Sensing properties of lattice resonances of 2D metal nanoparticle arrays: An analytical model

Barbora Špačková; Jiří Homola

doi:10.1364/OE.21.027490

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 $1 / α^{*} = 1 / α - G$ , 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 polarizabilityα. In the electrostatic approximation $1 / α^{e s}$ can be expressed as

1 / α^{e s} = \frac{3}{r^{3}} (L + \frac{ε_{m}}{ε - ε_{m}}),

where

r = \sqrt[3]{a b c}

is the equal-volume-sphere radius, L is a shape factor (

L = 1 / 3

for a sphere) [13] [see, Fig. 1(d)], and ε and ε_m 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{ε_{m}} = 1.33

).

Fig. 1 Dependence of the (a) real and (b) imaginary part of the inverse polarizability on wavelength for NPs of three different sizes. The solid lines represent the electrostatic approximation and the dotted line represents the Mie solution. (c) Dependence of the real and imaginary part of the inverse polarizability on wavelength for three different shape factors L of a spheroid with r = 40 nm. (d) Dependence of the shape factor L on the major/minor axis ratio a/b for prolate and oblate NPs excited by a light wave with different polarizations.

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To account for radiative damping, the expression for the electrostatic polarizability can be expanded as follows: $1 / α = {1 / α}^{e s} - 2 i k^{3} / 3$ [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

G = \sum_{n \neq 0} (k^{2} + \nabla \nabla) \frac{\exp (i k R_{n})}{R_{n}},

where R_n 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 Λ is presented in Fig. 2.

Fig. 2 The real and imaginary parts of the lattice sum as a function of the normalized wavelength calculated for two arrays with a different number of NPs.

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

G \approx \frac{4 π^{2} \sqrt{2}}{Λ^{3} \sqrt{2 π / k Λ - 1}} + i (\frac{2 π k}{Λ^{2}} - \frac{2 k^{3}}{3} + \frac{2 π^{2}}{{(Λ \sqrt{π N})}^{3} {(2 π / k Λ - 1)}^{2}}) - \frac{118}{Λ^{3}} .

Using the optical theorem [13], the extinction cross section $c_{e x t}$ can be written as $c_{e x t} = 4 π k Im {α^{*}}$ and the zero-order transmittance T under normal incidence can be expressed as

T = {| 1 + \frac{2 π i k / Λ^{2}}{1 / α - G} |}^{2} .

Figure 3 shows the transmittance through the array as a function of wavelength calculated using Eq. (4). When the lattice sum reaches a maximum, the transmittance increases to one and at this wavelength the array of NPs becomes invisible. This behavior is referred to as Rayleigh anomaly (

λ_{R A} = Λ n_{m}

), and represents the onset of diffraction on the periodic structure. According to Eq. (4), the transmittance exhibits resonances that are described by the following condition

Re {1 / α} = Re {G} .

The broad resonance that appears near

Re {1 / α} = Re {G} \approx 0

is associated with the excitation of LSP (λ_LSP). Since

Re {G}

values are close to zero, the contribution of the collective coupling is negligible and

α^{*} \approx α

near λ_LSP. The narrow resonance is associated with so-called lattice resonances (λ_r) and can be excited only when the Rayleigh anomaly occurs at longer wavelengths than LSP (

λ_{R A} > λ_{L S P}

), where

Re {1 / α} > 0

. When

λ_{R A} < λ_{L S P}

, the lattice resonance cannot be excited [Fig. 3(b)] and only the maxima corresponding to a Rayleigh anomaly are visible in the spectra.

Fig. 3 Transmission spectrum having characteristic resonances for three different gold NP arrays. The vertical lines indicate the lattice resonance λ_r ( $Re {1 / α} = Re {G}$ ), the localized surface plasmon resonance λ_LSP ( $Re {1 / α} \approx 0$ ) and the Rayleigh anomaly ( $λ_{R A} = Λ n_{m}$ ). Parameters of the arrays: N × N = 2000 × 2000, L = 1/6, (a) Λ = 500 nm, r = 40 nm, (b) Λ = 400 nm, r = 40 nm (since $λ_{R A} < λ_{L S P}$ , lattice resonance is not excited), (c) Λ = 500 nm, r = 15 nm (since $Re {1 / α} >max (Re {G})$ , lattice resonance is not excited).

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The spectral position of the lattice resonance is shifted from the Rayleigh anomaly to longer wavelengths by $δ = λ_{r} / λ_{R A} - 1$ . Using Eq. (3), for $δ ≪ 1$ , δ can be approximated by

δ = \frac{λ_{r}}{Λ n_{m}} - 1 \approx {(\frac{4 π^{2} \sqrt{2}}{Re {1 / α}_{λ_{r}} Λ^{3}})}^{2} .

In order to satisfy Eq. (5) for lattice resonances, two conditions must be fulfilled. The first condition,

Re {G} > 0

, is satisfied when

δ < 1 / 4

(Fig. 2). The second condition,

Re {1 / α} < max (Re {G})

, is fulfilled when

δ > 1 / (4 N)

. If the size and number of NPs are not sufficient to satisfy the second condition, only shallow dips corresponding to lattice resonance may appear in the spectrum [Fig. 3(c)]. For an infinite number of NPs, the second condition is always satisfied. When

δ > 1 / \sqrt[4]{{(π N)}^{3}}

, the last term in the lattice sum [Eq. (3)] can be neglected and the array starts to behave as an infinite array. This “infinite array behavior” can be accomplished for larger NPs, smaller periods of an array (to excite the lattice resonance closer to the LSP wavelength), or higher numbers of NPs.

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:

T = {| 1 + \frac{i 4 π^{2}}{Λ^{3}} \frac{λ_{R A} / λ_{r}}{\frac{d Re {1 / α - G}_{λ_{r}}}{d λ} (λ - λ_{r}) + i Im {1 / α - G}_{λ_{r}}} |}^{2}

with the full width at half maximum, W:

W = 2 {| \frac{Im {1 / α - G}}{d Re {1 / α - G} / d λ} |}_{λ_{r}} .

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 $Re {1 / α}$ , 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)].

Fig. 4 Transmission as a function of wavelength calculated for (a) three different radii of a spherical NP, for Λ = 450 nm, N × N = 2.10⁴ × 2.10⁴, (b) three different shape factors L of a spheroid, for Λ = 450 nm, N × N = 2.10⁴ × 2.10⁴, r = 40 nm, (c) three different periods of an array, for N × N = 2.10⁴ × 2.10⁴, r = 40 nm, and (d) three different numbers of spherical NPs, for Λ = 500 nm, r = 40 nm. The solid lines represent the analytical solutions obtained using the electrostatic approximation and the dotted lines represent the Mie solution.

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

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

S_{B} = \frac{d λ_{r}}{d n_{m}} \approx {(\frac{d Re {G} / d n_{m}}{d Re {G - 1 / α} / d λ})}_{λ_{r}},

where the sensitivity of the real part of the lattice sum can be written as

{(\frac{d Re {G}}{d n_{m}})}_{λ_{r}} = - \frac{1}{n_{m}} \frac{d Re {G}}{d δ} (δ + 1),

and for

δ ≪ 1

can be expressed as

{(\frac{d Re {G}}{d n_{m}})}_{λ_{r}} \approx \frac{2 π^{2}}{n_{m} Λ^{3}} \sqrt{\frac{2}{δ^{3}}} .

As shown in Fig. 5(b), a refractive index change within a dielectric shell of finite thickness Δ surrounding a NP produces a change in the polarizability of a NP while the lattice sum remains unchanged. This change in polarizability produces a change in the resonant condition and a shift of the lattice resonance. Since the lattice sum is nearly constant at λ_RA 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

S_{s} = \frac{d λ_{r}}{d n_{S}} \approx {(\frac{d Re {1 / α} / d n_{S}}{d Re {1 / α - G} / d λ})}_{λ_{r}} .

The inverse polarizability of a core-shell ellipsoid NP with semiaxes a, b, c, and a shell thickness Δ can be described by the following equation

1 / α^{e s} = \frac{3 f}{r^{3}} \frac{[ε_{2} + (ε - ε_{s}) (L^{(1)} - f L^{(2)})] [ε_{m} + (ε_{s} - ε_{m}) L^{(2)}] + f L^{(2)} ε_{s} (ε - ε_{s})}{(ε_{s} - ε_{m}) [ε_{s} + (ε - ε_{s}) (L^{(1)} - f L^{(2)})] + f ε_{s} (ε - ε_{s})},

where

r = \sqrt[3]{a b c}

is an equal-volume-sphere radius of the inner ellipsoid,

L^{(1)}

and

L^{(2)}

are the geometrical factors for the inner and outer ellipsoids, and ε, ε_m, and ε_s 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^{(1)} \approx L^{(2)} = L

. The factor

1 / f = (a + Δ) (b + Δ) (c + Δ) / (a b c)

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

Fig. 5 (top) Real part of the lattice sum and inverse polarizability and (bottom) transmission as a function of wavelength for two refractive indices (a) of a bulk medium (solid line n_m = 1.33, dashed line n_m = 1.335) and (b) within a layer of thickness Δ = 10nm (solid line n_s = 1.33, dashed line n_m = 1.35). Parameters of the array: Λ = 450 nm, N × N = 2.10⁴ × 2.10⁴, r = 40 nm.

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{(\frac{d Re {1 / α}}{d n_{S}})}_{λ_{r}} = \frac{2}{3 n_{m}} (1 - 1 / f) (r^{3} Re {1 / α}_{λ_{r}}^{2} + \frac{9 f}{r^{3}} (L - L^{2})) .

4. Figure of merit

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

F O M_{S} = \frac{1}{3 n_{m}} \frac{B}{g_{i} / g_{r} - A},

F O M_{B} = \frac{1}{2 n_{m}} \frac{d g_{r}}{g_{r} d δ} \frac{δ + 1}{A - g_{i} / g_{r}},

where

g_{i} = Λ^{3} (Im {G} - G_{0})

,

g_{r} = Λ^{3} Re {G}

, and A and B are

A = \frac{Im {1 / α^{e s}}}{Re {1 / α^{e s}}}, B = (1 - 1 / f) (r^{3} Re {1 / α^{e s}} + \frac{9 f (L - L^{2})}{r^{3} Re {1 / α^{e s}}}) .

Using Eq. (3), for

δ ≪ 1

, Eqs. (15) and (16) can be reduced to the following analytical form

F O M_{B} \approx {[4 n_{m} δ (\sqrt{\frac{δ}{2}} + \sqrt{\frac{1}{{(2 π N δ)}^{3}}} - A)]}^{- 1},

F O M_{S} \approx {[\frac{3 n_{m}}{B} (\sqrt{\frac{δ}{2}} + \sqrt{\frac{1}{{(2 π N δ)}^{3}}} - A)]}^{- 1} .

Figure 6 illustrates the dependence of FOM_S and FOM_B on δ for different resonant wavelengths, calculated using Eq. (15) and Eq. (16). The parameter δ was assumed to lie within the range $1 / (4 N) < δ < λ_{r} / λ_{L S P} - 1 < 1 / 4$ 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_S and FOM_B exhibit a maximum at the optimized values of δ

δ_{S, B}^{o p t} \approx K_{S, B} / {(π N)}^{3 / 4},

where

K_{S} = \sqrt{3 / 2}

and

K_{B} = \sqrt{1 / 6}

. Furthermore, FOM_B increases with the resonant wavelength and shape factor L (which decreases with A) and FOM_S exhibits a maximum at a specific resonance wavelength that depends on the number and shape of the NPs.

Fig. 6 (a) FOM_S and (b) FOM_B as a function of δ calculated for different resonant wavelengths. Parameters of the array: N × N = 2000 × 2000, L = 1/6 (λ_LSP = 590 nm).

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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 $F O M_{B} = 0.79 / δ_{ν} - 1.38$ , where $δ_{ν} = 1 - ν_{r} / ν_{R A}$ and υ_r and υ_RA 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.

Fig. 7 FOM_B as a function of parameter δ for an array of gold NPs. The solid lines represent the theoretical values calculated using Eq. (16) for different radii of NP. The dotted lines represent the empirical dependence determined experimentally [11]. Parameters of the arrays used in simulations: height 50 nm, period 300 – 600 nm, n_m = 1.45.

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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_B 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 FOM_S and FOM_B correspond to the optimized value of parameter δ and according to Eq. (6) can be expressed as

r_{S, B}^{o p t} \approx \frac{λ_{r}}{n_{m}} \sqrt[3]{\frac{3}{4 π^{2}} (L + \frac{ε_{m}}{Re {ε}_{λ_{r}} - ε_{m}}) \sqrt{\frac{δ_{S, B}^{o p t}}{2}}} .

As apparent from Fig. 8, when the NP size increases above the optimum value, both the types of 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

δ > 1 / (4 N)

is no longer satisfied. With a decreasing shape factor, the maximum value of FOM_S increases while the FOM_B decreases [Figs. 8(c) and 8(d)]. Further decrease in the shape factor results in disappearance of the lattice resonance, where the condition

λ_{R A} > λ_{L S P}

is no longer satisfied. In addition, both FOM_S and FOM_B decrease with decreasing number of particles [Figs. 8(a) and 8(b)].

Fig. 8 FOM_S and FOM_B as a function of NP radius for (a, b) different numbers of spherical NPs, and (c, d) different shape factor L of a spheroid for N × N = 2000 × 2000. The period of the array is set according to Eq. (6) to satisfy the resonance condition at λ_r = 800 nm. The solid lines represent the analytical solutions using Eqs. (15) and (16), dots represent the exact solution using the electrostatic approximation, and crosses represent the exact solution using the Mie theory.

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These findings suggest that both FOM_S and FOM_B can be improved by decreasing the size and increasing the number of NPs. In addition, FOM_S 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.

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 $Δ n_{m} = 0.01 RIU$ , and also to a surface refractive index change $Δ n_{s} = 0.1 RIU$ within a dielectric shell of the thickness $Δ = 10 n m$ . 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_B with wavelength is also rather rapid, it does not reach a local maximum and continues to increase albeit slowly with wavelength.

Fig. 9 (a) FOM_S and (b) FOM_B at the optimized NP size as a function of the resonant wavelength for different NP aspect ratios, N × N = 2000 × 2000. (Inset) The optimized NP radius as a function of the resonant wavelength. Dots represent FOM_S and FOM_B of a LSP supported by a non-ordered array of non-interacting particles.

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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_B is size-independent [17]. To enable direct comparison, we set the optimized radius of NPs supporting LSP as 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_B which are larger by about two orders of magnitude than FOM_B figures offered by LSP. On the other hand, the difference in the values for FOM_S, which is the relevant quantity for most of the biosensing applications, seems to be rather minor.

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 = \frac{i}{2 π} \sum_{n \neq 0} \int_{Q}^{} \frac{k^{2} - Q_{x}^{2}}{\sqrt{k^{2} - Q^{2}}} \exp (i Q \cdot R_{n}) d^{2} Q .

For an infinite number of particles, the dipole sum leads to a sum over the reciprocal-lattice vectors g

G^{\infty} = \frac{2 π i}{Λ^{2}} \sum_{g} \frac{k^{2} - g_{x}^{2}}{\sqrt{k^{2} - g^{2}}} - G_{0},

where the first term correspond to summation of (22) over

n > 0

and G₀ represents the n = 0 term that can be expressed as

G_{0} = \frac{i}{2 π} \int_{Q}^{} \frac{k^{2} - Q_{x}^{2}}{\sqrt{k^{2} - Q^{2}}} d^{2} Q = - \frac{2 i k^{3}}{3} .

When all diffracted waves other than the zero-th order are evanescent

(k < g_{1})

, the dipole sum (23) can be directly summarized into

G^{\infty} \approx \frac{4 π^{2} \sqrt{2}}{Λ^{3} \sqrt{2 π / k Λ - 1}} + i (\frac{2 π k}{Λ^{2}} - \frac{2 k^{3}}{3}) - \frac{118}{Λ^{3}},

where the last term is a fit to the non-divergent terms for the part of the spectrum where

Re {G^{\infty}} > 0

, and the other terms are derived analytically from the divergent terms of the sum [16].

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

G^{N} = \frac{i N^{2}}{2 π Λ^{2}} \sum_{g}^{} \int_{Q}^{} \frac{k^{2} - Q_{x}^{2}}{\sqrt{k^{2} - Q^{2}}} \sin c ((Q - g) \cdot \frac{N Λ}{2 π}) d^{2} Q - G_{0},

where the sinc function is derived from the restriction of the infinite sum by a rectangular function. For k close to g₁ , the imaginary part can be written as

Im {G^{N}} \approx \frac{π}{2 N Λ^{2}} \int_{0}^{k} \frac{k^{2} Q_{y}}{\sqrt{k^{2} - Q_{y}^{2}}} \frac{1}{{(Q_{y} - g_{1})}^{2} + \frac{{(π / N Λ)}^{2}}{2}} d Q_{y} + Im {G^{\infty}},

where the sinc function was approximated by the Lorentz curve to eliminate the effect of fast oscillation. The condition for lattice resonance excitation can be satisfied for

k Λ < 2 π - π / (2 N)

. Under this condition, the real part of the lattice sum is independent of the number of particles and the imaginary part of lattice sum obeys

Im {G^{N}} \approx \frac{2 π^{2}}{{(Λ \sqrt{π N})}^{3} {(2 π / k Λ - 1)}^{2}} + Im {G^{\infty}} .

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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Sensing properties of lattice resonances of 2D metal nanoparticle arrays: An analytical model

Abstract

1. Introduction

2. Transmission spectrum of arrays on nanoparticles

3. Sensitivity of the lattice resonance

4. Figure of merit

5. Conclusion

Appendix

Acknowledgments

References and links

Cited By

Figures (9)

Equations (28)

Optics Express