## Abstract

We examine the excitation of plasmonic resonances in arrays of periodically arranged gold nanoparticles placed in a uniform refractive index environment. Under a proper periodicity of the nanoparticle lattice, such nanoantenna arrays are known to exhibit narrow resonances with asymmetric Fano-type spectral line shape in transmission and reflection spectra having much better resonance quality compared to the single nanoparticle case. Using numerical simulations, we first identify two distinct regimes of lattice response, associated with two-characteristic states of the spectra: Rayleigh anomaly and lattice plasmon mode. The evolution of the electric field pattern is rigorously studied for these two states revealing different configurations of optical forces: the first regime is characterized by the concentration of electric field between the nanoparticles, yielding to almost complete transparency of the array, whereas the second regime is characterized by the concentration of electric field on the nanoparticles and a strong plasmon-related absorption/scattering. We present electric field distributions for different spectral positions of Rayleigh anomaly with respect to the single nanoparticle resonance and optimize lattice parameters in order to maximize the enhancement of electric field on the nanoparticles. Finally, by employing collective plasmon excitations, we explore possibilities for electric field enhancement in the region between the nanoparticles. The presented results are of importance for the field enhanced spectroscopy as well as for plasmonic bio and chemical sensing.

© 2012 OSA

## 1. Introduction

Metallic nanostructures are now in the research focus of many studies due to their capability of supporting collective electron oscillations (plasmons), yielding to a number of effects including spectrally tunable absorption and scattering, electromagnetic coupling to nanoscale objects, local enhancement and strong confinement of electromagnetic field at length scales much smaller than the optical diffraction limit [l]. Such unique properties can provide decisive advantages for many important applications such Surface Enhanced Raman Scattering [2,3], imaging and guiding beyond the diffraction limit [4,5], plasmonic emitters [6], active plasmonic devices [7,8], biosensors [9–11].

Regular plasmonic arrays are of particular interest for these applications as they enable controllable collective mechanisms of interaction between array constituents yielding to a local concentration/manipulation of electric field. It is well known for many years that periodicity may be employed for excitation of surface waves (later called surface plasmon polaritons (SPPs)) on metallic gratings. A century ago Wood observed anomalous patterns in the reflection spectrum of metallic gratings [12]. The spectra presented rapid variations of intensity with a succession of maxima and minima in certain narrow wavelength bands. A first explanation of these anomalies was proposed by Rayleigh [13], who derived the characteristic wavelength that corresponds to the edge of diffraction when one diffraction order passes from evanescent to radiative state. Fano [14] explained the asymmetric shape of observed spectral features by the manifestation of two distinct anomalies: (i) Rayleigh anomaly (RA) corresponding to the edge of diffraction [13]; and (ii) Resonant anomaly associated with the excitation of surface waves, which is red-shifted with respect to RA. A rigorous theory of grating anomalies on the basis of numerical treatment was presented by Hessel and Oliner [15]. They developed the general theoretical approach that allowed to derive the locations and detailed shapes for variety of cases of resonant anomalies. It was shown that at the resonance the evanescent diffraction order couples to the complex guided wave that is supported by the periodic structure. At this condition the amplitude of this order undergoes a sharp increase while the other orders exhibit a maximum on one side of the resonance and a minimum on the other. An asymmetric shape of the spectral profile at the resonance is called Fano-type resonance [16,17,23]. Anomalies with Fano spectral profile in optical response of metallic gratings have been investigated for many years [12,14,15,18]. However, more recently, Fano-like resonances have been identified in various types of periodic plasmon structures including nanohole arrays [19–23], photonic crystal slabs [24,25] etc.

Collective plasmonic effects in ensembles of metal nanoparticles present a particular case. The arrangement of these nanoparticles in periodic arrays under conditions of diffractive coupling can lead to a drastic improvement of the resonant quality, as it was first suggested theoretically [26–29] and then observed experimentally [30–39], as well as higher electric field enhancement factors [30,33,39] can be achieved compared to single nanoparticle case due to controllable collective mechanisms of optical interaction. Most studies observed narrow resonances in transmission spectra for metal nanoparticle array placed in uniform refractive index environment with characteristic asymmetric Fano profile under the normal incidence [32,34,38,43] (see, also [40–42] where the effect of environment asymmetry is addressed and [44] where inclined incidence and effect of finite array are considered). This spectral line shape can be explained by the interference between directly transmitted light and light scattered by the array [32]. Although conditions of excitation and properties of collective plasmon resonances in metallic arrays were already studied, mechanisms and regimes of interaction of nanoparticles in the lattice are not yet well understood. In particular, it is still important to clarify the interference effects and electric field enhancement distributions under different array parameters, as well as to optimize resonances for particular applications.

In this paper, we examine near and far field distributions of electromagnetic field for characteristic states of Fano spectral profile of ordered gold nanoparticle arrays: Rayleigh Anomaly and Lattice Plasmon Mode (LPM). In addition, we investigate the electric field behavior for different spectral positions of Rayleigh anomaly with respect to single nanoparticle resonance.

## 2. Method

In this work FDTD realization on the basis of sine/cosine plane wave method with Flocket type phase shift periodic boundary conditions (PBC) [45] is employed to study periodically arranged sub-wavelength nanoparticles. Computational domain is excited by two simultaneously incident plane waves: the first one $({E}_{1},{H}_{1})$ with a $\mathrm{cos}(\omega t)$ time dependence and the second one ${E}_{2},{H}_{2}$ with a $\mathrm{sin}(\omega t)$ dependence. The conventional Yee algorithm is used to update these two sets of fields separately. To apply the PBC the fields at the boundary are combined as follows ${E}_{c}={E}_{1}+i{E}_{1},$, ${H}_{c}={H}_{1}+i{H}_{2}$. According to Flocket theory for periodic scattering problems in frequency domain, PBC along the *x* and *y* directions for electric and magnetic field:

*x*and

*y*directions, ${k}_{x}$ and ${k}_{y}$ are horizontal wave vector projections. These conditions are applicable since the combined fields are excited by ${e}^{i\omega t}=\mathrm{cos}(\omega t)+i\mathrm{sin}(\omega t)$. Finally two sets of fields are extracted from the combined field: This procedure is repeated each time step until the steady state is reached.

In each simulation, monochromatic electromagnetic plane wave is launched into-a computational domain in the positive direction of the *z* axis. Periodic boundary conditions are applied along the *x* and *y* axis. The gold nanoparticles are arranged in periodic manner with unit cell of rectangular shape, as shown in the inset of Fig. 1(a)
. Perfectly Matched Layers (PML) are used above top and below bottom z-boundaries of the unit cell simulation domain to absorb the transmitted and reflected electromagnetic waves (we define simulation domain as a volume of space where E and H fields are calculated excluding region of PMLs). To find reflected E and H fields values, the incident fields are subtracted from the total fields. Fraction of the power that is reflected or transmitted into individual order of diffraction is defined as ratio of its energy flux through the simulation area in the (*x*, *y*) plane to the incident field energy flux through the same area. The results are normalized such that summation of all these power fractions gives unity. The integrated transmitted and reflected power is the sum of individual components above and below nanoparticle array respectively. Convergence is reached calculating the fields at narrow width resonances after less than 100 cycles (each cycle means that the wave oscillates one period in time). In our simulations refractive index of surrounding medium is equal to 1.5 and optical constants of gold are taken from [46]. In all simulations, the gold nanoparticles have an ellipsoidal shape with the dimensions of $120nm,120nm,$and $75nm$ along *x*, *y* and *z* directions respectively. The structures are studied for the light polarized along *y* axis under the condition of normal incidence.

## 3. Results

#### 3.1 Transmission and reflection spectra of nanoparticle array

A periodic 2D array of nanoparticles generates a number of diffracted orders if the wavelength of incident light in the medium is comparable with the spacing between array constituents. The wave vector projection in the (*x, y*) plane determined as: ${\overrightarrow{k}}_{\parallel {m}_{1},{m}_{2}}={k}_{x}{\overrightarrow{e}}_{x}+{k}_{y}{\overrightarrow{e}}_{y}+{m}_{1}{G}_{x}{\overrightarrow{e}}_{x}+{m}_{2}{G}_{y}{\overrightarrow{e}}_{y}$, where ${k}_{x}$ and ${k}_{y}$ are wave number components along *x* and *y* directions, ${G}_{x}=2\pi /{a}_{x}$ and ${G}_{y}=2\pi /{a}_{y}$ are reciprocal lattice vectors along *x* and *y* directions, ${m}_{1}$ and ${m}_{2}$ are integers that correspond to the order of diffraction. For the modeled structure only the orders characterized by ${m}_{1}=\pm 1,{m}_{2}=0$ change their state from evanescent to propagating in the spectral vicinity of the undisturbed single particle localized plasmon resonance (LPR). In the case of normal incidence the value of ${k}_{\parallel \pm 1,0}$ is determined only by the reciprocal lattice vector ${G}_{x}$ and these two orders have equal amplitudes due to the degeneracy in frequency. The component along *z* axis is determined as: ${\kappa}_{z}=\sqrt{{k}^{2}-{k}_{\parallel {m}_{1},{m}_{2}}^{2}}$, where $k=n\frac{2\pi}{\lambda}$ is the magnitude of the wave vector in a medium having refractive index *n*. There is a threshold wavelength ${\lambda}_{RA}=n{a}_{x}$ (Rayleigh anomaly wavelength), above which the order is attenuating along *z* due to the change of the sign in the expression under square root. To understand the physics of collective resonance phenomena in regular nanoparticle arrays, we present in Fig. 1 calculated fraction of light power reflected and transmitted into the 0-order and ± 1-order, as well as total reflection and transmission, for an array with lattice constants of ${a}_{x}$ = 450 nm and ${a}_{y}$ = 250 nm. As shown in Fig. 1(a), the excitation of collective resonances in such array leads to the appearance of asymmetric Fano-like line shape in transmission and reflection spectra, which is absent in the case of single nanoparticle LPR (the latter was calculated by replacing PBC with PML along the *x* and *y* axis). In this case, one can see a clear correlation in the evolution of spectral features in transmission and reflection. The first maximum in transmission, which coincides with minimum in reflectivity is observed at $\lambda $ = 675 nm, which corresponds to RA for the ± 1 diffraction order. Under this condition ± 1-order passes from evanescent to propagating state and the slope of its power function is almost 90 degrees (Fig. 1(b)). In addition, under a certain wavelength ($\lambda =$ 730 nm), red-shifted with respect to ${\lambda}_{RA}$, a clearlydistinguishable resonant dip in transmission and peak in reflection take place. At this condition, according to modeling, localized plasmons of the nanoparticle array are excited collectively (this plasmon wave will be later denoted as the Lattice Plasmon Mode (LPM)). In this case, the in-plane light momentum of the evanescent diffraction order matches the one of LPM and the energy is coupled into the array of metal nanoparticles [34]. For $\lambda >{\lambda}_{RA}$ all diffraction orders (except 0-order) are evanescent waves bounded to the lattice. Therefore, under this condition only 0-order contributes to the total transmitted and reflected power. In contrast, for $\lambda <{\lambda}_{RA}$ the total transmitted and reflected power is redistributed between 0- and ± 1-orders in considered wavelength range (Fig. 1).

#### 3.2 Electric field patterns for RA and LPM

Let us now consider distributions of electric field amplitude corresponding to main features of Fano-like spectra: RA and LPM. We imply that the pumping wave comes from the bottom and its amplitude is equal to the unity. The fields are viewed in the (*x, z*) and (*y, z*) planes that pass through the center of the unit cell. We consider only *y* projection of electric field since two other projections are strongly confined to the surface of nanoparticle and do not play a significant role in far field coupling under the considered excitation geometry and nanoparticle parameters. Figures 2(a)
-2(d) show the amplitude of the *y* component of the electric field |${E}_{y}$| under RA condition when ± 1 diffraction orders emerge at the grazing angle related to the lattice plane. The resulting electric field in the domain is formed by the superposition of the field pattern generated by these two orders of diffraction and the incident wave. Since the ensemble of the nanoparticles is embedded in uniform medium, the edge of diffraction has the same spectral position for the transmitted and reflected orders. Therefore the interference pattern is generated above and below the periodic array (Fig. 2(a)). As shown in Fig. 1(a) transmission reaches almost unity at the RA. This can be explained by interference phenomena. First, the amplitude of the electric field is almost zero at metal nanoparticles sites: this leads to the negligible electric force acting on free electrons of the metal. As a result, energy of the incident wave is not dissipated due to the excitation of the surface plasmons. Second, the interference at the diffraction edge is accompanied by a vanishing reflection. Figure 2(c) shows |*Ey*| as a function of *z* along the line passing through the center of a nanoparticle. It is visible that electric field is strongly modulated and has almost zero amplitude at the nanoparticle position. It is also visible that the modulation depth decreases under the increase of the distance from the lattice. As follows from plot in Fig. 2(d), calculated using near-to-far-field transformation [47], the envelope converges to the constant amplitude value at distance of about 10λ in a far field simulation region (the region of space that extends to infinity from upper boundary of simulation domain). Figure 3
shows the distribution of |${E}_{y}$| in the condition of LPM excitation. Here, due to the normal incidence, two lattice modes are excited having the same frequency and travelling in opposite directions. The superposition of these waves provides the generation of a standing wave pattern along the *x* axis. This situation is comparable with the excitation of Surface Plasmon Polaritons (SPPs) on metallic gratings under normal incidence when standing wave is generated by two contra-propagating SPPs. However, it is obvious that in the case of an ensemble of metal nanoparticles that sustain localized surface plasmons the underlying properties of the excitation are different. In addition, there are some interesting effects that arise from an additional degree of freedom in a case of 2D array compared to 1D geometry of SPP using metallic gratings. Figure 3(b) shows the behavior of |${E}_{y}$| in the (*y, z*) plane, that contains the electric field vector of the incident wave. Similar to the case of LPR the electric near field is concentrated mainly at the edges of nanoparticles, which are perpendicular to the direction of the external electric field. Since LPM is excited along *x* direction, the behavior of the propagating components of light can be understood if one examines field distribution in the (*x, z*) plane. As shown in Fig. 3(a), |${E}_{y}$| is periodically patterned in *x* direction due to the superposition of the scattered waves. This pattern does not extend far from the array due to the confinement of the lattice modes. Figure 3(c) shows the variation of |${E}_{y}$| along the line (vertical arrows) which crosses the point of field enhancement maxima at the nanoparticle boundary. It is visible that in the region below the array the incident and reflected waves are superimposed resulting in a modulation pattern, while in the vicinity of the nanoparticle boundary electric field significantly increases due to collective plasmon excitation. From this point, electric field decreases exponentially into ambient environment (Fig. 3(c)). As shown in Fig. 3(d), after a number of oscillations the amplitude of electric field becomes constant.

#### 3.3 Transmission and reflection spectra at different regimes of electromagnetic far field coupling

In order to clarify regimes of collective interaction in the nanoparticle lattice and their dependence on lattice periodicity, we calculated transmission and reflection spectra for different lattice constants along the *x* axis (${a}_{x}$ changes from 300 nm to 750 nm with 50 nm step), whereas the constant along the *y* axis was fixed at ${a}_{y}$ = 250 nm (Fig. 4
). Our calculations show that both coupling efficiency and resonant line shape critically depend on spectral position of ${\lambda}_{RA}$ with respect to the one of the single nanoparticle LPR ${\lambda}_{LPR}$ (as shown in Fig. 1(a), the single particle LPR is reached at ${\lambda}_{LPR}$ = 625 nm for the used nanoparticles).

If ${\lambda}_{RA}<{\lambda}_{LPR}$, which take place at relatively small lattice constants ${a}_{x}$ = 300 nm, 350 nm, the resonance is relatively broad and, for example at ${a}_{x}$ = 300 nm the full width at half maximum (FWHM) is 115 nm. The far field coupling between nanoparticles is almost absent in this case, as all propagating diffraction orders are at smaller wavelengths than single nanoparticle LPR. In contrast if $\lambda \ge {\lambda}_{RA}$ the coupling becomes strong and the line shape alters significantly. Here, the increase of the lattice constant ${a}_{x}$ causes a red shift of the resonance and a significant narrowing of its line shape [32]. In the extreme coupling regime reached at ${a}_{x}$ = 650 nm (it will be shown further that this regime provides the highest value of electric field enhancement) the FWHM is 7 nm. It is worth noting that such results are in good qualitative agreement with previous theoretical and experimental studies [28,29,32]. Notice that starting from ${a}_{x}$ = 350 nm one can observe the second dip [28] in the transmission (and a corresponding peak in reflection), which is blue shifted with respect to RA and takes place in the region where ± 1 diffraction order is propagating. In contrast to the LPM this mode is not confined to the lattice as it can match ($\omega ,k$) values of the propagating photons in the medium.

#### 3.4 Electric field patterns at characteristic regimes of electromagnetic far field coupling

Figures 5(a)
-5(d) show distributions of electric field amplitude in (*x, y*) plane sliced in the middle of nanoparticles for different characteristic regimes of lattice behavior: at the RA condition for the structure of ${a}_{x}$ = 450 nm (a), and at the LPM excitation for the arrays with lattice parameters ${a}_{x}$ = 300 nm (b), 450 nm (c), 650 nm (d) (transmission and reflection spectra of these arrays were considered in Fig. 4).The plots of |${E}_{y}$| along the line through the centers of the two adjacent nanoparticles in *x* direction (Figs. 5(a)-5(d)) show in details the linear cross section of the standing wave pattern generated by two contra-propagating lattice modes, while the linear plot along *y* direction (Fig. 5(e)) depicts the behavior of evanescent components of electric field. Let us first consider RA condition. According to Fig. 5(a), |${E}_{y}$| has a maximum magnitude in the space between nanoparticles and almost zero magnitude at nanoparticle sites. For the electromagnetic coupling regime realized at ${a}_{x}$ = 300 nm, the corresponding plasmon resonance possesses a wide profile as shown in Fig. (4). In this regime (Figs. 5(b) and 5(e)) electric field is concentrated at the edges of nanoparticles sharply decreasing into ambient environment. Such characteristic spatial distribution is almost identical to the one of the undisturbed single particle LPR (further we will refer to this regime as regime of weak (far field) coupling). Note, that |${E}_{y}$| becomes discontinuous at the nanoparticle boundary (showed by a dashed line in Fig. 5(e)). Such behavior is due to the fact that the *y* component of electric field is normal to the metal dielectric interface. As shown in Figs. 5(c) and 5(d), the character of the resonances at ${a}_{x}$ = 450 nm and 650 nm, differs from the pure localized surface plasmon case. An additional field maximum can be found in between two adjacent nanoparticles in *x* direction for ${a}_{x}$ = 450 nm and three related maxima can be found for ${a}_{x}$ = 650 nm. It is obvious that the appearance of these maxima is due to the interference of the propagating components of electromagnetic field. As can be seen in Fig. 5(e), for these two regimes evanescent component of electric field is enhanced at the boundaries as well as in the space between nanoparticles. In fact, it means that under appropriate parameters of the periodic structure, the electromagnetic far field coupling can result in a significant modification of pattern of evanescent (non propagating) electric field components. This issue was previously pointed out in literature [30,35]. In order to study this effect for various lattice parameters of the periodic arrays, in Fig. 5(f) we depict the values of |${E}_{y}$| as a function of ${a}_{x}$ in two characteristic points: (i) in the point of maximum enhancement at the nanoparticle boundary ${\left|{E}_{y}\right|}_{\mathrm{max}}$ and, (ii) in the middle point (at *y* = 250 nm in Fig. 5(e)) between nanoparticles ${\left|{E}_{y}\right|}_{mid}$ Here, the largest magnitudes of ${\left|{E}_{y}\right|}_{\mathrm{max}}$ = 26.5 and ${\left|{E}_{y}\right|}_{mid}$ = 9.5 are obtained for the structure with ${a}_{x}$ = 650 nm. As depicted in Fig. 5(d), corresponding distribution of electric field shows both the delocalization and strong enhancement of |${E}_{y}$| at the boundaries (we denote this regime of excitation as regime of strong (far field) coupling). Next, we compare in Table 1
the behavior of evanescent components of |${E}_{y}$| in the strong coupling regime with the one in regime of weak coupling. We fix parameter ${a}_{x}$ to maintain the corresponding regime of LPM excitation, while ${a}_{y}$ is changed (three characteristic values 450 nm, 250 nm and 120 nm are considered).

As follows from Table 1, in the limit of large separations (${a}_{y}$ = 450 nm) the enhancement between nanoparticles is negligible for strong and weak coupling regimes as near field coupling is almost absent. In contrast, for ${a}_{y}$ below 250 nm, ${\left|{E}_{y}\right|}_{mid}$ in the regime of strong coupling rise above ${\left|{E}_{y}\right|}_{max}$ of the weak coupling regime. Under the lattice parameter of ${a}_{y}$ = 120 nm, the gap between metal surfaces in *y* direction becomes very small (10 nm) and electric field can be localized in tiny volume between two nanoparticles. As it was shown for 1D arrays of nanoparticle dimers [30], long range (far field) coupling can lead to the improvement of electric field enhancement factor in the gap between two metal nanoparticles compared to single dimer case. In this paper, we observed a similar effect for 2D infinite arrays. This case may be described as far field coupling between infinite chains (external electric field is parallel to the axis of the chain). According to our results the far field coupling leads to a near 2-fold increase of ${\left|{E}_{y}\right|}_{mid}$ in the gap in comparison with uncoupled chains (Table 1).

## 4. Discussion

Thus, we determined conditions of excitation of ultra-narrow Fano-type resonances (the smallest width of the resonance is 7 nm compared to 115 nm that is of the single nanoparticle resonance) of plasmonic nanoantenna lattices and obtained electric field distributions for different lattice parameters in characteristic points of light-lattice interaction. Our calculations show that for the given size of nanoparticles the maximal amplitude of electric field on the nanoparticles is reached under relatively large lattice parameter (600-650nm) and this phenomenon is accompanied by the extension of enhancement region to the space between nanoparticles. The effect of delocalization of field enhancement due to non-locality of electron response in nanoscale metallic structures has been discussed recently [48]. Indeed, charges cannot be squeezed into a layer of infinitesimal thickness along the nanoparticle surface but occupy some volume in a case of real metal structures. This effect should be taken into consideration in theoretical modeling of field enhancements near sharp corners of metal nanostructures or within the subnanometer gap formed between plasmonic metal nanoparticle aggregates. However delocalization of enhancement region in diffractive metallic arrays studied here has different origin and arises from photonic-like behavior of the studied modes [34]. We believe that these data can be important for many plasmonic applications relying on locally enhanced electric field.

One prominent example is Surface Enhanced Raman Scattering (SERS). The electric field enhancement near rough metal surfaces and metal nanoparticles determines the electromagnetic enhancement in SERS [49,50]. The enhancement factor in SERS is proportional to the forth power of the localized field intensity. Two types of enhancement present interest: the average of electric field over the region of enhancement and the peak value of electric field which is important in single molecule SERS [51]. Although the latter phenomenon have received recently a lot of attention, the design and optimization of variety of SERS based chemical sensors for which the average electric field enhancement plays an essential role is still an important issue [3]. For this case, large volume of electric field enhancement is desirable, and, therefore, collective plasmon excitation that results in an increase of electric field enhancement region (characteristic length is a few hundred nanometers) may be useful.

Another example is plasmonic biosensing, which relies on the detection of biological binding recognition events on gold through refractive index monitoring. Classical plasmonic biosensors imply the employment of Attenuated Total Reflection geometry to excite SPPs over a 50 nm gold film, while angular (or spectral) position of a dip in reflection corresponding to the SPP excitation is used to monitor refractive index changes due to biological interactions. Here, a high sensitivity of the method is due to a strong probing electric field related to plasmons and a resonance character of plasmon excitation. Despite successful use of the method to study a variety of selective biological interactions (antigen-antibody, DNA-DNA capture, protein-protein, protein-ligand etc.), SPP do not look well adapted for modem trends of nanobiotechnology, including the creation of novel nanoarchitectures (beacons, complex organic-inorganic structures etc), resolution beyond the diffraction limit and manipulation in the nanoscale. With a series of novel options including spectral tuneability, size selectivity, large local fields etc, localized plasmons are much better adapted to match these trends, but localized plasmon are much less sensitive compared to SPP [10]. Indeed, the sensitivity of such sensors is normally 100-300 nm/RIU compared to 5 000 - 10 000 nm/RIU of classical SPR, while the Figure of Merit characterizing the ratio of the response to the width of the resonance FOM = (Δλ/Δn)/FWHM (where FWHM is the full-width at half-maximum (nm) and Δλ is the resonance shift for an refractive-index change Δn) is about 8-10 compared to 23-25, respectively [10]. It is clear that with the increase of the electric field and narrowing the plasmonic feature due to the excitation of resonant lattice modes, one can expect the improvement of both sensor sensitivity and FOM. Of particular interest, we see the implementation of phase sensitivity [51–53] in diffractively coupled plasmonic arrays. As an example, Kravets [36] recently showed that the excitation of ultra-narrow resonances under the reflection of light from metal nanoparticle arrays at large angles (50-70 Deg.), yielding to extremely sharp phase jumps. The improvement of contour sharpness resulted in phase sensitivity of 5∙10^{5} deg./RIU that was 5-10 times better compared to classical gold film SPR geometry. As phase singularities are related to undefined phase in the point of minimum of electric field rather than to optical properties of a concrete system [54], we expect similar effects under a normal incidence of light onto the lattice. It should be noted that under normal light incidence the lattice resonances are ultra-narrow when the nanoparticles are placed in symmetric environment having the same index of refraction [32,34]. This may somehow complicate the implementation of sensing design as sensing geometry implies a slight asymmetry of the system with refractive index of the nanoparticle-supporting substrate being slightly higher than that of water (1.33). As a solution of this problem, we see the implementation of lattice resonances using low refractive index substrates (teflon, porous glass or coatings etc…).

## 5. Conclusion

We report a study of optical response of ordered ensembles of gold nanoparticles employing FDTD method with Bloch Flocket boundary conditions. We first analyzed the evolution of the electric near and far field for two characteristic states of Fano-like transmission and reflection spectra: RA and LPM. The former regime of optical excitation is characterized by almost zero amplitude of the electric field in the sites of the metal nanoparticles, while the latter regime is characterized by the collective excitation of localized plasmons in the nanoparticle lattice. Then, we investigated the behavior of electric field at different regimes of LPM excitation, obtaining the parameters of the periodic array that provide the highest factor of electric field enhancement. Finally, we studied the conditions of optical coupling that provide the extension of electric near field enhancement region to the space between metal nanoparticles.

## Acknowledgments

This work is funded through the Nanobioplasmon ANR project.

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