Cirrus clouds consisting mainly of ice crystals playan important role in the radiative budget of theEarth. Consequently, their radiative properties needto be incorporated in up-to-date numerical modelsof climate prediction and change. These radiativeproperties have been calculated by many authorsassuming three-dimensional (3D) randomly orientedcrystals; a list of the proper references can be found,e.g., in [1,2]. However, experimental data obtainedfrom both the ground [3–5] and satellites [6–9]prove that the ice crystals often reveal theirpreferentially horizontal orientation. For such crys-tals, the radiative properties have been poorlystudied both experimentally [3–9] and theoretically[10–12].Light scattered by preferentially oriented ice crys-
tals reveals a specific property. Namely, it is dividedinto two qualitatively different parts, called the spec-ular and the diffuse components (e.g., [13]). There-fore it is convenient to introduce the terms of
specular and diffuse scattering by a crystal if thecrystal obeys a sharp probability distribution overits spatial orientations. To some extent, the specularscattering is formed by reflection of light from thosecrystal facets that are horizontally oriented, whilethe other facets create the diffuse component.Radiative properties of the specular component aresimply expressed through the microphysical para-meters of the crystals, that is, size, shape, and orien-tation. Such expressions are subjects of directscattering problems. And vice versa, from the pointof view of inverse scattering problems, this is thespecular component that is most informative for re-trieving the microphysical parameters from scat-tered light. The purpose of this paper is to presenta simple and rather general theory for the specularscattering that can be effectively used for both directand inverse scattering problems. It is worthwhile tonote that such a theory can be applied not only tolight scattering in the atmosphere but also to anyscattering media consisting of large, aligned parti-cles as compared to the incident wavelength.Examples of such media are special paints, biologicalstructures and tissues, and forest canopy.
Oscillations of orientation of the preferentially or-iented ice crystals near the horizon are called flutter.In the atmosphere, the flutter is usually confined to anarrow cone with a maximum flutter tilt T of about afew degrees. The value T for cirrus clouds was stu-died from the ground by use of scanning polarizationlidars [3–5]. Also, the value T was estimated from thedata obtained for the specular scattering componentby the satellite radiometer POLDER [6–8]. In thispaper, we show that not only the maximum fluttertilt T, but also the tilt probability density, can in prin-ciple be retrieved from certain specular scatteringpatterns.The paper is organized as follows. The concepts of
specular and diffuse scattering are introduced inSection 2. Then the problem of light scattering bya fluttering plate is solved analytically in Section 3,the fluttering plate being a good model for the spec-ular scattering. Section 4 presents display of crystalflutter in the geometric optics pattern of the specularscattering. And, finally, Section 5 presents numericalcalculations of the total specular scattering takinginto account both flutter and diffraction.
2. Specular and Diffuse Scattering by an Ice Crystal
Let us begin with some general notes. For any pro-blem of light scattering by a particle, it is commonto consider the scattered light in the wave zone ofthe particle, i.e., at distances z ≫ kb2, where k ¼2π=λ, λ being the wavelength, and b is a characteristicsize of the particle. Such a consideration is justifiedby the argument that the scattered light is usuallydetected just in the wave zone. However, if the par-ticle is much larger than the wavelength, it is oftenpreferable to study the scattered field in the nearzone z ≪ kb2. In the near zone, the building up ofthe scattered field becomes physically obvious, anda lot of analytical expressions that are valid at arbi-trary distances from the particle can be writtendirectly without any analytical or numerical calcula-tions (e.g., [14]). Then, if the near-zone field isobtained, the wave-zone scattered field is expressedthrough the near-zone field by well known wave pro-pagation equations resulting, e.g., in the Fraunhoferdiffraction patterns.Therefore, in the problem of light scattering by an
atmospheric ice crystal, it is expedient to considerthe near zone at first. A geometric optics descriptionof the electromagnetic field in the near zone becomesa rather exact approximation. It is no wonder thatpractically all numerical calculations of this problemare made by means of the ray-tracing method, corre-sponding exactly to geometric optics. Within the fra-mework of geometric optics, it is obvious that thenear-zone scattered light is produced by illuminatedfacets of the crystal. Every facet illuminated by aplane electromagnetic wave creates two plane-parallel beams with known transversal shapes andpropagation directions. One of them is the reflectedbeam, which becomes a component of the desirednear-zone scattered field. Another, i.e., refracted,
beam propagates in the crystal until it meets otherfacets. Then each of these facets produces new com-ponents of the near-zone field because of refraction,and so on. The total near-zone scattered field consistsof a lot of plane-parallel beams with different sizes,shapes, polarization, and propagation directions. It isworth noting that such a facet-tracing computer codefor calculating scattering matrices for ice crystalswas developed by us [15,16], and it has been success-fully employed.
Let us go to horizontally oriented crystals. We saythat a crystal has a perfect horizontal orientation ifone or more of its facets are always horizontal whenthe crystal rotates about the vertical. For such a ro-tating crystal, a horizontal facet reflects the incidentlight in the same direction, while nonhorizontalfacets smooth their reflected beams over changingpropagation directions. As a result, the scatteredlight averaged over the crystal rotation consists ofa bright dot and a fuzzy, wan pattern correspondingto the specular and diffuse scattering, respectively.
In practice, every horizontally oriented crystal isexposed to the flutter, or, equivalently, it revealsits preferentially horizontal orientation, where therotation axis oscillates slightly about the vertical.In this case both the bright dot of the specular com-ponent and a fuzzy pattern of the diffuse componentare expanded as compared to those of a perfectlyoriented crystal. It is well known that the preferen-tially horizontal orientation of ice crystals in theatmosphere is inherent mainly to platelike crystals.Here, two basic facets of the platelike crystals tend tobe horizontal. Columnlike crystals sometimes revealsuch a type of orientation, too. In particular, the so-called Parry orientation for a hexagonal column [17]means that the long axis of a column rotates in thehorizontal plane about the vertical, while two rectan-gular crystal facets are horizontal.
The split of the scattered electromagnetic field Einto the specular and diffuse components
E ¼ Es þ Ed ð1Þ
is strictly defined as follows. Among the horizontallyoriented facets of a crystal, it is easy to distinguish amain oriented facet that contributes directly to thespecular component by reflection of the incidentlight. In addition to this plane-parallel beam re-flected by the main oriented facet, there are a num-ber of other beams leaving the crystal in the samedirection at a given crystal orientation. For example,two basic facets of a platelike crystal create a numberof such beams of different transversal sizes becauseof multiple rereflection of light between these facets.Other facets can be involved in creation of beamswith the same propagation direction, too. By defini-tion, superposition of all the beams leaving the crys-tal in the same direction as the beam reflected fromthe main oriented facet is called the specular compo-nent Es. The rest of the total near-zone scattered fieldcorresponds to the diffuse scattering component Ed.
We should emphasize that the superposition ofEq. (1) is quite general. It retains its validity at arbi-trary distance from the particle including the wavezone. Indeed, at any distance from the crystal eachcomponent of Eq. (1) can be obtained independentlyof the other by certain integral transform of the near-zone fields. By definition, the split of Eq. (1) is pro-vided by appearance of the preferentially orientedfacets of crystals. Otherwise, for example, in the caseof 3D randomly oriented crystals, this split becomesmeaningless.
3. Scattering by a Fluttering Plate
The main physical regularities inherent to the spec-ular component can be obtained within a simplifiedproblem of light scattering by a fluttering plate.Consider the scattering scheme shown in Fig. 1. Herea thin plate with its orientation N is illuminated byan incident plane electromagnetic wave propagatingin the direction i. In the near zone, the scattered fieldcorresponding to the specular scattering is a plane-parallel beam propagating in the reflection directionr. These three unit vectors, r, i, and N, are connectedwith each other by the following equations:
r ¼ i − 2ði · NÞN; ð2Þ
N ¼ ði − rÞ=ji − rj: ð3ÞA dot between two vectors means their scalar pro-
duct. Equation (2) is easily obtained by decomposingthe incidence direction i on the longitudinal ijj ¼ði · NÞN and transversal i⊥ ¼ i − ijj components rela-tive to the normalN. As is well known, the transversecomponent in the reflection direction r is conserved,r⊥ ¼ i⊥, and the longitudinal component becomesopposite, rjj ¼ −ijj, which results in Eq. (2).Let us determine the shape of a plate. We consider
a plane coinciding with the plate of a given orienta-tion N and choose arbitrarily the Cartesian coordi-nates ρ on this plane. A shape of the plate isdescribed by the function SðρÞ taken as unity inside
and zero outside the plate, the plate area A beingequal to
A ¼Z
SðρÞdρ: ð4Þ
Then we determine the shape of the reflectedbeam. For this, we consider a plane perpendicularto the reflection direction r and determine its Carte-sian coordinates ρ0 as a projection of the previous co-ordinates ρ onto the new plane, the projection beingdescribed by the direct ρ0ðρÞ and inverse ρðρ0Þ func-tions. On the new plane, the transverse shape of thereflected plane-parallel beam is determined by thefunction
S0ðρ0Þ ¼ Sðρðρ0ÞÞ; ð5Þ
which differs from the initial shape SðρÞ by a com-pression of β ¼ ji · Nj along the direction nβ ¼ ½N − ðN ·rÞr�=jN − ðN · rÞrj on this plane. Here the transversearea of the beam A0 is equal to A0 ¼ ji · NjA.
At a large distance from the plate, the reflectedplane-parallel beam spreads about the reflection di-rection r because of diffraction. In the wave zone, thisbeam is transformed into a spherical diverging wave.Although polarization of the scattered field caneasily be taken into account, the scalar approxima-tion is used in this paper for brevity. Then the differ-ential cross section of the spherical scattered waveσðs; iÞ is generally presented as
σðs; iÞ ¼ Aji · NjRði · NÞFi;Nðs − rÞ; ð6Þ
where R is the reflection coefficient that is wellknown for any given polarization of incident light[18], and the dimensionless function Fi;N with thenormalization
RFi;Nðs − rÞds ¼ 1 describes diffrac-
tion. In particular, in the Fraunhofer approximation[18], the function Fi;N, is the 2D Fourier transformfrom the shape function of Eq. (5),
Fi;Nðs − rÞ ≈ jðk=2πÞZ
expð−ik½ðs − rÞ · ρ0�Þ
× S0ðρ0Þdρ0j2=A0; ð7Þ
where k ¼ 2π=λ is the wave number.We have to average the scattering differential
cross section given by Eq. (6) over a probability den-sity of plate orientations pðNÞ, where the density isnormalized as
RpðNÞdN ¼ 1. Denoting this averaging
by angular brackets, we get
hσðs; iÞi ¼ AZ
ji · NjRði · NÞFi;Nðs − rðNÞÞpðNÞdN
¼ AZ
ji · NðrÞjRði · NðrÞÞFi;NðrÞðs − rÞ
× pðNðrÞÞðDN=DrÞdr: ð8ÞFig. 1. Scattering by a thin plate.
In Eq. (8), the functions rðNÞ and NðrÞ, determinedby Eqs. (2) and (3), respectively, are introduced. Thefunction rðNÞ has the meaning of mapping the plateorientations N into the reflection directions r, and thefunction NðrÞ describes the inverse mapping. In thefinal integral of Eq. (8), the variableN is changed by r,which leads to the appearance of the JacobianDN=Dr. This Jacobian is easily calculated in spheri-cal coordinates, where the zenith angles are countedfrom the incidence direction i. We denote such zenithangles as θ1 and θ2 for the vectors N and r, respec-tively. Their azimuth angles φ1 and φ2 are accountedfor from the same arbitrary chosen meridian. Withthese variables, Eq. (2) corresponds to θ2 ¼ π þ 2θ1and φ2 ¼ φ1. As a result, we have
Substitution of this Jacobian into Eq. (8) cancelsthe scalar product that finally yields
hσðs; iÞi ¼ ðA=4ÞZ
Rði · NðrÞÞFi;NðrÞðs − rÞpðNðrÞÞdr:ð10Þ
To clarify a physical meaning of Eq. (10), let usignore diffraction by replacing the diffraction func-tion Fi;Nðs − rÞ by the Dirac delta function δðs − rÞ.In this case, Eq. (10) is reduced to the geometricoptics differential cross section
hγðs; iÞi ¼ ðA=4ÞRði · NðsÞÞpðNðsÞÞ: ð11Þ
Equation (11) reveals a remarkably simple physi-cal meaning. It means that the differential cross sec-tion of a fluttering plate within the framework ofgeometric optics at any scattering direction s is equal(with the trivial factor of AR=4) to the probabilitydensity of flutter at the corresponding plate orienta-tion NðsÞ.Now Eq. (11) allows us to write Eq. (10) as the
physically obvious 2D convolution:
hσðs; iÞi ¼Z
hγðr; iÞiFi;NðrÞðs − rÞdr: ð12Þ
Thus, the problem of light scattering by a flutter-ing plate is strictly reduced to a 2D convolution of twofunctions. One of the functions is responsible for flut-ter, and the other describes diffraction. If a scale ofthe diffraction function Fi;N is less than the scaleof the geometric optics pattern, the total specularscattering can be treated as a smoothing of a rathersharp geometric optics pattern by diffraction. In theopposite case, we can say that a diffraction pattern isbroadened by flutter.
4. Properties of the Geometric Optics PhaseFunctions
In nature, flutter of atmospheric ice crystals whenthey are preferentially oriented in the horizontalplane is confined to a narrow cone around the verticalorientation N0, with the maximum flutter tilt T. Nowlet us use for all vectors N, r, i, and s the sphericalcoordinates where zenith angles are counted fromthe vertical. The probability density of the tiltsbecomes an azimuthally symmetric function pðNÞ ¼pðθN ;φNÞ ¼ p0ðθNÞ, where the zenith tilt densityp0ðθNÞ is normalized as 2π
RT0 p0ðθNÞ sin θN dθN ¼ 1.
According to experimental data [3–9] the maximumzenith tilt T is usually of about a few degrees. In thiscase, the scattered light is also localized within asmall domain on the scattering direction sphere,as was recently discussed in our paper [19]. Let uscall the reflection direction appearing at perfect hor-izontal orientation r0 ¼ rðN0Þ as the center of thescattering domain. Then by changing incident direc-tion i, the center r0 moves on the scattering directionsphere according to the trivial reflection law ofEq. (2), while the shape of the scattering domain ischanged in a complicated manner as shown in Fig. 2.In particular, at the vertical incident direction θi ¼ 0°the scattering domain is a trivial circle with thezenith diameter of 4T. Then, for increasing incidentangle θi, the zenith diameter is kept the sameconstant of 4T, but the azimuth size decreases.Moreover, if θi ≥ ðπ=2Þ − T, a grazing incidence takesplace that deforms the scattering domain in a loopshown in Fig. 2, where a node corresponds to thegrazing incidence.
Instead of the differential cross section, certaindimensionless quantities are often used. Thus, theconventional bidirectional phase functions are thevalues
Pgðs; iÞds ¼ 1 and hSðiÞi is thescattering cross section at a given incident direction.
Fig. 2. Shapes and location of the scattering domains [T ¼ 15°,θi ¼ 0° (solid), 40° (dashed), and 85° (thin solid)]. The dots indicatethe domain centers. The right figure shows only shapes.
The value hSðiÞi has a simple physical meaning of theplate projection area ði · NÞA averaged over the orien-tation probability density pðNÞ. The dimensionlessquantity QðiÞ ¼ hSðiÞi=A can be called the scatteringefficiency. Thus, the scattering efficiency is equal tothe following integrals over either the plate orienta-tions N or scattering directions s:
QðiÞ ¼Z
ji · NjRði · NÞpðNÞdN
¼ ð1=4ÞZ
Rði · NðsÞÞpðNðsÞÞds: ð14Þ
As illustrations, Fig. 3 shows the scattering effi-ciencies for the simplest case of the uniform tiltdistributions p0ðθNÞ ¼ ½2πð1 − cosTÞ�−1 with the max-imum tilt T. Figure 3(a) presents the efficienciesQðiÞ ¼ Qðθi;φiÞ ¼ Q0ðθiÞ for a model case of absolutelyreflecting plates R ¼ 1. Here the functions Q0ðθiÞmonotonically decrease with the incidence angle θi.It is interesting to note that in this case the secondintegral of Eq. (14) means just the area of the scat-tering domain drawn in Fig. 2. Thus, Fig. 3(a) showsquantitatively the decrease of the scattering domainarea with incidence angle. For increasing maximumtilts T, the curves of Fig. 3(a) converge to the limitcase of Q0ðθiÞ ¼ ½, corresponding to random plateorientation ðT ¼ π=2Þ. A realistic magnitude of thereflection coefficient R essentially distorts thesefunctions. Figure 3(b) shows them calculated forthe coefficient R corresponding to an interface withthe refraction index of 1.31 and unpolarized incidentlight [18]. In Fig. 3(b), we see a sharp peak at θi ≈ 80°.This peak explains why sun pillars are seen in thesky usually at sunsets and sunrises.According to Eq. (11), the geometric optics values
hγðs; iÞi and Pgðs; iÞ are just a mapping of the tilt prob-ability density pðNÞ on the scattering directions s.This property gives us a unique opportunity toretrieve the tilt probability density from the specularpatterns. In particular, for the azimuthally sym-metric tilt density pðNÞ ¼ pðθN ;φNÞ ¼ p0ðθNÞ, themaximum tilt angle T can be found from the zenith
diameter Δθs ¼ 4T of the scattering domains shownin Fig. 2. The zenith tilt probability density p0 canbe directly retrieved, for example, from the simpleequation corresponding to Eq. (11):
where the scattered light is taken along the meridianφs ¼ φ0 crossing the scattering domain centerr0 ¼ ðθ0;φ0Þ.
5. Total Phase Functions
Shapes of the plates do not affect the geometric opticsequations obtained above, but the plate shapes re-veal themselves in the wave-zone-scattered fieldthrough diffraction as follows from the initialEq. (6). In general, a plate can rotate about a givennormal N resulting in an additional variable for astatistical averaging. In this paper, for simplicity,we ignore such a complication and assume thatthe plates are circles of the radius a. In this case,shapes of reflected beams introduced by Eq. (5)become ellipses where the circle is compressedby the factor β ¼ ji · Nj along the directionnβ ¼ ½N − ðN · rÞr�=jN − ðN · rÞrj. A diffraction patternfrom an ellipse can be described analytically bythe well known Fraunhofer approximation equationfor a circle [18]. As a result, we get
where J1 is the Bessel function. Here we denote aprojection of the vector s − r on the plane ρ0 orthogo-nal to the reflection direction r as the vector w and itslongitudinal and transverse components relative tothe direction nβ are denoted wjj and w⊥, respectively.The diffraction pattern Fi;N has its maximum ofβπa2=λ2 in the scattering direction s ¼ r, then itrapidly decreases approaching zero at the angular
Fig. 3. Scattering efficiencies for the uniform tilt distribution with the maximum tilt T : (a) absolutely reflecting interface R ¼ 1, and (b)the reflection coefficient R corresponding to an interface with the refraction index of 1.31 and unpolarized incident light.
radius of ρd ∼ λð2aβÞ ≪ 1. It is worthwhile to notethat the Fraunhofer approximation is not satisfac-tory if large diffraction angles are essential, i.e., ifλ=ð2aβÞ ≤ 1. Besides, the normalization integral forEq. (16) turns into unity if only the vector w is for-mally assumed to run over total infinite plane. Thisassumption is justified by the small domain of theessential diffraction angles ρd ≪ 1 for the functionFi;N (see, e.g., [14]). Thus, Eq. (16) is acceptable forpractical applications if only small diffraction anglesare of importance.According to the general equation (12), the total
phase function is spread about the center r0 ¼
rðN0Þ with the angle radius ρf ∼ 2T because of flutterand, independently, with the angle radius ρd ≈ λð2aβÞbecause of diffraction. The ratio
t ¼ ρf =ρd ð17Þ
determines a resulting shape of the total phase func-tion Pðs; iÞ. As illustration, Fig. 4 shows a schemewhere a contour of the geometric optics phase func-tion is drawn by a solid curve. Then every point in-side the contour should be spread by a diffractionfunction Fi;N of Eq. (16), drawn in gray scale in sev-eral points of the geometric optics scattering domain.It is essential that oblateness of the diffraction pat-terns and their orientation are different in all points,which makes difficulties for analytical calculations.Figures 5–8 show some examples of the total phasefunctions calculated numerically for both theuniform p0ðθNÞ ¼ ½2πð1 − cosTÞ�−1 and Gaussian tiltdistributions. Here the Gaussian distribution wasconstructed from the Gaussian functionexp½−θ2N=ð2σ2Þ� with σ2 ¼ 3° both by subtraction ofits magnitude at the boundary T ¼ 5° and by furthernormalization. It is interesting to note that, in addi-tion to trivial smoothing of contours of the geometricoptics phase function Pgðs; iÞ, a kind of cumulativeeffect for the total phase functions in the center r0 ¼rðN0Þ can appear, as seen in Figs. 5–8. This cumula-tive effect is explained by the fact illustrated in Fig. 4that the elongated diffraction functions Fi;N areoriented predominantly to the center r0.
6. Conclusions
A concept of specular scattering by ice crystals withpreferential orientations proves to be efficient forboth direct and inverse scattering problems of opticsof cirrus clouds. It is shown that the specular scatter-ing component is a 2D convolution of the geometricoptics scattering pattern and the diffraction function.Here the geometric optics pattern is determinedby only flutter parameters, while the diffraction
Fig. 4. Scheme illustrating the forming of the total scatteredlight. The solid curve shows a shape of the geometric opticsscattering domain (θi ¼ 60°, T ¼ 15°), and the half-tone picturesare the diffraction functions at λ=a ¼ 0:05 drawn for several pointsinside the domain.
Fig. 5. Total phase functions for (a) the uniform and (b) the Gaussian tilt distributions (T ¼ 5°, θi ¼ 50°, λ=a ¼ 0:027, and the reflectioncoefficient R corresponds to an interface with the refraction index of 1.31 and unpolarized incident light).
function depends only on the ratio of wavelength/(crystal size). The geometric optics specular patternis found analytically as a mapping of the probabilitydensity of plate orientations into scattering direc-tions. In the inverse scattering problems, the specu-lar patterns measured at several wavelengths of
incident light are informative to retrieve particlesizes, as was recently demonstrated in [13]. To re-trieve flutter parameters, a shortest wavelengthshould be used to measure the geometric optics pat-tern. In this case, not only the maximum flutter tilt,but also a probability density for the crystal tilts can
Fig. 6. Profiles of the phase functions of Fig. 5 along the meridian crossing the scattering domain center of (φ0 ¼ 0°, θ0 ¼ 130°). Thecorresponding geometric optics profiles are shown by dashed curves.
Fig. 7. Same as in Fig. 5 at the incident angle of 70°.
Fig. 8. Same as in Fig. 6 at the incident angle of 70°.
be retrieved. The results obtained in this paper arerather general, and they can also be applied to var-ious scattering media with aligned and large, as com-pared to incident wavelength, particles, such asspecial paints, biological media, and forest canopy.
This research is supported by the Russian Founda-tion for Basic Research under grant 09-05-00051 andby the International Association for the Promotion ofCooperation with Scientists through the New Inde-pendent States of the Former Soviet Union (INTAS)under grant 05-1000008-8024.
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