Experimental and theoretical study of light scattering by individual
  mature red blood cells by use of scanning flow cytometry and discrete dipole
  approximation
Microsoft Word - RBC_preprint.doc
Experimental and theoretical study of light scattering by 
individual mature red blood cells by use of scanning flow 
cytometry and discrete dipole approximation 
Maxim A. Yurkin, Konstantin A. Semyanov, Peter A. Tarasov,  
Andrei V. Chernyshev, Alfons G. Hoekstra, and Valeri P. Maltsev 
M. A. Yurkin, K. A. Semyanov, P. A. Tarasov, A. V. Chernyshev, and V. P. Maltsev are with 
the Institute of Chemical Kinetics and Combustion, Siberian Branch of the Russian Academy 
of Sciences, Institutskaya 3, Novosibirsk 630090 Russia 
A. G. Hoekstra is with the Faculty of Science, Section Computational Science, of the 
University of Amsterdam, Kruislaan 403, 1098 SJ, Amsterdam, The Netherlands 
M. A. Yurkin is also with the Faculty of Science, Section Computational Science, of the 
University of Amsterdam, Kruislaan 403, 1098 SJ, Amsterdam, The Netherlands 
A. V. Chernyshev and V. P. Maltsev are also with Novosibirsk State University, Pirogova 2, 
Novosibirsk 630090 Russia 
V. P. Maltsev’s e-mail is maltsev@ns.kinetics.nsc.ru 
Abstract 
Elastic light scattering by mature red blood cells (RBCs) was theoretically and 
experimentally analyzed with the discrete dipole approximation (DDA) and the 
scanning flow cytometry (SFC), respectively. SFC permits measurement of angular 
dependence of light-scattering intensity (indicatrix) of single particles. A mature RBC is 
modeled as a biconcave disk in DDA simulations of light scattering. We have studied 
the effect of RBC orientation related to the direction of the incident light upon the 
indicatrix. Numerical calculations of indicatrices for several aspect ratios and volumes 
of RBC have been carried out. Comparison of the simulated indicatrices and 
indicatrices measured by SFC showed good agreement, validating the biconcave disk 
model for a mature RBC. We simulated the light-scattering output signals from the SFC 
with the DDA for RBCs modeled as a disk-sphere and as an oblate spheroid. The 
biconcave disk, the disk-sphere, and the oblate spheroid models have been compared for 
two orientations, i.e. face-on and rim-on incidence. Only the oblate spheroid model for 
rim-on incidence gives results similar to the rigorous biconcave disk model. 
Keywords: red blood cell, discrete dipole approximation, non-spherical particle light 
scattering, scanning flow cytometry 
OCIS code: 170.0170, 170.1530, 170.1470, 290.5850, 120.5820 
Introduction 
The scattering of electromagnetic waves by dielectric particles is a problem of great 
significance for a variety of applications ranging from particle sizing and remote sensing to 
radar meteorology and biological sciences.1,2 In the area of medical diagnostics, 
understanding how a laser beam interacts with blood suspensions or a whole-blood medium is 
of paramount importance in quantifying the inspection process in many commercial devices 
and experimental setups that are used widely for in vivo or in vitro blood measurements.3 
These measurements are focused mainly on electromagnetic scattering properties of red blood 
cells (RBCs), the type of cell that are most numerous in the blood. 
Usually, RBCs measure from 6.6 to 7.5 µm in diameter, however, cells with a diameter 
greater than 9 µm (macrocytes) or less than 6 µm (microcytes) have been observed. These 
cells are nonnucleated (among vertebrates, only the red cells of mammalians lack nuclei) 
biconcave discs that are surrounded by thin, elastic membranes and filled with hemoglobin. 
They are soft, flexible, and elastic and therefore move easily through the narrow blood 
capillaries. The primary function of these cells is to carry oxygen from the lungs to the body 
cells.3 
The light scattering properties of blood are based on a solution of the single-
electromagnetic-scattering problem for a RBC. Moreover RBCs can play an important role in 
verification of solutions of the direct light-scattering problem for nonspherical particles 
because of their simple internal structure and stable biconcave discoid shape. Most of the 
relevant work solves the single-electromagnetic-scattering problem by a RBC analytically 
either basically by use of Mie4, Fraunhofer5 and anomalous diffraction theories5,6 or 
numerically with the aid of the T-matrix approach7 by treating a real RBC as a volume 
equivalent spherical4, spheroidal,7 or ellipsoidal5,8 dielectric particle. In these cases the 
assumptions can be considered exact only because of the special experimental conditions 
when RBCs become spheroidal or ellipsoidal. In contrast, only a few papers have dealt with 
light scattering by a real nondeformed RBC.9,10 Shvalov et al.10 made use of the single- and 
double-wave Wentzel-Kramers-Brillouin approximations and were able to reproduce 
analytically the experimental results for forward scattering angles in the range of 15° to 35°. 
Mazeron and Muller,11 with the aid of a physical optics approximation, presented small-angle 
forward-scattering patterns by a RBC whose axisymmetric geometry was obtained through 
complete rotation of a Cassini curve. 
At present RBCs form the border of the size range where effective methods of light 
scattering simulation can be applied. Tsinopoulos and Polyzos12 studied scattering by 
nondeformed, average-sized RBCs illuminated by a He-Ne laser beam. They examined 
various orientations of the RBC with respect to the direction of the incident light and 
computed scattering patterns in the forward, sideways, and backward directions. They used a 
boundary-element method appropriately combined with fast-Fourier-transform (FFT) 
algorithms. 
Another method to simulate light scattering by arbitrary shaped particles is the discrete 
dipole approximation (DDA).13 The latest improvements in the DDA14 and its implementation 
on parallel supercomputers15 permit simulation of light scattering by particles with the sizes 
of RBCs. 
The next generation of flow cytometers, the scanning flow cytometer (SFC),16 permits 
measurement of the angular dependence of the light-scattering intensity of single particles at a 
speed of O(102) particles/s. In the research reported in this manuscript, light scattering by 
mature RBCs was theoretically and experimentally analyzed with the DDA and a SFC, 
respectively. In the numerical simulations we modeled a mature RBC as a biconcave disk.17 
The effect of RBC orientation relative to the direction of the incident light was studied. We 
carried out numerical calculations of indicatrices for several axis ratios and volumes of RBC. 
Comparison of the simulated indicatrices with those measured with the SFC showed a good 
agreement, validating the biconcave disk model for a mature RBC. We simulated light-
scattering output signals from a SFC with DDA for a RBC modeled as a disk-sphere (a 
symmetric discoid layer cut out of a sphere) and an oblate spheroid and compared those 
results with results of biconcave disk model for two orientations, face-on and rim-on 
incidence. 
Theory 
Optical Model of the RBC 
A mature red blood cell (RBC) can be modeled as a biconcave discoid. A RBC is composed 
of hemoglobin (32%), water (65%), and membrane tissue (3%) and does not contain any 
nucleus.18 The shape of the RBC was described by Fung et al.17: 
( )422 8579.05262.11583.0165.0)( xxxdxT −+−= , (1)
where T is a thickness of RBC (along the axis of symmetry), x is a relative radial cylindrical 
coordinate x = 2ρ /d (−1 ≤ x ≤ 1), ρ is a radial cylindrical coordinate, and d is the diameter of 
the RBC. In order to vary the ratio of maximum thickness and diameter independently on the 
diameter of a RBC we have rewritten Eq. (1) in the following form: 
( )422 8579.05262.11583.012)( xxxdxT −+−= ε , (2)
where ε = Tmax/d is an aspect ratio of maximum thickness and diameter. The RBC shape is 
shown schematically in Fig. 1. 
Discrete Dipole Approximation 
The DDA permits simulation of light scattering by a particle modeled as a finite array of 
polarizable points. The points acquire dipole moments in response to the local electric field. 
The dipoles interact by means of their electric fields,19,20 and the DDA is also sometimes 
referred to as the coupled dipole approximation.21 The theoretical base for the DDA, including 
Fig. 1. Shape of a mature RBC and its orientation with respect to the incident radiation. xyz is a 
laboratory reference frame and x′y′z′ is a reference frame tied to the RBC (z′ is the axis of symmetry); 
inc and sca are propagation vectors for incident and scattering radiation respectively. 
radiative reaction corrections, was summarized by Draine and Flatau.13 The performance of 
the DDA was significantly improved by introducing complex-conjugate gradient methods for 
solving systems of linear equations and a FFT algorithm for matrix-vector multiplication 
(summarized by Draine and Flatau13). Hoekstra et al.15 parallelized the DDA and showed that 
the algorithm runs efficiently on distributed memory computers, provided that the number of 
dipoles per processor is large enough. They run DDA simulations with O(107) dipoles, limited 
only by the available amount of memory.15 
The size of a sub-volume that relates to the number of dipoles that form a particle must 
be small enough to ensure that response to an electromagnetic field is the response of an ideal 
induced dipole. The size should be in range λ/20 < d < λ/10.13,22 In some cases we can even 
use d ≥ λ/10. Comparison of light-scattering calculation performed with the DDA 
approximation and with exact Mie theory for a homogeneous sphere is a traditional way to 
determine the accuracy of the DDA. Using the parallel DDA code produced by Hoekstra et 
al.,15 we compared DDA simulations with Mie calculations for a sphere with a diameter of 
4.56 μm and a relative refractive index of 1.10; the wavelength was 0.6328 μm. The relative 
errors in DDA simulations as a function of the scattering angle are presented in Fig. 2 for 
three different discretization sizes: λ/5, λ/10, and λ/20. The relative error for discretization 
size of λ/5 is ∼10% for scattering angles smaller then 50°. This degree of precision is good 
enough for comparison with experimental curves for which the scattered intensity was 
measured in a range of scattering angles from 10° to 50°. In the sequel of the paper, the size 
of the dipoles was varied in the range λ/11–λ/8 for different sizes of RBCs because of 
requirements for lattice regularity in the current implementation of the parallel FFT 
algorithm.15 
Experimental Equipment and Procedures 
The experimental part of this study was carried out with a SFC that permits measurement of 
the angular dependency of light-scattering intensity (indicatrix) in a region ranging from 5° to 
100°. The design and basic principles of the SFC were described in detail elsewhere.16 The 
laser beam of the SFC is directed coaxially with the hydrodynamically focused flow which 
0 30 60 90 120 150 180
Scattering angle θ, deg
 a = λ/5
 a = λ/10
 a = λ/20
Fig. 2. Relative errors of DDA simulations for several discretization sizes. The DDA simulations were 
compared with the exact Mie solution for a sphere with diameter d = 4.56 μm and relative refractive 
index m = 1.10. 
carries analyzed cells. Most disklike-shaped RBCs attain the testing zone of the SFC in rim-
on orientation relative to the direction of the incident laser beam. The current set-up of the 
SFC provides a measurement of the following combination of Mueller matrix elements23: 
ϕθϕθϕθ
1411s )],(),([d)( SSI , (3)
where Is(θ ) is the output signal of the SFC and θ and ϕ are the polar and azimuthal angles, 
respectively. After integration over azimuthal angle ϕ the second term in Eq. (3) vanishes, 
because of the axisymmetry of RBCs (see Eq. (A5) of Appendix A for details; we also 
obtained the same result numerically for all simulations). Therefore, the SFC output signal 
will be proportional to S11 integrated over the azimuthal angle. To compare the experimental 
and theoretical light scattering from RBCs we used the DDA for calculation of the 
indicatrices. Our SFC setup allowed reliable measurements in the angular range 10°–50° 
(because of the operational range of the analog-digital converter).  
A sample containing approximately 106 cells/ml was prepared from fresh blood with 
buffered saline used for dilution. We used the SFC to continuously measure 3000 indicatrices 
of RBCs. Each of them was compared with each of the calculated theoretical indicatrices (for 
details see next section) by a χ 2 test with a weighting function: 
10 15 20 25 30 35 40 45 50
Scattering angle θ, deg
 β = 90°
 β = 70°
 β = 50°
 β = 30°
 β = 0°
Fig. 3. (a) Initial and (b) modified indicatrices calculated by DDA for five orientations of the single 
biconcave disk relative to the direction of incident light beam. 
theorexp
)()]()([ θθθ
where Iexp(θi) is the experimental indicatrix measured at N angular values θi (θ1 = 10°, 
θN = 50°), Itheor(θi) is the theoretical indicatrix calculated at the same angles, w(θi) is a 
weighting function, defined as: 
sin)( 2 iiw
πθ , (5)
which is intended to suppress experimental errors (a detailed discussion is presented in 
Ref. 24). An experimentally measured RBC is said to have characteristics of the closest (by 
χ 2) theoretical indicatrix, if their χ 2 distance is less than a threshold. The threshold was set 
empirically to 40. 
The optical model of the RBC was used in calculations with the profile described by 
Eq. (2). The RBC diameter was varied from 6 to 9 µm. We assume that the imaginary part of 
the refractive index is negligible at the wavelength λ of 0.6328 µm. The refractive index of 
10 15 20 25 30 35 40 45 50
V = 100 μm3
β = 90°
 d = 8.28 μm, ε = 0.294
 d = 7.60 μm, ε = 0.380
 d = 6.75 μm, ε = 0.542
Scattering angle θ, deg  
Fig. 4. Modified indicatrices of biconcave disks with different diameters and fixed volume. 
10 15 20 25 30 35 40 45 50
 V = 110 μm3, ε = 0.595
 V = 100 μm3, ε = 0.541
 V = 92 μm3,   ε = 0.498 
d = 6.75 μm
β = 90°
Scattering angle θ, deg  
Fig. 5. Modified indicatrices of biconcave disks with different volumes and fixed diameter. 
the surrounding medium (saline) was 1.333. The RBC refractive index was fixed at 1.40 that 
falls into the region of typical variation of a RBC refractive index (1.39 – 1.42)25 and results 
in a relative refractive index of 1.05. 
Results and discussions 
Using the DDA, we studied the effect of the biconcave disk orientation on the 
indicatrices. Angular orientation β is the angle between the axis of the biconcave disk 
symmetry and the direction of the incident beam (Fig. 1). We used the profile defined by 
Eq. (2) with the following typical RBC characteristics17: diameter of 8.28 µm, aspect ratio of 
0.323 (resulting in a volume of 110 μm3), and relative refractive index of 1.05. The 
indicatrices for five orientations are shown in Fig. 3(a). To provide an effective comparison of 
experimental and theoretical light-scattering data we modified the indicatrices by 
multiplication by weighting function w(θi) [Eq. (5)]. The multiplication corresponds to the 
standard Hanning window procedure that strongly reduces the effects of the discontinuities at 
the beginning and the end of the sampling period of the SFC.24 Moreover, this function 
resembles the SFC instrument function16 and improves visual inspection of indicatrices, as the 
logarithmic scale can be replaced by a linear one. The modified indicatrices for the five 
orientations are shown in Fig. 3(b). The orientation of the biconcave disk modifies the 
indicatrix substantially. For instance, the mean scattering intensity, in the angular range from 
15° to 50°, has a reduced intensity for symmetric orientation (β = 0°), whereas the oscillating 
structure vanishes for increasing orientation angle. Therefore the mean light-scattering 
intensity and indicatrix structure are good indicators of a mature RBC orientation in scanning 
flow cytometry. 
Practically, it is interesting to study the sensitivity of indicatrix structure to variations in 
the volume and diameter of the biconcave disk when the orientation corresponds to 
orthogonal directions of the biconcave disk symmetry axis and incident beam (i.e. for 
β = 90°). The last requirement is caused by the orientating effect of the hydrofocussing head 
of the SFC in which nonspherical particles are oriented with a long axis along the flow lines.9 
We calculated the indicatrices of RBC with a typical volume of 100 μm3 while we varied the 
biconcave disk diameter. The results of the DDA simulation are shown in Fig. 4. Variation of 
the diameter does not change the intensity substantially, whereas the visibility of the 
oscillating structure, which characterizes the relative difference between maximum and 
minimum values,16 is reduced with increasing biconcave disk diameter. 
Additionally, we varied the thickness of the biconcave disk, which resulted in variation 
of volume with a constant diameter. The RBC volume is the most important hematological 
index measured with modern automatic hematology analyzers.3 Flow cytometry25 and 
Coulter’s cell3 are the instrumental solutions utilized for measurement of RBC volume 
distribution. The modified indicatrices of biconcave discs calculated for three volumes with a 
typical diameter of 6.75 µm are shown in Fig. 5. The light-scattering intensity does not 
depend on variation of the biconcave disk volume or thickness. There is the same tendency in 
the indicatrix structure as for Fig. 4: the visibility of the oscillating structure increases for 
increasing aspect ratio (of thickness and diameter). 
Our simulation of light scattering by mature RBCs allows us to conclude that the 
orthogonal orientation of a biconcave disk relative to the direction of incident beam does not 
provide sufficient sensitivity of the indicatrix to the RBC characteristics. Coincidence of 
biconcave disk symmetry axis and direction of incident beam (i.e. β = 0°) gives an advantage 
in the solution of the inverse light-scattering problem because in that case the oscillating 
structure of the indicatrix is much more sensitive to variation of RBC characteristics. 
Unfortunately, as it is shown below, such orientation is improbable in SFC. 
Table 1. The parameters of RBCs for preliminary calculations.a 
 V, μm3 
d, μm 86 92 100 105 110 
6.08 0.638 – – – – 
6.33 0.565 0.605 – – – 
6.51 – 0.556 0.604 – 0.665 
6.75 0.466 0.499 0.542 0.569 0.596 
6.84 – – 0.521 – 0.573 
7.01 – – 0.484 – 0.532 
7.60 0.327 0.349 0.380 0.399 0.418 
8.28 – – 0.294 0.418 0.323 
a Values are aspect ratios for selected diameter and volume. Dashes indicate that indicatrices were not calculated. 
10 15 20 25 30 35 40 45
15 20 25 30 35 40 45 50
V = 100 μm3 
d = 6.84 μm
β = 80°
 2 = 30
 theory
 experiment
V = 100 μm3 
d = 8.28 μm
β = 90°
 2 = 30
V = 110 μm3 
d = 6.75 μm
β = 80°
 2 = 36
V = 92 μm3 
d = 6.75 μm
β = 90°
 2 = 33
V = 92 μm3 
d = 6.33 μm
β = 90°
 2 = 31
Scattering angle θ, deg
Scattering angle θ, deg
V = 110 μm3 
d = 7.6 μm
β = 90°
 2=28
Fig. 6. Experimental and theoretical modified indicatrices of individual mature RBCs. Values of χ 2 
differences are shown. 
Practically, the DDA cannot be used to fit experimental indicatrices because the DDA 
calculations require approximately 10 h for a fixed biconcave disk orientation (the 
computations where carried out on 8 nodes of a Beowulf computer with nodes running at 750 
MHz). To solve this problem we made calculations of indicatrices for several biconcave disks 
with different diameters and volumes, presented in Table 1, to fill a small database to be used 
in the inverse problem. Values in the cells as listed in the table are aspect ratios for selected 
diameter and volume, whereas dashes indicate that indicatrices of RBC with these 
characteristics were not calculated. For each set of diameter and volume, four indicatrices 
were calculated for orientation angles β (= 60°, 70°, 80°, 90°). Calculated theoretical 
indicatrices were used for determination of characteristics of experimentally measured RBCs 
by the χ 2 test as explained in the previous section. 
A few representative results of our comparison by the χ 2 test are presented in Fig. 6. 
One can see that theoretical indicatrices fit experimental curves well. The mean scattering 
intensity (intensity integrated over the angular interval) of the measured indicatrices is in a 
good agreement with the mean intensity of theoretical indicatrices calculated for orientation 
angles β of 70° and 90° [Fig. 3(b)], and exceeds the mean intensity for smaller orientation 
angles. This fact allows us to conclude that orientation of RBCs in the capillary of a SFC is 
close to orthogonal (β = 90°), which agrees with our previous results.9 This result justifies our 
choice of orientation angles for preliminary calculations of RBCs’ indicatrices. We used all 
the experimental indicatrices that passed the χ 2 test to plot a distribution of mature RBCs over 
the orientation angle, as presented in Fig. 7. This distribution proves once again that 
orthogonal orientation is much preferable for RBCs in the capillary of a SFC and gives an 
estimation of deviation from orthogonal orientation. 
Contrary to the DDA, the T-matrix method allows one to reduce a time of light-
scattering calculation substantially.2 However the biconcave disk shape complicates the light-
scattering simulation in the T-matrix method. This method can be effectively applied to a 
particle with disk-sphere or oblate spheroid shapes that can be used as models of RBCs. 
Therefore, we have compared the indicatrices of particles shaped according to Eq. (2) and of 
particles with the following shape geometries: disk-sphere and oblate spheroid. The 
comparison was performed for the diameter-volume-equal particles. The parameters of the 
biconcave disk were as follows: d = 7.60 μm, ε = 0.380 (V = 100 μm3), and m = 1.05. The 
light scattering for two orientations of the particles relative to the direction of the incident 
beam, rim-on and face-on incidence, was computed. The indicatrices of the biconcave disk 
and disk-sphere are shown in Fig. 8. This figure allows us to conclude that RBC can be 
modeled by a disk-sphere only for simulation of light scattering in the angular interval 
ranging from 10° to 15°. 
An oblate spheroid is the most popular model in simulation of light scattering of 
individual RBCs. This model was used in simulation of light scattering by the T-matrix 
method.7 We have calculated the indicatrices of diameter-volume-equal spheroids in rim-on 
and face-on orientations of the oblate spheroid and RBC. The results of these calculations, 
shown in Fig. 9, allow us to conclude that RBC can be modeled by an oblate spheroid over a 
wide angular interval only for rim-on incidence. This conclusion is in agreement with the 
boundary-element methodology12 and discrete sources method26 applied to study of light 
scattering of red blood cell. However the validity of such substitution for calculation of RBC 
indicatrices should be further studied for different RBC sizes with respect to the certain 
problem, where these indicatrices are to be used. 
60 70 80 90
Orientation angle β, deg  
Fig. 7. Distribution of mature RBCs over an orientation angle obtained using χ 2 test. 
10 15 20 25 30 35 40 45 50
 disk
 Scattering angle θ, deg
Fig. 8. The modified indicatrices of the biconcave disk and the diameter-volume-equivalent disk-
sphere: (a) rim-on and (b) face-on incidence. 
Conclusion 
Our simulation of light scattering of a mature RBC has shown that the indicatrix is 
sensitive to the RBC shape and the DDA (or some other method for which no simplifications 
of the RBC shape are assumed) should be used in a study of formation of the indicatrix for 
RBCs with various characteristics. However, light scattering of RBCs can be simulated with 
the T-matrix method for rim-on incidence by use of the oblate spheroid model. Fortunately, 
the hydrodynamic system of the SFC delivers mature RBCs into the testing zone in this 
specific orientation. This performance of the SFC gives a chance of solving the inverse light-
scattering problem for mature RBCs, e.g. by parameterization or use of a neural network 
because T-matrix simulation requires substantially less computing time than the DDA 
algorithm. However, the precision of such algorithms when developed should be tested using 
realistic indicatrices obtained e.g. by DDA simulations. 
Therefore, need for improvement of current DDA code arises. Such an improvement can 
not only provide enough testing indicatrices for the problem described above but also make 
feasible such tasks as solving an inverse light-scattering problem for any orientation of RBCs 
relative to the incident beam. 
10 15 20 25 30 35 40 45 50
Scattering angle θ, deg
 oblate sheroid
Fig. 9. The modified indicatrices of the biconcave disk and the diameter-volume-equivalent oblate 
spheroid: (a) rim-on and (b) face-on incidence. 
This research was supported by Russian Foundation for Basic Research through the 
grant 02-02-08120-inno and 03-04-48852-a, by Siberian Branch of the Russian Academy of 
Sciences through the grant 115-2003-03-06, and by the NATO Science for Peace program 
through grant SfP 977976. 
Appendix A 
Let incident radiation propagate along the z-axis, and assume that a particle has a 
symmetry plane containing this axis. We will investigate then the properties of the Mueller 
matrix integrated over complete azimuthal angle ϕ at a fixed polar angle θ. 
Not restricting generality (as we consider the integral over the complete azimuthal 
angle), we can consider that the x-axis resides in the symmetry plane of the particle. We 
divide the integral into two parts and then group them: 
∫∫∫∫ ),−+)=)+)=),
ϕθϕθϕϕθϕϕθϕϕθϕ
](,([d,(d,(d(d SSSSS . (A1)
Let us consider two scattering problems for angles (θ,ϕ) and (θ,−ϕ). We rotate the laboratory 
reference frame about the z-axis by angles ϕ and −ϕ for the first and the second problems 
respectively (it is the same as rotating everything else – the particle, the incident and 
scattering direction, and the electric field vectors – in the backward direction). These 
operations do not change the scattering matrices, therefore 
),=), − 0(( θϕθ ϕSS  and ),=),− 0(( θϕθ ϕSS , (A2)
where Sϕ, S−ϕ are scattering matrices for particles (pϕ and p−ϕ) rotated about the z-axis by 
angle ϕ and −ϕ respectively, relative to its initial orientation (p0). Let us denote operator of 
plane zx reflection as Pzx and operator of rotation around the axis z by angle ϕ as Rϕ. Then 
( ) ( ) ϕϕϕϕϕϕϕ ppRpPRpRPRpP zxzxzx ==== −− 00ooo , (A3)
where the first equation is an identity for any operand (which can be easily verified since 
operators involved do not affect z coordinates) and the third one exploits the assumption that 
particle has a zx symmetry plane in its initial orientation. Equations (A2) and (A3) imply that 
S(θ,ϕ) + S(θ,−ϕ) can be considered as a sum of scattering matrices for the same scattering 
geometry but for particles which are mirror images of each other with respect to the scattering 
plane. It was shown6 that such a sum gives a scattering matrix of the form: 
. (A4)
Then Eq. (A1) obviously implies that the Mueller matrix integrated over the complete 
azimuthal angle will be of the same form. Therefore we have proved that, if particle has a 
symmetry plane containing the z-axis (coincident with the propagation vector of the incident 
radiation), then 
0=∫ )(d ϕϕ ijS , for 2,1=i  and 4,3=j  or vice versa. (A5)
It is clear that, for a body of rotation, any plane containing symmetry axis is as well a 
symmetry plane. Therefore a plane containing both the symmetry axis and the axis z is 
symmetric for the body of rotation regardless of its orientation. Hence Eq. (A5) is 
automatically proved for axisymmetric particles. 
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