1. Introduction

The ED of rotating structures as observed in the laboratory (inertial) Frame of Reference (FoR) has been investigated extensively. Pioneering studies can be found, e.g. in [1–3]. The main difficulty in their generalization and application to metamaterials stems from the need to deal with moving boundaries and heterogeneities - a challenge of overwhelming complexity in rotating metamaterials that usually consist of a very large number of small inclusions. Indeed, previous studies of rotating structures deal mainly with bodies of revolution that rotate around their own symmetry axis, thus dealing with boundaries and heterogeneities that appear to be fixed in the laboratory (inertial) FoR [4–6]. To compare, when a rigid structure of arbitrary non-symmetric geometry and heterogeneity that rotates at an angular (radian) velocity $\boldsymbol{\Omega }$ is observed in its rest FoR–denoted here by $\mathcal {R}^\Omega$–its boundaries and heterogeneities are stationary, thus the difficulty pointed above is completely alleviated. This approach has been used recently in several studies. The work in [7] develops a quasi-static EM theory for rotating systems and circuits. In [8,9] a discrete dipole approximation and polarizability theory for rotating arrays of small inclusions is developed. The inclusions may have arbitrary locations with respect to the rotation axis and no symmetry of the structure is assumed. To observe the rotation footprint in a rigorous and systematic framework, the analysis there is based on the rest-frame 2D Green’s function theory in slowly rotating medium, developed in [10]. We note that $\mathcal {R}^\Omega$ is the natural FoR in many applications, e.g. rotation sensors and gyroscopes used in navigation systems where the entire platform rotates and an inertial FoR is not available. However, even if the rotating structure excitation and response-observation are executed in the laboratory inertial frame, casting the problem in $\mathcal {R}^\Omega$ still suggest great simplification since the actual solution is determined under stationary geometry while the passage from the laboratory FoR to $\mathcal {R}^\Omega$ and back can be achieved by applying explicit field transformations.

The purpose of the present study is to use the formulation developed in [8,9] to further explore the effect of rotation on complex arrays consisting of electrically small dielectric scatterers, as observed in the rotating structure rest-frame. We examine how rotation affects the metamaterial response to an electromagnetic excitation, and suggest a rigorous quantitative definition for the metamaterial sensitivity to rotation. We then explore the underlying physical mechanisms that govern its sensitivity and suggest several ways to enhance it. We show that the suggested measure of the metamaterial response with respect to rotation can be used to extract the rotation rate, thus offering a new paradigm for metamaterial based passive gyroscopes. Furthermore, metamaterial array design offers an extremely large number of degrees of freedom; namely the design of individual inclusions, their placement, and the choice of specific inclusions that serve as probes of the array response. The latter two are used here not only for sensitivity enhancement, but also for reducing the effect of signal noise, thus reducing the smallest detectable rotation rate under noisy conditions.

In the context of rotation sensors, it is instructive to contrast the traditional passive rotation sensing technology with the concepts suggested by rotating metamaterials explored here. The former is based on the Sagnac effect [11], in which the rotation induced phase accumulated by a light signal that propagates along a slowly rotating closed circular path is $\phi =2\Omega k_0 c^{-2} S_L$ where $c$ is the speed of light in vacuum, $\Omega$ is the rotation rate, and $S_L$ is the area enclosed by the path. By interfering two counter propagating light signals, their mutual phase difference $\Delta \phi = 4\Omega k_0 c^{-2} S_L$ results in an interference pattern from which $\Omega$ can be extracted. From a somewhat wider point of view, the Sagnac effect can be viewed as a rotation induced breach of mode degeneracy of the otherwise degenerate modes of the corresponding non-rotating system [12]. This physical picture shades light on the effect of parasitic scattering in optical gyroscopes; parasitic scattering processes, that are practically unavoidable, alleviate mode degeneracy as well, thus competing with the rotation induced breach of mode degeneracy and leading to the formation of the gyroscope dead zone: a range of rotation rates $\Omega$ within which the breach of mode degeneracy due to scattering overwhelms that induced by rotation, resulting in unmeasurable rotations [13,14]. Since scattering is an unavoidable phenomenon, dead-zones in traditional gyroscopes are unavoidable.

However, if you can’t beat them try to join them. Thus, to contrast, the metamaterials based approach exploits the practically infinite number of scattering events that exist in the metamaterial. A schematic picture is shown in Fig. 1. Any sequence of three or more scattering events forms a closed path. In arrays that consist of $N$ scatterers the number of these loops scales as $N!$. They are non-symmetric thus no mode degeneracy is supported. We don’t look for this degeneracy to start with since the basic principle doesnot seek for the phase difference between counter and co-rotating signals of the same loop. Rather, the extremely large number of different loops interfering in each single scatterer, render its excitation $\Omega$-dependent. The more scattering events, the more different loops interfere in each inclusion, hence the more intense is the change of its excitation as $\Omega$ changes and the larger is its sensitivity to rotation. Thus, here the scattering events are potentially a blessing in disguise.

 

Fig. 1. Sagnac loops defined by a set of $N$ point-scatterers. (a) Different Sagnac loops. (b) Different Sagnac loops that share at least one common point-scatterer. (c) Different Sagnac loops that share all their point scatterers (direction counts).

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Finally, two facts are worth pointing out,

2. Formulation and polarizability theory

We set the following terminology. Here and henceforth, we always observe the electromagnetic system in its rest frame of reference, i.e. in a frame where its boundaries do not move. Thus, by a “$\mathcal {R}^0$ problem” we mean a system that does not rotate, observed in the inertial (laboratory) Frame of Reference (FoR). Likewise, by “$\mathcal {R}^\Omega$ problem” we mean a system that rotates rigidly at an angular velocity $\boldsymbol{\Omega }=\boldsymbol{\hat {z}}\Omega$, observed in the non-inertial FoR $\mathcal {R}^\Omega$ where it appears at rest.

It has been shown that if the $\mathcal {R}^0$ problem is described by the scalar permittivity and permeability $\epsilon (\boldsymbol{\rho })=\epsilon _0\epsilon _r(\boldsymbol{\rho }),\,\mu (\boldsymbol{\rho })=\mu _0\mu _r(\boldsymbol{\rho })$, then in the limit of slow rotation the corresponding $\mathcal {R}^\Omega$ problem is still governed by the conventional set of Maxwell’s equations (ME), where the rotation is manifested only via the modified constitutive relations [15]

In both TE and TM polarizations the complete electromagnetic field can be derived from the corresponding $z$-directed field satisfying a modified Helmholtz equation [10] ($e^{-i\omega t}$ time dependence is assumed and suppressed)

Thus, for a thin wire electric [magnetic] current $\boldsymbol{J}=\boldsymbol{\hat {z}}I_z\delta (\boldsymbol{\rho }-\boldsymbol{\rho '})$ [$\boldsymbol{J}^{\textrm{M}}=\boldsymbol{\hat {z}}I_z^{\textrm{M}}\delta (\boldsymbol{\rho }-\boldsymbol{\rho '})$] the excited field is only of the TM [TE] polarization, with the electric [magnetic] field $E_z=i\omega \mu I_z G(\boldsymbol{\rho },\boldsymbol{\rho '})$ [$H_z=i\omega \epsilon I_z^{\textrm{M}} G(\boldsymbol{\rho },\boldsymbol{\rho '})$]. This Green’s function is given by

The properties above imply that Eq. (5) is in fact a uniform approximation of our Green’s function, that encapsulates all the essential physics contained in the exact expression (4). The advantages of the representation in Eq. (5) are clear; the need to sum slowly converging series (especially in the near field) has been alleviated. Furthermore, the simplicity of this expression enables one to investigate interference and diffraction phenomena by using essentially the same tools used for interference in $\mathcal {R}^0$.

In the framework of polarizability theory and discrete dipole approximation (DDA), the response of a set of $N$ electrically small scatterers to an incident field $\boldsymbol{E}^{\textrm{inc}}(\boldsymbol{r})$ is governed by

These properties are demonstrated by the typical array response shown in Fig. 2. The array, shown in Fig. 2(a), is excited by a $\boldsymbol{\hat {z}}$ directed line source of electric current with $\lambda =1\mu m$, located at the center $(x,y)=(0,0)$. It excites the TM mode. The array consists of $N=500$ randomly located dielectric cylinders with $\epsilon _r=11.4$ and radius $a=0.02\lambda$, in a vacuum background. Figure 2(b) shows $\left | I_n(\Omega )/I_n(0) \right |$ - the polarization currents magnitudes normalized to their respective values at no rotation, vs the normalized rotation rate $\Omega /\omega$, for $N_p=20$ cylinders. We define the slow rotation sensitivity of the $n$-th cylinder as the slope of the relative change in the current

 

Fig. 2. A random metamaterial array (a) and its response vs $\Omega$ (b),(c). The array consists of 500 dielectric cylinders, and is excited by a line source with $\lambda =1\mu m$ located at its center. (b) The polarization currents $I_n(\Omega )$ excited in the 20 cylinders with the highest slopes in the $\Omega \rightarrow 0$ limit. The red circle in (a) depicts the most sensitive cylinder according to Eq. (7a). (c) Zoom into the origin. The functions $I_n(\Omega )$ are approximately linear with $\Omega$ at the slow rotation regime.

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3. Array construction

The number of different Sagnac loops $N_{\textrm{SL}}$ inside an array of electrically small scatterers may serve as a qualitative estimate for its potential sensitivity to rotation, as shown qualitatively in Fig. 1. $N_{\textrm{SL}}$ equals the number of ordered sets (direction w.r.t. $\Omega$ counts!) that can be chosen out of $N$ elements, excluding the empty set, the set of a single element, and the set of two elements. Hence we have

For $N\gg 1$ $N_{\textrm{SL}}\approx eN!-(N^2+1)$ (recall the series expansion of $e^1$). We now slightly refine our physical picture. Assume that the structure is excited by some source, and the response of the $n$-th point scatterer is $E_n\equiv E_z(\boldsymbol{\rho }_n)$. The sensitivity of $I_n$ to rotation is due to the total of different Sagnac loops that share the $n$-th point scatterer. This is due to the fact that different loops experience different $\Omega$-dependent phase-shifts, and they interfere at their common point-scatterers. Examples are shown in Fig. 1(b)-(c). Thus, we look for $N_{\textrm{SL}1}$: the number of different SL that share at least one point-scatterer. This is nothing but $N_{\textrm{SL}1}=N(N-1)_{\textrm{SL}}$. Clearly, this number increases very fast with $N$. It represents, in fact, the number of waves with random phases that interfere at each point scatterer. Therefore, we anticipate that it may serve as a first measure of the potential sensitivity to rotation.

The mere number of loops, however, is not sufficient. Not all Sagnac loops are equal. Consider the ordered subset of $M$ non-repeated scatterers, $2<M\le N$, with the sequence of scattering events $1\Rightarrow 2\Rightarrow \ldots \Rightarrow M\Rightarrow 1$ such as those shown in Fig. 3. A necessary condition for the existence of Sagnac effect in this series of events is that the enclosed area does not vanish. This is satisfied only if the $M$ scatterers do not reside on a single straight line, as schematized in Fig. 3. It is clear that the condition above is satisfied for any choice of $M$ scatterers $2<M\le N$ iff it is satisfied for any choice of $M=3$ scatterers. Thus, for a given array, it is sufficient to verify that every set of 3 scatterers does not reside on a straight line. Equivalently - define a triangle of non-vanishing area. There are $\left(\begin{array}{c}{N}\\{3}\end{array}\right)$ possible triangles. We term this as the no-three-in-line - N3IL - condition.

 

Fig. 3. A closed loop that consists of four point scatterers, with the ordered interactions $1\Rightarrow 2\Rightarrow 3\Rightarrow 4\Rightarrow 1$. (a) All four scatterers reside on a straight line, hence the enclosed area is zero. (b) The four scatterers cannot be aligned along a single straight line, hence the enclosed area does not vanish. (c) Scattering events ordered as $1\Rightarrow 2\Rightarrow 3\Rightarrow 4\Rightarrow 3\Rightarrow 2\Rightarrow 1$, are never counted in our analysis since they have repeated elements.

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By the very basic definition of periodicity, a periodic array does not satisfy the N3IL condition. Thus, a large number of scattering events in periodic structures are in fact “waisted” in the sense that they do not produce the aforementioned Sagnac interferences. As we show in the simulations below, their sensitivity as defined in Eq. (7b) is indeed inferior when compared to other arrays. The use of non-periodic arrays potentially provide better sensitivity. One may suggest a random placement of points since in this case the probability to find three points on the same line is zero. However, a mere realization of random array doesnot guarantee the absence of three points sets that nearly reside on a straight line. Thus, the systematic construction of metamaterials satisfying the above properties constitute a new challenge to array design, that is by no means a trivial one. A formulation of the issues pointed above is suggested by the following two intimately connected and century-old problems in number theory [17]

No-three-in-line (N3IL) problem originally introduced in 1917 (see “A puzzle with pawns” in [18]). Given a grid of $N\times N$ in the plane, what is the maximal number of points that can be placed on the grid such that no three points reside on a straight line, and what is the corresponding distribution? In our case $1/N$ may represent limitations due to fabrication and/or due to the DDA. An obvious upper bound is $2N$, but it is tight only for relatively small $N$. Some bounds and specific distributions are provided in [19] but no general optimal solution is known - see Sec. F4 in [17]. For $N=p$ a prime, a construction for placement of $p$ points proposed by Erdős is $x_n,y_n=n,an^2\mbox {mod}(p)$ for any integer $a$ - see [20] and appendix there. Hall [19] proposed an array made of 12 boxes, each with integer locations on Hyperbolas given by the solutions of linear congruence equations $xy\equiv k\mbox {mod}(p)$ for any prime $p$ and integer $k$. See examples in Fig. 4.

 

Fig. 4. Examples of N3IL arrays. (a) Centered Erdős and (b) Hall-box $B_{01}$ arrays, in a domain of $20\lambda \times 20\lambda$.

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Heilbronn triangle problem [17,20] Given a set of $N$ points in a given convex area, we denote by $\Delta _N$ the smallest area of all possible $\left(\begin{array}{c}{N}\\{3}\end{array}\right)$ triangles. The Heilbronn problem asks for the distribution of points that maximizes $\Delta _N$. It can be viewed as a measure of how far are all the possible triples of the N3IL problem from residing on a straight line. Specific validated optimal solutions are known only up to $N=6$, and numerical solutions for $N=22$ mostly with no confirmation of global optimum. Non constructive upper and lower bounds are available - see Sec. F4 in [17]. The works in [21,22] provide the theoretical upper and lower bounds of the optimal construction $cN^{-2}\ln N\le \Delta _N\le CN^{-\mu },\,\mu =(8/7)-\epsilon$. Specific constructions of $N$-points arrays for which $\Delta _N$ is within these bound are not yet available. It is interesting to note, however, that for Erdös construction of the N3IL $\Delta _N\simeq 0.5(N-1)^{-2}$ [23], nearly touching from below the lower optimal construction bound. As pointed above it is clear that N3IL arrays can be generated merely by a random placement of points. However, it has been shown that for a random array of $N$ points the minimum triangle area $\Delta _N$ scales on the average as $cN^{-3}<\langle \Delta _N\rangle < CN^{-3}$ [24]. This is well below the bounds pointed above, thus such arrays are anticipated to be inferior on the average.

4. Examples

We simulated the response of several arrays and tested their performances both under ideal excitation (i.e. in the absence of signal noise) and under noisy conditions. In the latter case, we show that one may exploit the extra degrees of freedom offered by the metamaterial approach to reduce the effect of noise and thus to increase the smallest detectable rotation rate. In all the examples below, the arrays are excited by a $\boldsymbol{\hat {z}}$-directed line source at the array center.

4.1 Ideal excitation

We start by simulating the ideal response of several arrays and tested their performances. Figure 5 shows $S$ defined in Eq. (7b) vs the number of scatterers $N$ for square periodic array, Erdős and Hall arrays, and for Golden angle (GA) spiral array that has been used previously in photonic applications [25]. All arrays have the same properties as those of the array shown in Fig. 2, i.e. area of $400\lambda ^2$, and dielectric cylinders with $\epsilon _r=11.4$ and radius $a=0.02\lambda$. The minimal distance between cylinders is forced to be $d_{\textrm{min}}=0.1\mu m$. Thus, in the Erdős and Hall array realizations, the distances of the $n$-th generated location from the previously generated ones, $d_{n,m},\, m=1,\ldots n-1$, was checked. If $d_{n,m}<d_{\textrm{min}}$ for some $m<n$, the corresponding $n$-th cylinder was deleted. In Fig. 5 $N$ is the actual number of scatterers in the arrays, after the above dilution procedure. It is seen that the Erdős and Hall arrays generally have superior performances as the number of scatterer increases. Furthermore, they outperform the sensitivity of a conventional Sagnac loop of similar ares.

 

Fig. 5. Sensitivities $\bar {S}$ vs $N$ of the N3IL Erdős and Hall-box $B_{01}$ arrays in a domain of $20\lambda \times 20\lambda$, periodic and golden-angle spiral arrays, and Sagnac loop, all of similar area. The Sagnac loop sensitivity is defined here similar to Eq. (3): $S=I^{-1}\frac {\partial }{\partial \bar \Omega }I(\Omega =0)$ at the maximum slope of the loop interference pattern.

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While the N3IL and Heilbronn problems have contributed to the design concepts of rotation-sensitive materials by suggesting a systematic generation procedure, they do not yield the best possible design. In addition, the random arrays on the average yield poor performances as can be anticipated from the corresponding average Heilbronn problem bounds $\langle \Delta _N\rangle$ pointed above, but some rare realizations may yield exceptional sensitivities; one may win the lottery if he tries hard enough. We have conducted a set of numerical experiments. For a given number of scatterers $N$, we have generated $50$ realizations and computed the sensitivity $\bar{S}_i, \,i=1,\ldots,50$ for each of them. Figure 6 shows $\max _i|\bar{S}_i|$, $\min _i|\bar{S}_i|$, $\langle \bar{S}_i\rangle$ and $\sigma |\bar{S}_i|$. As anticipated, the average sensitivity is well below the Erdös and Hall arrays sensitivity shown in Fig. 5. However, among these 50 random realizations for each $N$, one can find a “lucky strike” that is at least as good, and even better, than the values shown in Fig. 5.

 

Fig. 6. Sensitivities vs. N for randomly generated arrays.

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4.2 Effect of noise

It is interesting to examine the metamaterial-based rotation sensor performance under noisy conditions. There are several different physical processes of noise generation, and we assume that the rotation itself does not have any significant effect on these processes. We concentrate here on shot-noise generated by the fluctuations in the number of electrons $N_{\textrm{e}}$ reaching the probe assigned to a specific cylinder, within the time interval $T=1/f$ where $f$ is the current sampling rate. $N_{\textrm{e}}$ follows the Poisson distribution with the average and variance $\langle N_{\textrm{e}} \rangle =\langle I \rangle /(q_{\textrm{e}}f)=\sigma ^2\left (N_{\textrm{e}}\right )$, where $q_{\textrm{e}}$ is the electron charge, and $I$ is the current. $\langle I \rangle$ here is in fact the theoretical or computed current in the absence of noise. Clearly, the random variable $I=N_{\textrm{e}}q_{\textrm{e}}f$ does not follow the Poisson distribution; specifically,

This current, in the probe attached, say, to scatterer #$n$ is proportional to its polarization current. For simplicity we assume below that this proportionality factor is 1. To get a feeling of its statistics, we have simulated the above process by generating $10^5$ different realizations of the poisson process that corresponds to the most sensitive cylinder of the array shown in Fig. 2, with the sampling rate $f=100$Hz. The resulting PDF is shown in Fig. 7. Its average is precisely the theoretically computed $I(0)$ of this cylinder. To a very good approximation this PDF is a Gaussian with $\sigma ^2$ given by Eq. (9).

 

Fig. 7. Simulated PDF of the polarization current in the most sensitive cylinder in the array of Fig. 2. Here $f=100$Hz.

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In accordance to the results shown, e.g. in Fig. 2 and Eqs. (7a)–(7b), in the limit of slow rotation the current and the $n$-th scatterer satisfies the relation

The estimator under noisy conditions can be improved by the use of more than a single cylinder, even if this cylinder is the one that maximizes $F_n$. This is achieved by generalizing the formulas above to the case of vector estimation. Assume now that the currents are measured in $N_p$ cylinders, $N_p<<N$. Towards this end, we define the zero-average RV of the polarization currents (shifted by their average). Then the vector RV of currents satisfies the linear vector equation

We further assume that the noise in each cylinder is independent of the rest. Then $\boldsymbol{\rho }$ is a multivariate RV with zero average and a diagonal covariance matrix $\Sigma$ with elements

We can invoke now a standard procedure; solve this system by weighting each equation according to the corresponding $1/\sigma _{nn}$ and solve by weighted least squares [26]

We turn now to show some examples. In all simulations, the specific scatterers used for estimations were chosen as those maximizing the figure of merit $F$ in Eq. (15). However, once these were chosen, we used a random generator of Poisson distribution to generate $10^5$ different realizations of $N_e$ from which the currents RVs were synthesized, from which the complete PDF of $\hat {\Omega }$ was computed and its standard deviation was extracted. Figure 8(a) shows the PDF of $\hat {\Omega }$ for the random array of 500 cylinders shown in Fig. 2 using the $N_p=20$ best $F_n$ cylinders. Figure 8(b) shows the standard deviation for $N_p$ ranging from 1 (a single cylinder estimation) to 20. It is seen that the addition of cylinders reduces $\sigma$ considerably, and eventually around $N_p=20$ it levels off. Two tests are shown there. One by using the $N_p$ cylinders with best noiseless sensitivity parameters of Eq. (7b), and by using the cylinders that maximize the figure of merit $F$ in Eq. (15). It is seen that $F$ provides a better criteria for $\sigma$ minimization, but the difference is not very significant. The estimation accuracy increases (i.e $\sigma$ decreases) as the number of scatterers $N$ increases. In Fig. 9 we show the same data of Fig. 8(b) but for Erdős array of Fig. 4(a) with $N=2518$, and for best selected random array with $N=2500$. We have generated an ensemble of $10900$ random arrays, out of which we choose the one that minimizes $\sigma$. It is seen that despite the difference between the noiseless sensitivity criteria $\bar {S}$ of Eqs. (7a)–(7b) and the figure of merit criteria derived specifically for noisy conditions for choosing “best cylinders”, the difference between the resulting estimation (in terms of smallest detectable $\Omega$) is not significant. Furthermore, the Erdős array immunity to noise is nearly that of the best array chosen out of a very large ensemble. Thus, the systematic N3IL construction encapsulates most of the significant factors leading to enhanced rotation-sensitive metamaterials.

 

Fig. 8. $\Omega$ estimation in the random array of Fig. 2 in the slow rotation regime $\bar {\Omega }=\Omega /\omega \rightarrow 0$. (a) The PDF obtained by using $N_p=20$ scatterers chosen to maximize $F$ of Eq. (15). (b) The estimation standard deviation vs $N_p$ chosen by maximizing the sensitivity as defined in Eq. (7b) (blue line), and by maximizing $F$ in Eq. (15).

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Fig. 9. $\Omega$ estimation standard deviation. (a) as Fig. 8(b) but for the Erdős array of Fig. 4(a). (b) as Fig. 8(b) but for the best random array selected from an ensemble of 10,900 random arrays with 2500 scatterers.

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5. Conclusions

A new approach for rotation sensing using passive metamaterials was introduced. It is based on the multiple-scattering events that take place inside an array that consists of a large number, $N\gg 1$, of electrically small scatterers. The number of different Sagnac loops $N_{\textrm{SL}}$ created by all sets of $n_s\ge 3$ scattering events scales as $N!$, and so does the number of different Sagnac loops that share any given common scatterer. Thus, the excitation intensity of each scatterer in the array is a result of interferences of a practically infinite number of different Sagnac loops. This physical effect paves the way to a new generation of sensors based on the paradigm of rotation-sensitive metamaterials. A surprising connection to century-old problems in number theory, namely the no-three-in-line (N3IL) problem and Heilbronn triangle problem, is pointed out and Erdős solution was tested. It is shown that the huge number of degrees of freedom allows not only to increase the sensitivity to rotation by a proper design of the metamaterial array, but also to reduce the effect of noise and thus to reduce the smallest detectable rotation rate under noisy conditions. General guidelines for the design of these novel materials were discussed and tested.