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In this work we propose two novel sensing principles of detection that exploit the magnetic dipolar Mie resonance in high-refractive-index dielectric nanospheres. In particular, we theoretically investigate the spectral evolution of the extinction and scattering cross sections of these nanospheres as a function of the refractive index of the external medium (next). Unlike resonances in plasmonic nanospheres, the spectral position of magnetic resonances in high-refractive-index nanospheres barely shifts as next changes. Nevertheless, there is a drastic reduction in the extinction cross section of the nanospheres when next increases, especially in the magnetic dipolar spectral region, which is accompanied with remarkable variations in the radiation patterns. Thanks to these changes, we propose two new sensing parameters, which are based on the detection of: i) the intensity variations in the transmitted or backscattered radiation by the dielectric nanospheres at the magnetic dipole resonant frequency, and ii) the changes in the radiation pattern at the frequency that satisfies Kerker’s condition of near-zero forward radiation. To optimize the sensitivity, we consider several semiconductor materials and particles sizes.
© 2013 Optical Society of America
1. Introduction
The interest of semiconductors photonic microstructures, and in particular silicon photonics, has been demonstrated in the development of several devices, such as waveguides, optical interconnections on chip, lasers, or integrated sensors [1–12]. With the aim of improving their performance and miniaturization capabilities, it has been recently shown that silicon and germanium subwavelength spheres present interesting electric and magnetic Mie resonances [13–16], which could add new functionalities to semiconductor photonics. Semiconductor nanoparticles can behave as photonic nanoresonators due to their high refractive index (n~3-4) and to the relationship between the particle size and the incident wavelength [17]. Although the electric dipolar resonance excitation and behavior in high refractive index dielectric nanoparticles is quite similar to that of the localized surface plasmon resonance (LSPR) in metal nanoparticles, the existence of intense magnetic resonances offers new optical features [13,18]. As the optical absorption of several semiconductor materials is substantially lower than that of plasmonic metals [19], some authors has suggested that high refractive index dielectric nanoparticles can be used as the counterpart of metal nanostructures in several applications, as in metamaterials [20,21], to overcome the problems associated to the inherent absorption of plasmonic nanostructures. Additionally, the possibility to excite both electric and magnetic modes can add crucial features to achieve unusual electromagnetic phenomena, such as negative refraction [22], electromagnetic cloaking [23,24], or directionality [26–28]. In this context, it has been recently demonstrated [15] that coherent effects between the electric and the magnetic modes in germanium particles enables control over the scattering direction, as it was predicted by Kerker et al. [29] several years ago for hypothetical magneto-dielectric particles. The experimental verification of these predictions has been recently reported in the microwave [30] and visible [31,32] regimes.
In addition to the development of metamaterials, one of the main applications of metal nanostructures has been sensing, due to the high sensitivity of the LSPR to the local changes in the refractive index of the surrounding medium, leading to the development of myriad sensing applications [33–39]. Herein, we have numerically explored the behavior of Mie resonances in high refractive index dielectric nanoparticles as the base of a new generation of optical sensors. We have analyzed the variations of these Mie resonances under changes of the refractive index of the external medium for several particle sizes and semiconductor materials. Finally, we have studied the effect of the external medium on the directionality conditions predicted by Kerker [29,30,40].
2. Methods
The electromagnetic field scattered by a homogeneous and isotropic sphere of radius a, illuminated by a linearly polarized plane wave of wavelength λ, can be described via the Lorenz-Mie theory. The scattering and extinction efficiencies of the sphere can be expressed as a multipolar expansion given by [41]:
where an and bn are, respectively, the electric and magnetic n-polar Mie coefficients, and x is the size parameter defined as x = k·a, k being the light wavenumber. In addition, the angular distribution of the scattered energy by the sphere is usually described via the differential scattering efficiency [41,42]:where τn and πn introduce the angular dependence through Legendre functions of first kind. The particular cases of the differential scattering efficiencies at the forward and backward directions are known as forward scattering (QFS) and radar backscattering (QRBS) efficiencies, and are given by [41]:For subwavelength particles (a/λ<<1), as we consider in this work, the multipolar expansion can be well approximated by the sum of the first two electric and magnetic terms (a1, a2, b1 and b2). Under this approximation, the scattering and extinction efficiencies of the particle are given by:whereas the differential scattering efficiency is described by:Finally, the QRBS and QFS for such small particle have the following expressions [41,42]:Equations (5) and (6) predict that coherent effects can occur between different dipolar and quadrupolar terms in the forward and backward directions. In particular, the electric and the magnetic dipoles can interfere constructively or destructively producing minima in either the backward or the forward scattered intensity at certain light frequencies [13–15]. The relationships between the first two Mie coefficients to achieve the zero-backward or a minimum forward scattering are denoted as generalized Kerker’s conditions, and are described as [14,15]:In the following sections we analyze the evolution of the extinction, differential scattering efficiencies and Kerker’s conditions under variations of the external refractive index.3. Results and discussion
3.1 Spectral analysis of the extinction cross-section
Firstly, we analyze the extinction cross-section of nanospheres with radii (R) of 200 and 100 nm, which are composed of several semiconductor materials whose refractive index ranges from 2.9 to 5.
Table 1 gathers the refractive index of these materials in the visible and the near-infrared (NIR) [19]. The majority of these materials show small variations in the real part of the refractive index from the visible to the NIR. Only germanium (Ge) presents a relative variation of the refractive index larger than 20%. Semiconductors present an important advantage over plasmonic materials: the imaginary part of their refractive index (k) is lower, and therefore, their absorption is weaker and most of the light is mainly scattered in nanoparticles. Indeed, Table 1 shows that, with the exception of CdTe and PbS, k is null in the infrared and it weakly increases in the visible, due to the presence of electronic transitions. In the case of CdTe and PbS, the imaginary part of their refractive index reaches non-negligible values in both visible and NIR regions.
Table 1. Semiconductor Materials with High Refractive Index
From previous works [13,15,16], we expect that subwavelength particles of these materials present similar spectral behavior as that shown for Si and Ge. Figures 1 and 2 show the extinction efficiency (Qext) spectra of an isolated sphere of the different semiconductor materials listed in Table 1. To tune the Mie resonances in the NIR or in the visible, we analyze two particle sizes,R = 200nm (Fig. 1) and R = 100nm (Fig. 2), respectively. In the NIR region, where absorption losses are negligible, most of the particles with R = 200 nm present extinction spectra characterized by prominent resonant peaks [Fig. 1]. In the case of Si, GaP, GaAs, InP, AlSb, AlAs and TiO2¸whos spectra are very similar to each other, the peak at larger wavelengths corresponds to the magnetic dipolar resonance, the central broad peak, which is difficult to observe in AlAs and TiO2, is related to the electric dipolar mode, and the sharp peak at shorter wavelengths corresponds to the magnetic quadrupolar mode [13,15,16]. Other less-intense peaks at even shorter wavelengths are due to higher-order multipolar contributions. Although the dipolar contributions in Ge also resemble those of previous materials, its higher order multipoles are weak due to the increased absorption at short wavelengths. On the other hand, all resonant peaks blue-shift as the refractive index of the semiconductor material (nsc) decreases (see Table 1), which is accompanied with an electric dipole peak broadening. Such broadening makes difficult to distinguish the electric dipolar peak for the lowest value of nsc, which correspond to TiO2.
Fig. 1 Extinction efficiency spectra of spherical particles of different semiconductor materials in vacuum (next = 1). Particle size is R = 200nmto achieve Mie resonances in the NIR range.
Fig. 2 Extinction efficiency spectra of spherical particles of different semiconductor materials in vacuum (next = 1). Particle size is R = 100nm to achieve Mie resonances localized in the visible range.
The exceptions to this general behavior are nanospheres composed of PbS and CdTe. The non-negligible absorption of these materials in the NIR range generates very broad resonant peaks, thereby limiting the interest of these nanospheres for technological applications.
In contrast, for R = 100nm [Fig. 2], the resonant peaks are in the visible, where the absorption losses are more important. The non-negligible k values in Ge, GaAs and InP result in less intense and broader resonant peaks than those in the infrared. However, the electric and magnetic dipolar modes are still well defined and prominent. On the contrary, Si, AlSb, AlAs and GaP have very weak or null absorption in this region, and therefore, their spectra show very intense, well-defined and narrow resonant modes, as in the NIR regime. Nevertheless, the quadrupolar magnetic mode is only barely detected in materials with very low absorption. An example of this behavior is the TiO2 nanosphere, whose spectrum only shows a well-defined magnetic dipolar peak. Therefore, the higher absorption and the smaller particle size inhibit the excitation of the quadrupolar mode, while the electric dipolar resonance appears as a “shoulder” of the magnetic mode. Likewise larger particles, CdTe and PbS nanospheres show extinction spectra without well-defined peaks.
From this analysis, we conclude that semiconductors with values of nsc similar to Si or Ge, present the expected resonant scattering behaviors [13,15,16]. Furthermore, CdTe and PbS spectra do not present interesting resonances for practical applications due to their high absorption losses. For that reason, we will not consider them in the following analysis. Interestingly, the magnetic dipolar mode, which appears at larger wavelengths, is the most stable resonant peak, as it is intense and well-defined for most of the materials, and for resonances in the visible and NIR. Consequently, we focus our attention on this mode for the potential sensing applications.
3.2 External refractive index dependence
Nanoplasmonic sensors are typically based on the sensitivity of the LSPR wavelength to the changes in the refractive index of the surrounding medium (next) [35,36,42]. Nanoplasmonic sensors commonly detect the shift of the LSPR wavelength to monitor in real-time the local changes of refractive index induced by biochemical interactions.
In contrast, Fig. 3 shows the variations of Mie resonances in Si nanoparticles as a function of next, to investigate if this system can be used as refractometric sensing transducer. We consider again two different particles sizes, R = 200nm and R = 100nm.The magnetic dipolar mode presents an almost constant spectral position, as it has been recently demonstrated experimentally in silicon colloids [11]. The origin of these phenomena is the localization of the magnetic dipolar mode inside the nanoparticle [13]. The magnetic dipolar peak intensity drastically decreases and broadens as next increases. Indeed, for the highest next values, the peak is hardly distinguished from the background. The magnetic quadrupolar resonance has a similar behavior, whereas the electric dipolar peak shows a substantial red-shift, but it rapidly broadens and becomes undetectable for next> 1.2.
Fig. 3 Extinction efficiency spectra of a spherical silicon particle for several next values. Particle sizes are (a) R = 200nm and (b) R = 100nm. The three different modes, electric dipolar (ED), magnetic dipolar (MD) and magnetic quadrupolar (MQ), are labeled. Arrows show the evolution of the spectral position of resonances as the external refractive index increases. Vertical lines point the wavelengths that satisfy Kerker’s conditions in vacuum. Curves with next>1 have been shifted upwards (by adding a constant background) for clarity.
Interestingly, the drastic drop of the extinction efficiency of the magnetic dipolar resonance as next increases could be used as principle of detection for sensing applications. To analyze this potential application, we show in Fig. 4 the extinction efficiency values at the magnetic dipolar resonant wavelength as a function of next for nanospheres of the semiconductor materials previously studied, and for two different particle sizes (R = 200nm and R = 100nm). As comparison, Fig. 4 also shows the evolution of the extinction efficiency of a gold nanoparticle of the same dimensions, when the wavelength is fixed at the electric dipole resonance wavelength in vacuum. As can be observed, the Qext evolution of the gold nanoparticle exhibits a complex pattern, which is governed by the redshift of the plasmon electric dipolar and quadrupolar resonances. In contrast, the extinction efficiency of semiconductor nanoparticles at the resonant wavelength of the magnetic dipolar mode in vacuum, decreases monotonously with next in most of the materials. The sensitivity to the next changes can be calculated via the derivative of latter trend:
Figure 5(a) summarizes the sensitivity at next = 1.33, which corresponds to the typical background index in biosensing applications. All the materials present similar sensitivities when their magnetic dipolar mode is in the NIR, but there are more drastic variations among materials when this resonance is in the visible. Ge and AlAs are the most sensitive materials in the NIR and in the visible ranges, respectively, but the values of the sensitivity are comparable. Germanium suffers the most important drop of the sensitivity between NIR and visible ranges as a consequence of the large increase of its absorption in the visible (Table 1).Fig. 4 Extinction efficiency at the magnetic dipolar resonance wavelength in vacuum as a function of the next, for nanospheres of different semiconductor materials. The particle sizes are (a) R = 200nm and (b) R = 100nm. The evolution of the extinction efficiency of a gold nanosphere at the electric dipolar resonance wavelength is also included as a dashed line.
Fig. 5 (a) Sensitivity of the extinction efficiency, at next = 1.33, for the considered semiconductor materials and for both particle sizes. (b) Spectral evolution of Qext in a silicon (solid black line) and in a gold (dashed blue line) nanoparticle of radius R = 100nm, embedded in water. (c) Spectral sensitivity of Qext, at next = 1.33for a silicon (red curve) and a gold (blue curve) nanoparticle.
Interesting information can be extracted from Figs. 5(b) and 5(c), which compares the spectra of Au and Si nanoparticles when the external medium is water, and their sensitivities to changes of refractive index. Firstly, Fig. 5(b) highlights the substantially higher extinction efficiency of the Si nanoparticle, despite its lower absorption cross section. Additionally, the sensitivity to changes in refractive index displayed in Fig. 5(c) demonstrates the higher sensitivity of the Si nanoparticles and the different spectral behavior. Whereas the maximum sensitivity in Si coincides with the resonant positions of the magnetic dipolar and quadrupolar modes, the maximum sensitivities in Au are observed when the slope of Qext is maximized, i.e. close to the electric dipolar and quadrupolar plasmon resonances. The different behavior is produced by the fact that the magnetic dipolar resonant wavelength of Si is almost insensitive to any change of next, while the Au plasmon peak strongly shifts as next changes. Overall, the two-fold higher sensitivity expected in the Si nanospheres is accompanied with a 3-fold larger extinction cross section with respect to Au in the most sensitive regions, which can have remarkable implications to improve the limit of detection in refractometric sensors.
For actual sensing applications, measurements at the single particle level are not practical due to the generally low signal-to-noise and high background-to-signal ratios of the measurements. An interesting implementation of the presented effects could be achieved via an array of silicon nanospheres with random distribution and mean edge-to-edge separation distance larger than their diameter (equivalent to a surface density of 4-5 nanoparticles/μm2 for R = 100 nm spheres) to simultaneously avoid diffraction and near-field interaction effects [36]. The size of the array could be in the order of 100-500μm2 to be easily illuminated with low numerical aperture lenses. Since the total extinction efficiency is a complex parameter to be determined experimentally, the experimental detection could be based on the analysis of the forward scattering or radar backscattering cross sections, which can be extracted from extinction measurements either in transmission or in reflection, respectively.
Figure 6 shows both scattering cross sections (QFS, QRBS) of a Si nanoparticle (R = 100 nm) as a function of the external refractive index at the magnetic dipolar resonance (λ~773nm). As can be observed, for next< 1.5 the forward scattering remains approximately constant, and the reduction of Qext mainly comes from the drastic attenuation of the backward scattering efficiency. Consequently, to achieve a high sensitivity in this region, the measurement should be based on the detection of intensity changes in the backscattered light, which in addition will benefit from a low background signal. In contrast, detection of the variations in the forward scattering cross section is the most interesting alternative for next> 1.5, due to the very low backscattering efficiency in this region.
Fig. 6 Forward scattering and radar backward scattering cross sections of a silicon nanoparticle with R = 100nm at the magnetic dipolar resonance wavelength (λ = 773nm).
3.3 Radiation pattern variations for sensing
As can be deduced from the previous sections, a very interesting aspect of Mie resonances is related to the ability to control and modify the radiation patterns thanks to the coherent interaction of electric and magnetic dipolar modes, giving rise to zero backward or near-zero-forward scattering radiation patterns [14,15,29,30,43]. In this context, we explore the effect of the external medium on the directionality effects.
Figure 7 shows the variation of the differential scattering efficiency of a Si nanoparticle of R = 100nm at the incident wavelengths that satisfy Kerker’s conditions in vacuum (i.e. λ = 843.6nm for the first Kerker’s condition or zero-backward scattering, and λ = 717.8nm for the second condition or near-zero-forward scattering) when next increases. To observe the changes in detail, we also plot a zoom of the polar plots in the bottom part. As can be seen, while the zero-backward condition can be kept for any value of the external refractive index, the minimum forward condition is very sensitive to next. In fact, although the extinction cross section does not significantly change as next increases, there is rapid transformation of the radiation pattern from near zero forward to near zero backward. Such effect is a consequence of the red-shift of the electric dipolar resonance that also modifies the wavelength at which the second condition is satisfied. For practical applications, detection of the forward scattering and radar backscattering efficiencies can be an interesting alternative. Figure 8 plots the evolutions of these parameter with next, showing that the forward scattering linearly increases for next<1.4, whereas the radar backscattering shows a region with high sensitivity to refractive index changes for next comprised between 1.2 and 1.6. Therefore, analysis of the forward or backward scattering of an arrays of subwavelength dielectric particles illuminated by a light beam whose wavelength satisfies Kerker’s condition in vacuum, can also be used as a new sensing principle of detection. This effect clearly shows potential experimental benefits, since this measurement could be experimentally performed by simply detecting extinction variations in transmission or reflection at a fixed wavelength, in an array of silicon nanospheres similar to that previously described.
Fig. 7 Polar representation of the differential scattering efficiency, in arbitrary units, of a Si nanoparticle (R = 100nm) which is illuminated by a plane wave whose wavelength satisfies (a) the zero-backward or (b) the minimum-forward condition in vacuum, for several values next. The incident light beam is linearly polarized with the electric field perpendicular to the scattering plane. A zoom of the central area is shown in the bottom part of the figure to observe in detail the evolution as next changes.
Fig. 8 Forward scattering and radar backscattering efficiencies, at the wavelength that satisfies Kerker’s condition of minimum forward scattering in vacuum for a Si nanoparticle (R = 100nm) as a function of the external refractive index
4. Conclusions
We have investigated the spectral and radiation pattern responses of Mie resonances in subwavelength semiconductor spherical particles under changes of the external refractive index. We have shown that the behavior of the magnetic dipolar resonance has remarkable potential sensing applications in two different configurations. The first configuration exploits the strong reduction of the extinction cross-section peak associated to the magnetic dipolar resonance as next increases. When the resonance is in the NIR, the most sensitive material is Ge due to its very high refractive index and negligible losses, whereas the highest sensitivities in the visible are achieved by Si and AlSb, as a consequence of the general increase of the absorption in all semiconductor material. The second configuration is based on the variation of the directionality of the scattered radiation at the Kerker’s conditions. While zero-backward scattering condition is very stable and does not vary as next changes, the minimum-forward condition shows a fast conversion from near zero forward to near zero backward scattering as the refractive index of the external medium increases. Thus, the change in the scattering radiation pattern could also be used as a new sensing parameter, via analysis of the forward scattering or radar backscattering efficiencies.
The use of these parameters could involve important advantages with respect to conventional plasmon sensors. Firstly, the drastic reduction of the absorption losses with respect to the plasmonic nanostructures eliminates the problems and risks associated to plasmonic heating at moderate or high light intensity. This feature can be crucial to ensure the stability of biomolecules or even live cells, while enabling the use of high intensity to improve the signal-to-noise ratio (SNR) of the measurements. Secondly, nanoplasmonic sensors typically monitor the LSPR wavelength shifts induced by the local changes of refractive index generated by biochemical interactions. A high signal-to-noise ratio involves the necessity of spectrometers with high spectral resolutions along a large wavelength range, in which the time resolution is in the millisecond range.In contrast, our proposal only needs the detection of the intensity variations at the magnetic dipolar resonance or at the Kerker’s condition wavelength, thus reducing technological requirements and costs, and opening the path to ultra-fast detection, which is especially interesting for single molecule detection applications.
Acknowledgments
B.G.-C. acknowledges support from the JAE-Doc program of the Spanish Council of Research (CSIC). This research has been funded by Ministerio de Ciencia e Innovación, through grants: Consolider NanoLight (CSD2007-00046), FIS2009-13430-C02, as well as by the Comunidad de Madrid (Microseres-CM, S2009/TIC-1476)
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