light scattering by absorbing hexagonal ice crystals in cirrus clouds

Light scattering by absorbinghexagonal ice crystals in cirrus clouds

Jianyun Zhang and Lisheng Xu

An improved ray-optics theory for single scattering and polarization of hexagonal columns and platesrandomly oriented in space has been developed by considering absorption and by using the Chebyshevsolution for diffraction integrals. The vector-tracing method and statistics technique of randomsampling are employed. The equivalent forms of Snell’s law and Fresnel formulas for absorbing icecrystals are derived, and two equivalent optical constants,m8 andm9, are obtained. Comparison is madeof the computed results of our model and the Takano and Liou model for asymmetry factors,single-scattering albedos, and scattering phase matrix elements. Some characteristics of our model arediscussed, and these analyses demonstrate that our ray-optics model is practical and much improved.Key words: Scattering, absorption, diffraction integral, Chebyshev solution, vector tracing, random

sampling, hexagonal ice crystals.

1. Introduction

Light scattering of nonspherical particles has becomea significant focus of activity in optics, especially sincethe 1980’s. Theoretically, to solve the scatteringproblem we are faced with the vector partial differen-tial equation of irregular boundary conditions. Solu-tions were obtained recently by performing extensivetheoretical studies and numerical calculations; see,for example, Mishchenko1 and Mishchenko et al.2Furthermore, the ray-optics theory has been used tostudy scattering by hexagonal ice crystals.3–12 Ab-sorption by ice crystals has been noted and consideredby Takano and Liou6 and by Yang et al.,12 but thisproblem still has not been resolved.We extend the ray-optics theory by considering ice

crystal absorption and by using an improved Cheby-shev solution for the diffraction integral. In addi-tion, to optimize the computations we used thevector-tracing technique 1VTT2 and the statisticsmethod of random sampling.10 Two crystal forms,i.e., hexagonal column and plate, randomly orientedin space, are assumed. We demonstrate some charac-teristics in our improved model for the ray-opticstheory.

The authors are with the Atmospheric Radiation and SatelliteRemote Sensing Laboratory, Chengdu Meteorological College,Chengdu, Sichuan 610041, China.Received 26 April 1994; revised manuscript received 27 March

1995.0003-6935@95@255867-08$06.00@0.

r 1995 Optical Society of America.

2. Fundamental Theory

Let OXYZ be the particle coordinate system that isattached to the particle and let OX 8Y 8Z 8 be the raycoordinate system so that the OZ 8 axis is along theincident direction. Thus, the relative orientation ofthese two Cartesian coordinate systems indicates theorientation of the particle with respect to the incidentray.4In the ray-optics theory, light scattering from par-

ticles may be treated approximately as a combinationof classical diffraction and geometric reflection andtransmission. To consider the absorption of particlestheoretically, the law of reflection and refraction1Snell’s law2 and Fresnel formulas need to be reformu-lated.As shown in Fig. 1, we define a set of orthonormal

vectors, ai,r,t, b1, and ei,r,t, in which ei,r,t denotes theincident, reflective, and transmission directions, re-spectively, and b1 is perpendicular to the plane ofincidence. The ai,r,t, b1, and ei,r,t, form a right-handed system, and n is the outer normal vector ofpoint P on the surface.

A. Reflection and Refraction of Absorbing Scatterers

1. Plane Wave Propagation from an Optically LessDense Medium into an Optically Denser MediumAccording to Born andWolf,13

sin ui 5 m sin ut, 112

where m 5 m11 1 ika2 is the complex index of refrac-

1 September 1995 @ Vol. 34, No. 25 @ APPLIED OPTICS 5867

tion, and ui is the incident angle; ut is complex and nolonger has the simple meaning of a refractive angle.Let the plane of incidence be the X–Z plane. The

space-dependent part of the phase of the wave in thecrystal is given by k3r · s1t24, where k is the propagationconstant of the wave in the crystal and s1t2 denotes aunit vector in the direction of propagation of thetransmitted wave. It can be shown that

sin ut 5sin ui

3sin2 ui 1 m2q21cos g 2 ka sin g2241@2, 122

where ut is the real angle and q and g are real

Fig. 1. Reflection and refraction of a plane wave in the VTT.Subscript n denotes the number of reflections or refractions.

numbers given by

q 5 511 21 2 ka2

m211 1 ka222sin2 ui2

2

1 32ka

m211 1 ka222sin2 ui4

2

61@4

, 132

g 51

2arctan3 2ka sin2 ui

m211 1 ka222 2 11 2 ka22sin2 ui4 . 142

Let

m8 5 3sin2 ui 1 m2q21cos g 2 ka sin g2241@2. 152

5868 APPLIED OPTICS @ Vol. 34, No. 25 @ 1 September 1995

Then

sin ui 5 m8 sin ut, 162

where all the quantities are real numbers, m8 can bereferred to as an equivalent optical constant, and Eq.162 still has the form of Snell’s law. When the opticalconstant of ice crystal is complex, the reflection andtransmission coefficients of Fresnel formulas are alsocomplex.For TE polarization let

Rb 5 RR' 1 iRI', Tb 5 TR' 1 iTI', 172

from which the real and imaginary parts can bederived as

RR' 5cos2 ui 2 u2 2 v2

1cos ui 1 u22 1 v2, RI' 5

22v cos ui

1cos ui 1 u22 1 v2,

182

TR' 52 cos ui1cos ui 1 u2

1cos ui 1 u22 1 v2, TI' 5

22v cos ui

1cos ui 1 u22 1 v2.

192

For TM polarization let

Ra 5 RR 1 iRI, Ta5 TR 1 iTI, 1102

where

RR 5m411 1 ka222cos2 ui 2 1u2 1 v22

3m211 2 ka22cos ui 1 u42 1 12m2ka cos ui 1 v22,

RI 52m2 cos ui32kau 2 11 2 ka22v4

3m211 2 ka22cos ui 1 u42 1 12m2ka cos ui 1 v22, 1112

TR 52m2 cos ui3m211 1 ka222cos ui 1 11 2 ka22u 1 2kav4

3m211 2 ka22cos ui 1 u42 1 12m2ka cos ui 1 v22,

TI 52m2 cos ui32kau 2 11 2 ka22v4

3m211 2 ka22cos ui1 u42 1 12m2ka cos ui 1 v22. 1122

In Eqs. 182, 192, 1112, and 1122, we have

u 5 C1@2Am211 2 ka22 2 sin2 ui

1 53m211 2 ka22 2 sin2 ui42 1 4m4ka261@2BD1@2,

v 5 C21@2Am211 2 ka22 2 sin2 ui

2 53m2112 ka222 sin2 ui42 1 4m4ka261@2BD1@2. 1132

2. Plane Wave Propagation from an OpticallyDenser Medium into an Optically LessDense MediumAccording to Snell’s law plane wave propagation froman optically denser medium into an optically less

dense mediummay be written as

sin ut 5 m sin ui, 1142

where ut is a real number and ui is complex. What isof concern for the ray-tracing method is angle ui 1realnumber2.As in Subsection 2.A.1., when ka fi 0, we can deduce

that

sin ut

sin ui5

s1@2 sin ui

3211 2 sin2 ui241@2

, 1152

where

s 5 2m211 2 ka2211 2 1@sin2 ui2

1 3m411 2 ka22211 2 1@sin2 ui22

1 4m4ka211@sin 4 ui 2 1@sin2 ui241@2.

The right-hand side of Eq. 1152 can be referred to asm9. Obviously, when the medium is not completelytransparent, i.e., when there is absorption, the rela-tionship between the real angle of incidence and thereal angle of refraction is considerably more complex.When total reflection does not occur, the Fresnelformulas can be given as follows.For TE polarization, the real and imaginary parts

of Eq. 172were derived as

RR' 5u2 1 v2 2 cos2 ut

1cos ut 1 u22 1 v2, RI' 5

2v cos ut

1cos ut 1 u22 1 v2,

1162

TR' 521u2 1 v2 1u cos ut2

1cos ut 1u22 1 v2, TI' 5

2v cos ut

1cos ut 1u22 1 v2.

1172

For TM polarization, according to Eq. 1102we have

RRE 5u2 1 v2 2 m411 1 ka222cos2 ut

3m211 2 ka22cos ut 1 u42 1 12m2ka cos ut 1 v22,

RIE 52m2 cos ut311 2 ka22v 2 2uka4


1182

TRE 52m3u2 1 v2 1 m211 1 ka221u 1 kav2cos ut4


TIE 52m3ka1u2 1 v22 1 m211 1 ka221v 2 kau2cos ut4

3m211 2 ka22cos ut 1 u42 1 12m2ka cos ut 1 v22.

1192

In Eqs. 1162–1192 there are formulas for u and v that aresimilar to those in Eq. 1132 except that we substitutedut for ui. When total reflection takes place, theFresnel formula for TE polarization is

Rb 5 RR' 1 iRI' 1202

where

RR' 5u2 1 v2 1 1 2 sin2 ut

3v 1 1sin2 ut 2 121@242 1 u2,

RI' 522u1sin2 ut 2 12

3v 1 1sin2 ut 2 121@242 1 u2. 1212

For TM polarization the Fresnel formula is

Ra 5 RRE 1 iRIE, 1222

where

RRE 5u2 1 v2 2 1sin2 ut 2 1234m4ka2 1 m411 2 ka2224

3m211 2 ka221sin2 ut 2 121@2 1 v42 1 3u 2 2m2ka1sin2 ut 2 121@242,

RIE 522m21sin2 ut 2 12311 2 ka22u 1 2kav4

3m211 2 ka221sin2 ut 2 121@2 1 v42 1 3u 2 2m2ka1sin2 ut 2 121@242. 1232

So far we have introduced two equivalent opticalconstants, m8 and m9. The former corresponds toplane wave propagation from an optically less densemedium into an optically denser medium, whereasthe latter corresponds to the opposite case. Equa-tion 152 shows that m8 depends not only on thequantities that specify the medium, but also on angleof incidence ui. It is of interest to see in Fig. 21a2 that,when the imaginary part of optical constants ka do notvanish, m8 increases with an increase in ui. More-over, the larger the imaginary part ka, the moremarked the change of m8 with ui. When ui 5 0°, m8

equals the real part of an optical constant. Figure21b2 shows the change ofm9 with ui. Obviously, whenui . 80°, m9 increases sharply, and m9 is moresensitive to the imaginary part ka thanm8.

B. Vector-Tracing Analyses

1. External Reflection and First Refraction 1n 5 12Following Cai and Yang,10 we performed vector-tracing analyses of the absorption of ice crystals.To use the VTT we must first relate the set of


orthonormal vectors shown Fig. 1 to the OX 8Y 8Z 8

system. Through this coordinate transformation, theelectric field vector 1EFV2 is given by

Eab1i 5 P1Ex8y8, Eab1

r 5 R1Eab1i,

Eab1t 5 T1Eab1

i, 1242

where

Eab1i,r,t 5 3

Ea1i,r, t

Eb1i,r, t4 ,

Ex8y8 5 3Ex8

Ey84 , P1 5 31b1 · ey82 21b1 · ex82

1b1 · ex82 1b1 · ey82 4 ,

R1 5 3Ra1

0

0 Rb14 , T1 5 3

Ta10

0 Tb14 . 1252

In Eqs. 1252 the superscripts i, r, t denote the EFV’sthat correspond to incidence, reflection, and transmis-sion, respectively.

Fig. 2. Variations of equivalent optical constants m8 and m9 withthe angle of incidence ui. Plane wave propagation from 1a2 anoptically less dense medium to an optically denser one, 1b2 anoptically denser medium to an optically less dense one.


Now for ice crystal scatter and a two-dimensionalrotation transformation, we can represent the exter-nal reflection and incident EFV’s as

E1s 5Q1Eab1

r 5Q1R1P1S1Elr1i, Ex8y8 5S1Elr1

i, 1262

where

E1s 5 3El1

s

Er1s4 , Elr1

i 5 3Eli

Eri4 ,

Q1 5 3 21u1s · b12 1u1

s · ar12

1u1s · ar12 1u1

s · b124 ,

S1 5 321ey8 · u1s2 1ex8 · u1

s2

1ex8 · u1s2 1ey8 · u1

s24 ,

u1s 5

1ez8 3 er12

sin u1

,

where u1s is the unit vector perpendicular to the

scattering plane, and scattering angle u1 satisfiescos u1 5 1ez8 · er12. When sin u1 5 0, u1

s 5 ex8 is used.The contribution of external reflection 1u1, w12 to the

scattering field averaged over the solid angle unitDv1u1, w12 is given by

E1m1u1, w12 5

1

r 1DSm

Dv 21@2

E1s, 1272

A1m 5

1

r 1DSm

Dv 21@2

Q1R1P1S1, 1282

where DSm is themth cross-sectional unit that the rayis incident upon and represents the energy carried bythe ray. A1

m is the amplitude function that corre-sponds to the scattering field.

2. One Internal Reflection and TwoRefractions (n 5 2)An internally reflected EFV can be written as

E2r 5 R2P2T1P1Ex8y8 exp122pkaDd1@l2. 1292

The contribution of the two refractions to the ampli-tude function of the scattering field within Dv1u2, w22can be obtained as

A2m 5

1

r 1DSm

Dv 21@2

Q2T2P2T1P1S2 exp122pkaDd1@l2,

1302

whereDd1 denotes the ray propagation distancewithinthe crystal and l is the wavelength of the medium.Note that, to transformOX 8Y 8Z 8 toai1b1ei1 , Eqs. 162

and 152 must be used; for one internal reflection andtwo reflections, Eq. 1152must be employed.For n $ 3, the recurrence formulas for amplitude

functions of the scattering field can be written as

Anm 5

1

r 3DSm

Dv1un, wn241@2

QnTnPnUn21Sn

3 exp122pkaDd@l2 1312

with

Un21 5 RnPnUn22, U1 5 T1R1,

U0 5 R1@T1, n 5 1, 2, . . . ,

where Dd denotes the total propagation distance ofthe wave within the crystal.It is worth pointing out that, with the ray-tracing

technique,4 the surface of a particle was divided intomany parts and the rays impinged upon each part ofthe surface. Thus, the numerical treatments weretedious and the computations were expensive. Withthe VTT, however, the rays impinge upon the particlesurface at random, so that the numerical computa-tions were simpler and more easily accessible thanthose obtained with the ray-tracing technique.

C. Diffraction Contribution

Particles much larger than the incident wavelengthalso scatter light by means of diffraction. In the farfield, the diffracted component of the scattered lightcan be approximated by Fraunhofer diffraction, andthe wave disturbance of the light at an arbitrary pointP can be expressed as

up1u, w2 5 2iuorl ee

B8

exp12ikro2dx8dy8, 1322

where uo is the disturbance in the original wave front.For the hexagonal crystal, we have

up1u, w2 5 2iuorl

ip1u, w2exp12ikro2,

ip1u, w2 5 eeB8

exp32ik1x8 cos wp 1 y8 sin wp2

3 sin up4dx8dy8, 1332

where r and ro denote the distances from point P topoint P8 1x8, y82 on projection area B8 and origin 0,respectively; see Fig. 4 in Ref. 4.In Eqs. 1332, the integral is a rapidly oscillating

function, which is the main problem encountered forthe computation of the diffraction integral. Re-cently, Toker et al.14 presented an approach based on adifferential equation that satisfied the diffractionintegral as a function of an off-axis coordinate, and itssolution was obtained by expansion of the Chebyshevpolynomials. In this study we present anothermethod to solve the diffraction integral.El-gendi15 developed a technique based on the

Clenshaw and Curtis quadrature scheme for approxi-mating the integral e

21x8 f 1x2dx 121 # x8 # 12, from

which we have

e21

1

f 1x2dx 5 os50

N

bNsfs. 1342

It is easy to verify that forN even we obtain

bNs 54

N oj50

N@2

9

1

1 2 4 j2cos

2 jps

N,

s 5 1, 2, . . . , N 2 1, 1352

bNO 5 bNN 51

N2 2 1,

fs 5 f 12 cossp

N 2 , s 5 0, 1, . . . , N. 1362

In Eq. 1352 the summation symbol with double primesdenotes a sum with the first and last terms halved.For limits 3a, b4 instead of just 321, 14, the computa-tions can be changed as follows:

x 5xs 2 1@21b 1 a2

1@21b 2 a2, 1372

xs 5 b@2311 cos1sp@N241a@2312 cos1sp@N24,

s 5 0, 1, . . . , N, 1382

ea

b

f 1x2dx 5 1b 2 a2@2 e21

11

f1x2dx 5 os50

N

bNsfs1x2, 1392

where bNs is the same as in Eq. 1352, and

fs1x2 5 f 3xs 2 1@21b 1 a2

1@21b 2 a2 4 . 1402

This numerical procedure is employed to solve thediffraction integral in Eqs. 1332, for which the diffrac-tion contribution is16

AF 5 2i

rlexp12ikro 2ip1u, w231 0

0 1 4 . 1412

It is worth pointing out that the El-gendi method isin fact a Chebyshev solution of the integral, but herewe calculated the diffraction integral at point xs of Eq.1382 instead of using the Chebyshev coefficients. ThebNs coefficients are easy to calculate and to store in thefiles for different values of N. Thus, this methodprovides an economic numerical procedure that isconvenient and extremely efficient for the solution ofdiffraction integrals.

D. Scattering Phase Matrix for a Random OrientedParticle System

The contribution of ray beams to the elements of theStokes transformation matrix Fjk 1see van de Hulst162can be obtained from matrix A, as was done by Cai


and Yang,10 giving

Fjk1u, w, up, wp2 5 om

on51

N

djk1u 2 un, w 2 wn2 Fjk,nm

3 1un, wn, p, wp2 1 djk Ff1u, w, up, wp2,

1422

where up, wp denote the angles of particle orientation.The first and second terms on the right-hand side ofEq. 1422 denote the contributions of reflection, refrac-tion, and diffraction. The normalized scatteringphase matrix is given by

P1u, w, up, wp2 54pra2

ssF 1u, w, up, wp2, 1432

where ss is the scattering cross section and ra is theparticle size. After taking the average of particleorientation 1up, wp2 and accounting for the symmetry ofw, one can obtain the phase matrix that relates to onlyscattering angle u.As mentioned in Subsection 2.B., a ray that im-

pinges upon the particle at random is assumed.Thus, for the numerical computations we used thestatistical technique of random sampling3,10 and aMonte Carlo method for the average of the particleorientation.

3. Computation Results and Discussions

Table 1 lists a comparison of our 1ZX2 results withthose of Takano and Liou 1TL2 for the single-scattering albedo 11 2 v02 and the asymmetry factor 1g2of hexagonal ice crystals at l 5 0.55 µm 1negligibleabsorption2 and l 5 2.2 µm 1moderate absorption2.For l 5 0.55 µm, the g values of our model are closeto those of the TL model, no matter what size theparticle. For l 5 2.2 µm, however, the g values ofour model were slightly less than those of the TLmodel, and the differences are especially evident forlarge particles. In general, the values of 1 2 v0 forour model at 2.2 µm are somewhat larger than those

Table 1. Comparison of the Single-Scattering Albedo and the AsymmetryFactor for Hexagonal Ice Crystals with Different Models at

l 5 0.55 and 2.20 mm

Particle SizeL@2a

1µm@µm2la

1µm2

1 2 v0 g

TLModel

ZXModel

TLModel

ZXModel

20@20 0.55 0 0 0.7704 0.77132.20 0.0409 0.0452 0.8185 0.8157

50@40 0.55 0 0 0.7780 0.78182.20 0.0818 0.0900 0.8416 0.8300

120@60 0.55 0 0 0.8155 0.81322.20 0.1246 0.1402 0.8829 0.8640

300@100 0.55 0 0 0.8429 0.83792.20 0.1871 0.2103 0.9165 0.8827

750@160 0.55 0 0 0.8592 0.85652.20 0.2503 0.2277 0.9380 0.8987

aThe refractive indices of ice are from Ref. 17.


for the TL model. The comparison of both modelsshows that the phasematrix elementsP22@P11,P33@P11,P44@P11, P43@P11 of our model agree well with those ofthe TLmodel, even though there are some differencesbetween the two, which occur mainly in the largescattering angles, i.e., u . 120° 1the figures wereomitted to save space2.The computed patterns for the degree of linear

polarization are depicted in Fig. 31a2 for plate crystalsand in Fig. 31b2 for columnar crystals. When L@2a 50.1 3Fig. 31a24, there are some noticeable differences inlinear polarization between the two models in the60° , u , 150° region; when L@2a 5 0.2, however, thedifferences become small. Figure 31b2 shows that thelinear polarizations of L@2a 5 2.5 and 5.0 for the twomodels are in good agreement. Therefore, the differ-ences in linear polarization between the two modelsfor columnar crystals are smaller than those for platecrystals.This analysis demonstrates that, when the wave-

length of an incident wave is short, the computedscattering phase matrix elements made with ourmodel are generally close to those of the TL model,with a few minor differences. This is due mainly tothe use of different techniques for each model. TheVTT and statistical technique of random sampling

Fig. 3. Degree of linear polarization, 2P12@P11, for randomlyoriented ice crystals at l 5 0.55 µm for 1a2 plate crystals and 1b2columnar crystals.

are both used in our model but not in the TL model.In addition, the particle surface is uniformly dividedfor the TL model so that the beam of the ray has afinite width and the central line of the beam wasconsidered as the beam propagation direction, whichis a flow that does not exist in our random samplingmodel. Furthermore, the divided density in the TLmodel is independent of the energy received by thesurface.It is worth noting that statistical fluctuations in the

random sampling technique really exist. However,when the photon number N that impinges upon theparticle surface is greatly increased, the fluctuationsare reduced. In addition, when N $ Nc, which is acritical photon number impinging upon the particlesurface, the fluctuations are small and can be ignored1see Table 22. The critical numberNc depends mainlyon the incident wavelength, the imaginary part of theoptical constant of ice, and the particle size. Asshown in Table 2, Nc 5 3 3 105. Table 2 also showsthat, for l 5 2.2 µm, if the imaginary part mi wereomitted, even though it is small noticeable errorswould be introduced. It is reasonable to expect that,when wavelength l is longer, with larger absorptionof ice, the effect of absorption on the computedsingle-scattering properties would bemore significant.Thus, the absorption and its perfect treatment in theray-optics theory is important.A detailed analysis and discussion of the improved

diffraction computations will be given in anotherpaper.18 Our improved ray-optics theory has beenused in the study of remote sensing of macrophysics,microphysics, and radiative properties of cirrusclouds,19–22 which further confirms the validity of ourmodel. Additional characteristics of the ray-opticstheory and its applications will be discussed in an-other paper.22

4. Conclusions

An improved ray-optics theory for single scatteringand polarization of ice crystals has been developed byconsidering absorption of ice crystals theoreticallyand by using the El-gendi method for solving thediffraction integral, which in fact is the Chebyshevsolution of the integral. In addition, the vector-tracing technique and statistics method of random

Table 2. Influence of Photon Number N that Impinges upon the ParticleSurface and Absorption of Ice Crystals on the Scattering Properties at

l 5 2.20 mm

ParticleSizeL@2a

1µm@µm2 N

mi 5

mr 5 7.9973

1.263 1024mr 5

1.263 mi 5 0

v0 g v0 g

300@100 3 3 105 0.7884 0.8828 1.0000 0.79853.5 3 105 0.7886 0.88284 3 105 0.7888 0.8829

750@160 3 3 105 0.7230 0.8988 1.0000 0.83003.5 3 105 0.7231 0.89894 3 105 0.7263 0.8990

sampling have been employed. The equivalent formsof Snell’s law and the Fresnel formulas for absorbingice particles have been derived, and two equivalentoptical constants, m8 and m9 were obtained, whichdepend not only on the quantities specified by themedium but also on the angle of incidence of the ray.Comparison of our model with the TL model shows

that the values of single-scattering albedo and theasymmetry factor that were calculated by both mod-els for l 5 0.55 µm are in close agreement. For l 52.2 µm, the asymmetry factors for our model areslightly less than those for the TL model. Thedifferences are especially evident for large particles.In addition, when the wavelength of the incidentwave is short, the computed scattering phase matrixelements, P22@P11, P33@P11, P44@P11, P43@P11, and2P12@P11, of our model are generally close to those ofthe TL model although some differences exist in thebackscattering directions because different treat-ments and techniques were used for each model.When the wavelength is longer and there is a largerabsorption of ice crystals, the effect of absorption onthe computed single-scattering properties is moresignificant.The statistical fluctuations in the random sampling

technique have been discussed, and it has been shownthat, when the photon number that impinges uponthe particle surface is equal to or larger than criticalnumber Nc, the fluctuations can be ignored. Finally,the El-gendi method15 provides a convenient andextremely efficient numerical technique to solve dif-fraction integrals.

The authors thank P. Yang and Q. Cai for theiruseful help and discussions. The authors also thankthe three anonymous reviewers for their commentsand English corrections. The original version of thisresearch was supported by the National NaturalScience Foundation of China under grant NSFC-4880212; the revision is supported by the Foundationunder grants NSFC-49375234 and 49275239.

References1. M. I. Mishchenko, ‘‘Light scattering by randomly oriented

axially symmetric particles,’’ J. Opt. Soc. Am. A 8, 871–882119912.

2. M. I. Mishchenko, L. D. Travis, and D. W. Mackowski, ‘‘T-matrix computations of light scattering by large nonsphericalparticles: recent advances,’’ in Passive Infrared Remote Sens-ing of Clouds and the Atmosphere, D. K. Lynch, ed. Proc. Soc.Photo-Opt. Instrum. Eng. 2309, 72–83 119942.

3. P. Wendling, R. Wendling, and H. K. Weickmann, ‘‘Scatteringof solar radiation by hexagonal ice crystals,’’ Appl. Opt. 18,2663–2671 119792.

4. Q. Cai and K. N. Liou, ‘‘Polarized light scattering by hexagonalice crystals: theory,’’ Appl. Opt. 21, 3569–3580 119822.

5. Y. Takano and K. Jayaweera, ‘‘Scattering phase matrix forhexagonal ice crystals computed from ray optics,’’ Appl. Opt.24, 3254–3263 119852.

6. Y. Takano and K. N. Liou, ‘‘Solar radiative transfer in cirrusclouds. Part I: Single scattering and optical properties ofhexagonal ice crystals,’’ J. Atmos. Sci. 46, 3–19 119892.

7. K. N. Liou, Y. Takano, S. C. Ou, A. J. Heymsfield, and W.Kreiss, ‘‘Infrared transmission through cirrus clouds: a radia-


tive model for target detection,’’ Appl. Opt. 29, 1886–1896119902.

8. M. Bottlinger and H. Umhauer, ‘‘Modeling of light scatteringby irregularly shaped particles using a ray-tracing method,’’Appl. Opt. 30, 4732–4738 119912.

9. E. A. Hovenac, ‘‘Calculation of far-field scattering from non-spherical particles using a geometrical optics approach,’’ Appl.Opt. 30, 4739–4746 119912.

10. Q. Cai and P. Yang, ‘‘Scattering phase matrices of ice crystalswith hexagonal prism and triangular pyramid form a vectorray tracing method,’’ Acta Meteorol. Sin. 5, 515–526 119912.

11. Y. Takano and K. N. Liou, ‘‘Infrared polarization signaturefrom cirrus clouds,’’ Appl. Opt. 31, 1916–1919 119922.

12. P. Yang, Q. Cai, X. Jiang, and Z. Han, ‘‘Single scattering of icecrystals in ice clouds with various ambient temperature,’’ Sci.Atmos. Sin. 17, 477–488 119932.

13. M. Born and E. Wolf, Principles of Optics 1Pergamon, NewYork, 19802, Chaps. 13 and 14.

14. G. Toker, A. Brunfeld and J. Shamir, ‘‘Diffraction of aperturedGaussian beams: solution by expansion of Chebyshev polyno-mials,’’ Appl. Opt. 32, 4706–4712 119932.

15. S. E. El-Gendi, ‘‘Chebyshev solutions of differential, integraland integro-differential equations,’’ Comput. J. 12, 282–287119692.


16. H. C. van de Hulst, Light Scattering by Small Particles 1Dover,NewYork, 19812, Chap. 5.

17. S. G. Warran, ‘‘Optical constants of ice from the ultraviolet tothe microwave,’’ Appl. Opt. 23, 1206–1225 119842.

18. L. Xu and J. Zhang, ‘‘Light scattering by randomly orientedhexagonal ice crystals: improved diffraction computationsand diffraction properties of ice crystals,’’ Optik, Int. J. LightElectron Opt. 119952, submitted to.

19. L. Xu and J. Zhang, ‘‘Simulation of retrieval of the base heightand thickness of cirrus clouds from satellite data,’’ J. Opt. Soc.Am.A 9, 1536–1546 119922.

20. L. Xu and J. Zhang, ‘‘Radiative properties and microphysics ofcirrus clouds in VAS IR channels,’’ in 1992 InternationalRadiation Symposium: Current Problems in AtmosphericRadiation, S. Keevallik and O. Karner, eds. 1Deepak Publish-ing, Hampton, Vir., 19932, pp. 310–314.

21. L. Xu, J. Zhang, G. Zhang, G. Luo, and H. Chen, ‘‘Radiativeproperties of cirrus and water clouds in visible, near-IR, andIR spectra and their applications in remote sensing,’’ inAtmospheric Propagation and Remote Sensing III,W. A. Floodand W. B. Miller, eds., Proc. Soc. Photo-Opt. Instrum. Eng.2222, 109–118 119942.

22. L. Xu and J. Zhang, ‘‘Simulation study of the remote sensing ofoptical and microphysical properties of cirrus clouds fromsatellite IR measurements,’’ Appl. Opt. 34, 2724–2736 119952.

light scattering by absorbing hexagonal ice crystals in cirrus clouds

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