Karsten Schmidt,1,* Jochen Wauer,2 Tom Rother,1 and Thomas Trautmann3
1Remote Sensing Technology Institute, German Aerospace Center, Kalkhorstweg 53, D-17235 Neustrelitz, Germany2Institute for Meteorology, University of Leipzig, Stephanstrasse 3, D-04103 Leipzig, Germany
3Remote Sensing Technology Institute, German Aerospace Center, Münchner Straße 20, D-82234 Wessling, Germany
Scattering of electromagnetic waves on nonsphericaldielectric particles becomes of growing importance inremote sensing of the Earth’s atmosphere as well asin technical and medical diagnostics. Studying theinfluence of dustlike particles on our climate systemis only one example in this context, which is of par-ticular interest to us. For instance, Markowicz et al.[1] pointed out that ground-based interferometer ob-servations in the Cape Verde region clearly show thesignal of Saharan mineral dust particles in the atmo-spheric infrared window region. Since the typical sizeof airborne dust particles is in the range between2 μm and 10 μm, it can be expected that particle scat-tering is most sensitive to particle shape in the ther-mal infrared window region. The impact of the shapeof mineral dust particles on the transmitted and re-flected solar radiation has been recently investigatedduring the Saharan Mineral Dust Experiment
(SAMUM). Otto et al. [2] showed that modeled down-welling spectral irradiances are less sensitive to thenonsphericity of assumed spheroidal mineral dustparticles. But in contrast to this, prolate or oblatedust particles lead to a much stronger hemisphericbackscattering of solar radiation as compared to aparticle population consisting of surface-equivalentspheres with the same refractive indices.
Recent and upcoming instruments measure inwavelength regions on the order of the size of aero-sols so that deviations from the spherical particleshape may represent major uncertainties in the datainterpretation. For example, the balloon-borne Tera-hertz Limb Sounder (TELIS) instrument, operatingin the range 500GHz to 1:8THz, has the capability todetect particles from polar stratospheric clouds aswell as from cirrus clouds. Scattering and emissionfrom these particles interferes with the feature-richemission spectra caused by various atmospherictrace gases. Therefore, for cirrus particles having ef-fective radii ranging typically from 20 μm to several100 μm, shape and composition information will bekey parameters that determine the single scattering
properties of these particles in the far-infrared wave-length region.The instruments Superconducting Submillimeter-
wave Limb-emission Sounder (SMILES; see Kasaiet al. [3]) and First Light from the Far-Infrared Spec-troscopy of the Troposphere (FIRST; see Mlynczaket al. [4]) are able to study many aspects of theEarth’s climate, including water vapor feedbacks, cir-rus radiative properties, and the natural greenhouseeffect of the planet. SMILES is a new technology ofsubmillimeter-wave sensors for sounding the middleatmosphere. Its operation is planned for 2009 aboardthe International Space Station. SMILES will pro-vide the scientific community with global data notonly for trace gases but also for high-altitude clouds.Because the typical size of cirrus cloud particles isalso in the submillimeter region, scattering on parti-cles with nonspherical shapes will be a major ingre-dient for a physically consistent interpretation ofspectra measured by instruments such as TELIS,SMILES, and FIRST (see also Emde et al. [5]).Since scattering of electromagnetic waves on small
particles represents a fundamental physical process,it plays an important role not only in remote sensingbut also in many other fields of science and technol-ogy. Increasingly, modern optical measurement tech-niques have emerged that make it necessary to takenonspherical particles into account. However, thereare two particular aspects that make this necessitya complex task. First, the numerical effort is muchhigher than that for spherical particles within Mietheory. This effort strongly depends on the morphol-ogy of the particle and can be performed online onlyin very specific situations. Second, the convergenceprocedures of the existing approaches are muchmorecomplex as compared to spheres. To obtain reliableresults requires a detailed knowledge of the methodbehind a certain approach. Otherwise one can runinto a lot of pitfalls.Databases containing scattering quantities of non-
spherical particles, either precalculated or experi-mentally determined, release the user to deal withthese problems. They offer fast access to the data,which is important for, e.g., near real-time proces-sing. Of course, effort and care have to be spent inadvance when generating trustworthy data. Depend-ing on the aim of the database, different data require-ments result. For example, the database by Yanget al. [6] comprises complex ice particles to modelthe radiative properties of cirrus clouds in the near-through far-infrared spectral region. Dubovik et al.[7] considered spheroids to account for the aerosolparticle nonsphericity in remote sensing of desertdust. In order to cope with the wide size parameterregions, different exact and approximate methodswere used to fill both databases. The database byVolten et al. [8] contains experimental scattering ma-trix elements measured at 441:6nm and 632:8nm fora large collection of micrometer-sized mineral parti-cles in random orientation to provide the light scat-tering community with easily accessible data that
are useful for a variety of applications. Informationon the accuracy of the data are given wheneveravailable.
In the paper at hand we present a databasecontaining light scattering quantities of randomly or-iented dielctric spheroidal particles in the resonanceregion. Its primary aim is to provide benchmarks forelectromagnetic and light scattering computations ofspheroids. Thus the data were generated by using athoroughly tested T-matrix method implementation[9] briefly outlined in Section 2. In doing so greatstore was set by a defined accuracy. Due to restric-tions of our implementation [9] and due to the dataaccuracy aspired, we were bounded in the parameterrange (see Section 3). But within the region coveredthe database may also be applied as a fast tool to in-vestigate the scattering properties of nonsphericalparticles and to verify assumptions or statementsconcerning their scattering behavior. Of course, thedatabase can be used in certain applications, too, ifits parameters meet their requirements. The resolu-tion with respect to the size parameter and the scat-tering angles of the phase function, discussed indetail in Section 3, allows for a reliable interpolationof all data within a defined accuracy. A user interface,described in Section 4, was developed to facilitate thedata access. It also offers some additional functional-ities such as the interpolations between data and thecomputation of size-averaged scattering quantities.In Section 5, examples are provided to illustrate thecapabilities of the database and the handling of theuser interface in the different operation modes.
2. Method and Accuracy Aspects
Some years ago Wauer et al. [9] presented two soft-ware tools to deal with the plane wave scattering onnonspherical particles. One of these tools, the pro-gram Mieschka, has been developed to treat axisym-metric scatterers in spherical coordinates. It is basedon the T-matrix method of Waterman [10], which hasbeen substantially advanced through the analyticalorientation averaging and intensively investigatedand applied by Mishchenko et al. (e.g., [11,12]). Thisprogram has been used to compute the data in thedatabase. Within the T-matrix method all fields areexpanded in terms of finite series of vector sphericalwave functions RgΨτnl and Ψτnl, respectively. In par-ticular the incident (Einc) and the scattered (Escat)field read as follows:
EincðkrÞ ¼X2τ¼1
Xlcutl¼−lcut
Xncut
n¼jljaτnl · RgΨτnlðkrÞ; ð1Þ
EscatðkrÞ ¼X2τ¼1
Xlcutl¼−lcut
Xncut
n¼jljf τnl ·ΨτnlðkrÞ: ð2Þ
The expansion coefficients f τnl of the scattered fieldare calculated from the known incident field
expansion coefficients aτnl by use of the T-matrixrelation
f ¼ T · a: ð3ÞThe T-matrix itself is the product of two matrices,
T ¼ −Q−1 · RgQ; ð4Þthe elements of which are integrals over the scattererboundary surface considered and contain combina-tions of the vector spherical wave functions (formore details see, e.g., [9,10]). In order to computepolarimetric scattering quantities, Einc and Es are de-composed into vertically (v) and horizontally (h) po-larized parts with respect to the x–z plane as thereference plane within a Cartesian coordinate sys-tem ðx; y; zÞ. Considering an incident plane wave,
EincðkrÞ ¼ e0 · expðikzÞ; ð5Þpropagating along the z-direction, the unit vector e0characterizes its polarization state. In the case of avertically polarized field, e0 is defined to be ey. Onthe other hand, we set e0 ¼ ex for a horizontally po-larized incident field. k ¼ 2 π=λ denotes the wave-number of the incident plane wave with thewavelength λ. The scattered field polarization isappointed to be
Esv ¼ Es
ϕ; ð6Þ
Esh ¼ Es
θ; ð7Þ
in spherical coordinates ðr; θ;ϕÞ. The relation be-tween the polarized scattered field in the far fieldapproximation and the polarized incident field isgoverned by the scattering amplitude matrix F (see,e.g., [13,14]):
�Es
v
Esh
�¼ expðikrÞ
r·�Fvvðθ;ϕÞFvhðθ;ϕÞFhvðθ;ϕÞFhhðθ;ϕÞ
�·�Einc
v
Einch
�:
ð8ÞNote that the azimuthal angle ϕ is either 0° or 180° inthe reference plane. All scattering quantities of inter-est are computed on the basis of the scattering am-plitudes Fαβ, ðα; βÞ ¼ ðv;hÞ. If the particle is randomlyoriented then orientation-averaged quantities areconsidered to characterize its scattering properties.Let M be any scattering quantity. Its average hMioris defined according to
hMior : ¼1
8π2 ·Z
2π
0dϕp
Z π
0dθp sin θpZ
2π
0dψpMðϕp; θp;ψpÞ ð9Þ
with the Eulerian angles (ϕp; θp;ψp) describing theorientation of the particle with respect to the
incident plane wave. The orientation averagingcan be accomplished either numerically or analyti-cally according to, e.g., [11,12,15]. Note furthermorethat efficiencies are defined to be the correspondingcross sections divided by the average geometricalcross section q of the spheroid in random orientation.According to van de Hulst [13] the latter one is one-fourth of the surface area.
The following scattering quantities were consid-ered for the database:
– the orientation-averaged extinction efficiencyvia the optical theorem
σexteff ¼σextq
¼ 4πkq
· hImFααðθ¼ 0°;ϕ¼ 0°Þior; α¼ ðv;hÞ;ð10Þ
– the orientation-averaged (total) scatteringefficiency
σscaeff ¼ σscaq
¼ 1q
Z2π
0dϕ
Z π
0dθ sin θðhjFααðθ;ϕÞj2ior
þ hjFβαðθ;ϕÞj2iorÞ; α ≠ β; ð11Þ
– the orientation-averaged absorption efficiency
σabseff ¼ σabsq
¼ σexteff − σscaeff ; ð12Þ
– the orientation-averaged single scatteringalbedo
ω ¼ σscaσext ; ð13Þ
– the phase function according to van de Hulst[13]
– the orientation-averaged direct-polarized back-scattering efficiency
σbackdir;eff ¼1q· σbackdir ¼ k2
q· hjFααðθ ¼ 180°;ϕ ¼ 0°Þj2ior;
α ¼ ðv;hÞ; ð17Þ
– the orientation-averaged cross-polarized back-scattering efficiency
σbackx;eff ¼1q· σbackx ¼ k2
q· hjFαβðθ ¼ 180°;ϕ ¼ 0°Þj2ior;
α ≠ β: ð18ÞFrom the numerical point of view there are four
aspects that influence the accuracy of the computa-tional results. These are
– the determination of the truncation para-meters ncut and lcut of the expansions [Eqs. (1)and (2)],– the performance of the surface integrations in
the elements of the matrices Q and RgQ in Eq. (4),– the performance of the inversion of the matrix
Q in Eq. (4), and– the performance of the orientation averaging
procedure.
Detailed information about the procedures to dealwith the above points and some benchmarks are pro-vided in [9]. It should however be mentioned that thetruncation value ncut was determined for a givenspheroid by inspecting its differential scatteringcross sections at a plane wave incidence along the ro-tational axis. If a maximum relative error of 2% wasachieved at 80% of all scattering angles while ncutwas increased then ncut was fixed to the current va-lue. This convergence strategy follows an idea of Bar-ber andHill [16], and it has been approved inmany ofour applications, too. Once ncut was fixed lcut was de-termined by applying the above convergence strategyto an effective phase function averaged only over 9different angles of orientation. The surface integralsforming the elements of the matrices Q and RgQ inEq. (4) were computed with a maximum relative er-ror of 0.1%. Note finally that the program Mieschkahas been tested in many applications and comparedwith results obtained by other software packages.
3. Database Content and Parameter Range
The database contains the orientation-averagedscattering quantities, listed in Section 2, of spher-oids. Their geometry is described in Cartesian coor-dinates by �
xb
�2þ�yb
�2þ�za
�2¼ 1: ð19Þ
Here the z axis represents the rotational axis, anda and b denote the semi-axes. The data are stored interms of the aspect ratio av ¼ a=b and the size
parameter kreqvðvÞ of the volume-equivalent spherewith the radius reqvðvÞ ¼ ðab2Þ1=3. An aspect ratioav > 1 belongs to prolate spheroids, and av < 1 cor-responds to oblate ones. Table 1 summarizes the re-fractive indices n and the aspect ratios av for whichscattering data are present in the database. Thesevalues were chosen to fit into the needs of lidar mea-surements within SAMUM. The present resolutionswith respect to n and av do not allow for a reliableinterpolation of the scattering data. To meet the pre-defined accuracy requirements the size parameterkreqvðvÞ had to be restricted to the maximum value 40.
A. Resolution with Respect to the Size Parameter
Regarding the resolution with respect to the sizeparameter, ΔkreqvðvÞ ¼ 0:2 was chosen. This resolu-tion enables interpolations between adjacent datasets and was determined by means of the backscat-tering efficiency. The latter represents a differentialscattering quantity that is known to be very sensitiveto any change of the particle morphology (size, shape,and refractive index), in general. Moreover it is ofspecial importance in lidar measurements. Likeother scattering quantities it exhibits, for nonspheri-cal particles in random orientation, a much smootherbehavior as a function of the size parameter thanthat of spheres. Thus the investigations were doneon spheres as the worst case. The resolution obtainedin this case was also used for the spheroids.
Figure 1 shows an example of the backscatteringand the extinction efficiency of a sphere with a refrac-tive index n ¼ 1:33 as a function of the size para-meter at a resolution of ΔkreqvðvÞ ¼ 0:025. It is seenthat the backscattering efficiency has much strongeroscillations than the extinction efficiency, thusrevealing its sensitivity mentioned above. Thereforewe considered the backscattering efficiency fordetermining the size parameter resolution. Themorphology-dependent resonances are seen inboth plots. Since we do not intend to resolve themorphology-dependent resonances by interpolationwe focused on the size parameter range up to 12.5to determine ΔkreqvðvÞ. Figure 2 provides an exampleof these investigations. It shows the backscatteringefficiency of the sphere of Fig. 1 in the size parameterrange [7.5,12.5] at different resolutions (ΔkreqvðvÞ ¼0:6, 0.4 and 0.2). It turned out that a resolution ofΔkreqvðvÞ ¼ 0:2 is sufficient to reproduce the originalreference curve, generated with ΔkreqvðvÞ ¼ 0:025, byinterpolation. The maximum error of 23% seen inFig. 2(d) was only reached in the deepest troughs.In Section 5, Fig. 5, below, we will present thenormalized phase function pðθÞ of a randomly or-iented spheroid with n ¼ 1:8 and av ¼ 1:5 at a size
Table 1. Refractive Indices n and Aspect Ratios av Presentin the Database
parameter kreqvðvÞ ¼ 39:52. It is compared with aphase function obtained by a linear interpolationof the phase functions at kreqvðvÞ ¼ 39:4 andkreqvðvÞ ¼ 39:6. The agreement between both curvesimplies that the size parameter resolutionΔkreqvðvÞ ¼ 0:2, originally determined within the sizeparameter range up to 12.5, can be used throughout
in the chosen region up to 40. Note that the corre-sponding interpolation is performed in the user inter-face (Section 4).
B. Angular Resolution of the Phase Functions
Table 2 shows the angular resolution used to gener-ate the phase function data for the database. This re-solution was chosen again in such a way that areliable interpolation of the phase functions with re-spect to the scattering angle is possible. At first, thewhole size parameter range was divided into the tworegions (0,20] and (20,40] since the phase function ofany scatterer is in general smoother at lower sizeparameters than at higher ones. Then the angularresolution was investigated separately in each range.This allowed us to optimize the numerical effort be-cause the computational time depends on the chosenangular resolution that is due to our convergencestrategy. In both regions the resolution was deter-mined at the highest size parameter as the worstcase, i.e., at kreqvðvÞ ¼ 20 and kreqvðvÞ ¼ 40. Further-more the phase functions of spheres oscillate muchmore than those of randomly oriented spheroids.Thus the investigations were done on spheres again.The resolution obtained in this case was also takenfor the spheroids.
In view of different applications of the databasewhich may require higher resolutions in certain an-gular regions we also divided the scattering planeinto three regions, a forward scattering region ½0°;10°�, a backscattering region ½173°; 180°�, and a sidescattering region ½10°; 173°�. In this way forward orbackscattering peaks can be better resolved, forexample, which is important in transmission mea-surements or lidar applications. Examples of theinvestigations concerning the angular resolutionand interpolation of the phase function are providedin Figs. 3 and 4. The maximum errors were onlyreached in the deep troughs and amount to about16% and 20%, respectively. Note that the correspond-ing interpolation is again performed in the userinterface.
4. User Interface to the Database
All data in the database are arranged with respect totheir refractive indices and aspect ratios in a file sys-tem. Of course, they are accessible via the files in thedirectory tree. But this way is rather cumbersome, ingeneral, due to the number of data sets, about 59,000.Therefore a user interface was developed to facilitatethe data access. The user interface also providessome additional functionalities going beyond thesimple readout of data. These are consistency tests
Fig. 1. Extinction efficiency σexteff (top) and direct-polarized back-scattering efficiency σbackdir;eff (bottom) as a function of the volume-equivalent size parameter kreqvðvÞ for a spherical scatterer witha refractive index n ¼ 1:33. The resolution is ΔkreqvðvÞ ¼ 0:025.
0
200
400
600
800
σback
eff
200
400
600
800
σback
eff
7.5 10 12.5size parameter kreqv(v)
200
400
600
800
σback
eff
0.1
0.2
rela
tive
erro
r
(a) step size ∆kreqv(v)=0.6
(b) step size ∆kreqv(v)=0.4
(c) step size ∆kreqv(v)=0.2
for ∆kreqv(v)=0.2(d)
Fig. 2. Comparison of different resolutions of the backscatteringefficiency σbackdir;eff as a function of the volume-equivalent size para-meter kreqvðvÞ within the range [7.5,12.5] for a spherical scattererwith n ¼ 1:33. The solid curves in Figs. 2(a)–2(c) represent the re-ference with a size parameter resolution ΔkreqvðvÞ ¼ 0:025. Thecrosses are the points at which the computations have been per-formed for the different resolutions ΔkreqvðvÞ ¼ ð0:6; 0:4; 0:2Þ, andthe corresponding dashed curves are their cubic spline interpola-tions. The relative error of the interpolation of Fig. 2(c) is shown inFig. 2(d).
Table 2. Angular Resolution of the Phase Function Depending on theVolume-Equivalent Size Parameter kreqvðvÞ and the Region of the
of the users requests, interpolations between dataand the computation of size-averaged scatteringquantities. The latter two functionalities are accom-plished in different operation modes described below.They are intended for supporting a variety of appli-cations of the database such as in radiative transfersimulations or in lidar inversion routines.
A. First Operation Mode: Generating a Single Data Set
In this mode the user can search for one specificdata set of a randomly oriented spheroidal particle
characterized by the radius reqvðvÞ of the volume-equivalent sphere, the aspect ratio av, the real partand the imaginary part of the complex refractive in-dex n. Additionally to these parameters the fre-quency has to be fixed by the user so that thecorresponding volume-equivalent size parameterkreqvðvÞ can be computed internally. In general, thissize parameter does not match any size parameterfor which data are stored in the database. This isdue to the finite resolution of all scattering quantitieswith respect to kreqvðvÞ. But to allow for continuouslyvarying scattering data with respect to reqvðvÞ orkreqvðvÞ, a linear interpolation scheme is used.
If krl and kru are the neighboring size parametersof the chosen kreqvðvÞ with krl ≤ kreqvðvÞ ≤ kru and if weindicate the corresponding scattering quantitieswith the indices l (“lower”) and u (“upper”), respec-tively, then any total cross section σ ¼ ðσext; σscat;σabs; σbackdir ; σbackx Þ belonging to the size parameterkreqvðvÞ is linearly interpolated according to
σ ¼ wl · σl þwu · σu: ð20ÞHere the weighting functions wl and wu of the lin-
ear interpolation are given by
wl ¼kru − kreqvðvÞkru − krl
; ð21Þ
wu ¼ kreqvðvÞ − krlkru − krl
: ð22Þ
They take the “distance” of the desired data set tothe adjacent ones, which are present in the database,into account. Note that the corresponding efficienciesare simply obtained by dividing the cross sections[Eq. (20)] by the average geometrical cross sectionq. The single scattering albedo follows from the inter-polated scattering and extinction cross sections. Thephase function and the asymmetry parameter are in-terpolated according to
In this mode the user has also the possibility to al-ter the given angular resolution of the phase func-tion. If requested then a cubic spline interpolationis started, in general, to interpolate between the dis-crete data of the database.
The data set generated by the user interface in thismode contains the orientation-averaged scatteringquantities, listed in Section 2, of the chosen spheroid.Additionally the total cross sections σext, σscat, σabs,σbackdir , and σbackx are provided.
0 45 90 135 180scattering angle [degree]
10−2
10−1
100
101
102
103
phas
e fu
nctio
n
0.05
0.1
0.15
0.2
rela
tive
erro
r
(a)
(b)
Fig. 3. (a) Phase function pðθÞ of a sphere (n ¼ 1:6, kreqvðvÞ ¼ 20)at an angular resolution of 0:25° An approximation of this phasefunction is obtained if started from a lower resolution of 0:75° inthe forward and backward direction and 2:5° in the side scatteringdirection and interpolated via cubic splines to the higher resolu-tion of 0:25°. The resulting relative error is plotted in (b).
0 45 90 135 180scattering angle [degree]
10−2
10−1
100
101
102
103
phas
e fu
nctio
n
0.05
0.1
0.15
0.2
0.25
rela
tive
erro
r
(a)
(b)
Fig. 4. (a) Phase function pðθÞ of a sphere (n ¼ 1:6, kreqvðvÞ ¼ 40)at an angular resolution of 0:25°. An approximation of this phasefunction is obtained if started from a lower resolution of 0:5° in theforward and backward direction and 1:5° in the side scattering di-rection and interpolated via cubic splines to the higher resolutionof 0:25°. The resulting relative error is plotted in (b).
B. Second Operation Mode: Generating a Range of Data
In this mode scattering quantities for a certain para-meter range are generated. The quantities are storedin the respective files as a function of the effectiveradius reqvðvÞ, a function of the effective radius reqvðvÞ,and the aspect ratio av, or as a function of the effec-tive radius reqvðvÞ and the real part of the refractiveindex n. Alternatively reqvðvÞ can be replaced by thevolume-equivalent size parameter kreqvðvÞ. Moreoverthe user has to decide which orientation-averagedscattering quantity is provided. Possible choices are
– one of the efficiencies ðσexteff ; σscateff ; σabseff ;σbackdir;eff ; σbackx;eff Þ,– one of the cross sections ðσext; σscat; σabs; σbackdir ;
σbackx Þ,– the single scattering albedo ω,– the asymmetry parameter g,– the lidar ratio S ¼ σext=ðσbackdir þ σbackx Þ,– the direct-polarized backscattering coefficient
Note that they are taken directly from thedatabase and computed internally in the user in-terface, respectively, without any size parameterinterpolation.
C. Third Operation Mode: Generating an Averaged DataSet
In this mode the user interface allows to calculate thesize and aspect ratio average of all scattering quan-tities of the first mode (Subsection 4.A). In doing sothe factorized distribution function
NðreqvðvÞ;avÞ ¼ NrðreqvðvÞÞ ·NavðavÞ ð25Þ
is used. Here NrðreqvðvÞÞ is one of the followingsize distribution functions predefined in the userinterface:
The parameters of the different size distributionfunctions have to be specified by the user on request.A discrete aspect ratio distribution functionsNavðavÞof the form
NavðavÞ ¼
8>>>><>>>>:
Nav1 ¼ Navð0:67Þ
Nav2 ¼ Navð0:77Þ
..
.
Nav7 ¼ Navð1:5Þ
ð29Þ
is assumed. Navi represents the number of particles
with the corresponding aspect ratio given in Table 1.The user has to provide these numbers on request.Any averaged scattering quantity X is then calcu-lated according to
hXir;av ¼1
hNir;avXi
Navi ·
ZNrðreqvðvÞÞ
· XðreqvðvÞ;avÞdreqvðvÞ ð30Þ
with
hNir;av ¼Xi
Navi ·
ZNrðreqvðvÞÞdreqvðvÞ: ð31Þ
The above integration is performed in applying anEulerian integration rule with a step size defined bythe user.
5. Examples
In the following we want to present some examples toillustrate the usage of the database in the differentoperation modes and its capabilities. Figure 5 showsa typical application of the first operation mode inSubsection 4.A. We are looking for the phase functionof a randomly oriented spheroid with n ¼ 1:8, av ¼1:5, and kreqvðvÞ ¼ 39:52. Since this is not presentin the database it is linearly interpolated by use ofEq. (23) between the phase functions belonging tothe size parameters kreqvðvÞ ¼ 39:4 and kreqvðvÞ ¼39:6, which exist in the database. The relative errorof the interpolated phase function to the one calcu-lated at the desired size parameter [in the presentexample kreqvðvÞ ¼ 39:52] is shown in Fig. 5(b). It isseen that the relative error does not exceed 3% inthe forward scattering direction up to a scatteringangle of about 50°. The maximum error amountsto about 15% and is obtained only in deepest troughof the phase function. Thus this example demon-strates two important things. First, let us recall thatthe size parameter resolution ΔkreqvðvÞ ¼ 0:2 in theunderlying database was originally determined byinvestigating the backscattering efficiencies ofspheres (see Subsection 3.A). Using this resolutionit is seen that it is also possible to linearly interpolatephase functions of randomly oriented spheroids with-in a sufficient accuracy for most applications. Second,the resolution was fixed within the size parameter
range up to 12.5 (see again Subsection 3.A). It turnsout that it can be used up to kreqvðvÞ ¼ 40.Figure 6 shows three examples belonging to the
second operation mode in Subsection 4.B. Theseare the backscattering efficiency σbackdir;eff þ σbackx;eff, theextinction efficiency σexteff , and the lidar ratio S of arandomly oriented oblate spheroid with av ¼ 0:67(dotted curves), a sphere (solid curves), and of aprolate spheroid with av ¼ 1:5 (dashed curves), all
having a refractive index n ¼ 1:5þ i · 0:05, as a func-tion of the volume-equivalent size parameter kreqvðvÞ.These quantities play an important role in lidar ex-periments. One aim of such experiments is to char-acterize cloud and aerosol particles in terms oftheir shape by distinguishing between sphericaland nonspherical ones (e.g., Sassen [17]). It is seenthat the extinction efficiency is not sensitive withrespect to the particle shapes chosen. In contrastto this the backscattering efficiency and the lidar ra-tio reveal a quite different behavior for the differentparticle shapes.
0 45 90 135 180scattering angle [degree]
10−2
10−1
100
101
102
103
phas
e fu
nctio
n
0.05
0.1
0.15
0.2
0.25
rela
tive
erro
r
(a)
(b)
Fig. 5. (a) Phase functions pðθÞ of a randomly oriented spheroidwith n ¼ 1:8, av ¼ 1:5, and kreqvðvÞ ¼ 39:52 (reqvðvÞ ¼ 2:23 μm) line-arly interpolated by use of Eq. (23) between the volume-equivalentsize parameters kreqvðvÞ ¼ 39:4 and kreqvðvÞ ¼ 39:6. The relative er-ror of this phase function to the one calculated at the desired sizeparameter kreqvðvÞ ¼ 39:52 is shown in (b).
Fig. 6. Backscattering efficiency σbackdir;eff þ σbackx;eff, extinction effi-ciency σexteff , and lidar ratio S of a randomly oriented oblate spheroidwith av ¼ 0:67 (dotted curves), a sphere (solid curves), and a pro-late spheroid with av ¼ 1:5 (dashed curves), all having a refractiveindex n ¼ 1:5þ i · 0:05, as a function of the volume-equivalent sizeparameter kreqvðvÞ.
Fig. 7. Monomodal log-normal size distribution function (27) withr1 ¼ 1:3 μm and σ1 ¼ 1:3.
Fig. 8. Phase functions pðθÞ of spheres (solid line) and randomlyoriented spheroids (dashed line) averaged over the particle size inusing the size distribution function of Fig. 7 at a wavelength λ ¼0:355 μm (volume-equivalent size parameter up to 40). All aspectratios of Table 1 with the sameweight are present in themixture ofthe spheroids. The scatterers have a refractive index n ¼ 1:6throughout.
In order to present an example of the third opera-tion mode in Subsection 4.C the monomodal log-normal size distribution function of Fig. 7 was usedto compute phase functions of mixtures of spheresand randomly oriented spheroids. In doing so all as-pect ratios of Table 1 with the same weight weretaken into account in the mixture of the spheroids.Furthermore all scatterers have a refractive indexn ¼ 1:6 at a wavelength λ ¼ 0:355 μm. Figure 8 showsboth normalized phase functions pðθÞ. One basic ef-fect of nonspherical particles—the enhanced side
scattering at about 120° as compared to spheres—can be clearly seen.
Another monomodal log-normal size distributionfunction (27) with r1 ¼ 1:3 μm and σ1 ¼ 1:2 and arough discretization of this distribution function toa size class distribution function are plotted in Fig. 9.
Fig. 9. Monomodal log-normal size distribution function (27) withr1 ¼ 1:3 μm and σ1 ¼ 1:2 (solid line) and a rough discretization ofthis distribution function to a size class distribution function(dashed line).
0 45 90 135 180scattering angle [degree]
10−2
10−1
100
101
102
103
phas
e fu
nctio
n
0.05
0.1
0.15
0.2
rela
tive
erro
r
(a)
(b)
av=1
Fig. 10. Phase functions pðθÞ of spheres averaged over the parti-cle size in using the monomodal log-normal size distribution func-tion of Fig. 9 [solid line in (a)] and the size class distributionfunction of Fig. 9 [dashed line in (a)]. The spheres have a refractiveindex n ¼ 1:5þ i · 0:001 at a wavelength λ ¼ 0:355 μm (volume-equivalent size parameter up to 40). In (b) the relative error ofthe phase function based on the size class distribution functionis shown.
0 45 90 135 180scattering angle [degree]
10−2
10−1
100
101
102
103
phas
e fu
nctio
n
0.05
0.1
0.15
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tive
erro
r
(a)
(b)
av=1.3
Fig. 11. Phase functions pðθÞ of randomly oriented spheroidsaveraged over the particle size in using the monomodal log-normalsize distribution function of Fig. 9 [solid line in (a)] and the sizeclass distribution function of Fig. 9 [dashed line in (a)]. The spher-oids have an aspect ratio av ¼ 1:3 and a refractive index n ¼ 1:5þi · 0:001 at a wavelength λ ¼ 0:355 μm (volume-equivalent sizeparameter up to 40). In (b) the relative error of the phase functionbased on the size class distribution function is shown.
0 45 90 135 180scattering angle [degree]
10−2
10−1
100
101
102
103
phas
e fu
nctio
n
0.05
0.1
0.15
0.2
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tive
erro
r
(a)
(b)
av=1.5
Fig. 12. Phase functions pðθÞ of randomly oriented spheroidsaveraged over the particle size in using the monomodal log-normalsize distribution function of Fig. 9 [solid line in (a)] and the sizeclass distribution function of Fig. 9 [dashed line in (a)]. The spher-oids have an aspect ratio av ¼ 1:5 and a refractive index n ¼ 1:5þi · 0:001 at a wavelength λ ¼ 0:355 μm (volume-equivalent sizeparameter up to 40). In Fig. 12(b) the relative error of the phasefunction based on the size class distribution function is shown.
Both functions were used to investigate the influenceof uncertainties in the knowledge of the size distribu-tion function for computing size-averaged phasefunctions. Figures 10–12, provide examples of theseinvestigations. The integrations over the size distri-butions were performed in applying an Eulerian rulewith a step size of 0:02 μm. It is seen that the max-imum error, when approximating the correspondingdistribution function (27) by a size class distribution
function, is about 18% in the case of sphere mixtures.The maximum error does not exceed 12% for prolatespheroid mixtures. Such an accuracy is sufficient formany applications. Thus the results obtained sup-port the customary practice to use experimentallydetermined size class distribution functions for com-puting the scattering behavior of particles. On theother hand, this means that the size distributionfunction can be deduced from phase function
Fig. 13. Backscattering depolarization σbackx;eff =ðσbackdir;eff þ σbackx;eff Þ as a function of the aspect ratio av and the volume-equivalent size parameterkreqvðvÞ for different refractive indices n.
Fig. 14. Asymmetry parameter g as a function of the aspect ratio av and the volume-equivalent size parameter kreqvðvÞ for differentrefractive indices n.
measurements only up to a certain degree when allother particle parameters are known.Figures 13 and 14 provide examples of scattering
quantities over wide parameter ranges. Figure 13shows the backscattering depolarization σbackx;eff =ðσbackdir;eff þ σbackx;eff Þ as a function of the aspect ratio avand the volume-equivalent size parameter kreqvðvÞfor different refractive indices n. The bluish regionsmark parameter sets with a lower depolarization,while the reddish ones designate larger depolariza-tion values. In general, randomly oriented prolatespheroids (av > 1) exhibit larger depolarizationsthan oblate ones (av < 1). In each tile, both regionsare clearly separated by the zero-depolarizationband originating from the spheres (av ¼ 1). It isfurthermore seen that, within the chosen parameterrange, the largest depolarization differences occurfor small real parts (ReðnÞ ¼ 1:33) and smallimaginary parts (ImðnÞ ¼ 0 − 0:05) of the refractiveindex. Otherwise spherical and spheroidal particlesbecome increasingly indistinguishable by means ofbackscattering depolarization measurements. Thesame holds also if the particles are small enough(kreqvðvÞ ≲ 2).Figure 14 shows the asymmetry parameter g as a
function of the aspect ratio av and the size parameterkreqvðvÞ for different refractive indices n. Large valuesof g, i.e., large differences between the forward andthe backscattering region of the phase function, canbe observed for stronger absorbing particles and lar-ger size parameters, in general. This is due to the factthat the forward scattering peak increases with in-creasing size parameter while side and backscatter-ing features are damped by absorption. Smallerscatterers with kreqvðvÞ ≲ 2 yield the smallest asym-metry parameters.Figures such as these give experimenters and
modelers a quick overview to identify interesting re-gions for going deeper in their investigations, fordesigning new experiments or for planning future sa-tellite missions. The expense to generate such plotsis relatively low when using the database.
6. Conclusions
We have presented a database of light scatteringquantities of randomly oriented dielectric spheroidalparticles in the resonance region. A well-tested T-matrix method was used to generate the data. Thedata possess a defined accuracy so that they can beideally used as benchmarks for electromagnetic andlight scattering computations of spheroids. The cho-sen fine resolution with respect to the size parameterand the scattering angle enables reliable interpola-tions between the data to generate continuouslyvarying scattering quantities with a sufficient accu-racy. A user interface was developed to facilitate thedata access and to process the data. On request, thedatabase, including the documentation, is available,free of charge, on a CD-ROM.
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