_________ __________ __________ ___
MATLAB Functions for Mie
Scattering and Absorption
Version 2
Christian Mätzler
_________ __________ __________ ___
Research Report No. 2002-11
August 2002
Institut für Angewandte Physik Mikrowellenabteilung
__________________________________________________________
Sidlerstrasse 5 Tel. : +41 31 631 89 11
3012 Bern Fax. : +41 31 631 37 65
Schweiz E-mail : matzler@iap.unibe.ch
1
MATLAB Functions for Mie Scattering and Absorption
Version 2
Christian Mätzler, Institute of Applied Physics, University of Bern, August 2002
List of Contents
Abstract..............................................................................................................1
1 Introduction.....................................................................................................2
2 Overview of changes with respect to Version 1..................................................2
3 Functions and computational procedures.........................................................3
3.1 Mie Coefficients for homogeneous spheres.................................................................... 3
3.2 Mie Coefficients for coated spheres ............................................................................... 5
3.3 Computation and plot of Mie Efficiencies...................................................................... 6
3.4 The scattered far field ..................................................................................................... 7
3.5 The internal field............................................................................................................. 8
3.6 Computation of Q
abs
, based on the internal fields ........................................................... 8
3.7 Dielectric functions for water and ice............................................................................. 10
4 Examples and Tests .......................................................................................10
4.1 The situation of x=1, m=1000+1000i ............................................................................. 10
4.2 Magnetic sphere with x=2, eps1=2+i, mu1=0.8+0.1i ..................................................... 11
4.3 Water-coated ice sphere.................................................................................................. 13
4.4 Ice-coated water sphere .................................................................................................. 13
4.5 Water bubble................................................................................................................... 14
4.6 Freezing rain ................................................................................................................... 14
4.7 Melting graupel............................................................................................................... 15
5 Conclusions...................................................................................................16
References ........................................................................................................17
Appendix: Behaviour of Riccati-Bessel Functions...............................................18
Abstract
A set of MATLAB Functions for Mie calculations (Mätzler, 2002a) and for applica-
tions to microwave radiation in rain (Mätzler, 2002b) has been improved and
expanded by including magnetic and metal-like media and coated spheres. The
appendix includes a discussion of the basic behaviour or the Riccati-Bessel and
related Functions needed in the computations of Mie Coefficients.
The applications of the Mie Functions are directed toward the study of radiative
properties of precipitation. Functions have been developed to compute propagation
parameters for freezing rain and melting graupel, assuming Marshall-Palmer drop-
size distribution, including functions to compute the complex dielectric permittivi-
ties of ice and water. Other applications can be envisaged if the dielectric or refrac-
tive properties of the particles and their size distributions are known.
2
1 Introduction
This report is an extension of Mie-scattering and -absorption programs (Mätzler,
2002a) and applications to propagation, scattering and emission of microwave
radiation in precipitation (Mätzler, 2002b) written in the numeric computation and
visualisation software, MATLAB (Math Works, 1992). Mie Theory is based on the
formulation of Bohren and Huffman (1983), in short BH. There and here the
assumed time variation of the fields is exp(-i!t), leading to positive imaginary parts
of the refractive index for absorbing media. For corresponding equations, equation
numbers refer to those in BH or to page numbers of BH. In addition, for absorption
by the internal electrical and magnetic fields, see Section 3.6 of the present report.
For descriptions of computational problems in the Mie calculations, see the notes
on p. 126-129, in Appendices A and B of BH and in the Appendix of this report
which includes a description of the relevant functions (Riccati-Bessel functions and
combinations thereof) and of their numerical behaviour.
Microwave interaction with precipitation mainly refers to Sauvageot (1992) and to
Mätzler (2002b), including references therein.
Descriptions of the functions are given in Section 3, followed by some examples in
Section 4.
2 Overview of changes with respect to Version 1
First of all, with respect to Mätzler (2002a), the parameter range was extended to
larger size parameters in case of large imaginary refractive index (metal-like
spheres) by replacing the function Mie_abcd for the Mie Coefficients a
n
, d
n
, c
n
, d
n
by
two separate functions with improved algorithms, Mie_ab to compute a
n
, b
n
and
Mie_cd for c
n
, d
n
.
Second, the new version allows for magnetic materials, by using the alternative
functions Mie2_ab for a
n
, b
n
, and Mie2_cd for c
n
, d
n
. Functions starting with the
name Mie2... refer to magnetic spheres with media properties given by mu1= "
1
/"
and eps1=#
1
/#, where # and " are the permittivity and permeability of the ambient
medium and #
1
and "
1
are the parameters of the sphere. For nonmagnetic spheres
i.e. for "
1
=", there is a single medium-parameter (as in Version 1), the refractive
index m relative to the ambient medium. This is the case for MATLAB functions
starting with Mie... (i.e. without 2). The old function, Mie_abcd, of Version 1 is still
included to enable numerical comparisons.
Third, the function Mie(m,x) returns the Mie Efficiencies (Qext, Qsca, Qabs, Qb,
<costeta>), but no more the input parameters. The same is true for the respective
functions, Mie2 and Miecoated. In this way the output of these three functions gets
the same format.
Forth, the new version also includes functions for coated spheres with inner radius
a (size parameter x=ka) and outer radius b (size parameter y=kb). The kernel has a
refractive index m
1
, and the coating has m
2
, both relative to the ambient medium.
Non-magnetic media are assumed as in BH. The name of such functions start with
‘Miecoated’.
Finally, freezing rain and melting graupel, both with Marshall-Palmer drop-size
distribution, are also included, see the functions Miecoated_raini, where i is a
number. These functions expand the microwave applications of rain (Mätzler,
2002b) to other forms of precipitation. Without difficulty, other size distributions
can be introduced (e.g. Sauvageot, 1992, Section 2.2).
涂层
3
3 Functions and computational procedures
3.1 Mie Coefficients for homogeneous spheres
MATLAB Functions:
Mie_ab(m, x) produces a
n
and b
n
, for n=1 to n
max
for non-magnetic spheres
Mie_cd(m, x) produces c
n
and d
n
, for n=1 to n
max
for non-magnetic spheres
Mie2_ab(eps1, mu1, x) produces a
n
and b
n
, for n=1 to n
max
for magnetic spheres
Mie2_cd(eps1, mu1, x) produces c
n
and d
n
, for n=1 to n
max
for magnetic spheres
Mie_abcd(m, x) produces a
n
, b
n
, c
n
and d
n
, for n=1 to n
max
for non-magnetic spheres (from Version 1)
The key parameters for Mie calculations are the Mie Coefficients a
n
and b
n
to com-
pute the amplitudes of the scattered field, and c
n
and d
n
for the internal field,
respectively. The coefficients are given in BH on p.100. First, the coefficients of the
scattered electrical field are:
)]'()[()]'()[(
)]'()[()]'()[(
)]'()[()]'()[(
)]'()[()]'()[(
)1()1(
1
1
)1(
1
)1(2
1
2
mxmxjxhxxhmxj
mxmxjxjxxjmxj
b
mxmxjxhxxhmxjm
mxmxjxjxxjmxjm
a
nnnn
nnnn
n
nnnn
nnnn
n
""
""
""
""
$
$
%
$
$
%
(4.53)
where prime means derivative with respect to the argument; similar expressions
exist for the coefficients c
n
and d
n
of the internal field (see below). The Index n runs
from 1 to &, but the infinite series occurring in Mie formulas can be truncated at a
maximum, n
max
; for this number Bohren and Huffman (1983) proposed
24
3/1
max
''% xxn (p.477)
and this value is used here as well. The size parameter is given by x=ka, a is the
radius of the sphere and k =2(/) is the wave number, ) the wavelength in the
ambient medium, m=(#
1
"
1
)
1/2
/(#")
1/2
is the refractive index with respect to the ambi-
ent medium, #
1
and "
1
are the permittivity and permeability of the sphere and # and
"
are the permittivity and permeability of the ambient medium. The functions j
n
(z)
and
)(
)1(
zh
n
=j
n
(z)+iy
n
(z) are spherical Bessel functions of order n and of the given
arguments, z= x or mx, respectively. The derivatives follow from the spherical Bessel
functions themselves, namely
)()()]'([);()()]'([
)1()1(
1
)1(
1
znhzzhzzhznjzzjzzj
nnnnnn
$%$%
$$
(p.127)
Relationships exist between Bessel and spherical Bessel functions:
)(
2
)(
5.0
zJ
z
zj
nn '
%
(
(4.9)
)(
2
)(
5.0
zY
z
zy
nn '
%
(
(4.10)
Here, J
*
and Y
*
are Bessel functions of the first and second kind; for n=0 and 1 the
spherical Bessel functions are simply given (BH, p. 87) by
zzzzzyzzzy
zzzzzjzzzj
/sin/cos)(;/cos)(
/cos/sin)(;/sin)(
2
10
2
10
$$%$%
$%%
and the recurrence formula can be used to obtain higher orders
