Pis'ma v ZhETF, vol.95, iss.3, pp. 138-142
© 2012 February 10
Effects of plasmonic resonances and transparency of nanoshells for
optical filtering
Y. B. Martynov'. R. G. Nazmitdinov*xl\ I. A. Tanachev+°, P. P. Gladyshev+°
+ Institute of Applied Acoustics, 141980 Dubna, Russia *Departament de Fisica, Universitat de les Illes Baleais, E-07122 Palma de Mallorca, Spain x Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, 141980 Dubna, Russia
° Dubna University, 141980 Dubna, Russia Submitted 13 December 2011
We show that narrow optical band pass filters can be created by means of nanoparticles which consist of a dielectric sphere and a metallic shell. The components can be adjusted such that there is a remarkable transparency at the desired wavelength range, while a strong absorption takes place outside of this region.
Nowadays, narrow band pass filters to be used in dense wavelength division multiplexing are among topical problems in the fiber optics communications field. Although the design of these filters is relatively straightforward, the classical techniques suffers from uncontrolled errors [1]. For near-infrared astronomical imaging, the efficient suppression of some emission lines in the Earth's sky together with an optimised transmit-tance band is a strategic need to increase the sensitivity of ground-based instruments for a valuable improvement in observing efficiency [2]. It is noteworthy that the advent of nanotechnology gives impetus to the field of plas-monics (cf. [3,4]) which enables one to operate with light at the nanoscale, well below the scale accessible for the classical techniques. For example, a high sensitivity can be achieved in biosensing technology using a plasmonic material [5]. Various applications of plasmonics appear for nanoscale switches [6], imaging below the diffraction limit [7], materials with negative refractive index [8], to name just a few. It is a challenge to understand how plasmon excitations and light localization at nanoscale might be used advantageously for a narrow band pass filtering.
Among well-known narrow band pass optical filters are those that are based on the Christiansen effect [9]. This effect is due to scattering by small particles in a transparent medium at a wavelength for which the refractive index of the particle material and that of the medium are equal. These filters transmit unscattered light at this wavelength and incoherently scatter light of other wavelengths. A change of the transmission behavior of this dispersion filter can be achieved by variation of the composed materials and external conditions. A
4 e-mail: rashidetheor.jinr.ru
fundamental constraint in manipulating light with such filters is a strong relationship between the transmission bandwidth and the detector aperture [10]. The major goal of the present paper is to demonstrate that the efficiency of such filters can be drastically improved by means of small nanoparticles with metallic nanoshells. Note that in such nanoparticles the absorption is greatly enhanced in contrast to the scattering at plasmon resonance frequency. Plasmon resonances are, in many cases, advantageous because the light scattering using localized plasmon resonances is relatively insensitive to the angle of incidence [11].
A particular interesting object is a nanoparticle composed of a dielectric core and a homogenous metallic nanoshell. Mie scattering theory predicts that by varying the ratio of the shell thickness with the respect to the overall diameter of the particle it is possible to obtain the invisibility of the nanoparticle at a specific wavelength [12-14]. In this case a scattering cancellation is based on the negative local polarizability of a metallic nanoshell with respect to the positive dielectric core polarizability. It was shown that the dipolar term is dominant in the Mie expansion for light scattering from a spherical small plasmonic particle with a radius а < Л/10, where Л is a wavelength [14]. Below we employ this fact to elucidate the effect produced by a nanoparticle composed of a dielectric spherical core and a homogenous metallic shell on filtering phenomena at nanoscale.
In the dipole approximation the extinction cross section creXt = crsc+craj is defined by means of the scattering crsc and absorption ааъ ones:
= f fc4H2, (1)
iтаЬ = 4тгЫт(а). (2)
Here, k = 2iTi/e^f/c is a wave number and / is a frequency of incident photon; em is a permittivity of a surrounding medium, and a dipole polarizability reads
as
a = a:
(ss — £m)(^c 2es) + ß(£m 2es)(ec — £s) (e. + 2e m )(ec + 2e
(3)
where ec, es are permittivities of the core and the shell, respectively; ¡3 = (ac/as)3 and CLq , CLg <1X6 the core and the shell radiuses, respectively. Hereafter, for the sake of discussion we present results for the cross sections in conventional units (c.u.).
Note that the permittivities are complex, in general. It is instructive, however, to consider first only real permittivities (no absorption). We assume that nanopar-ticles are embedded randomly in typical optical plastics (with the permittivity em « 2.25). We do not discuss a particular device which is beyond the scope of the present paper but rather focus on the filtering effect produced by a small nanoparticle with a metallic nanoshell.
Evidently, if the polarization is zero a = 0, the scattering cross section of nanoparticle becomes zero as well. It results in the drastic increase of the transparency of a medium composed of such nanoparticles.
The requirement a = 0 defines an analytical dependence of the shell properties on the size and dielectric characteristics of the nanoparticle core as
,2X1,2) _
-iHi-fMf}
(4)
where we have introduced the notations: t = £c/£m, x = 2((3 + 2)/(2/3 + 1 ),y = 4(1 - (3)/(2(3 + 1). Thus, the orientation of a polarization vector in the dielectric core (ec > 0) is opposite to a local polarization vector in the nanoshell cover. As a result, a cancellation of the scattering caused by the nanoparticle may occur. The first solution of Eq. (4) (being always negative) displays a strong dependence on the ratio ¡3 (see Fig. la). The second solution of Eq. (4) is always positive and, thus, another possibility to have a zero polarizability occurs. As follows from Fig. lb, the nanoparticle should be invisible at: 1) em = ec = es; 2) es > em > ec; 3) ec > em > Eg. In other words, either the core or the shell acts as a void with the effective "negative" dielectric permittivity. This solution displays a weak dependence on the ratio ¡3 (see Fig. lb).
Once the denominator becomes equal to zero, the conditions are realized for the surface plasmons to be
excited by incident light (an electromagnetic wave with a wave frequency w = 2nf). This process depends on the dielectric constant of the metal's surface, and this effect is exploited in surface plasmon resonance spectroscopy. For nanoparticles with the radius on the order of the plasmon resonance wavelength, the surface plasmon dominates the electromagnetic response of the structure.
In fact, two plasmon frequencies are produced by the inner and outer surfaces of the shell (cf. [15]). Indeed, these resonances are excited, if the shell permittivity is subject to the condition
1/2,
rP( 1,2) _
y
(t + x)±
(t-
y2t
I
(5)
In the Drude approximation (without absorption) (wp/w)2 = 1 — ef (w), where wp is a plasmon frequency of the bulk metal; and two solutions for the shell permittivity provide two plasmon frequencies. The plasmon frequencies can be varied widely with the variation of the inner to outer shell radius ratio (or the ratio /3) (see Fig. lc). Evidently, the scattering cross section (1) is greatly enhanced.
The above analysis implies that it is possible to select such a set of parameters which may bring together two essential ingredients, transparency and absorption of the complex nanoparticle, to produce a desired filtering effect. It seems that for a visible and infra-red optical spectra metals with a large negative permittivity (Re(es) < —10) and a small absorption Im(es) (close to zero) would provide the best fit to the above requirement. According to available sources (cf. [16]), silver, gold, copper and, probably, aluminum are possible candidates for metallic nanoshells. Figure 2 demonstrates that simple formulas Eqs. (4), (5) provide a reliable evaluation of the maximal and minimal absorption, calculated rigorously with the aid of Eqs. (1), (2), and (3); when the measured values Re(es), Im(es) are considered [16]. For example, for a spherical nanoparticle with the gold nanoshell (see Fig. 2a and the figure caption for the parameters) the plasmon oscillations, as follows from Eq. (5), should occur at ei1'2^ = —35, —0.5. Taking into account the measured values Im(es) [16] for this calculated Re(es) values, we obtain that the first solution predicts the onset of the plasmon resonance at the incident light wavelength A « 890 nm, while for the second solution it should occur at A « 355 nm. From rigorous results, based on the tabulated values for Re(ec), Im(ec) for different wavelengths, one observes a strong extinction maximum at the first predicted solution, although there is only the extinction growth for the second solution. The
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Y. В. Martynov, R. G. Nazmitdinov, I. A. Tanachev, P. P. Gladyshev
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Fig. 1. (Color online) Shell permittivity as a function of the ratio ¡3 = (ac/as)3 for various values of the core permittivity. Different core permittivities are indicated by symbols: squares connected by solid line are used for ec = 1; circles connected by dashed line are used for ec = 2.25; up triangles connected by dotted line are used for ec = 4; down triangles connected by dash-dotted line are used for ec = 10; diamonds connected by dash-two dotted line are used for ec = 20. The permittivity of the surrounding was set to be 2.25. The top panels display shell pe
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