Evanescent wave in multiple slit diffraction and n-array antennas in metamaterial using Cesàro convergence

In this paper, we study Multiple slit diffraction and n- array linear antennae in negative refractive index material. We show that the evanescent wave plays a vital role in the near-field term. The evanescent wave grows significantly, unlike in conventional materials, and satisfies a novel kind of convergence known as Cesàro convergence. We calculate the intensity of multiple slits and the antenna’s amplification factor (AF) in terms of the Riemann zeta function. We further demonstrate that the Riemann zeta function gives rise to additional nulls. We deduce that all the diffraction scenarios in which the traveling wave satisfies the geometric series in the medium of the positive refractive index will enhance the evanescent wave, which satisfies Cesàro convergence in the medium of the negative refractive index.

www.nature.com/scientificreports/ effect, and the reversed Cherenkov radiation, etc. The second feature can find essential applications in the subwavelength focusing, super-resolution imaging, and high-capacity storage. The amplification of evanescent waves has been shown to be able to enhance the interaction between wave and matter and then increase the sensitivity of sensors. The experimental verification of evanescent wave amplification (EWA) has been studied in 21,22 . In order to observe the exponentially increasing-decreasing field distribution in EWA, a series-shunt capacitor (CC) and series-shunt inductor (LL) to simulate artificial magnetic plasma and electric plasma has been proposed 23 .
In this paper, we show that applying cesàro convergence enhances the amplification of evanescent waves in n-array negative refractive material antennas. Two of the authors, in their previous work, have shown that the linear dielectrics and magnetic materials in the matter 24 and the restoration of the evanescent wave in a perfect lens 25 obey a unique convergence called Cesàro convergence.
Multiple slit diffraction. The underlying physical principle of image formation is diffraction. In a perfect lens, an evanescent wave plays an important role in image formation. A natural question that emerges is whether an evanescent wave enhances negative refractive materials during diffraction. To this, we study multiple slit diffraction. Consider the geometry for diffraction with multiple slits. For N slits, the net contribution to the field E P 26,27 is given by substituting α value in the Eq. (8) and finally (see the calculations in Supplementary Information 2), we get  Linear antenna arrays. The other application, which is directly related to diffraction, is the antenna. One example that resembles multiple-slit diffraction is n-linear antenna arrays. We study its effect in the presence of negative refractive index materials or metamaterial. The metamaterial antennas, their production, application, and research are gaining prominence rapidly. Several strategies have been explored during the past decade to enhance antenna performance. The employment of metamaterials in antenna design is one such method. The metamaterial form antenna optimizes antenna performance by employing the innovative functionality of the metamaterials. The peculiar properties of these materials enable the creation of high-performance antennas, filters, and microwave devices that can not be attained with traditional antennas. Negative refractive indices, chiral left-handed materials, single negative metamaterials, electromagnetic bandgap left-handed materials, double positive medium, bi-isotropic, and anisotropic backwardwave media are some of the electromagnetic metamaterial compositions available for use in the antenna field 28,29 Leaky-wave antennas (LWAs) 30 and small resonator-type antennas(SRAs) are the two most common types of metamaterial-based antennas. Metamaterial-associated small antennas are suggested to offer a mechanism for modifying the dispersion relationship or conditions at the near-field boundary, which might lead to antenna size reduction while retaining decent radiation efficacy. Metamaterial-associated antennas facilitate small antennas to circumvent the confined efficiency and bandwidth restriction. However, this technique is still far from being robust. LWA's executed by the composite right/left-handed (CRLH) transmission line (TL) left-handed materials stimulate the backward to forward beam scanning, incorporating the broadside radiation, which is tough to accomplish with traditional LWAs 30-34 .

Two-element array
The most basic geometry of an array is a linear array, where every component is oriented along a straight line. The minimal span linear array is the two-element array. Consider electric fields E 1 and E 2 in the distant locality of the array elements [35][36][37][38] here M 1 and M 2 , -size of field (magnitudes), E n1 and E n2 -uniform (normalized) field patterns, r 1 and r 2 -distances from P, β-phase difference(between 2array elements), and ρ ρ ρ 1 and ρ ρ ρ 2 are far-zone E fields polarization vectors. Assuming the array elements are similar, have identical polarizations ρ ρ ρ 1 =ρ ρ ρ 2 =ρ ρ ρ , and have excitations of the same amplitude. By taking E n 1 (θ, φ) = E n 2 (θ, φ) = E n (θ, φ) , and M 1 = M 2 = M , the electric field is given by The array's full field is equivalent to the product of the field created by a single element at the origin (element factor) and the array factor (AF): In the Eq. (13), we use normalized array factor, E n (θ, φ) , and single element normalized field pattern.

n-Array.
A uniform array is a group of related objects that possess the same magnitudes and progressive phases. It is possible to determine the AF of the uniform array by seeing each element as a single point (isotropic) source. The AF of an N-element linear array of isotropic sources is [35][36][37][38] Eq. (14) can be re-written as where � = kd cosθ + β . The normalized AF and neglecting phase effect result in For small values of Eq. (16) leads to (10) Array factor (13) AF n = cos kd cosθ + β 2 (14) AF = 1 + e j(kd cosθ +β) + e j2(kd cosθ +β) + · · · + e j(N−1)(kd cosθ+β) .  (19) is referred to as Grandi's series in the literature. Grandi's series doesn't quite meet the standard geometric convergence; that is, the sum to infinity ∞ 0 x n = 1 1−x is not specified. Typically, for a geometric series to converge, x should fall between −1 and 1, i.e.,−1 < x < 1 ; in this instance, − 1 and 1 are also excluded. In Grandi's series, the value of x is x = −1 . It is interesting to note that Ramanujan 40 has used the value of x = −1 in the geometric series sum to infinity ∞ 0 x n = 1 1−x and obtained the value of ∞ 0 x n = 1/2 . The value of 1/2 as the sum to infinite series in Eq. (19) is justified if we assume that the Grandi's series obeys Cesàro convergence.
Here is a brief description of Cesàro convergence: For a geometric series to converge, the sequence of partial sums should converge to a real number. The sequence of partial sums for Grandi's series gives It is clear from Eq. (20) that the sequence of partial sums does not converge to a real number. But, the sum to infinity of geometric series gives ∞ 0 (−1) n = 1 1−(−1) = 1 2 , a real number. The RHS converges, and LHS diverges, hence for consistency, we consider the averages of partial sums, that is, and so on. Sequence of the average of partial sums gives The Eq. (22) is re-casted as as n goes to infinity, it converges to 1 2 . From the Eq. (22), as the average of partial sums converges to a real number, it's a Cesàro Convergence. A series n j=0 a j is Cesàro summable if this satisfies the following theorem: Theorem 1 Suppose n j=0 a j is a series that converges and it has sum, say L. Consequently n j=0 a j is Cesàro summable to L.
The proof is given in 41 . The properties of Cesàro sums: If n a n = A and n b n = B are convergent series, then a. Sum-Difference Rule: n (a n ± b n ) = n a n ± n b n = A ± B. b. Constant Multiple Rule : n c a n = c n a n = cA for any real number c. c. The product of AB = n a n n b n also as Cesàro sums.
Applying theorem 1 we get For more details, readers may refer to 24 . A surprising result on summability asserts that statistical convergence and high Cesàro convergence are inextricably linked 42,43 . Within the strong p-Cesàro convergence framework, a new version of the Orlicz-Pettis theorem is proposed 44 .

Application of Cesàro convergence to linear antenna arrays
An antenna array is a series of equivalent miniature antennas that collectively generate a signal comparable to that of a giant antenna. We will now examine the evanescent wave in an n-array metamaterial antenna using Cesàro convergence. Consider the Eq. (17) By defining the refractive index in complex form and re-writing the Eq. (26), we get The Eq. (27) can be rewritten into the following form if 'i' is included inside the brackets: Let Rearranging, Let ψ = (κ cosθ + β) Euler obtained Riemann Zeta ( ζ ) function using the following sequence which converges for any y > 0. For y = 0 , the Eq. (33) simplifies to Grandi's series. Using y = −(n im ψ) we can write Eq. (32) as When we differentiate the Eq. (33) (n − 1) times, we obtain which, for every y> 0, converges once more. Then by extending the function 1/(e y + 1) around y = 0 using Taylor series, we obtain Euler, by taking k to be complex in Eq. (36), obtained the following functional version of Riemann Zeta function 45,46 From (35) The intensity takes the form

Results and discussion
We have investigated two scenarios: multiple slit and n-array linear antennas in a negative refractive index where the evanescent wave enhances and will not die. In the first case, we computed the AF(Amplification Factor) in Eq. (40), and in the second case, we calculated the intensity in Eq. (43). In the case of linear antenna, as it is clear from Eq. (27) when it is placed in negative refraction, the evanescent wave does not dampen. Similarly, in the case of multiple slit diffraction, it is clear from Eq. (41) evanescent wave does not dampen and will grow indefinitely. By applying Cesàro convergence, we demonstrate that the evanescent wave does indeed exhibit convergence as indicated by Eq. (40) and Eq. (43). We have analyzed both scenarios and plotted them below. From Fig. 1, it is evident that the evanescent wave is amplified in the negative refractive medium, where we recover the diffraction pattern as we vary N(N=-10 to 10) for a fixed value of (� = π ) and different values of l(l = − 1.5 + 0.5i, − 1 + 0.5i, − 0.5 + 0.5i, 0 + 0.5i, 0.5 + 0.5i, 1.0 + 0.5i, 1.5 + 0.5i) in Eq. (40). In Fig. 2, we observe www.nature.com/scientificreports/ that for the fixed value of N (N = 4) and l(l = − 1.5 + 0.5i, − 1 + 0.5i, − 0.5 + 0.5i, 0 + 0.5i, 0.5 + 0.5i, 1.0 + 0.5i, 1.5 + 0.5i), the pattern repeats after every 2π by varying ( =−π to π ) in Eq. (40). In Fig. 3, we vary l(l = − 7 to 7 + 3i) by fixing N(N=2) and ( = π ). It should be clear to the readers that the evanescent wave amplifies the AF. Even it also vanishes on the imaginary line, giving rise to new nulls absent in the natural materials. The nulls occur when AF is zero, which gives the condition [35][36][37][38] . From Eq. (40), we get additional nulls from the Riemann zeta function, which vanishes at l = 2 as shown in Fig. 1 and also at − 6, − 4, − 2, 0, 2, 4, and 6. From Fig. 4, intensity versus θ(θ = − π/2 to π/2 ) plot, the evanescent wave clearly enhances the electric field's intensity for fixed values of l(l = − 1 + 0.5i) and different values of N (N = 1 to 5) in Eq. (43) (where φ is taken 0). Readers should note that the diffraction condition does not change for the traveling wave. But we get new conditions for the minimum as the intensity equation in terms of the Riemann zeta function gives zero for particular values of l, i.e., l = −2s , where s is an integer, the Riemann zeta function also has a unique zero on the critical line, also known as the Riemann hypothesis. The demonstration of these zero is the restatement of the Riemann hypothesis.
The enhancement of evanescent waves is a well-studied problem in the case of superlens or perfect lenses. t'Hooft commented 39 on Pendry's perfect lens 10 , where he addressed the geometric series of a convergence problem. To this, Pendry still needs to give a satisfactory reply. The authors of this paper have shown that one can recover Pendry's results using Cesàro Convergence. Readers should note that historically convergence has always led to physical reality. A well-known example is the Zero paradox, which was solved by calculus. Our approach to solving the enhancement of evanescent waves glorifies metamaterials. The electric field of Several coherent oscillators 26 satisfies geometric series, is given by

Conclusion
In this paper, we studied Multiple slit diffraction and n-array linear antennae in negative refractive index material. We have shown that the evanescent wave plays a vital role in the near-field term. The evanescent wave grows significantly, unlike in conventional materials, and satisfies a novel kind of convergence known as Cesàro convergence. We calculated the intensity of multiple slits and the antenna's amplification factor (AF) in terms of the Riemann zeta function. We further demonstrated that the Riemann zeta function gives rise to additional nulls for l being real and has zeros on the critical line, known as the Riemann hypothesis. In the case of diffraction, the diffraction condition does not change for the traveling wave. But we get new conditions for the minimum as the intensity equation in terms of the Riemann zeta function gives zero for particular values of l, i.e., l = −2s , where s is an integer, and the Riemann zeta function also has a unique zero on the critical line when the real part is 1/2, also known as the Riemann hypothesis. The demonstration of these zero is the restatement of the Riemann hypothesis. We have also shown that the electric field of Several coherent oscillators also satisfies Cesàro convergence in the negative refractive medium. Therefore, we conclude all the diffraction scenarios in which the traveling wave satisfies the geometric series in the medium of the positive refractive index will enhance the evanescent wave, which satisfies Cesàro convergence in the medium of the negative refractive index.

Data availability
The datasets used and analyzed during the current study are available from the corresponding author upon reasonable request.