## Abstract

Theoretical study of sensing properties of lattice resonances supported by arrays of gold nanoparticles expressed in terms of the figure of merit (FOM) is reported. Analytical expressions for the FOM for surface and bulk refractive index changes are derived to establish the relationship between the sensing performance and design parameters and to allow for the design of nanoparticle arrays with optimal sensing performance. It is demonstrated that lattice resonances exhibit about two orders of magnitude higher bulk FOM than localized surface plasmon (LSP) resonance and that the surface FOM provided by lattice resonances and LSP resonances are comparable.

© 2013 Optical Society of America

## 1. Introduction

Metal nanoparticles (NPs) exhibit a variety of electromagnetic modes that provide distinct optical properties [1]. An isolated metal NP can support localized surface plasmons (LSP), which exhibit an electromagnetic field highly confined at the surface of the NP and give rise to a characteristic peak in the extinction spectrum. When metal NPs are arranged into a periodic array, they can support collective electromagnetic modes, so-called lattice resonances. In contrast to LSPs, lattice resonances are delocalized propagating waves and manifest themselves in the extinction spectrum by considerably narrower features.

The concept of lattice resonances was initially developed in the context of surface-enhanced Raman scattering [2]. Subsequently, the excitation spectra of 1-D [3] and 2-D arrays [4, 5] were theoretically studied using the coupled-dipole approximation (CDA). Using the CDA method, the lattice resonances on arrays of metal NPs were linked with those supported by both arrays of holes as well as dielectric NPs [6]. The effect of the substrate supporting the array was studied and it was demonstrated that lattice resonances are suppressed when the refractive index of the medium above the array departs from that of the substrate [7]. The sharp extinction peaks due to lattice resonances were first experimentally demonstrated on a 1-D array [8], and later observed on 2-D arrays of nanodisks [9] and nanorods [10]. As the properties of lattice resonances are sensitive to changes in refractive index of the surrounding medium, they can serve as a tool for refractive index sensing [11]. While the potential of lattice resonances for bulk refractive index sensing has been established [11], the investigation of potential of lattice resonances for sensing of refractive index changes occurring only at the surface of the NPs, which is of great interest in label-free affinity biosensing [12], is still lacking.

In this paper, we present a theoretical analysis of the sensing properties of metal NP arrays based on the coupled-dipole approximation approach. Analytical expressions for the figure of merit (FOM) regarding arrays of metal NPs exposed to bulk and surface refractive index changes are derived and used to design arrays with an optimal performance.

## 2. Transmission spectrum of arrays on nanoparticles

In order to describe optical properties of an array of metal NPs using the CDA, we assumed that the size of the NPs is small compared to both the wavelength and period of the array. In the CDA approach, each NP in the array is treated as an electric dipole with an electric polarizability *α*. When a dipole is excited by an electromagnetic wave, it reradiates a scattered wave with an amplitude proportional to its dipole moment. The field acting on a dipole is then a sum of the incident field plus the field radiated by all other dipoles. Assuming that the induced polarization for each NP is the same, an analytical expression for the effective polarizability is $1/{\alpha}^{*}=1/\alpha -G$, where *G* is the lattice sum, which accounts for the field scattered by the array [4].

In this study, we consider a NP with a general ellipsoid shape having semiaxes *a,b,c*, and polarizability*α*. In the electrostatic approximation $1/{\alpha}^{es}$can be expressed as

*L*is a shape factor ($L=1/3$ for a sphere) [13] [see, Fig. 1(d)], and

*ε*and

*ε*are the dielectric constants of the NP and surrounding medium, respectively [13]. In these simulations, we assume that the NPs are made of gold (dielectric constant was taken from [14]) and that the surrounding medium is aqueous (${n}_{m}=\sqrt{{\epsilon}_{m}}=1.33$).

_{m}To account for radiative damping, the expression for the electrostatic polarizability can be expanded as follows: $1/\alpha ={1/\alpha}^{es}-2i{k}^{3}/3$ [15], where $k={k}_{0}{n}_{m}$ is the wavenumber of light. As follows from Fig. 1(a), which shows the inverse polarizability for NPs of different sizes, the electrostatic approximation provides a good agreement with the exact solution by Mie [13].

Under normal incidence of the incident electromagnetic wave, the lattice sum can be expressed as

**R**

*are the positions of NPs [16]. The rigorous summation of Eq. (2) for a different number of NPs arranged in a two-dimensional square array with period*

_{n}*Λ*is presented in Fig. 2.

As follows from Fig. 2, the finite number of NPs gives rise to oscillations in the spectrum of *G* and $\text{Re}\left\{G\right\}$and $\text{Im}\left\{G\right\}$exhibit maxima at the wavelengths $\Lambda {n}_{m}\left(1+1/\left(4N\right)\right)$ and $\Lambda {n}_{m}\left(1-1/\left(4N\right)\right)$, respectively. The maximal values of$\text{Re}\left\{G\right\}$and $\text{Im}\left\{G\right\}$ increase with the number of NPs in the array. For an infinite number of NPs, the dipole sum diverges at $\Lambda {n}_{m}$. In the Appendix it is shown that for $1.25>\lambda /\Lambda {n}_{m}>1+1/\left(4N\right)$, Eq. (2) can be reduced to

Using the optical theorem [13], the extinction cross section ${c}_{ext}$ can be written as ${c}_{ext}=4\pi k\text{Im}\left\{{\alpha}^{*}\right\}$ and the zero-order transmittance *T* under normal incidence can be expressed as

*λ*). Since $\text{Re}\left\{G\right\}$ values are close to zero, the contribution of the collective coupling is negligible and ${\alpha}^{*}\approx \alpha $ near

_{LSP}*λ*. The narrow resonance is associated with so-called lattice resonances (

_{LSP}*λ*) and can be excited only when the Rayleigh anomaly occurs at longer wavelengths than LSP (${\lambda}_{RA}>{\lambda}_{LSP}$), where $\mathrm{Re}\left\{1/\alpha \right\}>0$. When ${\lambda}_{RA}<{\lambda}_{LSP}$, the lattice resonance cannot be excited [Fig. 3(b)] and only the maxima corresponding to a Rayleigh anomaly are visible in the spectra.

_{r}The spectral position of the lattice resonance is shifted from the Rayleigh anomaly to longer wavelengths by $\delta ={\lambda}_{r}/{\lambda}_{RA}-1$. Using Eq. (3), for $\delta \ll 1$, *δ* can be approximated by

The transmission can be expanded around the lattice resonance into a Taylor series. By neglecting the higher-order terms, the transmittance can be expressed in the Lorentz form:

*W*:

Figure 4 illustrates the effect of the parameters associated with an array on the lattice resonance. It can be seen that an increase in the NP size or aspect ratio results in (i) a shift in the resonance to longer wavelengths due to a decrease of $\text{Re}\left\{1/\alpha \right\}$, and (ii) broadening of the transmission feature (due to increased losses) [Figs. 4(a) and 4(b)]. Similarly, an increase in the period of the array shifts the resonance to longer wavelengths and reduces the width of the resonance feature (due to decreased losses) [Fig. 4(c)]. When the number of NPs increases, the resonance feature becomes narrower and deeper (due to decreased losses) [Fig. 4(d)]. For a sphere, the results obtained using the electrostatic approximation of polarizability were compared with the Mie solution [Figs. 4(a)-4(d)] and found to be in qualitative agreement. The observed differences in the positions of the lattice resonance are due to the imperfect approximation of the inverse polarizability [see, Fig. 1(a)].

## 3. Sensitivity of the lattice resonance

Two types of sensitivity to refractive index changes are considered here. The bulk sensitivity *S _{B}* is defined as the sensitivity to changes in the refractive index of an infinite homogeneous medium surrounding the NP. The surface sensitivity

*S*is defined as the sensitivity to changes in the refractive index within a limited distance from the surface of the NP.

_{S}As shown in Fig. 5(a), a change in the bulk refractive index produces a small change in the polarizability and a large change in the lattice sum, which results in a shift of both the Rayleigh anomaly and lattice resonance to longer wavelengths. By differentiating the lattice resonance condition [Eq. (5)], the sensitivity to bulk refractive index changes can be expressed as

*λ*for a sufficient number of NPs, changes in the surface refractive index do not produce a shift of the Rayleigh anomaly. By differentiating the lattice resonance condition [Eq. (5)], the sensitivity to surface refractive index changes can be expressed as

_{RA}*a, b, c,*and a shell thickness

*Δ*can be described by the following equation

*ε, ε*, and

_{m}*ε*are the permittivity of the NP, surrounding medium, and dielectric shell, respectively [13]. When the thickness of the shell is much smaller than the size of the NP, ${L}^{\left(1\right)}\approx {L}^{\left(2\right)}=L$. The factor $1/f=\left(a+\text{\Delta}\right)\left(b+\text{\Delta}\right)\left(c+\text{\Delta}\right)/\left(abc\right)$ denotes the fraction of the total NP volume occupied by the inner ellipsoid. Using Eq. (13), the real part of inverse polarizability can then be expressed as

_{s}## 4. Figure of merit

Two types of figures of merit are investigated: the figure of merit corresponding to bulk refractive index changes ($FO{M}_{B}={S}_{B}/W$) and the figure of merit corresponding to surface refractive index changes ($FO{M}_{S}={S}_{S}/W$). Assuming the parameters of the array satisfy the lattice resonance condition [Eq. (5)], the figures of merit can be expressed as

where ${g}_{i}={\Lambda}^{3}\left(\mathrm{Im}\left\{G\right\}-{G}_{0}\right)$, ${g}_{r}={\Lambda}^{3}\mathrm{Re}\left\{G\right\}$, and*A*and

*B*are

Figure 6 illustrates the dependence of *FOM _{S}* and

*FOM*on

_{B}*δ*for different resonant wavelengths, calculated using Eq. (15) and Eq. (16). The parameter

*δ*was assumed to lie within the range $1/\left(4N\right)<\delta <{\lambda}_{r}/{\lambda}_{LSP}-1<1/4$ to satisfy the lattice resonant condition [Eq. (5)]. To reduce the effect of the fast oscillation on the resonant feature, lattice sums were smoothened by means of the moving average function. As follows from Eq. (19),

*FOM*and

_{S}*FOM*exhibit a maximum at the optimized values of

_{B}*δ*

*FOM*increases with the resonant wavelength and shape factor

_{B}*L*(which decreases with

*A*) and

*FOM*exhibits a maximum at a specific resonance wavelength that depends on the number and shape of the NPs.

_{S}The dependence of *FOM _{B}* on the spectral distance between the Rayleigh anomaly and the lattice resonance was explored in an experimental study involving arrays of gold NPs with the radius of 47.5 – 82.5 nm, the height of 50 nm, and the periods of 300 – 600 nm [11]. Based on the experimental data, an empirical formula was postulated as$FO{M}_{B}=0.79/{\delta}_{\nu}-1.38$, where ${\delta}_{\nu}=1-{\nu}_{r}/{\nu}_{RA}$and

*υ*and

_{r}*υ*were the lattice resonance and Rayleigh anomaly frequencies [11]. To allow for the comparison of our theory with this work, Fig. 7 collects results obtained using the empirical formula and our theory [Eq. (16)] for four different NP radii and a range of the periods. It can be seen that there is a good agreement between the presented theory and empirical values.

_{RA}Figure 8 shows the dependence of the two types of *FOM* on the radius of the NP for different shape factors and numbers of NPs. The period of the array was set to satisfy the resonance condition for a given wavelength according to Eq. (6). Both *FOM _{S}* and

*FOM*exhibit a strong dependence on the radius of NP, having a single maximum at a specific radius. The values of the optimized radius of NP yielding the highest values of

_{B}*FOM*and

_{S}*FOM*correspond to the optimized value of parameter

_{B}*δ*and according to Eq. (6) can be expressed as

*FOM*decrease (due to the broadening of the resonance feature) and the array starts to behave as an infinite array. Both

*FOM*values also decrease when the size of NPs is smaller than the optimum value. This decrease is associated with the fact that losses introduced by the finiteness of an array start to dominate and the width of the resonance feature increases. As the size of the NP continues to decrease, the resonance feature eventually disappears, where the condition $\delta >1/\left(4N\right)$ is no longer satisfied. With a decreasing shape factor, the maximum value of

*FOM*increases while the

_{S}*FOM*decreases [Figs. 8(c) and 8(d)]. Further decrease in the shape factor results in disappearance of the lattice resonance, where the condition ${\lambda}_{RA}>{\lambda}_{LSP}$ is no longer satisfied. In addition, both

_{B}*FOM*and

_{S}*FOM*decrease with decreasing number of particles [Figs. 8(a) and 8(b)].

_{B}These findings suggest that both *FOM _{S}* and

*FOM*can be improved by decreasing the size and increasing the number of NPs. In addition,

_{B}*FOM*can be improved by decreasing the size and shape factor. However, it should be noted that decreasing the size of the particles reduces the depth (contrast) of the resonance feature [Fig. 4(a)], rendering arrays of small particles rather impractical.

_{S}Finally, the analytical results obtained for the two types of *FOM* using Eqs. (18) and (19) were compared with the exact values obtained by calculating the width of the transmission dip using Eq. (4), and furthermore, calculating the surface and bulk sensitivities by determining the shift of the resonance wavelength in response to a bulk refractive index change $\Delta {n}_{m}=0.01\text{RIU}$, and also to a surface refractive index change $\Delta {n}_{s}=0.1\text{RIU}$ within a dielectric shell of the thickness $\Delta =10nm$. As follows from Fig. 8, the presented analytical theory is in a good agreement with the exact calculations for the entire parameter range. In order to evaluate the limitations imposed by the use of the electrostatic approximation, an exact calculation using the polarizability calculated from Mie theory was also performed, confirming the trend of the dependencies. The minor differences are probably a consequence of somewhat overestimated values of the real part of inverse polarizability [Fig. 1(a)].

Figure 9 shows the optimized *FOM* as a function of wavelength for different NP shape factors *L*. The size of NP was optimized to achieve the maximum *FOM* with at least 10% contrast of the resonance feature for each wavelength. The number of particles in the array was set to 2000 × 2000 to correspond with the area of a sensing substrate used typically in plasmonic sensing. The results suggest that both the surface and bulk refractive index *FOM* can be maximized by choosing the optimum resonance wavelength. The optimal value of *FOM _{S}* initially increases with wavelength and, after reaching a maximum, slowly decreases. While the initial growth of

*FOM*with wavelength is also rather rapid, it does not reach a local maximum and continues to increase albeit slowly with wavelength.

_{B}Finally, the optimized *FOM* for lattice resonance on ordered arrays of metal NPs were compared with the case of a LSP supported by a non-ordered array of non-interacting NPs (dots in Fig. 9). According to electrostatic theory, *FOM _{S}* decreases with the size of NP and

*FOM*is size-independent [17]. To enable direct comparison, we set the optimized radius of NPs supporting LSP as

_{B}*r =*10 nm (smaller NPs are challenging to fabricate). The comparison suggests that the lattice resonances on ordered arrays of metal NPs can provide

*FOM*which are larger by about two orders of magnitude than

_{B}*FOM*figures offered by LSP. On the other hand, the difference in the values for

_{B}*FOM*, which is the relevant quantity for most of the biosensing applications, seems to be rather minor.

_{S}## 5. Conclusion

The theoretical analysis of the sensing properties of gold nanoparticle arrays supporting lattice resonances is presented. The sensing performance is studied in terms of the figure of merit (FOM), defined as a ratio of the width of the resonance feature and the sensitivity of the position of the resonant feature to changes in the refractive index. The analysis is based on the coupled-dipole approximation, which together with an electrostatic approximation makes it possible to derive an analytical expression for the bulk and surface refractive index FOM as a function of the design parameters of the array. The derived expression for the bulk refractive index FOM provides results that agree very well with the experimental data presented in [11]. Furthermore, our study suggests that for a defined number of particles in the array, there is an optimal resonant wavelength, size, and shape of the particle that yields a maximum FOM. Finally, the FOM of lattice resonance excited by a 2000 × 2000 gold nanoparticle array was compared with the FOM of localized surface plasmon (LSP) resonance on a non-ordered array of non-interacting gold nanoparticles. It was concluded that (a) the FOM related to bulk refractive index changes offered by the lattice resonance is about two orders of magnitude higher than that of the LSP, and (b) the optimal values of FOM for surface refractive index changes are rather similar.

## Appendix

By representing the dipole-dipole interaction in 2D momentum space in the plane of the array, the dipole sum can be expressed by integration over the reciprocal vector **Q** [4],

**g**

*G*represents the

_{0}*n = 0*term that can be expressed as

For a finite number of particles *N*, the sum in Eq. (22) can be reduced to

*k*close to

*g*, the imaginary part can be written as

_{1}## Acknowledgments

This research was supported by Praemium Academiae of the Academy of Sciences of the Czech Republic, the Czech Science Foundation (contract P205/12/G118) and by the Ministry of Education, Youth and Sports (contract LH11102).

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