Page 1
Extension of geometrical-optics approximation to
on-axis Gaussian beam scattering.
spheroidal particle with end-on incidence
II.By a
Feng Xu, Kuan Fang Ren, Xiaoshu Cai, and Jianqi Shen
On the basis of our previous work on the extension of the geometrical-optics approximation to Gaussian
beam scattering by a spherical particle, we present a further extension of the method to the scattering
of a transparent or absorbing spheroidal particle with the same symmetric axis as the incident beam. As
was done for the spherical particle, the phase shifts of the emerging rays due to focal lines, optical path,
and total reflection are carefully considered. The angular position of the geometric rainbow of primary
order is theoretically predicted. Compared with our results, the Möbius prediction of the rainbow angle
has a discrepancy of less than 0.5° for a spheroidal droplet of aspect radio ? within 0.95 and 1.05 and less
than 2° for ? within 0.89 and 1.11. The flux ratio index F, which qualitatively indicates the effect of a
surface wave, is also studied and found to be dependent on the size, refractive index, and surface
curvature of the particle. © 2006 Optical Society of America
OCIS codes:
140.0140, 290.4020, 290.5850, 200.0200.
1.
The rigorous theory of plane-wave scattering by a
spheroidal particle was derived several decades ago by
AsanoandYamamoto1,2bysolvingMaxwell’sequation
with a variable separation method under given bound-
ary conditions. On the basis of their method, Vosh-
chinnikov and Farafonov3improved the calculation
efficiency by combining both Debye and Hertz poten-
tials to describe the electromagnetic fields; Han and
Wu4and Barton5have studied Gaussian beam scat-
tering by a spheroid within the framework of general-
ized Lorenz–Mie theory by using the beam-shape
coefficients gn
method. However, because of the poor convergence of
radial spheroidal harmonics of the second kind for a
Introduction
mor by using the surface integral
large aspect ratio or size parameter,6these rigorous
methods can be hardly applied to large spheroids
typically of size parameters larger than 50 or 100.
Furthermore, for a relatively large particle, to ensure
the precision of calculation, a large but ill-conditioned
coefficient matrix is required to determine the un-
known coefficients from boundary conditions, which
might invoke numerical instabilities in solving the
linear equations.
In addition to the above-mentioned rigorous meth-
ods, among others, surface-based methods, such as the
generalized multipole technique7and the T-matrix
method8have been developed for the study of spheroi-
dal particle scattering. A review of this issue has been
carried out by Wriedt.9On the other hand, some ap-
proximation approaches have also been of interest to
researchers because of their advantages in calculation
efficiency and clear physical interpretation of scatter-
ing phenomena. For example, within the framework of
geometricaloptics,Hovenac10hasappliedraytheoryto
study the forward scattering of a plane wave by a
spheroid, and Lock11,12has given a general formula for
calculating the contribution of specularly reflected and
directly transmitted rays of an arbitrarily oriented
spheroid. In rainbow studies, geometrical optics has
also found many applications, particularly in the stud-
ies of droplet and elliptical cylinder scattering.13–16
However, because of the difficulties in ray tracing and
divergence factor calculation for three-dimensional ob-
F. Xu (f3_xu@yahoo.com), X. Cai, and J. Shen are with the In-
stitute of Particle and Two-Phase Flow Measurement Technology,
University of Shanghai for Science and Technology, 516 Jungong
Road, Shanghai 200093, China. F. Xu and K. F. Ren are with the
Unité Mixte de Recherche 6614, Complex de Recherche Interpro-
fessionnel en Aéro-thermochimie, Centre National de la Recherche
Scientifique, Université et Institut National des Sciences Appli-
qués de Rouen, Site du Madrillet, Avenue de l’Université, BP12
76801 Saint Etienne du Rouvray, France.
Received 15 November 2005; revised 16 January 2006; accepted
26 January 2006; posted 2 February 2006 (Doc. ID 66042).
0003-6935/06/205000-10$15.00/0
© 2006 Optical Society of America
5000APPLIED OPTICS ? Vol. 45, No. 20 ? 10 July 2006
Page 2
jects, these methods can hardly handle the emergent
rays experiencing more than one internal reflection.
In fact, once the incident field and the three-
dimensional scatterer have the same symmetric axis,
the calculation can be simplified, since the divergence
factor, the phase shift, and the amplitude of the emer-
gentraysexperiencinganarbitrarynumberofinternal
reflections can be evaluated in the two-dimensional
domain.
In this paper, the far-field scattering of a sphe-
roidal particle illuminated by a Gaussian beam will
be treated by geometrical optics as a superposition
of the contributions from the diffracted, specularly
reflected, and refracted rays. The involved scatterer
is a prolate or oblate spheroid formed by rotating an
ellipse around its major axis a or the minor axis b,
as illustrated in Figs. 1(a) and 1(b), respectively.
Since the beam propagates along the rotational axis
of the spheroid and the situation of end-on inci-
dence is brought in, the incident and scattered rays
are all symmetric to the scattering plane; therefore
the TE and TM polarizations always stay separate
and the mathematical handling can be simplified.
For any other orientation of the spheroid or off-axis
incidence, there is no such symmetry; the plane of
incidence changes at every interaction of a ray with
the spheroid surface, leading to TE–TM polariza-
tion mixing that is difficult to unravel. As was done
for the spherical particle in the companion paper,17
we still use the straight trajectory approximation
for all the rays, both inside and outside the sphere.
The paper is organized as follows. A description of
the method is presented in Section 2. The numerical
results as well as the analysis of the surface wave
effect and the prediction of the geometric primary
rainbow are given in Section 3. Conclusions are given
in Section 4.
2.
We consider a TEM00Gaussian beam of waist radius
w0and wavelength ?, polarized in the x direction and
propagating along the z axis, on which is located a
spheroid of refractive index m ? mr? mii. The center
of the particle OP is located at the origin of the
coordinate system and the center of the beam OGat
(0, 0, d). The aspect ratio ? of the spheroid is defined
as the ratio of its semiaxis along the symmetric axis
(z direction) and the semiaxis in the transverse
direction (in the x, y plane), so ? ? 1 for a prolate
particle and ? ? 1 for an oblate one. For convenience,
we use a and b to denote the semimajor and semim-
inor axes of the spheroid, respectively, and R to de-
note the projection radius of the spheroid in the x, y
plane. So R ? a for the oblate spheroid and R ? b for
the prolate one.
As for the spherical particle, the scattered field is
considered as a superposition of the contributions of
all the modes of the rays, including specularly re-
flected rays, Sj,0, and refracted rays of order p, Sj,p,
which undergo p ? 1 internal reflections as well as
the diffraction field Sd. The scattered field is then
calculated by the summation of the complex ampli-
tude of the diffracted, the reflected, and all the modes
of the refracted light:
Description of the Method
Sj? Sd??
p?0
?
Sj,p,(1)
where the subscript j is 1 or 2, indicating, respec-
tively, the component perpendicular ?? ? 90°? or par-
allel ?? ? 0°? to the scattering plane. The far-field
scattering intensity Ijat an observation point with
distance r from the particle center is calculated by
Ij?
I0
?kr?2ij????
I0
?kr?2 ?Sj????
2,(2)
where I0is the intensity at the center of the Gaussian
beam.
The diffraction part and the description of the
propagation of the Gaussian beam remain the same
as for the spherical particle. We neglect also the
so-called climbing wave18in geometrical diffrac-
tion theory for simplicity and treat the three-
dimensional spheroid as a two-dimensional disk
Fig. 1.
located on the z axis of the incident Gaussian beam. Rotation axis of
thespheroidisonthezaxis.(a)Inaprolatespheroid,(b)inanoblate
spheroid.
Schema of ray tracing in a homogeneous spheroid that is
10 July 2006 ? Vol. 45, No. 20 ? APPLIED OPTICS5001
Page 3
with radius R.19,20The reflected and refracted rays
can be treated in the same manner as in the com-
panion paper17for a spherical particle, but we no
longer have analytical expressions to evaluate the
deviation angles, the phase shifts, or the number of
focal lines passed by the rays. On the basis of the
theoretical consideration of the phase shift and the
divergent and the attenuation factors, a sophisti-
cated numerical algorithm is developed to predict
the scattering intensity.
Since van de Hulst’s definition of the focal points21
still holds for the Gaussian beam scattering by the
rotational symmetric spheroid, the phase shift due to
the focal lines ?p,FL, is still numerically determined
by the number of intersections both inside and out-
side the spheroid, as was done for the sphere. How-
ever, it is worth noting that for a spheroid, both the
refractive index mrand the aspect ratio ? decide the
number of the focal lines a ray encounters,12whereas
for a sphere, only refractive index mrtakes effect.
A.
For a spherical particle, once the incident angle is
known, the incident angle of the ray inside the sphere
always stays the same and the length of each optical
path between two successive internal reflection points
is always constant. For a spheroid, however, each time
the ray hits the boundary of the spheroid, the incident
angle changes and the length of the optical path inside
the spheroid varies. No analytical expression is found
for them. Then we will find the relation of the path for
each internal reflection and evaluate the path numer-
ically step by step.
We consider a prolate illuminated by a Gaussian
beam as shown in Fig. 2. The center of the beam is
located at OGwith a distance d from the center of the
particle OP. First, similar to the case of a sphere,
three kinds of phase shifts are taken into account: the
phase shift due to the optical path ?p,PH, the phase
shift due to the focal lines (inside and outside of
the sphere) ?p,FL, and the phase shift due to the wave-
front curvature of the Gaussian beam ?G:
Phase Shift Due to the Optical Path
?p??
2??p,PH??p,FL??G. (3)
We take the ray passing by the center of the particle
OPwith the same directions as the incident and the
emergent rays as our reference,21i.e., ROPR1, where
R and R1are the intersection points between the radii
and the reference circle, which has the same radius
as the semimajor axis a. Then the phase shift of the
reflected ray is
?0,PH?k?ROP?HS0?OPR0?S0H0?. (4)
The phase shift for the directly transmitted ray with-
out any internal reflection can be evaluated by (see
Fig. 2)
?1,PH?k?ROP? HS0??kmrS0S1?k?OPR1?S1H1?.
(5)
Similarly, the phase shift for the ray undergoing
p ? 1 internal reflections is determined by
?p,PH?k?ROP?HS0??kmrLp?k?OPRp?SpHp?,
(6)
where
?i?1
p, OPRp? SpHpis calculated from the coordinate of
the incident point of the ray on the surface of the
particle and the deviation angle ?p:
the totalpath in thespheroid
Lp
?
p
Si?1Siis evaluated numerically. For all orders of
OPRp?SpHp??zp
2?yp
2?HpRp
2
??zp
2?yp
2??yp?zptan ?p?2
tan2?p?1
.(7)
And for ROP? HS0, we have
ROP?HS0? ?z0
2?y0
2??y0?z0tan ??2
tan2??1
,(8)
where ?y0, z0? is the intersection point of the incident
ray and particle surface.
Then phase shift due to the curvature of the wave-
front should be taken into account. As for the spher-
ical case, this phase difference is defined by
?G??HS0??PQ
??HS0??k?d?a???i?
?k?a??z0
??k?d?a???i?.
2?y0
2??y0?z0tan ??2
tan2??1?
(9)
The phase shift for the reflected ray is then given by
Fig. 2.Scheme for the calculation of phase shifts.
5002APPLIED OPTICS ? Vol. 45, No. 20 ? 10 July 2006
Page 4
?0,PH?k?z0
2?y0
2??y0?z0tan ??2
tan2??1
?k?z0
2?y0
2??y0?z0tan ?0?2
tan2?0?1
, (10)
and the phase shift for the refracted ray is given by
?p,PH??z0
2?y0
2??y0?z0tan ??2
tan2??1
?kmrLp
?k??zp
2?yp
2??yp?zptan ?p?2
tan2?p?1?,(11)
where ? is the angle between incident ray HS0and
the z axis, and ?pis the scattering angle of the emer-
gent ray. ?p, Lp, and ?yp, zp? are determined from the
ray-tracing program similar to that in an elliptical
cross section.22It is proved that when ? → 1,
Lp?2pa cos ?r, (12)
k?z0
2?y0
2??y0?z0tan ??2
tan2??1
?ka cos ?i,(13)
k?zp
2?yp
2??yp?zptan ?p?2
tan2?p?1
?ka cos ?i. (14)
Then Eqs. (8)–(10) will recover to the corresponding
expressions of the spherical particle scattering.
B.
Particularly for the ray tracing in a spheroid, total
reflection may be encountered and it should be taken
into account carefully, since it results in two effects.
One is the absence of the emergent rays beyond the
critical angle ?cand the other is additional phase
shifts ??p,T,j, where the subscript j ? 1 or 2 stands for
the perpendicular or parallel component. Then we
have23
??p,T,1?2 tan?1??sin2?i,p?1?mr
??p,T,2?2 tan?1?
Phase Shift Due to Total Reflection
2?1?2
cos ?i,p
?,(15)
mr
2?sin2?i,p?1?mr
cos ?i,p
2?1?2
?. (16)
Although such a phase change can be incorporated
into a complex number expression of reflection coef-
ficients, Eqs. (15) and (16) are more convenient for
later superposition24of amplitude and phase sepa-
rately. Accounting for all these effects, the final ex-
pression of the phase shift should be revised as
?p,j??
2??G??p,PH??p,FL???p,T,j.(17)
The above equations for a prolate spheroid are all
valid for an oblate one.
C.
When a ray encounters a surface, a fraction of the
energy is reflected and the rest is refracted. The frac-
tion is determined by the Fresnel coefficients rj,p:
Amplitude of the Scattered Field
r1,p?cos ?i,p?mrcos ?r,p
cos ?i,p?mrcos ?r,p,(18)
r2,p?mrcos ?i,p?cos ?r,p
mrcos ?i,p?cos ?r,p,(19)
where ?i,pand ?r,pare the incidence and refraction
angle of the ray. After p ? 1 internal reflections, the
amplitude of the ray, ?j,p, can then be calculated by
?j,p??
rj,p
?1?rj,0
for p?0
2?1?2?1?rj,p
2?1?2?
n?1
p?1??rj,n? for p?1.
(20)
Obviously, ?j,p
gent ray.
On the other hand, when a bundle of rays arrive at
a surface, it is diverged or converged according to the
local curvature of the surface. As for the spherical
scattering, the divergence factor DGis defined by
2indicates the intensity of the emer-
DG?
cos ?isin ?
sin ?p?
d?p?
d??
,(21)
where ? is the position angle between the vector S0OP
and the z axis. It can be calculated by
??tan?1?
y0
z0?.(22)
Similar to the spherical particle scattering, the diver-
gence factor |d?p??d?| for the spheroid is evaluated
numerically by
?
?
d?p?
d???
??p,l?1???p,l?1??
2?2rl?1rl?1cos??l?1??l?1?
rl
rl?1
2?rl?1
2
?
l?2,
(23)
where ?p? is the deviation angle of the emergent ray
and the subscripts l ? 1 and l ? 1 designate the two
incident rays adjacent to the lth ray. rlis the distance
from the intersection point S0?y0, z0? of the lth inci-
dent ray and the particle surface incident to the par-
ticle center OP(0, 0), namely,
rl??y0,l
2?z0,l
2?1?2.(24)
10 July 2006 ? Vol. 45, No. 20 ? APPLIED OPTICS5003
Page 5
For an absorbing spheroid, an attenuation factor
should also be taken into account. When the path
inside a spheroid Lpis known, the attenuation factor
?pcan be easily obtained from
?p?kmiLp. (25)
Through the energy balance identity,10,21the ampli-
tude of an emergent ray can then be calculated by
Sj,p?kR?SG??j,pDG
1?2exp???p?exp?i?p?, (26)
where SGis the amplitude of the Gaussian beam at
the incident point.
3.
According to the method presented in Section 2, we
can predict the scattering intensity of a prolate or
oblate spheroid illuminated by a Gaussian beam. The
particle can be transparent or absorbing. In our cal-
culation, the diffraction of the spheroid is approxi-
mated by a simple disk of the same section as the
projection section of the spheroid in the plane per-
pendicular to the incident beam. But the effect of the
surface wave has not been counted. However, the
influence of the surface wave on a scattering scheme
has been carefully carried out by the evaluation of
the so-called impact factor that gives a qualitative
analysis of the surface wave effect.
Numerical Results and Discussion
A.
For all the numerical calculations presented, the
number of internal reflections, pmax, is taken to be 20,
and 2000 equidistant incident rays are adopted to
carry out the ray tracing in the spheroid. These val-
ues are enough to ensure the precision of the final
interpolation and the convergence of amplitudes
summation.
To check our algorithm and the code, the calcu-
lation is carried out for a prolate or oblate spheroid
with an aspect ratio near 1 to be compared with the
result ofthegeneralized
(GLMT). In Figs. 3 and 4, the scattering intensities
of a transparent water droplet ?m ? 1.33? and a
slightly absorbing one ?m ? 1.33 ? 0.001i? with
a projection radius of R ? 100 ?m illuminated by a
Gaussian beam are compared with that of a sphere
predicted by the GLMT. We find that, if the aspect
ratio ? approaches 1, the scattering diagrams tend
to those of the GLMT and extended geometrical-
optics approximation (EGOA) for the sphere. We
also calculate the forward scattering of a prolate
spheroid of semimajor axis a ? 15 ?m and semiminor
axis b ? 10 ?m illuminated by a plane wave (Fig. 5)
to compare with the result obtained by Hovenac.10
The same intensity profiles are found.
Then the dependence of the scattering pattern on
the aspect ratio of the spheroid is shown in Figs. 6
and 7. The particle is a water droplet (refractive
index m ? 1.33) and is illuminated by the Gaussian
beam of waist radius w0? 50 ?m. The projection
Scattering Diagrams of a Spheroid
Lorenz–Mietheory
radius R of the particle remains at 100 ?m in all the
calculations. With an increase of the aspect ratio, a
remarkable backward movement of the primary
rainbow position is observed except for ? ? 2 for
which neither primary- nor higher-order rainbows
are clearly observable. It is noteworthy that, as the
beam radius is half of the projection radius of an
oblate spheroid, as illustrated in Fig. 6, both
primary- and third-order rainbows are remarkable
and they locate at 100° and 70°, respectively. The
secondary-order rainbow, however, is hardly per-
ceivable. The dependence of the rainbow position on
the aspect ratio and the focalization of the beam as
well as the particle location will be examined in
Subsection 3.B.
Fig. 3.
GLMT for a sphere and by the EGOA for a spherical prolate
(? ? 1.0005) and oblate (? ? 0.9995) spheroid. The projection
radius of the spheroid R is equal to the radius of the sphere
(R ? a ? 100 ?m). The particle of refractive index m ? 1.33 is
located at the center of a Gaussian beam of waist radius w0? 50
?m and wavelength ? ? 0.6328 ?m. The curves of the EGOA have
been, respectively, offset by the factors of 102, 104, and 106for
clarity.
Comparison of the scattering intensity calculated by
Fig. 4. Same parameters as Fig. 3, but with m ? 1.33 ? 0.001i.
5004 APPLIED OPTICS ? Vol. 45, No. 20 ? 10 July 2006
Page 6
For a spheroid, the way of propagation for the sur-
face wave rays might be different from that for a
sphere because the surface curvature of the spheroid
varies from place to place and the incident angle
changes each time the ray hits the surface of the
spheroid. Up to now, we have not been able to develop
a numerical or theoretical approach to evaluate the
surface wave effect in our calculation. However, in
this paper we try to discuss qualitatively the surface
wave effect from the point of view of the energy inci-
dent on the surface at grazing angle and neglect the
concrete propagation behavior of the surface wave
inside the particle as well as the newly created sur-
face wave by internal grazing incidence. Such a sim-
plification permits a similar flux analysis of the
surface wave as in the companion paper by the so-
calledfluxratioindexF,whichrevealssomepotential
influence of the surface wave on the scattering. For
the spheroid, we use the same definition of the flux
ratio index F as for the sphere in the companion
paper.17
In Fig. 8 are plotted the F curves versus the as-
pect ratio for a Gaussian beam ?w0? 50 ?m? and
quasi-plane-wave incidence ?w0? 10 cm in the cal-
culation). The refractive index and the projection ra-
dius of the spheroid are, respectively, m ? 1.33 and
R ? 100 ?m. It is evident that for the same aspect
ratio ?, when the beam waist is half of the radius of
the spheroid, the flux ratio index F is much smaller
than that for the plane because the incident intensity
at the surface wave zone is so weak that it can be
neglected. This means that the surface wave has a
Fig. 5.
projection radius R ? b ? 10 ?m and aspect ratio ??1.5 illumi-
nated by a plane wave. The scattering pattern is the same as that
given by Hovenac in Fig. 8 of Ref. 10.
Forward-scattering intensity of a spheroidal droplet of
Fig. 6.
droplet of projection radius R ? 100 ?m, different aspect ratios,
and located at the center of a Gaussian beam of w0? 50 ?m and
? ? 0.6328 ?m. The results of the cases ? ? 0.8, 0.95, and 0.99 have
been, respectively, offset by the factors of 102, 104, and 106for
clarity.
Scattering intensity calculated by the EGOA for an oblate
Fig. 7.
droplet of projection radius R ? 100 ?m, different aspect ratios,
and located at the center of a Gaussian beam of w0? 50 ?m and
? ? 0.6328 ?m. The results of the cases ? ? 1.05, 1.2, and 2.0 have
been, respectively, offset by the factors of 102, 104, and 106for
clarity.
Scattering intensity calculated by the EGOA for a prolate
Fig. 8.
a spheroid of projection radius R ? 100 ?m illuminated by a
Gaussian beam of waist radius w0? 50 ?m (dashed line) and a
quasi-plane wave of w0? 10 cm (solid curve).
Impact factor as a function of aspect ratio calculated for
10 July 2006 ? Vol. 45, No. 20 ? APPLIED OPTICS 5005
Page 7
limited potential to influence the scattering when the
ratio of the beam-waist radius w0to the projection
radius of the particle R is small.
Meanwhile, Fig. 8 shows that the flux ratio index F
also increases with the aspect ratio. This reveals the
fact that the surface wave has a more remarkable
influence on the scattering of a prolate spheroid than
on that of an oblate one.
Combined with the discussions of the surface wave
effect for the spherical particle in the companion pa-
per, we end this subsection with the following re-
marks on the surface wave effect.
1. The surface wave effect depends on the ratio of
the size of the particle to the beam-waist radius; the
smaller the ratio, the less important the effect. As
found in the companion paper, the flux ratio index F
is inversely proportional to x2?3for a plane wave in-
cident on a large sphere.17
2. The surface wave effect depends on the local
curvature of the surface of the particle, the aspect
ratio, and the propagation directions of the incident
Gaussian rays, or the focalization of the beam. These
parameters decide the size of the surface wave zone.
For example, a prolate spheroid has a larger surface
wave zone than an oblate one for the same projection
radius R. The same effect can be achieved by a spher-
oid located before or behind the beam waist.
B.
The surface wave rays, tunneling rays, and complex
rays are not included in our geometrical-optics ap-
proximation because they bring some inaccuracies to
the scattering calculation. The prediction of the rain-
bow position ?rgfor a Gaussian beam incident on a
spheroid can still be achieved within the framework
of geometrical optics, as was done by Möbius25for
plane-wave incidence. We are especially interested in
the angular location of the primary rainbow of a
spheroid because of its more practical applications in
thermometry and particle sizing technologies.26,27
From the point of view of geometrical optics, the
stationary deflection of the emergent rays experienc-
ing one internal reflection with respect to the varia-
tion of position angles, i.e., d?2??d? → 0, produces the
start point of the primary geometrical rainbow. On
the basis of such a definition, rainbow positions ver-
sus the beam, the aspect ratio, the waist radius, the
refractive index, and the particle location in the beam
can be predicted.
As pointed out in Subsection 3.A, the surface wave
effect increases with the aspect ratio of the spheroid.
Geometrical optics might not be accurate enough in
such a case, and its validity needs further verifica-
tion. However, with the decrease of the aspect ratio,
the rays impinge on a flatter surface and the observed
backscatter in the experiment is also quite different
from the prediction of geometrical optics.10Therefore,
in the present paper, we discuss only the rainbow of
a spheroid that is not greatly deformed from the
sphere, i.e., 0.5 ? ? ? 2.0.
Geometric Rainbow Angle of Primary Order
1.
Comparison with Möbius’s Results
The geometric rainbow is found to be sensitive to the
aspect ratio. As illustrated in Fig. 9, when a spheroid
water droplet of refractive index 1.33 and projection
radius R ? 100 ?m is located at the center of a beam
of different waist radii, the geometric angular posi-
tion of the rainbow ?rgmoves backward from 98.7° for
? ? 0.5 to 180° for ? ? 1.42. The rainbow disappears
for the aspect ratio between 1.42 and 1.58, then re-
appears and moves forward until 152° for ? ? 2. Such
a result is obtained for all four beam waists of radii
25, 75, 200, and 3000 ?m.
To explore the rainbow phenomenon for a spheroid
of different aspect ratios, we perform ray tracing for
a prolate spheroid of aspect ratios ? ? 1.35 (case
? ? 1.42), ? ? 1.5 (case 1.42 ? ? ? 1.58), and
? ? 1.65 (case ? ? 1.58) with same projection radius
R ? 100 ?m. Three parallell incident rays, labeled by
a, b, and c are used for the ray tracing. They corre-
spond to the emergent rays a?, b?, and c?, respectively.
As illustrated in Fig. 10, after experiencing one in-
ternal reflection, the emergent ray b? corresponds to
the rainbow angles of ?rg? 177.1 ° and 175.5°, re-
spectively, for a spheroid of ? ? 1.35 and ? ? 1.65,
and the other emergent rays a? and c? are located at
the same side of ray b?, which indicates that there is
a turn angle for the emergent rays at the geometrical
rainbow angle. For the spheroid of aspect ratio
? ? 1.5, a turn angle does not exist; therefore there is
no rainbow phenomenon. This can also be seen
clearly from the variation of the deviation angle ?p? of
emergent rays versus the incident angle in Fig. 11. It
is noteworthy that for these three aspect ratios, after
experiencing one internal reflection, all the incident
rays are refracted out from the spheroid.
In Fig. 12 we can find that, for the beam of waist
radii w0? 25 and 75 ?m, the intensity magnitude of
the incident rays associated with the rainbow angle
grows drastically when the aspect ratio increases.
Rainbow Position versus Aspect Ratio and
Fig. 9.
dicted by the EGOA method for a spheroidal droplet of projection
radius R ? 100 ?m and illuminated by the Gaussian beam of waist
radii w0? 25, 75, 200, and 3000 ?m, respectively.
Primary-order rainbow position versus aspect ratio, pre-
5006APPLIED OPTICS ? Vol. 45, No. 20 ? 10 July 2006
Page 8
This reveals that the rays causing primary-order
rainbows move from the edge region of the particle
toward the beam axis. Such a comment is confirmed
by adopting 2000 equidistant rays to perform the ray
tracing in a prolate droplet of projection radius
R ? 100 ?m and refractive index m ? 1.33 and illu-
minated by a plane wave. When ? increases from 1.01
to 1.25 and then to 1.40, the rays at y1652? 826 ?m,
y1471? 735.5 ?m, and y642? 321 ?m are found to be
associated with d?2??d? ? 0, respectively.
A comparison of Möbius’s prediction of a primary
rainbow is made with that of the EGOA method for
aspect ratio ? between 0.8 and 1.2 in Fig. 13. The
droplet with refractive index m ? 1.33 is illuminated
by a plane wave with wavelength ? ? 0.6328 ?m. We
find that the discrepancy of Möbius’s result with that
of the EGOA is less than 0.5° for |? ? 1| ? 0.05 and
less than 2° for |? ? 1| ? 0.11. The more the aspect
ratio differs from 1, the more the discrepancy. This is
because Möbius theory is only an O??? approximation
to the exact ray tracing, which can be applied only for
the aspect ratio near unity.
It is noteworthy that, according to the definition
of a rainbow, the emergent rays of p ? 2 might
have more than one stationary deflection angle if
the spheroid deviates much from the sphere, e.g.,
? ? 0.89 or ? ? 1.68. Within the framework of geo-
metrical optics, this means that there exist several
revolutions of the emergent rays corresponding to
more than one primary rainbow outside the spheroid.
However, in Fig. 9 the first stationary deflection an-
gle is looked on as the occurrence position of the
primary rainbow.
2.
Except for some tiny differences when ? approaches
2.0, nearly the same primary rainbow positions are
Rainbow Position versus Waist Radius
Fig. 10.
projection radius of R ? 100 ?m, but with aspect ratio ? ? 1.35,
? ? 1.5, and ? ? 1.65, respectively. The three incident rays are
parallel to the z axis.
Ray tracing in three prolate spheroids with the same
Fig. 11.
three prolate spheroids with the same projection radius of R ? 100
?m, but with aspect ratios ? ? 1.35, ? ? 1.5, and ? ? 1.65,
respectively (2000 equidistant incident rays are used).
Deviation angle ?p= of the emergent rays (p ? 2) from
Fig. 12.
position ?rgin Fig. 9.
Intensity of the incident ray associated with the rainbow
Fig. 13.
aspect ratio) predicted by the EGOA method and the Möbius for-
mula when a spheroidal droplet of projection radius R ? 100 ?m is
illuminated by a plane wave.
Comparison of primary-order rainbow position (versus
10 July 2006 ? Vol. 45, No. 20 ? APPLIED OPTICS5007
Page 9
predicted for the different beam-waist radii, as shown
in Fig. 9. The difference of ?rgbetween w0? 25 and
3000 ?m is found to be less than 0.06° when ? is
between 0.5 and 1.42. This is because the divergence
angle of the beam is very small, less than 0.46° for
w0? 25 ?m and ? ? 0.6328 ?m.
3.
The primary rainbow position is sensitive to the re-
fractive index. A relationship of geometric rainbow po-
sition ?rgversus the refractive index mris shown in
Fig. 14 for a prolate spheroid of the same projection
radius of R ? 100 ?m illuminated by a plane wave
?w0→ ?? of wavelength ? ? 0.6328 ?m. We find that
for all the ? values, the rainbow position moves back-
ward when the refractive index increases, until 180°.
ThisisconsistentwiththephenomenoninFigs.6and
7, in which the position of the primary rainbow ex-
hibits a backward moving phenomenon, from nearly
98.7° ?? ? 0.5? to 161° ?? ? 1.2? and then disappears
?? ? 2?.
Rainbow Position versus Refractive Index
4.
The preceding discussions are based on the assump-
tion that the particle is located at the center of the
beam. Actually, when the beam is not too focused,
the dependency of ?rgon the particle’s location in the
beam is not quite remarkable. For example, when
w0? R ? 100 ?m and the prolate droplet of ? ? 1.1
and m ? 1.33 moves along the z axis from d ? 0 to
d ? 0, the primary rainbow position varies slightly at
the vicinity of 148.2936° (see Fig. 15), which corre-
sponds to plane-wave incidence.
It should be noted that the behavior of higher-order
rainbows of a spheroid is much more complicated
than that of the primary order. Although, in princi-
ple, our method can be employed to predict the posi-
tions of these higher-order rainbows for a spheroid
illuminated by Gaussian beam, they will be discussed
elsewhere.
Rainbow Position versus Particle Location
4.
On-axis Gaussian beam scattering by a transparent or
absorbing spheroid has been studied within the frame-
work of the extended geometrical optics. On the basis
of the EGOA, the position of a primary-order rainbow
angle was theoretically predicted and compared with
the prediction of Möbius’s formula. It shows high de-
pendence on the aspect ratio of the spheroid and its
refractive index, but little on the beam-waist radius as
well as particle location in the beam when its waist
radius is much larger than the wavelength. A compar-
ison between the prediction of rainbow angle for a
transparent water droplet of refractive index m
? 1.33 by the EGOA method and the Möbius formula
hasshownadeviationoflessthan0.5°whentheaspect
ratio ? is between 0.95 and 1.05. The deviation be-
comes 2° when ? is between 0.89 and 1.11. Meanwhile,
a nonrainbow region is found when the aspect ratio ?
is larger than 1.42 and smaller than 1.58.
The surface wave for spheroid scattering plays a
role different from that for a sphere since the incident
angle varies each time the ray hits the surface of the
spheroid. However, by introducing the flux ratio in-
dex F to qualitatively analyze the surface wave, we
found that its effect depends mainly on the refractive
index, the size, and the surface curvature of the par-
ticle, but little on the focalization of the beam and the
position of the particle in the beam.
The authors acknowledge support from the French
Embassy in China for the joint thesis of Feng Xu
(project 4B2-007) and the Natural Science Founda-
tion of China (50336050).
Conclusion
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