Using linear polarization for sensing and sizing dielectric nanoparticles

Ángela I. Barreda; Juan M. Sanz; Francisco González

doi:10.1364/OE.23.009157

1. Introduction

Nanotechnology has revolutionized science with important theoretical and practical developments. Particularly, the interaction of light with metallic nanoparticles (NPs) has been a very active field that has impacted many different areas. For instance, the development of new sensing techniques especially has attracted the attention of researchers in science and engineering [1]. When incident light illuminates a metallic NP, electron plasma oscillations are produced and, consequently, localized surface plasmons (LSPs) are generated. These coherent oscillations of the electron plasma depend on the material properties, the particle size and shape, and also on the wavelength of the incoming radiation and result in particular surface charge distributions [2]. At specific frequencies, resonances can be observed and strong enhancements of the electric field in the surroundings of the NPs may occur. Although most of the plasmonic studies of metallic NPs take advantage of the significant response of the plasma in the VIS-NIR range, their metallic nature is also the cause of their main disadvantage, ohmic losses.

High refractive index (HRI) dielectric NPs have been proposed as a solution for this problem because light can interact with these materials with negligible absorption [3]. In fact, instead of electronic plasma oscillations, certain distributions of displacement currents leading to whispering gallery-like modes are responsible for resonances in dielectric NPs [4]. These are located in well-defined spectral ranges, depend on the NPs size and shape, and can be of either electric or magnetic nature, although the particle magnetic permeability μ has a value of 1 [5]. This magneto-dielectric behaviour has been vastly explored for some elements, such as Silicon and Germanium [6–8], while the study of other semiconductor compounds has begun only recently [9, 10]. There has been much work trying to exploit the experimental capabilities of HRI dielectric NPs in the last years [11–14].

The resonance position depends on the NP size: For a given material, resonances are red-shifted as NP size increases, and they also are influenced by the refractive index of the surrounding medium, m_med. Particularly, for HRI dielectric materials, resonances are red-shifted as m_med increases, in addition to being strengthened [10]. In this work, the extinction efficiency (Q_ext) of HRI isolated NPs was proposed as a sensing parameter, based on its variations with m_med. These variations in Q_ext can be analysed in a straight-forward way from a theoretical point of view. However, at high incident frequencies (VIS-NIR), the experimental measurement of Q_ext is not trivial due to its intrinsic characteristics. Parameters other than extinction may be measured more directly and easily and present an alternative for sensing applications.

In previous works [9, 15, 16], it was demonstrated that the degree of linear polarization of scattered light in a right-angle scattering configuration, P_L (90°), is a suitable parameter for extracting specific scattering properties of a system with a high degree of accuracy. It also has been shown to be a suitable parameter for determining particle size, also constituting a good sensor for studying multiple scattering problems of metallic systems. For dielectric particles and, in particular, for those with high values of electric permittivity, the characteristics of P_L (90°) allow one to distinguish between their intrinsic electric or magnetic responses [9, 15].

In this research, isolated spherical NPs of representative HRI materials are analysed as a function of the refractive index of its surrounding medium (m_med ∈ [1, 2]) and the NP size (R ∈ [100, 250] nm). The spectral evolution of P_L(90°) is proposed as a polarimetric parameter for sensing, providing an estimate of m_med, and for nanosizing purposes. As we will show, it allows both dependencies to be quantified simultaneously because their effects on P_L (90°) are concomitant but independent. Although the primary focus of this study is on Silicon (Si) NPs, we consider the performance of four other HRI dielectric materials: Germanium (Ge), Aluminum Arsenide (AlAs), Aluminum Antimonide (AlSb) and Gallium Phosphide (GaP) [17]. The most important common feature of these HRI materials is their low absorption in the spectral range analysed (VIS-NIR) that can be considered insignificant in most cases.

This work is organized as follows: In Section 2 we will introduce the theoretical background. In Section 3, we will present the main results of this research, specifically, P_L (90°) will be compared with Q_ext in order to establish its validity for sensing purposes. In addition, the capabilities of P_L (90°) in sizing NPs will be discussed from a realistic point of view. Finally, results will be summarized in Section 4, where our concluding remarks will be presented.

2. Theoretical methods

Following the Lorenz-Mie formalism for scattering and absorption of light by small particles, the extinction and scattering efficiencies by a spherical particle are given by [18]:

\begin{array}{l} Q_{e x t} = \frac{2}{x^{2}} \sum_{n = 1}^{\infty} (2 n + 1) R e {a_{n} + b_{n}} \\ Q_{s c a} = \frac{2}{x^{2}} \sum_{n = 1}^{\infty} (2 n + 1) ({| a_{n} |}^{2} + {| b_{n} |}^{2}), \end{array}

where a_n and b_n are the well-known scattering coefficients of the Lorenz-Mie theory. These depend on the particle size and on the electric and magnetic properties of the NP relative to its surrounding medium. The size parameter x is defined as (2πm_medR)/λ, where λ is the wavelength of incident light in the surrounding medium. Physically, coefficients a_n and b_n represent the weighting factors of the different multipolar contributions of order n, electric or magnetic respectively. In particular, a₁ and b₁ correspond to the electric and magnetic dipolar modes, respectively, and a₂ and b₂ correspond to the quadrupolar modes, respectively.

In the same way, the intensities of light scattered by NPs are related to a_n and b_n through the scattering matrix elements, which depend on the scattering direction. For linearly polarized light, the components parallel (E_p(θ)) and perpendicular (E_s(θ)) to the scattering plane are related to the incident fields by a diagonal scattering matrix:

{(\begin{matrix} E_{p} (θ) \\ E_{s} (θ) \end{matrix})}_{s c a} \propto (\begin{matrix} S_{2} (θ) & 0 \\ 0 & S_{1} (θ) \end{matrix}) {(\begin{matrix} E_{p} \\ E_{s} \end{matrix})}_{i n c},

where S₁(θ) and S₂(θ) are the diagonal elements of the scattering matrix, which are defined from the Lorenz-Mie scattering coefficients a_n and b_n and the angle-dependent functions π_n(θ) and τ_n(θ) as [18]

\begin{array}{l} S_{1} (θ) = \sum_{n = 1}^{\infty} \frac{(2 n + 1)}{n (n + 1)} (a_{n} π_{n} (θ) + b_{n} τ_{n} (θ)) \\ S_{2} (θ) = \sum_{n = 1}^{\infty} \frac{(2 n + 1)}{n (n + 1)} (a_{n} τ_{n} (θ) + b_{n} π_{n} (θ)), \end{array}

The total scattered intensities with polarization parallel, I_p(θ), and perpendicular, I_s(θ), to the scattering plane are proportional to |S₂(θ)|² and |S₁(θ)|², respectively, and the linear polarization degree of the scattered light in a given direction θ can be defined as

P_{L} (θ) = \frac{I_{s} (θ) - I_{p} (θ)}{I_{s} (θ) + I_{p} (θ)}

In particular, in a right-angle scattering configuration (θ = 90°), and for a NP size smaller than the incident wavelength, we retain only the first two orders in Eq. (3) [9]:

P_{L} (90 °) = \frac{9 {| a_{1} |}^{2} + 25 {| b_{2} |}^{2} + 30 R e {b_{1} a_{2}^{*}} - 9 {| b_{1} |}^{2} - 25 {| a_{2} |}^{2} - 30 R e {a_{1} b_{2}^{*}}}{9 {| a_{1} |}^{2} + 25 {| b_{2} |}^{2} + 30 R e {b_{1} a_{2}^{*}} + 9 {| b_{1} |}^{2} + 25 {| a_{2} |}^{2} + 30 R e {a_{1} b_{2}^{*}}}

Indirectly, Eq. (5) may provide information about both the NP size and its surrounding medium based on the value of the polarimetric observation P_L (90°). This is because the scattering coefficients are related to the particle size and the relative refractive index. From an experimental point of view, right-angle-detection measurements are simple to execute and were used recently to analyse the spectral behaviour of P_L (90°) for a single particle made of an HRI material [9]. Although in experimental measurements the scattered light is collected and the signal is integrated over the solid angle of detection, in this work it was shown that P_L (90°) could be measured accurately in an experimentally feasible way. Figure 1 shows a set-up of the scattering geometry to perform right-angle-detection measurements.

Fig. 1 Configuration for performing right-angle light-scattering measurements.

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3. Results

In this section we present results of sensing by means of P_L (90°) to illustrate the concepts of the previous sections. Results about nanosizing will be shown and analyzed subsequently in section 3.3. These results will be related to the previous ones to present an overview of the experimental capabilities of P_L(θ = 90°).

3.1. P_L (90°) vs. Q_ext

In a recent study, García-Cámara et al. [10] evaluate both the evolution and sensitivity of Q_ext on m_med for HRI dielectric NPs. In order to compare these results with those obtained in this research, we present Fig. 2, which shows the spectral behaviour of Q_ext [4] and P_L (90°) for a Si NP (R = 200 nm, m_med = 1.0). The dipolar electric and magnetic, and quadrupolar magnetic resonances can be identified in both patterns for given R and m_med, so that P_L (90°) provides the information contained in Q_ext. The maxima in Q_ext identify the positions of the resonances in a₁, b₁ and b₂ exactly because the contribution of the other coefficients to that of the resonance is small. Thus, P_L (90°) is a relative estimate of the resonance strength. Not only is it able to reproduce exactly the location of the resonance peak, but its sign also contains information that can be used to identify the resonance [15].

Fig. 2 Spectral response of P_L(90°) and Q_ext (m_med = 1) for a Si spherical NP (R = 200 nm). Resonances a₁, b₁ and b₂ are labeled in the analysed spectrum. Refractive index values are taken from Ref. [17].

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Such spectra can be produced for other HRI dielectric materials to evaluate their behavior and capabilities in sensing. In Figs. 3(a)–3(d), we present Q_ext and P_L(90°) for isolated NPs (R = 200 nm) of some of the most common materials suggested for sensing applications: Ge, AlAs, AlSb and GaP. In order to allow a better comparison, Figs. 3(e)–3(h) represents the real (n) and imaginary (k) part of the refractive index of the selected materials. As shown in Figs. 3(a)–3(d), the primary resonance features of Q_ext can be associated with analogous features in P_L (90°), suggesting that the polarization curves contain the same information, especially in the spectral region in which the dipolar behaviour dominates.

Fig. 3 Left Column: Spectral pattern of P_L (90°) (m_med = 1) for spherical R = 200 nm NPs of a) Ge, b) AlAs, c) AlSb and d) GaP. The a₁, b₁ and b₂ resonance locations are labeled where they occur. Right Column: Real (n) and imaginary (k) parts of the refractive index of e) Ge, f) AlAs, g) AlSb and h) GaP. Values taken from Ref. [17].

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In reality, the majority of the analysed materials may show absorption, either intrinsic or due to contamination, and depending on the spectral range where they would be analysed [17]. The presence of absorption can severely dampen the efficiency of resonances. Nevertheless, even in the worst case, it has been demonstrated that electric resonances are less sensitive to absorption than magnetic resonances [19], so that their fingerprint in P_L (90°) can be followed in sensing. For instance, in the case of Ge shown in Fig. 3(a), absorption appears significant for wavelengths up to 1500 nm (k > 0.01, see Fig. 3(e)) and partially extinguishes the quadrupolar magnetic resonance. This resonance, b₂ (located at 1235 nm), is strongly influenced by its presence. However, the electric (a₁) and magnetic (b₁) dipolar resonances still show their well-determined peaks located at 1325 and 1740 nm, respectively. In addition, it has been demonstrated that interference terms between quadrupolar and dipolar modes in Eq. (5) produce special scattering conditions [9] that result in well-defined patterns of P_L (90°), such as the minima in Fig. 2 at λ ∼ 980 nm, and in Fig. 3(a) at λ ∼ 1220 nm. In this way, resonances can be identified even in a spectral range where the HRI dielectric material has low, but significant, absorption.

In addition to the ratio of the NP size to wavelength, the resonance position depends on the refractive index of the NP. For a given NP size, materials with higher refractive index red-shift the light-scattering response, and resonances. We can see this effect in the spectral evolution of P_L(90°) and Q_ext for AlAs, AlSb and GaP of Figs. 3(b), 3(c) and 3(d), respectively. For instance, the pattern described by GaP shows the b₂ resonance located at over 900 nm. Alternatively, AlAs shows the b₂ resonance located at a lower wavelength, and AlSb at a higher wavelength, than 900 nm. Thus, the refractive index of AlAs in the analysed range is lower than the refractive index of GaP, and that of AlSb is higher. That said, the right choice of materials and sizes, and their corresponding spectral ranges for each sensing application, will lead to better results in sensing devices.

3.2. P_L(90°) in sensing

Figure 4(a) shows the spectral evolution of P_L(90°) as a function of m_med for a Si spherical NP (R = 200 nm). As can be seen, values of all resonances evolve with m_med, so that an estimate of m_med can be obtained through the evolution of P_L(90°), for those wavelengths where resonances occur (λ_res). As Fig. 4(b) shows, the resonances spectral position λ_res red-shifts as m_med increases for both a₁ and b₂, whereas both the red-shift and the minimum in b₁ are hardly visible for m_med > 1.5. As can be seen, the red-shift is more significant for the electric dipolar resonance a₁ than for the magnetic quadrupolar resonance b₂.

Fig. 4 Si spherical R = 200 nm NP: a) Spectral behaviour of P_L(90°) as a function of λ (resonances a₁, b₁ and b₂ are labeled), and b) Resonance spectral position (λ_res) as a function of m_med.

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In order to evaluate the accuracy of P_L(90°) as an estimator of m_med, it is necessary to analyse the positions of the resonances as a function of the wavelength at which they occur λ_res, i.e., $\frac{δ P_{L} (90 °)}{δ λ} = 0$ . The sensitivity of P_L(90°) to m_med can be obtained through the parameter S_m(90°), defined as

S_{m} (90 °) = | {(\frac{δ P_{L} (90 °)}{δ m_{m e d}})}_{λ_{r e s}} |

The value of S_m(90°) shows the relative change in P_L(90°) as a function of the refractive index of the medium. This parameter is measured in inverse refractive index units (RIU⁻¹) [20] and can be used to evaluate how the values of P_L(90°) in the resonances are influenced by m_med variations. Accordingly, the greater the value of S_m(90°), the more accurate the measurement of m_med. To clarify this point, the evolution of P_L(90°) with respect to m_med is shown in Fig. 5(a), for a Si R = 200 nm NP and at those wavelengths where a₁, b₁ and b₂ resonances take place (λ_res). It is clear that the greater the slope, the greater the value of S_m(90°); i.e., small changes in m_med result in large variations in P_L(90°). On this basis, we can evaluate S_m(90°) (expressed in RIU⁻¹) averaged over a given m_med range. Figure 5(b) provides an example of how this parameter can be used to quantify the resonance behaviour of different materials. It shows S_m(90°) values for all semiconductors shown in Fig. 3 at λ_res and averaged over the range m_med ∈ [1, 2].

Fig. 5 a) Values of P_L(90°) for a Si spherical R = 200 nm NP on resonance as a function of m_med. b) P_L(90°) sensitivity (S_m(90°)) to m_med (m_med ∈ [1, 2]) at resonances for the selected materials (particle radius R = 200 nm).

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Fig. 6 Spectral evolution of P_L (90°) for a Si spherical m_med = 1.0 NP as a function of particle size R. The a₁, b₁ and b₂ resonance locations are labeled.

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The sensitivity values of P_L(90°) for all the materials and resonances mentioned thus far are large enough to be used to make experimental estimates of m_med. As shown in Fig. 5(b), most of the S_m(90°) values are larger than 0.5 RIU⁻¹ for both electric and magnetic dipolar resonances, and increase toward 1.5 RIU⁻¹ for the quadrupolar magnetic resonance. Obviously, S_m(90°) values on the order of 1.0 RIU⁻¹ imply similar changes in both P_L(90°) and in m_med. From a realistic point of view, the error margin of most polarimetric arrangements is smaller than 1% at present [21], so that S_m(90°) ∼ 1.0 RIU⁻¹ suggests an accuracy in sensing around 0.01 RIU at worst. By way of comparison, theoretically sensitivities of Q_ext are around 5 RIU⁻¹ [10], and Q_ext measurements include inherent problems that can lead to sensitivity losses, such as the evaluation of the amount of forward scattered light that has been scattered by the NP. Finally, apart from these experimental considerations, it is worth mentioning that the change of sign in P_L(90°) values, at either the a₁ or b₂ resonance positions, can provide a rapid estimation of m_med. As Fig. 5(a) shows, both values are positive for m_med < 1.40; whereas, they are negative for m_med > 1.45.

3.3. P_L(90°) in NP sizing

The primary factor determining the resonance location is particle size. As the NP size increases, the resonance is red-shifted toward larger wavelengths. Figure 6 shows the spectral behavior of P_L(90°) for different sizes of Si spherical NPs, whose radii R range from 100 to 250 nm. The exterior medium refractive index m_med = 1.0. This figure shows a nearly linear dependence on resonance position with NP size. The slope of this dependence is not constant, but depends on the resonance position on the spectra. What results is a stretching of the spectra as particle size increases. In other words, the NP size preserves its shape-material fingerprint in P_L(90°). This feature can be used to characterize NPs, but unfortunately, it breaks down as particle size increases and more resonances appear in the spectra [22].

In addition to the resonance location, it is important to know how strong the resonance and whether it can be measured. We consider in Fig. 7 the evolution of (a) P_L (90°) and (b) Q_ext with NP size R. We note that the strength of the polarization measurements is more stable, potentially making them easier to locate experimentally, compared with those of the extinction measurements. We can define a new sensitivity parameter, which relates the variation in P_L (90°) with changes in R, as

S_{R} (90 °) = | {(\frac{δ P_{L} (90 °)}{δ R})}_{λ_{r e s}} |

Fig. 7 Spectral evolution of the values of local maxima and minima in a) P_L(90°), and b) Q_ext, as a function of the particle size R at resonance locations for a spherical Si NP with m_med = 1.0.

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The sensitivity of P_L(90°) on R is measured in nm⁻¹ and it demonstrates that P_L(90°) is relatively insensitive to the size of the NP. Typical values are less than 0.001 nm⁻¹. The magnitude of the P_L(90°) values for the three NP resonances of interest are shown in Fig. 7(a). There is little change in the values and this change is almost linear over a large span of particle size R. In Fig. 7(b), Q_ext shows more significant changes in the resonances magnitudes and the response is not always linear, especially for the magnetic quadrupolar resonance.

The magnitude of P_L(90°), being almost independent of particle size, has high potential as a sensing parameter, especially NPs submerged in liquids. Since the magnitude of P_L(90°) only has a strong dependence on m_med, it can be used for characterizing the external medium. With m_med now a known quantity, the spectral position of the resonance can be used to determine NP size R, as Fig. 8 suggests.

Fig. 8 Spectral displacement of the local maximum or minimum in P_L (90°) for a Si spherical NP (m_med = 1.0) as a function of the particle size (R).

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Determining NP size R is a common characterization problem. Suspensions of NPs in a liquid medium are typically polydisperse, with size deviations ranging from 10 to 0.1 times a mean value R. The ability to characterize both the particle size R and to estimate the medium refractive index m_med simultaneously is what makes P_L(90°) a promising experimental tool.

4. Conclusion

In this manuscript, we have discussed the use of the degree of linear polarization of light scattered in a right-angle configuration, P_L(90°), as a tool to characterize NPs and the medium they are suspended within. Lorenz-Mie calculations of spherical NPs made of Si, Ge, AlAs, AlSb and GaP were made at several radii (R ∈ [100, 250] nm) and refractive indices of the surrounding medium (m_med ∈ [1.0, 2.0]).

Graphs of P_L(90°) suggest that this quantity contains the same information about resonance locations and strengths as Q_ext, but there are several advantages to using P_L(90°). First and foremost, as opposed to Q_ext, which requires a complicate measurement, P_L(90°) can be measured easily and accurately at a single detector position. Second, the magnitude of this resonance has only a small, nearly linear dependence on the size of the NP (R), and a much larger, nearly linear dependence on m_med; whereas, their influence on Q_ext is much more complicated. This makes characterization of the external medium a much simpler process using polarization.

While the absorption of some materials, such as Ge and Si in the NIR and VIS respectively, is able to make unsightly its P_L(90°) fingerprint, with the use of appropriate materials, resonances and sizes may allow monitoring m_med by means of P_L(90°), regardless of the spectral range. Furthermore, the change of sign in P_L(90°) values, at either a₁ or at b₂ resonance spectral position, can provide a rapid estimate of m_med at first glance.

Acknowledgments

This research was supported by MICINN (Spanish Ministry of Science and Innovation, project FIS2013-45854-P) and USAITC-A (United States Army International Technology Center Atlantic, grant W911NF-13-1-0245). The authors would like to thank Fernando Moreno and Gorden Videen for their insightful comments.

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Using linear polarization for sensing and sizing dielectric nanoparticles

Abstract

1. Introduction

2. Theoretical methods