Monte Carlo for multiple scattering and non-spherical particles
P.Bruscaglioni* a, G.Zaccantia, S.Del Biancoa, G.Bazzinib, M.Gaib
a
Dept.of Physics, University of Florence, Italy and INFM
b
ASPER, Florence
ABSTRACT
This note describes a possible Monte Carlo procedure to deal with propagation of optical radiation in a medium
containing non-spherical particles, without the limit of isotropic orientation of their axes.
It is based on the choice of two polarization states which does not change during the propagation in the turbid
medium. An approximate procedure is considered for permitting the use of only two polarization states.
Key-words : Multiple scattering, Monte Carlo, non-spherical particles.
1. Introduction
A typical feature of a Monte Carlo code for dealing with beam propagation in turbid media (e.g. for lidar problems) is
the consideration of light trajectories forming broken lines with corners at the scattering points and linear tracts with
lengths determined by statistical laws regarding the medium scattering-absorption properties.
When, properly, the polarization characteristics of radiation is taken into account, the medium properties are
represented by phase matrices M, in addition to linear extinction coefficients  and albedo.
By assuming a homogeneous medium  is determined by the average extinction cross section Cs of the turbidity
particles and by their numerical density N.
For spherical particles and non-spherical particles with completely random orientation Cs is independent of the
direction of arrival of incident radiation: Cs can be maintained the same along the trajectories. (As is well known the
procedure can be easily extended to inhomogeneous media).
However, the case of non-spherical particles with preferred orientation of their axes (such as the case of crystal
particles in high altitude atmosphere) is of interest.
It is necessary to take into account that along a trajectory, even during the rectilinear tracts, the particles cross
sections vary, since polarization varies, unless particular types of radiation polarization are present.
It is well known that in these cases two polarization states are such as to keep themselves unchanged. These are
generally names as field eigenvectors. The relevant particle average extinction cross sections are named as
eigenvalues.
The mathematics to search for these states and cross sections can be found in basic texts, and are based on
considering Stokes vectors, with the relevant phase eigen matrices and Stokes eigenvectors.
These note has the aim of indicating a simple (though approximate) way of dealing with this problem. The procedure
is shown in the following sections, where also a comparison of the formal results with those of basic texts will be
presented.
2. The scattering matrix for fields and the particles cross section
The scattering properties of a particle are represented by the F matrix.
We will make reference to incident (i) and scattered (s) field, with components 1 and 2 respectively normal and
perpendicular to the scattering plane. One can write:
1)
*
 E s  exp( ikr)  E i 
 1
F  1 
 s 
r
Ei 
 E2 
 2
piero.bruscaglioni@unifi.it; phone +39 055 4572 282
 f11

F 
f
 21
f12 


f 22 
From this formalism one can use the Optical Theorem to derive the particle cross section Cs.
It reads
2)
Cs

=
4


Im f i, i  u *

where f (i,i) is the vector scattering amplitude and u is the unit vector representing the polarization state of incident
radiation. In Eq.2 f has to be evaluated in the forward direction, that is in the propagation direction of the incident
radiation represented by the unit vector i.
Let us assume the incident radiation has the components E1 and E2. (complex numbers).
Its incident field polarization state is therefore represented as:
u  a x x  ay y
3)
with:
E1
ax 
2
E1
E2
ay 
 E2
2
E1
2
 E2
2
where x, y are the unit vectors along the axes 1 and 2 respectively.
Then, by making reference to Eqs 1 and 2, one has:
Cs = 
4)
4
2

Im  f11 a x 2  f12 a x a y  f 21a x a y   f 22 a y 



Where f11, f22, f12, f21 are evaluated in the forward direction, and ax is taken as reference for phase.
For a comparison with another formalism we make reference to the procedure shown in the book by Tsang, Kong
and Shin 1 (Sect. 4.3). When different particles or different orientations are present, they consider the average value
of the matrix elements. The linear extinction coefficient  for the medium is obtained by averaging the contributions
to the matrix elements of the different particles (index l) whose numerical density is Nl
The matrix F of Eq.1 is modified if the scattering properties per unit volume of the medium are .needed : that is the
elements fmn changes :
5)
fmn
changes to gmn
:
gmn =  f mnlNl .
l
Then the linear extinction coefficient has the form of Eq.4 with the substitution of Eq.5, and in the place of F one
has:
6)
g
F'   11
 g 21
g12 

g 22 
One can compare Eq.4 with the substitution 5, with what is shown by Tsang, Kong and Shin 1 (Sect. 4.3, Eqs. 7,8 of
Ref. [1]), where the differential equations for the first two elements of the modified Stokes Vector are presented For
the comparison one has to take into account that in Ref.[1] the matrix corresponding to F' is written as:
Es 
 v




Es 
 h
 M vv
exp( ikr) 

r
M
 hv

M vh 


M hh 
Ei 
 v




Ei 
 h
and the axes v, h correspond to our axes y and -x respectively. (Note a change of the imaginary axis)
The expression for Cs obtained by considering these equations corresponds to that of our Eqs. 4,5.
3. Propagation without polarization change
As mentioned in the introduction an essential part of a Monte Carlo code refers to determining the length L of the
rectilinear tracts of the trajectories. For a homogeneous isotropic medium L is given according to the Lambert Beer
law of probability.
However, in general for a non-isotropic medium the polarization of the trajectory changes during propagation, thus
preventing this simple determination of the rectilinear tract to be followed.
To avoid this difficulty, one has to consider particular types of polarization, which are not altered by scattering in the
forward direction. Given as E1 and E2 the field components of the incident radiation, in order to obtain these
polarization states, one has to fulfil the conditions, relating the incident field to that scattered in the forward direction
(gmn in the forward direction)
7)
E1s

E 2s



exp( ikr)
g11E1i  g12 E 2i
r
exp( ikr)
g 21E1i  g 22 E 2i
r

.

with the condition
8)
E 2s
E1s
E 2i
=
E1i
Thus the scattered field in the forward direction add coherently to the incident field and the overall polarization is
maintained.
The second order equation gives the following two conditions for the incident field.
9)
E2i
E1i

1 
 g 22  g11 
2 g12 
g
22  g11
2  4g12 g21
.
Apart from an inessential factor the two searched for polarization states are then :
E1

x  2y
g 21
g 22  g11
R
E2

y  2x
g12
g 22  g11
R
10)
with:
R
g22  g11 2  4g12g21
One obtains the two extinction coefficients for the two opportune polarization states from Eq.1, modified by
inserting gmn in the place of fmn. The two polarization states are now defined by the unit complex vectors :
a'

11)
a" 
with
b1
=
E1
2 g 21
y
b1 g 22  g11  R
2 g12
y
x

b 2 g 22  g11  R
b2
x

b1
and
b2 =
E2
4. Use in a Monte Carlo code
Given an incident field, represented by a Stokes Vector, one obtains the Stokes vector in a certain scattering direction
by using the relevant phase matrix.
(In general it is a 4X4 matrix).
An analysis of the different intensity of radiation scattered in the different directions provides the probability of
finding the scattered radiation in each direction.
One has to determine the length of the rectilinear tract before a subsequent scattering point occurs.
This can be done if previously one divides the field emerging from the scattering point into constant polarization
states.
When the decomposition of the scattered field in the opportune polarization states is obtained, the Monte Carlo code
can proceed by splitting the trajectorie into different trajectories, each carrying a weight according the relative
intensity.
1




 2 g 21


It can be seen that the two polarization vector states of Eq.10
:
and


 g 22  g11  R 
 2 g12




 g11  g 22  R 


1


(where the first line refers to the x axis (-h of Ref.1) and the second one to the y axis (v)) correspond to those of
Ref.[1] Sect. 4.3 (note the different sign of the unit vectors along the directions x and h, and the corresponding
change of sign for g12 and g21 ). In Ref. [1] a warning is given regarding the sign of the root term R. It must be taken
so that, when the g21 and g12 are small, the polarization states tend to linear polarization.
However, the outlined procedure must be commented.
be scattered field Es.
Exx  Eyy
Let us try to decompose it by means of the two vectors E1 and E2 of Eq. 10, and put:
Let
Es = A E1  B E 2 .
12)
One has for the components of the scattered field:
E sx

A E1x  B E 2 x
E sy
 A E1 y  B E 2 y
and the constants A and B are obtained.
The polarization is maintained by the decomposition. However one can see that in general the power connected to
the original field is not equal to the sum of the powers connected to each of the two component fields. An
interference term is missed by this sum.
One can accept this limitation only if particular types of situations occur.
This is the case of Optically Active particles, such as those described by 2 Bohren and Huffman [Ref.2, Sect. 8.3].
For these particles in Ref.[2] it is shown that the off-diagonal elements of the field matrix (1) are equal and of
opposite sign, while the diagonal ones are equal.. One then sees that the vectors a' and a" of Eq.11 represent the
elementary left and right circular polarization states. They are orthogonal, and the decomposition of Eq. 12 is
therefore sufficient, also from the point of view of power. This particular case does not offer difficulty. The splitting
of trajectories is sufficient for the Monte Carlo procedure. (One can also decide to follow one of the two trajectories,
2
2
with the decision statistically made according to the two different intensities, A and B )
For the general case, if particular symmetries simplifying the problem are absent, we can consider the following
approximation procedure: One can take into account that this kind of Monte Carlo procedure considers independent
propagation of the field components:
Then let us add a factor  to the coefficients A and B in Eq. 12. The summed power of the component fields can thus
be put equal to that of the original field.
The summed fields , which are thus obtained, represent polarization and power of the total field.
During propagation the different phase constants of the two components decreases the average interference effects.
This effect is, however, of less importance.
5. Conclusions
In this paper we have outlined a possible Monte Carlo procedure for dealing with propagation of optical radiation in a
medium containing non spherical particles. For these particles one needs to use the elements of the field matrices (1),
(6), which must be previously calculated. The propagating field must be divided into components maintaining their
polarization states during propagation. Only in particular cases a couple of components are sufficient. However, for a
general case we suggest an approximate procedure permitting one to split the scattered field in only two components.
Acknowledgements
We acknowledge the support by EU, through INTAS contract 01-0239, and by the Istituto Nazionale per la Fisica
della Materia (INFM).
References
1.
2.
L.Tsang, J.A. Kong, R.T.Shin, Theory of Microwave Remote Sensing, Wiley, New York, 1985.
C.F.Bohren, D.R.Huffman, Absorption and Scattering of Light by Small Particles, Wiley, New York, 1983.
A comment.
As explained in the paper, In Sections 2 and 3 we have followed a procedure which is parallel to that of Reference 1,
in order to have possible comparison.
There could be a concern about the implicit insertion of a factor in eqs.7 (N: total number of particles per unit
volume). However in the subsequent equations the ratios of the relevant quantities are considered.