Linear polarization degree for detecting
magnetic properties of small particles
Braulio García-Cámara,* Francisco González, and Fernando Moreno
Grupo de Óptica, Departamento Física Aplicada, Universidad de Cantabria, Avenida los Castros s/n, Santander, Spain
*Corresponding author: garciacb@unican.es
Received September 14, 2010; revised October 20, 2010; accepted October 29, 2010;
posted November 2, 2010 (Doc. ID 135066); published November 30, 2010
Motivated by the recent advances with magnetic nanoparticles, in this research we propose a new technique for
their characterization based on themeasurement of certain polarimetric parameters of the scattered light, such as the
linear polarization degree when it is determined at a “right-angle” scattering configuration. We will show the sen-
sitivity of its spectral evolution with the magnetic properties of the particle. © 2010 Optical Society of America
OCIS codes: 160.4236, 290.5855.
Current technological advances, mainly in the field of
information storage, are due to the increasing interest
of researchers in the magnetic properties of materials
[1]. During the past few years, technological efforts have
been directed to the miniaturization of data storage sys-
tems, and much research has been done on the prepara-
tion, manipulation, and characterization of magnetic
nanoparticles. Several applications based on these nano-
particles have been proposed, in particular, for biomedi-
cal tasks [2] suchasmolecular imaging [3,4] or therapeutic
treatments [5,6].
At the same time, metamaterials have emerged with
the objective of being able to control the optical proper-
ties of the medium, both electric and magnetic. This kind
of engineered media allow us to observe magnetic per-
meabilities different from 1, even in the visible range
of the spectrum [7,8]. The possibility to manipulate both
the electric permittivity and the magnetic permeability of
a material is the base of several applications, such as in
the field of optical communications [9–11].
Oneway todetermine themagnetic properties of nanos-
tructures is through the spectral characteristics of the
scattered radiation: the sample is illuminated by an elec-
tromagneticwave, and the scattered intensity ismeasured
as a function of the incident wavelength. The analysis
of certain spectral resonances can reveal magnetic re-
sponses of the scatterer. However, other techniques can
be used for detecting the magnetic character of a system.
In this Letter, we propose the use of polarimetric tech-
niques and, in particular, we show the efficiency of the
linear polarization degree, measured at θsca ¼ 90° [right-
angle scattering configuration (RASC)] [12], as a param-
eter sensitive to magnetic conducts.
The linear polarization degree [P
L
ðθscaÞ] of the scat-
tered intensity is defined by [13]
P
L
ðθscaÞ ¼
I
⊥
ðθscaÞ − I∥ðθscaÞ
I
⊥
ðθscaÞ þ I∥ðθscaÞ
; ð1Þ
where I
⊥
and I
∥
are the components of the scattered in-
tensity linearly polarized perpendicular and parallel to
the scattering plane, respectively, and are measured at
the scattering angle θsca.
For particles with conventional optical properties,
μ ¼ 1, whose size is smaller than the incident wavelength,
the electric dipolar response dominates. Thus, in the far
field, the angular distribution of the scattered intensity
follows the typical “figure-eight shape” for the parallel
component and the isotropic distribution for the perpen-
dicular one. Consequently, P
L
measured at θsca ¼ 90° is
equal to 1, because I
∥
ð90°Þ ¼ 0 [seeEq. (1)]. Any deviation
from the dipolar behavior, such as the excitation of multi-
polar modes, induces changes in the distribution of the
scattered intensity, which involves a decrease in the value
ofP
L
ð90°Þ [12].On theother hand, if the scatterer ismainly
magnetic and the dipolar magnetic character dominates,
the angular distribution of the scattered intensity is oppo-
site to the previous one: the isotropic distribution appears
for a parallel incident polarization and the figure-eight-
shaped one for the perpendicular component. After
checking Eq. (1), we can observe that for a dominantmag-
netic dipolar response, the linear polarization degree at
RASC tends to be equal to −1. Hence, we can conclude
that while positive values of P
L
ð90°Þ are associated to an
electric behavior, negative values of this polarimetric
parameter correspond to a magnetic one.
To check this, we have considered a spherical particle
whose optical constants are such that the scatterer pre-
sents either electric or magnetic dipolar characters at dif-
ferent incident frequencies. In particular, following the
work of Ruppin [14,15]:
εðωÞ ¼ 1 −
ω
2
p
ωðωþ iγÞ
ð2Þ
for the electric permittivity and
Fig. 1. (Color online) Real part of the electric permittivity
(black squares) and the magnetic permeability (red circles)
considered in our calculations. This material example could
be left-handed between 500 and 750 nm.
4084 OPTICS LETTERS / Vol. 35, No. 23 / December 1, 2010
0146-9592/10/234084-03$15.00/0 © 2010 Optical Society of America
μðωÞ ¼ 1 − Fω
2
ω
2
− ω
2
0 þ iωΓ
ð3Þ
for the magnetic permeability, where ω
p
is the plasma
frequency, ω0 is the magnetic resonance frequency,
ω is the frequency of the incident electromagnetic
radiation, and γ and Γ are damping parameters. In our
numerical calculations, we have rescaled the values
of [14] to the optical range in such a way that:
ω
p
¼ 10 × 1014 rad=s, ω0 ¼ 4 × 1014 rad=s, γ ¼ 0:03ωp,
Γ ¼ 0:03ω0, and the parameter F ¼ 0:56. Following these
expressions, the optical constants are plotted in Fig. 1 in
the considered wavelength range.
In Fig. 2, we plot the linear polarization degree mea-
sured at 90° [P
L
ð90°Þ—solid curve] and the extinction
efficiency [ðQextÞ—dashed curve] of a spherical particle
(R ¼ 100 nm) as a function of the wavelength of the in-
cident field. The considered size of the particle is such
that dipolar scattering is dominant. However, a quadru-
polar resonance can be observed in Qext around
λ ¼ 750 nm. In this case, resonances in Qext are related
to maximum values of Mie coefficients of the first two
orders (a1, b1, and a2 [13]), which have been shown in
the figure. Also, we have plotted the angular distribution
of light scattered by the particle for certain wavelengths
of the incident electromagnetic field. Out of resonance or
when the electric dipolar resonance (associated to a1
[13]) is excited (λ < 650 nm or λ > 800 nm), the scatterer
behaves like an electric dipole, as can also be seen in the
scattering patterns. This means that the linear polariza-
tion degree is close to 1. When a magnetic response
dominates due to the excitation of a magnetic dipolar re-
sonance (associated to b1 [13]), the angular distribution
reverses the shape of their components and P
L
ð90°Þ
reaches negative values, as was predicted above. Theo-
retically, P
L
ð90°Þ ¼ −1 when the magnetic dipolar char-
acter emerges; however, in Fig. 2, the linear polarization
degree reaches the value P
L
ð90°Þ ¼ −0:8 only because of
the influence of the finite size of the particle. As the evo-
lution of the linear polarization degree is directly related
to the dipolar response of the scatterer, the influence of
the particle size is crucial. In this sense, for large values
of the wavelength in Fig. 2, where the relation R=λ rein-
forces the dipolar character of the scatterer, P
L
ð90°Þ is 1,
but for smaller λ the linear polarization degree differs
slightly from 1.
To show the influence of particle size, in Fig. 3, we plot
the spectral evolution of the linear polarization degree
measured at RASC ðP
L
ð90°ÞÞ of a spherical scatterer with
four different sizes: (a)R ¼ 100 nm (previously discussed
in Fig. 2), (b) R ¼ 200 nm, (c) R ¼ 300 nm, and (d)
R ¼ 400 nm.We have also included the spectral evolution
of the extinction efficiency (Qext) of the particles. As par-
ticle size increases [Figs. 3(b)–3(d)], other modes appear
due to multipolar contributions: in particular, electric
quadrupolar resonances (associated to maximum values
of a2 [13]). These resonances produce important distur-
bances on the angular distribution of the scattered light,
which involves sharp decreases in the value of P
L
[12],
such as for wavelengths around 500 and 750 nm. How-
ever, the electric character of these resonances means
that P
L
ð90°Þ does not reach negative values. The size also
has influences on the minimum of P
L
when the
magnetic resonance is excited. As can be seen, as particle
Fig. 2. (Color online) Linear polarization degree at
RASC ðP
L
ð90°ÞÞ [solid line] and extinction efficiency ðQextÞ
[dashed line] of a spherical particle (R ¼ 100 nm) as a function
of the incident wavelength. Mie coefficients associated to reso-
nances of the extinction efficiency are shown. Also, we have
included the angular distribution of the scattered intensity
for several cases (λ ¼ 400, 526, 674, and 950 nm).
Fig. 3. (Color online) Linear polarization degree measured
at RASC, P
L
ð90°Þ [solid curve] of a spherical particle with
radius (a) R ¼ 100 nm, (b) R ¼ 200 nm, (c) R ¼ 300 nm, and
(d) R ¼ 400 nm as a function of the wavelength of the incident
field. The corresponding extinction efficiencies (Qext) are also
plotted [dashed curve]. Mie resonances are marked with the
Mie coefficients, whose maximum values are associated to each
resonance.
December 1, 2010 / Vol. 35, No. 23 / OPTICS LETTERS 4085
size increases, the value of the linear polarization degree
at this point is less negative [from P
L
ð90°Þ≃ −0:8 for R ¼
100 nm to P
L
ð90°Þ≃ −0:3 for R ¼ 400 nm].
In summary, in this research we have analyzed the pos-
sibility of using polarimetric parameters to detect mag-
netic responses in small particles. In particular, we
have shown that the linear polarization degree at RASC
remains equal to 1when particles scatter light like an elec-
tric dipole. However, if the particle presents magnetic
properties, P
L
ð90°Þ can get negative values. In the past
few years, interest in magnetic nanoparticles has grown
and this technique could be considered as an alternative
method to characterize their optical properties. Also, we
have checked the influence of particle size on the sensi-
tivity of the method: as the ratioR=λ increases, new drops
in the values ofP
L
ð90°Þ appear due tomodes of order high-
er than the dipolar. Also, the value of the linear polariza-
tion degree when the magnetic dipolar resonance is
excited is less negative. However, this can still be easily
distinguished from the effect of multipolar modes.
This research has been supported by the Ministerio de
Ciencia e Innovacion under project FIS2007-60158.
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