Analysis of rainbow scattering by a chiral sphere

Abstract

Based on the scattering theory of a chiral sphere, rainbow phenomenon of a chiral sphere is numerically analyzed in this paper. For chiral spheres illuminated by a linearly polarized wave, there are three first-order rainbows, with whose rainbow angles varying with the chirality parameter. The spectrum of each rainbow structure is presented and the ripple frequencies are found associated with the size and refractive indices of the chiral sphere. Only two rainbow structures remain when the chiral sphere is illuminated by a circularly polarized plane wave. Finally, the rainbows of chiral spheres with slight chirality parameters are found appearing alternately in E-plane and H-plane with the variation of the chirality.

Full text

Analysis of rainbow scattering by a chiral sphere
Qing-Chao Shang,1,2 Zhen-Sen Wu,1,* Tan Qu,1 Zheng-Jun Li,1 Lu Bai,1 and Lei Gong3
1School of Physics Optoelectronic Engineering, Xidian University, Xi’an, 710071, China
3School of Photoelectric Engineering, Xi’an Technological University, Xi’an 710021, China
2chaoxidian@foxmail.com
*wuzhs@mail.xidian.edu.cn
Abstract: Based on the scattering theory of a chiral sphere, rainbow
phenomenon of a chiral sphere is numerically analyzed in this paper. For
chiral spheres illuminated by a linearly polarized wave, there are three first-
order rainbows, with whose rainbow angles varying with the chirality
parameter. The spectrum of each rainbow structure is presented and the
ripple frequencies are found associated with the size and refractive indices
of the chiral sphere. Only two rainbow structures remain when the chiral
sphere is illuminated by a circularly polarized plane wave. Finally, the
rainbows of chiral spheres with slight chirality parameters are found
appearing alternately in E-plane and H-plane with the variation of the
chirality.
©2013 Optical Society of America
OCIS codes: (290.4020) Mie theory; (290.5825) Scattering theory; (160.1585) Chiral media.
References and links
1. V. D. Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).
2. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1998).
3. R. T. Wang and H. C. van de Hulst, “Rainbows: Mie computations and the Airy approximation,” Appl. Opt.
30(1), 106–117 (1991).
4. J. A. Lock, “Contribution of high-order rainbows to the scattering of a Gaussian laser beam by a spherical
particle,” J. Opt. Soc. Am. A 10(4), 693–706 (1993).
5. J. A. Lock, J. M. Jamison, and C. Y. Lin, “Rainbow scattering by a coated sphere,” Appl. Opt. 33(21), 4677–
4690 (1994).
6. Z. S. Wu, L. X. Guo, K. F. Ren, G. Gouesbet, and G. Gréhan, “Improved algorithm for electromagnetic
scattering of plane waves and shaped beams by multilayered spheres,” Appl. Opt. 36(21), 5188–5198 (1997).
7. G. Kaduchak, P. L. Marston, and H. J. Simpson, “E(6) diffraction catastrophe of the primary rainbow of oblate
water drops: observations with white-light and laser illumination,” Appl. Opt. 33(21), 4691–4696 (1994).
8. J. P. A. J. van Beeck and M. L. Riethmuller, “Rainbow phenomena applied to the measurement of droplet size
and velocity and to the detection of nonsphericity,” Appl. Opt. 35(13), 2259–2266 (1996).
9. J. P. A. J. van Beeck, Rainbow Phenomena: Development of a Laser-Based, Non-Intrusive Technique for
Measuring Droplet Size, Temperature and Velocity (Technische Universiteit Eindhoven, 1997).
10. X. Han, K. F. Ren, Z. Wu, F. Corbin, G. Gouesbet, and G. Gréhan, “Characterization of Initial Disturbances in a
Liquid Jet by Rainbow Sizing,” Appl. Opt. 37(36), 8498–8503 (1998).
11. J. P. A. J. van Beeck, L. Zimmer, and M. L. Riethmuller, “Global Rainbow Thermometry for Mean Temperature
and Size Measurement of Spray Droplets,” Particle & Particle Systems Characterization 18(4), 196–204 (2001).
12. M. R. Vetrano, J. P. van Beeck, and M. L. Riethmuller, “Global Rainbow Thermometry: Improvements in the
Data Inversion Algorithm and Validation Technique in Liquid-Liquid Suspension,” Appl. Opt. 43(18), 3600–
3607 (2004).
13. J. Wang, G. Gréhan, Y. Han, S. Saengkaew, and G. Gouesbet, “Numerical study of global rainbow technique:
sensitivity to non-sphericity of droplets,” Exp. Fluids 51(1), 149–159 (2011).
14. D. Jaggard, A. Mickelson, and C. Papas, “On electromagnetic waves in chiral media,” Appl. Phys., A Mater. Sci.
Process. 18, 211–216 (1979).
15. L. D. Barron, Molecular light scattering and optical activity (Cambridge Univ Pr, 2004).
16. A. Lakhtakia, V. K. Varadan, and V. V. Varadan, Time-Harmonic Electromagnetic Fields in Chiral Media,
Lecture Notes in Physics (Springer, 1989), Vol. 335.
17. D. L. Jaggard and X. Sun, “Theory of chiral multilayers,” J. Opt. Soc. Am. A 9(5), 804–813 (1992).
18. L. John, “Optical properties of isotropic chiral media,” Pure and Applied Optics: Journal of the European Optical
Society Part A 5(4), 417–443 (1996).
19. S. Bassiri, C. Papas, and N. Engheta, “Electromagnetic wave propagation through a dielectric-chiral interface
and through a chiral slab,” J. Opt. Soc. Am. A 5(9), 1450–1459 (1988).
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(C) 2013 OSA
23 September 2013 | Vol. 21, No. 19 | DOI:10.1364/OE.21.021879 | OPTICS EXPRESS 21879

20. M. Silverman, “Reflection and refraction at the surface of a chiral medium: comparison of gyrotropic
constitutive relations invariant or noninvariant under a duality transformation,” J. Opt. Soc. Am. A 3(6), 830–
837 (1986).
21. A. Lakhtakia, V. V. Varadan, and V. K. Varadan, “Field equations, Huygens's principle, integral equations, and
theorems for radiation and scattering of electromagnetic waves in isotropic chiral media,” J. Opt. Soc. Am. A
5(2), 175–184 (1988).
22. L. Le-Wei, K. Pang-Shyan, L. Mook-Seng, and Y. Tat-Soon, “A general expression of dyadic Green's functions
in radially multilayered chiral media,” IEEE Trans. Antenn. Propag. 43(3), 232–238 (1995).
23. F. Bohren, “Light scattering by an optically active sphere,” Chem. Phys. Lett. 29(3), 458–462 (1974).
24. Z.-S. Wu, Q.-C. Shang, and Z.-J. Li, “Calculation of electromagnetic scattering by a large chiral sphere,” Appl.
Opt. 51(27), 6661–6668 (2012).
25. Q.-C. Shang, Z.-S. Wu, T. Qu, Z.-J. Li, L. Bai, and L. Gong, “Analysis of the radiation force and torque exerted
on a chiral sphere by a Gaussian beam,” Opt. Express 21(7), 8677–8688 (2013).
26. D. Sarkar and N. J. Halas, “General vector basis function solution of Maxwell's equations,” Phys. Rev. E Stat.
Phys. Plasmas Fluids Relat. Interdiscip. Topics 56(1), 1102–1112 (1997).
27. Z. S. Wu and Y. P. Wang, “Electromagnetic scattering for multilayered sphere: Recursive algorithms,” Radio
Sci. 26(6), 1393–1401 (1991).
28. A. L. Aden and M. Kerker, “Scattering of electromagnetic wave from concentric sphere,” J. Appl. Phys. 22(10),
1242–1246 (1951).
29. Y. L. Geng, C. W. Qiu, and N. Yuan, “Exact solution to electromagnetic scattering by an impedance sphere
coated with a uniaxial anisotropic layer,” IEEE Trans. Antenn. Propag. 57(2), 572–576 (2009).
30. X. e. Han, “Study of refractometry of rainbow and applications to the measurement of instability and
temperature gradient of a liquid jet,” thesis (Rouen University, 2000).
1. Introduction
There has been a long history of research on rainbow phenomenon. The earliest theoretical
study can date back to several centuries ago when Descartes and other researchers explained
the rainbow by using the classical geometrical optics. Later, Yong and Airy, respectively,
proposed their theories after taking into account the effect of optical interference. With the
help of electromagnetic theory, rainbow phenomenon can be described precisely by using the
Lorenz-Mie theory [1–3]. In recent decades, study on rainbows has been expanded to cases of
Gaussian laser beams incidence [4], multilayered particles [5, 6], and ellipsoids [7]. Based on
the rainbow phenomenon, rainbow techniques [8–10] and global rainbow techniques [11–13]
have been developed and applied in measurements of particle sizes and temperature.
Natural chiral media are called “optical active media”, due to that linearly polarized light
changes its polarization plane after traveling through them. Representatives of natural chiral
media are solutions of substances with handed microstructure, such as grape sugar and tartaric
acid [14]. Researches on optical activity help a lot in exploring the structure of some
biological molecules [15]. Later, scientists explained optical activity by using electromagnetic
theory and determined chiral media by defining the constitutive relations [16]. Interactions of
electromagnetic waves with chiral media, including reflection and transmission [17–20],
radiation and scattering [21, 22] are well studied by many researchers. Bohren first solved
light scattering by chiral spheres in Lorenz-Mie framework [23]. Based on his work, the
authors researched on scattering from large chiral spheres and found very different rainbow
phenomenon for chiral particles [24].
This paper is devoted to studying the rainbow phenomenon of a chiral sphere by analyzing
the numerical results generated by Lorenz-Mie theory. We mainly focus on the new character
of rainbows and the effects of chirality on them, as the other parameters, such as size and loss
of the sphere, affect rainbows of chiral spheres and isotropic spheres in similar ways. The
following is the arrangement of this paper. In section 2, we review the work of
electromagnetic scattering from a large chiral sphere. Circularly polarized plane wave
incidences are considered here as the rainbow phenomenon is different for these cases. In
section 3, the scattering intensity distributions of rainbows for chiral spheres are presented.
We examine the effects of chirality parameters on rainbow structures and make an attempt to
analyze their spectrum. Then rainbows for circularly polarized wave incidence are analyzed.
In section 4, we analyze rainbows of chiral spheres with slight chirality as common chiral
media at optical frequencies in nature, i.e., the optical active solutions, seem to have very
#193499 - $15.00 USD
Received 9 Jul 2013; revised 26 Aug 2013; accepted 3 Sep 2013; published 10 Sep 2013
(C) 2013 OSA
23 September 2013 | Vol. 21, No. 19 | DOI:10.1364/OE.21.021879 | OPTICS EXPRESS 21880

slight chirality parameters. Section 5 is a summary of our work. In the following analysis, a
time dependence of exp( )it
ω
− is assumed.
2. Scattering by a large chiral sphere
2.1. Scattering coefficients
In electromagnetics, chiral medium is characterized by their special constitutive relations. The
constitutive relations of chiral medium in this paper are adopt as 000ri
εε κ εμ
=+DE H and
00 0r
i
κε
μμμ
=− +BEH, where r
ε
, r
μ
, and
κ
are the relative permittivity, relative
permeability, and chirality parameter of the medium, respectively. 0
ε
and 0
μ
represent the
permittivity and permeability of free space, respectively. The problem of plane wave
scattering by a chiral sphere was solved by Bohren [23]. Based on his work, we extended the
theory to calculate scattering by a large chiral sphere [24]. Consider a chiral sphere of radius
a with chirality parameter κ illuminated by a beam. As we discussed in [24, 25], the incident
field, scattered field and internal field of a chiral sphere can be expanded in terms of spherical
vector wave functions (SVWFS) [26], respectively, as follows:
(1) (1)
0
1
(, ) (, ),
n
ip ip ip
mn mn mn mn
nmn
Eakbk
∞
==−


=+



EMrNr (1)
(1) (1)
0
1
(, ) (, ) ,
n
ip ip ip
mn mn mn mn
nmn
kE akb k
i
ωμ
∞
==−


=+



HNrMr (2)
0
(3) (3)
1
(, ) (, ),
n
sss
mn mn mn mn
nmn
EAkBk
∞
==−


=+



EMrNr
(3)
0(3) (3)
1
(, ) (, ),
n
sss
mn mn mn mn
nmn
kE
A
kB k
i
ωμ
∞
==−


=+



HNrMr (4)
int (1 ) (1) (1) (1)
11 2 2
1
(, ) (, ) (, ) (, ) ,
n
mn mn mn mn mn mn mn mn
nmn
AkAkBkBk
∞
==−

=++−


EMrNrMrNr (5)
int (1) (1) (1) (1)
0
1122
1
0
(, ) (, ) (, ) (, ),
n
r
mn mn mn mn mn mn mn mn
nmn
r
iAkAkBkBk
εε
μμ
∞
==−
=− + + −


HNrMrNrMr(6)
where 0
E represents the amplitude of electric field;
ω
is the angular frequency of the
incident wave;
ε
,
μ
and k
ωε
μ
= denote the permittivity, permeability and wave number
in the surrounding medium, respectively.
()
100rr
k
ω
μ
εκε
μ
=+ and
()
200rr
k
ω
μ
εκε
μ
=− represent, respectively, wave number of the right-handed
circularly polarized (RCP) wave and the left-handed circularly polarized (LCP) wave in chiral
medium. The superscript ip in the equations above indicates the x-polarized (linearly
polarized in the x-direction), y-polarized (linearly polarized in the y-direction), RCP and LCP
wave incidences when ip is ix, iy, iR, and iL, respectively. ip
mn
a and ip
mn
b represent expansion
coefficients of the incident wave. mn
A
and mn
B are expansion coefficients of internal field of
the chiral sphere.
According to the boundary conditions at the spherical surface, scattering coefficients of
scattered field
s
mn
A
and
s
mn
B can be obtained as [24]:
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Received 9 Jul 2013; revised 26 Aug 2013; accepted 3 Sep 2013; published 10 Sep 2013
(C) 2013 OSA
23 September 2013 | Vol. 21, No. 19 | DOI:10.1364/OE.21.021879 | OPTICS EXPRESS 21881

,,
s
sa ip sb ip s sa ip sb ip
mn n mn n mn mn n mn n mn
A
Aa Ab B Ba Bb=+ =+ (7)
where,
(1) (1) (1) (1)
1020
(1) (3 ) (1) (3)
010 20
(3) (1) (3) (1)
001 02
(1) (3 ) (1) (3)
10 20
() () () ()
() () () () ()
,
() () () () ()
() () () ()
nrn nrn
sa nrn n rn n
n
nrn n rn n
rn n rn n
Dx Dx Dx Dx
x
Dx Dx Dx D x
A
x
Dx Dx Dx Dx
Dx Dx Dx D x
ηη
ψη η
ξη η
ηη
−−
+
−−
=−−
+
−−
(8)
(1) (1) (1) (1)
10 20
(1) (3 ) (1) (3)
010 20
(3) (1) (3) (1)
001 02
(1) (3 ) (1) (3)
10 20
() () () ()
() () () () ()
,
() () () () ()
() () () ()
rn n rn n
sb nrn n rn n
n
nrn n rn n
rn n rn n
Dx Dx Dx Dx
x
Dx Dx Dx Dx
A
x
Dx Dx Dx Dx
Dx Dx Dx Dx
ηη
ψη η
ξη η
ηη
−−
−
−−
=−−
+
−−
(9)
,
s
asb
nn
BA= (10)
(1) (1) (1) (1)
10 20
(1) (3) (1) ( 3)
01 0 2 0
(3) (1) (3) (1)
00 1 0 2
(1) (3) (1) ( 3)
1020
() () ( ) ()
() () () () ()
.
() () () () ()
() () () ()
rn n rn n
sb nn rn n rn
n
nn rn n rn
nrn nrn
Dx Dx Dx Dx
x
Dx Dx Dx Dx
BxDx Dx Dx Dx
Dx Dx Dx Dx
ηη
ψη η
ξηη
ηη
−−
+
−−
=−−
+
−−
(11)
In the formulas above, 0
x
ka=, 11
x
ka=, 22
x
ka=, and 0
// /( )
rrrr
η
ε
μ
εε
μμ
=;
()
()
i
n
Dz and ()/ ()
nn
zz
ψξ
are functions associated with Bessel functions and both can be
calculated efficiently by using the recurrence relations [27, 28].
2.2. Scattered field for differently polarized incident plane waves
By substituting the scattering coefficients
s
mn
A
and
s
mn
B into Eq. (3), the scattered field in the
far field can be readily obtained as:
0
1
exp( ) () ,
n
snssim
mn mn mn mn
nmn
ikr
EE iAm B e
kr
φ
θπτ
∞
==−

=−+

 (12)
0
1
exp( ) () ,
n
snssim
mn mn mn mn
nmn
ikr
EiE iA Bm e
kr
φ
φ
τπ
∞
==−

=−+

 (13)
where mn
π
and mn
τ
are defined as:
(cos ) (cos )
,.
sin
mm
nn
mn mn
PdP
d
θθ
πτ
θθ
== (14)
For the case of plane wave incidence, Eqs. (12) and (13) can be simplified a lot as the
expansion coefficients of plane waves are much easier than those of beams. The following
presents the expansions of differently polarized plane waves and the simplification of
scattered field. For an x-polarized plane wave propagating along the z-axis with the form

0
ix ikz
Ee x=E, the expansion coefficients ix
mn
a and ix
mn
b are [29]:
11 11
,1 , 1 ,1 , 1
21 21 21 21
,,
2( 1) 2 2( 1) 2
ix n n ix n n
mn m m mn m m
nn nn
ai i bi i
nn nn
δδ δδ
++ ++
−−
++ ++
=+ =−
++
(15)
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Received 9 Jul 2013; revised 26 Aug 2013; accepted 3 Sep 2013; published 10 Sep 2013
(C) 2013 OSA
23 September 2013 | Vol. 21, No. 19 | DOI:10.1364/OE.21.021879 | OPTICS EXPRESS 21882

with ,
s
l
δ
being the Kronecker delta. Thus, for an x-polarized plane wave incidence, scattered
field can be simplified after substituting Eq. (15) into Eqs. (12) and (13) (See appendix in [24]
for details), as follows:
()()
0
1
exp( ) 2 1 cos sin ,
(1)
ssasbsbsa
nn nn nn nn
n
ikr n
EiE A B i A B
kr n n
θφπ τ φπ τ
∞
=
+


=+++


+
 (16)
()()
0
1
exp( ) 2 1 cos sin ,
(1)
ssasbsbsa
nn nn nn nn
n
ikr n
EE B A i B A
kr n n
φφπ τ φπ τ
∞
=
+


=− + + +


+
(17)
where n
π
and n
τ
are the angle-dependent functions with the following relations with
Legendre functions:
11
11
(cos ) (cos )
,.
sin
nn
nn nn
PdP
d
θθ
ππ ττ
θθ
== == (18)
Now consider the case of circularly polarized plane wave incidence. A circularly polarized
plane wave can be regarded as a superposition of two linearly polarized waves with vibration
direction perpendicular to each other. A right-handed circularly polarized plane wave with the
form 0ˆˆ
()
iR ikz
Ee x iy=+E can be decomposed into two parts: 0ˆ
ix ikz
Ee x=E and 0ˆ
iy ikz
Ee y=E.
By taking a curl of Eq. (1), for the case of incidence 0ˆ
ix ikz
Ee x=E, we have

(1) (1)
1
(, ) (, ) ( ).
n
ix ix ikz
mn mn mn mn
nmn
akbkex
∞
==−

∇× + =∇×

 Mr Nr (19)
As


()
ikz ikz
ex ikey∇× = , (k=∇×NM) / and (k=∇×MN) / , we get expansion coefficients
of a y-polarized plane wave from Eq. (19): iy ix
mn mn
aib=− and iy ix
mn mn
bia=− . Hence, the
expansion coefficients of RCP wave with the form 0ˆˆ
()
iR ikz
Ee x iy=+E can be readily
obtained as
,.
iR ix iy ix ix iR ix iy ix ix
mn mn mn mn mn mn mn mn mn mn
a a ia a b b b ib b a=+ =+ =+ =+ (20)
By substituting Eq. (15) into Eq. (20), the expressions of expansion coefficients iR
mn
a and iR
mn
b
are obtained as:
1
,1
21
(1)
iR iR n
mn mn m
n
abi
nn
δ
++
== + (21)
Hence, the scattered fields for a RCP wave incidence can be simplified as
()
0
1
exp( ) 2 1 ,
(1)
s i sa sb sa sb
nn nn nn nn
n
ikr n
EE ie A A B B
kr n n
φ
θ
ππττ
∞
=


+
=+++


+


 (22)
()
0
1
exp( ) 2 1 .
(1)
s i sa sb sa sb
nn nn n n n n
n
ikr n
EE e A A B B
kr n n
φ
φ
ττππ
∞
=


+
=− +++


+


 (23)
For the case of a LCP plane wave with the form 0ˆˆ
()
iL ikz
Ee x iy=−E, the expansions
coefficients can be obtained in a similar way to RCP case:
1
,1
(2 1) .
iL iL n
mn mn m
abin
δ
+−
=− =− + (24)
Hence, we can derive the simplified expressions for a LCP incident wave.
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Received 9 Jul 2013; revised 26 Aug 2013; accepted 3 Sep 2013; published 10 Sep 2013
(C) 2013 OSA
23 September 2013 | Vol. 21, No. 19 | DOI:10.1364/OE.21.021879 | OPTICS EXPRESS 21883

()
0
1
exp( ) 2 1 ,
(1)
s i sa sb sa sb
nn nn nn nn
n
ikr n
EE i eA A B B
kr n n
φ
θ
ππττ
∞−
=


+
=− −−+


+


 (25)
()
0
1
exp( ) 2 1 .
(1)
s i sa sb sa sb
nn nn n n n n
n
ikr n
EE eA A B B
kr n n
φ
φ
ττππ
∞−
=


+
=−−+


+


 (26)
Finally, the scattering intensity can be calculated by using the following expressions:
()
22 2
22
0
lim / .
ss
sr
IkrEEE
θφ
→∞
=+
(27)
3. Rainbow phenomenon of chiral spheres
3.1. Rainbow structures of chiral spheres
We calculated the scattering intensity distributions of a large chiral sphere and gave a brief
introduction to its rainbow phenomenon in [24]. Figure 1(a) and Fig. 1(b) show rainbow
scattering intensities by chiral spheres with chirality parameter 0.10 and 0.01 for an x-
polarized incident plane wave with wavelength λ, respectively. As the radius is as large as
500λ, step of the scattering angle θ is set to 0.01 to avoid the distortion. As we know,
rainbow phenomenon of isotropic spheres illuminated by an x-polarized plane wave can be
observed in only H-plane ( 90
φ
=). However, three first-order rainbows and slight second-
order rainbows can be found in both E-plane ( 0
φ
=) and H-plane in Fig. 1(a) for a chiral
sphere with chirality 0.10. In the following depiction, we name the three rainbow structures,
respectively, the Left, the Middle, and the Right rainbow, according to their relative positions.
It can be seen that the Left rainbow and the Middle rainbow have a good Airy structure.
However, the Airy structure of the Right rainbow is not so obvious, which might be due to
that the Right rainbow is affected by backward scattering or by the two rainbows before it.
For a chiral sphere with chirality 0.01, as shown in Fig. 1(b), rainbow phenomenon can be
also observed in both E-plane and H-plane. However, it’s difficult to identify Airy structure
from the curves. Neither the three Airy structures as shown in Fig. 1(a), nor a single Airy
structure as the isotropic one’s can be observed. We infer that the three rainbows are too close
to each other and superposed seriously by each other. The following numerical results will
account for the assumption and make it more understandable.
100 110 120 130 140 150 160
0
1x10
6
2x10
6
3x10
6
4x10
6
5x10
6
Right
Middle
Left
Second order
Is
Scattering Angle
θ
E-plane
H-plane
κ
= 0.10
a = 500
λ
ε
r
= 1.7689
μ
r
= 1
(a)
120 125 130 135 140 145 150
0
1x10
6
2x10
6
3x10
6
4x10
6
5x10
6
6x10
6
κ
= 0.01
a = 500
λ
ε
r
= 1.7689
μ
r
= 1
(b)
Is
Scattering Angle
θ
E-plane
H-plane
Fig. 1. Rainbow phenomenon of a chiral sphere. (a) κ = 0.10; (b) κ = 0.01.
3.2. Variation of rainbow structure with chirality parameters
A group of results for a chiral sphere with different chirality parameters is calculated to
examine the effects of chirality on rainbow structures. Figure 2(a) and Fig. 2(b) show,
respectively, the variation of peak angles (the scattering angle corresponding to the peak) and
peak intensities (the scattering intensity of the peak) of the three rainbows with the chirality
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Received 9 Jul 2013; revised 26 Aug 2013; accepted 3 Sep 2013; published 10 Sep 2013
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23 September 2013 | Vol. 21, No. 19 | DOI:10.1364/OE.21.021879 | OPTICS EXPRESS 21884

parameters. The peak angle can generally represent rainbow angle as it is just a little larger
than rainbow angle. It can be seen from Fig. 2(a) that as the chirality approaches zero, the
three peak angles come close to each other and approach the peak angle of the isotropic one.
While as the chirality increases, peak angle of the Left rainbow decreases rapidly; peak angle
of the Middle rainbow slightly decreases; and peak angle of the Right rainbow increases,
moving towards backward direction. Thus, it’s quite understandable that for a small chirality,
such as 0.01 in Fig. 1(b), the three rainbows are too close that their structures are affected
seriously by each other. In Fig. 2(b), the intensities of the Left rainbow and the Middle
rainbow decrease roughly to a very small value; and intensity of the Right rainbow does not
seem to reduce too much. However, for a larger chirality, the Right rainbow is closer to
backward direction and affected more by the strong backward scattering. Finally for a chiral
sphere with chirality large enough, the Left rainbow and Middle rainbow disappear; the Right
rainbow is buried in backward scattering; and no rainbow phenomenon can be observed. It
seems quite reasonable to associate the three rainbows of a chiral sphere with its refractive
indices. There are two refractive indices for a chiral medium [2] as only RCP and LCP waves
propagate in it. The refractive index of a chiral medium is R
nn
κ
=+ for the RCP wave and
L
nn
κ
=− for the LCP wave, where rr
n
ε
μ
=. Considering the relation between rainbow
angle and refractive index of an isotropic sphere, it’s readily to associate the Left rainbow
with nL, and the Right rainbow with nR. We presented a rough physical interpretation in [24]
that after once internal reflection in the chiral sphere there may be three emitted rays for a
linearly polarized ray incidence. However, more work is necessary if we want to explain the
curves in Fig. 2(a) from viewpoint of geometric optics.
0.00 0.04 0.08 0.12 0.16 0.20
90
100
110
120
130
140
150
160
(a)
a = 500
λ
ε
r
= 1.7689
μ
r
= 1
Peak Angle
κ
E-plane Left-rainbow
E-plane Middle rainbow
E-plane Right-rainbow
H-plane Left-rainbow
H-plane Middle-rainbow
H-plane Right-rainbow
0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20
0
1x10
6
2x10
6
3x10
6
4x10
6
5x10
6
6x10
6
7x10
6
(b)
a = 500
λ
ε
r
= 1.7689
μ
r
= 1
H-plane Left-rainbow
H-plane Middle-rainbow
H-plane Right-rainbow
Peak Intensity
κ
E-plane Left-rainbow
E-plane Middle-rainbow
E-plane Right-rainbow
Fig. 2. Effects of chirality on (a) Peak angle; (b) Peak intensity.
3.3. Spectrums of the rainbow structures
Rainbow phenomenon can be applied to measuring the size and temperature of a particle [8,
9]. One method to achieve it is to analyze angular spectrum of the rainbow structures. The
spectrum of the three rainbow structures in E-plane of Fig. 1(a) is calculated and shown in
Fig. 3, respectively. To suppress effects of the mean value and low-frequency component, we
conduct fast Fourier transform (FFT) on derivation of the rainbow structure. According to the
properties of Fourier transform, the processed result is proportional to ()
f
Ff , where ()
F
f
is the original spectrum. It can be seen that the spectrum structure of the Left rainbow and
Middle rainbow are similar to that of an isotropic one, which can be generally divided into
three sections [30]. As shown in Fig. 3, section A corresponds to spectrum of the Airy
structure; section B corresponds to spectrum of the ripple structure; and section C results from
the interference between surface waves. It is found that ripple frequency of the Left rainbow
and Middle rainbow is 12.5977 and 11.3281, respectively. As the Right rainbow structure is
different from the ordinary rainbows, its spectrum is strange compared with the other two.
However, the ripple frequency of Right rainbow can still be identified at 9.9609 easily.
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Received 9 Jul 2013; revised 26 Aug 2013; accepted 3 Sep 2013; published 10 Sep 2013
(C) 2013 OSA
23 September 2013 | Vol. 21, No. 19 | DOI:10.1364/OE.21.021879 | OPTICS EXPRESS 21885

119 120 121 122 123 124 125 126 127
0.0
5.0x105
1.0x106
1.5x106
2.0x106
2.5x106
(a)
Intensity
Scattering angle θ
Left
135 136 137 138 139 140 141 142 143
0.0
5.0x105
1.0x106
1.5x106
2.0x106
2.5x106
3.0x106
(e)
(d)
(c)
(b) Middle
Intensity
Scattering angle θ
149 150 151 152 153 154 155 156 157
0.0
5.0x105
1.0x106
1.5x106
2.0x106
2.5x106
3.0x106
3.5x106
4.0x106
Right
Intensity
Scattering angle θ
0 2 4 6 8 101214161820
0.0
2.0x107
4.0x107
6.0x107
8.0x107
1.0x108
1.2x108
1.4x108
C
B
A
Left
Amplitude
Frequency (1/deg.)
0 2 4 6 8 101214161820
0.0
2.0x107
4.0x107
6.0x107
8.0x107
1.0x108
1.2x108
C
B
A
Middle
Amplitude
Frequency (1/deg. )
0 2 4 6 8 101214161820
0.0
2.0x107
4.0x107
6.0x107
8.0x107
1.0x108
1.2x108
1.4x108
1.6x108
1.8x108
(f) Right
Amplitude
Frequency (1/deg.)
Fig. 3. Rainbow structures and spectrums.
To make a simple analysis on the three ripple frequencies, we assume that the Left
rainbow, Middle rainbow and Right rainbow correspond to refractive index 1.23, 1.33 and
1.43, respectively. Then we can use the empirical relation in [30] proposed by Han to estimate
the size of the chiral sphere. The diameter d can be calculated by the following formulas after
considering the effect of wavelength: 1.4944
/ 58.5505 ripple
dnf
λ
=, where n represents the
refractive index and fripple is the corresponding ripple frequency. Our calculations show that
the empirical relation is valid when the refractive index is in the range 1.23-1.43. By using the
relation above, the diameter calculated by the ripple frequency of the Left rainbow, Middle
rainbow, and Right rainbow is, respectively, 1005.0λ, 1015.7λ, and 995.3λ. It can be seen that
results estimated according to the Left rainbow and Right rainbow are very close to1000λ, the
actual diameter of the chiral sphere. Although the Right rainbow has a strange rainbow
structure and spectrum structure, its ripple frequency can still be used to estimate the size of
the chiral sphere. And the result is as good as that of the Left rainbow, which has an ordinary
rainbow structure and spectrum structure as isotropic ones. For the Middle rainbow, the errors
may be caused by the refractive index we adopted. In fact, according to the depiction in
section 3.2, it is inappropriate to associate the Middle rainbow with a refractive index n =
1.33. All the upper limits of the spectrums of the three rainbows are almost the same, which
can be readily understood. The corresponding components result from the interferences of the
surface waves, which depend on only the particle size and the wavelength, and have nothing
to do with the medium of the particle.
3.4. Rainbows for circularly polarized plane wave incidences
Scattering characteristics of a chiral sphere illuminated by a RCP wave and LCP wave are
different. Therefore, different circularly polarized waves generate different rainbow
phenomenon for a chiral sphere. Figure 4(a) and Fig. 4(b) show, respectively, rainbows of a
chiral sphere with chirality 0.05 and −0.05 illuminated by circularly polarized plane waves.
As scattering intensity distributions in E-plane and H-plane are the same for a circularly
polarized plane wave incidence, rainbows in E-plane are presented in Fig. 4. Compared with
an x-polarized wave incidence, only two rainbow structures can be found. As shown in Fig.
4(a), for a chiral sphere with chirality 0.05
κ
= illuminated by a RCP plane wave, the Left
rainbow disappears and only the Middle and Right rainbows exist. Conversely, for a LCP
incidence case, the Left and Middle rainbows remain, and the Right rainbow disappears. In
Fig. 4(b), for a chiral sphere with negative chirality parameter −0.05, the results are converse
to those with chirality 0.05 in Fig. 4(a). Symmetry can be found between rainbows with
opposite chirality. It is quite reasonable if we note that the scattering intensity distributions of
a chiral sphere with chirality κ illuminated by a RCP wave are symmetric physically to those
of a chiral sphere with chirality -κ illuminated by a LCP wave. Additionally, no second order
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Received 9 Jul 2013; revised 26 Aug 2013; accepted 3 Sep 2013; published 10 Sep 2013
(C) 2013 OSA
23 September 2013 | Vol. 21, No. 19 | DOI:10.1364/OE.21.021879 | OPTICS EXPRESS 21886

rainbow occurs for chirality 0.05 in the case of a LCP incidence and chirality −0.05 in the
case of a RCP incidence.
110 120 130 140 150 160
0.0
2.0x10
6
4.0x10
6
6.0x10
6
8.0x10
6
1.0x10
7
1.2x10
7
(a)
Right
Middle
Is
Scattering Angle
θ
RCP
LCP
E-plane
κ
=0.05
a = 500
λ
ε
r = 1.7689
μ
r = 1
Left
110 120 130 140 150 160
0.0
2.0x10
6
4.0x10
6
6.0x10
6
8.0x10
6
1.0x10
7
1.2x10
7
(b)
Right
MiddleLeft
Is
Scattering Angle θ
RCP
LCP
E-plane
κ
= - 0.05
a = 500
λ
ε
r
= 1.7689
μ
r
= 1
Fig. 4. Rainbow structures for circularly polarized wave incidences. (a) κ = 0.05; (b) κ = −0.05.
4. Rainbows of chiral spheres with slight chirality
Section 3 and 4 focus on the rainbow phenomenon of chiral spheres with sufficient chirality,
which is more accordant with the chirality of the man-made chiral material in the microwave
region. However, most of chiral media at optical frequencies, such as the optically active
media in nature, do not possess so large a chirality parameter. According to the
electromagnetic characteristics of chiral media, it can be readily to derive the relationship
between the optical activity and chirality parameters. For a linearly polarized plane wave
propagating through a chiral slab of thickness d with chirality κ, the polarization plane of the
incident wave is rotated an angle of κk0d, where k0 is wave number of the plane wave. Thus, it
can be estimated that the common optically active media such as sugar solutions possess
slight chirality parameters at about 10−5, which in fact are very close to isotropic media.
Based on the curves presented in Fig. 2(a), there should be only one rainbow structure for
these media and the rainbow angles are the same as those for isotropic spheres. Rainbows of a
chiral sphere with slight chirality 5 × 10−5 and 1.5 × 10−4 are shown in Fig. 5(a) and Fig. 5(b),
respectively. In Fig. 5(a), rainbow structures similar to isotropic ones can be observed in both
E-plane and H-plane. However, the intensity of the rainbow in E-plane is weaker than that in
H-plane. In Fig. 5(b) rainbow occurs in E-plane but almost disappears in H-plane. Obviously
the intensity at peak angle in E-plane and H-plane depend on the chirality. Besides, all peak
angles in Fig. 5 are 37.75°, identical to that of an isotropic one with the same parameters
except the vanished chirality.
120 125 130 135 140 145 150
0
1x10
7
2x10
7
3x10
7
4x10
7
5x10
7
6x10
7
(a)
κ
=5.0e-5
a =1000
λ
ε
r
= 1.7689
μ
r
= 1
Is
Scattering angle
θ
E-plane
H-plane
120 125 130 135 140 145 150
0
1x10
7
2x10
7
3x10
7
4x10
7
5x10
7
6x10
7
7x10
7
8x10
7
(b)
κ
=1.5e-4
a =1000
λ
ε
r
= 1.7689
μ
r
= 1
Is
Scattering angle
θ
E-plane
H-plane
Fig. 5. Rainbows for sphere with slight chirality. (a) κ = 5 × 10−5; (b) κ = −1.5 × 10−4.
In order to investigate the variation of rainbow intensity with chirality, a group of
rainbows for chiral spheres with chirality from 1.0 × 10−5 to 8.0 × 10−4 are calculated, at an
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Received 9 Jul 2013; revised 26 Aug 2013; accepted 3 Sep 2013; published 10 Sep 2013
(C) 2013 OSA
23 September 2013 | Vol. 21, No. 19 | DOI:10.1364/OE.21.021879 | OPTICS EXPRESS 21887

incidence of x-polarized plane wave. As peak angle and shape of the rainbow structure almost
do not change with the slight chirality, the rainbow can be represented basically by the
scattering intensities at the peak angle. Figure 6 presents the scattering intensities at peak
angle in both E-plane and H-plane as a function of chirality, respectively. Phenomenon shown
in Fig. 5 can be interpreted exactly by the curves presented in Fig. 6. It can be seen that the
intensity varies periodically, just like a sine function of chirality. And a maximum intensity in
E-plane corresponds to a minimum intensity in H-plane. Rainbow occurs and disappears
alternately in E-plane and H-plane as the chirality slightly increases. Besides, the maximum
intensity in E-plane is larger than the maximum value in H-plane. Though it’s difficult to give
an interpretation, it’s obvious that the phenomenon is related to the rotation of the
polarization plane after the incident wave propagating through the sphere.
012345678
0
1x107
2x107
3x107
4x107
5x107
6x107
7x107
8x107
a =1000λ, εr = 1.7689, μr = 1
Intensity at peak angle
κ (x104)
E-plane
H-plane
Fig. 6. Intensity at peak angle versus chirality.
5. Conclusion
Rainbow scattering by a chiral sphere is investigated in this paper based on the previous work
about scattering by a large chiral sphere. It is found that for a chiral sphere with proper
chirality, three rainbow structures with their peak angles varying with chirality occur in both
E-plane and H-plane. All rainbows disappear when the chirality increases to a certain value.
As the chirality decreases to zero, the three rainbow structures move close to each other and
approach to structure of the isotropic ones. A FFT analysis of the three rainbow structures
shows that there are different ripple frequencies for each rainbow structure. However, all of
them can be used to estimate the size of the chiral sphere. Only two rainbow structures remain
for a circularly polarized plane wave incidence. And symmetry on rainbow structures is found
between chiral spheres with opposite chirality parameters. Finally the rainbows generated by
chiral spheres with slight chirality are analyzed as the common optical active solutions may
be regarded as chiral medium with slight chirality. The rainbows occur and disappear
alternately in E-plane and H-plane and their intensities vary with the chirality periodically
like sine functions.
Acknowledgment
The authors gratefully acknowledge supports from the National Natural Science Foundation
of China under Grant No. 61172031, No. 61308025, No. 61308071 and the Fundamental
Research Funds for the Central Universities.
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Received 9 Jul 2013; revised 26 Aug 2013; accepted 3 Sep 2013; published 10 Sep 2013
(C) 2013 OSA
23 September 2013 | Vol. 21, No. 19 | DOI:10.1364/OE.21.021879 | OPTICS EXPRESS 21888