Journal of Quantitative Spectroscopy & Radiative Transfer 112 (2011) 1492–1508
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Journal of Quantitative Spectroscopy &
Radiative Transfer
journal homepage: www.elsevier.com/locate/jqsrt
Scattering and absorption of light by ice particles: Solution by a new
physical-geometric optics hybrid method
Lei Bi a,n, Ping Yang a,b, George W. Kattawar a, Yongxiang Hu c, Bryan A. Baum d
a
Department of Physics and Astronomy, Texas A&M University, College Station, TX 77843, USA
Department of Atmospheric Sciences, Texas A&M University, College Station, TX 77843, USA
Climate Science Branch, NASA Langley Research Center, Hampton, VA 23681, USA
d
Space Science and Engineering Center, University of Wisconsin-Madison, Madison, WI 53706, USA
b
c
a r t i c l e in f o
abstract
Article history:
Received 9 December 2010
Received in revised form
19 February 2011
Accepted 21 February 2011
Available online 4 March 2011
A new physical-geometric optics hybrid (PGOH) method is developed to compute the
scattering and absorption properties of ice particles. This method is suitable for
studying the optical properties of ice particles with arbitrary orientations, complex
refractive indices (i.e., particles with significant absorption), and size parameters
(proportional to the ratio of particle size to incident wavelength) larger than ! 20,
and includes consideration of the edge effects necessary for accurate determination of
the extinction and absorption efficiencies. Light beams with polygon-shaped cross
sections propagate within a particle and are traced by using a beam-splitting technique.
The electric field associated with a beam is calculated using a beam-tracing process in
which the amplitude and phase variations over the wavefront of the localized wave
associated with the beam are considered analytically. The geometric-optics near field
for each ray is obtained, and the single-scattering properties of particles are calculated
from electromagnetic integral equations. The present method does not assume
additional physical simplifications and approximations, except for geometric optics
principles, and may be regarded as a ‘‘benchmark’’ within the framework of the
geometric optics approach. The computational time is on the order of seconds for a
single-orientation simulation and is essentially independent of the size parameter. The
single-scattering properties of oriented hexagonal ice particles (ice plates and hexagons) are presented. The numerical results are compared with those computed from the
discrete-dipole-approximation (DDA) method.
& 2011 Elsevier Ltd. All rights reserved.
Keywords:
Light scattering
Geometric optics
Physical optics
Hexagonal ice crystal
1. Introduction
The scattering and absorption of light by nonspherical ice
particles has been of great interest in the atmospheric
radiation research community. Over the past several decades,
steady improvements have been made in the numerical
modeling of the scattering and absorption of light by nonspherical particles [1–5]. Rigorous techniques developed for
n
Corresponding author. Tel.: þ 1 979 862 1722.
E-mail address: [email protected] (L. Bi).
0022-4073/$ - see front matter & 2011 Elsevier Ltd. All rights reserved.
doi:10.1016/j.jqsrt.2011.02.015
various applications involving electromagnetic scattering by
nonspherical particles include the T-matrix method [6–8],
the finite-difference time-domain (FDTD) method [9–11],
and the discrete-dipole-approximation (DDA) method
[12–15]. Although these exact methods are efficient for the
calculation of the optical properties of ice particles with small
size parameters (k ¼2pr/l, where r is radius and l is
wavelength), they are impractical and computationally inefficient for large size parameters (k 420). In practice, the
most effective approach to deriving the solution to light
scattering by ice particles over a large size parameter range is
a combination of exact numerical techniques for small
L. Bi et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 112 (2011) 1492–1508
particles and other methods based on the geometric optics
approximation for moderate-to-large sized particles [16,17].
The conventional geometric optics method (CGOM) is
based on a straightforward combination of the ray-tracing
technique and Fraunhofer diffraction to calculate the
angular distribution of the far-field scattered energy and
polarization state of the radiation field [18–21]. The
advantages of the CGOM are its formalism simplicity
and numerical simulation efficiency. However, there are
some inherent limitations in the CGOM that limit its
applicability to randomly oriented particles with large
( 4100–200) size parameters. The CGOM is not well
suited for calculations of oriented ice particles. Another
limitation is that the ray spreading effect in the forward
scattering angles is not considered. Additionally, the edge
effects [22] are not properly considered in deriving
extinction and absorption efficiencies. Research to
improve the accuracy of the geometric optics method
and to incorporate semi-classical scattering effects in the
computation of the single-scattering properties of nonspherical particles has been reported in the literature
[23–25].
To circumvent the limitations in the CGOM, several
physical-geometric optics hybrid (PGOH) methods have
been suggested [26–30]. The PGOH methods calculate the
scattered or total near-field (i.e., the electromagnetic field
in or near the particle) based on geometric optics principles. The near-field is mapped to the far-field to obtain
the single-scattering properties of particles through electromagnetic integral equations. The PGOH methods have
been demonstrated to have a better approximation than
the CGOM and are applicable to moderate size parameters. However, the applicability of the PGOH methods
has not been well quantified because of the lack of a
rigorous and efficient numerical algorithm.
The present study employs a beam-splitting technique
[28,30] to enhance the PGOH ice particle modeling
capabilities. Instead of adopting a large number of
straight-line rays (or numerical photons) assumed by
most ray-tracing algorithms, the beams that propagate
within the particle have well defined polygonal-shaped
cross sections. In the beam-tracing process, several facets
may intercept a beam, and subsequently the wavefront of
the localized wave associated with the beam splits and
undergoes Fresnel reflections and refractions on different
facets. The advantage of tracing beams with well-defined
cross sections is that the total number of beams in the
numerical simulation is independent of the size parameter. A distinct advantage of this approach is that the
computational time is significantly reduced for particles
with large size parameters.
In the present PGOH formulation, the geometric-optics
near-field is obtained analytically based on Snell’s law
and the Fresnel formulae. We further develop a theoretical formalism to calculate the scattered far-field based
on an exact near-to-far field transformation. The extinction and absorption efficiency factors are derived based
on the optical theorem and an electromagnetic volume
integral equation. Note that the near-to-far field transformation is based on an exact electromagnetic integral
equation more theoretically rigorous than the Fraun-
1493
hofer diffraction approximation assumed in the previous
studies [28,30]. In addition, the variation of the amplitude
of the electric field [28] over a beam’s cross section is
considered. Therefore, the present PGOH algorithm is
applicable to absorptive particles.
Briefly stated, the PGOH method herein is suitable for
studying the optical properties of ice particles with arbitrary
orientations, complex refractive indices, and size parameters
larger than ! 20. To examine the accuracy of the present
method, we compare results with those computed from the
DDA method. From an application perspective, ice particle
orientation and ice habit geometry are the two issues of the
most concern to studies associated with ice cloud radiative
properties [31–33]. To demonstrate the benefits of the
present method for studying the two issues, we model the
effect of preferentially oriented ice plates on the backscattering properties and the effect of hexagonal ice particle
geometries on the angular distribution of scattered light.
This paper is organized into six sections. Section 2
outlines the theory of the present PGOH method including
the beam-splitting algorithm, the representation of geometric-optics near-field, and the PGOH formalism for the
single-scattering properties. Section 3 compares the
representative results simulated from the PGOH and
DDA methods, and discusses the accuracy of the PGOH
simulations and the deviations from their DDA counterparts. In Section 4, the present PGOH algorithm is applied
to model the effects of preferable orientations of hexagonal ice particles on their optical properties, and we focus
on the two quantities, backscattering efficiency and color
ratio. In Section 5, we discuss the approach to modeling
imperfect hexagonal ice particles and how the lack of ice
particle perfection affects the phase function. The concluding remarks are given in Section 6.
2. Theoretical basis
The three components of the present PGOH method
include: (1) tracing beams within a particle based on the
beam-splitting algorithm, (2) specifying the electric field
and polarization state of each beam, and (3) calculating
the optical properties of a particle (phase matrix, extinction efficiency, and absorption efficiency) based on electromagnetic integral equations. For algorithm simplicity,
the particle is assumed to be dielectric, isotropic, and
homogenous, and the geometry of a particle is assumed to
be convex and faceted. Concave particles are not considered in this study. Moreover, the time dependence of a
harmonic electromagnetic wave is assumed to be
exp( $iot), leading to a positive imaginary part of the
refractive index in the case of absorptive particles.
2.1. Beam-splitting algorithm
When a plane wave of light is incident on a faceted
particle, the portion of the wavefront of the incident
electromagnetic wave that is intercepted by the projected
geometric cross section of the particle subsequently splits
into several parts. As a wavefront (or localized wave)
impinges on a given facet, the subsequent electromagnetic interaction leads to outgoing reflected and inwardly
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L. Bi et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 112 (2011) 1492–1508
propagating refracted beams. Subsequently, the refracted
beams undergo multiple internal reflections within the
particle, leading to various higher-order outgoing
refracted beams. The first-order refracted beams and
higher order internally reflected beams may split during
their propagation within the particle. In this study a
beam-splitting algorithm is developed to describe how
the internal beams split and is aimed at specifying the
geometries of internal ray paths (or ray tubes). Because
the geometry of the scattering particle is assumed to be
convex, any externally reflected beams and higher-order
refracted beams cannot be blocked by the particle itself.
Therefore, the beam-splitting algorithm is unaffected by
beams propagating outside the particle.
An internal beam is specified by its propagating direction and initial cross section. Let the subscript index
p( ¼1,2,3y) indicate the pth order reflection/refraction
event. The direction of one internal beam leaving some
interface of the pth order reflection/refraction is specified
by e^ p , and the vertices of the beam cross section on the
interface of electromagnetic interaction is denoted as
*
r p,i ði ¼ 1,NV Þ, where NV is the number of vertices. When
*
p ¼1 (i.e., external reflection and refraction), r 1,i ði ¼ 1,NV Þ
are the straightforward coordinates of the vertices of
the corresponding facet where external reflection takes
place.
To describe the splitting of an internal beam (specified
*
by e^ p and r p,i ), the first step is to determine the intercepting particle facets. We assume NV to be the number of
straight-line rays (with no cross section) starting from the
positions of NV vertices and propagating in the direction
e^ p . The facets of the particle surface are convex shapes,
and if, for example, the NV number of rays strike Mv
number of different facets, the beam cross section is
divided into Mv parts with each part impinging on a
single facet. t^ i ði ¼ 1,Mv Þ are assigned to denote the normal
directions of the facets. Fig. 1(a) shows an example of a
first-order refracted beam split into three sub-beams
(rays starting from four vertices are incident on three
different facets) leading to three first-order ray tubes.
We must mathematically separate the initial beam
cross section of the internal beam into Mv parts. To this
end, we let an arbitrary position within the initial cross
section be written as
*
*
*
r ¼ cu u þcv v ,
ð1Þ
*
where cu and cv are two arbitrary coefficients. Vectors u
*
and v are defined by
*
*
*
u ¼ r p,2 $ r p,1 ,
*
*
*
v ¼ r p,N $ r p,1 :
ð2Þ
The coordinates (cu, cv) of those points on the initial
beam cross section are along the common edge of two
*
*
facets (outward normal directions are t 1 and t 2 ) and
satisfy the following condition:
cu
wu
wv
þ cv
¼ 1,
d1 $d2
d1 $d2
ð3Þ
*
where d1 and d2 represent the distances from r p,1 to the
planes of two aforementioned facets, and wu and wv are
given by
* *
* * !
* *
* * !
u Ut 1
u Ut 2
v Ut 1
v Ut 2
ð4Þ
wu ¼ * * $ * * , wv ¼ * * $ * * :
e pUt 1 e pUt 2
e pUt 1 e pUt 2
Eq. (3) defines a straight line, which splits the original
beam cross section into two sub-beams. After mathematical manipulation, the intersection points between the
straight line given by Eq. (3) and the polygon-shaped
boundary can be written in the form of
*
*
*
*
r ¼ r p,j þ ð r p,j þ 1 $ r p,j Þlj , if lj 2 ½0,1(,
ð5Þ
where lj(j ¼1,Nv) are defined as follows
l1 ¼
d1 $d2
,
wu
lN ¼ 1$
d1 $d2
,
wv
ð6Þ
Fig. 1. (a) The first order refracted beam is divided into three sub-beams with each impinging on a single facet. (b) An example of splitting a rectangular
beam cross section into two parts.
L. Bi et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 112 (2011) 1492–1508
lj ¼
8
*
*
* *
n^ U½ð r
þ l u $ r Þ)q (
>
>
> p *p,1 1 * p,j * ,
<
n^ p U½ð r p,j þ 1 $ r p,j Þ)q Þ(
*
*
*
9wv 9 r9wu 9
,
*
n^ p U½ð r p,1 þ ð1$lN Þv $ r p,j Þ)q (
>
>
>
: n^ p U½ð*r p,j þ 1 $*r p,j Þ)q*Þ( ,
9wv 9 49wu 9
j ¼ 2,Nv $1,
ð7Þ
where
*
q¼
8*
*
v
< v$ w
w u,
u
*
: wu *
wv v $u ,
9wv 9 r9wu 9
9wv 9 49wu 9
1495
subroutines requires more computer memory and is
unnecessary in the traditional ray-tracing algorithm,
where, for one incident ray, only one internal ray emerges
at each subsequent reflection and refraction event. To
terminate the beam-tracing process, a necessary condition is required in the recursive subroutine and is
addressed in Section 2.2.
2.2. Geometric-optics near-field
:
ð8Þ
In Eq. (7), n^ p is the normal direction of the initial beam
cross section. As the beam cross section is convex, there
are only two lj in the 0–1 range, for example see the case
shown in Fig. 1(b). At this point, it is straightforward to
split the original beam into two sub-beams by regrouping
the vertices of the original beam cross section and two
intersection points. When Mv 42, each sub-beam may
impinge on multiple facets, and, thus, the process is
repeated for each sub-beam until each next-order subbeam impinges on a single facet. After the initial beam
cross section is divided, the vertex coordinates of the end
cross section of each sub-beam can be obtained in a
straightforward manner. All sub-beams undergo internal
reflections at different facets, corresponding to the emergence of the next order reflected beams.
Similar to a data family tree, all the internal beams are
revealed in a recursive data structure. For the pth order
refraction/reflection, there are a number of the pth order
internal beams, fundamentally determined by the particle
orientation and refractive index. Each pth order internal
beam would generate several next order internal beams.
As the computer program allows for tracing a single beam
at each step, a recursive subroutine is most appropriate to
implement the beam-splitting algorithm. The recursive
subroutine contains the algorithm of splitting the input
beam and a loop defined in terms of calling the recursive
subroutine itself with each next-order reflected beam as
the input. The programming feature based on recursive
To calculate the electric field within the particle
i i i
through a beam-tracing process, we define ðb^ , a^ , e^ Þ,
s
s s
ðb^ p , a^ p , e^ p Þ and ðb^ p , a^ p , e^ p Þ, as shown in Fig. 2, to specify
three local coordinate systems associated with the incident light, the pth order inwardly propagating beam, and
the pth order outwardly propagating beam. Based on
defined local coordinate systems, Snell’s law, and Fresnel’s formulas, the geometric-optics near-field within the
particle can be expressed as the superposition of electromagnetic fields in conjunction with various internal
ray tubes.
Once the electric field at a specific point (e.g., the first
vertex) in the initial beam cross section is known, the
electric field at an arbitrary position in the ray tube can be
obtained by taking into account the variation of the phase
and the amplitude. For one of the pth order ray tubes, the
two components of the electric field associated with the
first vertex of the initial cross section is given by,
2
3
" i #
*
Ea
Ep, a ð r p,1 Þ
4
5
¼
U
ð9Þ
expðikdp,1 Þexpð$kNi dp,1 Þ,
p
*
i
E
b
Ep, b ð r p,1 Þ
where Eia and Eib are two components of the incident field
i
i
along two polarization vectors a^ and b^ ; Up is a matrix
associated with Snell’s law, Fresnel formulas, and necessary coordinate transformations; k is the wave number; Ni
is the imaginary part of the effective refractive index
N ¼Nr þ iNi [16]; and dp,1 and dp,1 account for the phase
delay and the decrease in the amplitude of the electric
s
i
Fig. 2. Coordinate systems defined in the ray-tracing process: (a) external reflection (p¼ 1) and (b) internal reflection (p 41). Note that b^ p ¼ b^ p , and b^
j
i
^
^
^
^
and a are rotated to b1 and e ) b1 , respectively.
1496
L. Bi et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 112 (2011) 1492–1508
field due to absorption. When p¼1, it can be verified that
*i *
d1,1 ¼ e U r 1,1 , d1,1 ¼ 0:
*
*
r p,1 þ w p ,
*
ð10Þ
At the position denoted by
where w p is a
vector lying in the beam *cross section, the two components of the electric field E p are given by
2
3
" i #
*
*
Ea
Ep, a ð r p,1 þ w p Þ
4
5
¼ Up i exp
*
*
E
b
E ðr þw Þ
p, b
p
p,1
*
*
*
½ikðNr e^ p Uw p þ dp,1 Þ( ) exp½$Ni kðA p Uw p þdp,1 Þ(,
ð11Þ
*
where Nr e^ p Uw p is associated with the variation of the
*
phase, and A p is a vector defined to account for the
*
variation of the amplitude in the beam cross section. A p
is found to be determined by an iterative formula
*
*
*
*
A 1 ¼ 0, A p ¼ A p$1 þð1$e^ p$1 UA p$1 Þ
n^ p$1
:
e^ p$1 Un^ p$1
ð12Þ
*
A 1 ¼ 0 as the field has a phase variance but no
amplitude variance on the external reflection interface.
A p ðp 41Þ is obtained by considering the differences in
total path lengths associated with the first vertex and any
other position in the beam cross
section.
*0
At an arbitrary position r in the ray tube, we have
*
*0
*
*
*
r ¼ r p,1 þ w p þle p ,
ð13Þ
where l is a variable associated with the propagating
*
*
distance from the position r p,1 þw p . Therefore, after
considering the phase variation and the decrease of the
*
amplitude along the propagation direction e p , the electric
field at any position within the ray tube can be written as
2
$ i %
*0 3
Ea
Ep, a ð r Þ
* *
4
5
¼ Up i exp½ikðNr e p Uw p þ dp,1 Þ(
*0
E
b
Ep, b ð r Þ
*
*
)exp½$Ni kðA p Uw p þ dp,1 Þ(expðikNlÞ,
*
ð14Þ
and at the specific position of r p þ 1,1 , is given by
2
3
$ i %
*
Ea
Ep, a ð r p þ 1,1 Þ
4
5
¼ Up i exp½ikdp þ 1,1 (exp½$Ni kdp þ 1,1 (,
*
Eb
Ep, b ð r p þ 1,1 Þ
ð15Þ
where
*
*
dp þ 1,1 ¼ dp,1 þNr 9 r p þ 1,1 $ r p,1 9,
*
*
dp þ 1,1 ¼ dp,1 þ 9 r p þ 1,1 $ r p,1 9:
ð16Þ
ð17Þ
Thus far, the information of the electric field in the
considered ray tube is completely specified.
After the reflection of the pth order ray tube, depending upon the beam splitting, several next-order ray tubes
may exist. We let the position vectors of one of the sub*0
beam cross sections be r p þ 1,i ði ¼ 1,2,. . .Þ to be distinguished with the notations of the original beam cross
section. Obtaining the electric field in the corresponding
next-order ray tube requires the information of the
*0
electric field associated with r p þ 1,1 , which is represented
in a similar form to Eq. (9) and given by
3
*0
$ i %
E
ð
r
Þ
Ea
p
þ
1,
a
p
þ
1,1
6
7
0
4
5 ¼ Up þ 1 Ei exp½ikdp þ 1,1 (exp½$Ni kd0p þ 1,1 (,
*0
b
Ep þ 1, b ð r p þ 1,1 Þ
2
ð18Þ
where Up þ 1 is calculated from Up and the Fresnel reflection matrix, and d0 p þ 1,1 and d0 p þ 1,1 are given by
*0
*
d0p þ 1,1 ¼ dp þ 1,1 þ Nr e^ p þ 1 Uð r p þ 1,1 $ r p þ 1,1 Þ,
*
*0
ð19Þ
*
d0p þ 1,1 ¼ dp þ 1,1 þ A p þ 1 Uð r p þ 1,1 $ r p þ 1,1 Þ:
ð20Þ
Up to this point, the electric field information in the
next-order ray tube can be obtained by applying a similar
procedure described for the pth order ray tube. The
internal electric field in all the ray tubes can be determined with the help of the beam-tracing technique.
In principle, each beam propagating within the particle
undergoes an infinite number of internal reflections, and
the electric field amplitude of beams decreases during the
interactions with a particle. Therefore, the contribution
from those ray tubes, after a number of internal reflections to the total radiation scattered and absorbed by a
particle, can be neglected. In the numerical algorithm, the
beam-tracing process is terminated when the energy
associated with the internal reflected beam is smaller
than a user-defined number (e.g., less than 10 $ 5). The
energy of the pth order internal reflected beam is given by
F¼
1
2
2
2
2
~ p 9e^ p Un^ p 9,
ð9Up11 9 þ 9Up12 9 þ 9Up21 9 þ 9Up22 9 Þ exp ð$2kNi dp ÞD
2
ð21Þ
~ p is an integral over the beam cross section and
where D
given as follows:
~p¼
D
¼
Z
*
*
*
*
*
expð$2kNi w p UA p Þd2 w p
*
*
*
*
N
ð r p,j þ 1 $ r p,j ÞUðA p ) n^ p Þ sin½ikNi A p Uð r p,j þ 1 $ r p,j Þ(
1 X
*
*
*
*
*
2
2kNi j ¼ 1
9A p 9 $ðA p Un^ p Þ2
ikNi A p Uð r p,j þ 1 $ r p,j Þ
*
*
*
*
)exp½$kNi A p Uð r p,j þ 1 $ r p,j $2 r p,1 Þ(:
ð22Þ
The calculation of the integral in Eq. (22) is based on
Stokes’ theorem.
The present algorithm of the near-field calculation
based on the beam-splitting technique can be applied to
arbitrary convex faceted particles. For non-absorbing
particles, the efficiency of the algorithm depends on the
orientation and shape of the particle and is essentially
independent of the particle size. For absorptive particles,
the algorithm speed increases with increase in size parameter because the higher order beams can be neglected
within the limits of acceptable accuracy. Thus, this algorithm can be applied to very large size parameters. The
computational time necessary is found to be on the order
of seconds for a simulation involving a single particle
orientation.
Another advanced feature of the present algorithm is
that the beam-tracing process for a single size can be
repeatedly employed to compute the near-field of particles with a series of size parameters for a specified
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L. Bi et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 112 (2011) 1492–1508
refractive index. Matrices associated with Fresnel formulas and coordinate transformations are independent of
size. The size dependent part in the beam-tracing process
is the geometry of internal ray-tubes, which would be
scaled according to the size parameter. As the ray number
is significantly reduced, the information of each ray-tube
could be saved in the computational program. The geometric optics near-field within the particle of a different
size parameter can be obtained by scaling the geometric
parameters of each ray-tube. This feature will save the
computational time when a series of size parameters are
involved in the computation.
*s *
scattered field E ð r Þ in the radiation region is transverse
with respect to the scattering direction r^ and can be
decomposed into two components in the form of
*s *
*
*
s
E ð r Þ ¼ Esa ð r Þa^ þ Esb ð r Þb^ ,
s
ð24Þ
s
where a^ and b^ are unit vectors parallel and perpendicular to the scattering plane, respectively, as shown in
Fig. 3(a). Taking dot products on both sides of Eq. (23)
s
s
with respect to vectors a^ and b^ yields
s
$
Esa
s
Eb
%
kr-1
¼
k2 exp ðikrÞ
4pr
ZZZ
v
2
ðm2 $1Þ4
* *0
3
a^ s UE ð r Þ 5
*0
*0
expð$ikr^ U r Þd3 r :
0
s *
*
b^ UE ð r Þ
ð25Þ
2.3. PGOH single-scattering properties
Once the electric field within the particle is known, the
single-scattering properties of the dielectric particle can be
obtained based on fundamental electromagnetic theory. The
procedure is similar to those in the DDA and FDTD methods,
but the PGOH allows the amplitude scattering matrix, the
extinction efficiency, and the absorption efficiency factors to
be in analytical form with respect to each reflection/
refraction event.
We use a volume integral equation to obtain the
amplitude scattering matrix, which relates the total
electric field within the particle to the induced scattered
field in the radiation zone, i.e., the far-field [34]
ZZZ
*s *
* *0
k2 exp ðikrÞ
E ð r Þ9kr-1 ¼
ðm2 $1ÞfE ð r Þ
4pr
v
* *0
*0
*0
$r^ ½r^ UE ð r Þ(g exp ð$ikr^ U r Þd3 r ,
ð23Þ
where v is the volume of the particle, r^ is the scattering
direction to the observation position, and m is the
refractive index which can be a complex number. The
In the geometric optics based PGOH, the internal field
in Eq. (25) can be formally written as a summation with
each term arising from different orders of reflection/
refraction events
* *0
Eðr Þ ¼
1
X
p¼1
0
0
*
*
Ep, a ð r Þa^ p þ Ep, b ð r Þb^ p :
ð26Þ
Substituting Eq. (26) into Eq. (25), we obtain
$
s
Ea
Esb
%
kr-1
¼
2
*0 3
1 ZZZ
Ep, a ð r Þ
*0
*0
k2 exp ðikrÞ X
5expð$ikr^ U r Þd3 r ,
ðm2 $1ÞKp 4
*0
4pr
vp
E ðr Þ
p¼1
p, b
ð27Þ
where vp is the volume associated with the pth order
internal ray tube as shown in Fig. 3(b), and Kp is a matrix
given by
2
Kp ¼ 4
a^ s Ua^ p a^ s Ub^ p
s
b^ Ua^ p
3
5:
s
b^ Ub^ p
Fig. 3. (a) Scattering coordinate systems and (b) volume associated with a ray tube.
ð28Þ
1498
L. Bi et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 112 (2011) 1492–1508
Substituting the geometric-optics near-field given by
Eq. (14) into Eq. (27), we obtain the following equation:
$
Esa
Esb
%
kr-1
¼
$ i %
1
Ea
k2 expðikrÞ X
ðm2 $1ÞKp Up expðikdp,1 $Ni kdp,1 ÞIp i ,
Eb
4pr
p¼1
ð29Þ
The amplitude scattering matrix associated with the
scattered field in Eq. (37) is given by
"
#
1
X
S2 S3
K p Up G
¼ ð1$m2 Þ
S4 S1
N$r^ Ue^ p
p¼1
)½9e^ p Un^ p þ 1 9Dp þ 1 exp ðikdp þ 1 $Ni kdp þ 1 Þ
$9e^ p Un^ p 9Dp exp ðikdp $Ni kdp Þ(,
where Ip is an integral defined by
Ip ¼
ZZZ
*
*
vp
*0
*
*0
exp½ikðNr e p þ iNi A p ÞUw p ( exp ðikNlÞ exp ð$ikr^ U r Þd3 r :
ð30Þ
or
$
S2
S4
S3
S1
%
¼ ðm2 $1Þ
Eq. (30) must be analytically solved before additional
numerical computations are considered. Recalling
Eq. (13), we transform Eq. (30) into the following form:
Ip ¼
ZZ
*
2*
*
*
w p 9e^ p Un^ p 9 exp ½ikðNr e^ p þiNi A p ÞUw p (exp ½$ikr^ Uð r p,1 þ w p Þ(
d
s
!
! *
Z !*r p þ 1 $*r p ! þ w p þ 1 Un^ p
^ ^
ep Unp
)
0
*
ð31Þ
After solving the integration in Eq. (31) in terms of l
and employing the following identities:
*
*
*
wp þ 1 ¼ wp þ
*
w p þ 1 Un^ p
e^ p ,
e^ p Un^ p
*
*
*
*
ð32Þ
*
A p þ 1 Uw p þ 1 ¼ A p Uw p þ
w p þ 1 Un^ p
,
e^ p Un^ p
ð33Þ
*
*
s
s
e^ p þ 1 Uw p þ 1 ¼ e^ p Uw p þ Nr
w p þ 1 Un^ p
,
e^ p Un^ p
ð34Þ
we obtain an explicit expression for Ip, given as follows
Ip ¼
h
*
*
4p
1
9e^ p Un^ p þ 1 9Dp þ 1 exp ðikN9 r p þ 1 $ r p 9Þ
*
2
k ikðN$r^ Ue p Þ
$9e^ p Un^ p 9Dp (,
ð35Þ
where
Dp ¼
*
k2
exp ð$ikr^ U r p,1 Þ
4p
*
*
Z
*
s
*
*
exp fikðe^ p $r^ þ iNi A p ÞUw p gd2 w p
*
s
N
ð r p,j þ 1 $ r p,j ÞU½ðe^ p $r^ þ iNi A p Þ ) n^ p (
ik X
¼
*
*
2
4p j ¼ 1 ^ s ^
s
9ep $r þ iNi A p 9 $½ðe^ p $r^ þ iNi A p ÞUn^ p (2
*
s
)
*
*
sin ½kðe^ p $r^ þ iNi A p ÞUð r p,j þ 1 $ r p,j Þ=2(
*
s
*
*
kðe^ p $r^ þ iNi A p ÞUð r p,j þ 1 $ r p,j Þ=2
*
p¼2
)G9e^ p$1 Un^ p 9Dp expðikdp,1 Þexpð$Ni kdp,1 Þ:
*
exp ½ikðN$r^ Ue p Þl(dl:
*
*
*
*
s
)exp½ikðe^ p þ iNi A p ÞUð r p,j þ 1 þ r p,j $2 r p,1 Þ=2(:
ð36Þ
The scattered far-field can be written in an analytical
form
" s#
" i #
1
Ea
X
Ea
exp ðikrÞ
Kp U p
2
ð1$m
¼
Þ
s
i
*
Eb
$ikr
^ Ue p Eb
p ¼ 1 N$r
ð39Þ
G is a rotational matrix that transforms the two components of the incident field to their counterparts
parallel and perpendicular to the scattering plane and
given by
6 i s
7
s7
6 ^ ^
6 b Ub
a^ i Ub^ 7
4
5
ð40Þ
G¼
s
i
s :
i
$a^ Ub^
b^ Ub^
Note, a number of beams are associated with the pth
order reflection/refraction not explicitly indicated in
Eq. (39), but are actually in the numerical algorithm
summation. Once the amplitude scattering matrix is
obtained, the phase matrix elements are straightforward
to compute [2].
The physical meaning implied in Eq. (39) is clearer
than in Eq. (38). The first term in Eq. (39) accounts for the
diffraction and external reflection contributions, and the
second term arises from higher order outgoing refracted
beams. Note the shape factor Dp is the largest when the
observation position vector is aligned with the direction
of the relevant outgoing beam. This feature partially
explains why the angular scattering pattern is dominant
around the scattered beam direction when the size parameter tends to be large.
We further derived the extinction and absorption cross
sections based on the following two integral equations
[35]:
6
7
6
7
ZZZ * 0 *inc* 0
6 k
07
*
*
6
2
3* 7
sext ¼ Im4 *inc ðm $1Þ
E ð r ÞUE
ð r Þd r 5,
ð41Þ
v
2
9E 9
sabs ¼
*
)exp½$ikr^ Uð r p,j þ 1 þ r p,j Þ=2(
K1 U1
G9e^ 1 Un^ 1 9D1 expðid1,1 Þ
N$r^ Ue^ 1
"
#
1
X
Kp$1 Up$1
Kp Up
$
þ ð1$m2 Þ
N$r^ Ue^ p$1 N$r^ Ue^ p
ð38Þ
k
*inc 2
9E
9
ei
ZZZ
* *0
v
** *0
*0
E ð r ÞUE ð r Þd3 r :
ð42Þ
A similar procedure is applied to derive the amplitude
scattering matrix based on the beam-splitting algorithm,
and the extinction cross section obtained from Eq. (41)
can be proven to be the same as that derived from the
optical theorem given by
kr-1
)½9e^ p Un^ p þ 1 9Dp þ 1 exp ðikdp þ 1 $Ni kdp þ 1 Þ
$9e^ p Un^ p 9Dp exp ðikdp $Ni kdp Þ(:
ð37Þ
sext ¼
h
i
2p
i
i
Re S11 ðe^ Þ þS22 ðe^ Þ :
k2
ð43Þ
L. Bi et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 112 (2011) 1492–1508
3. Accuracy of the PGOH simulations
The absorption cross section is given by
sabs ¼
1499
1
X
1
2
2
2
2
Nr expð$2Ni kdp Þð9Up11 9 þ 9Up12 9 þ 9Up21 9 þ 9Up22 9 Þ
2p¼1
*
~ p $exp ð$2Ni k9*
~ p þ 1 Þ:
r p þ 1 $ r p 9Þ9e^ p þ 1 Un^ p þ 1 9D
)ð9e^ p Un^ p 9D
ð44Þ
The physical process implied in Eq. (44) is evident,
because each term in the summation represents the
energy difference between the energy entering the ray
tube and that leaving the ray tube. The energy entering
the ray tube is given by Eq. (21). The real part of the
effective refractive index in Eq. (44) accounts for the
difference between the speed of light in the particle and
its surrounding medium.
The integration of the geometric-optics near-field
through electromagnetic integral equations in a ray tube
are carried out in an exact manner. This is the essential
difference between the present algorithm and that
reported in Yang and Liou [29], where an internal ray
tube of light is assumed to be in the shape of a circular
cylinder with a very small cross section. Therefore, a large
number of rays are required by Yang and Liou [29] to
guarantee the appropriateness of the assumption. In the
present algorithm, the number of rays is different for each
order of internal reflection and is determined by the
particle shape, but not to its size. As previously mentioned, the optical properties of a series of size parameters
can be obtained based on the beam-tracing process for a
specific size by scaling the geometry of internal ray tubes.
In principle, the PGOH is an approximate method. The
accuracy of the PGOH simulations can be estimated by
comparing the results with their counterparts simulated
from other exact methods. In this study, we select the
DDA method as a reference and use the Amsterdam DDA
(ADDA) code developed by Yurkin and Hoekstra [15] for
‘‘benchmark’’ simulations. The DDA method discretizes
the volume of the particle into various sub-volumes,
termed ‘‘dipoles’’, to solve an exact electromagnetic
volume integral equation. The numerical accuracy of the
DDA method depends on the number of dipoles used to
represent the geometry of the particle. The DDA method is
essentially an ‘‘exact’’ method as it directly solves the
equations in the context of electrodynamics, and can be
employed as a reference to test the accuracy of results
computed from the PGOH method. The accuracy of the
DDA method has been reported in the literature [15,36].
When an ice particle is strongly absorptive, the contribution to the scattering matrices from outgoing refracted
rays can be neglected. The amplitude scattering matrices
associated with the diffraction and external reflection can
be semi-analytically derived in the PGOH [37]. The PGOH
results closely agree with their hexagonal ice particle
counterparts computed from the DDA method. In this
section, we present some results for transparent and
semi-transparent particles.
Fig. 4 shows the phase functions simulated from both
the ADDA and PGOH. The aspect ratio of a hexagonal
Fig. 4. Comparison of the phase functions computed from the DDA method and the PGOH method for three selected refractive indices. The size
parameter defined in terms of the length is 50. The aspect ratio is 1.0.
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L. Bi et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 112 (2011) 1492–1508
particle is L/D¼1.0, where L and D are the length and
width of the particle, respectively, and the size parameter,
defined in terms of particle length, is 50. The three rows
correspond to refractive indices of 1.3, 1.3þ i0.01, and
1.3 þi0.1. The first and second columns are for the two
fixed orientations indicated in the figure. The third column is the average phase function for 10 orientations
with an interval of 101 between 01 and 901. As illustrated
in the figure, the PGOH results and those computed from
the ADDA compare well. The agreement is better for the
strong absorption case where diffraction and external
reflection are dominant. The general agreement between
the results computed from the PGOH and ADDA suggest
that the PGOH provides a reasonably accurate estimation
of the optical properties of ice particles including those of
moderate sizes. From the comparison, the averaging
process seems to improve the accuracy of the phase
function near the backward scattering directions. Note
the peak of the phase function due to external reflection is
evident for oriented particles.
Fig. 5 shows the logarithms of 2-D phase functions
with respect to scattering and azimuthal angles simulated
from the ADDA and PGOH. The computational parameters
including the shape, the size, and the orientation of the
hexagonal particle are the same as those in the first
column of Fig. 4. The general angular patterns of PGOH
simulated scattering are similar to those for the ADDA for
all the selected refractive indices. For cases with stronger
absorption, the PGOH results are much closer to their
ADDA counterparts. Two elements of the phase matrix,
P12 and P22, are simulated from the ADDA and PGOH with
results shown in Figs. 6 and 7, respectively. Note the
similarity between the PGOH and ADDA results for both
the P12 and P22 components.
Fig. 8 shows the phase function of ice particles with large
size parameters. Fig. 8(a) compares the phase functions
computed from the ADDA and PGOH. For this case, the
DDA code is computationally expensive. Four orientations
are assumed for an ice particle with respect to the symmetry
axis. The PGOH results have similar oscillations to those from
the ADDA, but there are some differences noted in the
scattering angle range from 901to 1501. The size parameter
for a hexagonal particle in Fig. 8(b) is very large, moving it
beyond the computational capability of the DDA method. As
expected, two halos are observed in the phase function
computed from the PGOH method. The results are calculated
for 1000 different ice particle orientations with respect to the
symmetry axis and subsequently averaged. In this simulation,
we find that increasing the number of orientations does not
diminish the oscillations in the PGOH simulated phase
functions. One possible explanation for the oscillation is that
it may be caused by interference between the various
scattered beams.
Fig. 9 shows the extinction efficiency factor and the
absorption efficiency factor simulated from the ADDA and
PGOH for three typical refractive indices. The particle orientation has a 201 incident angle between the 6-fold symmetry
axis and the incident direction. The figure shows the extinction efficiency factors computed from the PGOH can be larger
than those computed from the ADDA when the size parameter is small. In this size parameter region, the geometricoptics approximation method is expected to fail as the ‘‘ray’’
is not a proper conceptualization of the process when the
particle size is small or comparable with the wavelength of
incident light. When the size parameter is larger than 10, the
extinction efficiency factors simulated from the PGOH
demonstrate similar behavior to their ADDA counterparts;
however, the ADDA results are larger than the PGOH results.
The physical reason for the difference is that the edge effect
has not been considered. The existence of the edge effect
contribution to the extinction of light for particles with no
‘‘profile’’ curvature has been investigated by using the
localization principle [25]. As a safe estimation of the applicability of the PGOH method, the lower limit of the size
Fig. 5. Comparison of the logarithms of 2-D phase functions computed from the DDA method (upper panels) and the PGOH method (lower panels) for
three selected refractive indices. The aspect ratio, size and orientation of the particle are the same as those of the first column in Fig. 4.
L. Bi et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 112 (2011) 1492–1508
1501
Fig. 6. Same as Fig. 5, but for the P12/P11. The upper panels show the DDA simulations whereas the lower panels show the PGOH results.
Fig. 7. Same as Fig. 6, but for the P22/P11. The upper panels show the DDA simulations whereas the lower panels show the PGOH results.
parameter is about 20, although some simulated results may
have acceptable accuracy for even smaller sizes.
To bridge the gap between the ADDA results and their
PGOH counterparts, two semi-empirical formulae to
incorporate the edge effect contribution to the extinction
and absorption efficiency factors are used in the present
study and given by
Qext,edge ¼
Qabs,edge ¼
fe
ðkLÞ2=3
fa
ðkLÞ2=3
,
ð45Þ
,
ð46Þ
where the two factors fe and fa are determined by the
difference between the values of the efficiency factor
computed from the ADDA and the PGOH at the size
parameter where the two methods are unified. Fig. 10
shows the extinction efficiency and the absorption efficiency factor results after the incorporation of edge effect
contribution given by Eqs. (45) and (46). As evident from
Fig. 10, the curves of the extinction and absorption
efficiency factor are now continuous over the range of
size parameters. As a rigorous treatment of the edge
effects for ice particles using Maxwell’s equations is not
available at present, the semi-empirical method is essential to obtain the efficiency factors over a complete range
of size parameters. The oscillation of the extinction
efficiency factors results from interference between the
forward transmission and diffraction. For non-absorptive
particles, the oscillations do not diminish with large size
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L. Bi et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 112 (2011) 1492–1508
Fig. 8. (a) Comparison of the scattering phase function computed from the ADDA and the PGOH for the size parameter of 200. (b) Phase function
computed from the PGOH for hexagonal ice particles randomly oriented with respect to the 6-fold symmetry axis.
Fig. 9. The extinction and absorption efficiency factors simulated from the ADDA and the PGOH for oriented hexagonal ice particles. These results
exclude consideration of the particle edge effect. Three typical refractive indices are selected.
parameters. The oscillation phenomenon is not observed
for either spheres or randomly oriented nonspherical
particles. The non-convergent asymptotic value of the
oriented ice particle extinction efficiency factor is demonstrated in the PGOH method results, but to the best of our
knowledge, is neither justified through exact methods nor
by measurements. Further investigation of this issue is
warranted.
4. Oriented ice plates
Large ice particles in the atmosphere may not be
randomly oriented in space, but reveal some preferable
orientations and flutter relative to a horizontal plane.
Straightforward evidence to support the existence of
preferably oriented ice particles in the atmosphere is
various optical phenomena such as parhelia, sub sun,
and sun pillars [38]. Their existence is confirmed based
on observations from satellite instruments and groundbased lidar [39–46]. Aerodynamic microphysical processes to determine ice particle orientation and fall
characteristics have been investigated (e.g., [42]).
The optical properties of oriented ice particles are
quite different from those of randomly oriented particles,
and cause a different radiative impact on the atmosphere.
An accurate modeling of the single-scattering properties
of oriented ice particles has important implications to
climate study and remote sensing applications. As previously stated, the CGOM has inherent flaws, and is thus
inappropriate for studying the optical properties of particles with fixed orientations. For example, the phase
function of oriented ice particles is not a continuous
L. Bi et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 112 (2011) 1492–1508
1503
Fig. 10. Similar to Fig. 9, but with the edge effect incorporated in the PGOH method. Note how smoothly the ADDA results transition to those from the
PGOH method.
curve, but a set of singular points. Some attempts to study
the optical properties of oriented ice particles based on
the PGOH method can be found in [32].
We apply the new PGOH method to study the backscattering of oriented ice particles, and subsequently
analyze the size dependence of two quantities: the backscattering cross section and the backscatter color ratio at
0.532 and 1.064 mm. We assume the incident light is
x-polarized, the lidar backscattering cross section is
defined as,
i
4p h
3 3 2
3 3 2
9S
ð180
,0
Þ9
þ
9S
ð180
,0
Þ9
2
4
k2
4p
¼ 2 ½P11 ð1803 ,03 Þ þ P12 ð1803 ,03 Þ(
k
sb ¼
ð47Þ
The backscattering efficiency is defined by Qb ¼ sb/G,
where G is the projected area of the particle. Note, P11 and
P12 are not normalized, and P12 is included as it may not
be zero for oriented particles.
In the formulation of CGOM, the backscattering radiation is associated with scattered beams propagating in the
backscattering direction. Therefore, for a specific orientation with unidentified backscattered beams, the backscattering cross section is zero. However, in the PGOH, the
backscattered radiation can still be considered. The physical reason is the spreading effect of beams propagating
near the backscattered angle. However, backscattered
beams in the CGOM could also spread some energy into
other directions. In the following discussion, beam
spreading and interference are crucial concepts in understanding the properties of backscattered radiation for
oriented particles. Two effects associated with the beam
spreading and interference can be understood based on
Eq. (39). For non-absorptive particles with different size
parameters, Dp exp(ikdp,1) accounts for the beam spreading and phase variance.
To understand the beam spreading effect on the backscattering radiation, we now investigate the diffraction
and external reflection by a hexagonal plate. The contribution due to higher-order refraction is separated to
avoid interference among scattered beams. Two orientations of a plate are considered, the incident light normal
to the top facet and the incident light at a 51 angle from
the symmetry axis. Fig. 11 illustrates the 2-D phase
functions for the two cases, and the spreading of externally reflected beams can be observed. Fig. 12 shows the
backscattering efficiency with respect to the particle size.
The reflected beam directly contributes the backscattering
when the incident light direction is normal to the basal
face. As the size parameter increases, the degree of ray
spreading decreases and backscatter results increase.
When the incident direction makes a 51 angle with the
six-fold symmetry axis, the scattering angle associated
with the reflected beam in the CGOM is 1701. In this case,
the observed backscattering physically originates from
the spreading of the reflected beam, as shown in Fig. 11.
As the size parameter increases, less energy is spread into
the backscattering direction. Similar to the case of Fraunhofer diffraction, the backscattering efficiency generally
decreases but oscillates locally. As can be seen in Fig. 12,
the backscattering efficiency dependence on the size
parameter differs for various plate particle orientations.
The results in Fig. 13 are similar to those in Fig. 12 but
include consideration of all higher-order scattered beams.
The interference between scattered beams depends on
the phase delay associated with total path length. As a
result, the backscatter cross section oscillates significantly
with respect to the particle size. However, the physics of
the ray spreading effect determines the behavior of backscatter in terms of the size parameter.
For its potential application to the active remote
sensing of ice cloud properties, the lidar color ratio,
1504
L. Bi et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 112 (2011) 1492–1508
Fig. 11. Display of the phase function as a function of scattering angle and azimuthal angles. The direction of incident light makes a 01 angle (left) and 51
angle (right) with the 6-fold symmetric axis.
Fig. 12. Backscattering efficiency for diffraction and external reflection.
defined as the ratio of backscatter coefficients between
the wavelengths of 1.064 and 0.532 mm, is investigated.
The lidar color ratio value associated with ice particle
backscattering for a randomly oriented hexagonal particle
was simulated in a previous study [17] and found to be
less than unity. The physical explanation is the size
parameter of the particle at the 0.532 mm wavelength is
twice that at 1.064 mm. Because the degree of the ray
spreading backscattered beam effect differs for the two
size parameters, the value of the lidar color ratio deviates
from and is less than unity. The results from the simulations are found to be consistent with those from observations. For oriented particles, the backscattered radiation
may arise from the spreading of scattered beams, and the
lidar color ratio is expected to be larger than unity
because the behavior of backscattering efficiencies with
respect to the particle size is reversed (Figs. 12 and 13).
To calculate the backscattering color ratio, we use the
Gamma function to specify the size distribution [47],
nðDÞ ¼ N0 Dg exp ð$gD=Dm Þ,
ð48Þ
where N0 is the total number of plate particles in a unit
volume, D is the width of the plate, and Dm is the modal
size. Given a size distribution, the effective size of the
ensemble of particles is defined [48,49],
R d2
3 d VðDÞnðDÞdD
:
ð49Þ
De ¼ R d1
2
2
d1 AðDÞnðDÞdD
A value of the lidar color ratio can be derived for a
given effective diameter. Fig. 14 shows the backscattering
coefficients at two wavelengths and the dependence of
the color ratio values with respect to the effective plate
size. The aspect ratio of the plate is selected to be 5.
L. Bi et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 112 (2011) 1492–1508
1505
Fig. 13. Similar to Fig. 12, all scattered beams are taken into account.
Fig. 14. Backscattering efficiency and color ratio at wavelengths of 0.532 and 1.064 mm.
N0 and g are assumed to be 1.4882 ) 10 $ 4 and 2.1921,
respectively. When the light is incident normal on the top
of the particle, the color ratio is approximately 0.23, and
almost independent of the effective size. For oblique
incidence, the color ratio is larger than unity, and
decreases when the effective size is smaller than ! 600
and slightly increases for larger effective sizes.
5. Imperfect hexagonal ice particles
The optical properties of imperfect hexagonal ice
particles are investigated to explain why halos are not
observed more often. Observed ice particle habits generally reveal various geometric characteristics due to complex temperature and humidity conditions during their
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L. Bi et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 112 (2011) 1492–1508
growth. Because of complex atmospheric conditions
encountered during particle growth, the top or bottom
facets of ice particles are not generally regular hexagons
[3] and often reveal surface texture. The complexity of
modeling ice particle imperfections poses a challenge for
realistic numerical simulations of optical properties. To
simplify the modeling procedure, Macke [21] and Hess
et al. [33] developed a method to model ice particle
imperfections through statistical ray path deviations in a
regular hexagonal particle with the CGOM. To modify the
PGOH method to model ice particle imperfections, we
distort regular hexagonal ice particles instead of using
either the complex polycrystal method [21] or by changing the ray path in the beam-tracing process [33]. The
advantages of the present method are simplicity and
efficiency, which allow calculating consistent single-scattering properties over a wide range of size parameters.
Fig. 15 shows a set of ice particle habits including the
basic hexagonal column or plate. To model imperfect ice
particles, we distort the regular hexagonal shape using
two different procedures. Ice particles are given irregular
top and bottom faces but the right angle is kept between
the top and six side facets. Specifically, the shape of the
Fig. 15. Model particles chosen to represent both regular and imperfect
hexagonal ice particles.
top face is obtained by randomly choosing two points on
each side of a right triangle to be vertices of a hexagon. To
model ice particles with more complex characteristics
such as roughness, we randomly tilt the normal directions
of each face of a regular hexagonal prism. Considering the
top face as an example, the normal direction could be
defined through two random numbers x1 and x2 between
0 and 1 given by,
%
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi&
1$ðx1 Þ2 :
ð50Þ
x1 x2 , x1 1$ðx2 Þ2 ,
By using the aforementioned procedure, an ensemble
of imperfect ice particles can be generated. The average
scattering properties of an ensemble of irregular ice
particles might be expected to represent the realistic
optical properties of imperfect ice particles, although the
morphology of model particles is quite different from that
of realistic ice particles. The optical properties of the
imperfect model particles can be easily computed using
the present PGOH algorithm. As shown in the following
results, the adoption of irregular habit choices effectively
diminishes halo phenomena.
Fig. 16 shows the phase functions for hexagonal
particles with irregular bases computed from the PGOH
method. For convenience, each side is parallel to its facing
side and the ratio of its longer side to its shorter side is
assumed to 1, 2 or 3. When the ratio is 1, the particle is a
regular hexagonal ice particle, the size parameter defined
in terms of the length is 500, and the aspect ratio is 0.5
(diameter divided by length). Other assumptions are that
the surface area of two particles with irregular bases is
the same as for a regular ice particle, the direction of the
incident light is normal to the side faces, and the particle
is randomly oriented with respect to a symmetric axis.
We observe some differences in the phase functions
simulated from the PGOH method for the three particles.
The irregular base has three 601 vertex angles, and the 221
halo can be observed. The 1541 scattering maximum is
reduced for the two irregular hexagonal columns. The
1541 scattering maximum for a regular hexagonal particle
Fig. 16. Phase functions computed from the PGOH method for hexagonal particles with irregular bases.
L. Bi et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 112 (2011) 1492–1508
1507
Fig. 17. Phase functions computed from the PGOH method for hexagonal particles with tilted facets.
is due to the refracted beams undergoing several internal
reflections [19]. Due to less symmetry associated with the
irregular ice particles, the 1541 scattering maximum
found for regular hexagonal particles is not observed here.
Fig. 17 shows the scattering phase function computed
from the PGOH for randomly oriented hexagonal particles
with tilted facets. As evident in the figure, the halo peaks
observed from the regular hexagonal particles are diminished. The present method of tilting the facets of hexagonal ice particles can be employed to simulate the presence
of surface roughness. A similar approach to modeling the
optical properties of mineral dust by tilting the facets of
regular hexahedra can be found in [50].
6. Summary and conclusions
We develop a new PGOH method to approximately
compute the single-scattering properties of any general
dielectric faceted particle, and focus attention on the
optical properties of ice particles. Our method is suitable
for studying the optical properties of ice particles with
arbitrary orientations, complex refractive indices (i.e.,
particles with significant absorption), and size parameters
(proportional to the ratio of particle size to incident
wavelength) larger than ! 20, and includes consideration
of edge effects necessary for accurate determination of
the extinction and absorption efficiencies.
The near field is computed based on geometric optics
in conjunction with a beam-splitting technique and the
characteristics associated with inhomogeneous waves
inside of the absorptive particle are taken into account.
In comparison with the DDA method, the PGOH method is
found to provide reasonable accuracy. The major advantages and benefits of our new formulation of the PGOH
method is that it permits:
+ more rigorous studies of the optical properties of large
ice particles. No limitations exist on the maximum
particle size parameter,
+ simulations for particles with complex refractive
+
+
+
indices, so it is applicable to particles with significant
absorption,
simulations of oriented ice particles,
more accuracy in the description of backscattering
properties, and
calculation of optical properties of particles with a
range of size parameters based on the beam-tracing
process for a single size.
The present algorithm can be easily extended to any
faceted dielectric particle (e.g., non-symmetric hexahedra
dust model [50]).
We apply the present algorithm to study the backscattering efficiencies of preferentially oriented ice
particles and the effects of irregular hexagonal ice particles. The dependence of the ice particle backscattering
efficiency on the particle size and orientation is demonstrated. The ray spreading effect is found to play a major
role in the backscattering properties. Two kinds of
imperfect ice particles are studied. The method of tilting
the facets of hexagonal ice particles can be effectively
employed to model the particle surface roughness,
although the morphology of a model particle is quite
different from its realistic counterpart. Imperfect ice
particles tend to reduce the halo features and backscatter.
Furthermore, the single-scattering properties of ice particles can be computed based on a combination of the DDA
method and the new PGOH method for a wide range of
size parameters.
Acknowledgments
The authors thank M. A. Yurkin and A. G. Hoekstra for
their ADDA code. A major portion of numerical computation
was conducted by using the NASA High-End Computing
(HEC) resources under award SMD-09-1413. Ping Yang
acknowledges support from NASA grants NNX08AF68G and
NNX08AI94G and the National Science Foundation (NSF)
1508
L. Bi et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 112 (2011) 1492–1508
grant ATM-0803779. George Kattawar acknowledges support
by the Office of Naval Research under contract N00014-06-10069. Bryan Baum acknowledges support through NASA
ROSES grant NNX08AF81G.
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Scattering and absorption of light by ice particles: Solution by... physical-geometric optics hybrid method