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Abstract
We investigate effective medium properties of nanoparticles (NPs) in surface plasmon (SP) resonance detection. Attention was paid to the effective medium characteristics with respect to the particle distribution in equally spaced, aggregated, and intermediate models, although effects of other parameters such as size, material, and concentration were also explored. The results suggest that the distribution may cause significant measurement deviation by as much as 20% for gold NPs and less than 5% for silica. Particle concentration showed complicated dependence in the effective medium. Different mechanisms were observed to govern effective medium properties of dielectric and metal NPs, SP mode transition and multiple scattering for silica NPs. In contrast, metal damping dominated resonance characteristics for gold NPs. The results are expected to provide fresh insights on how to apply an effective medium and interpret measured data in SP resonance sensors and beyond.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Complex optical structures on micro and nanometric scales have emerged as an integral part in all aspects of daily human lives. Precise yet simple ways to analyze complex structures have thus been highly desired with growing complexity. Among various approaches to this goal, an effective medium is a tool which allows simplified analysis of an inhomogeneous composite structure by approximating one with a homogeneous anisotropic layer and thereby enables analytical computation of complex optical structures [1–6]. Effective medium performs homogenization and therefore may be viewed as a concept opposite to metamaterial, including photonic crystals in a larger scope, which implements property not found in nature by meso/nanoscale engineering [7,8].
Effective medium is provided by various effective medium theories (EMTs) and has found uses in many applications ranging from simple spectroscopy to biosensor modalities. In biosensing experiments, molecular interactions involve a large number of binding events between participating molecules. An optical model of these interactions typically employs an effective medium, a good example of which is one corresponding to DNA immobilization and hybridization that was experimentally determined [9]. One of the areas that EMTs can be particularly useful for is surface plasmon resonance (SPR) detection, because an EMT may help reduce the complexity of analysis and thus predict the detection characteristics in an intuitive manner, as more complicated nanoscale structures emerge in SPR biosensing to circumvent classical limitations. For example, use of patterned nanoarrays is one application which would benefit directly from an EMT to improve the moderate sensitivity of SPR detection and to produce enhanced sensor properties. Effectiveness of EMT in SPR has also been evaluated for propagating surface plasmon (SP) localized into spoof SPs [10] and particle plasmon in metal-coated dielectric nanospheres [11], while EMT has also been applied to plasmonic solar cells [12], superlattices [13], plasmon-enhanced fluorescence [14], nanostructured gratings [15,16], optical force on plasmonic shells [17], and nanoantenna [18]. Shape-dependence of metallic nanoparticles (NPs), nanorods, and polydisperse nanoellipsoids on an effective medium was investigated in SPR sensing [19–22].
Despite great utilities of EMTs, many parameters of an interaction are ignored in the assessment of an effective medium, for instance, hybridization efficiency and molecular configurations are largely disregarded when approximating DNA interactions. Structural effects have long been neglected in EMT and were found to need correction [23]. If we take Maxwell-Garnet EMT, one of the most widely used EMT for a bi-anisotropic medium, the permittivity of an effective medium εeff = (neff + iκeff)2 is obtained from
where εamb and εnp represent the permittivity of ambiance and NPs [24–28]. fv denotes the volume fraction concentration of NPs. Equation (1) clearly shows that an effective medium, in this case, depends on only a few parameters and, as a matter of fact, only on the particle concentration once material permittivities are fixed. More exhaustive investigation of the parameter effects on an effective medium, therefore, can be enlightening. Here, we are mainly interested in the effect of the distribution because the distribution is typically assumed to be random and a specific configuration is ignored in most EMTs with an exception of Rytov EMT which provides an effective medium for periodic structures [29], i.e.,for p-polarized light. In general, molecules participating in a chemical interaction do not form periodic patterns. In this sense, the distribution remains as a parameter that may affect an effective medium with significant experimental noise.In this paper, we investigate the properties of the effective medium in SPR detection using the propagation matrix retrieval procedure. In particular, we focus on an effective medium of nanoscale molecular labels, such as metallic or dielectric NPs and nanorods, that amplify optical and/or physiochemical signatures of an interaction by more than an order of magnitude [30–32]. In fact, an effective medium has been employed rarely, if ever, for such labels, which we call nanolabels (NLs), presumably because the labels are relatively large in size compared to the interacting molecules and thus tend to make an approximation with an effective medium potentially limited, although EMTs, in general, remain valid for a fairly wide size range of NLs. Also note that NLs are often used to localize light fields for efficient amplification of optical signatures, while an effective medium describes the average properties of NLs. For this reason, we first check the validity of an effective medium of NLs in SPR detection. Then we examine effective medium properties for the label distribution and other parameters such as size, material, and label concentration.
2. Numerical models and method
2.1 Numerical models
Figure 1(a) illustrates a simplified model based on binding NLs to approximate a molecular interaction involving NLs after a post-binding rinsing step. For simplicity, we have performed two-dimensional calculation because NLs show most of the induced electric fields in the plane perpendicular to the principal axis along the polarization direction [33,34].
Fig. 1 (a) Schematic illustration of the calculation models with a label number N = 4: scattering NLs are assumed to be aggregated (AG model) and equally spaced (ES model). Intermediate model distribution appears between AG and ES model. x denotes the label-to-label distance. For AG and ES model, x = ϕ and Λ/N. All the distribution models correspond to an identical effective medium on the right. (b) Histograms of nearest-neighbor distances between NLs (ϕ = 100 nm) corresponding to the AG and ES model distributions and intermediate models with x = 1.25, 2.50, and 3.75 μm at a concentration of CNL = 4/Λ (Λ = 20 μm is assumed with N = 4). Each distribution consists of two components except for the ES model. Arrows represent the transition from the AG to the ES model. (c) Effective medium approximation of SPR detection characteristics compared with exact results calculated by RCWA. NLs were assumed to form ES and AG model distribution with ϕ = 50, 100, 250, and 500 nm. Both silica and gold NLs were considered (top two vs. bottom two rows for silica and gold). Label concentration is fixed at CNL = 1/μm with Λ = 20 μm and N = 20. The standard deviation σ with respect to the exact results for various distributions of (d) silica and (e) gold NLs with ϕ = 10 nm ~-700 nm. CNL = 1/μm with Λ = 20 μm and N = 20. The data were lineated by Bezier interpolation.
Of particular interest is a configuration of aggregated NLs with center-to-center inter-label distance x = ϕ (ϕ: NL size in diameter), in which the effects of higher-order multiple scattering modes would become severe (AGgregation or AG model). On the other hand, NLs may be separated by an equal distance as long as the concentration remains the same (Equally Spaced or ES model), i.e., x = Λ/N (Λ: period and N: NL number per period). We have also considered intermediate models with ϕ < x < Λ/N in Fig. 1(a) between those of aggregated and equally spaced distribution. For simplicity, we assumed all the models of NLs to be monodisperse in equilibrium. To the first degree, under experimental conditions, the ES model is likely to prevail at a low concentration of NLs while it is the AG model at a high concentration [35]. For the evaluation of the distribution dependence of SPR detection characteristics, we have used the AG and ES model as the potential two extremes of an interaction mediated by NLs.
For comparison of the SPR detection performance, we have calculated angular resonance spectra for the AG and ES model. In each model, a 2-nm thick chromium and a 50-nm gold film (refractive index n = 3.48 + 4.36i and 0.18 + 3i for λ = 632.8 nm) were assumed on SF10 glass substrate (n = 1.723) with buffer ambiance (n = 1.33). A 24-mer single-stranded DNA (ssDNA) probe layer that is 8.16-nm thick with n = 1.449 was modeled as a biomolecular interaction [9]. For hybridization, complementary ssDNA molecules are conjugated with gold or silica NLs in the diameter range of ϕ = 10 ~ 700 nm. For silica NLs, n = 1.457 [36]. Gold NLs may induce and localize particle plasmon that interacts with propagating SP, while silica NLs have light scattering interfere with momentum-matching of SP modes and alter SPR characteristics. In many applications, gold and silica are used together to form a core-shell structure by employing silica NPs as a core coated with gold shells or vice versa [37–39].
The concentration of NLs (CNL) was measured with a number density defined as the number per calculation period, i.e., N/Λ [unit: #/μm]. Note that the refractive index was considered not to change as a result of hybridization between complementary ssDNA molecules other than the change produced by the conjugation of NLs. This is for two reasons: first, the refractive index is not well defined on a molecular scale and secondly, we intend to focus on the effect of particle scattering and potential absorption which can dominate the effect of DNA hybridization per se. Incident light was assumed to be monochromatic and p-polarized with λ = 632.8 nm. We assume that all the materials are non-magnetic and thus μeff = μ0.
2.2 Numerical method
For light wave calculation in the far-field, rigorous coupled-wave analysis (RCWA) has been used with 50 spatial harmonic orders under periodic boundary conditions. The periodicity inherent in the calculation can be minimized for Λ >> λ. Near-fields produced by various NL distribution models were also calculated using RCWA. Once field intensity was found with RCWA, it was fitted to the effective medium models. Using the propagation matrix approach,
where the index m = 0 ~ 5 represents model layers corresponding to SF10 substrate, Cr, Au, ssDNA, effective medium of NLs, and ambiance [40,41]. E0,i denotes an incident electric field and E0,r the field obtained by RCWA. E5,i is an evanescent field formed in the ambiance. Elements of propagation matrix Sm,m + 1 are given bytm,m + 1 and rm,m + 1 are transmission and reflection coefficients of light propagating from the m-th layer to the (m + 1)-th layer. Phase delay δm = kmdmcosθm with km, dm, and θm respectively as wave number, thickness, and propagation angle.3. Results and discussion
3.1 Distribution characteristics
Small particles behave either as well-separated optically isolated objects, which we use the ES model to address, or as an ensemble of the whole (AG model). The response of an N-particle system can be described by the property of an individual particle, only if the particles are well-separated in optically thin samples. In general, however, the single-particle approximation collapses in an ensemble of many particles, in which case the electromagnetic interactions play a crucial role in explaining SP oscillations [42].
In this sense, understanding the distance distribution between NLs can be helpful. The histograms of inter-label distances for N = 4 corresponding to the models described in Fig. 1(a) are shown in Fig. 1(b) for Λ = 20 μm and ϕ = 100 nm. We assumed that strong optical coupling exists only between nearest-neighbor NL particles without the long-range correlation [43]. A random distribution of NLs in a more realistic model would exhibit a similar transition, although each histogram is to be broadened and may not be as binary as in Fig. 1(b) with broad distribution characteristics. In this regard, AG and ES model configurations using an effective medium may reflect characteristics of random distribution despite the simplicity of the model.
The histograms clearly show that the transition between AG and ES distribution of NLs, as represented by the arrow for the distribution changing from the AG to the ES model. Except for the ES model, each distribution consists of two components: the one within a period is more dominant than the other between neighboring periods. Note the significant disparities in the histogram distribution among various NL models. Although optical characteristics may be vastly affected by different model configurations, those presented in Fig. 1(a) are all approximated with an identical effective medium. In this sense, the distribution-dependent variation in the effective medium does not show up explicitly in conventional EMTs.
3.2 Validity of an effective medium
Figure 1(c) shows the validity of an effective medium based on the propagation matrix retrieval procedure for estimating SPR detection characteristics of silica and gold NLs at CNL = 20/20 μm = 1/μm, as the label size is varied. For both ES and AG models, the effective medium produces data in excellent agreement with exact results calculated by RCWA. In fact, an improved agreement is observed for the ES model over the AG model and with the smaller NL size due primarily to the reduced multiple scattering, which EMTs using simple thin film models as an effective medium are difficult to reflect [44]. An exception to these trends is the ES model of gold NLs with ϕ = 50 nm. In this case, the disparity between EMT and exact results becomes more significant than using larger labels. This is manifested by increased damping characteristics. The disparity is reduced for smaller NLs of ϕ < 50 nm.
The validity of EMT can be understood more quantitatively based on standard deviation σ, which we define as σ = [Σi = 1(RRCWA − REM)2/Nθ]1/2, where RRCWA and REM represent reflectance calculated by RCWA and effective medium models with i running over the angular sweep θ = 50° ~ 80° with a step size of 0.01° (angular sample number Nθ = 3001). Standard deviation presented for each model in Figs. 1(d) and 1(e) overall confirms the trends observed in Fig. 1(c), i.e., EMT shows improved performance using the ES model. Note the exception for gold NLs with ϕ ~ 50 nm in connection with the specific resonant size determined by the SP dipole scattering mode [45]. The degree of agreement was comparable between silica and gold.
3.3 Effective medium properties
In this section, we investigate the effect of label distribution and other secondary parameters, such as its concentration, size, and material, on the effective medium determined of plasmonic detection.
3.3.1 Effect of label distribution
Effective medium refractive index neff and κeff are presented in Fig. 2. Clearly, effective medium properties vary significantly depending on the distribution models. The graded area represents the difference in the effective index and absorption constant between various models, i.e., Δneff = neff,max – neff,min and Δκeff = κeff,max – κeff,min. Δneff and Δκeff describe the uncertainty experienced under experimental conditions in which one cannot control the distribution of NLs. Overall, Δneff and Δκeff are much larger for gold NLs than for silica (note the different scales on the y-axis in Fig. 2). The maximum index disparity associated with the distribution Δneff = 0.033 for silica with ϕ = 400 nm vs. 0.279 for gold with ϕ = 600 nm. In the case of absorption constant, Δκeff = 0.030 with ϕ = 400 nm (silica) vs. 0.254 with ϕ = 150 nm (gold). This strongly suggests that the index variation arising from the stochastic nature of the label distribution be much larger with metallic NLs, by 8.5 times ( = 0.297/0.033), almost an order-of-magnitude larger, than the case of dielectric labels. The influence of the distribution can be evaluated on a quantitative basis by introducing relative error (RE), which is defined as RE = Δneff/neff,ave for an effective refractive index. Here, the average effective index neff,ave = (neff,max + neff,min)/2. Maximum RE was obtained as 2.4% and 19.2% for silica and gold NLs, when ϕ = 400 and 600 nm, respectively, as denoted with arrows in Figs. 2(a) and 2(b).
Fig. 2 Effective medium refractive index neff (a) for silica and (b) gold NLs and absorption constant κeff (c) for silica and (d) gold with ϕ = 10 nm ~ 700 nm. Effective permittivity is given by εeff = (neff + iκeff)2. Arrows in (a) and (b) represent the largest index difference Δneff. Label concentration is fixed at CNL = 1/μm with Λ = 20 μm and N = 20. The gray shade represents the difference in the effective medium refractive index and absorption constant Δneff and Δκeff between the maximum and the minimum of the various distribution models. The data in solid lines were Bezier interpolated.
Although silica has no absorption, κeff may not be zero as a result of extinction by scattering [46,47]. Figure 2(c) shows that the AG model distribution was associated with absorption due to the inter-particle coupling and multiple scattering when an individual NL comes into close proximity to neighboring ones. This trend is reversed for gold NLs, as shown in Fig. 2(d), i.e., the ES model presents much higher absorption than the AG model, because the damping characteristics of individual NLs tend to dominate. Electromagnetic coupling in the aggregated clusters of metallic NPs has been studied rather extensively and is known to govern the extinction spectra for an inter-particle distance x < 5ϕ [48–51]. Also, squeezed optical fields were observed in an aggregated gold NP chain [52] and the effects of local aggregation formation on the infrared absorption spectrum were studied [53].
Insights on the absorption characteristics of gold NLs can be obtained from the near-field distribution produced in the ES and AG model, as shown in Fig. 3. In the ES model, fields are localized between NLs when the size is small, indicating low metal absorption. In contrast, fields do exist and tend to be localized in NPs of a larger size, causing significant absorption in agreement with Fig. 2(d). On the other hand, for the AG model, near-fields are largely extinguished within aggregated gold NLs and distributed mostly in the space between aggregated labels with little absorption leading to very low κeff.
Fig. 3 Near-field intensity distribution (|E|2): (a) ϕ = 30 nm and (b) 100 nm in the ES model (magnified view of two NLs). (c) ϕ = 200 nm and (d) 700 nm in the AG model. For both models, Λ = 20 μm.
Interestingly and quite expectedly, extremely small silica NLs, i.e., for ϕ < 50 nm ~ λ/10, exhibit very little distribution-dependent disparities in the effective medium. In other words, an effective medium is affected little by the distribution in this range, which confirms the conventional wisdom that an EMT remains valid in the quasistatic long-wavelength limit for ϕ << λ [54]. Obviously, this is not true in general when the size of NLs grows and also for metallic ones regardless of the size. For gold NLs, the disparity between the models, especially between the ES and the other models, is stark even with a small size.
3.3.2 Size and material dependence
In Fig. 2(a), effective refractive index neff tends to increase with larger NLs of silica regardless of the distribution models, although the degree of the increase depends on the model. The change of effective index is far from being linear with respect to the label size except for the ES model, in which case the influence of multiple light scattering from neighboring NLs remains to be the least. The effective index also increases with size for gold, as shown in Fig. 2(b), although the dependence is much more complicated due to the interaction between propagating SP and localized particle plasmon modes, which manifests itself as metallic damping coupled with the characteristics of higher-order scattering. In either case, increased effective refractive index neff represents an amplification effect by employing larger labels, which was confirmed experimentally [31]. In the case of metallic NLs, a high refractive index is observed in the ES model with ϕ ~ 50 nm, which is related to the SP dipole scattering mode, as mentioned earlier. In the ES model, the overall increase of neff is much larger with gold NLs than with silica: this reflects more efficient amplification of optical signatures for SPR detection as well as higher near-field efficiency using metallic NPs over dielectric ones if the distribution can be controlled [45]. On the other hand, effective absorption constant κeff for silica in Fig. 2(c) shows drastically different behavior with the highest absorption at ϕ = 400 nm (marked by arrow) and decreases with size above 400 nm, a characteristic of mode transition and resonant scattering for aggregate NLs.
The overall nature becomes clear in the resonance characteristics presented in Fig. 4. Interestingly, when NLs are apart in the ES model, SP modes propagating in the film dominate the resonance characteristics, as shown in Fig. 4(a). However, if NLs are aggregated, effective index increases with the size and the aggregate effect eventually governs the resonance characteristics (see Figs. 4(b)-4(d)), i.e., the AG model creates the transition between SP modes in the film and aggregates. The transition into SP modes corresponding to higher momentum (or blue-shift in terms of wavelength) with larger NLs reflects increased light extinction by scattering as well as higher index arising from the aggregation. The SP mode transition is not observed for gold NLs, except that absorption by damping is significantly larger, especially in the case of the ES model observed in Fig. 2(d) and 4(e), than for silica. This is caused largely by the absorption of gold vs. little absorption for silica by comparison.
Fig. 4 Resonance characteristics with silica NLs: (a) ES model, intermediate models with (b) x = ϕ + 100 nm and (c) x = ϕ + 50 nm, and (d) AG model. For gold NLs, (e) ES model, (f) x = ϕ + 100 nm, (g) x = ϕ + 50 nm, and (h) AG model. Label concentration fixed at CNL = 1/μm with Λ = 20 μm and N = 20. Black dotted lines show the evolution of modes with the label size. Arrows represent an increase in diameter (larger NLs) from ϕ = 10 to 700 nm.
3.3.3 Effective medium dependence on concentration
Effective refractive index neff and absorption constant κeff tend to increase with the higher concentration of NLs, as shown in Fig. 5. This is conceptually obvious because larger label volume per unit surface area is directly connected with a higher likelihood of light waves to interact with labels. Interestingly, gold NLs give rise to a ripple structure in the effective medium, similar to interference, at high concentration. The ripple does not appear with silica and originates from resonant SP modes of a metallic sphere in the measured extinction cross-section, which translates into the effective medium [46].
Fig. 5 Concentration dependence of effective medium properties with respect to NL size (ϕ) for CNL = 0.25 ~ 1/μm with Λ = 20 μm and N = 20: (a) neff and (b) κeff for silica NLs. For gold NLs, (c) neff and (d) κeff. The labels are assumed in the ES and the AG model (ES: filled and AG: open symbol).
For quantitative evaluation of concentration dependence of an effective medium, we have introduced the index slope Sn and Sκ of effective medium index and absorption constant relative to the concentration, Sn = δneff/CNL and Sκ = δκeff/CNL, where δneff = neff(ϕ) – neff(ϕ = 0) and δκeff = κeff(ϕ) – κeff(ϕ = 0). ϕ = 0 represents the absence of labels in the course of plasmonic detection. The index slope obtained from four-point linear regression shows that the slope is in general positive, although it is highly nonlinear. Interestingly, Sκ for silica NLs in the ES model remains zero, i.e., no effective absorption regardless of the label concentration. This suggests that inter-particle coupling is minimum even at the highest concentration that we considered. Furthermore, correlation analysis shows that the correlation coefficient between effective indices and the concentration is positive for the most part. The concentration dependence in the ES distribution tends to be much more linear than in the AG model. Likewise, effective indices for silica NLs change linearly compared to gold due to non-linear damping characteristics for the latter.
4. Conclusion
As a summary, we have investigated the effective medium dependence of NP labels in SPR detection on the distribution and other parameters such as size, material, and concentration. The distribution of NLs was explored in the ES, AG, and intermediate models.
Effective medium showed diverse behavior, the implication of which reaches beyond effective medium properties. Most importantly, it is suggested that the distribution of labels, unless it is controlled, may cause additional deviation and thereby be responsible for a sizeable portion in the measured errors. The results in Fig. 2 suggest that the deviation may, in fact, be quite significant to be almost as high as 20% for the case of gold NLs attributed to the anisotropic attenuation and damping through SP absorption. The deviation was smaller for silica to be less than 5%.
Effective medium properties of dielectric and metal NPs are governed by different mechanisms: for silica, the transition between modes was observed when SP modes are matched at higher momentum. In contrast, metal damping tends to dominate resonance characteristics for gold NLs.
Funding
National Research Foundation of Korea (NRF) (No. 2018R1D1A1B07042236).
Acknowledgments
KK acknowledges the support by the BK21 program of School of Electrical and Electronic Engineering of Yonsei University.
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