1. Introduction

Recently so-called left-handed materials [1] also known as negative index materials have attracted a lot of attention [1–18]. An interest to the original Veselago’s work [1] (and even earlier suggestions by Schuster [2]) has been renewed by the seminal Pendry’s paper [3] proving a “perfect” nature of Veselago’s lens [1] and by an experimental realization of negative index of refraction at microwaves [4,5] based on a clever idea to generate magnetic response from nonmagnetic (metallic) elements [6,7]. The extension of the concept of left-handedness to visible domain turned out to be a tricky problem because even an introduction of magnetic permeability into visible optics is not straightforward [8]. The solution to this problem was found in simplification of resonators providing magnetic response at high frequencies [9,10] which resulted in realization of μ first at infrared [11–13] and then visible-light frequencies [14]. Later, negative refractive indices in visible domain have been deduced from optical spectra for samples of different geometries [15–18]. In our earlier works [14,16] we have demonstrated optomagnetic [9] behavior of arrays of double dots covered with an optically thin layer of glycerol. Here we present a detailed analysis of denser samples of analogous geometry which do not require glycerol to achieve high values of permeability. The main task of this work will be an experimental study of plasmonic resonances of double-dot samples based on combined analysis of samples reflection, transmission, ellipsometry and phase measurements. We also provide conditions at which effective medium theory can be applied to a composite optical medium (metamaterial) and discuss general difficulties in extracting optical constants of thin films.

2. Optical constants of a two-dimensional array of particles

First, we make several general remarks on the refractive index of metamaterials.

2.1 Extraction of bulk optical constants from optical properties of two-dimensional arrays (2D) of nanoparticles

The “bulk” optical constants of 2D arrays of nanoparticles do not exist since there is no averaging of fields in the direction perpendicular to a sample [19,20] (in other words it is impossible to introduce sensible values of averaged dipole and magnetic moments of a unit volume). One can still introduce physically sensible notions of magnetic and dipole moments of structural elements (nanomolecules), which can be used for calculation of reflection and transmission of 2D samples [21] and would yield optical constants of a bulk medium made of studied nanomolecules [14]. It is customary, nevertheless, to apply an effective medium theory to 2D samples by replacing an actual 2D array by a film of some thickness, permittivity and permeability with an idea that optical properties of such a film would give the best fit to the measured reflection and transmission data [22]. There exist a number of different methods for assigning optical constants to a thin film. These methods are based on reflectometry, ellipsometry, phase measurements, etc., for a review see Ref. 22.

Quite popular appears to be a method based on normal light reflection and transmission measurements. In 1951 Schopper derived simple expressions which provide a complex refractive index of a thin sample and its thickness in terms of light amplitude and phase measured for three cases: light reflection from a sample, light transmission through a sample and light reflection from the backside of a sample [22–25]. Later, it was suggested that it is not necessary to extract the effective optical thickness of a sample from optical measurements; instead it can be fixed at a value somehow connected to geometry of a structure. This allowed one to simplify calculations of a complex refractive index of a thin film considerably and reconstruct a complex refractive index of a thin sample on the basis of only two measurements of normal light (amplitude and phase) reflection and transmission through a sample [26]. We will refer to this method [26] as modified Schopper method (MSM).

MSM is an elegant and adequate method for finding optical constants of thick flat films with well-defined boundaries. However, it is necessary to stress that an unreserved application of MSM to 2D arrays of nanomolecules is incorrect [27]. Indeed, the definition of a “geometrical” thickness of a nanostructured or corrugated film is not clear (is it a maximal height of a structure, averaged geometrical thickness or mass thickness of a sample?). More importantly, an effective optical thickness t_eff of a 2D array of nanomolecules is usually governed by distribution of near-fields of nanomolecules and by effective volumes of induced magnetic and dipole moments of array elements (this volume V can be defined as $d=VE,$ where d is the induced dipole moment and E is the driving electric field) and is not known a priory. As an example we consider a 2D array of carbon atoms - graphene layer - which presents a 2D system with well-defined optical reflection and transmission spectra [28]. It is clear that there is no predefined “geometrical” thickness of a graphene layer: one can argue with different amount of credibility that optical thickness of a graphene sheet is equal either to a size of the atomic nucleus (where all mass is concentrated), or a size of s-orbitals electrons of carbon atoms, or a size of p-orbitals, or twice the size of p-orbitals. This implies that the effective optical thickness of 2D nanoparticle array has to be determined from experiments contrary to MSM. (In parenthesis we note that the fact that “optical” sizes of a system could be different from “geometrical” one is well-known in fiber optics and has been observed for metamaterials [14,29]).

Another problem of MSM application to metamaterials comes from its simplicity. The complex reflection and transmission coefficients at the normal incidence exist for any optical system. As a result, MSM might be used to assign and calculate a refractive index for systems which cannot be described by effective media theory (EMT) such as some layered materials, arrays of holes in the metal, media with strong scattering, etc. It is easy to see that MSM itself does not have additional experimental data to check if a system under study can be described as a single optical film (as is supposed by MSM theory). Therefore, in order to apply MSM to a thin sample one needs i) to verify that the optical response of a sample can be described as a single Fresnel layer (see also point 2.2 below) and ii) to confirm the optical thickness of the sample with the help of optical measurements. This can be done by performing optical measurements under different angles of incidence. Indeed, oblique angle measurements would allow one to extract effective optical thickness t_eff directly and to check if a studied sample can be described by EMT (as a single optical layer). (One should check that the optical constants calculated from normal incidence data provide correct values of reflection and transmission for oblique angles, there are no diffractive maxima or minima, etc.). The best method for extracting optical constants of thin films with the help of optical measurements under oblique angles is variable angle spectroscopic ellipsometry [22,30,31]. As a result, effective negative refractive indices of metamaterials calculated in [15,17] with an ad hoc restriction on t_eff require confirmation by optical measurements under an oblique angle of incidence.

2.2 Non-local response and limitations of effective medium approach

Magnetic permeability is produced by a long-wavelength limit of a spatial dispersion [20,32]. As a result, optomagnetic metamaterials are inherently spatially dispersive. It is important to note that it makes no problem to produce “negative” refractive index in a material with strong spatial dispersion, i.e., stratified media. For example, the effective refractive index, n_eff, of a structure consisting of 50nm layer of MgF₂ (n = 1.37) and a 30nm layer of silver (n = 0.48 + 10.7i at λ = 1.5μm) calculated with MSM is negative Re(n_eff)≈-0.4 at λ = 1.5μm (this implies that holes in the fishnet structures [15,17] are not actually necessary to produce negative index of refraction using MSM). The effective refractive index of a sandwich consisting of 50nm layer of glass (n = 1.5) and a 10nm layer of gold (n = 0.24 + 3.6i at λ = 632.8nm) calculated with MSM is not only negative but possesses extremely small dissipation n_eff = −0.66 + 0.05i at λ = 632.8nm. (The effective refractive index here has been calculated using MSM from the Fresnel coefficients of a two-layered film with the parameters given above.) Even larger values of negative Re(n_eff) can be obtained by applying MSM to an optical response of a Fabry-Perrault resonator or a multilayered film of a general form. These bogus results come from the fact that the electromagnetic response of stratified samples generally cannot be modelled within a simple effective medium theory. Indeed, the Abeles characteristic matrix [33] (§1.6) for a single optical layer of thickness t reads as

It is easy to check, e.g., that the characteristic matrix of a system consisting of two different layers of thickness t ₁ and t ₂, $M(t_1+t_2)=M(t_1)\cdot M(t_2),$ does not satisfy (2) (when $p_1\ne p_2$) and hence such a system cannot be (generally speaking) described with the help of effective mean field theory and does not possess viable effective refractive index. As another example, let us consider a layered structure of alternative dielectrics with n ₁, h ₁ and n ₂, h ₂ such that n ₁ h ₁ = n ₂ h ₂ = λ/4 and n ₂>n ₁. Characteristics matrix of this structure at the normal incidence and large N is $M_{2N}(N(h_1+h_2))=\left(\begin{array}{cc}\multicolumn{1}{c}{{\left(-\genfrac{}{}{0.1ex}{}{n_2}{n_1}\right)}^N}& \multicolumn{1}{c}{0}\\ \multicolumn{1}{c}{0}& \multicolumn{1}{c}{{\left(-\genfrac{}{}{0.1ex}{}{n_1}{n_2}\right)}^N}\end{array}\right)\approx \left(\begin{array}{cc}\multicolumn{1}{c}{{(-1)}^N}& \multicolumn{1}{c}{0}\\ \multicolumn{1}{c}{0}& \multicolumn{1}{c}{0}\end{array}\right),$ see §1.6.5 of [33]. Note that m ₁₁≈1 and m ₂₂≈0. This matrix breaks the condition (2) and cannot be written as the effective medium theory matrix (1) and hence one should not apply MSM to this structure. Although the period of the structure is rather large and one should not apply MSM to the structure on this basis already, the result ($m_{11}\ne m_{22}$) also holds for metallic structures where the structure period could be an order of magnitude smaller than the light wavelength. Analogously, regular 2D arrays of nanostructures possess narrow plasmonic features produced by diffractive coupling of localized plasmons [34] in the vicinity of which the condition (2) is not satisfied. It is worth mentioning that even in case of pronounced spatial dispersion one may try to introduce refractive index of metamaterials using various approaches. For example, a possibility to define refractive index in case of Bloch bands present in a metamaterial is discussed in Ref. 35, while spatial dispersion in terms of bianisotropy is discussed in [36,37]. Some scientists claim that bianisotropy is not based on spatial dispersion, see [38–40]. Since Maxwell equations written for plane waves interconnect electric and magnetic fields (and therefore the electric and magnetic fields are not independent variables of the theory), one might exercise different views on material equations for a macroscopic medium and introduce different versions of macroscopic electrodynamics. A particular choice of a version is probably as sound as a choice of the system of units in physics. (From an esthetic point of view, an introduction of new additional parameters required by bianisotropy to describe bound currents and charges seems to be ungrounded for standard static media.)

Having established that an application of MSM to 2D arrays of nanoparticles requires great care, we briefly discuss the case of a periodic medium. By developing the cosines and sinuses of Eq. (1) in series, Abeles has shown [41] that a periodic medium can be described by EMT up to the terms of the second order, see also [33] §1.6. Analogous calculations have been performed in [42], where (in contrast to Abeles) the possibility to describe periodic system by EMT was postulated and a heuristic limit on the EMT application was obtained such that a period of a medium should be smaller than 30 wavelengths. However, even in the simplest case of a periodic medium produced by two alternating dielectric layers the exact transfer matrix can be modelled by EMT only when $p_1=p_2$. It is easy to show that the matrix elements for a periodic structure consisting of N periods with n ₁, h and n ₂, h, (where h is supposed to be sufficiently small) are $m_{11}\approx 1-\left(\genfrac{}{}{0.1ex}{}{n_1^2+n_2^2}{2}+n_1{n}_2\sqrt{\genfrac{}{}{0.1ex}{}{n_2^2-{\sin }^2(\theta )}{n_1^2-{\sin }^2(\theta )}}\right)\genfrac{}{}{0.1ex}{}{{\pi }^2}{{\lambda }^2}h\cdot 2Nh$ and $m_{22}\approx 1-\left(\genfrac{}{}{0.1ex}{}{n_1^2+n_2^2}{2}+n_1{n}_2\sqrt{\genfrac{}{}{0.1ex}{}{n_1^2-{\sin }^2(\theta )}{n_2^2-{\sin }^2(\theta )}}\right)\genfrac{}{}{0.1ex}{}{{\pi }^2}{{\lambda }^2}h\cdot 2Nh,$ where θ is the angle of incidence. These elements are completely different when $n_1\approx \sin (\theta )$ and $n_2\ne n_1$ which implies the breakdown of EMT (Eq. (2) is not satisfied) even in this simplest case. The same conclusion has been obtained in [43] where macroscopic theory of resonant dielectrics with inhomogeneities much smaller than the light wavelength in vacuum has been discussed and it was shown that macroscopic description breaks down near resonances where additional optical vibrations (analogous to optical phonons) might be excited. Treatment of these optical vibrations requires introduction of a spatial dispersion [43].

The fact that multilayered structures cannot be described with just a single layer EMF is well-known in ellipsometry [30,31] and only signifies the importance of oblique angle measurements for MSM. These measurements would allow one to check if an Abeles characteristic matrix of a system under study can be described within EMT and would provide a firm basis for application of MSM expressions to a system.

3. Plasmonic resonances of the double dot arrays

After these general remarks, we turn our attention to gold dot arrays. Optical properties of 2D (~10-20nm thick) gold dot pairs have been extensively studied and are well understood [44]. In 2D case one can neglect fields produced by surface edge charges. Hence, it is relatively easy to model the plasmonic resonances of 2D dots and there is generally a good agreement between experimental and calculated data. At the same time, localized plasmon resonances (LPRs) of 3D (~80-90nm thick which is comparable with the skin-depth) gold dots have not been studied in such details. Only recently a number of thorough works devoted to this problem appeared in literature, for recent review see Ref. 45. Here we identify LPR modes of nanopillars using the combined analysis of sample’s reflection, transmission and ellipsometry. The fabrication of our samples is described in [14,34,46,47].

Figure 1(a) shows the three main LPRs possible in a single metallic nanopillar placed on a glass substrate (the solid arrows show the dipole vibrations in the pillar, dotted arrows show the induced dipole moment in the substrate [48] and ${\lambda }_x={\lambda }_y$ due to symmetry). For a pair of nanopillars each of these resonances splits into two – symmetric and antisymmetric ones shown in Fig. 1(a). The symmetric resonances contribute to the electric permittivity while antisymmetric resonances correspond to zero net dipole moment and therefore contribute either to magnetic permittivity ${\mu }_y$ for LPR with the resonance wavelength ${\lambda }_z^a,$ ${\mu }_z$ for LPR with ${\lambda }_y^a,$ or quadrupole non-local response for LPR with ${\lambda }_x^a$ [14]. (It is easy to evaluate the induced magnetic moment of the nanomolecule consisting of the nanopillar pair as $m_y\sim \genfrac{}{}{0.1ex}{}{s}{\lambda }{d}_z,$ where s is the separation of pillars in the pair and $d_z$ is the dipole moment of an individual pillar.) The symmetry group of a nanopillar pair is C_2v which implies that only LPR modes with ${\lambda }_x^s$ and ${\lambda }_z^a$ are coupled to normal incident TM light (with electric field along the x-axis) while LPR with ${\lambda }_y^s$ is coupled to TE light (with electric field along the y-axis) [14]. Light with an oblique angle of incidence (${\theta }_{inc}$) is locally coupled to ${\lambda }_z^s$ (for p-polarization) and non-locally to ${\lambda }_x^a$ and ${\lambda }_y^a$ (by non-local coupling we imply coupling that depends on the wavenumber of the light).

 

Fig. 1 Localized plasmon resonances of gold nanodots evaluated with Mie theory. (a) Main LPR modes in single and double dots (the arrows show currents in the dots, the dotted arrows show image currents in the substrate). Insets show the axes and an example of incident electromagnetic wave for the x-resonance. Dependence of LPR on the dot diameter D for (b) single dots and (c) double dots. The dot height is h = 90nm, the dot separation in the pair s = 140nm.

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Figure 1 also plots a simulated dependence of plasmon resonance wavelengths on the dot diameter, D, for single dots, Fig. 1(b), and double dots, Fig. 1(c). The dot height was fixed at h = 90nm and the dot separation for the dot pairs was s = 140nm. The data in Fig. 1 have been evaluated using the analytical Mie theory [33] with pillars being replaced by spherical particles. The coupling between the pillars and their images [48] has been taken into account using the coupled dipole approximation [49]. The permittivity of the gold ε_Au (used in the calculations) has been taken from [47], where the optical constants of 90nm Au film deposited in our system has been extracted using variable angle spectroscopic ellipsometry. Since the permittivity of nanostructured gold may differ from that of the film and our dots are not exactly spherical, we regard the data presented in Fig. 1 as a rough (and useful) evaluation. Figure 1 demonstrates that LPRs of single and double pillars of D~80-140nm should be in the visible spectrum (e.g., ${\lambda }_x^s$≈600nm and ${\lambda }_z^a$≈530nm). We note that, in addition to the resonances shown in Fig. 1(a), gold possesses another important resonance feature connected to its band structure. Indeed, in noble metals, transitions are possible between occupied states of the d-band into unoccupied states of the s-p conduction band (above the Fermi energy). This leads to a pronounced increase in absorption at an interband threshold frequency. In bulk gold, this absorption happens at about ≈500-600nm and ultimately is responsible for its yellowish colour. The Mie theory calculations (with the gold permittivity taken from Ref. 47) suggest that the interband transitions result in an additional resonance peak for the studied gold nanoparticles at the wavelength of λ_IB≈510nm, see Fig. 1(b) and (c). It is important to stress that the calculated position of this additional interband peak does not depend on the particle diameter for our dots.

To identify LPRs of 3D dots and coupled dot pairs we fabricated a set of square arrays of nanoparticles (more than 100 samples, each array had a size of 200μm × 200μm) in which we changed the average dot diameter at a fixed lattice constant a = 320nm, h = 90nm and s = 140nm. (To see if the extracted plasmonic resonances depends on the lattice constant in the array we have also studied samples with a = 400nm and a = 500nm. These samples showed plasmonic resonances close to those presented in the paper. The spread of resonance positions was larger than a possible resonance shift caused by a change of the lattice constant.) We measured optical reflection and transmission under angles of incidence from 0° to 75°, as well as the ellipsometry data for all samples.

Figures 2(a), (c) and (e) show examples of optical transmission through the samples with single dots under θ_inc of 0°, 15° and 45°, respectively, for D of 90nm, 110nm, 130nm; Figs. 2(b), (d) and (f) show optical transmission for dot pairs at incident angles of 0°, 15° and 10°, respectively, for D of 90nm, 110nm, 130nm. (The pair orientation was such that the light of s-polarization had electric field directed along the line connecting the dots which corresponded to the TM mode of Ref. 14). The plasmonic resonances in the system correspond to the drop in transmission. By comparing transmission of single dots and double dots measured under different angles of incidence, we have been able to identify all LPR excited in our system, see Fig. 2. and 3.

 

Fig. 2 Optical transmission through the nanodot samples. (a), (c), (e) - single dots and (b), (d), (f) - double dots. Dot sizes are 90nm (black squares), 110nm (red circles) and 130nm (blue triangles). The insets show the unit cell of the arrays. Resonant features are marked by the dotted arrows.

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It is worth noting that, in addition to LPR, the regular arrays of nanoparticles also demonstrate so-called Wood anomalies that happen close to Rayleigh cut-off wavelengths in our case [34]. These sharp features are also marked in Fig. 2(e) and denoted by ${\lambda }_R$ and ${\lambda }_R^{sub}.$ Fig. 3(a) shows examples of the single and double dot reflection spectra (D = 110nm) under the normal light incidence, Fig. 3(b) provides an example of the ellipsometry data [34] for the double dots D = 110nm observed at λ = 500nm.

 

Fig. 3 Optical properties of the arrays. (a) Normal reflection from a single dot array (black squares), and a double dot array – TM light (red circles) and TE light (blue triangles). Dot sizes are D = 110nm. Insets show the unit cell of arrays. (b) Ellipsometry data for the double-dot array of (a): Ψ (green circles) and Δ (blue triangles). The solid line show the best fit to the combined ellipsometry data (including data for the sample rotated onto 90° in the plane) with t_eff = 35nm. The dotted line show the best fit with fixed t_eff = 90nm. Experimental values of LPR wavelengths for (c) single dots and (d) dot pairs (${\lambda }_z^a$ – green triangles, ${\lambda }_y^s$ – olive hexagons).

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By analysing transmission, reflection and ellipsometry data, we extract experimental dependences of LPR wavelengths on D for single dots, Fig. 3(c), and double dots, Fig. 3(d). These dependences represent one of the main results of this paper. There are several important features associated with them. First, we see that the single dot resonances for 90nm dots fall into the region of ${\lambda }_x$ = 500-600nm (green circles of Fig. 3(c)) and ${\lambda }_z$~400nm (blue squares of Fig. 3(c)) which is above the values obtained with the approximate theory of Fig. 1(b). This is not surprising since the pillar geometry is different from the spherical geometry used in the approximation. Second, some resonance wavelengths depend on the amount of Cr left after chromium etch (5nm chromium was used to avoid charging during lithography [14,34,46,47]). As a result, spread of LPR wavelengths in our samples was rather large. Third, the interband resonance wavelength ${\lambda }_{IB}$ was indeed observed at ≈510-520nm and showed little dependence on dot diameter, see rhombi of Fig. 3(c), (d). Finally, we found the symmetric resonance of double dots ${\lambda }_x^s$ (red circles of Fig. 3(d)) experiences a very strong red shift when the dots start touching themselves at the pillar bases (while the antisymmetric resonance ${\lambda }_z^a$ exhibits a blue shift from this point) connected to a change in the plasmon coupling. It happens at dot diameter of D≈100nm which is smaller than the pillar separation s = 140nm. This is due to the conical shape of the pillars [14] and the difficulty removing chromium from the area between pillar bases. It may also imply that the electron beam micrographs (used to evaluate average dot diameters) underestimate the dot sizes. The behaviour of other LPRs is not strongly affected by pillar touching or overlapping, compare Fig. 3(c), (d) and Fig. 1(b), (c).

Figure 3(d) shows that the double dots possess both symmetric ${\lambda }_x^s$ and antisymmetric ${\lambda }_z^a$ plasmon resonances necessary for realization of optomagnetic phenomena [14,16]. To check this we directly measured the phase of light propagating through the optomagnetic samples based on double-dot arrays. The phase was determined using the fringe mode of a Mach-Zehnder interferometer shown in Fig. 4(a) .

 

Fig. 4 Interferometry of optomagnetic structures produced by double-dot arrays. (a) Schematics of the installation. Insets show typical interferograms of our samples for TM light and TE light at λ = 543nm. Phase shift as a function of the dot diameter for (b) TM light and (c) TE light. Red circles corresponds to light of λ = 650nm, green squares to λ = 543nm.

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Insets of Fig. 4 provide interferograms of the double-dot samples for TM light (coupled to the antisymmetric mode ${\lambda }_z^a$) where the fringes inside the sample show phase lead (shifted to the left) and the TE light (decoupled from the antisymmetric mode) where the fringes inside the sample show phase lag (shifted to the right). Figures 4(b) and (c) plot the phase difference $\Delta \varphi$ extracted from interferograms for TM and TE light respectively at λ = 543nm and 650nm as a function of the diameter D of the dot. We see that in case of TM light $\Delta \varphi$ is negative for most of our samples in green and for samples with overlapping dots in red light, while for TE light $\Delta \varphi$ stays positive for all samples. A large scatter of the data is probably connected to the final chromium etch. Disregarding interface phase shifts and multiple reflections, we can evaluate the real part of the effective refractive index as $\mathrm{Re}(n_{eff})=1+\genfrac{}{}{0.1ex}{}{\lambda \cdot \Delta \varphi }{2\pi t_{eff}}.$ From spectroscopic ellipsometry measurements we found that for our samples t_eff~30-40nm, see the best fit of Fig. 3(b) which is presented here as an example. This fit was obtained by using combined ellipsometry, transmission and reflection data with the help of Wvase software. The ellipsometry model consisted of a biaxial anisotropic single layer (representing an optomagnetic metamaterial) placed on a glass substrate. The optical constants of the substrate have been extracted from spectroscopic ellipsometry measurements performed on the bare substrate close to the sample. Figure 3(b) shows for comparison the best fit to combined data in case when t_eff is fixed at the height of the dots as in MSM, dotted line. It is clear from Fig. 3 (b) that the model with prefixed t_eff = 90nm does not provide a good fit to our data. Using the effective thickness of t_eff~30-40nm we obtain Re(n_eff)≈-0.5 for overlapping dot at λ = 543nm. It is worth noting that interface and multiple reflection contributions to the phase could be sizeable for optomagnetic samples. Extraction of the effective refractive index for our samples with the help of the combined data of spectroscopic ellipsometry, transmission and reflection as well as influence of a Cr sub-layer on plasmonic resonances will be described elsewhere. We stress again that the effective refractive index of 2D arrays of nanoparticles is not a “real” refractive index but rather an effective number assigned with some theory to describe some optical properties of 2D arrays (see the discussion in the section 2 above).

3. Conclusions

To conclude, we describe LPR wavelengths and phase shifts for the gold nanoparticle arrays made of “thick” 3D nanodots (nanopillars). We measure the size dependence of LPR modes in single and double-dot arrays. We show that double-dot arrays demonstrate a phase lead for light of TM polarization which is coupled to antisymmetric LPR mode (responsible for magnetic permeability) and a phase delay for light of TE polarization decoupled from this mode. We also show that care is needed when applying an effective medium theory to optomagnetic metamaterials and suggest checking effective optical constants by optical measurements with oblique light.