Journal of Quantitative Spectroscopy & Radiative Transfer 112 (2011) 1492–1508
Contents lists available at ScienceDirect
journal homepage: www.elsevier.com/locate/jqsrt
a ,
n
a , b
a
c
d
a
Department of Physics and Astronomy, Texas A&M University, College Station, TX 77843, USA b d
Department of Atmospheric Sciences, Texas A&M University, College Station, TX 77843, USA c
Climate Science Branch, NASA Langley Research Center, Hampton, VA 23681, USA
Space Science and Engineering Center, University of Wisconsin-Madison, Madison, WI 53706, USA a r t i c l e i n f o
Article history:
Received 9 December 2010
Received in revised form
19 February 2011
Accepted 21 February 2011
Available online 4 March 2011
Keywords:
Light scattering
Geometric optics
Physical optics
Hexagonal ice crystal a b s t r a c t
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.
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: bilei@tamu.edu (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
¼ 2 p r / l , where r is radius and l is wavelength), they are impractical and computationally inefficient for large size parameters ( k 4 20). 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
( 4 100–200) 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( $ i o t ), 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

1494 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 ( ¼ 1,2,3 y ) indicate the p th order reflection/refraction event. The direction of one internal beam leaving some interface of the p th order reflection/refraction is specified by ^ p
, and the vertices of the beam cross section on the interface of electromagnetic interaction is denoted as r p p , i
¼
ð i ¼ 1 , N
V
Þ , where N
V is the number of vertices. When
1 (i.e., external reflection and refraction), r
1 , i
ð i ¼ 1 , N
V
Þ are the straightforward coordinates of the vertices of the corresponding facet where external reflection takes place.
by
To describe the splitting of an internal beam (specified
^ p and
* r p , i
), the first step is to determine the intercepting particle facets. We assume N
V to be the number of straight-line rays (with no cross section) starting from the positions of N
V vertices and propagating in the direction e p
. The facets of the particle surface are convex shapes, and if, for example, the N
V number of rays strike M v number of different facets, the beam cross section is divided into M v single facet.
t i parts with each part impinging on a
ð i ¼ 1 , M v
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 M v parts. To this end, we let an arbitrary position within the initial cross section be written as
¼ c u
*
þ c v
*
, ð 1 Þ where c u and v and c v
* are two arbitrary coefficients. Vectors u are defined by
¼ r p , 2
$ r p , 1
, v ¼ r p , N
$ r p , 1
: ð 2 Þ
The coordinates ( c u
, c v
) 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: c u w u d
1
$ d
2
þ c v w v d
1
$ d
2
¼ 1 , ð 3 Þ where d
1 and d
2 represent the distances from planes of two aforementioned facets, and w u r p , 1 to the and w v are given by w u
¼
*
U t t
1
* e p
U
*
1
$
*
U t t
2
* e p
U
*
2
!
, w v
¼
*
U
* e p
U t t
1
*
1
$
*
U
* e p
U t t
2
*
2
!
: ð 4 Þ
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
*
þð r
* p , j þ 1
$ r p , j
Þ l j
, if l j
2 ½ 0 , 1 ( , ð 5 Þ where l j
( j ¼ 1, N v
) are defined as follows l
1
¼ d
1
$ d
2 w u
, l
N
¼ 1 $ d
1
$ d
2 w v
, ð 6 Þ
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 l j
¼
8
>
>
: n p
*
U ½ð r p , 1 n p
*
U ½ð r p , j
þ l
1
þ
* u
*
$ r
*
1
$ r p , j p , j
*
Þ) q (
*
Þ) q Þ(
, n p
*
U ½ð r p n p
, 1
*
U ½ð r
*
þ ð 1 $ l
N
Þ v
* p , j þ 1
$ r p , j
$
* r p , j
*
Þ) q Þ(
*
Þ) q (
9 w v
9 r 9 w u
9
, 9 w v
9 4 9 w u
9
, j ¼ 2 , N v
$ 1 ,
ð 7 Þ where
¼
8
<
:
* v $ w u w v w v w u
* v
*
$ u
* u , 9 w v
9 r 9 w u
9
:
, 9 w v
9 4 9 w u
9
ð 8 Þ
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
In Eq. (7), ^ p is the normal direction of the initial beam cross section. As the beam cross section is convex, there are only two l j 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 M v
4 2, 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 p th order refraction/reflection, there are a number of the p th order internal beams, fundamentally determined by the particle orientation and refractive index. Each p th 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 through a beam-tracing process, we define ð
ð b p
, a p
, ^ p
Þ and ð b s p
, a s p
, ^ s p b i
, a i
, ^ i
Þ
Þ , as shown in
Fig. 2 , to specify
, three local coordinate systems associated with the incident light, the p th order inwardly propagating beam, and the p th 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 p th order ray tubes, the two components of the electric field associated with the first vertex of the initial cross section is given by,
2
4
E
E p , a p , b
ð r
ð r p , 1 p , 1
Þ
Þ
3
5
¼ U p
"
E
E i b i a
# exp ð ik d p , 1
Þ exp ð$ kN i d p , 1
Þ , ð 9 Þ where E i a and E i b are two components of the incident field along two polarization vectors a i and b ; U p is a matrix associated with Snell’s law, Fresnel formulas, and necessary coordinate transformations; k is the wave number; N i is the imaginary part of the effective refractive index
N ¼ N r
þ iN i
[16] ; and
d p ,1 and d p ,1 account for the phase delay and the decrease in the amplitude of the electric
Fig. 2.
Coordinate systems defined in the ray-tracing process: (a) external reflection ( p ¼ 1) and (b) internal reflection ( p 4 1). Note that and a j are rotated to b
1 and ^ i
) b
1
, respectively.
b p
¼ b s p
, and b i

1496 field due to absorption. When p ¼ 1, it can be verified that d
1 , 1
¼ e i
U r
1 , 1
, d
1 , 1
¼ 0 :
At the position denoted by r p , 1
þ w p
, where w p
ð 10 Þ is a nents of the electric field E p
2 3
4
E p , a ð
E p , b
*
ð r p , 1 p , 1
þ w p
Þ
*
þ w p
Þ
5 are given by
¼ U p
"
E
#
E i b i a exp
½ ik ð N r e p
*
U w p
þ d p , 1
Þ( ) exp ½$ N i
* k ð A p
*
U w p
þ d p , 1
Þ( , ð 11 Þ where N r e p
*
U w 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
þð 1 $
^ p $ 1
*
U A p $ 1
Þ e n p $ 1 p $ 1
U ^ p $ 1
: ð 12 Þ
*
A
1
¼ 0 as the field has a phase variance but no
A p
ð p 4 1 Þ is obtained by considering the differences in total path lengths associated with the first vertex and any other position in the beam cross section.
At an arbitrary position r in the ray tube, we have r
0
¼ p , 1
þ w p
þ p
, ð 13 Þ where l is a variable associated with the propagating
* distance from the position r p , 1
*
þ w 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
4
E p ,
* 0 a ð r Þ
E p , b
* 0
ð r Þ
3
5
¼ U p
$
E
%
E i b i a * exp ½ ik ð N r e p
*
U w p
þ d p , 1
Þ(
) exp ½$ N i
* k ð A p
*
U w p
þ d p , 1
Þ( exp ð ikNl Þ , ð 14 Þ and at the specific position of r
2
4
E p ,
E a ð p , b
*
ð r p þ 1 , 1 p þ 1 , 1
Þ
Þ
3
5
¼ U p
$
E
E i b i a
% exp ½ ik p þ 1 , 1 d p þ 1 , 1
, is given by
( exp ½$ N i kd p þ 1 , 1
( ,
ð 15 Þ where d p þ 1 , 1
¼ d p , 1
þ N r
9
* r p þ 1 , 1
*
$ r p , 1
9 , ð 16 Þ d p þ 1 , 1
¼ d p , 1
þ 9
* r p þ 1 , 1
*
$ r p , 1
9 : ð 17 Þ
Thus far, the information of the electric field in the considered ray tube is completely specified.
After the reflection of the p th order ray tube, depending upon the beam splitting, several next-order ray tubes may exist. We let the position vectors of one of the subbeam cross sections be r
0 p þ 1 , i
ð i 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 electric field associated with r
0 p þ 1 , 1
, which is represented
L. Bi et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 112 (2011) 1492–1508 in a similar form to Eq. (9) and given by
2 3
E p
E p
þ 1 ,
þ 1 ,
*
0 a ð r p þ
* 0 b
ð r p þ
1 , 1
Þ
1 , 1
Þ
7
¼ U p þ 1
$
E
%
E i a i b exp ½ ik d 0 p þ 1 , 1
( exp ½$ N i kd 0 p þ 1 , 1
( ,
ð 18 Þ where U p þ 1 is calculated from tion matrix, and d 0 p þ 1,1 and d 0 p
U p
þ 1,1 and the Fresnel reflecare given by d 0 p þ 1 , 1
¼ d p þ 1 , 1
þ N r e p þ 1
U ð r
0 p þ 1 , 1
$ r p þ 1 , 1
Þ , ð 19 Þ d 0 p þ 1 , 1
¼ d p þ 1 , 1
*
þ A p þ 1
U ð r
0 p þ 1 , 1
$ r p þ 1 , 1
Þ : ð 20 Þ
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 p th 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 p th order internal reflected beam is given by
F ¼
1
2
ð 9 U
11 p
9
2
þ 9 U
12 p
9
2
þ 9 U
21 p
9
2
þ 9 U
22 p
9
2
Þ exp ð$ 2 kN i d p
Þ
~ p
9 ^ p
U ^ p
9 ,
ð 21 Þ where D p is an integral over the beam cross section and given as follows:
Z
~ p
¼ exp ð$ 2 kN i
* w p
*
U A p
Þ d 2
* w p
¼
1
2 kN i j ¼ 1
*
ð r p , j þ
*
1
9 A p
*
$ r p , j
9
2
Þ
*
$ð A
U p
ð
U
*
A
^ p p
Þ
)
2
^ p
Þ sin ½ ikN ikN i i
*
A p
*
U ð r p , j þ
*
A p
*
U ð r p , j þ 1
1
$ r
*
$ r
* p , j p , j
Þ
Þ(
*
) exp ½$ kN i
A p
*
U ð r p , j þ
*
1
$ r p , j
*
$ 2 r p , 1
Þ( : ð 22 Þ
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

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.
2.3. PGOH single-scattering properties
1497
* 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
E ð r Þ ¼ E s a
ð r Þ a s
þ E s b
ð r Þ s
, ð 24 Þ where a s and b s 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)
with respect to vectors a s and b s yields
2 3
$
E
E s a s b
% kr 1
¼ k 2 exp ð ikr Þ
4 p r
ZZZ v
ð m 2
$ 1 Þ
4 s
U
*
E
* 0
ð r Þ s
U
*
E
* 0
ð r Þ
5 exp ð$ ik ^ U r
0
Þ d 3 r
0
:
ð 25 Þ
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]
* s
E
*
ð r Þ 9 kr 1
¼ k 2 exp ð ikr Þ
4 p r
ZZZ
ð m
2
*
$ 1 Þf E
* 0
ð r Þ v
$ ^ ½ ^
* r U E
* 0
ð r Þ(g exp ð$ ik ^
* 0 r U r Þ d 3
* 0 r , ð 23 Þ where v is the volume of the particle, ^ 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
*
E ð r
0
Þ ¼ p ¼ 1
E p , a ð r
0
Þ a p
þ E p , b
ð r
0
Þ b p
: ð 26 Þ
Substituting Eq. (26) into Eq. (25), we obtain
$
E s a
%
E s b kr 1
¼ k 2 exp ð ikr Þ
4 p r p ¼ 1
ZZZ v p
2
E
ð m 2
$ 1 Þ K p
4
E p , a ð r
0
Þ p , b
* 0
ð r Þ
3
5 exp ð$ ik r ^ U r
0
Þ d 3 r
0
,
ð 27 Þ where v p is the volume associated with the p th order internal ray tube as shown in
Fig. 3 (b), and
K given by p is a matrix
2
K p
¼
4 s
U a p s
U a p s
U b p s
U b p
3
5
: ð 28 Þ
Fig. 3.
(a) Scattering coordinate systems and (b) volume associated with a ray tube.

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:
$
E
E s b s a
% kr 1
¼ k 2 exp ð ikr Þ
4 p r p ¼ 1
ð m
2
$ 1 Þ K p
U p exp ð ik d p , 1
$ N i kd p , 1
Þ I p
$
E
%
E i a i b
,
ð 29 Þ where I p is an integral defined by
I p
¼
ZZZ v p
* exp ½ ik ð N r e p
þ iN i
*
A p
*
Þ U w p
( exp ð ikNl Þ exp ð$ ik ^ U
* 0 r Þ d 3
* 0 r :
ð 30 Þ
Eq. (30) must be analytically solved before additional numerical computations are considered.
Recalling
Eq. (13), we transform Eq. (30) into the following form:
I p
¼
ZZ s d
2
* w p
9 ^ p
U ^ p
9 exp ½ ik ð N r e p
þ iN i
*
A p
*
Þ U w p
( exp ½$ ik ^
* r U ð r p , 1
*
þ w p
Þ(
)
Z
0
* r
* p þ 1
$ r p
þ
* p þ 1
U ^ np
U np exp ½ ik ð N $ r ^ U
* e p
Þ l ( dl : ð 31 Þ
After solving the integration in Eq. (31) in terms of l and employing the following identities:
* w p þ 1
*
¼ w p
þ p þ 1
U ^ p e p
U ^ p e p
, ð 32 Þ
*
A p þ 1
U w p þ 1
¼
*
A p
U w p
þ p þ 1
^ p
U
U ^ p
^ p
, ð 33 Þ s p þ 1
U w p þ 1
¼ ^ s p
U w p
þ N r
* w p þ 1
U n p e p
U ^ p
, ð 34 Þ we obtain an explicit expression for I p
, given as follows
I p
¼
4 p k 2
1 ik ð N $ ^
* r U e p
Þ h
9 ^ p
U ^ p þ 1
9 D p þ 1 exp ð ikN 9 r p þ 1
$
$ 9 ^ p
U ^ p
9 D p
( , p
9 Þ
ð 35 Þ where
D p
¼
¼ k 2
4 p ik
4 p exp ð$ ik r U r p , 1
Þ j ¼ 1
ð r p , j þ 1
$ r
Z exp f ik ð ^ s p
$ ^ þ iN i
*
A p
Þ U w p g d 2 w p p , j
Þ U ½ð ^ s p
$ ^ þ iN i
*
A p
Þ ) ^ p
(
9 ^ s p
$
^
þ iN i
*
A p
9
2
$½ð
^ s p
$
^
þ iN i
*
A p
Þ U n p
(
2
) sin ½ k ð
^ s p k ð ^ s p
$
^
þ iN
$ ^ r þ iN i i
*
A p
*
A p
Þ U
Þ
ð
U r
ð
* r p , j þ 1
$ r
* p , j þ 1
$ r p , j p , j
Þ = 2 (
Þ = 2
) exp ½$ ik ^ U ð r
) exp ½ ik ð
^ s p
þ p , j þ 1
* iN i
A p
þ
Þ U ð r p , j
Þ = 2 ( p , j þ 1
þ p , j
$ 2 r p , 1
Þ = 2 ( : ð 36 Þ
The scattered far-field can be written in an analytical form
"
E
E a s b
# kr 1
¼ exp ð ikr Þ
$ ikr
ð 1 $ m
2
Þ p ¼ 1
K p
U p
N $ ^
* r U e p
"
E
E i b i a
#
)½ 9 ^ p
U ^
$ 9 ^ p
U n p p þ 1
9 D p þ 1
9 D p exp ð exp ð ik ik d p
$ N i d p þ 1 kd p
Þ( :
$ N i kd p þ 1
Þ
ð 37 Þ
The amplitude scattering matrix associated with the scattered field in Eq. (37) is given by
"
S
2 or
$
S
2
S
4 given by
G
S
4
¼
6
4 s ext
¼
S
3
#
S
1
S
3
%
S
1 b
$ i a
2 p k 2
U i b
U
^
¼ ð 1 $ m 2
Þ
K p
U p
G p ¼ 1
N $ ^ U ^ p s b s
)½ 9 ^ p
U ^ p þ 1
9 D p þ 1
$ 9 ^ p
U n p
9 D p exp ð exp ð ik ik d p
$ N i d p þ 1 kd p
Þ( ,
$ N i kd p þ 1
Þ a i
U b s 7
5
: b i
U b s h
Re S
11
ð
^ i
Þþ S
22
ð
^ i
Þ i
:
ð 38 Þ
¼ ð m 2
$ 1 Þ
þð 1 $ m 2
K
1
U
1
N $
^ U e
1
"
Þ p ¼ 2
G 9 ^
1
U ^
1
K p $ 1
U p $ 1
N $
^ U e p $ 1
9 D
1
$ exp
K p
ð
U i p d
1 , 1
N $
^ U e p
#
Þ
)
G 9 ^ p $ 1
U ^ p
9 D p exp ð ik d p , 1
Þ exp ð$ N i kd p , 1
Þ : ð 39 Þ
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
ð 40 Þ
Note, a number of beams are associated with the p th 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 D p 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]
: s ext
¼ Im
6
6 k
* inc
9 E 9
2
ð m 2
$ 1 Þ
ZZZ v
*
E ð r
0 * inc *
Þ U E ð r
0
Þ d 3 r
0
7
7
, ð 41 Þ s abs
¼ k
* inc
9 E 9
2 e i
ZZZ v
*
E
* 0
ð r
* *
Þ U E
* 0
ð r Þ d 3
* 0 r : ð 42 Þ
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
ð 43 Þ

L. Bi et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 112 (2011) 1492–1508
The absorption cross section is given by s abs
¼
1
2 p ¼ 1
N r exp ð$ 2 N i kd p
Þð 9 U 11 p
9
2
þ 9 U 12 p
9
2
þ 9 U 21 p
9
2
þ 9 U 22 p
9
2
Þ
)ð 9 ^ p
U ^ p
9 ~ p
$ exp ð$ 2 N i k 9
* r
* p þ 1
$ r p
9 Þ 9 ^ p þ 1
U n p þ 1
9 ~ p þ 1
Þ :
ð 44 Þ
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.
1499
3. Accuracy of the PGOH simulations
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.

1500 L. Bi et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 112 (2011) 1492–1508 particle is L/D ¼ 1.0, 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, 1.3
þ i 0.01, and
1.3
þ i 0.1. 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 10 1 between 0 1 and 90 1 . 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,
P
12 and P
22
, 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 P
12 and P
22 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 90 1 to 150 1 . 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 20 1 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 applic-
ability 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
P
12
/ P
11
. The upper panels show the DDA simulations whereas the lower panels show the PGOH results.
Fig. 7.
Same as
Fig. 6 , but for the
P
22
/ P
11
. 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
Q ext , edge
¼ f e
ð kL Þ
2 = 3
,
Q abs , edge
¼ f a
ð kL Þ
2 = 3
,
ð
ð
45
46
Þ
Þ where the two factors f e and f a 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

1502 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 pro-
cesses 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
m m. We assume the incident light is x -polarized, the lidar backscattering cross section is defined as, s b
¼
¼
4 p k 2 h
9 S
2
ð 180 3 , 0 3
Þ 9
2
þ 9 S
4
ð 180 3 , 0 3
Þ 9
2 i
4 p k 2
½ P
11
ð 180
3
, 0
3
Þþ P
12
ð 180
3
, 0
3
Þ( ð 47 Þ
The backscattering efficiency is defined by Q b
¼ where G is the projected area of the particle. Note, P
11 s b
/ G , and
P
12 are not normalized, and P
12 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, D p exp( ik d p ,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 5 1 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 5 1 angle with the six-fold symmetry axis, the scattering angle associated with the reflected beam in the CGOM is 170 1 . 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 0 1 angle (left) and 5 1 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
m m, 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
twice that at 1.064
m m wavelength is m m. 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] ,
n ð D Þ ¼ N
0
D g exp ð$ g D = D m
Þ , ð 48 Þ where N
0 is the total number of plate particles in a unit volume, D is the width of the plate, and D m is the modal size. Given a size distribution, the effective size of the ensemble of particles is defined
[48 , 49]
,
D e
¼
3
2
R d
2
R d
1 d
2 d
1
V ð D Þ n ð D Þ dD
A ð D Þ n ð D Þ dD
: ð 49 Þ
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
m m.
N
0 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

1506 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 x
1 and x
2 between
0 and 1 given by,
% x
1 x
2
, x
1 q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 $ð x
2
Þ
2
, q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 $ð x
1
Þ
2
&
: ð 50 Þ
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 60 1 vertex angles, and the 22 1 halo can be observed. The 154 1 scattering maximum is reduced for the two irregular hexagonal columns. The
154 1 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 154 1 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 N00014-06-1-
0069. Bryan Baum acknowledges support through NASA
ROSES grant NNX08AF81G.
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