¼0

Xð Þ2k2

k2!G nþ k2 þ ð3=2Þ� �

� RgSh121mnm0n0 ;k1þk2þ

X 2

nþ k2 þ ð3=2ÞRgSh122mnm0n0 ;k1þk2

, ðA:2Þ


n0�1X1k1¼0

Xm0ð Þ2k1

k1!G n0 þ k1 þ ð3=2Þ� �X1

k2¼0

Xð Þ2k2

k2!G nþ k2 þ ð3=2Þ� �


Xm0ð Þ2

n0 þ k1 þ ð3=2ÞRgSh212mnm0n0 ;k1þk2

, ðA:3Þ

Rg J22mnm0n0 X ;m0ð Þ ¼ X nþn0 m0ð Þ

n0�1X1k1¼0

Xm0ð Þ2k1

k1!G n0 þ k1 þ ð3=2Þ� �X1

k2¼0

Xð Þ2k2

k2!G nþ k2 þ ð3=2Þ� �


Xm0ð Þ2

n0 þ k1 þ ð3=2ÞRgSh222

mnm0n0 ;k1þk2

þX 2

nþ k2 þ ð3=2ÞRgSh223mnm0n0 ;k1þk2

þX 2 Xm0ð Þ

2

n0 þ k1 þ ð3=2Þ� �

nþ k2 þ ð3=2Þ� �RgSh224mnm0n0 ;k1þk2

!, ðA:4Þ

J11mnm0n0 X ;m0ð Þ ¼ Rg J11

mnm0n0 X ;m0ð Þ þ X n0�nþ1 m0ð Þn0

�X1k1¼0

Xm0ð Þ2k1

k1!G n0 þ k1 þ ð3=2Þ� �X1

k2¼0

Xð Þ2k2

k2!G k2 � nþ ð1=2Þ� �Sh11mnm0n0 ;k1þk2

, ðA:5Þ


mnm0n0 X ;m0ð Þ þ X n0�n m0ð Þn0X1k1¼0

Xm0ð Þ2k1

k1!G n0 þ k1 þ ð3=2� �X1

k2¼0

Xð Þ2k2

k2!G k2 � nþ ð1=2Þ� �

� Sh121mnm0n0 ;k1þk2þ

X 2

k2 � nþ ð1=2ÞSh122

mnm0n0;k1þk2

, ðA:6Þ


mnm0n0 X ;m0ð Þ þ X n0�n m0ð Þn0�1

X1k1¼0

Xm0ð Þ2k1

k1!G n0 þ k1 þ ð3=2Þ� �

�X1k2¼0

Xð Þ2k2

k2!G k2 � nþ ð1=2Þ� � Sh211mnm0n0;k1þk2

þXm0ð Þ

2

n0 þ k1 þ ð3=2ÞSh212mnm0n0 ;k1þk2

, ðA:7Þ


mnm0n0 X ;m0ð Þ þ X n0�n�1 m0ð Þn0�1

X1k1¼0

Xm0ð Þ2k1

k1!G n0 þ k1 þ ð3=2Þ� �

�X1k2¼0

Xð Þ2k2

k2!G k2 � nþ ð1=2Þ� � Sh221mnm0n0 ;k1þk2

þXm0ð Þ

2

n0 þ k1 þ ð3=2ÞSh222

mnm0n0 ;k1þk2

þX 2

k2 � nþ ð1=2ÞSh223

mnm0n0;k1þk2þ

X 2 Xm0ð Þ2

n0 þ k1 þ ð3=2Þ� �

k2 � nþ ð1=2Þ� �Sh224

mnm0n0;k1þk2

!.

ðA:8Þ


Since we here deal with the axially symmetrical particles, the Sh-matrices can be written as

RgSh11mnm0n0k ¼ �2p2dmm0 i

�1ð Þm0�mþk

22kþn0þnþ2Ann0

Z p

0

dy sin y pm0n0 yð Þtmn yð Þ þ pmn yð Þtm0n0 yð Þ½ � R0ð Þ2kþnþn0þ2

n o,

(A.9)

RgSh121mnm0n0k ¼ 2p2dmm0�1ð Þm

0�mþk

22kþn0þnþ2Ann0

�

R p0 dy sin y pm0n0 yð Þpmn yð Þ þ tmn yð Þtm0n0 yð Þ½ � R0ð Þ

2kþnþn0þ1 nþ 1ð Þ

þR p0 dy sin ydmn yð Þtm0n0 yð Þ R0ð Þ

2kþnþn0 qR0

qy

24

35, ðA:10Þ

RgSh122mnm0n0k ¼ �2p

2dmm0�1ð Þm

0�mþk

22kþn0þnþ3Ann0

Z p

0

dy sin y pm0n0 yð Þpmn yð Þ þ tmn yð Þtm0n0 yð Þ½ � R0ð Þ2kþnþn0þ3

� �,

(A.11)

RgSh211mnm0n0k ¼ � 2p2dmm0�1ð Þm

0�mþk

22kþn0þnþ2Ann0

�


2kþnþn0þ1 n0 þ 1ð Þ

þR p0 dy sin ydm0n0 yð Þtmn yð Þ R0ð Þ

2kþnþn0 qR0

qy

24

35, ðA:12Þ

RgSh212mnm0n0k ¼ 2p2dmm0

�1ð Þm0�mþk

22kþn0þnþ3Ann0

Z p

0

dy sin y pm0n0 yð Þpmn yð Þ þ tmn yð Þtm0n0 yð Þ½ � R0ð Þ2kþnþn0þ3

� �,

(A.13)

RgSh221mnm0n0k ¼ � i2p2dmm0�1ð Þm

0�mþk

22kþn0þnþ2Ann0

�

R p0 dy sin y pm0n0 yð Þtmn yð Þ þ pmn yð Þtm0n0 yð Þ½ � R0ð Þ

2kþnþn0 n0 þ 1ð Þ nþ 1ð Þ

þR p0 dy sin ydmn yð Þpm0n0 yð Þ R0ð Þ

2kþnþn0�1 qR0

qy n0 þ 1ð Þ

þR p0 dy sin ydm0n0 yð Þpmn yð Þ R0ð Þ

2kþnþn0�1 qR0

qy nþ 1ð Þ

26664

37775, ðA:14Þ

RgSh222mnm0n0k ¼ i2p2dmm0

�1ð Þm0�mþk

22kþn0þnþ3Ann0

�


2kþnþn0þ2 nþ 1ð Þ


2kþnþn0þ1 qR0

qy

24

35, ðA:15Þ

RgSh223mnm0n0k ¼ i2p2dmm0

�1ð Þm0�mþk

22kþn0þnþ3Ann0

�


2kþnþn0þ2 n0 þ 1ð Þ


2kþnþn0þ1 qR0

qy

24

35, ðA:16Þ


RgSh224mnm0n0k ¼ �i2p2dmm0

�1ð Þm0�mþk

22kþn0þnþ4Ann0

Z p

0

dy sin y pm0n0 yð Þtmn yð Þ þ pmn yð Þtm0n0 yð Þ½ � R0ð Þ2kþnþn0þ4

� �,

(A.17)

Sh11mnm0n0k ¼ RgSh11mnm0n0k þ 2p2dmm0

�1ð Þm0�mþn�1þk

22kþn0�nþ1Ann0

�

Z p

0

dy sin y pm0n0 yð Þtmn yð Þ þ pmn yð Þtm0n0 yð Þ½ � R0ð Þ2k�nþn0þ1

n o, ðA:18Þ

Sh121mnm0n0k ¼ RgSh121mnm0n0k þ i2p2dmm0


22kþn0�nþ1Ann0

�

�R p0 dy sin y pm0n0 yð Þpmn yð Þ þ tmn yð Þtm0n0 yð Þ½ � R0ð Þ

2k�nþn0n

þR p0 dy sin ydmn yð Þtm0n0 yð Þ R0ð Þ

2k�nþn0�1 qR0

qy

24

35, ðA:19Þ

Sh122mnm0n0k ¼ RgSh122

mnm0n0k � i2p2dmm0�1ð Þm

0�m�nþ1þk

22kþn0�nþ2Ann0

�

Z p

0

dy sin y pm0n0 yð Þpmn yð Þ þ tmn yð Þtm0n0 yð Þ½ � R0ð Þ2k�nþn0þ2

� �, ðA:20Þ

Sh211mnm0n0k ¼ RgSh211mnm0n0k � i2p2dmm0

�1ð Þm0�m�n�1þk

22kþn0�nþ1Ann0

�


2k�nþn0 n0 þ 1ð Þ

þR p0 dy sin ydm0n0 yð Þtmn yð Þ R0ð Þ

2k�nþn0�1 qR0

qy

24

35, ðA:21Þ


mnm0n0k þ i2p2dmm0�1ð Þm

0�m�n�1þk

22kþn0�nþ1Ann0

�

Z p

0

dy sin y pm0n0 yð Þpmn yð Þ þ tmn yð Þtm0n0 yð Þ½ � R0ð Þ2k�nþn0þ2

� �, ðA:22Þ

Sh221mnm0n0k ¼ RgSh221mnm0n0k þ 2p2dmm0


22kþn0�nþ1Ann0

�


2k�nþn0�1 n0 þ 1ð Þ �nð Þ


2k�nþn0�2 qR0

qy n0 þ 1ð Þ

�R p0 dy sin ydm0n0 yð Þpmn yð Þ R0ð Þ

2k�nþn0�2 qR0

qy n

26664

37775, ðA:23Þ

Sh222mnm0n0k ¼ RgSh222mnm0n0k � 2p2dmm0


22kþn0�nþ2Ann0

�

�R p0 dy sin y pm0n0 yð Þtmn yð Þ þ pmn yð Þtm0n0 yð Þ½ � R0ð Þ

2k�nþn0þ1n


2k�nþn0 qR0

qy

24

35, ðA:24Þ


Sh223mnm0n0k ¼ RgSh223mnm0n0k � 2p2dmm0


22kþn0�nþ2Ann0

�


2k�nþn0þ1 n0 þ 1ð Þ


2k�nþn0 qR0

qy

24

35, ðA:25Þ


mnm0n0k þ 2p2dmm0�1ð Þm

0�m�nþ1þk

22kþn0�nþ3Ann0

�

Z p

0

dy sin y pm0n0 yð Þtmn yð Þ þ pmn yð Þtm0n0 yð Þ½ � R0ð Þ2k�nþn0þ3

� �, ðA:26Þ

where R0 ¼ ðRðyÞ=X Þ.

Appendix B

The Sh-elements (given by Eqs. (A.9)–(A.26)) can be integrated for a capsule-shaped particle. Theirexpressions are given in this appendix.

RgSh11mnm0n0 ;k ¼ �2p2dmm0 i

�1ð Þm0�mþk

22kþn0þnþ2Ann0I

ð1Þmnm0n0 2k þ nþ n0 þ 2ð Þ, (B.1)

RgSh121mnm0n0 ;k ¼ 2p2dmm0�1ð Þm

0�mþk

22kþn0þnþ2Ann0 I

ð2Þmnm0n0 2k þ nþ n0 þ 1ð Þ nþ 1ð Þ þ L

ð01Þmnm0n0 2k þ nþ n0ð Þ

h i, (B.2)

RgSh122mnm0n0 ;k ¼ �2p2dmm0

�1ð Þm0�mþk

22kþn0þnþ3Ann0 I

ð2Þmnm0n0 2k þ nþ n0 þ 3ð Þ

h i, (B.3)

RgSh211mnm0n0 ;k ¼ �2p2dmm0

�1ð Þm0�mþk

22kþn0þnþ2Ann0 I

ð2Þmnm0n0 2k þ nþ n0 þ 1ð Þ n0 þ 1ð Þ þ L

ð00Þmnm0n0 2k þ nþ n0ð Þ

ih, (B.4)

RgSh212mnm0n0 ;k ¼ 2p2dmm0�1ð Þm

0�mþk

22kþn0þnþ3Ann0 I


h i, (B.5)

RgSh221mnm0n0 ;k ¼ � i2p2dmm0

�1ð Þm0�mþk

22kþn0þnþ2Ann0 I

ð1Þmnm0n0 2k þ nþ n0ð Þ n0 þ 1ð Þ nþ 1ð Þ

hþLð11Þmnm0n0 2k þ nþ n0 � 1ð Þ n0 þ 1ð Þ þ L

ð10Þmnm0n0 2k þ nþ n0 � 1ð Þ nþ 1ð Þ

i, ðB:6Þ

RgSh222mnm0n0 ;k ¼ i2p2dmm0�1ð Þm

0�mþk

22kþn0þnþ3Ann0 I

ð1Þmnm0n0 2k þ nþ n0 þ 2ð Þ nþ 1ð Þ þ L


ih,

(B.7)

RgSh223mnm0n0 ;k ¼ i2p2dmm0�1ð Þm

0�mþk

22kþn0þnþ3Ann0 I

ð1Þmnm0n0 2k þ nþ n0 þ 2ð Þ n0 þ 1ð Þ þ L


ih,

(B.8)

RgSh224mnm0n0 ;k ¼ �i2p2dmm0�1ð Þm

0�mþk

22kþn0þnþ4Ann0 I

ð1Þmnm0n0 ; 2k þ nþ n0 þ 4ð Þ

ih, (B.9)

Sh11mnm0n0 ;k ¼ RgSh11mnm0n0 ;k þ 2p2dmm0


22kþn0�nþ1Ann0I

ð1Þmnm0n0 2k � nþ n0 þ 1ð Þ, (B.10)


Sh121mnm0n0;k ¼ RgSh121mnm0n0 ;k

þ i2p2dmm0�1ð Þm

0�mþn�1þk

22kþn0�nþ1Ann0 �nI

ð2Þmnm0n0 2k � nþ n0ð Þ þ L

ð01Þmnm0n0 2k � nþ n0 � 1ð Þ

h i, ðB:11Þ

Sh122mnm0n0;k ¼ RgSh122mnm0n0 ;k � i2p2dmm0

�1ð Þm0�m�nþ1þk

22kþn0�nþ2Ann0 I

ð2Þmnm0n0 2k � nþ n0 þ 2ð Þ

h i, (B.12)

Sh211mnm0n0;k ¼ RgSh211mnm0n0 ;k

� i2p2dmm0�1ð Þm

0�m�n�1þk


ð2Þmnm0n0 2k � nþ n0ð Þ n0 þ 1ð Þ þ L

ð00Þmnm0n0 2k � nþ n0 � 1ð Þ

ih,

ðB:13Þ

Sh212mnm0n0;k ¼ RgSh212mnm0n0 ;k þ i2p2dmm0




h i, (B.14)

Sh221mnm0n0;k ¼ RgSh221mnm0n0 ;k þ 2p2dmm0


22kþn0�nþ1Ann0 �I

ð1Þmnm0n0 2k � nþ n0 � 1ð Þn n0 þ 1ð Þ

hþLð11Þmnm0n0 2k � nþ n0 � 2ð Þ n0 þ 1ð Þ � L

ð10Þmnm0n0 2k � nþ n0 � 2ð Þn

i, ðB:15Þ

Sh222mnm0n0;k ¼ RgSh222mnm0n0 ;k � 2p2dmm0


22kþn0�nþ2Ann0

� �nIð1Þmnm0n0 2k � nþ n0 þ 1ð ÞFð0Þ2k�nþn0þ1;m0�m þ L

ð11Þmnm0n0 2k � nþ n0ð Þ

ih, ðB:16Þ



22kþn0�nþ2Ann0

� Ið1Þmnm0n0 2k � nþ n0 þ 1ð Þ n0 þ 1ð Þ þ L

ð10Þmnm0n0 2k � nþ n0ð Þ

ih, ðB:17Þ


�1ð Þm0�m�nþ1þk



ih. (B.18)

The following designations are used in formulae (B.1)–(B.18):

Ið1Þmnm0n0 zð Þ ¼

Z p

0

dy sin y pm0n0 yð Þtmn yð Þ þ pmn yð Þtm0n0 yð Þ½ � R0ð Þz

� �

¼ mn0

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffin0 þ 1ð Þ

2�m02

q2n0 þ 1

IðyÞmnm0n0þ1 zð Þ �

n0 þ 1ð Þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffin02 �m02p

2n0 þ 1IðyÞmnm0n0�1 zð Þ

24

35

þm0n

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffinþ 1ð Þ

2�m2

q2nþ 1

IðyÞmnþ1m0n0 zð Þ �

nþ 1ð Þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffin2 �m2p

2nþ 1IðyÞmn�1m0n0 zð Þ

24

35, ðB:19Þ


Ið2Þmnm0n0 zð Þ ¼

Z p

0

dy sin y pm0n0 yð Þpmn yð Þ þ tmn yð Þtm0n0 yð Þ½ � R0ð Þz

� �

¼ mm0 I ðyÞmnm0n0 zð Þ þ

1

2n0 þ 1ð Þ 2nþ 1ð Þn0 þ 1ð Þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffin02 �m02

pnþ 1ð Þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffin2 �m2p

IðyÞmn�1m0n0�1 zð Þ

"

þ n0ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffin0 þ 1ð Þ

2�m02

qn


2�m2

qIðyÞmnþ1m0n0þ1 zð Þ

� n0 þ 1ð Þ


pn


2�m2

qIðyÞmnþ1m0n0�1 zð Þ

�n0ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffin0 þ 1ð Þ

2�m02

qnþ 1ð Þ


IðyÞmn�1m0n0þ1 zð Þ

#, ðB:20Þ

IðyÞmnm0n0 zð Þ ¼

Z p

0

dy R0ð Þz dn

0m yð Þdn0

0m0 yð Þsin y

¼ �1ð Þnþn0XmXm0n!ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffin� mj jð Þ! nþ mj jð Þ!

pn0!

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffin0 � m0j jð Þ! n0 þ m0j jð Þ!

p�Xn� mj j

k¼0

�1ð Þk

k! n� kð Þ! n� mj j � kð Þ! mj j þ kð Þ!

Xn0� m0j j

k0¼0

�1ð Þk0

k0! n0 � k0� �

! n0 � m0j j � k0� �

! m0j j þ k0� �

!

�

IðC1Þmnm0n0 2n� 2k � mj j þ 2n0 � 2k0 � m0j j � 1; 2k þ mj j þ 2k0 þ m0j j � 1; z

� �þIðSÞmnm0n0 2n� 2k � mj j þ 2n0 � 2k0 � m0j j � 1; 2k þ mj j þ 2k0 þ m0j j � 1; z

� �þIðC2Þmnm0n0 2n� 2k � mj j þ 2n0 � 2k0 � m0j j � 1; 2k þ mj j þ 2k0 þ m0j j � 1; z

� �

26664

37775, ðB:21Þ

Ið/1Þmnm0n0 m; n; zð Þ ¼

1

2

Z p=4

0

siny2

n

cosy2

m

cos yð Þz dy ¼

2P1k¼0

C0 z; kð Þ �2ð ÞkOC1 m; nþ 2kð Þ; zo0;

�1ð ÞzPzk¼0

Ckz �2ð Þ

kOC1 m; nþ 2kð Þ; zX0;

8>>><>>>:

(B.22)

IðSÞmnm0n0 m; n; zð Þ ¼

1

2zþ1

Z 3p=4

p=4sin

y2

n

cosy2

m1

sin yð Þz dy ¼

1

22zOS m� z; n� zð Þ, (B.23)


�1ð Þz

2

Z p

3p=4sin

y2

n

cosy2

m

cos yð Þz dy

¼

2 �1ð ÞzP1k¼0

C0 z; kð Þ �2ð ÞkOC2 m; nþ 2kð Þ; zo0;

Pzk¼0

Ckz �2ð Þ

kOC2 m; nþ 2kð Þ; z � 0;

8>>>><>>>>:

ðB:24Þ


OC1 m; nð Þ ¼

Z p=8

0

dyðsin yÞnðcos yÞm

¼

�ð2þ

ffiffi2pÞðmþ1Þ=2

mþnð Þ2mþ1ð2�

ffiffi2pÞðn�1Þ=2

2n�1þPn�1k¼1

n�1ð Þ!! mþn�2k�2ð Þ!!n�2k�1ð Þ!! mþn�2ð Þ!!

ð2�ffiffi2pÞðn�2k�1Þ=2

2n�2k�1

þn�1ð Þ!!m!!mþnð Þ!! ZðCÞ m; p

8; 0

� �; n ¼ 2n; ma� 2;�4; . . .

�

ð2�ffiffi2pÞðnþ1Þ=2

�m�1ð Þ2nþ1ð2þ


2mþ1þPm�1k¼1

�m�n�2ð Þ� �m�n�4ð Þ�:::� �m�n�2kð Þ

�m�3ð Þ� �m�5ð Þ�:::� �m�2k�1ð Þ

ð2þffiffi2pÞmþ2kþ1

2

2mþ2kþ1

þ�m�n�2ð Þ� �m�n�4ð Þ�...� �nð Þ

�m�1ð Þ!! ZðSÞ m; p8; 0

� �;

2664

3775 n ¼ 2n; �m ¼ 2m

�ð2þ




2n�1þPnk¼1

2kG nð Þ mþn�2k�2ð Þ!!G n�kð Þ mþn�2ð Þ!!


2n�2k�1

; n ¼ 2nþ 1; ma� 1;�3; . . .

� 12

Pmk¼0

ð�1ÞkCkm ð2þ

ffiffi2pÞð2k�nþ1Þ=2

�1½ �n�kð Þ22k�nþ1

� �; n ¼ 2nþ 1;�m ¼ 2mþ 1;

8>>>>>>>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>>>>>>>:

ðB:25Þ

OS m; nð Þ ¼

Z 3p=8

p=8dyðsin yÞnðcos yÞm

¼

�ð2�


mþnð Þ2mþ1ð2þ


2n�1þPn�1k¼1


ð2þffiffi2pÞðn�2k�1Þ=2

2n�2k�1

þð2þ




2n�1þPn�1k¼1



2n�2k�1

þ

n�1ð Þ!!m!!mþnð Þ!! ZðCÞ m; 3p

8; p8

� �;

n ¼ 2n; ma� 2;�4; . . .

ð2þffiffi2pÞðnþ1Þ=2

�m�1ð Þ2nþ1ð2�


2mþ1þPm�1k¼1

�m�n�2ð Þ� �m�n�4ð Þ�...� �m�n�2kð Þ

�m�3ð Þ� �m�5ð Þ�...� �m�2k�1ð Þ

ð2�ffiffi2pÞðmþ2kþ1Þ=2

2mþ2kþ1

�ð2�

ffiffi2pÞðnþ1Þ=2

�m�1ð Þ2nþ1ð2þ


2mþ1þPm�1k¼1

�m�n�2ð Þ� �m�n�4ð Þ�...� �m�n�2kð Þ

�m�3ð Þ� �m�5ð Þ�...� �m�2k�1ð Þ

ð2þffiffi2pÞðmþ2kþ1Þ=2

2mþ2kþ1

þ�m�n�2ð Þ� �m�n�4ð Þ�...� �nð Þ

�m�1ð Þ!! ZðSÞ m; 3p8; p8

� �;

266666664

377777775n ¼ 2n;�m ¼ 2m

ð2þffiffi2pÞðmþ1Þ=2



2n�1þPnk¼1



2n�2k�1

�ð2�


mþnð Þ2mþ1ð2þ


2n�1þPnk¼1



2n�2k�1

; n ¼ 2nþ 1; ma� 1;�3; . . .

� 12

Pmk¼0

ð�1ÞkCkm ð2�

ffiffi2pÞð2k�nþ1Þ=2

�ð2þffiffi2pÞð2k�nþ1Þ=2½ �

n�kð Þ22k�nþ1

� �; n ¼ 2nþ 1; �m ¼ 2mþ 1;

8>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>:

ðB:26Þ


OC2 m; nð Þ ¼

Z p=2

3p=8dyðsin yÞnðcos yÞm

¼

ð2�ffiffi2pÞðmþ1Þ=2

mþnð Þ2mþ1ð2þ


2n�1þPn�1k¼1



2n�2k�1

þ

n�1ð Þ!!m!!mþnð Þ!! ZðCÞ m; p

2; 3p8

� �;

n ¼ 2n; ma� 2;�4; . . .

ð2þffiffi2pÞðnþ1Þ=2

mþ1ð Þ2nþ1ð2�


2mþ1þPm�1k¼1

�m�n�2ð Þ� �m�n�4ð Þ�...� �m�n�2kð Þ

�m�3ð Þ� �m�5ð Þ�...� �m�2k�1ð Þ

ð2�ffiffi2pÞðmþ2kþ1Þ=2

2mþ2kþ1

þ�m�n�2ð Þ� �m�n�4ð Þ�...� �nð Þ

�m�1ð Þ!! ZðSÞ m; p2; 3p8

� �;

2664

3775 n ¼ 2n; �m ¼ 2m

ð2�ffiffi2pÞðmþ1Þ=2

mþnð Þ2mþ1ð2þ


2n�1þPnk¼1



2n�2k�1

; n ¼ 2nþ 1; ma� 1;�3; . . .

� � 12

Pmk¼0

ð�1ÞkCkm ð2�

ffiffi2pÞð2k�nþ1Þ=2½ �

n�kð Þ22k�nþ1

� �; n ¼ 2nþ 1; �m ¼ 2mþ 1;

8>>>>>>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>>>>>>:

ðB:27Þ

ZðCÞ m; b; að Þ ¼

Z b

a

ðcos yÞm dy

¼

Cnm

b�a2mþ 1

2m�1

Pn�1k¼1

Ckmsin m�2kð Þb½ ��sin m�2kð Þa½ �

m�2k; m ¼ 2n; mX0

Pnk¼0

Ckn�1ð Þk

2kþ1sin bð Þ

2kþ1� sin að Þ

2kþ1� �

; a ¼ 2nþ 1; a40

sin amþ1 cos að Þ

mþ1þPn�1k¼1

2kn n�1ð Þ... n�kþ1ð Þ

2n�3ð Þ 2n�5ð Þ... 2n�2k�1ð Þcos að Þ

mþ2kþ1

� sin bmþ1 cos bð Þ

mþ1þPn�1k¼1


2n�3ð Þ 2n�5ð Þ... 2n�2k�1ð Þcos bð Þ

mþ2kþ1

266664

377775; �m ¼ 2n; mo0

sin amþ1 cos að Þ

mþ1þPn�2k¼1

2n�1ð Þ 2n�3ð Þ... 2n�2kþ1ð Þ

2kn n�1ð Þ... n�kþ1ð Þcos að Þ

mþ2kþ1

� sin bmþ1 cos bð Þ

mþ1þPn�2k¼1

2n�1ð Þ 2n�3ð Þ... 2n�2kþ1ð Þ

2kn n�1ð Þ... n�kþ1ð Þcos bð Þ

mþ2kþ1

þ2n�1ð Þ!!2nn! ln

tan b=2ð Þtan a=2ð Þ

26666666664

37777777775; �m ¼ 2nþ 1; mo0;

8>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>:

ðB:28Þ


ZðSÞ n; b; að Þ ¼

Z b

a

ðsin yÞn dy

¼

n�1ð Þ!! b�að Þ

2n=2 n=2ð Þ!þ cos a

n sin að Þn�1þ 1

G n=2ð Þ

Pn�1k¼1

G n2� k þ 1

� � n�1ð Þ n�3ð Þ... n�2kþ1ð Þ sin að Þn�2k�1

2k

� cos bn sin bð Þ

n�1þ 1

G n=2ð Þ

Pn�1k¼1

G n2� k þ 1� � n�1ð Þ n�3ð Þ... n�2kþ1ð Þ sin bð Þ

n�2k�1

2k

266664

377775;

n ¼ 2n; n40Pnk¼0

Ckn�1ð Þkþ1

2kþ1cos bð Þ

2kþ1� cos að Þ

2kþ1� �

; n ¼ 2nþ 1; n40

cos bnþ1 sin bð Þ

nþ1þPn�1k¼1


2n�3ð Þ 2n�5ð Þ... 2n�2k�1ð Þsin bð Þ

nþ2kþ1

� cos anþ1 sin að Þ

nþ1þPn�1k¼1


2n�3ð Þ 2n�5ð Þ... 2n�2k�1ð Þsin að Þ

nþ2kþ1

266664

377775; �n ¼ 2n; no0

cos bn�1 sin bð Þ

nþ1þPn�2k¼1

2n�1ð Þ 2n�3ð Þ... 2n�2kþ1ð Þ

2kn n�1ð Þ... n�kþ1ð Þcos bð Þ

nþ2kþ1

� cos an�1 sin að Þ

nþ1þPn�2k¼1

2n�1ð Þ 2n�3ð Þ... 2n�2kþ1ð Þ

2kn n�1ð Þ... n�kþ1ð Þcos að Þ

nþ2kþ1

þ2n�1ð Þ!!2nn! ln

tan b=2ð Þtan a=2ð Þ

26666666664

37777777775; �n ¼ 2nþ 1; np0:

8>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>:

ðB:29Þ

Further

Lð00Þmnm0n0 zð Þ ¼

Z p

0

dy sin ydm0n0 yð Þtmn yð Þ R0ð Þz qR0

qy

¼�n0

2nþ 1n


2�m2

qI 0ðyÞmnþ1m0n0 zð Þ � nþ 1ð Þ


I 0ðyÞmn�1m0n0 py; z

� �� , ðB:30Þ


Z p

0

dy sin ydmn yð Þtm0n0 yð Þ R0ð Þz qR0

qy

¼�n

2n0 þ 1n0


2�m02

qI 0ðyÞmnm0n0þ1 py; z

� �� n0 þ 1ð Þ


pI 0ðyÞmnm0n0�1 py; z

� �� , ðB:31Þ


Z p

0

dy sin ydm0n0 yð Þpmn yð Þ R0ð Þz qR0

qy

¼�n0

2nþ 1n


2�m2

qI 0ðyÞmnþ1m0n0 py; z

� �� nþ 1ð Þ


I 0ðyÞmn�1m0n0 py; z

� �� , ðB:32Þ



Z p

0

dy sin ydmn yð Þpm0n0 yð Þ R0ð Þz qR0

qy

¼�n

2n0 þ 1n0


2�m02

qI 0ðyÞmnm0n0þ1 py; z

� �� n0 þ 1ð Þ


pI 0ðyÞmnm0n0�1 py; z

� �� , ðB:33Þ

where

I 0ðyÞmnm0n0 zð Þ ¼

Z p

0

dydn0m yð Þdn0

0m0 yð Þ R0ð Þz qR0

qy


pn0!


p�Xn� mj j

k¼0

�1ð Þk


Xn0� m0j j

k0¼0

�1ð Þk0

k0! n0 � k0� �

! n0 � m0j j � k0� �

! m0j j þ k0� �

!

�

�4IðC1Þmnm0n0 2n� 2k � mj j þ 2n0 � 2k0 � m0j j þ 1; 2k þ mj j þ 2k0 þ m0j j þ 1; z

� �þIðSÞmnm0n0 2n� 2k � mj j þ 2n0 � 2k0 � m0j j þ 1; 2k þ mj j þ 2k0 þ m0j j � 1; z

� ��IðSÞmnm0n0 2n� 2k � mj j þ 2n0 � 2k0 � m0j j � 1; 2k þ mj j þ 2k0 þ m0j j þ 1; z

� �þ4I

ðC2Þmnm0n0 2n� 2k � mj j þ 2n0 � 2k0 � m0j j þ 1; 2k þ mj j þ 2k0 þ m0j j þ 1; z

� �

266666664

377777775, ðB:34Þ

Xm ¼1; mX0;

ð�1Þm; mo0;

((B.35)

where dn;m ¼1; n ¼ m

0; nam

(is Kronecker’s delta, Cm

n ¼ n!=ðm! n�mð Þ!Þ is the binomial coefficients, and

C0mn ¼

�1ð Þn nþm� 1ð Þ!

n! m� 1ð Þ!. (B.36)

Appendix C

The Sh-elements for a bi-sphere are very similar to those of a capsule (given in Appendix B). The onlydifferences occur in Eq. (B.21) for I

ðyÞmnm0n0 zð Þ and (B.34) for I 0

ðyÞmnm0n0 (see Appendix C):

IðyÞmnm0n0 zð Þ ¼

Z p

0

dy R0ð Þz dn

0m yð Þdn0

0m0 yð Þsin y


pn0!


p�Xn� mj j

k¼0

�1ð Þk


Xn0� m0j j

k0¼0

�1ð Þk0

k0! n0 � k0� �

! n0 � m0j j � k0� �

! m0j j þ k0� �

!

� IðC1Þmnm0n0 2n� 2k � mj j þ 2n0 � 2k0 � m0

� 1; 2k þ mj j þ 2k0 þ m0 � 1; z

� �h i, ðC:1Þ

where


1

2

Z p

0

dy siny2

n

cosy2

m

cos yj jz¼

P1n¼0

Gð1�zÞ2n

Gðnþ1ÞGðn�zÞOðm; nþ 2n; zÞ; zo0;

Pzn¼0

Cnzð�1Þ

n2nOðm; nþ 2n; zÞ; z40;

8>>><>>>:

(C.2)


O m; n; zð Þ ¼

Z p=2

0

dy sin yð Þn cos yð Þ

m� �1ð Þz

Z p

p=2dy sin yð Þ

n cos yð Þm

¼

1mþn� 1� ð�1Þzð Þ 2

ð�m=2Þ�ð1=2Þ

mþn 2ð�n=2Þþð1=2Þ þPn�1k¼1

n�1ð Þ!! mþn�2k�2ð Þ!!n�2k�1ð Þ!! mþn�2ð Þ!! 2

ðk�ðn=2Þþð1=2Þ

;

n ¼ 2n; ma� 2;�4; . . .

1� ð�1Þzð Þ 2ð�n=2Þ�ð1=2Þ

�m�1 2ð�m=2Þ�ð1=2Þ þPm�1k¼1

�m�n�2ð Þ� �m�n�4ð Þ�...� �m�n�2kð Þ

�m�3ð Þ� �m�5ð Þ�...� �m�2k�1ð Þ2ð�kð�m=2Þ�ð1=2ÞÞ

;

n ¼ 2n; �m ¼ 2m

1mþn� 1� ð�1Þzð Þ 2

ð�m=2Þ�ð1=2Þ

mþn 2�n2þ

12 þ

Pnk¼1

G nð Þ mþn�2k�2ð Þ!!G n�kð Þ mþn�2ð Þ!! 2

2k�n2þ

12

; n ¼ 2nþ 1;ma� 1;�3; . . .

1� ð�1Þzð Þ 12

Pmk¼0

ð�1ÞkCkm2ðn=2Þ�k�ð1=2Þ

n�k� 1

2

Pmk¼0

ð�1ÞkCkm

n�k

� �; n ¼ 2nþ 1; �m ¼ 2mþ 1;

8>>>>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>>>>:

ðC:3Þ

I 0ðyÞmnm0n0 zð Þ ¼

Z p

0

dydn0m yð Þdn0

0m0 yð Þ R0ð Þz qR0

qy


pn0!


p�Xn� mj j

k¼0

�1ð Þk


Xn0� m0j j

k0¼0

�1ð Þk0

k0! n0 � k0� �

! n0 � m0j j � k0� �

! m0j j þ k0� �

!

� IðC2Þmnm0n0 2n� 2k � mj j þ 2n0 � 2k0 � m0

; 2k þ mj j þ 2k0 þ m0 ; z� �h i

, ðC:4Þ

IðC2Þmnm0n0 m; n; zð Þ ¼

Z p

0

dy siny2

n

cosy2

m

cos yj jz q cos yj j

qy

¼

4P1n¼0

Gð1�zÞ2n

Gðnþ1ÞGðn�zÞOðmþ 1; nþ 2nþ 1; zÞ; zo0;

4Pzn¼0

Cnzð�1Þ

n2nOðmþ 1; nþ 2nþ 1; zÞ; z40:

8>>><>>>:

ðC:5Þ

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Documents

Analytic T-matrix solution of light scattering from capsule and bi-sphere particles: Applications to spore detection